"{\"text\":{\"0\":\" Attention Is All You Need\\nAshish Vaswani\\u0003\\nGoogle Brain\\navaswani@google.comNoam Shazeer\\u0003\\nGoogle Brain\\nnoam@google.comNiki Parmar\\u0003\\nGoogle Research\\nnikip@google.comJakob Uszkoreit\\u0003\\nGoogle Research\\nusz@google.com\\nLlion Jones\\u0003\\nGoogle Research\\nllion@google.comAidan N. Gomez\\u0003y\\nUniversity of Toronto\\naidan@cs.toronto.edu\\u0141ukasz Kaiser\\u0003\\nGoogle Brain\\nlukaszkaiser@google.com\\nIllia Polosukhin\\u0003z\\nillia.polosukhin@gmail.com\\nAbstract\\nThe dominant sequence transduction models are based on complex recurrent or\\nconvolutional neural networks that include an encoder and a decoder. The best\\nperforming models also connect the encoder and decoder through an attention\\nmechanism. We propose a new simple network architecture, the Transformer,\\nbased solely on attention mechanisms, dispensing with recurrence and convolutions\\nentirely. Experiments on two machine translation tasks show these models to\\nbe superior in quality while being more parallelizable and requiring signi\\ufb01cantly\\nless time to train. Our model achieves 28.4 BLEU on the WMT 2014 English-\\nto-German translation task, improving over the existing best results, including\\nensembles, by over 2 BLEU. On the WMT 2014 English-to-French translation task,\\nour model establishes a new single-model state-of-the-art BLEU score of 41.8 after\\ntraining for 3.5 days on eight GPUs, a small fraction of the training costs of the\\nbest models from the literature. We show that the Transformer generalizes well to\\nother tasks by applying it successfully to English constituency parsing both with\\nlarge and limited training data.\\n1 Introduction\\nRecurrent neural networks, long short-term memory [ 13] and gated recurrent [ 7] neural networks\\nin particular, have been \\ufb01rmly established as state of the art approaches in sequence modeling and\\n\\u0003Equal contribution. Listing order is random. Jakob proposed replacing RNNs with self-attention and started\\nthe effort to evaluate this idea. Ashish, with Illia, designed and implemented the \\ufb01rst Transformer models and\\nhas been crucially involved in every aspect of this work. Noam proposed scaled dot-product attention, multi-head\\nattention and the parameter-free position representation and became the other person involved in nearly every\\ndetail. Niki designed, implemented, tuned and evaluated countless model variants in our original codebase and\\ntensor2tensor. Llion also experimented with novel model variants, was responsible for our initial codebase, and\\nef\\ufb01cient inference and visualizations. Lukasz and Aidan spent countless long days designing various parts of and\\nimplementing tensor2tensor, replacing our earlier codebase, greatly improving results and massively accelerating\\nour research.\\nyWork performed while at Google Brain.\\nzWork performed while at Google Research.\\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.arXiv:1706.03762v5  [cs.CL]  6 Dec 2017 transduction problems such as language modeling and machine translation [ 35,2,5]. Numerous\\nefforts have since continued to push the boundaries of recurrent language models and encoder-decoder\\narchitectures [38, 24, 15].\\nRecurrent models typically factor computation along the symbol positions of the input and output\\nsequences. Aligning the positions to steps in computation time, they generate a sequence of hidden\\nstatesht, as a function of the previous hidden state ht\\u00001and the input for position t. This inherently\\nsequential nature precludes parallelization within training examples, which becomes critical at longer\\nsequence lengths, as memory constraints limit batching across examples. Recent work has achieved\\nsigni\\ufb01cant improvements in computational ef\\ufb01ciency through factorization tricks [ 21] and conditional\\ncomputation [ 32], while also improving model performance in case of the latter. The fundamental\\nconstraint of sequential computation, however, remains.\\nAttention mechanisms have become an integral part of compelling sequence modeling and transduc-\\ntion models in various tasks, allowing modeling of dependencies without regard to their distance in\\nthe input or output sequences [ 2,19]. In all but a few cases [ 27], however, such attention mechanisms\\nare used in conjunction with a recurrent network.\\nIn this work we propose the Transformer, a model architecture eschewing recurrence and instead\\nrelying entirely on an attention mechanism to draw global dependencies between input and output.\\nThe Transformer allows for signi\\ufb01cantly more parallelization and can reach a new state of the art in\\ntranslation quality after being trained for as little as twelve hours on eight P100 GPUs.\\n2 Background\\nThe goal of reducing sequential computation also forms the foundation of the Extended Neural GPU\\n[16], ByteNet [ 18] and ConvS2S [ 9], all of which use convolutional neural networks as basic building\\nblock, computing hidden representations in parallel for all input and output positions. In these models,\\nthe number of operations required to relate signals from two arbitrary input or output positions grows\\nin the distance between positions, linearly for ConvS2S and logarithmically for ByteNet. This makes\\nit more dif\\ufb01cult to learn dependencies between distant positions [ 12]. In the Transformer this is\\nreduced to a constant number of operations, albeit at the cost of reduced effective resolution due\\nto averaging attention-weighted positions, an effect we counteract with Multi-Head Attention as\\ndescribed in section 3.2.\\nSelf-attention, sometimes called intra-attention is an attention mechanism relating different positions\\nof a single sequence in order to compute a representation of the sequence. Self-attention has been\\nused successfully in a variety of tasks including reading comprehension, abstractive summarization,\\ntextual entailment and learning task-independent sentence representations [4, 27, 28, 22].\\nEnd-to-end memory networks are based on a recurrent attention mechanism instead of sequence-\\naligned recurrence and have been shown to perform well on simple-language question answering and\\nlanguage modeling tasks [34].\\nTo the best of our knowledge, however, the Transformer is the \\ufb01rst transduction model relying\\nentirely on self-attention to compute representations of its input and output without using sequence-\\naligned RNNs or convolution. In the following sections, we will describe the Transformer, motivate\\nself-attention and discuss its advantages over models such as [17, 18] and [9].\\n3 Model Architecture\\nMost competitive neural sequence transduction models have an encoder-decoder structure [ 5,2,35].\\nHere, the encoder maps an input sequence of symbol representations (x1;:::;x n)to a sequence\\nof continuous representations z= (z1;:::;z n). Given z, the decoder then generates an output\\nsequence (y1;:::;y m)of symbols one element at a time. At each step the model is auto-regressive of continuous representations z= (z1;:::;z n). Given z, the decoder then generates an output\\nsequence (y1;:::;y m)of symbols one element at a time. At each step the model is auto-regressive\\n[10], consuming the previously generated symbols as additional input when generating the next.\\nThe Transformer follows this overall architecture using stacked self-attention and point-wise, fully\\nconnected layers for both the encoder and decoder, shown in the left and right halves of Figure 1,\\nrespectively.\\n2 Figure 1: The Transformer - model architecture.\\n3.1 Encoder and Decoder Stacks\\nEncoder: The encoder is composed of a stack of N= 6 identical layers. Each layer has two\\nsub-layers. The \\ufb01rst is a multi-head self-attention mechanism, and the second is a simple, position-\\nwise fully connected feed-forward network. We employ a residual connection [ 11] around each of\\nthe two sub-layers, followed by layer normalization [ 1]. That is, the output of each sub-layer is\\nLayerNorm( x+ Sublayer( x)), where Sublayer(x)is the function implemented by the sub-layer\\nitself. To facilitate these residual connections, all sub-layers in the model, as well as the embedding\\nlayers, produce outputs of dimension dmodel = 512 .\\nDecoder: The decoder is also composed of a stack of N= 6identical layers. In addition to the two\\nsub-layers in each encoder layer, the decoder inserts a third sub-layer, which performs multi-head\\nattention over the output of the encoder stack. Similar to the encoder, we employ residual connections\\naround each of the sub-layers, followed by layer normalization. We also modify the self-attention\\nsub-layer in the decoder stack to prevent positions from attending to subsequent positions. This\\nmasking, combined with fact that the output embeddings are offset by one position, ensures that the\\npredictions for position ican depend only on the known outputs at positions less than i.\\n3.2 Attention\\nAn attention function can be described as mapping a query and a set of key-value pairs to an output,\\nwhere the query, keys, values, and output are all vectors. The output is computed as a weighted sum\\nof the values, where the weight assigned to each value is computed by a compatibility function of the\\nquery with the corresponding key.\\n3 Scaled Dot-Product Attention\\n Multi-Head Attention\\nFigure 2: (left) Scaled Dot-Product Attention. (right) Multi-Head Attention consists of several\\nattention layers running in parallel.\\n3.2.1 Scaled Dot-Product Attention\\nWe call our particular attention \\\"Scaled Dot-Product Attention\\\" (Figure 2). The input consists of\\nqueries and keys of dimension dk, and values of dimension dv. We compute the dot products of the\\nquery with all keys, divide each bypdk, and apply a softmax function to obtain the weights on the\\nvalues.\\nIn practice, we compute the attention function on a set of queries simultaneously, packed together\\ninto a matrix Q. The keys and values are also packed together into matrices KandV. We compute\\nthe matrix of outputs as:\\nAttention(Q;K;V ) = softmax(QKT\\npdk)V (1)\\nThe two most commonly used attention functions are additive attention [ 2], and dot-product (multi-\\nplicative) attention. Dot-product attention is identical to our algorithm, except for the scaling factor\\nof1pdk. Additive attention computes the compatibility function using a feed-forward network with\\na single hidden layer. While the two are similar in theoretical complexity, dot-product attention is\\nmuch faster and more space-ef\\ufb01cient in practice, since it can be implemented using highly optimized\\nmatrix multiplication code.\\nWhile for small values of dkthe two mechanisms perform similarly, additive attention outperforms\\ndot product attention without scaling for larger values of dk[3]. We suspect that for large values of\\ndk, the dot products grow large in magnitude, pushing the softmax function into regions where it has\\nextremely small gradients4. To counteract this effect, we scale the dot products by1pdk.\\n3.2.2 Multi-Head Attention\\nInstead of performing a single attention function with dmodel-dimensional keys, values and queries,\\nwe found it bene\\ufb01cial to linearly project the queries, keys and values htimes with different, learned\\nlinear projections to dk,dkanddvdimensions, respectively. On each of these projected versions of\\nqueries, keys and values we then perform the attention function in parallel, yielding dv-dimensional\\noutput values. These are concatenated and once again projected, resulting in the \\ufb01nal values, as\\ndepicted in Figure 2.\\n4To illustrate why the dot products get large, assume that the components of qandkare independent random\\nvariables with mean 0and variance 1. Then their dot product, q\\u0001k=Pdk\\ni=1qiki, has mean 0and variance dk.\\n4 Multi-head attention allows the model to jointly attend to information from different representation\\nsubspaces at different positions. With a single attention head, averaging inhibits this.\\nMultiHead( Q;K;V ) = Concat(head 1;:::;head h)WO\\nwhere head i= Attention( QWQ\\ni;KWK\\ni;VWV\\ni)\\nWhere the projections are parameter matrices WQ\\ni2Rdmodel\\u0002dk,WK\\ni2Rdmodel\\u0002dk,WV\\ni2Rdmodel\\u0002dv\\nandWO2Rhdv\\u0002dmodel.\\nIn this work we employ h= 8 parallel attention layers, or heads. For each of these we use\\ndk=dv=dmodel=h= 64 . Due to the reduced dimension of each head, the total computational cost\\nis similar to that of single-head attention with full dimensionality.\\n3.2.3 Applications of Attention in our Model\\nThe Transformer uses multi-head attention in three different ways:\\n\\u000fIn \\\"encoder-decoder attention\\\" layers, the queries come from the previous decoder layer,\\nand the memory keys and values come from the output of the encoder. This allows every\\nposition in the decoder to attend over all positions in the input sequence. This mimics the\\ntypical encoder-decoder attention mechanisms in sequence-to-sequence models such as\\n[38, 2, 9].\\n\\u000fThe encoder contains self-attention layers. In a self-attention layer all of the keys, values\\nand queries come from the same place, in this case, the output of the previous layer in the\\nencoder. Each position in the encoder can attend to all positions in the previous layer of the\\nencoder.\\n\\u000fSimilarly, self-attention layers in the decoder allow each position in the decoder to attend to\\nall positions in the decoder up to and including that position. We need to prevent leftward\\ninformation \\ufb02ow in the decoder to preserve the auto-regressive property. We implement this\\ninside of scaled dot-product attention by masking out (setting to \\u00001) all values in the input\\nof the softmax which correspond to illegal connections. See Figure 2.\\n3.3 Position-wise Feed-Forward Networks\\nIn addition to attention sub-layers, each of the layers in our encoder and decoder contains a fully\\nconnected feed-forward network, which is applied to each position separately and identically. This\\nconsists of two linear transformations with a ReLU activation in between.\\nFFN(x) = max(0;xW 1+b1)W2+b2 (2)\\nWhile the linear transformations are the same across different positions, they use different parameters\\nfrom layer to layer. Another way of describing this is as two convolutions with kernel size 1.\\nThe dimensionality of input and output is dmodel = 512 , and the inner-layer has dimensionality\\ndff= 2048 .\\n3.4 Embeddings and Softmax\\nSimilarly to other sequence transduction models, we use learned embeddings to convert the input\\ntokens and output tokens to vectors of dimension dmodel. We also use the usual learned linear transfor-\\nmation and softmax function to convert the decoder output to predicted next-token probabilities. In\\nour model, we share the same weight matrix between the two embedding layers and the pre-softmax\\nlinear transformation, similar to [ 30]. In the embedding layers, we multiply those weights bypdmodel.\\n3.5 Positional Encoding\\nSince our model contains no recurrence and no convolution, in order for the model to make use of the\\norder of the sequence, we must inject some information about the relative or absolute position of the\\n5 Table 1: Maximum path lengths, per-layer complexity and minimum number of sequential operations\\nfor different layer types. nis the sequence length, dis the representation dimension, kis the kernel\\nsize of convolutions and rthe size of the neighborhood in restricted self-attention.\\nLayer Type Complexity per Layer Sequential Maximum Path Length\\nOperations\\nSelf-Attention O(n2\\u0001d) O(1) O(1)\\nRecurrent O(n\\u0001d2) O(n) O(n)\\nConvolutional O(k\\u0001n\\u0001d2)O(1) O(logk(n))\\nSelf-Attention (restricted) O(r\\u0001n\\u0001d)O(1) O(n=r)\\ntokens in the sequence. To this end, we add \\\"positional encodings\\\" to the input embeddings at the\\nbottoms of the encoder and decoder stacks. The positional encodings have the same dimension dmodel\\nas the embeddings, so that the two can be summed. There are many choices of positional encodings,\\nlearned and \\ufb01xed [9].\\nIn this work, we use sine and cosine functions of different frequencies:\\nPE(pos;2i)=sin(pos=100002i=d model)\\nPE(pos;2i+1)=cos(pos=100002i=d model)\\nwhereposis the position and iis the dimension. That is, each dimension of the positional encoding\\ncorresponds to a sinusoid. The wavelengths form a geometric progression from 2\\u0019to10000\\u00012\\u0019. We\\nchose this function because we hypothesized it would allow the model to easily learn to attend by\\nrelative positions, since for any \\ufb01xed offset k,PEpos+kcan be represented as a linear function of\\nPEpos.\\nWe also experimented with using learned positional embeddings [ 9] instead, and found that the two\\nversions produced nearly identical results (see Table 3 row (E)). We chose the sinusoidal version\\nbecause it may allow the model to extrapolate to sequence lengths longer than the ones encountered\\nduring training.\\n4 Why Self-Attention\\nIn this section we compare various aspects of self-attention layers to the recurrent and convolu-\\ntional layers commonly used for mapping one variable-length sequence of symbol representations\\n(x1;:::;x n)to another sequence of equal length (z1;:::;z n), withxi;zi2Rd, such as a hidden\\nlayer in a typical sequence transduction encoder or decoder. Motivating our use of self-attention we\\nconsider three desiderata.\\nOne is the total computational complexity per layer. Another is the amount of computation that can\\nbe parallelized, as measured by the minimum number of sequential operations required.\\nThe third is the path length between long-range dependencies in the network. Learning long-range\\ndependencies is a key challenge in many sequence transduction tasks. One key factor affecting the\\nability to learn such dependencies is the length of the paths forward and backward signals have to\\ntraverse in the network. The shorter these paths between any combination of positions in the input\\nand output sequences, the easier it is to learn long-range dependencies [ 12]. Hence we also compare\\nthe maximum path length between any two input and output positions in networks composed of the\\ndifferent layer types.\\nAs noted in Table 1, a self-attention layer connects all positions with a constant number of sequentially\\nexecuted operations, whereas a recurrent layer requires O(n)sequential operations. In terms of\\ncomputational complexity, self-attention layers are faster than recurrent layers when the sequence\\nlengthnis smaller than the representation dimensionality d, which is most often the case with\\nsentence representations used by state-of-the-art models in machine translations, such as word-piece\\n[38] and byte-pair [ 31] representations. To improve computational performance for tasks involving\\nvery long sequences, self-attention could be restricted to considering only a neighborhood of size rin\\n6 the input sequence centered around the respective output position. This would increase the maximum\\npath length to O(n=r). We plan to investigate this approach further in future work.\\nA single convolutional layer with kernel width k<n does not connect all pairs of input and output\\npositions. Doing so requires a stack of O(n=k)convolutional layers in the case of contiguous kernels,\\norO(logk(n))in the case of dilated convolutions [ 18], increasing the length of the longest paths\\nbetween any two positions in the network. Convolutional layers are generally more expensive than\\nrecurrent layers, by a factor of k. Separable convolutions [ 6], however, decrease the complexity\\nconsiderably, to O(k\\u0001n\\u0001d+n\\u0001d2). Even with k=n, however, the complexity of a separable\\nconvolution is equal to the combination of a self-attention layer and a point-wise feed-forward layer,\\nthe approach we take in our model.\\nAs side bene\\ufb01t, self-attention could yield more interpretable models. We inspect attention distributions\\nfrom our models and present and discuss examples in the appendix. Not only do individual attention\\nheads clearly learn to perform different tasks, many appear to exhibit behavior related to the syntactic\\nand semantic structure of the sentences.\\n5 Training\\nThis section describes the training regime for our models.\\n5.1 Training Data and Batching\\nWe trained on the standard WMT 2014 English-German dataset consisting of about 4.5 million\\nsentence pairs. Sentences were encoded using byte-pair encoding [ 3], which has a shared source-\\ntarget vocabulary of about 37000 tokens. For English-French, we used the signi\\ufb01cantly larger WMT\\n2014 English-French dataset consisting of 36M sentences and split tokens into a 32000 word-piece\\nvocabulary [ 38]. Sentence pairs were batched together by approximate sequence length. Each training\\nbatch contained a set of sentence pairs containing approximately 25000 source tokens and 25000\\ntarget tokens.\\n5.2 Hardware and Schedule\\nWe trained our models on one machine with 8 NVIDIA P100 GPUs. For our base models using\\nthe hyperparameters described throughout the paper, each training step took about 0.4 seconds. We\\ntrained the base models for a total of 100,000 steps or 12 hours. For our big models,(described on the\\nbottom line of table 3), step time was 1.0 seconds. The big models were trained for 300,000 steps\\n(3.5 days).\\n5.3 Optimizer\\nWe used the Adam optimizer [ 20] with\\f1= 0:9,\\f2= 0:98and\\u000f= 10\\u00009. We varied the learning\\nrate over the course of training, according to the formula:\\nlrate =d\\u00000:5\\nmodel\\u0001min(step_num\\u00000:5;step _num\\u0001warmup _steps\\u00001:5) (3)\\nThis corresponds to increasing the learning rate linearly for the \\ufb01rst warmup _steps training steps,\\nand decreasing it thereafter proportionally to the inverse square root of the step number. We used\\nwarmup _steps = 4000 .\\n5.4 Regularization\\nWe employ three types of regularization during training:\\nResidual Dropout We apply dropout [ 33] to the output of each sub-layer, before it is added to the\\nsub-layer input and normalized. In addition, we apply dropout to the sums of the embeddings and the\\npositional encodings in both the encoder and decoder stacks. For the base model, we use a rate of\\nPdrop= 0:1.\\n7 Table 2: The Transformer achieves better BLEU scores than previous state-of-the-art models on the\\nEnglish-to-German and English-to-French newstest2014 tests at a fraction of the training cost.\\nModelBLEU Training Cost (FLOPs)\\nEN-DE EN-FR EN-DE EN-FR\\nByteNet [18] 23.75\\nDeep-Att + PosUnk [39] 39.2 1:0\\u00011020\\nGNMT + RL [38] 24.6 39.92 2:3\\u000110191:4\\u00011020\\nConvS2S [9] 25.16 40.46 9:6\\u000110181:5\\u00011020\\nMoE [32] 26.03 40.56 2:0\\u000110191:2\\u00011020\\nDeep-Att + PosUnk Ensemble [39] 40.4 8:0\\u00011020\\nGNMT + RL Ensemble [38] 26.30 41.16 1:8\\u000110201:1\\u00011021\\nConvS2S Ensemble [9] 26.36 41.29 7:7\\u000110191:2\\u00011021\\nTransformer (base model) 27.3 38.1 3:3\\u00011018\\nTransformer (big) 28.4 41.8 2:3\\u00011019\\nLabel Smoothing During training, we employed label smoothing of value \\u000fls= 0:1[36]. This\\nhurts perplexity, as the model learns to be more unsure, but improves accuracy and BLEU score.\\n6 Results\\n6.1 Machine Translation\\nOn the WMT 2014 English-to-German translation task, the big transformer model (Transformer (big)\\nin Table 2) outperforms the best previously reported models (including ensembles) by more than 2:0\\nBLEU, establishing a new state-of-the-art BLEU score of 28:4. The con\\ufb01guration of this model is\\nlisted in the bottom line of Table 3. Training took 3:5days on 8P100 GPUs. Even our base model\\nsurpasses all previously published models and ensembles, at a fraction of the training cost of any of\\nthe competitive models.\\nOn the WMT 2014 English-to-French translation task, our big model achieves a BLEU score of 41:0,\\noutperforming all of the previously published single models, at less than 1=4the training cost of the\\nprevious state-of-the-art model. The Transformer (big) model trained for English-to-French used\\ndropout rate Pdrop= 0:1, instead of 0:3.\\nFor the base models, we used a single model obtained by averaging the last 5 checkpoints, which\\nwere written at 10-minute intervals. For the big models, we averaged the last 20 checkpoints. We\\nused beam search with a beam size of 4and length penalty \\u000b= 0:6[38]. These hyperparameters\\nwere chosen after experimentation on the development set. We set the maximum output length during\\ninference to input length + 50, but terminate early when possible [38].\\nTable 2 summarizes our results and compares our translation quality and training costs to other model\\narchitectures from the literature. We estimate the number of \\ufb02oating point operations used to train a\\nmodel by multiplying the training time, the number of GPUs used, and an estimate of the sustained\\nsingle-precision \\ufb02oating-point capacity of each GPU5.\\n6.2 Model Variations\\nTo evaluate the importance of different components of the Transformer, we varied our base model\\nin different ways, measuring the change in performance on English-to-German translation on the\\ndevelopment set, newstest2013. We used beam search as described in the previous section, but no\\ncheckpoint averaging. We present these results in Table 3.\\nIn Table 3 rows (A), we vary the number of attention heads and the attention key and value dimensions,\\nkeeping the amount of computation constant, as described in Section 3.2.2. While single-head\\nattention is 0.9 BLEU worse than the best setting, quality also drops off with too many heads.\\n5We used values of 2.8, 3.7, 6.0 and 9.5 TFLOPS for K80, K40, M40 and P100, respectively.\\n8 Table 3: Variations on the Transformer architecture. Unlisted values are identical to those of the base\\nmodel. All metrics are on the English-to-German translation development set, newstest2013. Listed\\nperplexities are per-wordpiece, according to our byte-pair encoding, and should not be compared to\\nper-word perplexities.\\nN d modeldffh d kdvPdrop\\u000flstrain PPL BLEU params\\nsteps (dev) (dev)\\u0002106\\nbase 6 512 2048 8 64 64 0.1 0.1 100K 4.92 25.8 65\\n(A)1 512 512 5.29 24.9\\n4 128 128 5.00 25.5\\n16 32 32 4.91 25.8\\n32 16 16 5.01 25.4\\n(B)16 5.16 25.1 58\\n32 5.01 25.4 60\\n(C)2 6.11 23.7 36\\n4 5.19 25.3 50\\n8 4.88 25.5 80\\n256 32 32 5.75 24.5 28\\n1024 128 128 4.66 26.0 168\\n1024 5.12 25.4 53\\n4096 4.75 26.2 90\\n(D)0.0 5.77 24.6\\n0.2 4.95 25.5\\n0.0 4.67 25.3\\n0.2 5.47 25.7\\n(E) positional embedding instead of sinusoids 4.92 25.7\\nbig 6 1024 4096 16 0.3 300K 4.33 26.4 213\\nTable 4: The Transformer generalizes well to English constituency parsing (Results are on Section 23\\nof WSJ)\\nParser Training WSJ 23 F1\\nVinyals & Kaiser el al. (2014) [37] WSJ only, discriminative 88.3\\nPetrov et al. (2006) [29] WSJ only, discriminative 90.4\\nZhu et al. (2013) [40] WSJ only, discriminative 90.4\\nDyer et al. (2016) [8] WSJ only, discriminative 91.7\\nTransformer (4 layers) WSJ only, discriminative 91.3\\nZhu et al. (2013) [40] semi-supervised 91.3\\nHuang & Harper (2009) [14] semi-supervised 91.3\\nMcClosky et al. (2006) [26] semi-supervised 92.1\\nVinyals & Kaiser el al. (2014) [37] semi-supervised 92.1\\nTransformer (4 layers) semi-supervised 92.7\\nLuong et al. (2015) [23] multi-task 93.0\\nDyer et al. (2016) [8] generative 93.3\\nIn Table 3 rows (B), we observe that reducing the attention key size dkhurts model quality. This\\nsuggests that determining compatibility is not easy and that a more sophisticated compatibility\\nfunction than dot product may be bene\\ufb01cial. We further observe in rows (C) and (D) that, as expected,\\nbigger models are better, and dropout is very helpful in avoiding over-\\ufb01tting. In row (E) we replace our\\nsinusoidal positional encoding with learned positional embeddings [ 9], and observe nearly identical\\nresults to the base model.\\n6.3 English Constituency Parsing\\nTo evaluate if the Transformer can generalize to other tasks we performed experiments on English\\nconstituency parsing. This task presents speci\\ufb01c challenges: the output is subject to strong structural\\n9 constraints and is signi\\ufb01cantly longer than the input. Furthermore, RNN sequence-to-sequence\\nmodels have not been able to attain state-of-the-art results in small-data regimes [37].\\nWe trained a 4-layer transformer with dmodel = 1024 on the Wall Street Journal (WSJ) portion of the\\nPenn Treebank [ 25], about 40K training sentences. We also trained it in a semi-supervised setting,\\nusing the larger high-con\\ufb01dence and BerkleyParser corpora from with approximately 17M sentences\\n[37]. We used a vocabulary of 16K tokens for the WSJ only setting and a vocabulary of 32K tokens\\nfor the semi-supervised setting.\\nWe performed only a small number of experiments to select the dropout, both attention and residual\\n(section 5.4), learning rates and beam size on the Section 22 development set, all other parameters\\nremained unchanged from the English-to-German base translation model. During inference, we\\nincreased the maximum output length to input length + 300. We used a beam size of 21and\\u000b= 0:3\\nfor both WSJ only and the semi-supervised setting.\\nOur results in Table 4 show that despite the lack of task-speci\\ufb01c tuning our model performs sur-\\nprisingly well, yielding better results than all previously reported models with the exception of the\\nRecurrent Neural Network Grammar [8].\\nIn contrast to RNN sequence-to-sequence models [ 37], the Transformer outperforms the Berkeley-\\nParser [29] even when training only on the WSJ training set of 40K sentences.\\n7 Conclusion\\nIn this work, we presented the Transformer, the \\ufb01rst sequence transduction model based entirely on\\nattention, replacing the recurrent layers most commonly used in encoder-decoder architectures with\\nmulti-headed self-attention.\\nFor translation tasks, the Transformer can be trained signi\\ufb01cantly faster than architectures based\\non recurrent or convolutional layers. On both WMT 2014 English-to-German and WMT 2014\\nEnglish-to-French translation tasks, we achieve a new state of the art. In the former task our best\\nmodel outperforms even all previously reported ensembles.\\nWe are excited about the future of attention-based models and plan to apply them to other tasks. We\\nplan to extend the Transformer to problems involving input and output modalities other than text and\\nto investigate local, restricted attention mechanisms to ef\\ufb01ciently handle large inputs and outputs\\nsuch as images, audio and video. Making generation less sequential is another research goals of ours.\\nThe code we used to train and evaluate our models is available at https:\\/\\/github.com\\/\\ntensorflow\\/tensor2tensor .\\nAcknowledgements We are grateful to Nal Kalchbrenner and Stephan Gouws for their fruitful\\ncomments, corrections and inspiration.\\nReferences\\n[1]Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint\\narXiv:1607.06450 , 2016.\\n[2]Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly\\nlearning to align and translate. CoRR , abs\\/1409.0473, 2014.\\n[3]Denny Britz, Anna Goldie, Minh-Thang Luong, and Quoc V . Le. Massive exploration of neural\\nmachine translation architectures. CoRR , abs\\/1703.03906, 2017.\\n[4]Jianpeng Cheng, Li Dong, and Mirella Lapata. Long short-term memory-networks for machine\\nreading. arXiv preprint arXiv:1601.06733 , 2016.\\n[5]Kyunghyun Cho, Bart van Merrienboer, Caglar Gulcehre, Fethi Bougares, Holger Schwenk,\\nand Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical\\nmachine translation. CoRR , abs\\/1406.1078, 2014.\\n[6]Francois Chollet. Xception: Deep learning with depthwise separable convolutions. arXiv\\npreprint arXiv:1610.02357 , 2016.\\n10 [7]Junyoung Chung, \\u00c7aglar G\\u00fcl\\u00e7ehre, Kyunghyun Cho, and Yoshua Bengio. Empirical evaluation\\nof gated recurrent neural networks on sequence modeling. CoRR , abs\\/1412.3555, 2014.\\n[8]Chris Dyer, Adhiguna Kuncoro, Miguel Ballesteros, and Noah A. Smith. Recurrent neural\\nnetwork grammars. In Proc. of NAACL , 2016.\\n[9]Jonas Gehring, Michael Auli, David Grangier, Denis Yarats, and Yann N. Dauphin. Convolu-\\ntional sequence to sequence learning. arXiv preprint arXiv:1705.03122v2 , 2017.\\n[10] Alex Graves. Generating sequences with recurrent neural networks. arXiv preprint\\narXiv:1308.0850 , 2013.\\n[11] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for im-\\nage recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern\\nRecognition , pages 770\\u2013778, 2016.\\n[12] Sepp Hochreiter, Yoshua Bengio, Paolo Frasconi, and J\\u00fcrgen Schmidhuber. Gradient \\ufb02ow in\\nrecurrent nets: the dif\\ufb01culty of learning long-term dependencies, 2001.\\n[13] Sepp Hochreiter and J\\u00fcrgen Schmidhuber. Long short-term memory. Neural computation ,\\n9(8):1735\\u20131780, 1997.\\n[14] Zhongqiang Huang and Mary Harper. Self-training PCFG grammars with latent annotations\\nacross languages. In Proceedings of the 2009 Conference on Empirical Methods in Natural\\nLanguage Processing , pages 832\\u2013841. ACL, August 2009.\\n[15] Rafal Jozefowicz, Oriol Vinyals, Mike Schuster, Noam Shazeer, and Yonghui Wu. Exploring\\nthe limits of language modeling. arXiv preprint arXiv:1602.02410 , 2016.\\n[16] \\u0141ukasz Kaiser and Samy Bengio. Can active memory replace attention? In Advances in Neural\\nInformation Processing Systems, (NIPS) , 2016.\\n[17] \\u0141ukasz Kaiser and Ilya Sutskever. Neural GPUs learn algorithms. In International Conference\\non Learning Representations (ICLR) , 2016.\\n[18] Nal Kalchbrenner, Lasse Espeholt, Karen Simonyan, Aaron van den Oord, Alex Graves, and Ko-\\nray Kavukcuoglu. Neural machine translation in linear time. arXiv preprint arXiv:1610.10099v2 ,\\n2017.\\n[19] Yoon Kim, Carl Denton, Luong Hoang, and Alexander M. Rush. Structured attention networks.\\nInInternational Conference on Learning Representations , 2017.\\n[20] Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR , 2015.\\n[21] Oleksii Kuchaiev and Boris Ginsburg. Factorization tricks for LSTM networks. arXiv preprint\\narXiv:1703.10722 , 2017.\\n[22] Zhouhan Lin, Minwei Feng, Cicero Nogueira dos Santos, Mo Yu, Bing Xiang, Bowen\\nZhou, and Yoshua Bengio. A structured self-attentive sentence embedding. arXiv preprint\\narXiv:1703.03130 , 2017.\\n[23] Minh-Thang Luong, Quoc V . Le, Ilya Sutskever, Oriol Vinyals, and Lukasz Kaiser. Multi-task\\nsequence to sequence learning. arXiv preprint arXiv:1511.06114 , 2015.\\n[24] Minh-Thang Luong, Hieu Pham, and Christopher D Manning. Effective approaches to attention-\\nbased neural machine translation. arXiv preprint arXiv:1508.04025 , 2015.\\n[25] Mitchell P Marcus, Mary Ann Marcinkiewicz, and Beatrice Santorini. Building a large annotated\\ncorpus of english: The penn treebank. Computational linguistics , 19(2):313\\u2013330, 1993.\\n[26] David McClosky, Eugene Charniak, and Mark Johnson. Effective self-training for parsing. In\\nProceedings of the Human Language Technology Conference of the NAACL, Main Conference ,\\npages 152\\u2013159. ACL, June 2006.\\n11 [27] Ankur Parikh, Oscar T\\u00e4ckstr\\u00f6m, Dipanjan Das, and Jakob Uszkoreit. A decomposable attention\\nmodel. In Empirical Methods in Natural Language Processing , 2016.\\n[28] Romain Paulus, Caiming Xiong, and Richard Socher. A deep reinforced model for abstractive\\nsummarization. arXiv preprint arXiv:1705.04304 , 2017.\\n[29] Slav Petrov, Leon Barrett, Romain Thibaux, and Dan Klein. Learning accurate, compact,\\nand interpretable tree annotation. In Proceedings of the 21st International Conference on\\nComputational Linguistics and 44th Annual Meeting of the ACL , pages 433\\u2013440. ACL, July\\n2006.\\n[30] O\\ufb01r Press and Lior Wolf. Using the output embedding to improve language models. arXiv\\npreprint arXiv:1608.05859 , 2016.\\n[31] Rico Sennrich, Barry Haddow, and Alexandra Birch. Neural machine translation of rare words\\nwith subword units. arXiv preprint arXiv:1508.07909 , 2015.\\n[32] Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc Le, Geoffrey Hinton,\\nand Jeff Dean. Outrageously large neural networks: The sparsely-gated mixture-of-experts\\nlayer. arXiv preprint arXiv:1701.06538 , 2017.\\n[33] Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdi-\\nnov. Dropout: a simple way to prevent neural networks from over\\ufb01tting. Journal of Machine\\nLearning Research , 15(1):1929\\u20131958, 2014.\\n[34] Sainbayar Sukhbaatar, Arthur Szlam, Jason Weston, and Rob Fergus. End-to-end memory\\nnetworks. In C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, editors,\\nAdvances in Neural Information Processing Systems 28 , pages 2440\\u20132448. Curran Associates,\\nInc., 2015.\\n[35] Ilya Sutskever, Oriol Vinyals, and Quoc VV Le. Sequence to sequence learning with neural\\nnetworks. In Advances in Neural Information Processing Systems , pages 3104\\u20133112, 2014.\\n[36] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jonathon Shlens, and Zbigniew Wojna.\\nRethinking the inception architecture for computer vision. CoRR , abs\\/1512.00567, 2015.\\n[37] Vinyals & Kaiser, Koo, Petrov, Sutskever, and Hinton. Grammar as a foreign language. In\\nAdvances in Neural Information Processing Systems , 2015.\\n[38] Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang\\nMacherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al. Google\\u2019s neural machine\\ntranslation system: Bridging the gap between human and machine translation. arXiv preprint\\narXiv:1609.08144 , 2016.\\n[39] Jie Zhou, Ying Cao, Xuguang Wang, Peng Li, and Wei Xu. Deep recurrent models with\\nfast-forward connections for neural machine translation. CoRR , abs\\/1606.04199, 2016.\\n[40] Muhua Zhu, Yue Zhang, Wenliang Chen, Min Zhang, and Jingbo Zhu. Fast and accurate\\nshift-reduce constituent parsing. In Proceedings of the 51st Annual Meeting of the ACL (Volume\\n1: Long Papers) , pages 434\\u2013443. ACL, August 2013.\\n12 Attention Visualizations\\nInput-Input Layer5\\nIt\\nis\\nin\\nthis\\nspirit\\nthat\\na\\nmajority\\nof\\nAmerican\\ngovernments\\nhave\\npassed\\nnew\\nlaws\\nsince\\n2009\\nmaking\\nthe\\nregistration\\nor\\nvoting\\nprocess\\nmore\\ndifficult\\n.\\n<EOS>\\n<pad>\\n<pad>\\n<pad>\\n<pad>\\n<pad>\\n<pad>\\nIt\\nis\\nin\\nthis\\nspirit\\nthat\\na\\nmajority\\nof\\nAmerican\\ngovernments\\nhave\\npassed\\nnew\\nlaws\\nsince\\n2009\\nmaking\\nthe\\nregistration\\nor\\nvoting\\nprocess\\nmore\\ndifficult\\n.\\n<EOS>\\n<pad>\\n<pad>\\n<pad>\\n<pad>\\n<pad>\\n<pad>\\nFigure 3: An example of the attention mechanism following long-distance dependencies in the\\nencoder self-attention in layer 5 of 6. Many of the attention heads attend to a distant dependency of\\nthe verb \\u2018making\\u2019, completing the phrase \\u2018making...more dif\\ufb01cult\\u2019. Attentions here shown only for\\nthe word \\u2018making\\u2019. Different colors represent different heads. Best viewed in color.\\n13 Input-Input Layer5\\nThe\\nLaw\\nwill\\nnever\\nbe\\nperfect\\n,\\nbut\\nits\\napplication\\nshould\\nbe\\njust\\n-\\nthis\\nis\\nwhat\\nwe\\nare\\nmissing\\n,\\nin\\nmy\\nopinion\\n.\\n<EOS>\\n<pad>\\nThe\\nLaw\\nwill\\nnever\\nbe\\nperfect\\n,\\nbut\\nits\\napplication\\nshould\\nbe\\njust\\n-\\nthis\\nis\\nwhat\\nwe\\nare\\nmissing\\n,\\nin\\nmy\\nopinion\\n.\\n<EOS>\\n<pad>\\nInput-Input Layer5\\nThe\\nLaw\\nwill\\nnever\\nbe\\nperfect\\n,\\nbut\\nits\\napplication\\nshould\\nbe\\njust\\n-\\nthis\\nis\\nwhat\\nwe\\nare\\nmissing\\n,\\nin\\nmy\\nopinion\\n.\\n<EOS>\\n<pad>\\nThe\\nLaw\\nwill\\nnever\\nbe\\nperfect\\n,\\nbut\\nits\\napplication\\nshould\\nbe\\njust\\n-\\nthis\\nis\\nwhat\\nwe\\nare\\nmissing\\n,\\nin\\nmy\\nopinion\\n.\\n<EOS>\\n<pad>Figure 4: Two attention heads, also in layer 5 of 6, apparently involved in anaphora resolution. Top:\\nFull attentions for head 5. Bottom: Isolated attentions from just the word \\u2018its\\u2019 for attention heads 5\\nand 6. Note that the attentions are very sharp for this word.\\n14 Input-Input Layer5\\nThe\\nLaw\\nwill\\nnever\\nbe\\nperfect\\n,\\nbut\\nits\\napplication\\nshould\\nbe\\njust\\n-\\nthis\\nis\\nwhat\\nwe\\nare\\nmissing\\n,\\nin\\nmy\\nopinion\\n.\\n<EOS>\\n<pad>\\nThe\\nLaw\\nwill\\nnever\\nbe\\nperfect\\n,\\nbut\\nits\\napplication\\nshould\\nbe\\njust\\n-\\nthis\\nis\\nwhat\\nwe\\nare\\nmissing\\n,\\nin\\nmy\\nopinion\\n.\\n<EOS>\\n<pad>\\nInput-Input Layer5\\nThe\\nLaw\\nwill\\nnever\\nbe\\nperfect\\n,\\nbut\\nits\\napplication\\nshould\\nbe\\njust\\n-\\nthis\\nis\\nwhat\\nwe\\nare\\nmissing\\n,\\nin\\nmy\\nopinion\\n.\\n<EOS>\\n<pad>\\nThe\\nLaw\\nwill\\nnever\\nbe\\nperfect\\n,\\nbut\\nits\\napplication\\nshould\\nbe\\njust\\n-\\nthis\\nis\\nwhat\\nwe\\nare\\nmissing\\n,\\nin\\nmy\\nopinion\\n.\\n<EOS>\\n<pad>Figure 5: Many of the attention heads exhibit behaviour that seems related to the structure of the\\nsentence. We give two such examples above, from two different heads from the encoder self-attention\\nat layer 5 of 6. The heads clearly learned to perform different tasks.\\n15\",\"1\":\" On the Bene\\ufb01ts of Biophysical Synapses\\nJulian Lemmel, Radu Grosu\\nFaculty of Informatics of Technische Universit \\u00a8at Wien, Austria.\\njulian.lemmel@tuwien.ac.at, radu.grosu@tuwien.ac.at\\nAbstract\\nThe approximation capability of ANNs and their RNN instan-\\ntiations, is strongly correlated with the number of parameters\\npacked into these networks. However, the complexity barrier\\nfor human understanding, is arguably related to the number\\nof neurons and synapses in the networks, and to the associ-\\nated nonlinear transformations. In this paper we show that\\nthe use of biophysical synapses, as found in LTCs, have two\\nmain bene\\ufb01ts. First, they allow to pack more parameters for\\na given number of neurons and synapses. Second, they allow\\nto formulate the nonlinear-network transformation, as a linear\\nsystem with state-dependent coef\\ufb01cients. Both increase inter-\\npretability, as for a given task, they allow to learn a system\\nlinear in its input features, that is smaller in size compared\\nto the state of the art. We substantiate the above claims on\\nvarious time-series prediction tasks, but we believe that our\\nresults are applicable to any feedforward or recurrent ANN.\\nIntroduction\\nInspired by spiking neurons, arti\\ufb01cial neurons (ANs) com-\\nbine in one unit, the additive behavior of biological neurons\\nwith the graded nonlinear behavior of their synapses (Bishop\\n1995; Goodfellow, Bengio, and Courville 2016). This makes\\nANs implausible from a biophysical point of view, and pre-\\ncluded their adoption in neural science.\\nArti\\ufb01cial neural networks (ANNs) however, correct this\\nbiological blunder. In ANNs it is irrelevant what is the mean-\\ning of a neuron, and what is that of a synapse. What mat-\\nters, is the mathematical expression of the network itself.\\nThis was best exempli\\ufb01ed by ResNets, which were forced,\\nfor technical reasons, to separate the additive transforma-\\ntion from the graded one, and introduce new state variables,\\nwhich are the outputs of the additive neural, rather than the\\nnonlinear synaptic units (He et al. 2016).\\nThis separation allows us to reconcile ResNets with liq-\\nuid time-constant neural networks (LTCs), a biophysical\\nmodel for nonspiking neurons, that shares architectural mo-\\ntifs, such as activation, inhibition, sequentialization, mutual\\nexclusion, and synchronization, with gene regulatory net-\\nworks (Lechner et al. 2019, 2020; Hasani et al. 2021; Alon\\n2007). LTCs capture the behavior of neurons in the retina of\\nlarge species (Kandel et al. 2013), and that of the neurons\\nCopyright \\u00a9 2022, Association for the Advancement of Arti\\ufb01cial\\nIntelligence (www.aaai.org). All rights reserved.in small species, such as the C.elegans nematode (Wicks,\\nRoehrig, and Rankin 1996). In LTCs, a neuron is a capaci-\\ntor, and its rate of change is the sum of a leaking current, and\\nof synaptic currents. The conductance of a synapse varies in\\na graded nonlinear fashion with the potential of the presy-\\nnaptic neuron, and this is multiplied with a difference of po-\\ntential of the postsynaptic neuron, to produce the synaptic\\ncurrent. Hence, the graded nonlinear transformation is the\\none that synapses perform, which is indeed the case in na-\\nture, and not the one performed by neurons.\\nIn contrast to ResNets, NeuralODEs and CT-RNNs (Chen\\net al. 2018; Funahashi and Nakamura 1993), LTCs multi-\\nply (or gate) the conductance with a difference of poten-\\ntial. This is dictated by physics, as one needs to obtain a\\ncurrent. Gating makes each neuron interpretable as a lin-\\near system with state-dependent coef\\ufb01cients (Alvarez-Melis\\nand Jaakkola 2018; C \\u00b8 imen 2008). Moreover, LTCs associate\\neach activation function to a synapse (like in nature) and not\\nto a neuron (like in ANNs). This allows LTCs to pack con-\\nsiderably more parameters in a network with a given number\\nof neurons and synapses. As the approximation capability\\nof ANNs and LTCs is strongly correlated with their num-\\nber of learnable parameters, LTCs are able to approximate of neurons and synapses. As the approximation capability\\nof ANNs and LTCs is strongly correlated with their num-\\nber of learnable parameters, LTCs are able to approximate\\nthe same behavior with a much smaller network, that is ex-\\nplainable in terms of its architectural motifs. We argue that\\nnonlinearity and the size of a neural network are the major\\ncomplexity barriers for human understanding.\\nMoving the activation functions to synapses can be ac-\\ncomplished in any ANN, with the same bene\\ufb01ts as for LTCs\\nin network-size reduction. The gating of sigmoidal activa-\\ntion functions can be replaced with hyperbolic-tangent acti-\\nvation functions. However, one looses the biophysical inter-\\npretation of a neural network, the linear interpretation of its\\nneurons, and the polarity of its synapses.\\nWe compared the expressive power of LTCs with that\\nof CT-RNNs, (Augmented) NeuralODEs, LSTMs, and CT-\\nGRUs, for various recurrent tasks. In this comparison, we\\nconsidered LTCs and CT-RNNs with both neural and synap-\\ntic activation functions. We also investigated the bene\\ufb01ts of\\ngating sigmoidal activation with a difference of potential.\\nOur results show that synaptic activation considerably re-\\nduces the number of neurons and associated synapses re-\\nquired to solve a task, not only in LTCs but also in CT-\\nRNNs. We also show that the use of hyperbolic-tangent ac-arXiv:2303.04944v1  [cs.NE]  8 Mar 2023 tivation functions in CT-RNNs has similar expressive power\\nas gating sigmoids with a difference of potential, but it\\nlooses the linear interpretation.\\nThe rest of the paper is structured as follows. First, we\\nprovide a fresh look into ANNs, ResNets, NeuralODEs, CT-\\nRNNs, and LTCs. This paves the way to then show the ben-\\ne\\ufb01ts of biophysical synapses in various recurrent tasks. Fi-\\nnally we discuss our results and touch on future work.\\nA Fresh Look at Neural Networks\\nArti\\ufb01cial Neural Networks\\nAn AN receives one or more inputs, sums them up in a linear\\nfashion, and passes the result through a nonlinear activation\\nfunction, whose bias bis the condition for the neuron to \\ufb01re\\n(spike). However, activation is graded (non-spiking), with\\nsmooth (e.g. sigmoidal) shape. Formally:\\nyt+1\\ni=\\u001b(nX\\nj=1wt\\njiyt\\nj+bt+1\\ni)\\u001b(x) =1\\n1 +e\\u0000x(1)\\nwhere as in Figure 1, yt+1\\niis the output of neuron iat layer\\nt+ 1,yt\\njis the output of neuron jat layert,wt\\njiis the weight\\nassociated to the synapse between neuron jat layertand\\nneuroniat layert+ 1,bt+1\\niis the bias (threshold) of neuron\\niat layert+ 1, and\\u001bis the activation function, e.g., the\\nlogistic function above. A network with one input layer, one\\noutput layer, and N\\u00152hidden layers, is called a deep neural\\nnetwork (DNN) (Goodfellow, Bengio, and Courville 2016).\\nANNs are universal approximators.\\nAlthough ANs are biophysically implausible, ANNs are\\nin fact closely related to nonspiking neural networks. To\\ndemonstrate this, let us look \\ufb01rst at ResNets (He et al. 2016).\\nResidual Neural Networks\\nDNNs with a large number of hidden layers suffer from the\\ndegradation problem, which persists even if the vanishing\\ngradients are curated. Intuitively, DNNs cannot accurately\\nlearn identities. Hence, they were simply added to the DNNs\\nin form of skip connections (He et al. 2016).\\nThe resulting architecture, as shown in Figure 1, was\\ncalled a residual neural network (ResNet)1. In ResNets, the\\noutputsxt\\niof the sums are distinguished from the outputs yt\\ni\\nof the sigmoids. Formally:\\nxt+1\\ni=xt\\ni+nX\\nj=1wt\\njiyt\\njyt\\nj=\\u001b(xt\\nj+bt\\nj) (2)\\nThis distinction is very important from a biophysical point of\\nview. The main idea is that neurons are just summation units,\\nand the sigmoidal transformation happens in synapses. In\\nfact, one can put the weights in the synaptic transformation,\\ntoo, which leads to the equivalent equations:\\nxt+1\\ni=xt\\ni+nX\\nj=1yt\\njiyt\\nji=wt\\nji\\u001b(xt\\nj+bt\\nj)(3)\\n1In (He et al. 2016), xt\\niskips the \\ufb01rst sum and it is added di-\\nrectly to xt+2\\ni. Hence, the architecture shown in Figure 1, can be\\nregarded as ResNets with \\ufb01nest skip granularity.\\nFigure 1: DNN (in black) and ResNet (in black and blue).\\nHerewt\\njican be thought of as the maximum conductance\\nof the input dependent synaptic transformation \\u001b(xt\\nj+bt\\nj).\\nThis transformation is indeed graded in nature, that is non-\\nspiking. Since ResNets are particular DNNs, with identity as\\na linear activation, they are also universal approximators.\\nNeural Ordinary Differential Equations\\nEquations (3) is the Euler discretization of a set of differ-\\nential equations, where the time step is simply taken to be\\none (E 2017; Chen et al. 2018). Mathematically:\\n_xi(t) =nX\\nj=1yji(t)yji(t) =wji(t)\\u001b(xj(t) +bj(t))(4)\\nIn these equations, x,y, and the parameters wandbchange\\ncontinuously in time. Now suppose we make the parameters\\nconstant. Are we still going to have a universal ODE approx-\\nimator? The answer is yes, as we will show in next section.\\nThe differential equations are as follows:\\n_xi(t) =nX\\nj=1yji(t)yji(t) =wji\\u001b(xj(t) +bj)(5)\\nThis is the form of Neural Ordinary Differential Equations\\n(NeuralODEs) (E 2017; Chen et al. 2018).2Taking the state\\nof the network as the sigmoid yof a sum is equivalent to\\ntaking the state as the sum xof sigmoids.\\nTheorem 1 (NeuralODEs) .Letxandybe state vectors.\\nThen _y=\\u001b(Wy+b)is equivalent to _x=W\\u001b(x+b).\\nProof. Takex=Wy. Then the following holds:\\n_x=W_y=W\\u001b(Wy+b) =W\\u001b(x+b)\\nA slight extension called ANODEs is given in (Dupont, Then _y=\\u001b(Wy+b)is equivalent to _x=W\\u001b(x+b).\\nProof. Takex=Wy. Then the following holds:\\n_x=W_y=W\\u001b(Wy+b) =W\\u001b(x+b)\\nA slight extension called ANODEs is given in (Dupont,\\nDoucet, and Teh 2019), which embeds the input in the inter-\\nnal state, and projects the state to the outputs S, as follows:\\nx(t0) = [x;0]Ty=\\u0019S(x(tN)) (6)\\nNeuralODEs are harder to learn than ResNets. For the train-\\ning purpose, one can use the adjoint equation, and employ\\nef\\ufb01cient numerical solvers (E 2017; Chen et al. 2018).\\n2Strictly speaking, NeuralODEs _x(t) =f(x)may have an arbi-\\ntrary number of neural layers for the function f. Figure 2: Synaptic-activation DNN and ResNet.\\nContinuous-Time Recurrent Neural Networks\\nAutonomous case. In this form of CT-RNNs, the input is\\nthe initial state. Let us call them ACT-RNNs, They extend\\nNeuralODEs with a leading term \\u0000wixi(t)(Funahashi and\\nNakamura 1993). Their mathematical form is as follows:\\n_xi(t) =\\u0000wixi(t) +Pn\\nj=1yji(t)\\nyji(t) =wji\\u001b(xj(t) +bj)(7)\\nThe leading term brings the system back to the equilibrium\\nstate, when no input is available. Hence, a small perturba-\\ntion is forgotten, that is, the system is stable. Like in Neu-\\nralODEs, one can interchange sumation and activation.\\nTheorem 2 (ACT-RNNs) .Letxandybe state vectors. Then\\n_y=\\u0000w\\u0003y+\\u001b(Wy+b)and_x=\\u0000w\\u0003x+W\\u001b(x+b)are\\nequivalent ODEs where \\u0003is the pointwise product of vectors.\\nProof. Letx=Wy. Then (Funahashi and Nakamura 1993):\\n_x=W_y\\n=W(\\u0000w\\u0003y+\\u001b(Wy+b)) =\\u0000w\\u0003x+W\\u001b(x+b)\\nACT-RNNs are universal approximators, and stabilization is\\nnot relevant in this respect (Funahashi and Nakamura 1993).\\nHence, NeuralODEs are universal approximators, too.\\nSynaptic activation. Like in ANNs, ACT-RNNs associate\\neach activation function to a neuron. We therefore call them\\nNA-ACT-RNNs, where NA stands for neural activation.\\nHowever, as shown in Figure 2, any ANN can be rewrit-\\nten, such that activation functions are associated to synapses.\\nWe call this form of ACT-RNNs, SA-ACT-RNN, where SA\\nstands for synaptic activation. Adding to each activation a\\nvariancea, too, one has the following explicit form:\\n_xi(t) =\\u0000wixi(t) +Pn\\nj=1yji(t)\\nyji(t) =wji\\u001b(ajixj(t) +bji)(8)\\nThe advantage of SA-ACT-RNNs is that they pack many\\nmore parameters in a network, for the same number of neu-\\nrons and synapses. For example, an SA-ACT-RNN with 32\\nneurons, connected in an all to all fashion, is able to pack\\n3104 parameters. This roughly corresponds to an NA-ACT-\\nRNN with 54 neurons which packs 3132 parameters.\\nWhile the approximation capability of a neural network\\nis strongly correlated with its number of parameters, we\\nstrongly believe that the complexity barrier for human un-\\nderstanding, is in the number of neurons and synapses.\\nFigure 3: The electric representation of a nonspiking neuron.\\nGeneral case. CT-RNNs have in general associated a time\\nvarying input signal u, too, that is, they are RNNs. The way\\nthe input is considered, plays a very important role.\\nA popular way of adding the input signal u, is to extend\\nthe sum within a sigmoid with a sum corresponding to the\\ninput. In vectorial form this looks as follows:\\n_y=\\u0000w\\u0003y+\\u001b(Wy+Vu+b) (9)\\nThis form has excellent convergence properties, but it cannot\\nbe extended to synaptic activations. We therefore prefer the\\nfollowing form, which has the same convergence properties:\\n_x=\\u0000w\\u0003x+W\\u001b(ax\\u0003x+bx)+V\\u001b(au\\u0003u+bu)(10)\\nwhereax;bxandau;burepresent the variance and the bias\\nvectors for the state and the input vectors, respectively. Fi-\\nnally, another popular way of adding the input to CT-RNNs\\nis in a linear fashion, as below.\\n_x=\\u0000w\\u0003x+W\\u001b(a\\u0003x+b)+Vu (11)\\nSynaptic activation. Like in SA-ACT-RNNs, the last two\\nCT-RNNs can be rewritten, by associating each activation to\\na synapse. To distinguish the two variants, we call them NA-\\nCT-RNNs and SA-CT-RNNs, respectively. In scalar form,\\nthe sigmoidal-input version can be written as below. The\\nlinear-input version is very similar:\\n_xi(t) =\\u0000wixi(t) +Pn\\nj=1yji(t) +Pm\\nj=1zji(t)\\nyji(t) =wji\\u001b(ax\\njixj(t) +bx\\nji)\\nzji(t) =vji\\u001b(au\\njiuj(t) +bu\\nji)(12)\\nNow consider an SA-CT-RNN with 32 neurons, connected\\nall to all, and with 32 inputs. It packs a total of 6176 param-\\neters. This roughly corresponds to an NA-CT-RNN with 54\\nneurons, which packs a total of 6102 parameters.\\nLiquid Time Constant Networks\\nAutonomous case. LTCs are a biophysical model for the\\nneural system of small species (Wicks, Roehrig, and Rankin\\n1996; Lechner et al. 2019, 2020; Hasani et al. 2021), and the\\nretina of large species (Kandel et al. 2013). Due to the small\\ndimension of these neural systems ( \\u00141mm), neural trans-\\nmission happens passively, in the analog domain, without retina of large species (Kandel et al. 2013). Due to the small\\ndimension of these neural systems ( \\u00141mm), neural trans-\\nmission happens passively, in the analog domain, without\\nconsiderable attenuation. Hence, the neurons do not need to\\nspike for an accurate signal transmission.\\nAs shown in Figure 3, the neuron\\u2019s membrane is an insu-\\nlator, with ions both on its inside and outside. Electrically, sigmpresynaptic  \\nactivation\\npostsynaptic  \\nactivationsynaptic  \\ncurrent\\nsigmpresynaptic  \\nactivation\\npostsynaptic  \\nactivationunsqueeze(x,-2)unsqueeze(x, -1)\\nsum(x,-2)\\nsynaptic  \\ncurrentFigure 4: Synaptic Layer in LTCs with synaptic (top) and neural (bottom) activation.\\nit is a capacitor. The difference between the inside-outside\\nionic concentrations de\\ufb01nes the membrane potential (MP)\\nx. The rate of change of xdepends on the currents ypass-\\ning through the membrane. These are either external currents\\n(ignored for ALTCs), a leakage current, and synaptic cur-\\nrents. For simplicity, we consider only chemical synapses.\\nThe capacitor equation is then as follows:\\nC_xi(t) =wli(eli\\u0000xi(t)) +Pn\\nj=1yji(t)\\nyji(t) =wji\\u001b(ajixj(t) +bji) (eji\\u0000xi(t))(13)\\nwhereCis the membrane capacitance, elithe resting poten-\\ntial,wlithe leaking conductance, and ejithe synaptic poten-\\ntials. These are either 0 mV for excitatory synapses (poten-\\ntialxiis negative so the current is positive), or -90 mV for\\ninhibitory synapses (the current is in this case negative).\\nEquations (18) are very similar to Equations (7) of an SA-\\nACT-RNN. They have a leaking current, which ensures the\\nstability of the ALTC, and a presynaptic-neuron controlled\\nconductance \\u001bfor the synapses, with maximum conduc-\\ntancewji. This conductance is multiplied with a difference\\nof potential eji\\u0000xi(t), to get a current. This biophysical\\nconstraint, makes them different from SA-ACT-RNNs. So\\nwhat is the signi\\ufb01cance of this gating term from the point of\\nview of machine learning? As we prove below, it has impor-\\ntant consequences for the interpretability of ALTCs.\\nTheorem 3 (Interpretability) .Each ALTC neuron is inter-\\npretable as linear regression of its inputs.\\nProof. Letx(0)be the input. This is propagated in time\\nasx(t). Letwji\\u001b(ajixj+bji)be the the state-dependent\\nweight from neuron jto neuroni. Then according to Equa-\\ntions (18), _xi(t)is a linear regression in x, for eachi. More-\\nover, small perturbations of xlead to small changes in _x.\\nALTCs are able to pack even more parameters than SA-\\nACT-RNNs. For example, an ALTC with 32 neurons, con-\\nnected in an all to all fashion, is able to pack 4192 parame-\\nters, whereas an SA-ACT-RNN packs only 3104 parameters.This roughly corresponds to an NA-ACT-RNN with 64 neu-\\nrons which packs 4224 parameters\\nNeural activation. While in nature each synapse has dis-\\ntinct dynamics, one may want to consider that in particu-\\nlar cases, all outgoing synapses of a neuron have the same\\nactivation parameters. Let us call this version of ALTCs as\\nNA-ALTCs, where NA stands for neural activation. We also\\ninterchangeably refer to ALTCs as SA-ALTCs. Formally:\\nC_xi(t) =wli(eli\\u0000xi(t)) +Pn\\nj=1yji(t)\\nyji(t) =wji\\u001b(ajxj(t) +bj) (ei\\u0000xi(t))(14)\\nIf one takes elito be zero and ajto be one, NA-ALTCs\\nare the same as NA-ACT-RNNs except for the gating term\\neji\\u0000xi(t). As discussed above, this term makes NA-ALTCs\\nlinear systems with state dependent coef\\ufb01cients, which not\\nonly makes them more interpretable, but allows the appli-\\ncation of state-dependent Ricatti equations in the automatic\\nsynthesis of nonlinear controllers (C \\u00b8 imen 2008).\\nIn our experiments, we found that CT-RNNs where the\\nactivation function is a hyperbolic tangent, has very similar\\nconvergence and learning-accuracy properties with LTCs,\\nwhere the activation function is a sigmoid. However, hyper-\\nbolic tangents fail to capture the opening degree of synaptic\\nchannels and their associated polarity, the same way gated\\nsigmoids do: sigmoids accurately capture this degree, and\\nthe difference of potential the gating polarity.\\nGeneral case. Like CT-RNNs, LTCs have in general asso-\\nciated a time-varying input signal u. As for CT-RNNs, we\\nconsider both a sigmoidal-input version and a linear-input\\nversion. In scalar form, the \\ufb01rst can be written as below:\\nC_xi(t) =wli(eli\\u0000xi(t)) +Pn\\nj=1yji(t) +Pm\\nj=1zji(t)\\nyji(t) =wji\\u001b(ax\\njixj(t) +bx\\nji)(eji\\u0000xi(t))\\nzji(t) =vji\\u001b(au\\njiuj(t) +bu\\nji)(eji\\u0000xi(t))(15) version. In scalar form, the \\ufb01rst can be written as below:\\nC_xi(t) =wli(eli\\u0000xi(t)) +Pn\\nj=1yji(t) +Pm\\nj=1zji(t)\\nyji(t) =wji\\u001b(ax\\njixj(t) +bx\\nji)(eji\\u0000xi(t))\\nzji(t) =vji\\u001b(au\\njiuj(t) +bu\\nji)(eji\\u0000xi(t))(15)\\nAn SA-LTC with 32 neurons, connected in an all to all fash-\\nion and with 32 inputs, packs a total of 8288 parameters, 600 2000 6000\\n# Params1.01.52.02.53.0test loss\\n600 2000 6000 20000\\n# Params1.01.52.02.53.0test lossNA-CT-RNN-S \\nNA-CT-RNN-T \\nNA-LTC \\nSA-CT-RNN-S \\nSA-CT-RNN-T \\nSA-LTC \\nLSTM\\nCT-GRU\\nANODEFigure 5: Results for the Walker2d kinematics-learning experiments. Left: synaptic inputs, Right: linear inputs. The size of the\\nmarker dots represent the number of neurons (or cells in case of LSTMs): 8, 16, 32 or 64 (from smallest to largest).\\n600 2000 6000\\n# Params0.0040.0060.0080.0100.0120.014test loss\\n600 2000 6000 20000\\n# Params0.0040.0060.0080.0100.0120.014test lossNA-CT-RNN-S \\nNA-CT-RNN-T \\nNA-LTC \\nSA-CT-RNN-S \\nSA-CT-RNN-T \\nSA-LTC \\nLSTM\\nCT-GRU\\nANODE\\nFigure 6: Results for the Half-Cheetah kinematics-learning experiments. Left\\/Right and marker size as before.\\nwhereas an SA-CT-RNN packs only 6176 parameters. This\\nroughly corresponds to an NA-CT-RNN with 63 neurons,\\nwhich packs a total of 8253 parameters.\\nThe linear-input version of SA-LTCs is very similar. For-\\nmally, it is described as below:\\nC_xi(t) =wli(eli\\u0000xi(t)) +Pn\\nj=1yji(t) +Pm\\nj=1zji(t)\\nyji(t) =wji\\u001b(ax\\njixj(t) +bx\\nji)(eji\\u0000xi(t))\\nzji(t) =vjiuj(t)(16)\\nAn SA-LTC with 32 neurons, connected in an all to all fash-\\nion and with 32 inputs, packs 5216 parameters. This roughly\\ncorresponds to a linear-input NA-CT-RNN with 51 neurons,\\nwhich packs a total of 5355 parameters.\\nNeural activation. Like in the autonous case, one can also\\nconsider NA-LTCs, where all outgoing synapses of a neuron\\nhave the same activation parameters. For the sigmoidal-input\\nversion one obtains the following equations:\\nC_xi(t) =wli(eli\\u0000xi(t)) +Pn\\nj=1yji(t) +Pm\\nj=1zji(t)\\nyji(t) =wji\\u001b(ax\\njxj(t) +bx\\nj)(ei\\u0000xi(t))\\nzji(t) =vji\\u001b(au\\njuj(t) +bu\\nj)(ei\\u0000xi(t))(17)\\nThe linear-input version is similar, but in this case zji(t) =\\nvjiuj(t). As for ALTCs, NA-LTCs are very similar to NA-\\nCT-RNNs, with the exception of the gating term eji\\u0000xi(t).As discussed before, one can get rid of gating, by using a\\nhyperbolic-tangent activation function, with its associated\\nloss of linearity and biophysical meaning.\\nLTCs are universal approximators (Hasani 2020; Hasani\\net al. 2021). This is true for both their autonomous and gen-\\neral form, and for synaptic and linear inputs.\\nExperimental Evaluation\\nSequential model structure\\nA three-layered sequential structure was used for all experi-\\nments in this section. Let us denote by u(t)the input at time\\ntand byyi(t)the output of layer iat timet. The output of\\nthe \\ufb01nal layer is the predicted output ^y(t) =y3(t).\\nThe \\ufb01rst layer maps the inputs to an RNN-layer with ei-\\nther a linear ( y1(t) =Ainu(t) +bin) or a synaptic (as dis-\\ncussed above) transformation - dubbed linear orsynaptic\\ninput mapping, respectively. In case of synaptic input map-\\nping these sensory synapses were implemented in accor-\\ndance with the RNN model used, i.e. they either used synap-\\ntic or neural activation, and did also incorporate the multi-\\nplication with a difference of potentials in case of LTCs.\\nThe second layer contains the RNN cells and its output\\nis computed by employing an ODE-solver. The actual ODE\\nbeing solved is determined by the model type. The different\\nvariants are explained in the preceding section. Unlike con-\\nventional NeuralODEs, the RNN cells in our model retain a\\nstate (and consequently information) after each time-step t.\\nThe third layer, irrespective of the speci\\ufb01c model type 600 2000 6000\\n# Params0.0050.0100.0150.0200.0250.030test loss\\n600 2000 6000 20000\\n# Params0.0050.0100.0150.0200.0250.030test lossNA-CT-RNN-S \\nNA-CT-RNN-T \\nNA-LTC \\nSA-CT-RNN-S \\nSA-CT-RNN-T \\nSA-LTC \\nLSTM\\nCT-GRU\\nANODEFigure 7: Results for the Half-Cheetah Behavioural Cloning modeling experiments. Left\\/Right and marker size as before.\\n600 2000 6000\\n# Params0.10.20.30.40.5test loss\\n600 2000 6000 20000\\n# Params0.10.20.30.40.5test lossNA-CT-RNN-S\\nNA-CT-RNN-T\\nNA-LTC\\nSA-CT-RNN-S\\nSA-CT-RNN-T\\nSA-LTC\\nLSTM\\nCT-GRU\\nFigure 8: Results for the Sequential MNIST classi\\ufb01cation experiments. Left\\/Right and marker size as before.\\nused, maps the \\ufb01nal RNN state y2(t)to the output vector\\ny3(t)in a linear fashion, that is, y3(t) =Aouty2(t) +bout.\\nNeural and synaptic dynamics were implemented as py-\\ntorch-lightning modules, ensuring re-usability and portabil-\\nity across different devices such as CPU and GPU. Upon\\ninstantiating the module, the desired model type is speci\\ufb01ed\\nand the parameters are initialized accordingly. Initialization\\nbounds were taken from (Hasani et al. 2021), and are given\\nin the Supplementary Materials. In order to reduce the pa-\\nrameter space, some parameters were \\ufb01xed at some value\\nand were not subject to training through backpropagation.\\nSince the resulting system of ODEs is stiff, the choice of\\nthe ODE-solver has a strong impact on the performance. We\\nchose the explicit Euler solver with 10 unfolds for all the\\nexperiments in this paper, as it gave good enough accuracy\\nwith low time-complexity, compared to more sophisticated\\nsolvers such as Runge-Kutta methods (rk4 or dopri5).\\nThe chain of computations for a layer of synapses with\\nneural and synaptic activation is shown in Figure 4 top and\\nbottom respectively. Here, xandyare the state of the pre-\\nsynaptic and post-synaptic neuron respectively. LTC-RNNs\\nextend CT-RNN synapses by multiplying the activation with\\na difference-of-potential term (bottom part of the \\ufb01gures).\\nSynaptic activation was realised by extending vectors to ma-\\ntrices throughout the computation graph while also replacing\\neach corresponding matrix-multiplication by the element-\\nwise (or Hadamard) product. Particularly, this means that\\nintermediary results are also represented as matrices while\\nthey are vector-valued in case of neural activation.Robotic Experiments\\nTo explore the parameter-packing and the linear-gating ben-\\ne\\ufb01ts of biophysical synapses we conducted four supervised-\\nlearning time-series experiments: Walker2d prediction,\\nHalf-Cheetah prediction, Half-Cheetah behavioural cloning,\\nand Sequential-MNIST classi\\ufb01cation.\\nThe parameter-packing bene\\ufb01ts are evaluated by using\\nCT-RNNs and LTCs with 8, 16, 32, and 64 neurons (con-\\nnected all-to-all) in the hidden layer, with both neural and\\nsynaptic activation, and with both synaptic and linear in-\\nputs. The gating bene\\ufb01ts are evaluated by using CT-RNNs\\nwith both sigmoid and tanh activation functions.\\nWe also provide results for LSTMs, CT-GRUs and AN-\\nODEs with 10 augmenting dimensions. We use a linear input\\nmapping as this is the default for the \\ufb01rst two. By lacking a\\nstabilization term, the simple NeuralODEs discussed before\\nperform worse than CT-RNNs, and we do not show them in\\nour results.\\nAlthough our results are mainly for robotic control and\\nthey are restricted to LTCs and CT-RNNs, we claim that they\\nare applicable to any feedforward or recurrent ANN.\\nThe CT-RNN and LTC models tested are abbreviated as\\nNA-CT-RNN and NA-LTC for neural activation, and SA-\\nCT-RNN and SA-LTC for synaptic activation. CT-RNNs\\nused either a sigmoidal or a hyperbolic-tangent activation,\\nmarked with the suf\\ufb01x S and T, respectively. LTCs always\\nused a sigmoidal activation, because a hyperbolic tangent\\nnot only fails to capture the biophysics of synapses, but also\\nrenders the network very unstable. For all models we did ex-\\nperiments with both linear- and synaptic-input mappings, as the latter more closely capture sensory neurons. All experi-\\nments used the Adam optimizer (Kingma and Ba 2014).\\nLearning Walker-2D kinematics. This robotic task is in-\\nspired by the physics simulation in (Rubanova, Chen, and\\nDuvenaud 2019), and implemented by using the gym envi-\\nronment in (Brockman et al. 2016). It evaluates how well\\nvarious RNNs are suited to learn kinematic dynamics.\\nTo create the training dataset, we used a non-recurrent\\npolicy, pretrained via proximal policy optimization (PPO)\\n(Schulman et al. 2017), and the Rllib (Liang et al. 2017) re-\\ninforcement learning framework. To increase the task com-\\nplexity, we used the pretrained policy at 4 different training\\nstages (between 500 to 1200 PPO iterations). We then col-\\nlected 17-dimensional observation vectors, performing 400\\nrollouts of 1000 steps each on the Walker2d-v2 OpenAI\\ngym environment and the MuJoCo physics engine (Todorov,\\nErez, and Tassa 2012). Note that there is no need to include\\nthe actions in the training set, because the policy is determin-\\nistic. We used 15% percent of the dataset for testing, 10%\\npercent for validation and the rest for training.\\nWe aligned the rollouts into sequences of length 20 and\\nthen trained each of the models three times for 200 epochs.\\nThis was done for 8, 16, 32, and 64 RNN cells.\\nFigure 5 shows for each model the median test loss and its\\nmin and max values, for three runs, with respect to the num-\\nber of neurons, and the associated number of parameters.\\nCT-RNNs perform better for the linear input-mapping,\\nwhereas SA-LTCs for synaptic input-mapping. The pack-\\ning bene\\ufb01t of biophysical synapses is seen in the fact that\\nSA-CT-RNNs and SA-LTCs pack essentially as many pa-\\nrameters as NA-CT-RNNs and NA-LTCs, with half of the\\nnumber of neurons. The gating bene\\ufb01t is exempli\\ufb01ed by the\\nfact that CT-RNN-Ts perform better than CT-RNN-Ss. LTCs\\nperform better than CT-RNNs in all instances. LSTMs and\\nCT-GRUs attain (even greater) parameter packing through\\na more elaborate concept of a structured cell, but not nec-\\nessarily with greater accuracy, when their number of cells\\nequals the number of neurons of SA-LTCs. Since LSTMs\\nand CT-GRUs have by default a linear input-mapping, they\\nappear only in the right \\ufb01gure. ANODEs, are also shown in\\nthe right \\ufb01gure. They perform comparable to NA-LTCs.\\nLearning Half-Cheetah kinematics. Similar to the Wal-\\nker-2D, we learned the kinematics of the Half-Cheetah.\\nFor this experiment we collected 100 rollouts with a con-\\ntroller that was trained using Truncated Quantile Critics\\n(TQC) (Kuznetsov et al. 2020). Just a single version pro-\\nvided by the stable-baselines zoo (Raf\\ufb01n 2018) was used\\nthis time, making the task relatively easier than the previ-\\nous one. Again, each rollout is composed of a series of 1000\\ndatapoints consisting of a 27-dimension observation vector\\ngenerated by the MuJoCo physics engine and a 6-dimension\\naction vector that is produced by the controller. The same\\ndata was used in the following two different tasks:\\n1.Kinematics modeling. Predicting the next observation af-\\nter having seen 20 preceding observations. The action\\nvectors are not used for this task since the observations\\nserve both as inputs and as labels.2.Behavioural cloning. Predicting the next action after hav-\\ning seen 20 preceding observations. In this task the ob-\\nservations serve as inputs while the actions are the labels.\\nFigure 6 shows the results for the Half-Cheetah kinematic\\nmodeling, and Figure 7 the ones for Half-Cheetah behavioral\\ncloning. The median, min and max test loss are represented\\nas before. The results in both \\ufb01gures follow a very similar\\npattern as the ones for Walker-2D. However, in this case\\nthe bene\\ufb01ts of CT-RNN-Ts are evident only for the synaptic\\ninput-mapping. LTCs remain more performant. In Figure 7,\\nLTSMs and CT-GRUs have a slightly better accuracy com-\\npared to SA-LTCs, at the expense of more parameters.\\nSequential-MNIST classi\\ufb01cation. The MNIST dataset LTSMs and CT-GRUs have a slightly better accuracy com-\\npared to SA-LTCs, at the expense of more parameters.\\nSequential-MNIST classi\\ufb01cation. The MNIST dataset\\nconsists of 70,000 gray-scale images of 28 \\u000228 pixels, con-\\ntaining hand-written digits (LeCun 1998). In order to make\\nthis task as a sequential one, the images are transformed into\\nsequences of length 28, by taking each row vector, as an in-\\nput in time. The desired output is a one-hot encoded vec-\\ntor representing integers from 0 to 9. Consequently, a cross-\\nentropy loss was used when training the models.\\nThe results shown in Figure 8 are as before, the median,\\nmin and max test loss of three runs each. They follow a\\nsimilar pattern with the previous \\ufb01gures, but the LTCs with\\nsynaptic inputs are less stable, and fail to properly converge\\nfor the largest number of neurons. The best accuracy is at-\\ntained by NA-CTRNNs and LSTMs.\\nDiscussion and Conclusion\\nThe main goal of this paper was to investigate the synaptic-\\nactivation and linear-gating bene\\ufb01ts of biophysical synapses,\\nas they occur in LTCs. To this end, we asked:\\n\\u2022 What happens if one uses neural activation in LTCs?\\n\\u2022 What happens if linear gating is dropped in LTCs?\\nThis resulted in two versions of LTCs, and four versions\\nof CT-RNNs, with either linear or synaptic input, and with\\nsigmoid or tanh activation, respectvely. We thoroughly ex-\\namined the accuracy and parameter-packing ability of these\\nnetworks, for an increasing number of neurons, and com-\\npared them to those of ANODEs, LSTMs, and CT-GRUs.\\nWe observed that LTCs and CT-RNNs with synaptic acti-\\nvation achieve essentially the same accuracy and parameter\\npacking, for half of the number of neurons, as LTCs and\\nCT-RNNs with neural activation. The linear gating of LTCs\\nfurther improved this accuracy. We also observed that the ac-\\ncuracy and packing bene\\ufb01ts of LTCs is comparable to those\\nof cells in LSTMs. However, the latter rely on a much more\\nelaborate concept of a structured cell.\\nWe claimed that the bene\\ufb01ts of biophysical synapses ap-\\nply to any ANN. However, we showed them explicitly for\\nLTCs and CT-RNNs, only. Hence, the full version of this pa-\\nper would have to substantiate this claim. For example, for\\nfeed-forward CNNs, one could use a standard CNN base,\\nand consistently replace its neural activations with synap-\\ntic ones. Similarly, for LTSMs and CT-GRUs, one could\\nmake their recurrent connections synaptic, by using a tanh-\\nactivation for each synapse, instead of one for each cell. For clarity, we kept NeuralODEs as simple as possible, by\\ncon\\ufb01ning their right-hand-side transformation to one layer.\\nHowever, one could have used more powerful transforma-\\ntions, which might have led to better results. Nevertheless,\\nwe think that the discussed bene\\ufb01ts would still apply.\\nFinally, biophysical synapses may better support sparse\\nnetworks too, as it was claimed in (Lechner et al. 2020).\\nHowever, for obvious space reasons, a thorough investiga-\\ntion of this claim had to be postponed to future work.\\nReferences\\nAlon, U. 2007. Network Motifs: Theory and Experimental\\nApproaches. Nature Reviews , 8.\\nAlvarez-Melis, D.; and Jaakkola, T. 2018. Towards Robust\\nInterpretability with Self-Explaining Neural Networks. In\\nProceedings of NIPS\\u201918, the 32nd Conference on Neural In-\\nformation Processing Systems . Montreal, Canada.\\nBishop, C. 1995. Neural Networks for Pattern Recognition .\\nClaredon Press, Oxford.\\nBrockman, G.; Cheung, V .; Pettersson, L.; Schneider, J.;\\nSchulman, J.; Tang, J.; and Zaremba, W. 2016. Openai gym.\\narXiv preprint arXiv:1606.01540 .\\nChen, T.; Rubanova, Y .; Bettencourt, J.; and Duvenaud, D.\\n2018. Neural Ordinary Differential Equations. In Advances\\nin Neural Information Processing Systems , 6571\\u20136583.\\nDupont, E.; Doucet, A.; and Teh, Y . 2019. Augmented Neu-\\nral ODEs. ArXiv, eprint 1904.01681, arXiv:1904.01681.\\nE, W. 2017. A Proposal on Machine Learning via Dynamical\\nSystems. Communications in Mathematics and Statistics , 5:\\n1\\u201311.\\nFunahashi, K.; and Nakamura, Y . 1993. Approximation of\\nDynamical Systems by Continuous Time Recurrent Neural\\nNetworks. Neural networks , 6(6): 801\\u2013806.\\nGoodfellow, I.; Bengio, Y .; and Courville, A. 2016. Deep\\nLearning . MIT Press.\\nHasani, R. 2020. Interpretable Recurrent Neural Networks\\nin Continuous-time Control Environments . Ph.D. thesis,\\nWien.\\nHasani, R.; Lechner, M.; Amini, A.; Rus, D.; and Grosu,\\nR. 2021. Liquid Time-constant Networks. Proceedings of\\nthe AAAI Conference on Arti\\ufb01cial Intelligence , 35(9): 7657\\u2013\\n7666.\\nHe, K.; Zhang, X.; Ren, S.; and Sun, J. 2016. Identity Map-\\npings in Deep Residual Networks. CoRR , abs\\/1603.05027.\\nKandel, E.; Schwartz, J.; Jessel, T.; Siegelbaum, S.; and\\nHudspeth, A. 2013. Principles of Neural Science . McGraw-\\nHill Education \\/ McGraw-Hill Medical, 5 edition.\\nKingma, D.; and Ba, J. 2014. Adam: A Method for Stochas-\\ntic Optimization. arXiv preprint arXiv:1412.6980 .\\nKuznetsov, A.; Shvechikov, P.; Grishin, A.; and Vetrov,\\nD. 2020. Controlling Overestimation Bias with Trun-\\ncated Mixture of Continuous Distributional Quantile Critics.\\narXiv:2005.04269 [cs, stat] .\\nLechner, M.; Hasani, R.; Amini, A.; Henzinger, T.; Rus, D.;\\nand Grosu, R. 2020. Neural circuit policies enabling au-\\nditable autonomy. Nature Machine Intelligence , 2: 642\\u2013652.Lechner, M.; Hasani, R.; Zimmer, M.; Henzinger, T.; and\\nGrosu, R. 2019. Designing Worm-Inspired Neural Networks\\nfor Interpretable Robotic Control. In Proceedings of the\\n2019 International Conference on Robotics and Automation\\n(ICRA) . Montreal, Canada.\\nLeCun, B., Cortes. 1998. MNIST Handwritten Digit\\nDatabase, Yann LeCun, Corinna Cortes and Chris Burges.\\nhttp:\\/\\/yann.lecun.com\\/exdb\\/mnist\\/.\\nLiang, E.; Liaw, R.; Nishihara, R.; Moritz, P.; Fox, R.; Gon-\\nzalez, J.; Goldberg, K.; and Stoica, I. 2017. Ray RLLib: A\\nComposable and Scalable Reinforcement Learning Library.\\nCoRR , abs\\/1712.09381.\\nPoli, M.; Massaroli, S.; Yamashita, A.; Asama, H.; and Park,\\nJ. 2020. TorchDyn: A Neural Differential Equations Library.\\narXiv preprint arXiv:2009.09346 .\\nRaf\\ufb01n, A. 2018. RL Baselines Zoo.\\nRubanova, Y .; Chen, R. T.; and Duvenaud, D. 2019. La-\\ntent odes for irregularly-sampled time series. arXiv preprint\\narXiv:1907.03907 .\\nSchulman, J.; Wolski, F.; Dhariwal, P.; Radford, A.; and\\nKlimov, O. 2017. Proximal policy optimization algorithms.\\narXiv preprint arXiv:1707.06347 .\\nTodorov, E.; Erez, T.; and Tassa, Y . 2012. Mujoco: A physics\\nengine for model-based control. In 2012 IEEE\\/RSJ Interna- arXiv preprint arXiv:1707.06347 .\\nTodorov, E.; Erez, T.; and Tassa, Y . 2012. Mujoco: A physics\\nengine for model-based control. In 2012 IEEE\\/RSJ Interna-\\ntional Conference on Intelligent Robots and Systems , 5026\\u2013\\n5033. IEEE.\\nWicks, S.; Roehrig, C.; and Rankin, C. 1996. A Dynamic\\nNetwork Simulation of the Nematode Tap Withdrawal Cir-\\ncuit: Predictions Concerning Synaptic Function Using Be-\\nhavioral Criteria. Journal of Neuroscience , 16(12): 4017\\u2013\\n4031.\\nC \\u00b8 imen, T. 2008. State-Dependent Riccati Equation (SDRE)\\nControl: A Survey. Proceedings of the 17th World Congress\\nof the International Federation of Automatic Control , 6\\u201311.\\nAppendix\\nOur implementation is using the torchdyn pack-\\nage (Poli et al. 2020) which in turn is based on\\npytorch-lightning . It comprises several different\\nODE-solvers supporting automatic differentiation. Since\\nthe package by itself was intended for implementing au-\\ntonomous NeuralODEs, a small trick was necessary for cre-\\nating CT-RNNs and LTC-RNNs that receive inputs. Au-\\ntonomous NeuralODEs encapsulate a non-trivial function\\n(such as a neural network) that is used for computing\\nthe derivative at each ODE-solver stepdx\\ndt=F(x). For\\nachieving the non-autonomous behaviordx\\ndt=F0(x;u)\\nthe input was appended to the state at each step when\\npassed to the ODE-solver and it\\u2019s derivative was set to zero\\n[dx\\ndt;0] =F([x;u]). Consequently, the solution computed\\nby the ODE-solver amounts to being the unaltered input ap-\\npended to the desired \\ufb01nal state.\\nC_xi(t) =wli(eli\\u0000xi(t)) +Pn\\nj=1yji(t)\\nyji(t) =wji\\u001b(ajixj(t) +bji) (eji\\u0000xi(t))(18) Table 2: Parameters of the recurrent layer and their initial-\\nization bounds.\\u0003only present in LTC-RNNs.\\nName Description Initialization\\nsynaptic parameters\\nC\\u0003Membrane capacitance 1 (\\ufb01xed)\\nw Synaptic strength 0.01 - 1.0\\nb Synaptic midpoint 0.3 - 0.8\\na Synaptic slope 3 - 8\\ne\\u0003Reversal potential -1 or +1\\ncell body parameters\\nel Resting potential 0 (\\ufb01xed)\\nwl Leakage conductance 0.01 - 1.0\\nTable 2 shows the initialization ranges used for the LTC\\nlayer parameters taken from previous work. Since LTC dy-\\nnamics tend to diverge rapidly if leaving the parameters un-\\nconstrained, the following conditions were enforced during\\ntraining:C\\u00150,w\\u00150,a\\u00150. These constraints are a\\ndirect consequence of the capacitor equation Eq. 18 from\\nwhich LTCs are derived wherein neither conductance ( w),\\ncapacitance ( Cm) nor synaptic gating ( a) may be negative.\\nSimilar constraints are also assumed in the proof of universal\\napproximation found in (Hasani et al. 2021).\\nType-descriptors\\nThe form of the derivative used whithin the NeuralODE is\\ncon\\ufb01gured by passing a type descriptor which is a string of\\nthe form:\\nctrnn [w-mode]act[factor][rec-type] in-mode [lis]\\nIn any case the derivative is computed using the following\\nformula: _x=Synapses (x;x) +Input (u;x)\\u0000decay (x).\\nThe type descriptor determines how the individual terms are\\ncalculated.\\nSizes ofE,band\\u000bdepend on the recurrence type used:Table 3: Type descriptor components and resulting formulas\\nw-mode Synapses (x;y)\\nNone a\\u001b(wx+b)\\u0003Factor (y)\\nr w\\u001b(x+b)\\u0003Factor (y)\\nv w\\u001b(a(x+b))\\u0003Factor (y)\\nact \\u001b(x)\\n\\u2019sigm\\u2019 1=(1 +e\\u0000x)\\n\\u2019tanh\\u2019 tanh(x)\\nfactor Factor (x)\\nNone 1\\n\\u2019*\\u2019 (1\\u0000x)\\n\\u2019+\\u2019 (e\\u0000x)\\nrec-type\\nNone neuronal activation\\n\\u2019s\\u2019 synaptic activation\\nin-mode Input (u;x)\\n\\u2019linear\\u2019 Iu+bi\\n\\u2019synaptic\\u2019 Synapses (u;x)\\nlis decay (x)\\nNone \\u001cx\\n\\u2019lis\\u2019 \\u001c(x\\u0000x0)\\nRecurrence type Shape of parameters\\nneuronal ( None ) ( model size)\\nsynaptic (\\u2019s\\u2019) ( model size,model size)\\nThe different w-mode s allow for adjusting the way\\nSynapses are parameterized, actdetermines the activation\\nfunction used, the factor distinguishes LTCs from CT-\\nRNNs, rec-type is used for switching between Neuronal and\\nSynaptic activation ,in-mode determines the input mode and\\nTable 1: Results for the INSERT experiments\\nModel n = 8 n = 16 n = 32 n = 64\\n# Par MSE \\u000210\\u00002# Par MSE \\u000210\\u00002# Par MSE \\u000210\\u00002# Par MSE \\u000210\\u00002\\nANODE 1663 0:62\\u00060:04 2263 0:46\\u00060:01 3463 0:40\\u00060:02 5863 0:31\\u00060:01\\nCT-GRU 5139 0:84\\u00060:05 12427 0:50\\u00060:02 - -\\nLSTM 1427 0:94\\u00060:00 3339 0:53\\u00060:02 8699 0:38\\u00060:00 -\\nNA-CT-RNN linear 555 1:10\\u0006nan 1211 0:60\\u0006nan 2907 0:41\\u00060:01 7835 0:45\\u00060:02\\nNA-CT-RNN synaptic 601 1:04\\u00060:04 1249 0:58\\u00060:02 2929 0:45\\u00060:08 7825 0:57\\u00060:04\\nNA-LTC linear 571 1:15\\u00060:07 1243 0:56\\u00060:01 2971 0:40\\u0006nan 7963 0:32\\u00060:07\\nNA-LTC synaptic 617 1:08\\u0006nan 1281 0:67\\u0006nan 2993 0:52\\u00060:01 7953 0:61\\u00060:06\\nSA-CT-RNN linear 667 1:01\\u00060:02 1691 0:54\\u00060:00 4891 0:36\\u0006nan -\\nSA-CT-RNN synaptic 1091 0:88\\u00060:04 2539 0:61\\u0006nan 6587 0:50\\u00060:04 -\\nSA-LTC linear 947 0:93\\u0006nan 2379 0:46\\u00060:02 6779 0:33\\u00060:03 -\\nSA-LTC synaptic 1371 0:82\\u00060:03 3227 0:47\\u00060:03 8475 0:28\\u0006nan - lisis used when the initial state (= leakage potential) should\\nbe learnable.\\nThe recurrence type determines how recurrenct connec-\\ntions are implemented:\\n\\u2022neuronal : corresponds to the ctrnn model. Connections\\ncan be thought of being only of linear nature, all incom-\\ning synapses of a particular cell share the same bias and\\nsummation happens before applying the activation func-\\ntion.\\n\\u2022synaptic : each synapse has a separate bias b, scalew\\nand reference potential E. An additional summation step\\nhappens after calculating individual synaptic activations.\\n_x=P[Synapses (x;x)] +Input (u;x)\\u0000decay (x)\\nExamples\\n\\u2022 Vanilla CT-RNN: ctrnn vtanh linear\\n\\u2022 SA-LTC: ctrnn vsigm+s synaptic\",\"2\":\" MOREA: a GPU-accelerated Evolutionary Algorithm for\\nMulti-Objective Deformable Registration of 3D Medical Images\\nGeorgios Andreadis\\nLeiden University Medical Center\\nLeiden, The Netherlands\\nG.Andreadis@lumc.nlPeter A.N. Bosman\\nCentrum Wiskunde & Informatica\\nAmsterdam, The Netherlands\\nPeter.Bosman@cwi.nlTanja Alderliesten\\nLeiden University Medical Center\\nLeiden, The Netherlands\\nT.Alderliesten@lumc.nl\\nABSTRACT\\nFinding a realistic deformation that transforms one image into\\nanother, in case large deformations are required, is considered\\na key challenge in medical image analysis. Having a proper im-\\nage registration approach to achieve this could unleash a number\\nof applications requiring information to be transferred between\\nimages. Clinical adoption is currently hampered by many exist-\\ning methods requiring extensive configuration effort before each\\nuse, or not being able to (realistically) capture large deformations.\\nA recent multi-objective approach that uses the Multi-Objective\\nReal-Valued Gene-pool Optimal Mixing Evolutionary Algorithm\\n(MO-RV-GOMEA) and a dual-dynamic mesh transformation model\\nhas shown promise, exposing the trade-offs inherent to image reg-\\nistration problems and modeling large deformations in 2D. This\\nwork builds on this promise and introduces MOREA: the first evo-\\nlutionary algorithm-based multi-objective approach to deformable\\nregistration of 3D images capable of tackling large deformations.\\nMOREA includes a 3D biomechanical mesh model for physical plau-\\nsibility and is fully GPU-accelerated. We compare MOREA to two\\nstate-of-the-art approaches on abdominal CT scans of 4 cervical\\ncancer patients, with the latter two approaches configured for the\\nbest results per patient. Without requiring per-patient configura-\\ntion, MOREA significantly outperforms these approaches on 3 of\\nthe 4 patients that represent the most difficult cases.\\nKEYWORDS\\ndeformable image registration, multi-objective optimization, smart\\nmesh initialization, repair method, GOMEA\\n1 INTRODUCTION\\nIn recent decades, the field of radiation oncology has experienced\\nrapid developments. Key to its modern practice are medical images\\nacquired before, during, and after treatment. Although these im-\\nages are already guiding clinical decision-making in many ways,\\nthe transfer of information between multiple images that feature\\nlarge deformations or content mismatches has proven to be a hard\\nchallenge and has eluded widespread clinical adoption. In general,\\nthe challenge of Deformable Image Registration (DIR) is to find a\\nrealistic transformation that matches two or more image spaces\\nto each other, as illustrated in Figure 1. Given this transformation,\\nother metadata could be transferred between images, such as anno-\\ntated contours [ 30] or 3D radiation dose distributions [ 33], opening\\nup opportunities to make radiation treatment more precise [16].\\nThe DIR problem consists of three main objectives: an image-\\nbased objective (for a visual comparison), a contour-based objective\\n(for an assessment of object contour overlap), and a realism-basedobjective (to measure the energy required to perform the defor-\\nmation). These objectives are conflicting, especially when large\\ndeformations and content mismatches are at play [ 1]. DIR is there-\\nfore an inherently multi-objective problem, making Evolutionary\\nAlgorithms (EAs) well-suited for its optimization [19].\\nA diverse set of approaches to DIR has emerged [ 5,17,45]. These\\nall take a single-objective approach, requiring the user to choose\\nthe weights associated with the optimization objectives for each\\nuse, a priori . This can however hinder clinical adoption, since it has\\nbeen shown that choosing good weights (and other parameters) for\\nspecific patients is difficult in general and can strongly influence\\nregistration quality [ 36]. Even when configured for the best results,\\nmany existing approaches struggle with large deformations and\\ncontent mismatches between images because of limitations of their registration quality [ 36]. Even when configured for the best results,\\nmany existing approaches struggle with large deformations and\\ncontent mismatches between images because of limitations of their\\nunderlying transformation models and (often gradient-descent-\\nbased) optimization techniques. This shortcoming forms a second\\nobstacle to their translation into clinical workflows. Therefore, there\\nstill is a need for a DIR approach that does not require a priori\\nobjective weight configuration andcan tackle large deformations.\\nThe need to configure objective weights a priori has previously\\nbeen addressed by taking a multi-objective approach [ 2]. This re-\\nmoves the need to select weights for the optimization objectives in a\\nscalarized problem formulation a priori , since a set of solutions can\\nbe produced that appropriately represents the trade-off between\\ndifferent conflicting objectives, allowing the user to select a solu-\\ntion from this set, a posteriori . To overcome the second obstacle, a\\nflexible dual-dynamic triangular mesh transformation model that\\nallows for inverse-consistent, biomechanical registration has been\\nintroduced [ 3]. This model can match structures on both images to\\ncapture large deformations. The Multi-Objective Real-Valued Gene-\\npool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA)\\nhas proven to be effective at performing DIR with this model for\\n2D images by decomposing the problem into local, partial evalu-\\nations [ 10]. The Graphics Processing Unit (GPU) is exceptionally\\nwell-suited to execute these partial evaluations in parallel, yielding\\n(a)Source image\\n (b)Target image\\n (c)Example registration\\nFigure 1: Illustration of two images with large deformations\\nand an example of a deformable image registration with\\nMOREA\\u2019s dual-dynamic mesh transformation model.arXiv:2303.04873v1  [cs.CV]  8 Mar 2023 Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\nsignificant speed-ups [ 12]. Recently, first steps have been taken to\\nextend this GPU-accelerated approach to 3D images [ 4], for which\\nthe benefits of partial evaluations may be even greater due to the\\nincrease in the amount of image information (from 65k pixels in\\n2D to more than 2 million voxels in 3D), leading to more, but also\\ncostlier partial evaluations. While this extended approach has been\\nshown to be capable of solving simple registration problems of sin-\\ngle objects, it misses several crucial components required to tackle\\nclinical problems that feature multiple interacting objects.\\nIn this work, we therefore introduce MOREA, the first EA-based\\nMulti-Objective Registration approach capable of registering 3D im-\\nages with large deformations using a biomechanical model, without\\nrequiring a priori configuration of objective weights. In MOREA, a\\n3D tetrahedral mesh is initialized on interesting structures using a\\nnovel custom mesh generation approach, and a repair mechanism\\nfor folded meshes is embedded. With MOREA we furthermore im-\\nprove on prior modeling strategies [ 4] for all objectives to ensure\\ndesirable deformations will be achieved.\\n2 DEFORMABLE IMAGE REGISTRATION FOR\\nLARGE DEFORMATIONS\\nIn this section, we define the DIR optimization problem (Section 2.1)\\nand examine existing approaches (Section 2.2).\\n2.1 Problem Definition\\nThe problem of DIR for a pair of 2 images is to find a non-rigid\\ntransformation \\ud835\\udc47that deforms a source image \\ud835\\udc3c\\ud835\\udc60to match a tar-\\nget image\\ud835\\udc3c\\ud835\\udc61as closely as possible [ 40]. We distinguish between\\nunidirectional andsymmetric registration: in unidirectional registra-\\ntion, only\\ud835\\udc47(\\ud835\\udc3c\\ud835\\udc60)\\u2248\\ud835\\udc3c\\ud835\\udc61is optimized, while in symmetric registration,\\n\\ud835\\udc47\\u2032(\\ud835\\udc3c\\ud835\\udc61)\\u2248\\ud835\\udc3c\\ud835\\udc60is also optimized [ 40]. This can improve the physical\\nviability of the registration. Another desirable distinction for reg-\\nistrations is inverse-consistency [40], guaranteeing a one-to-one\\ncorrespondence between any point in the source image and its\\ncorresponding point in the target image.\\nRegistrations can generally be evaluated according to three\\nclasses of quality metrics. Image intensity metrics compare the pre-\\ndicted voxel intensity values of \\ud835\\udc47(\\ud835\\udc3c\\ud835\\udc60)to the voxel intensity values\\nof\\ud835\\udc3c\\ud835\\udc61, using metrics such as cross-correlation or mutual informa-\\ntion [ 26].Contour metrics judge registration accuracy by applying\\n\\ud835\\udc47to pairs of sets of points, representing contours ( \\ud835\\udc36\\ud835\\udc60and\\ud835\\udc36\\ud835\\udc61), and\\ncomputing the distances between those point sets. One example is\\nthe Chamfer distance [ 22]: for each pair\\u27e8\\ud835\\udc36\\ud835\\udc60,\\ud835\\udc36\\ud835\\udc61\\u27e9, the longest mini-\\nmum distance is calculated between points in \\ud835\\udc47(\\ud835\\udc36\\ud835\\udc60)and any point\\nin\\ud835\\udc36\\ud835\\udc61. DIR approaches can also use these contours at initialization\\ntime, to build transformation models for use during optimization.\\nFinally, deformation magnitude metrics express registration realism\\nby measuring the force needed to apply the deformation, using a\\nphysical model of the image space [ 23]. This can serve as a regular-\\nization mechanism, discouraging the registration to overfit.\\n2.2 Related Work\\nThese three quality metrics are conflicting objectives that form a\\ntrade-off [ 1]. A number of single-objective registration approaches\\nhave emerged in recent years, typically attempting to deal with thistrade-off by exploring different objective scalarizations. This how-\\never has the downside of having to set objective weights, a priori .\\nWe categorize these existing approaches broadly according to the\\nabove defined classes of quality metrics, into classes of approaches\\nmainly optimizing for (1) intensity match, (2) contour match, and\\n(3) both matches simultaneously. These and other features are com-\\npared for selected prominent approaches in Table 1.\\nAn example of the first class, optimizing for intensity match, is\\nthe Elastix toolbox [ 28]. It uses a B-spline based transformation\\nmodel, which uses B\\u00e9zier curves to model physical space. With this\\nmodel, Elastix optimizes for intensity, regularized by deformation the Elastix toolbox [ 28]. It uses a B-spline based transformation\\nmodel, which uses B\\u00e9zier curves to model physical space. With this\\nmodel, Elastix optimizes for intensity, regularized by deformation\\nmagnitude metrics. While this is a good fit for many applications,\\nwe observe that registering more complex, large deformations with\\nlocal discontinuities (such as studied in this work) can be difficult.\\nThe ANTs SyN registration approach [ 5] was conceived to model\\nsuch large deformations, featuring symmetric, inverse-consistent,\\nand intensity-based registration using time-varying velocity fields.\\nA third intensity-based approach is the Demons algorithm [ 42], us-\\ning principles from optical flow and Maxwell\\u2019s Demons for inverse-\\nconsistent registration. A more recent version of this approach also\\nhas a mechanism to handle content mismatch between images [ 34].\\nBoth the ANTs and Demons approach can in theory flexibly model\\nlarge deformations, but lack biomechanical modeling capabilities\\nand only operate on image intensity. This can hamper reliably\\nproducing anatomically valid registrations [30].\\nThis is one reason to consider the second class of approaches.\\nOne of these approaches is the Thin-Plate Splines Robust Point\\nMatching approach (TPS-RPM), which deforms contours using a\\nthin-plate spline model [ 18]. Subsequent work validated this on\\nan abdominal test case, registering a deforming bladder and two\\nsurrounding organs [ 44]. There is also a symmetric version of TPS-\\nRPM, which improves robustness on large deformations [ 8]. Work\\nconducted in parallel also applies a similar model to contours for ab-\\ndominal registration problems [ 39]. While large deformations can\\nbe modeled, the biomechanical plausibility of the transformation is\\nnot guaranteed, and objective weights still require configuration.\\nAnother contour-based approach is MORFEUS [ 17], which registers\\na mesh representation of imaged objects using a Finite Element\\nMethod (FEM) solver. It has shown promising results on brachyther-\\napy applications in the abdomen [ 37]. Although MORFEUS uses\\na biomechanical model, which improves realism, it does not take\\nimage intensities into account, thus losing detail between object\\nsurfaces and relying too heavily on (user-supplied) contours.\\nRecent work has targeted this shortcoming by proposing a com-\\nbined contour-based and image-based approach: the ANAtomically\\nCONstrained Deformation Algorithm (ANACONDA) [ 45] optimizes\\na fixed scalarization of image and contour terms by using the quasi-\\nNewton algorithm. This approach however lacks biomechanical\\nmodeling, and also introduces yet another parameter to configure.\\nOther hybrid attempts have also emerged, such as a combination\\nof the Demons approach with local FEM meshes [ 48], or the use\\nof an image-based registration step to derive tissue elasticities that\\nare later used in an FEM-based registration approach [29].\\nIn general, we see a gap: an approach that includes all registra-\\ntion aspects in one model. As Table 1 shows, we target this gap\\nwith MOREA by being both image-based and contour-based, fea-\\nturing biomechanical modeling, and exploiting the multi-objective MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\nFeature Elastix [28] ANTs SyN [5] Demons [42] TPS-RPM [18] ANACONDA [45] MORFEUS [17] MOREA (this work)\\nImage-based \\u2713 \\u2713 \\u2713 \\u2717 \\u2713 \\u2717 \\u2713\\nContour-based \\u2717 \\u2717 \\u2717 \\u2713 \\u2713 \\u2713 \\u2713\\nBiomechanical \\u2717 \\u2717 \\u2717 \\u2717 \\u2717 \\u2713 \\u2713\\nMulti-objective \\u2717 \\u2717 \\u2717 \\u2717 \\u2717 \\u2717 \\u2713\\nTable 1: Comparison of selected prominent existing DIR approaches by supported registration features.\\nnature of the DIR problem. These novelties are made possible by\\nthe flexibility and robustness of EAs, which are well-suited to op-\\ntimize non-differentiable, multi-objective problems. Additionally,\\nthe objective functions include millions of image voxel values and\\nare therefore expensive to compute, calling for hardware acceler-\\nation. Modern model-based EAs such as MO-RV-GOMEA feature\\nexcellent GPU compatibility, making them a good fit for optimizing\\nthe DIR problem.\\n3 MO-RV-GOMEA\\nThe structure of Black-Box Optimization (BBO) problems only gets\\nrevealed through repeated function evaluations. Gray-Box Opti-\\nmization (GBO) problems, on the other hand, have a (partly) known\\nproblem structure, which can be exploited during optimization. The\\nGOMEA suite of EAs has proven to be exceptionally well suited\\nfor efficiently solving both benchmark and real-world GBO prob-\\nlems [ 41]. Its extension to multi-objective, real-valued problems,\\nMO-RV-GOMEA [ 11], has even found real-world adoption in clini-\\ncal practice for prostate brachytherapy treatment planning [ 7,13].\\nWe give an overview of the key working principles of MO-RV-\\nGOMEA here. A detailed description may be found in literature [ 14].\\nNon-dominated solutions are preserved across generations in an\\nelitist archive with a pre-specified capacity [ 31]. Each generation\\nstarts with the selection of a subset of non-dominated solutions from\\nthe current population. This selection is clustered into \\ud835\\udc58equally\\nsized clusters. For each cluster, MO-RV-GOMEA employs a linkage\\nmodel that describes dependence relations between variables using\\na set of dependent variable sets, called Family of Subset (FOS) ele-\\nments. This linkage model can be learned during optimization in a\\nBBO setting, but in MOREA, we employ a static linkage model based\\non topological proximity of variables (see Section 4.2.1). Variation\\nthen proceeds by considering variables in FOS elements jointly in\\na procedure called optimal mixing . In this step, distributions are\\nestimated for each FOS element in each cluster, and new, partial\\nsolutions are sampled from these distributions. Newly sampled\\npartial solutions are evaluated and accepted if their insertion into\\nthe parent solution results in a solution that dominates the parent\\nsolution or that is non-dominated in the current elitist archive.\\n4 APPROACH\\nThe approach outlined in this work builds on the recently pro-\\nposed multi-objective approach for 3D images [ 4]. In this section,\\nwe present the new techniques we have added, in modeling the\\nproblem (Section 4.1), initializing the population of solutions (Sec-\\ntion 4.2), and optimizing the deformations (Section 4.3).4.1 Modeling\\n4.1.1 Enhancing realism with tissue-specific elasticities. Adjacent\\nwork has indicated that using tissue-specific elasticities, instead\\nof assuming one homogeneous elasticity for the entire image re-\\ngion, can enhance the realism of resulting deformations [ 37,46].\\nFollowing this insight, we extend the deformation magnitude ob-\\njective used in existing work [ 4] by computing an elasticity factor\\nfor each tetrahedron, based on its underlying image region. Imple-\\nmentation details for this computation are provided in Appendix A.\\nWe observe in exploratory experiments that this leads to better\\nregistration outcomes (see Appendix Section C.3.1).\\nTo compute the deformation magnitude objective, we consider\\nall corresponding edges \\ud835\\udc52\\ud835\\udc60and\\ud835\\udc52\\ud835\\udc61of each tetrahedron \\ud835\\udeff\\u2208\\u0394, be-\\nlonging to the mesh on the source image and the mesh on the target To compute the deformation magnitude objective, we consider\\nall corresponding edges \\ud835\\udc52\\ud835\\udc60and\\ud835\\udc52\\ud835\\udc61of each tetrahedron \\ud835\\udeff\\u2208\\u0394, be-\\nlonging to the mesh on the source image and the mesh on the target\\nimage, respectively. This includes 4 spoke edges that better capture\\nflattening motion, giving a total of 10 edges per tetrahedron [ 4].\\nGiven the tetrahedron-specific elasticity constant \\ud835\\udc50\\ud835\\udeff, the objective\\nis computed as follows:\\n\\ud835\\udc53magnitude =1\\n10|\\u0394|\\u2211\\ufe01\\n\\ud835\\udeff\\u2208\\u0394\\uf8ee\\uf8ef\\uf8ef\\uf8ef\\uf8ef\\uf8f0\\u2211\\ufe01\\n(\\ud835\\udc52\\ud835\\udc60,\\ud835\\udc52\\ud835\\udc61)\\u2208\\ud835\\udc38\\ud835\\udeff\\ud835\\udc50\\ud835\\udeff(\\u2225\\ud835\\udc52\\ud835\\udc60\\u2225\\u2212\\u2225\\ud835\\udc52\\ud835\\udc61\\u2225)2\\uf8f9\\uf8fa\\uf8fa\\uf8fa\\uf8fa\\uf8fb\\n4.1.2 Robustly estimating image similarity. The intensity objective\\nwe use is defined as a voxel-to-voxel comparison by taking the sum\\nof squared intensity differences, with special handling for compar-\\nisons of foreground (i.e., non-zero intensity) and background (i.e.,\\nzero intensity) voxels. We use a random sampling technique that is\\nwell-suited for GPU acceleration (defined in detail in Appendix A).\\nUsing the set of all sampled image points on both images, \\ud835\\udc43\\ud835\\udc60and\\n\\ud835\\udc43\\ud835\\udc61, and image intensities of source image \\ud835\\udc3c\\ud835\\udc60and target image \\ud835\\udc3c\\ud835\\udc61, the\\nobjective is defined as follows:\\n\\ud835\\udc53intensity =1\\n|\\ud835\\udc43\\ud835\\udc60|+|\\ud835\\udc43\\ud835\\udc61|\\uf8ee\\uf8ef\\uf8ef\\uf8ef\\uf8ef\\uf8f0\\u2211\\ufe01\\n\\ud835\\udc5d\\ud835\\udc60\\u2208\\ud835\\udc43\\ud835\\udc60\\u210e(\\ud835\\udc5d\\ud835\\udc60,\\ud835\\udc47(\\ud835\\udc5d\\ud835\\udc60))+\\u2211\\ufe01\\n\\ud835\\udc5d\\ud835\\udc61\\u2208\\ud835\\udc43\\ud835\\udc61\\u210e(\\ud835\\udc5d\\ud835\\udc61,\\ud835\\udc47\\u2032(\\ud835\\udc5d\\ud835\\udc61))\\uf8f9\\uf8fa\\uf8fa\\uf8fa\\uf8fa\\uf8fb\\n\\u210e(\\ud835\\udc5d\\ud835\\udc60,\\ud835\\udc5d\\ud835\\udc61)=\\uf8f1\\uf8f4\\uf8f4\\uf8f4 \\uf8f2\\n\\uf8f4\\uf8f4\\uf8f4\\uf8f3(\\ud835\\udc5d\\ud835\\udc60\\u2212\\ud835\\udc5d\\ud835\\udc61)2\\ud835\\udc5d\\ud835\\udc60>0\\u2227\\ud835\\udc5d\\ud835\\udc61>0\\n0 \\ud835\\udc5d\\ud835\\udc60=0\\u2227\\ud835\\udc5d\\ud835\\udc61=0\\n1 otherwise\\n4.1.3 Approximating the guidance error. In contrast to previous\\nwork where an exact guidance measure was used as one of the ob-\\njectives [ 4], in this work we have opted to introduce a measure that\\nis an approximation thereof that can be much more efficiently com-\\nputed using the GPU-accelerated sampling method that we already\\nuse for the calculation of the values for the image similarity objec-\\ntive. Preliminary experiments showed very similar results (when\\nlooking at the voxel displacement fields), also because a perfect Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\n\\/gid01748\\n\\/gid01748\\/gid01748\\n\\/gid01748\\n\\/gid01748\\/gid01748\\n\\/gid01748\\/gid01748\\n\\/gid01748\\/gid01748\\/gid01748\\n\\/gid01748\\n(a)The initial configura-\\ntion, with positive area\\nsigns for each triangle.\\n\\/gid01748\\n\\/gid01748\\/gid01748\\n\\/gid01748\\n\\/gid01748\\/gid01748\\n\\/gid01748\\/gid01748\\n\\/gid01748\\/gid01748\\/gid01748\\n\\/gid01162(b)The fold, detected\\nby a sign change in the\\nfolded (red) triangle.\\n(c)The repair method,\\nresolving the fold by\\nmoving the red point.\\nFigure 2: 2D illustration of a mesh configuration with and\\nwithout a constraint violation (fold). One of the triangles is\\nfolded, due to the red point having moved outside the cen-\\ntral triangle, colored yellow. The folded area is colored red.\\nguidance error is not necessarily the best solution. In Appendix A,\\nwe provide details regarding the implementation.\\nMOREA\\u2019s guidance objective is computed at positions \\ud835\\udc43\\ud835\\udc60and\\n\\ud835\\udc43\\ud835\\udc61, using the set \\ud835\\udc3aof all point set pairs \\u27e8\\ud835\\udc36\\ud835\\udc60,\\ud835\\udc36\\ud835\\udc61\\u27e9\\ud835\\udc56and the minimal\\npoint-to-point-set distance \\ud835\\udc51(\\ud835\\udc5d,\\ud835\\udc36). The total number of guidance\\npoints is indicated as |\\ud835\\udc3a\\ud835\\udc60|and|\\ud835\\udc3a\\ud835\\udc61|, and a truncation radius as \\ud835\\udc5f.\\nThe guidance objective is now defined as follows:\\n\\ud835\\udc53guidance =1\\n|\\ud835\\udc43\\ud835\\udc60|+|\\ud835\\udc43\\ud835\\udc61|\\u2211\\ufe01\\n\\u27e8\\ud835\\udc36\\ud835\\udc60,\\ud835\\udc36\\ud835\\udc61\\u27e9\\u2208\\ud835\\udc3a\\\"\\n|\\ud835\\udc36\\ud835\\udc60|\\n|\\ud835\\udc3a\\ud835\\udc60|\\ud835\\udc54(\\ud835\\udc43\\ud835\\udc60,\\ud835\\udc47,\\ud835\\udc36\\ud835\\udc60,\\ud835\\udc36\\ud835\\udc61)+|\\ud835\\udc36\\ud835\\udc61|\\n|\\ud835\\udc3a\\ud835\\udc61|\\ud835\\udc54(\\ud835\\udc43\\ud835\\udc61,\\ud835\\udc47\\u2032,\\ud835\\udc36\\ud835\\udc61,\\ud835\\udc36\\ud835\\udc60)#\\n\\ud835\\udc54(\\ud835\\udc43,\\u03a6,\\ud835\\udc36,\\ud835\\udc36\\u2032)=\\u2211\\ufe01\\n\\ud835\\udc5d\\u2208\\ud835\\udc43\\n\\ud835\\udc51(\\ud835\\udc5d,\\ud835\\udc36)<\\ud835\\udc5f\\\"\\n\\ud835\\udc5f\\u2212\\ud835\\udc51(\\ud835\\udc5d,\\ud835\\udc36)\\n\\ud835\\udc5f(\\ud835\\udc51(\\ud835\\udc5d,\\ud835\\udc36)\\u2212\\ud835\\udc51(\\u03a6(\\ud835\\udc5d),\\ud835\\udc36\\u2032))2#\\n4.1.4 Rapidly computing constraints. MOREA\\u2019s solutions represent\\nmeshes with hundreds of points, which can easily get entangled\\ninto folded configurations. Such constraint violations should be\\nprevented, to uphold the guarantee of inverse-consistency. Prior\\nwork [ 4] used a strategy that proved error-prone in more complex\\nmeshes. MOREA includes a novel fold detection method that is\\nbased on an observed phenomenon: a mesh fold will cause the sign\\nof at least one tetrahedron\\u2019s volume to change, as illustrated in\\nFigure 2 (the figure is in 2D, but this also holds in 3D). Our method\\nuses this phenomenon to detect folds and to measure their severity,opening up repair opportunities (see Section 4.3.1). Implementation\\ndetails for our method are provided in Appendix A.\\n4.2 Initialization of Registration Solutions\\nSignificant performance gains can be obtained if the initial guesses\\ngiven to the optimizer are closer to desirable objective space regions\\nthan a random guess or grid-like initializations [ 9]. We introduce\\ntwo techniques that provide such initial guesses.\\n4.2.1 Exploiting problem structures with mesh initialization. We\\ninitialize the meshes to align with objects in the image, adapting\\nan existing method for 2D images [ 9] and expanding it to facilitate\\nparallelization on the GPU. First, we place points on the contours\\nof objects in the source image to capture their shape (see Fig. 3a).\\nWe choose these points by greedily taking a spread-out subset from\\nthe contour annotations also used for the guidance objective, as\\nwell as a small fraction of randomly chosen points across the image.\\nThen, we perform a Delaunay tetrahedralization on these points,\\nusing the TetGen suite [ 25] (see Fig. 3b). This yields a mesh that we\\nduplicate to the target image space to complete the dual-dynamic\\ntransformation model.\\nAs laid out in Section 3, MO-RV-GOMEA evaluates groups of\\nvariables (i.e., FOS elements) jointly during variation. Exploratory\\nexperiments have shown that using edges as FOS elements (i.e.,\\ngroups of two connected points, with the variables encoding their\\ncoordinates), is beneficial for this problem. If two FOS elements\\nare completely independent because their variables are not needed\\nfor the partial evaluation of each set, variation and evaluation for\\nthese FOS elements can be done in parallel. We conduct two further\\nsteps to facilitate parallel evaluation and optimization on the GPU.\\nFirst, we execute a greedy set cover algorithm1to find a subset\\nof edges that covers all points (see Fig. 3c), so that each variable\\n(point coordinate) undergoes variation. We could alternatively use First, we execute a greedy set cover algorithm1to find a subset\\nof edges that covers all points (see Fig. 3c), so that each variable\\n(point coordinate) undergoes variation. We could alternatively use\\nall edges, but this would lead to points being included in several\\nFOS sets and thus undergoing variation multiple times per genera-\\ntion. For parallelization purposes, it is more efficient to select an\\n(approximately) minimal set of edges.\\nGiven the edge subset found by the set cover, we now determine\\nwhich FOS elements can be safely optimized in parallel. For this,\\nwe build an interaction graph based on topological proximity [ 12],\\nwhere two elements are connected if their sets of dependent tetra-\\nhedra overlap, i.e., the tetrahedra that are reevaluated when an\\n1Source: https:\\/\\/github.com\\/martin-steinegger\\/setcover\\n(a)Points placed on poten-\\ntially interesting positions.\\n(b)Custom mesh derived\\nfrom these points.\\n(c)Edges selected for varia-\\ntion through set cover.\\n(d)Interaction graph (blue)\\nbetween selected edges.\\n(e)Graph coloring computed\\non interaction graph.\\nFigure 3: 2D illustration of the mesh initialization process, which produces a custom mesh and determines which groups of\\nedges (i.e., FOS elements) can be optimized in parallel. Selected edges are highlighted in red, interaction edges in blue. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\nelement is changed (see Fig. 3d). Given this graph, parallel groups\\nare created with the DSATUR graph coloring algorithm [ 15] (see\\nFig. 3e). The dependent tetrahedra of each parallel group can be\\nevaluated in parallel on the GPU, which has been proven to lead to\\nspeed-ups of more than 100x on 2D images [12].\\nTetrahedral mesh quality can further be improved by specifying\\nsurfaces that should be included in the generated mesh. We apply\\nthis principle to the bladder by generating a surface mesh using the\\nMarching Cubes algorithm. We then specify its triangular surfaces\\nas constraints to the mesh generation algorithm, ensuring that\\nbladder surface triangles are included in the mesh. Exploratory\\nexperiments show superior performance when using this step (see\\nAppendix B.3.1).\\n4.2.2 Ensuring diversity in initial population. To promote diversity\\nin the initial population, prior work generates random deviations\\nfor each point in the mesh, starting at a grid-initialized solution [ 4].\\nWe observe that this method can produce many folded mesh con-\\nfigurations in generated solutions, which get discarded and thus\\nhamper convergence speed. In this work, we use a radial-basis-\\nfunction approach to introduce large deformations free of mesh\\nfolds. Implementation details on how these fields are generated and\\napplied to solution meshes are provided in Appendix A.\\n4.3 Repairing and Steering\\nDuring optimization, we apply two techniques to improve the qual-\\nity of solutions obtained, and the time needed to reach them.\\n4.3.1 Repairing infeasible solutions. By default, infeasible solutions\\n(i.e., solutions with either of the two meshes having one or more\\nfolds) are discarded. This, however, can hamper the creation of\\nhigh-quality offspring, as infeasible solutions may still provide\\nuseful information for higher-quality search space regions. We\\ntherefore devise a repair method that attempts to reverse folds\\non a point-by-point basis. For each point in a folded tetrahedron,\\nthe method mutates the point using a Gaussian distribution scaled\\nby its estimated distance to the surrounding 3D polygon. After\\n64 samples, the change with the best constraint improvement is\\nselected, if present. If all samples result in a deterioration, repair is\\naborted. The repair process for one point is illustrated in Figure 2c.\\n4.3.2 Applying pressure with adaptive steering. In general, an ap-\\nproximation set should be as diverse as possible while resembling\\nthe Pareto set as closely as possible. In practice, however, not all\\nregions of the Pareto front are of equal interest to users. A user con-\\nducting medical DIR for images with large deformations is typically\\nnot interested in solutions with a small deformation magnitude.\\nThe user is actually most interested in solutions with good guid-\\nance objective values, and we would like the algorithm to steer its\\nsearch towards that region in the objective space. Following earlier\\nwork [ 1], we implement an adaptive steering strategy, which steers\\nthe front towards high-quality guidance solutions after an explo-\\nration period of 100 generations. Given the best guidance objective\\nvalue\\ud835\\udc60\\ud835\\udc3aof any solution in the elitist archive, we only preserve\\nsolutions with guidance objective values between [\\ud835\\udc60\\ud835\\udc3a; 1.5\\ud835\\udc60\\ud835\\udc3a], i.e.,\\nthis becomes a hard constraint.5 EXPERIMENTS\\nWe compare MOREA to existing state-of-the-art registration ap-\\nproaches. Due to the complexity of the problem, we do not impose\\none time limit on all approaches, but rather ensure that they have\\n(reasonably) converged. We repeat all approaches with all configu-\\nrations 5 times, seeded reproducibly. All MOREA registration runs\\nare run on Dell Precision 7920R machines with NVIDIA RTX A5000\\nGPUs. Additional information on experimental setup and results is\\nprovided in the appendix.\\n5.1 Registration Problems\\nWe test all approaches on 4 clinical registration problems with large GPUs. Additional information on experimental setup and results is\\nprovided in the appendix.\\n5.1 Registration Problems\\nWe test all approaches on 4 clinical registration problems with large\\ndeformations (see Table 2). We retrospectively select two successive\\nComputerized Tomography (CT) scans of the abdominal area of\\ncervical cancer patients, acquired for radiation treatment planning\\npurposes, with a Philips Brilliance Big Bore scanner. On the first CT\\nscan, the bladder of the patient is filled, and on the second scan, the\\nbladder is empty and thus has shrunken significantly. This large\\ndeformation is challenging to register correctly while respecting\\nthe surrounding organs (e.g., rectum and bowel) and bony anatomy.\\nPatients 1\\u20133 represent common cases in clinical practice, exhibiting\\nlarge deformations and little to no margin between bladder and\\nbowel in the full-bladder scan. The bladder of Patient 4 largely\\npreserves its shape and exhibits a wide margin between bladder\\nand bowel, making registration easier. This case, however, is also\\nrarer in practice, and therefore less representative.\\nThe axial slices of the CT scans have a thickness of 3 mm,\\nwith in-slice resolutions ranging between (0.86,0.86)mm and\\n(1.07,1.07)mm. Each scan is resampled to (1.5,1.5,1.5)mm for\\nconsistency. Afterward, each scan pair is rigidly registered (i.e.,\\ntranslated, rotated, or scaled linearly) to align the bony anatomies\\nof both scans, using bone contours delineated by a radiation\\ntherapy technologist (RTT). Each pair is cropped to an axis-aligned\\nbounding box surrounding the bladder with a 30 mm margin,\\ntaking the maximal bounds from both images. This restricts the\\nregistration to the region where treatment was delivered, including\\nthe surrounding organs at risk.\\nContours of key organs in each scan have been annotated by\\nan RTT and verified by a radiation oncologist. The sets of points\\ndefining these contours serve as input to the guidance objective\\nof MOREA. We also use these clinical contours to generate binary\\nmasks for each organ and the bones by filling 2D polygonal esti-\\nmates formed by contours on each slice. As is common in practice,\\nthese contours can overlap, since organs are delineated indepen-\\ndently and are often surrounded by a small safety margin. Registra-\\ntion approaches therefore need to be robust enough to handle this\\noverlap. Several anatomically relevant corresponding landmarks\\nhave been annotated by an RTT and verified by a radiation oncolo-\\ngist on both scans, for evaluation purposes (see Appendix D).\\n5.2 Registration Approaches\\nWe consider a number of existing, popular registration approaches\\nfor which executable code is available. For these approaches, we\\nfollow a two-phase configuration process. First, we explore relevant\\ncoarse-grained settings for a single patient scan pair (of Patient 1), Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\nInstance Source Target\\nPatient 1\\nPatient 2\\nPatient 3\\nPatient 4\\nTable 2: Sagittal slices of all registration problems, with or-\\ngans contoured in different colors.\\nto find a suitable configuration for the imaging modality and prob-\\nlem difficulty. Then, we conduct fine-grained configuration on the\\nremaining settings (e.g., objective scalarization weights) for each\\npatient scan pair. We describe the resulting configuration for each\\napproach below, including the general coarse-grained configuration\\nof MOREA. A detailed overview of how we reached these configu-\\nrations, with additional configuration experiments, can be found in\\nAppendix C.\\n5.2.1 Elastix. We configure Elastix to conduct a regularized, multi-\\nresolution [ 43] image registration. Recommended settings2did not\\nyield satisfactory results on our scans, therefore we first register\\ncomposite mask images onto each other for each patient. This\\nis used as starting point for optimization on the original image\\nintensities. As a fine-grained configuration step for each patient,\\nwe configure the weight assigned to the deformation magnitude\\n2Based on an official parameter settings database: https:\\/\\/elastix.lumc.nl\\/modelzoo\\/objective in a fixed sweep of exponentially increasing weights of\\n[0,0.001,0.01,..., 10.0], as is done in related work [8].\\n5.2.2 ANTs SyN. For the ANTs SyN algorithm, the recommended\\nsettings3for multi-resolution registration also were not satisfactory,\\nwhich led us to conduct initial configuration experiments with sev-\\neral key parameters, listed in Appendix C. We also add a composite\\nmask in an additional image channel that is registered alongside the\\nimage. For each patient, we test the same regularization weight of\\nthe overall deformation by testing the same weights as for Elastix.\\n5.2.3 This work: MOREA. MOREA uses a single-resolution ap-\\nproach and is configured to generate a mesh of 600 points (i.e., the\\nproblem is 3600-dimensional), using the strategies for mesh gen-\\neration described in Section 4.2. We set the elitist archive capacity\\nto 2000 and use 10 clusters during optimization, with a runtime\\nbudget of 500 generations, during which the EA converges (see\\nAppendix D). As MOREA is a multi-objective approach returning\\nan approximation set of registrations, we do not need to configure\\nit further for each patient.\\n5.3 Evaluation of Registrations\\nSolutions to complex registration problems, such as the problems\\nin this study, require a multi-faceted evaluation. Below, we outline\\ntwo main methods for evaluating registrations: surface-based ac-\\ncuracy and visual inspection. Additional methods are described in\\nAppendix Section B.2 and applied in Appendices C and D.\\n5.3.1 Surface-based registration accuracy. A key part of evaluating\\nregistration accuracy is to assess how well the surfaces (contours) of\\nobjects align [ 16]. We use the Hausdorff distance, which represents\\nthe largest minimal distance between any two points on two object\\nsurfaces. This can be interpreted as the severity of the worst surface\\nmatch. To account for potential deformation inaccuracies at the\\nborder regions of the image, we discard a margin of 15 mmon each\\nside for the computation of this metric. Since this is smaller than the\\nearlier cropping margin of 30 mm, the bladder and regions around\\nit are left untouched by this second crop.\\n5.3.2 Visual inspection. Surface-based accuracy analysis is com-\\nplemented by a visual inspection, since a registration with a good\\ncontour match can still have undesirable deformations in regions\\nbetween contours. This inspection includes viewing slices of the\\ntarget image overlaid with the source contours transformed using\\nthe computed forward DVF of the registration. To also inspect the\\ndeformation between contours, we also visualize the full deforma-\\ntion: First, we render the DVF itself with a quiver plot. Second,\\nwe overlay a regular grid onto a slice and deform it with the DVF, deformation between contours, we also visualize the full deforma-\\ntion: First, we render the DVF itself with a quiver plot. Second,\\nwe overlay a regular grid onto a slice and deform it with the DVF,\\nwhich gives a different perspective.\\n5.4 Comparison of Registrations\\nAll registration solutions from all approaches are compared using\\nthe same evaluation pipeline, to ensure a fair comparison. Each\\napproach is configured to output its registrations in the form of a\\nforward and an inverse DVF, which define the deformation on the\\nsource and the target image, respectively. Existing approaches either\\n3Based on technical documentation: https:\\/\\/github.com\\/ANTsX\\/ANTs\\/wiki\\/Anatomy-\\nof-an-antsRegistration-call MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\n(a)Patient 1\\n (b)Patient 2\\n (c)Patient 3\\n (d)Patient 4\\nFigure 4: A selection of the best predicted deformations for each patient, represented by deformed contours rendered onto the\\ntarget image with its reference contours (i.e., target in blue). Annotated slices showing all organs are provided in Table 2.\\n(a) Elastix\\n (b) ANTs\\n (c) MOREA\\nFigure 5: Forward deformation vector fields and deformed contours of selected predicted deformations on Patient 1, for all 3\\napproaches (down-sampled for visibility). Arrow colors represent deformation magnitudes, in voxels (1 voxel =1.5mm).\\ndirectly or indirectly can be configured to output such DVFs. For\\nMOREA, we rasterize the deformation encoded by the two deformed\\nmeshes of a solution, using an existing rasterization method [ 24].\\nSince we are comparing single-objective approaches to a multi-\\nobjective approach (MOREA), we need to select solutions from\\nMOREA\\u2019s approximation set. We conduct this a posteriori selection\\nby starting at the solution with the best guidance objective value\\nand manually navigating through the approximation front to find a\\nsolution with a good trade-off between contour quality and realism.\\nWe also conduct statistical testing using the two-sided Mann-\\nWhitney U test (a standard non-parametric test) to compare MOREA\\nto ANTs and Elastix. The Hausdorff distance of the bladder contour\\nis used as the test metric, as it describes the largest deforming organ.\\nTo correct for multiple tests in the pair-wise comparisons, we apply\\nBonferroni correction to the \\ud835\\udefc-level and reduce it from 0.05 to 0.025.\\n6 RESULTS AND DISCUSSION\\nFigure 4 shows selected outcomes from each per-patient fine-\\ngrained configuration experiment, along with a solution from\\nMOREA\\u2019s approximation front for each patient. For Elastix, we\\nselect the runs with regularization weights 1.0, 1.0, 10.0, and 10.0\\non Patients 1\\u20134, respectively, and for ANTs, we select all runs with\\nweight 0. The full results of our configuration experiments for bothProblem MOREA vs. Elastix MOREA vs. ANTs\\nPatient 1 0.011 (+) 0.007 (+)\\nPatient 2 0.007 (+) 0.007 (+)\\nPatient 3 0.012 (+) 0.007 (+)\\nPatient 4 0.007 (+) 0.195 ( -)\\nTable 3: p-values of pair-wise comparisons of Hausdorff dis-\\ntances for the bladder between approaches. A plus ( +) indi-\\ncates a better mean with MOREA, a minus ( -) the opposite.\\nSignificant results are highlighted.\\nexisting approaches can be inspected in Appendix Sections B.1.2\\nand B.2.2. Convergence plots for Patient 1, which show how all\\napproaches have converged to the results presented here, can\\nbe found in Appendix D. As described in Section 5.1, there is an\\nintrinsic difference in difficulty between the scans. In general, we\\nobserve MOREA generally outperforming other approaches on\\nthe more difficult patients (1\\u20133), as can be seen visually in the\\ndeformed contours shown in Figure 4 and in the additional renders\\nand analyses provided in Appendix D.\\nForPatient 1 , we also render DVF slices in Figure 5, showing the\\ntransformation computed for each region of one slice. We observe\\nthat the deformations returned by Elastix and ANTs only deform Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\nFigure 6: Approximation front produced by MOREA on Pa-\\ntient 1. We render 3 zoomed-in registration solutions.\\nthe top region of the bladder. MOREA is the only approach which\\ndistributes this deformation across the entire bladder, which is a\\nmore realistic deformation in this flexible volume. Figure 6 plots\\nthe approximation set that is produced by MOREA on Patient 1,\\nhighlighting 3 solutions with slightly different deformations. This\\nillustrates the range of solutions presented to the user, all of which\\nspread the deformation across the bladder.\\nPatient 2 , which features the largest volume change in the blad-\\nder, seems to prove the most difficult: MOREA comes closest to\\nmodeling its deformation (see Fig. 4), although this comes at the\\ncost of the bowel also being moved downwards. A probable cause\\nis the little space (i.e., margin) left between the two organs in the\\nsource image. Here, MOREA\\u2019s result exposes a more fundamental\\nproblem that affects all approaches: structures separated by little to\\nno margin in one image cannot be separated in the other image with\\na transformation model consisting of a single mesh. The change of\\nbladder shape in Patient 3 is less severe than for Patient 2, but still\\nproves challenging for Elastix and ANTs (see Fig. 4). Especially the\\nback region (located left of the image center) does not match the\\ntarget. Patient 4 represents a relatively easy registration problem,\\nwith little change in the shape of the bladder and a clear margin\\nbetween bladder and bowel (see Fig. 2). On this problem, visual\\ninspection shows that ANTs and MOREA both find a good bladder\\ncontour fit, while Elastix struggles with both bladder and bowel.\\nExamining these results quantitatively, we conduct significance\\ntests on the Hausdorff distance of the bladder, listed in Table 3.\\nIn all patients, the contour match of the bladder as deformed by\\nMOREA is significantly superior to Elastix\\u2019s contour match. ANTs\\nmodels the contour of the bladder significantly less accurately than\\nMOREA in 3 out of 4 cases, with the fourth case (Patient 4) not\\nhaving a significantly different result. Appendix D lists significance\\ntest results for all organs, which confirm these trends, but also show\\nthat MOREA\\u2019s Hausdorff distance can sometimes be significantlyhigher than that of ANTs or Elastix. This does not however need\\nto imply worse registration performance, as a qualitative analysis\\nshows. For example, the deformed shape of the sigmoid of Patient 2\\nfound by ANTs is strongly off (see Figure 4). However, its metric\\nvalue is deemed significantly better than MOREA\\u2019s, even though\\nMOREA is closer to the target in terms of general shape.\\n7 CONCLUSIONS\\nThis work uniquely brings multiple lines of research in the field of\\ndeformable image registration together. We have introduced a reg-\\nistration approach, MOREA, that is both contour-based and image-\\nbased, uses a biomechanical model, and performs multi-objective op-\\ntimization. This combination uniquely positions MOREA to tackle\\nchallenging 3D image registration problems with large deforma-\\ntions and content mismatches. MOREA was built on the MO-RV-\\nGOMEA model-based evolutionary algorithm with several problem-\\nspecific extensions, such as GPU acceleration, solution repair, and\\nobject-aligned mesh initialization. Our experiments have shown\\npromising results on 4 cervical cancer patient scans, reaching higher\\ncontour registration accuracy than two state-of-the-art approaches\\non 3 of the 4 patients, representing the most difficult cases. Impor-\\ntantly, the deformation produced by MOREA seems to be more\\nuniformly spread across objects than the deformations produced\\nby existing approaches, which is deemed to be more realistic.\\nSolutions obtained by MOREA still contain local inaccuracies\\nwhich does leave room for improvement, in particular in regions\\nwhere organs interface. In fact, the results of this study expose a Solutions obtained by MOREA still contain local inaccuracies\\nwhich does leave room for improvement, in particular in regions\\nwhere organs interface. In fact, the results of this study expose a\\nmore fundamental problem in DIR, which is the inability of typical\\nDIR models to capture local discontinuities and content mismatches.\\nThis motivates future research into the modeling of independent or-\\ngan motion, following recent work on this topic [ 35,38]. MOREA\\u2019s\\nextensible, biomechanical model could be well-positioned for ex-\\npansions to capture these phenomena. Given such an expanded\\napproach, a larger validation study, with more patients and involv-\\ning domain experts, could help close the gap to clinical practice.\\nACKNOWLEDGMENTS\\nThe authors thank W. Visser-Groot and S.M. de Boer (Dept. of Ra-\\ndiation Oncology, LUMC, Leiden, NL) for their contributions to\\nthis study. This research is part of the research programme Open\\nTechnology Programme with project number 15586, which is fi-\\nnanced by the Dutch Research Council (NWO), Elekta, and Xomnia.\\nFurther, the work is co-funded by the public-private partnership\\nallowance for top consortia for knowledge and innovation (TKIs)\\nfrom the Dutch Ministry of Economic Affairs. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\nA TECHNICAL IMPLEMENTATION DETAILS\\nFOR THE MOREA APPROACH\\nIn this appendix, we provide additional technical implementation\\ndetails for the MOREA approach proposed in Section 4.\\nA.1 Modeling the deformation magnitude\\nMOREA\\u2019s deformation magnitude objective models heterogeneous\\nelasticities for different image regions. For each tetrahedron \\ud835\\udeff, we\\nestablish the elasticity of its underlying image region by sampling\\nfrom object-specific binary masks (see Figure 7). These masks are\\ncomputed for each object by filling the interior of its contour (avail-\\nable as guidance), yielding a discrete object segmentation. We com-\\npute the overlap that each object mask has with the tetrahedron\\n\\ud835\\udeff, which produces one fraction per object. In the example given\\nin Figure 7, this would be a fraction of 0.4 for the object corre-\\nsponding to this mask. These object fractions are multiplied by\\npre-determined elasticity factors for different tissue types, yielding\\nan overall element-specific factor for \\ud835\\udeff. At present, only bones and\\nbladder are assigned custom factors. The magnitude objective value\\nfor\\ud835\\udeffis multiplied by this factor to better model the actual energy\\nrequired to deform this image region.\\nA.2 Modeling the image similarity\\nThe image intensity objective of MOREA is defined as a sum of\\nsquared intensity differences at certain sample points. Modeling\\nthe partial objective value of one tetrahedron requires determining\\nwhich image voxels to sample. The existing prototype [ 4] tries to\\nfind all voxels with center points lying inside the tetrahedron, us-\\ning a line-search-inspired method. We observe, however, that this\\ndiscrete association of voxels with tetrahedra leads to undesirable\\nbehavior around tetrahedral surfaces, with voxels sometimes be-\\ning associated with multiple or no neighboring tetrahedra. This\\nphenomenon can be used to improve the sampled value while not\\nimproving or even deteriorating the true value.\\nIn our approach, we therefore introduce a random-sampling\\nbased method which samples the image space continuously, in-\\nterpolating intensity values between voxel centers. This is also\\nbetter-suited for GPU acceleration, since there are less decision\\npoints at which execution needs to pause. We uniformly sample \\ud835\\udc41\\npoints in each tetrahedron using its barycentric coordinate system,\\nwith\\ud835\\udc41being determined by the volume of the tetrahedron. For\\neach point, we sample 4 random real numbers \\ud835\\udc5f\\ud835\\udc56\\u2208[0; 1]and take\\n\\u2212log(\\ud835\\udc5f\\ud835\\udc56)for a uniform spread. We then normalize the coordinates\\nby their sum, to ensure that they lie in the tetrahedron. Instead of a\\nconventional random number generator, we use the Sobol sequence,\\nfor a more even spread of sample points. We ensure reproducibility\\nby seeding the Sobol sequence for each tetrahedron with a seed de-\\nrived from its coordinates. Therefore, the same positions are always\\nsampled per tetrahedron configuration.\\nA.3 Modeling the guidance error\\nThe guidance error objective of MOREA approximates the contour\\nmatch of a solution. Previous work [ 4] computes the extent of a\\ncontour match by considering each point in \\ud835\\udc36\\ud835\\udc60and computing the\\ndistance of its corresponding version in target space to the closest\\npoint in the set \\ud835\\udc36\\ud835\\udc61. This requires iterating over all points in \\ud835\\udc36\\ud835\\udc60,\\nFigure 7: 2D illustration of how one tetrahedral element\\n(here: the red triangle) overlaps with the mask of an organ.\\nThe computed overlap fractions are used to establish the\\nelasticity factor for this tetrahedron\\u2019s deformation magni-\\ntude.\\n(a)Source contour point set.\\nT(ps)\\n 1\\n 1\\n 2 2\\n 3 4\\n 4 5\\n 5 6\\n 6  6 6 6\\n 6  6 7 7  7  7  7  7\\n 7\\n 7 8 8 (b)Target contour point set.\\nFigure 8: Two point sets of object contours in a source and\\ntarget image, with minimal distance maps visualized using\\nisolines. A randomly sampled point \\ud835\\udc5d\\ud835\\udc60is close to the source\\ncontour, but the transformed \\ud835\\udc47(\\ud835\\udc5d\\ud835\\udc60)is farther away from the target image, with minimal distance maps visualized using\\nisolines. A randomly sampled point \\ud835\\udc5d\\ud835\\udc60is close to the source\\ncontour, but the transformed \\ud835\\udc47(\\ud835\\udc5d\\ud835\\udc60)is farther away from the\\ntarget contour. The yellow shaded area represents the trun-\\ncation area beyond which sampled points are discarded.\\nestablishing which tetrahedron they are located in, and computing\\nthe transformation at that point. We introduce a new, continuous\\nguidance formulation that approximates point-wise distances and\\nproved to be faster and more robust, in preliminary experiments.\\nDuring the random sampling process used for the intensity ob-\\njective on the source image \\ud835\\udc3c\\ud835\\udc60, we also consider the same locations\\non a distance map of \\ud835\\udc36\\ud835\\udc60, which gives the closest point to the source\\ncontour (see Figure 8). The distance at that point in the map of \\ud835\\udc36\\ud835\\udc60\\nis subtracted from the distance at the corresponding point in the\\nmap of\\ud835\\udc36\\ud835\\udc61, and weighted inversely by the distance to the source\\ncontour. The distances are truncated to a radius around each guid-\\nance point, measuring 2.5% of the width of the image, so that far\\naway movements do not influence the guidance error of a point set.\\nWe normalize the guidance error of each point set by the number of\\npoints in that set compared to the total number of guidance points,\\nto counteract biases towards more well-defined or larger contours.\\nA.4 Accurately detecting mesh folds\\nA function detecting constraint violations needs to have high pre-\\ncision (i.e., accurately identify all violations) and low latency (i.e.,\\nquickly return its answer). It should furthermore be defined contin-\\nuously, so that the method can also assess the severity of violations.\\nThis is important for methods that repair violations.\\nPrior work on mesh-based 3D image registration [ 4] uses a ray-\\nintersection method, testing if a point is inside a so-called bounding Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\nFigure 9: A 2D vector field produced by our radial-basis-\\nfunction approach used to generate solutions. Red dots\\nmark attractors, with their size indicating their weight.\\npolygon. This method has proven error-prone in 3D in preliminary\\nexperiments, producing false positives and negatives. We therefore\\ndevelop a new method for detecting folds in a tetrahedral mesh,\\nbased on the signed volumes of its tetrahedra [ 21]. Our method\\ncalculates the signed volume of each tetrahedron in the initial mesh\\nconfiguration, to establish a set of reference signs. When a point is\\nmoved, we recalculate the signed volumes of all tetrahedra that this\\naffects and compare them to the respective reference signs. The\\nsigns of at least one tetrahedron will flip if a fold has occurred. We\\nuse this phenomenon to detect mesh constraint violations and to\\ncompute the severity of each violation, using the absolute value of\\nthe violating signed volume.\\nA.5 Ensuring diversity in the initial population\\nEven with a smartly initialized mesh, the diversity of the popula-\\ntion at generation 0 plays an important role [ 32]. Prior work uses\\none reference solution and generates random deviations by sam-\\npling around each mesh point with increasingly large variance [ 4].\\nFor low-resolution meshes, this method can be effective, but for\\nhigher-resolution meshes, this method can lead to many constraint\\nviolations in the generated solutions (i.e., folded mesh configura-\\ntions). We introduce a method for initialization noise that generates\\nlarge deformations free of constraint violations, inspired by ap-\\nproaches using radial basis functions in other domains [ 47]. Our\\nmethod places a number of Gaussian kernels on both source and\\ntarget images and models a sense of gravity from mesh points to-\\nwards these kernels. These forces are applied in incremental rounds,\\nas long as they do not cause constraint violations. A deformation\\nvector field generated by this strategy is depicted in Figure 9. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\nB EXTENDED PROBLEM SPECIFICATION\\nIn this appendix, we provide additional information on the regis-\\ntration problems used in this study and specify additional methods\\nfor evaluation and comparison of registration quality.\\nB.1 Additional Problem Information\\nTable 4 lists the in-slice resolutions of the CT scans used. This is\\nthe physical resolution of each slice prior to our resampling step\\nto(1.5,1.5)mm. We also provide additional views on each medical\\nimage: For each patient, Table 5 lists two slices per source and target\\nimage. This provides a useful additional perspective, since some\\nmovements are better visible from a different angle.\\nB.2 Additional Evaluation Methods\\nWe evaluate each solution with four types of methods, based on\\n(1) surface-based registration accuracy, (2) visual inspection using\\n2D and 3D visualizations, (3) volume-based registration accuracy,\\n(4) landmark registration accuracy. Method types (1) and (2) have\\nbeen described in Section 5.3. Here, we give an additional strategy\\nfor (1), and outline additional methods (3) and (4).\\nB.2.1 Surface-based registration accuracy. Alternatively to the\\nHausdorff distance, the 95th percentile of the Hausdorff distance is\\nanother indicator we use in our study. This represents the distance\\nfor which it holds that 95% of all surface point distances are smaller\\nthan this distance. Both Hausdorff and Hausdorff 95th percentile\\nmetrics are computed using the pymia PyPI package.\\nB.2.2 Volume-based registration accuracy. Adjacent to surface ac-\\ncuracy, we are interested in the accuracy of individual volumes\\n(e.g., organs, bones) represented in the images. A common metric\\nfor this is the Dice coefficient, which represents the fraction of\\nvolume overlap compared to total volumes. Using binary masks of\\neach annotated object in the images, we compute this metric on a\\nvoxel-by-voxel basis. We compare the binary masks corresponding\\nto the target image against binary masks of the source image trans-\\nformed using the computed deformation. With the same reasoning\\nas for surface-based evaluation (see Section 5.3), we discard the\\nsame border margin when evaluating volume-based metrics.\\nB.2.3 Landmark registration accuracy. A set of corresponding land-\\nmarks not provided to the algorithm during optimization can be\\nused to locally assess the accuracy of a registration. For each pair\\nof landmarks, we transform the source landmark using the forward\\ntransformation to target space, and compute landmark accuracy as\\nthe Euclidean distance between the transformed source landmark\\nand its corresponding target landmark. This is a common accuracy\\nmeasure in image registration studies [ 16,20], but can be less accu-\\nrate as an indicator of overall registration quality, since landmarks\\nare placed on visible anatomical structures that often have limited\\nmovement, as is the case in our scans.B.3 Comparing Multi-Object Metrics\\nThe metrics of individual organs cannot be adequately interpreted\\nin isolation, as organ motions are related and therefore form trade-\\noffs. We visualize these trade-offs by plotting scores for different\\norgans in one parallel coordinates plot, similar to the color-coded\\nheatmap comparison presented in [ 27]. These line plots help inform\\ndecisions that need to take registration quality across registration\\ntargets into account.\\nPatient Scan In-slice Resolution\\nPatient 1Full bladder (0.86,0.86)mm\\nEmpty bladder (0.98,0.98)mm\\nPatient 2Full bladder (1.04,1.04)mm\\nEmpty bladder (1.07,1.07)mm\\nPatient 3Full bladder (0.98,0.98)mm\\nEmpty bladder (0.98,0.98)mm\\nPatient 4Full bladder (1.04,1.04)mm\\nEmpty bladder (1.00,1.00)mm\\nTable 4: In-slice resolutions for the slices of each CT scan,\\nprior to resampling them to (1.5,1.5)mm. Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\nInstance Source image: sagittal Target image: sagittal Source image: coronal Target image: coronal\\nPatient 1\\nPatient 2\\nPatient 3\\nPatient 4\\nTable 5: Slices of all registration problems, with organs contoured. Sagittal: side view; coronal: front-to-back view. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\nC CONFIGURATION OF COMPARED\\nAPPROACHES\\nC.1 Elastix\\nWe use Elastix version 5.0.0. Based on parameter settings from the\\nElastix Model Zoo4, we apply multi-resolution Elastix registration\\nto our registration problems with a range of configurations, trying\\nto find the optimal configuration for each problem (see Section C.1.3\\nfor our parameter files). Inspired by an approach implementing sym-\\nmetric registration in Elastix using a group-wise methodology [ 6],\\nwe also experiment with a symmetric variant which registers both\\nimages to a common image mid-space. For all setups, we relax con-\\nvergence requirements by increasing the number of iterations per\\nresolution to 10,000, which is significantly larger (5 times) than\\nthe computational budget given in most reference files. This is\\ndone to give Elastix sufficient opportunity to model the large defor-\\nmations present. We also stabilize optimization by increasing the\\nnumber of image sampling points from the frequently used 10,000\\nto 20,000. Although increasing the computational complexity, this\\nshould make image intensity approximations used internally during\\noptimization more accurate and computed gradients more reliable.\\nElastix computes the inverse transform by default, meaning a vec-\\ntor field defined in fixed (target) space leading to moving (source)\\nspace. To compute the forward transform, which is needed to trans-\\nform annotations from moving (source) to fixed (target) space, we\\nrerun the registrations with the given parameter files and the com-\\nputed transform as initial transform, but replace the metric(s) with\\ntheDisplacementMagnitudePenalty metric. This effectively finds\\ntheforward transform of the computed inverse transform. Export-\\ning this forward transform in isolation, by removing the initial\\ntransform pointer from the parameter file, yields the desired DVF.\\nElastix does not support the optimization of object contour\\nmatches, which are optimized by the MOREA approach through\\nthe guidance objective. To ensure a fair comparison, we attempt\\nto input this information as a pair of composite mask images to\\nimplicitly pass on contour information. Each mask image is made\\nby combining the different binary object masks available for each\\nscan, giving each object segmentation a different homogeneous\\nintensity value. In runs where this feature is enabled, we precede\\nthe CT image registration run with a registration of these prepared\\ncomposite masks.\\nC.1.1 Coarse-grained configuration experiments. First, we conduct\\nan initial set of runs on Patient 1 to establish a suitable base con-\\nfiguration for this problem modality and difficulty. We explore the\\ninfluence of registration direction (unidirectional vs. symmetric)\\nand the use of a composite mask registration step (with vs. without),\\nassuming a regularization weight of 0.001, to give Elastix flexibility\\nfor large deformations (a large weight on the deformation magni-\\ntude weight can hinder large deformations).\\nIn Figure 10, we plot the performance of Elastix using symmet-\\nric and unidirectional registration, reporting two different metrics\\n(Dice score and 95th percentile of the Hausdorff distance). We ob-\\nserve that unidirectional registration generally performs similarly\\nor better compared to symmetric registration, except for the rec-\\ntum and anal canal, in terms of Dice score. Due to the relatively\\n4https:\\/\\/elastix.lumc.nl\\/modelzoo\\/\\n(a) Dice scores.\\n(b) 95th percentiles of the Hausdorff distance.\\nFigure 10: Comparison of symmetric and unidirectional reg-\\nistration in Elastix, for multiple runs. The baseline score af-\\nter rigid registration is plotted in blue.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 11: Visual renders of deformations predicted by\\nElastix configurations using unidirectional and symmetric\\nregistration, without mask registration step. (a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 11: Visual renders of deformations predicted by\\nElastix configurations using unidirectional and symmetric\\nregistration, without mask registration step.\\nlarge performance gain in the bladder (the most strongly deforming\\norgan), we choose unidirectional registration at this point. This\\nchoice is supported by visual inspection of Figure 11, which shows\\nslightly better performance on the bladder in the coronal slice.\\nWe now turn to the use of a composite mask registration step, in\\nan attempt to get larger deformations by simplifying the informa-\\ntion input to Elastix. Figure 12 shows the same metrics, but with Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\n(a) Dice scores.\\n(b) 95th percentiles of the Hausdorff distance.\\nFigure 12: Comparison of unidirectional registration with\\nand without a composite mask registration step in Elastix,\\nfor multiple runs. The baseline score after rigid registration\\nis plotted in blue.\\nand without the use of such a step (while using unidirectional reg-\\nistration). The results do not identify one clear superior approach,\\nsince the Dice score of the with-mask configuration is generally\\nsuperior but the Hausdorff 95th percentile is lower for the without-\\nmask configuration. Figure 13 indicates that adding a mask step\\nimproves the modeling of the base region of the bladder, but the\\nmiddle region is merely contracted sideways without moving the\\ntop region downwards, thereby not resulting in anatomically real-\\nistic deformations. Nevertheless, we choose this version over the\\nversion without mask registration step, since the large deformation\\nneeded is modeled more closely with the step added.\\nC.1.2 Fine-grained configuration experiments per patient. For each\\npatient, we try exponentially increasing regularization weights; an\\nexponential regularization weight sweep that is also used in similar\\nwork [ 8]. The Dice scores on each patient are reported in Figure 14\\nand the 95th percentiles of the Hausdorff distance in Figure 15.\\nRenders for each problem are provided in Figures 16\\u201319.\\nWe observe that the optimal regularization weight varies\\nstrongly between different registration problems. While the scans\\nof Patient 1 (Fig. 16) are best served with a weight of 1.0 out of the\\ntried settings, the scans of Patient 3 (Fig. 18) seem better off with a\\nweight of 10.0.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 13: Visual renders of deformations predicted by\\nElastix configurations with and without a composite mask\\nregistration step, using unidirectional registration. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\n(a) Patient 1.\\n (b) Patient 2.\\n(c) Patient 3.\\n (d) Patient 4.\\nFigure 14: Dice scores for per-patient fine-grained configuration runs in Elastix. The baseline score after rigid registration is\\nplotted in blue.\\n(a) Patient 1.\\n (b) Patient 2.\\n(c) Patient 3.\\n (d) Patient 4.\\nFigure 15: Hausdorff 95th percentiles for per-patient fine-grained configuration runs in Elastix. The baseline score after rigid\\nregistration is plotted in blue. Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 16: Visual renders of deformations predicted by\\nElastix with different regularization weights, on Patient 1.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 17: Visual renders of deformations predicted by\\nElastix with different regularization weights, on Patient 2.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 18: Visual renders of deformations predicted by\\nElastix with different regularization weights, on Patient 3.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 19: Visual renders of deformations predicted by\\nElastix with different regularization weights, on Patient 4. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\nC.1.3 Parameter files. Below, we list the parameter files that we\\nused for the different variants of Elastix registration. Tokens starting\\nwith the $character denote variables that are resolved before we\\npass the file to Elastix (e.g., a random seed that we increment at\\nevery repeat).\\nListing 1: Forward transform parameters for conventional,\\nunidirectional deformation.\\n\\/\\/ ImageTypes\\n(FixedImagePixelType \\\"short\\\")\\n(FixedImageDimension 3)\\n(MovingImagePixelType \\\"short\\\")\\n(MovingImageDimension 3)\\n\\/\\/ Multi resolution\\n(Registration \\\"MultiMetricMultiResolutionRegistration\\\")\\n(HowToCombineTransforms \\\"Compose\\\")\\n(NumberOfHistogramBins 32)\\n(NumberOfResolutions 4)\\n(MaximumNumberOfIterations 10000)\\n\\/\\/ Optimizer\\n(Optimizer \\\"AdaptiveStochasticGradientDescent\\\")\\n(AutomaticParameterEstimation \\\"true\\\")\\n(UseAdaptiveStepSizes \\\"true\\\")\\n(CheckNumberOfSamples \\\"true\\\")\\n(UseDirectionCosines \\\"true\\\")\\n(RandomSeed $random_seed)\\n\\/\\/ Metric\\n(Metric \\\"AdvancedMattesMutualInformation\\\"\\n\\\"TransformBendingEnergyPenalty\\\")\\n(Metric0Weight 1.0)\\n(Metric1Weight $regularization_weight)\\n\\/\\/ Components\\n(FixedImagePyramid \\\"FixedSmoothingImagePyramid\\\")\\n(MovingImagePyramid \\\"MovingSmoothingImagePyramid\\\")\\n(Interpolator \\\"BSplineInterpolator\\\")\\n(ResampleInterpolator \\\"FinalBSplineInterpolator\\\")\\n(Resampler \\\"DefaultResampler\\\")\\n(Transform \\\"BSplineTransform\\\")\\n\\/\\/ Transform\\n(FinalGridSpacingInPhysicalUnits 2.0)\\n\\/\\/ Sampling\\n(ImageSampler \\\"RandomCoordinate\\\")\\n(NewSamplesEveryIteration \\\"true\\\")\\n(NumberOfSpatialSamples 20000)\\n\\/\\/ Interpolation and resampling\\n(BSplineInterpolationOrder 1)\\n(FinalBSplineInterpolationOrder 3)\\n(DefaultPixelValue 0)\\n\\/\\/ Output and other\\n(WriteTransformParametersEachIteration \\\"false\\\" \\\"false\\\" \\\"false\\\"\\n\\\"false\\\" \\\"false\\\")\\n(WriteTransformParametersEachResolution \\\"true\\\" \\\"true\\\" \\\"true\\\" \\\"true\\\"\\n\\\"true\\\")\\n(ShowExactMetricValue \\\"false\\\" \\\"false\\\" \\\"false\\\" \\\"false\\\" \\\"false\\\")\\n(WriteResultImageAfterEachResolution \\\"false\\\")\\n(WriteResultImage \\\"true\\\")\\n(ResultImagePixelType \\\"short\\\")\\n(ResultImageFormat \\\"nii.gz\\\")Listing 2: Forward transform parameters for symmetric de-\\nformation.\\n\\/\\/ ImageTypes\\n(FixedImagePixelType \\\"short\\\")\\n(FixedInternalImagePixelType \\\"short\\\")\\n(FixedImageDimension 4)\\n(MovingImagePixelType \\\"short\\\")\\n(MovingInternalImagePixelType \\\"short\\\")\\n(MovingImageDimension 4)\\n\\/\\/ Multi resolution\\n(Registration \\\"MultiResolutionRegistration\\\")\\n(HowToCombineTransforms \\\"Compose\\\")\\n(NumberOfHistogramBins 32)\\n(NumberOfResolutions 4)\\n(MaximumNumberOfIterations 10000)\\n(MaximumNumberOfSamplingAttempts 10)\\n\\/\\/ Optimizer\\n(Optimizer \\\"AdaptiveStochasticGradientDescent\\\")\\n(AutomaticParameterEstimation \\\"true\\\")\\n(UseAdaptiveStepSizes \\\"true\\\")\\n(CheckNumberOfSamples \\\"true\\\")\\n(UseDirectionCosines \\\"true\\\")\\n(RandomSeed \\\\$random_seed)\\n\\/\\/ Metric\\n(Metric \\\"$metric\\\")\\n(NumEigenValues 1)\\n(TemplateImage \\\"ArithmeticAverage\\\" \\\"ArithmeticAverage\\\")\\n(Combination \\\"Sum\\\" \\\"Sum\\\")\\n(SubtractMean \\\"true\\\")\\n(MovingImageDerivativeScales 1.0 1.0 1.0 0.0)\\n\\/\\/ Components\\n(FixedImagePyramid \\\"FixedSmoothingImagePyramid\\\")\\n(MovingImagePyramid \\\"MovingSmoothingImagePyramid\\\")\\n(ImagePyramidSchedule 8 8 8 0 4 4 4 0 2 2 2 0 1 1 1 0)\\n(Interpolator \\\"ReducedDimensionBSplineInterpolator\\\")\\n(ResampleInterpolator \\\"FinalReducedDimensionBSplineInterpolator\\\")\\n(Resampler \\\"DefaultResampler\\\")\\n(Transform \\\"BSplineStackTransform\\\")\\n\\/\\/ Transform\\n(FinalGridSpacingInPhysicalUnits 2.0)\\n\\/\\/ Sampling\\n(ImageSampler \\\"RandomCoordinate\\\")\\n(NewSamplesEveryIteration \\\"true\\\")\\n(NumberOfSpatialSamples 20000)\\n\\/\\/ Interpolation and resampling\\n(BSplineTransformSplineOrder 1)\\n(FinalBSplineInterpolationOrder 3)\\n(DefaultPixelValue 0)\\n\\/\\/ Output and other\\n(WriteTransformParametersEachIteration \\\"false\\\" \\\"false\\\" \\\"false\\\"\\n\\\"false\\\")\\n(WriteTransformParametersEachResolution \\\"true\\\" \\\"true\\\" \\\"true\\\" \\\"true\\\")\\n(ShowExactMetricValue \\\"false\\\" \\\"false\\\" \\\"false\\\" \\\"false\\\")\\n(WriteResultImageAfterEachResolution \\\"false\\\")\\n(WriteResultImage \\\"true\\\")\\n(ResultImagePixelType \\\"short\\\")\\n(ResultImageFormat \\\"nii.gz\\\") Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\nC.2 ANTs SyN\\nWe use ANTs SyN algorithm version 2.4.2. We bootstrap a regis-\\ntration command using the antsRegistrationSyN.sh script and\\ncustomize it to fit our problem (see Section C.2.3 for our run com-\\nmands). Following official recommendations5, we consider the fol-\\nlowing settings to be left tunable for this problem: (1) what region\\nradius to use for the cross correlation metric, (2) whether to use\\ncomposite masks as an additional image modality channel during\\nregistration, (3) what gradient step size to use, (4) what regular-\\nization weight to assign to local deformations between time steps,\\nand (5) what regularization weight to assign to the total deforma-\\ntion. We configure the first four parameters for Patient 1, and then\\nconfigure the fifth parameter for each patient, separately.\\nIn our setup, we relaxed convergence limits compared to guide-\\nlines to allow for longer, and hopefully more accurate registration.\\nIn terms of metrics, we do not use the point set registration metric\\nthat is mentioned in the manual, as the manual states that this\\nmetric is not currently supported in ANTs SyN.\\nWe encountered that ANTs SyN random seed does not have any\\neffect on the outcome of registration with the Cross Correlation\\n(CC) measure, even with a random sampling strategy. The current\\nversion seems fully deterministic, but without taking the random\\nseed into account, therefore always producing the same output,\\nregardless of the seed. This is problematic, since we would like to\\nget multiple outputs that expose how the registration approach\\nreacts to slightly varying inputs. To mitigate the lack of control on\\nthe determinism of the registration, we slightly perturb the sigma\\nsmoothing factors (see Listing 3) with very small (deterministically\\nrandom) deltas. \\u03943is normally distributed and capped between\\n[\\u22120.1,0.1],\\u03942between[\\u22120.05,0.05], and\\u03941between[\\u22120.01,0.01].\\nC.2.1 Coarse-grained configuration experiments. We conduct an\\ninitial set of coarse-grained configuration experiments on Patient 1\\nwith the ANTs SyN algorithm. The officially recommended set-\\ntings serve as our baseline: a cross-correlation radius of 4 voxels,\\na gradient step size of 0.1, registration of only the image itself (no\\nadditional channels), and an update regularization weight of 3.0.\\nFor each of these settings, we experiment with different deviations\\nfrom the baseline.\\nCross correlation radius First, we investigate the impact of a different\\ncross correlation radius. Larger values should improve registration\\naccuracy, since more context information is taken into account\\nwhen computing the cross correlation of a sample. Figure 20 con-\\nfirms this expectation, although it shows little impact overall. Most\\norgans show little deviation in score, but the anal canal is registered\\nmore accurately in terms of Dice score when the radius is increased.\\nWe observe that there are diminishing returns here, e.g., a change\\nof radius from 7 to 8 provides only marginal improvement. Still, we\\ndecide to use the largest setting tested (8 voxels, meaning 12 mm\\nin the case of the clinical problems), since this setting provides\\nthe best outcome and there is no time limit on registration in our\\nstudy. The visual render in Figure 21 shows the visual impact of\\nthis setting, which can be described as limited.\\n5https:\\/\\/github.com\\/ANTsX\\/ANTs\\/wiki\\/Anatomy-of-an-antsRegistration-call\\n(a) Dice scores.\\n(b) 95th percentiles of the Hausdorff distance.\\nFigure 20: Comparison of registrations with different region\\nradii for the ANTs cross correlation metric. The baseline\\nscore after rigid registration is plotted in blue.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 21: Visual renders of deformations predicted by ANT\\nconfigurations with different CC radii.\\nComposite mask channel Second, we explore the effect of including\\na composite mask image channel during registration. Figure 22 configurations with different CC radii.\\nComposite mask channel Second, we explore the effect of including\\na composite mask image channel during registration. Figure 22\\nprovides evidence that including a mask channel has added value in\\nterms of Dice score for registration of all organs. The difference in\\nperformance is only slightly visible in Figure 23, but the difference\\nin metric values motivates our decision to use a mask channel in\\nthe upcoming patient-specific configuration steps.\\nGradient step size Third, we examine the impact of using a different\\ngradient step size on the registration performance of ANTs. A MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\n(a) Dice scores.\\n(b) 95th percentiles of the Hausdorff distance.\\nFigure 22: Comparison of registrations with and without a\\ncomposite mask channel in ANTs. The baseline score after\\nrigid registration is plotted in blue.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 23: Visual renders of deformations predicted by ANT\\nconfigurations with and without a composite mask channel.\\nlarger step size between time points in ANTs\\u2019 registration could\\nlead to larger deformations becoming feasible, since optimization\\nis less likely to get stuck in local minima. Figure 24 indicates that\\nchoosing a larger step size than the recommended value of 0.1 can\\nbe beneficial, with 1.0 providing a good trade-off for different organs.\\nLarger step sizes such as 5.0 cause the algorithm to overshoot the\\ntarget and strongly deform a number of organs, as can be seen in\\nthe contour renders (Figure 25). We choose a gradient step size of\\n1.0 for its good trade-off between performance targets.\\nUpdate regularization weight Finally, we use the deduced settings\\nfrom the previous three sweeps to test which update regularization\\n(a) Dice scores.\\n(b) 95th percentiles of the Hausdorff distance.\\nFigure 24: Comparison of ANTs registrations with different\\ngradient step sizes between time points. The baseline score\\nafter rigid registration is plotted in blue.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 25: Visual renders of deformations predicted by ANT\\nconfigurations with different gradient step sizes.\\nweight performs best. Figure 26 shows best overall performance\\nfor 4.0, in both metrics. Visually, Figure 27 indicates that weights\\n4.0 and 5.0 lead to the best registration outcomes, with little visible\\ndifference between the two. Based on visual and quantitative results,\\nwe choose an update regularization weight of 4.0 for the patient-\\nspecific configuration experiments.\\nC.2.2 Fine-grained configuration experiments per patient. We try\\nexponentially increasing total regularization weights for all prob-\\nlem instances. Figures 28 and 29 plot the Dice scores and Hausdorff\\n95th percentiles for each problem instance, and Figures 30\\u201333 show Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\n(a) Dice scores.\\n(b) 95th percentiles of the Hausdorff distance.\\nFigure 26: Comparison of ANTs registrations with differ-\\nent update regularization weights between time points. The\\nbaseline score after rigid registration is plotted in blue.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 27: Visual renders of deformations predicted by ANT\\nconfigurations with different update regularization weights.\\nrenders of the deformed contours that ANTs predicts for these in-\\nstances. We observe that regularization has a strong impact on per-\\nformance in all examined cases, but that often the (relatively) better\\noutcomes are still acquired without regularization. Figures 30\\u201332\\nshow ANTs failing to model the large deformation taking place in\\nthe bladder and its surrounding organs, regardless of the regular-\\nization. The Dice and Hausdorff metric results underscore these\\nobservations. In Figure 33, ANTs shows that it can model the blad-\\nder deformation quite closely, but it should be noted that this is\\nmorphologically also the easiest problem.C.2.3 Run commands. We list the two commands that we used for\\nregistration with ANTs. Tokens starting with the $character denote\\nvariables that are resolved before we execute these commands. Note\\nthat the random seed, even though given to the command, is not\\nfunctional and does not change the output.\\nListing 3: ANTs registration command for multivariate reg-\\nistration with composite masks.\\n$ANTSPATH\\/antsRegistration\\n--verbose 1\\n--random-seed $random_seed\\n--dimensionality 3\\n--float 0\\n--collapse-output-transforms 1\\n--output [ , Warped.nii.gz, InverseWarped.nii.gz ]\\n--interpolation Linear\\n--use-histogram-matching 0\\n--winsorize-image-intensities [ 0.005, 0.995 ]\\n--initial-moving-transform [ $fixed_composite_mask,\\n$moving_composite_mask, 1 ]\\n--transform SyN[ $gradient_step_size,\\n$update_regularization_weight,\\n$total_regularization_weight ]\\n--metric CC[ $fixed_composite_mask, $moving_composite_mask, 1,\\n$cross_correlation_radius ]\\n--metric CC[ $fixed_image, $moving_image, 1,\\n$cross_correlation_radius ]\\n--convergence [ 2000x1000x500x250, 1e-6, 10 ]\\n--shrink-factors 8x4x2x1\\n--smoothing-sigmas {3+delta_3}x{2+delta_2}x{1+delta_1}x0vox\\nListing 4: ANTs registration command for multivariate reg-\\nistration without composite masks.\\n$ANTSPATH\\/antsRegistration\\n--verbose 1\\n--random-seed $random_seed\\n--dimensionality 3\\n--float 0\\n--collapse-output-transforms 1\\n--output [ , Warped.nii.gz, InverseWarped.nii.gz ]\\n--interpolation Linear\\n--use-histogram-matching 0\\n--winsorize-image-intensities [ 0.005, 0.995 ]\\n--initial-moving-transform [ $fixed_image, $moving_image, 1 ]\\n--transform SyN[ $gradient_step_size,\\n$update_regularization_weight,\\n$total_regularization_weight ]\\n--metric CC[ $fixed_image, $moving_image, 1,\\n$cross_correlation_radius ]\\n--convergence [ 2000x1000x500x250, 1e-6, 10 ]\\n--shrink-factors 8x4x2x1\\n--smoothing-sigmas {3+delta_3}x{2+delta_2}x{1+delta_1}x0vox MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\n(a) Patient 1.\\n (b) Patient 2.\\n(c) Patient 3.\\n (d) Patient 4.\\nFigure 28: Dice scores for per-patient fine-grained configuration runs in ANTs, with the baseline after rigid registration in\\nblue.\\n(a) Patient 1.\\n (b) Patient 2.\\n(c) Patient 3.\\n (d) Patient 4.\\nFigure 29: Hausdorff 95th percentiles for per-patient fine-grained configuration runs in ANTs, with the baseline after rigid\\nregistration in blue. Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 30: Visual renders of deformations predicted by ANTs\\nwith different total regularization weights, on Patient 1.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 31: Visual renders of deformations predicted by ANTs\\nwith different total regularization weights, on Patient 2.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 32: Visual renders of deformations predicted by ANTs\\nwith different total regularization weights, on Patient 3.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 33: Visual renders of deformations predicted by ANTs\\nwith different total regularization weights, on Patient 4. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\nC.3 This Work: MOREA\\nWe describe several coarse-grained configuration experiments that\\nwe conducted with MOREA on Patient 1. The base parameter file\\nwe derived from these experiments can be found in Section C.3.2.\\nWe do not conduct fine-grained configuration steps, since MOREA\\nis a multi-objective approach.\\nFor MOREA\\u2019s guidance objective, we perform an additional pre-\\nprocessing step on each scan, to address the discrepancy between\\nresolutions in different dimensions. The initial resampling step\\nbringing each scan to a uniform voxel resolution of 1.5 mmleads to\\nthe between-slice dimension being over-sampled (originally, slices\\nare 3 mm apart). Contour annotations are placed only on slices,\\nwhich means that the new slices added by resampling to 1.5 mm,\\nbetween original slices, do not have contour information. These\\nslice \\u201cgaps\\u201d in the contours of objects can be exploited during\\noptimization. We address this with an intermediate step, building a\\n3D model of each object across slices and generating border points\\nfrom this model.\\nC.3.1 Coarse-grained configuration experiments.\\nHeterogeneous elasticity In Section 4.1, we describe a model that\\nenables capturing biomechanical properties of different tissue types\\nin the deformation magnitude objective. The core principle of this\\nbiomechanical model is to ascribe heterogeneous elasticities to\\ndifferent regions of image space, corresponding with objects (e.g.,\\norgans and bones) present. In this first configuration experiment,\\nwe compare the performance of this model with the performance of\\nthe model which is used by prior work [ 4], assuming homogeneous\\nelasticity of image space. This experiment was conducted without\\na contour on the body, later experiments do have this contour.\\nThe metric results in Figure 34 indicate that the heterogeneous\\nmodel generally receives higher Dice scores and similar Hausdorff\\n95th percentiles. Figure 35 shows renderings of selected solutions\\nwith the heterogeneous and homogeneous models, which confirm\\nthis trend. We observe in both slices that heterogeneous elasticity\\nespecially shows improved performance on the bladder deforma-\\ntion, potentially due to the increased elasticity that this models\\nassigns to the bladder.\\nMesh generation Using the biomechanical model that experiments\\nin the previous subsection covered, we now investigate the impact\\nof different mesh point placement strategies. The strategy used to\\ncreate meshes from these points is described in Section 4.2.1.\\nIn this experiment, compare how well a random (Sobol-sequence\\nbased) placement compares to a contour-based strategy where\\npoints are sampled per contour and a contour-based strategy which\\nhas special handling for the bladder\\u2019s surface. Figure 36a shows the\\nbladder being modeled best by the last strategy, with contour-based\\nstrategies in general performing better than random, across organs.\\nThe renders in Figure 37 indicate that a random placement method\\ncan model the general deformation, but is too coarse to accurately\\ntreat details of specific organs and parts of the bones. Both contour-\\nbased strategies perform well, but around the bladder\\u2019s surface, the\\nstrategy with special surface constraints excels.\\nSupplying guidance information The multi-objective line of reg-\\nistration approaches, which MOREA continues, can have a third\\n(a) Dice scores.\\n(b) 95th percentiles of the Hausdorff distance.\\nFigure 34: Comparison of the use of heterogeneous elas-\\nticities in the deformation magnitude objective of MOREA\\nagainst the prior use of a homogeneous elasticity model, for\\nmultiple runs. The baseline score after rigid registration is\\nplotted in blue.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 35: Visual renders of deformations predicted by\\nMOREA with a heterogeneous elastic deformation model\\nand a homogeneous model.\\nobjective that captures guidance (contour) match. In this experi- Figure 35: Visual renders of deformations predicted by\\nMOREA with a heterogeneous elastic deformation model\\nand a homogeneous model.\\nobjective that captures guidance (contour) match. In this experi-\\nment, we assess what the impact of this objective is on the quality\\nof registrations.\\nThe quantitative results in Figure 38 leave little doubt that the\\nadoption of a guidance objective is crucial to modeling large defor-\\nmations. Without it, the bladder remains largely in place, as can be\\nseen in Figure 39. It seems that in this problem, image information\\nis not sufficient to guide the optimization. Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\n(a) Dice scores.\\n(b) 95th percentiles of the Hausdorff distance.\\nFigure 36: Comparison of different mesh point placement\\nstrategies, for multiple runs. The baseline score after rigid\\nregistration is plotted in blue.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 37: Visual renders of deformations predicted by\\nMOREA with different mesh point placement strategies.\\n(a) Dice scores.\\n(b) 95th percentiles of the Hausdorff distance.\\nFigure 38: Comparison of MOREA registrations with and\\nwithout guidance information, for multiple runs. The base-\\nline score after rigid registration is plotted in blue.\\n(a) Sagittal slice.\\n (b) Coronal slice.\\nFigure 39: Visual renders of deformations predicted by\\nMOREA with and without guidance enabled. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\nC.3.2 Parameter file. We pass parameters to MOREA in a self-\\nwritten parameter file format. Below we list the parameter file used\\nas basis for the experiments listed in this work.\\nListing 5: Parameter file used as basis for the main MOREA\\nexperiments.\\nsweep_descriptor = \\\"$experiment_descriptor\\\"\\nnum_runs = 5\\nproblem_id = \\\"$problem_id\\\"\\nzip = true\\nproblem_guidance_enabled = true\\nproblem_guidance_selection = \\\"-1\\\"\\ncuda_compute_level = 80\\ncuda_gpu_id = 0\\nea_num_generations = 500\\nea_population_size = 700\\nea_num_clusters = 10\\nea_archive_size = 2000\\nea_adaptive_steering_enabled = true\\nea_adaptive_steering_activated_at_num_generations = 100\\nea_adaptive_steering_guidance_threshold = 1.5\\nmorea_init_noise_method = \\\"global-gaussian\\\"\\nmorea_init_noise_factor = 1.0\\nmorea_mesh_generation_method = \\\"annotation-group-random-bladder-10\\\"\\nmorea_mesh_num_points = 600\\nmorea_max_num_mesh_levels = 1\\nmorea_num_generations_per_level = 0\\nmorea_magnitude_metric = \\\"biomechanical\\\"\\nmorea_image_metric = \\\"squared-differences\\\"\\nmorea_guidance_metric = \\\"continuous-per-group\\\"\\nmorea_sampling_rate = 1.0\\nmorea_fos_type = \\\"edges\\\"\\nmorea_symmetry_mode = \\\"transform-both\\\"\\nmorea_dual_dynamic_mode = \\\"dual\\\"\\nmorea_repair_method = \\\"gaussian\\\"\\nmorea_ams_strategy = \\\"none\\\"\\nmorea_num_disruption_kernels = 0\\nmorea_disruption_frequency = 0 Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\nD FULL EXPERIMENTAL RESULTS\\nIn this appendix, we list more extensive results of the experiments\\npresented in Section 6. Figure 40 and 41 give full metric results for all\\npatients, comparing the three approaches with parallel coordinate\\nplots. Table 6 lists significance test results for all organ Hausdorff\\ndistances. A visual perspective is provided by Table 8, which shows\\nan additional slice per patient, overlaid with the predicted deforma-\\ntions. Below, we analyze convergence behavior (Section D.1) and\\nlandmark performance (Section D.2).\\nD.1 Convergence Behavior\\nWe plot the convergence behavior of each approach on Patient 1\\nin Figure 42 to show how each approach has converged before\\nyielding the results we show here. Elastix and ANTs both have\\na multi-resolution approach. To deal with the discontinuities in\\nmulti-stage resolution, we mark resolution switches in those plots\\nwith red vertical lines. Our configuration of Elastix also has a mask\\nregistration step, meaning that there are in total 8 segments (4 reso-\\nlutions of mask registration and 4 resolutions of image registration).\\nThe scaling of the value to be optimized is not always normalized\\nacross resolutions, which explains the jumps in value ranges be-\\ntween resolutions. Note that ANTs uses a separate \\u201cconvergence\\nvalue\\u201d to determine when it has converged, plotted in Figure 42d.\\nFor MOREA, we plot the achieved hypervolume and the best guid-\\nance objective value achieved. The sudden decrease in hypervolume\\nat generation 100 is related to the adaptive steering strategy used,\\nwhich purges any solutions with unfavorable guidance objective\\nvalues from the elitist archive.\\nD.2 Landmark Accuracy\\nWe list landmark registration accuracy on all 4 patients in Table 7.\\nWe aggregate all errors of all landmarks across repeats for one pa-\\ntient and approach, and compute the mean and standard deviation\\non this sample set. Since these landmarks are generally placed on\\nvisible, anatomically stable locations, and typically not in strongly\\ndeforming regions, this accuracy should be interpreted as a measure\\nof how well the method preserves certain anatomical structures.\\nThis measure is therefore less suitable as a measure of how well\\nthe registration problem is \\u201csolved\\u201d, for which visual (DVF and\\nrendered) inspection is still key. For some landmarks, the precise lo-\\ncation can be ambiguously defined or less visible on certain patients.\\nThese landmarks are, however, still accurately placeable between\\nscans by using the visual context they are situated in and taking\\nconsistent placement decisions for each pair of scans.\\nGenerally, we observe that Elastix performs worse than ANTs\\nand MOREA, and MOREA always improves or roughly maintains\\nthe baseline landmark registration error. We do not see a consis-\\ntent correlation between actual registration performance on large\\ndeforming objects and target registration error values, due to the\\naforementioned reasons.Problem Contour MOREA \\/ Elastix MOREA \\/ ANTs\\nPatient 1bladder 0.011 (+) 0.007 (+)\\nbones 0.009 (+) 0.006 (+)\\nrectum 0.007 (+) 0.007 (+)\\nanal canal 0.007 (+) 0.007 (+)\\nsigmoid 0.007 (+) 0.007 (+)\\nbowel 0.010 (+) 0.011 (+)\\nbody 0.006 (+) 0.006 (-)\\nPatient 2bladder 0.007 (+) 0.007 (+)\\nbones 0.007 (+) 0.007 (+)\\nrectum 0.118 (+) 0.007 (-)\\nanal canal 0.123 (-) 0.180 (-)\\nsigmoid 0.007 (+) 0.007 (-)\\nbowel 0.401 (+) 0.007 (+)\\nbody 0.655 (+) 1.000 (-)\\nPatient 3bladder 0.012 (+) 0.007 (+)\\nbones 0.007 (+) 0.007 (+)\\nrectum 0.290 (+) 0.007 (-)\\nanal canal 0.118 (-) 0.007 (+)\\nsigmoid 0.007 (+) 0.007 (+)\\nbowel 0.007 (+) 0.056 (+)\\nbody 0.007 (+) 0.118 (+)\\nPatient 4bladder 0.007 (+) 0.195 (-)\\nbones 0.007 (-) 0.007 (-)\\nrectum 0.010 (-) 0.007 (-)\\nanal canal 0.606 (+) 0.007 (-)\\nsigmoid 0.009 (+) 0.118 (+)\\nbowel 0.119 (+) 0.119 (-)\\nbody 0.020 (-) 0.020 (-)\\nTable 6: p-values of pair-wise comparisons of Hausdorff dis-\\ntances for all contours between approaches, computed by sigmoid 0.009 (+) 0.118 (+)\\nbowel 0.119 (+) 0.119 (-)\\nbody 0.020 (-) 0.020 (-)\\nTable 6: p-values of pair-wise comparisons of Hausdorff dis-\\ntances for all contours between approaches, computed by\\nthe two-sided Mann-Whitney U test. A plus ( +) indicates a\\nbetter mean with MOREA, a minus ( -) the opposite. Signifi-\\ncant results are highlighted according to an \\ud835\\udefcof 0.025.\\nProblem Baseline Elastix ANTs MOREA\\nPatient 1 4.8 \\u00b13.1 5.6\\u00b12.8 4.2\\u00b12.0 4.8\\u00b12.0\\nPatient 2 7.5 \\u00b14.0 11.8\\u00b17.3 7.7\\u00b14.3 7.8\\u00b13.8\\nPatient 3 9.5 \\u00b16.7 6.4\\u00b12.0 7.7\\u00b12.6 6.5\\u00b11.9\\nPatient 4 14.1 \\u00b19.5 8.1\\u00b14.3 6.3\\u00b13.4 6.8\\u00b14.0\\nTable 7: Target registration errors (mean and standard devi-\\nation) for the shown registrations of each approach on each\\npatient, across repeats. All errors are specified in mm. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\nInstance Transformed: sagittal Transformed: coronal\\nPatient 1\\nPatient 2\\nPatient 3\\nPatient 4\\nTable 8: A selection of the best predicted deformations of the compared registration approaches, represented by deformed\\ncontours compared to the target contours and image. Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\n(a) Patient 1.\\n (b) Patient 2.\\n(c) Patient 3.\\n (d) Patient 4.\\nFigure 40: Dice scores for all approaches on all patients. The baseline score after rigid registration is plotted in blue.\\n(a) Patient 1.\\n (b) Patient 2.\\n(c) Patient 3.\\n (d) Patient 4.\\nFigure 41: Hausdorff distances for all approaches on all patients. The baseline score after rigid registration is plotted in blue. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\n0 10000 20000 30000 40000 50000 60000 70000 80000\\nIterations2.12.01.91.81.71.61.51.41.31.2Objective value\\n(a) Elastix: objective value at each iteration.\\n0 10000 20000 30000 40000 50000 60000 70000 80000\\nIterations2.12.01.91.81.71.61.51.4Objective value (b) Elastix: best objective value achieved at each point.\\n0 10 20 30 40 50 60 70 80\\nTime steps0.960.940.920.900.880.860.840.820.800.78Objective value\\n(c) ANTs: objective value at each iteration.\\n0 10 20 30 40 50 60 70 80\\nTime steps0.0020.0000.0020.0040.0060.0080.010Convergence value (d) ANTs: convergence measure at each iteration.\\n0 100 200 300 400 500\\nGenerations1015202530354045Hypervolume\\n(e) MOREA: hypervolume at each generation.\\n0 100 200 300 400 500\\nGenerations0.00.10.20.30.40.5Guidance error (f) MOREA: best guidance objective value found at each generation.\\nFigure 42: Convergence plots for all 3 approaches on one run of Patient 1. Vertical red lines indicate a change of resolution.\\nFor ANTs, this leads to 4 optimization segments. For Elastix, we first run a mask registration step (with 4 segments) and then\\nan image registration step (with again 4 segments). Georgios Andreadis, Peter A.N. Bosman, and Tanja Alderliesten\\nREFERENCES\\n[1]T. Alderliesten, P. A. N. Bosman, and A. Bel. 2015. Getting the most out of\\nadditional guidance information in deformable image registration by leveraging\\nmulti-objective optimization. In SPIE Medical Imaging 2015: Image Processing .\\n94131R.\\n[2]T. Alderliesten, J. J. Sonke, and P. A. N. Bosman. 2012. Multi-objective optimization\\nfor deformable image registration: proof of concept. In SPIE Medical Imaging\\n2012: Image Processing , Vol. 8314. 831420.\\n[3]T. Alderliesten, J. J. Sonke, and P. A. N. Bosman. 2013. Deformable image reg-\\nistration by multi-objective optimization using a dual-dynamic transformation\\nmodel to account for large anatomical differences. In SPIE Medical Imaging 2013:\\nImage Processing , Vol. 8669. 866910.\\n[4]G. Andreadis, P. A. N. Bosman, and T. Alderliesten. 2022. Multi-objective dual\\nsimplex-mesh based deformable image registration for 3D medical images - proof\\nof concept. In SPIE Medical Imaging 2022: Image Processing . 744\\u2013750.\\n[5]B. B. Avants, C. L. Epstein, M. Grossman, and J. C. Gee. 2008. Symmetric dif-\\nfeomorphic image registration with cross-correlation: Evaluating automated\\nlabeling of elderly and neurodegenerative brain. Medical Image Analysis 12, 1\\n(2008), 26\\u201341.\\n[6]F. Bartel, M. Visser, M. de Ruiter, J. Belderbos, F. Barkhof, H. Vrenken, J. C. de\\nMunck, and M. van Herk. 2019. Non-linear registration improves statistical power\\nto detect hippocampal atrophy in aging and dementia. NeuroImage: Clinical 23\\n(2019), 101902.\\n[7]D. L. J. Barten, B. R. Pieters, A. Bouter, M. C. van der Meer, S. C. Maree, K. A.\\nHinnen, H. Westerveld, P. A. N. Bosman, T. Alderliesten, N. van Wieringen,\\nand A. Bel. 2023. Towards artificial intelligence-based automated treatment\\nplanning in clinical practice: A prospective study of the first clinical experiences\\nin high-dose-rate prostate brachytherapy. Brachytherapy In Press (2023).\\n[8]L. Bondar, M. S. Hoogeman, E. M. V\\u00e1squez Osorio, and B. J.M. Heijmen. 2010. A\\nsymmetric nonrigid registration method to handle large organ deformations in\\ncervical cancer patients. Medical Physics 37, 7 (2010), 3760\\u20133772.\\n[9]P. A. N. Bosman and T. Alderliesten. 2016. Smart grid initialization reduces\\nthe computational complexity of multi-objective image registration based on a\\ndual-dynamic transformation model to account for large anatomical differences.\\nInSPIE Medical Imaging 2016: Image Processing . 978447.\\n[10] A. Bouter, T. Alderliesten, and P. A. N. Bosman. 2017. A novel model-based\\nevolutionary algorithm for multi-objective deformable image registration with\\ncontent mismatch and large deformations: benchmarking efficiency and quality.\\nInSPIE Medical Imaging 2017: Image Processing , Vol. 10133. 1013312.\\n[11] A. Bouter, T. Alderliesten, and P. A. N. Bosman. 2021. Achieving highly scal-\\nable evolutionary real-valued optimization by exploiting partial evaluations.\\nEvolutionary Computation 29, 1 (2021), 129\\u2013155.\\n[12] A. Bouter, T. Alderliesten, and P. A. N. Bosman. 2021. GPU-Accelerated Par-\\nallel Gene-pool Optimal Mixing applied to Multi-Objective Deformable Image\\nRegistration. In IEEE Congress on Evolutionary Computation . 2539\\u20132548.\\n[13] A. Bouter, T. Alderliesten, B. R. Pieters, A. Bel, Y. Niatsetski, and P. A. N. Bosman.\\n2019. GPU-accelerated bi-objective treatment planning for prostate high-dose-\\nrate brachytherapy. Medical Physics 46, 9 (2019), 3776\\u20133787.\\n[14] A. Bouter, N. H. Luong, C. Witteveen, T. Alderliesten, and P. A. N. Bosman. 2017.\\nThe multi-objective real-valued gene-pool optimal mixing evolutionary algorithm.\\nInProceedings of the 2017 Genetic and Evolutionary Computation Conference . 537\\u2013\\n544.\\n[15] D. Br\\u00e9laz. 1979. New methods to color the vertices of a graph. Commun. ACM\\n22, 4 (1979), 251\\u2013256.\\n[16] K. K. Brock, S. Mutic, T. R. McNutt, H. Li, and M. L. Kessler. 2017. Use of image\\nregistration and fusion algorithms and techniques in radiotherapy: Report of 22, 4 (1979), 251\\u2013256.\\n[16] K. K. Brock, S. Mutic, T. R. McNutt, H. Li, and M. L. Kessler. 2017. Use of image\\nregistration and fusion algorithms and techniques in radiotherapy: Report of\\nthe AAPM Radiation Therapy Committee Task Group No. 132: Report. Medical\\nPhysics 44, 7 (2017), e43\\u2013e76.\\n[17] K. K. Brock, M. B. Sharpe, L. A. Dawson, S. M. Kim, and D. A. Jaffray. 2005. Accu-\\nracy of finite element model-based multi-organ deformable image registration.\\nMedical Physics 32, 6 (2005), 1647\\u20131659.\\n[18] H. Chui and A. Rangarajan. 2000. A new algorithm for non-rigid point matching.\\nInIEEE Conference on Computer Vision and Pattern Recognition . 44\\u201351.\\n[19] K. Deb. 2001. Multi-Objective Optimization using Evolutionary Algorithms . Wiley.\\n[20] B. Eiben, V. Vavourakis, J. H. Hipwell, S. Kabus, T. Buelow, C. Lorenz, T.\\nMertzanidou, S. Reis, N. R. Williams, M. Keshtgar, and D. J. Hawkes. 2016. Symmet-\\nric Biomechanically Guided Prone-to-Supine Breast Image Registration. Annals\\nof Biomedical Engineering 44, 1 (2016), 154\\u2013173.\\n[21] C. Ericson. 2004. Real-time collision detection (1 ed.). CRC Press.\\n[22] M. Faisal Beg, M. I. Miller, A. Trouv\\u00e9trouv, and L. Younes. 2005. Computing\\nLarge Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms.\\nInternational Journal of Computer Vision 61, 2 (2005), 139\\u2013157.\\n[23] B. Fischer and J. Modersitzki. 2008. Ill-posed medicine - An introduction to image\\nregistration. Inverse Problems 24, 3 (2008), 1\\u201316.\\n[24] J. Gascon, J. M. Espadero, A. G. Perez, R. Torres, and M. A. Otaduy. 2013. Fast\\ndeformation of volume data using tetrahedral mesh rasterization. In Proceed-\\nings - SCA 2013: 12th ACM SIGGRAPH \\/ Eurographics Symposium on ComputerAnimation . 181\\u2013186.\\n[25] S. Hang. 2015. TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator.\\nACM Trans. Math. Software 41, 2 (2015), 1\\u201336.\\n[26] F. Khalifa, G. M. Beache, G. Gimel\\u2019farb, J. S. Suri, and A. S. El-Baz. 2011. State-of-\\nthe-Art Medical Image Registration Methodologies: A Survey. In Multi Modal-\\nity State-of-the-Art Medical Image Segmentation and Registration Methodologies .\\nSpringer, 235\\u2013280.\\n[27] A. Klein, J. Andersson, B. A. Ardekani, J. Ashburner, B. Avants, M. C. Chiang,\\nG. E. Christensen, D. L. Collins, J. Gee, P. Hellier, J. H. Song, M. Jenkinson, C.\\nLepage, D. Rueckert, P. Thompson, T. Vercauteren, R. P. Woods, J. J. Mann, and\\nR. V. Parsey. 2009. Evaluation of 14 nonlinear deformation algorithms applied to\\nhuman brain MRI registration. NeuroImage 46, 3 (2009), 786\\u2013802.\\n[28] S. Klein, M. Staring, K. Murphy, M. A. Viergever, and J. P.W. Pluim. 2010. Elastix:\\nA toolbox for intensity-based medical image registration. IEEE Transactions on\\nMedical Imaging 29, 1 (2010), 196\\u2013205.\\n[29] M. Li, E. Castillo, X. L. Zheng, H. Y. Luo, R. Castillo, Y. Wu, and T. Guerrero. 2013.\\nModeling lung deformation: A combined deformable image registration method\\nwith spatially varying Young\\u2019s modulus estimates. Medical Physics 40, 8 (2013),\\n1\\u201310.\\n[30] G. Loi, M. Fusella, E. Lanzi, E. Cagni, C. Garibaldi, G. Iacoviello, F. Lucio, E.\\nMenghi, R. Miceli, L. C. Orlandini, Antonella Roggio, Federica Rosica, Michele\\nStasi, Lidia Strigari, Silvia Strolin, and Christian Fiandra. 2018. Performance\\nof commercially available deformable image registration platforms for contour\\npropagation using patient-based computational phantoms: A multi-institutional\\nstudy. Medical Physics 45, 2 (2018), 748\\u2013757.\\n[31] H. N. Luong and P. A. N. Bosman. 2012. Elitist Archiving for Multi-Objective\\nEvolutionary Algorithms: To Adapt or Not to Adapt. In Proceedings of the 12th\\nConference on Parallel Problem Solving from Nature . 72\\u201381.\\n[32] H. Maaranen, K. Miettinen, and A. Penttinen. 2007. On initial populations of\\na genetic algorithm for continuous optimization problems. Journal of Global\\nOptimization 37, 3 (2007), 405\\u2013436.\\n[33] R. Mohammadi, S. R. Mahdavi, R. Jaberi, Z. Siavashpour, L. Janani, A. S. Meigooni,\\nand R. Reiazi. 2019. Evaluation of deformable image registration algorithm for [33] R. Mohammadi, S. R. Mahdavi, R. Jaberi, Z. Siavashpour, L. Janani, A. S. Meigooni,\\nand R. Reiazi. 2019. Evaluation of deformable image registration algorithm for\\ndetermination of accumulated dose for brachytherapy of cervical cancer patients.\\nJournal of Contemporary Brachytherapy 11, 5 (2019), 469\\u2013478.\\n[34] S. Nithiananthan, S. Schafer, D. J. Mirota, J. W. Stayman, W. Zbijewski, D. D. Reh,\\nG. L. Gallia, and J. H. Siewerdsen. 2012. Extra-dimensional Demons: A method for\\nincorporating missing tissue in deformable image registration. Medical Physics\\n39, 9 (2012), 5718\\u20135731.\\n[35] D. F. Pace, M. Niethammer, and S. R. Aylward. 2012. Sliding Geometries in De-\\nformable Image Registration. In International MICCAI Workshop on Computational\\nand Clinical Challenges in Abdominal Imaging . 141\\u2013148.\\n[36] K. Pirpinia, P. A. N. Bosman, C. E. Loo, G. Winter-Warnars, N. N. Y. Janssen, A. N.\\nScholten, J. J. Sonke, M. van Herk, and T. Alderliesten. 2017. The feasibility of\\nmanual parameter tuning for deformable breast MR image registration from a\\nmulti-objective optimization perspective. Physics in Medicine and Biology 62, 14\\n(2017), 5723\\u20135743.\\n[37] B. Rigaud, A. Klopp, S. Vedam, A. Venkatesan, N. Taku, A. Simon, P. Haigron, R.\\nDe Crevoisier, K. K. Brock, and G. Cazoulat. 2019. Deformable image registration\\nfor dose mapping between external beam radiotherapy and brachytherapy images\\nof cervical cancer. Physics in Medicine and Biology 64, 11 (2019), 115023.\\n[38] L. Risser, F. X. Vialard, H. Y. Baluwala, and J. A. Schnabel. 2013. Piecewise-\\ndiffeomorphic image registration: Application to the motion estimation between\\n3D CT lung images with sliding conditions. Medical Image Analysis 17, 2 (2013),\\n182\\u2013193.\\n[39] B. Schaly, J. A. Kempe, G. S. Bauman, J. J. Battista, and J. van Dyk. 2004. Tracking\\nthe dose distribution in radiation therapy by accounting for variable anatomy.\\nPhysics in Medicine and Biology 49, 5 (2004), 791\\u2013805.\\n[40] A. Sotiras and N. Paragios. 2012. Deformable Image Registration: A Survey . Tech-\\nnical Report. Center for Visual Computing, Department of Applied Mathematics,\\nEcole Centrale de Paris, Equipe GALEN, INRIA Saclay.\\n[41] D. Thierens and P. A. N. Bosman. 2011. Optimal mixing evolutionary algorithms.\\nInProceedings of the 2011 Genetic and Evolutionary Computation Conference .\\n617\\u2013624.\\n[42] J.-P. Thirion. 1998. Image matching as a diffusion process: an analogy with\\nMaxwell\\u2019s Demons. Medical Image Analysis 2, 3 (1998), 243\\u2013260.\\n[43] M. Unser, A. Aldroubi, and C. R. Gerfen. 1993. Multiresolution image registra-\\ntion procedure using spline pyramids. In SPIE Mathematical Imaging: Wavelet\\nApplications in Signal and Image Processing , Vol. 2034. 160\\u2013170.\\n[44] E. M. V\\u00e1squez Osorio, M. S. Hoogeman, L. Bondar, P. C. Levendag, and B. J. M.\\nHeijmen. 2009. A novel flexible framework with automatic feature correspon-\\ndence optimization for nonrigid registration in radiotherapy. Medical Physics 36,\\n7 (2009), 2848\\u20132859.\\n[45] O. Weistrand and S. Svensson. 2015. The ANACONDA algorithm for deformable\\nimage registration in radiotherapy. Medical Physics 42, 1 (2015), 40\\u201353.\\n[46] S. Wognum, L. Bondar, A. G. Zolnay, X. Chai, M. C. C. M. Hulshof, M. S. Hoogeman,\\nand A. Bel. 2013. Control over structure-specific flexibility improves anatomical\\naccuracy for point-based deformable registration in bladder cancer radiotherapy. MOREA: a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images\\nMedical Physics 40, 2 (2013), 1\\u201315.\\n[47] W. Zhang, Y. Ma, J. Zheng, and W. J. Allen. 2020. Tetrahedral mesh deformation\\nwith positional constraints. Computer Aided Geometric Design 81 (2020), 1\\u201316.[48] H. Zhong, J. Kim, H. Li, T. Nurushev, B. Movsas, and I. J. Chetty. 2012. A finite\\nelement method to correct deformable image registration errors in low-contrast\\nregions. Physics in Medicine and Biology 57, 11 (2012), 3499\\u20133515.\",\"3\":\" What Performance Indicators to Use for Self-Adaptation in\\nMulti-Objective Evolutionary Algorithms\\nFurong Ye\\nf.ye@liacs.leidenuniv.nl\\nLIACS, Leiden University\\nLeiden, NetherlandsFrank Neumann\\nfrank.neumann@adelaide.edu.au\\nThe University of Adelaide\\nAdelaide, AustraliaJacob de Nobel\\nj.p.de.nobel@liacs.leidenuniv.nl\\nLIACS, Leiden University\\nLeiden, Netherlands\\nAneta Neumann\\naneta.neumann@adelaide.edu.au\\nThe University of Adelaide\\nAdelaide, AustraliaThomas B\\u00e4ck\\nT.H.W.Baeck@liacs.leidenuniv.nl\\nLIACS, Leiden University\\nLeiden, Netherlands\\nABSTRACT\\nParameter control has succeeded in accelerating the convergence\\nprocess of evolutionary algorithms. Empirical and theoretical stud-\\nies for classic pseudo-Boolean problems, such as OneMax ,Leadin-\\ngOnes , etc., have explained the impact of parameters and helped us\\nunderstand the behavior of algorithms for single-objective optimiza-\\ntion. In this work, by transmitting the techniques of single-objective\\noptimization, we perform an extensive experimental investigation\\ninto the behavior of the self-adaptive GSEMO variants.\\nWe test three self-adaptive mutation techniques designed for\\nsingle-objective optimization for the OneMinMax ,COCZ ,LOTZ ,\\nandOneJumpZeroJump problems. While adopting these techniques\\nfor the GSEMO algorithm, we consider different performance met-\\nrics based on the current non-dominated solution set. These metrics\\nare used to guide the self-adaption process.\\nOur results indicate the benefits of self-adaptation for the tested\\nbenchmark problems. We reveal that the choice of metrics signifi-\\ncantly affects the performance of the self-adaptive algorithms. The\\nself-adaptation methods based on the progress in one objective\\ncan perform better than the methods using multi-objective met-\\nrics such as hypervolume, inverted generational distance, and the\\nnumber of the obtained Pareto solutions. Moreover, we find that\\nthe self-adaptive methods benefit from the large population size\\nforOneMinMax andCOCZ .\\nKEYWORDS\\nMulti-objective evolutionary algorithm, self-adaptation, mutation,\\nperformance metric\\nACM Reference Format:\\nFurong Ye, Frank Neumann, Jacob de Nobel, Aneta Neumann, and Thomas\\nB\\u00e4ck. 2023. What Performance Indicators to Use for Self-Adaptation in\\nMulti-Objective Evolutionary Algorithms . In Genetic and Evolutionary\\nPermission to make digital or hard copies of all or part of this work for personal or\\nclassroom use is granted without fee provided that copies are not made or distributed\\nfor profit or commercial advantage and that copies bear this notice and the full citation\\non the first page. Copyrights for components of this work owned by others than the\\nauthor(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or\\nrepublish, to post on servers or to redistribute to lists, requires prior specific permission\\nand\\/or a fee. Request permissions from permissions@acm.org.\\nGECCO \\u201923, July 15\\u201319, 2023, Lisbon, Portugal\\n\\u00a92023 Copyright held by the owner\\/author(s). Publication rights licensed to ACM.\\nACM ISBN xxx. . . $15.00\\nhttps:\\/\\/doi.org\\/xxxComputation Conference (GECCO \\u201923), July 15\\u201319, 2023, Lisbon, Portugal.\\nACM, New York, NY, USA, 9 pages. https:\\/\\/doi.org\\/xxx\\n1 INTRODUCTION\\nEvolutionary algorithms (EAs) are capable of finding global optima\\nby creating offspring solutions via global search variators. Apart\\nfrom the guarantee of reaching a global optimum, a major concern\\nin designing algorithms is the convergence rate, as we can not\\nafford infinite running time in real-world applications. The study\\nof parameter control is essential in this, in order to understand the\\nrelation between variator parameters and the dynamic behavior of\\nalgorithms. For example, in the context of single-objective pseudo-\\nboolean optimization \\ud835\\udc53:{0,1}\\ud835\\udc5b\\u2192R, the mutation of EAs creates\\noffspring\\ud835\\udc66by flipping 0<\\u2113\\u2264\\ud835\\udc5bbits of the parent solution \\ud835\\udc65,\\nwhere\\u2113can be sampled from different distributions depending on\\nthe design of mutation operators. Previous studies [ 2,3,5,10] have offspring\\ud835\\udc66by flipping 0<\\u2113\\u2264\\ud835\\udc5bbits of the parent solution \\ud835\\udc65,\\nwhere\\u2113can be sampled from different distributions depending on\\nthe design of mutation operators. Previous studies [ 2,3,5,10] have\\ndemonstrated the impact of the choice of \\u2113onOneMax ,Leadin-\\ngOnes , and other classic problems, and self-adaptive methods have\\nbeen proven to have higher convergence rates than the ones with\\nstatic settings.\\nWhile there have been detailed studies for single-objective prob-\\nlems, detailed investigations complement existing theoretical stud-\\nies for multi-objective benchmark problems that are missing in the\\nliterature. In the area of runtime analysis, the most basic multi-\\nobjective benchmark problems that have been studied in a rigor-\\nous way are LOTZ ,COCZ , and OneMinMax . Furthermore, One-\\nJumpZeroJump has recently been introduced as a multi-modal multi-\\nobjective benchmark problem. Investigations in the area of runtime\\nanalysis have been started by [ 15]. In this work, the authors stud-\\nied a variant of the simple evolutionary multi-objective optimizer\\n(SEMO) which produces an offspring by flipping a single bit and\\nalways maintaining a set of non-dominated trade-offs according to\\nthe given objective functions. Runtime bounds of \\u0398(\\ud835\\udc5b3)for LOTZ\\nand\\ud835\\udc42(\\ud835\\udc5b2log\\ud835\\udc5b)for COCZ have been shown in [ 15].OneMinMax\\nhas been investigated in [ 4,13,16] and it has been shown that\\nthe global simple evolutionary multi-objective optimizer (GSEMO),\\nwhich differs from SEMO by apply standard bit mutations instead of\\nsingle bit flips, computes the whole Pareto front for OneMinMax in\\nexpected time \\ud835\\udc42(\\ud835\\udc5b2log\\ud835\\udc5b). Furthermore, hypervolume-based evolu-\\ntionary algorithms have been studied for OneMinMax in [4,16] and\\nthe computation of structural diverse populations has been investi-\\ngated in [ 4]. In [ 17], different parent selection methods have beenarXiv:2303.04611v1  [cs.NE]  8 Mar 2023 GECCO \\u201923, July 15\\u201319, 2023, Lisbon, Portugal Furong Ye, Frank Neumann, Jacob de Nobel, Aneta Neumann, and Thomas B\\u00e4ck\\nanalyzed for OneMinMax andLOTZ and their benefit has been\\nshown when incorporated into GSEMO. Recently, OneMinMax\\nhas also been used to study the runtime behavior of the NSGA-II\\nalgorithm [ 22]. All the bounds obtained are asymptotic ones, i.e.\\nthey are missing the leading constants. Therefore, it is interesting\\nto carry out more detailed experimental investigations of simple\\nevolutionary multi-objective algorithms from the area of runtime\\nanalysis alongside some variations such as different mutation oper-\\nators and the use of larger offspring population sizes. We carry out\\nsuch experimental investigations in this work with the hope that\\nit provides complementary insights that enable further progress\\nin the theoretical understanding of evolutionary multi-objective\\noptimization.\\nIn this work, we analyze several variation operators that flip \\u2113\\nrandomly selected distinct bits and study the impact of \\u2113for multi-\\nobjective evolutionary algorithms (MOEAs). More precisely, we\\nequip GSEMO with several self-adaptive mutation operators and\\ninvestigate its performance for OneMinMax ,LOTZ ,COCZ , and\\nOneJumpZeroJump . Based on the experimental results, we study\\nthe impact of the number of offspring (i.e., population size \\ud835\\udf06) for the\\nself-adaptive GSEMO variants. Moreover, we investigate the impact\\nof the metrics used to guide self-adaptation. For the single-objective\\noptimization problems, each solution maps to a fitness function\\nvalue\\ud835\\udc53, which self-adaptive EAs can use directly as a performance\\nindicator to guide the dynamic parameter control. However, for the\\nmulti-objective optimization problems \\ud835\\udc53:{0,1}\\ud835\\udc5b\\u2192R\\ud835\\udc5a, where\\n\\ud835\\udc5ais the number of the objectives (note that we consider only bi-\\nobjectives in this work), each solution maps to a set of objective\\nfunction values, which renders the choice of the appropriate per-\\nformance indicator to guide adaptation less obvious. Since MOEAs\\nusually maintain a set of non-dominated solutions, we can use a\\nmetric\\ud835\\udc52(\\ud835\\udc65):\\ud835\\udc65\\u2192Rto calculate the relative improvement gain\\nfrom a solution \\ud835\\udc65with the non-dominated set, which can be used\\nto apply the single-objective techniques to MOEAs.\\nOverall, this paper performs an extensive experimental investi-\\ngation on GSEMO variants, addressing whether self-adaptive muta-\\ntion can benefit MOEAs. In practice, this involves the translation of\\nself-adaptive mutation mechanisms of single-objective optimization\\nalgorithms to MOEAs. Empirical analysis is conducted to illustrate\\nthe behavior of self-adaptive mutation for MOEAs, focusing on\\ntwo principal factors, population size, and performance indicator.\\nWe hope the presented results can inspire future theoretical under-\\nstanding of self-adaptation MOEAs. Moreover, such fundamental\\nanalysis of the classic problems can provide guidelines for designing\\nMOEAs in practical applications.\\n2 PRELIMINARIES\\n2.1 Benchmark Problems\\nAs discussed in the Introduction section, the motivation of this\\nwork is to perform empirical analyses and investigate the behav-\\nior of MOEAs. Therefore, 4classic multi-objective optimization\\nproblems that attracted many runtime analysis studies are selected\\nfor benchmarking. The problem definitions are provided in the\\nfollowing.2.1.1 OneMinMax .OneMinMax introduced in [ 13] is a bi-objective\\npseudo-Boolean optimization problem that generalizes the classical\\nsingle-objective OneMax problem to the bi-objective case:\\nOneMinMax :{0,1}\\ud835\\udc5b\\u2192N2,\\ud835\\udc65\\u21a6\\u2192 \\ud835\\udc5b\\u2211\\ufe01\\n\\ud835\\udc56=1\\ud835\\udc65\\ud835\\udc56,\\ud835\\udc5b\\u2212\\ud835\\udc5b\\u2211\\ufe01\\n\\ud835\\udc56=1\\ud835\\udc65\\ud835\\udc56!\\n(1)\\nThe problem is maximizing the numbers of both oneandzero bits,\\nand the objective of maximizing the number of onebits is identical\\nto the classic pseudo-Boolean optimization problem OneMax :\\n{0,1}\\ud835\\udc5b\\u2192N,\\ud835\\udc65\\u21a6\\u2192\\u00cd\\ud835\\udc5b\\n\\ud835\\udc56=1\\ud835\\udc65\\ud835\\udc56. For the OneMinMax problem, all\\nsolutions locate at the optimal Parent front, and the goal is to\\nobtain the complete set of Pareto front {(\\ud835\\udc56,\\ud835\\udc5b\\u22121)|\\ud835\\udc56\\u2208[0..\\ud835\\udc5b]}\\n2.1.2 COCZ ..COCZ [15] is another extension of the OneMax solutions locate at the optimal Parent front, and the goal is to\\nobtain the complete set of Pareto front {(\\ud835\\udc56,\\ud835\\udc5b\\u22121)|\\ud835\\udc56\\u2208[0..\\ud835\\udc5b]}\\n2.1.2 COCZ ..COCZ [15] is another extension of the OneMax\\nproblem which is called Count Ones Count Zeroes . Its definition is\\ngiven below.\\nCOCZ :{0,1}\\ud835\\udc5b\\u2192N2,\\ud835\\udc65\\u21a6\\u2192\\u00a9\\u00ad\\n\\u00ab\\ud835\\udc5b\\u2211\\ufe01\\n\\ud835\\udc56=1\\ud835\\udc65\\ud835\\udc56,\\ud835\\udc5b\\/2\\u2211\\ufe01\\n\\ud835\\udc56=1\\ud835\\udc65\\ud835\\udc56+\\ud835\\udc5b\\u2211\\ufe01\\n\\ud835\\udc56=\\ud835\\udc5b\\/2(1\\u2212\\ud835\\udc65\\ud835\\udc56)\\u00aa\\u00ae\\n\\u00ac(2)\\nwhere\\ud835\\udc5b=2\\ud835\\udc58,\\ud835\\udc58\\u2208N. The Pareto front of COCZ is a setPconsisting\\nof the solutions with \\ud835\\udc5b\\/2ones in the first half of the bit string. The\\nsize ofPis\\ud835\\udc5b\\/2. Differently from OneMinMax , of which all possible\\nsolutions locate at the Pareto front, many solutions that are strictly\\ndominated by others exist in the search space of COCZ .\\n2.1.3 LOTZ .The Leading Ones, Trailing Zeroes (LOTZ ) introduced\\nin [15] maximizes the numbers of leading onebits and trailing zero\\nbits, simultaneously. The problem can be defined as\\nLOTZ :{0,1}\\ud835\\udc5b\\u2192N2,\\ud835\\udc65\\u21a6\\u2192\\u00a9\\u00ad\\n\\u00ab\\ud835\\udc5b\\u2211\\ufe01\\n\\ud835\\udc56=1\\ud835\\udc56\\u00d6\\n\\ud835\\udc57=1\\ud835\\udc65\\ud835\\udc57,\\ud835\\udc5b\\u2211\\ufe01\\n\\ud835\\udc56=1\\ud835\\udc5b\\u00d6\\n\\ud835\\udc57=\\ud835\\udc56(1\\u2212\\ud835\\udc65\\ud835\\udc57)\\u00aa\\u00ae\\n\\u00ac(3)\\nThe Pareto front of LOTZ is{(\\ud835\\udc56,\\ud835\\udc5b\\u2212\\ud835\\udc56\\u22121)|\\ud835\\udc56\\u2208[1..\\ud835\\udc5b\\u22122]}\\u222a{(\\ud835\\udc56,\\ud835\\udc5b\\u2212\\ud835\\udc56)|\\n\\ud835\\udc56\\u2208 {0,\\ud835\\udc5b}}, given by the set of \\ud835\\udc5b+1solutions\\ud835\\udc65={1\\ud835\\udc560(\\ud835\\udc5b\\u2212\\ud835\\udc56)|\\n\\ud835\\udc56\\u2208[0..\\ud835\\udc5b]}. One property of LOTZ is that for all non-dominated\\nsolutions, the neighbors that are with 1Hamming distance are\\neither better or worse but incomparable.\\n2.1.4 OneJumpZeroJump .TheOneJumpZeroJump problem, which\\nwas originally proposed in [ 7], obtains objectives that are isomor-\\nphic to the classic single-objective problem Jump\\ud835\\udc58:{0,1}\\ud835\\udc5b\\u2192\\nN.Jump\\ud835\\udc58(\\ud835\\udc65)=\\ud835\\udc5b+|\\ud835\\udc65|1if|\\ud835\\udc65|1\\u2208[0..\\ud835\\udc5b\\u2212\\ud835\\udc58]\\u222a{\\ud835\\udc5b}, and Jump\\ud835\\udc58(\\ud835\\udc65)=\\n\\ud835\\udc5b\\u2212|\\ud835\\udc65|1otherwise , where|\\ud835\\udc65|\\ud835\\udc56=\\u00cd\\ud835\\udc5b\\n\\ud835\\udc56=0\\ud835\\udc65\\ud835\\udc56and\\ud835\\udc58\\u22652is a given\\nparameter. OneJumpZeroJump is defined as\\nOneJumpZeroJump \\ud835\\udc58:{0,1}\\ud835\\udc5b\\u2192N2,\\ud835\\udc65\\u21a6\\u2192(\\ud835\\udc661,\\ud835\\udc662),\\n\\ud835\\udc661=\\ud835\\udc58+|\\ud835\\udc65|1if|\\ud835\\udc65|1\\u2264\\ud835\\udc5b\\u2212\\ud835\\udc58or\\ud835\\udc65=1\\ud835\\udc5b,otherwise\\ud835\\udc5b\\u2212|\\ud835\\udc65|1,\\n\\ud835\\udc662=\\ud835\\udc58+|\\ud835\\udc65|0if|\\ud835\\udc65|0\\u2264\\ud835\\udc5b\\u2212\\ud835\\udc58or\\ud835\\udc65=0\\ud835\\udc5b,otherwise\\ud835\\udc5b\\u2212|\\ud835\\udc65|0(4)\\nThe Pareto front of OneJumpZeroJump is{(\\ud835\\udc4e,2\\ud835\\udc58+\\ud835\\udc5b\\u2212\\ud835\\udc4e)|\\ud835\\udc4e\\u2208\\n[2\\ud835\\udc58,\\ud835\\udc5b]\\u222a{\\ud835\\udc58,\\ud835\\udc5b+\\ud835\\udc58}}. We set\\ud835\\udc58=2for the experiments in this work.\\nOneJumpZeroJump is a multimodal problem obtaining a valley of\\nlow fitness values in the search space, and the size of this valley\\ndepends on \\ud835\\udc58.\\n2.2 The GSEMO Algorithm\\nThe simple evolutionary multi-objective optimizer (SEMO) [ 15]\\nuses a population \\ud835\\udc43of solutions that do not dominate each other. What Performance Indicators to Use for Self-Adaptation in Multi-Objective Evolutionary Algorithms GECCO \\u201923, July 15\\u201319, 2023, Lisbon, Portugal\\nThe population is initialized with a random solution. Then, the\\nalgorithm creates offspring by selecting a solution \\ud835\\udc65from\\ud835\\udc43uni-\\nformly at random (u.a.r.) and flipping one bit of \\ud835\\udc65. If any solutions\\nof\\ud835\\udc43that are dominated by \\ud835\\udc65, those solutions will be removed, and\\n\\ud835\\udc65will be added to \\ud835\\udc43. The algorithm terminates until reaching ter-\\nminate conditions, e.g., the budget is used out, or the full Parent\\nfront is reached. Global SEMO (GSEMO) [ 12] differs from SEMO by\\napplying the standard bit mutation instead of flipping exact one-bit\\nwhile creating offspring. As shown in Algorithm 1, GSEMO applies\\nthe standard bit mutation forcing to flip at least one bit each time.\\nMore precisely, \\u2113is sampled for a conditional binomial distribution\\nBin>0(\\ud835\\udc5b,\\ud835\\udc5d)following the suggestion in [ 18], and offspring is cre-\\nated by flip\\u2113(\\ud835\\udc65)flipping\\u2113bits chosen u.a.r. of \\ud835\\udc65. For the comparison\\nto the self-adaption methods, we denote the static GSEMO using\\n\\ud835\\udc5d=1\\/\\ud835\\udc5b.\\nAlgorithm 1: Global SEMO\\n1Input: mutation rate \\ud835\\udc5d=1\\/\\ud835\\udc5b;\\n2Initialization: Sample\\ud835\\udc65\\u2208{0,1}\\ud835\\udc5buniformly at random\\n(u.a.r.), and evaluate \\ud835\\udc53(\\ud835\\udc65);\\n3\\ud835\\udc43\\u2190{\\ud835\\udc65};\\n4Optimization: while not stop condition do\\n5 Select\\ud835\\udc65\\u2208\\ud835\\udc34u.a.r.;\\n6 Sample\\u2113\\u223cBin>0(\\ud835\\udc5b,\\ud835\\udc5d), create\\ud835\\udc66\\u2190flip\\u2113(\\ud835\\udc65), and\\nevaluate\\ud835\\udc53(\\ud835\\udc66);\\n7 ifthere is no\\ud835\\udc67\\u2208\\ud835\\udc34such that\\ud835\\udc66\\u2aaf\\ud835\\udc67then\\n8\\ud835\\udc34={\\ud835\\udc67\\u2208\\ud835\\udc34|\\ud835\\udc67\\u2aaf\\u0338\\ud835\\udc66}\\u222a{\\ud835\\udc66}\\n3 MULTI-OBJECTIVE SELF-ADAPTATION\\n3.1 The Self-Adaptive GSEMO Variants\\nAs discussed in previous studies [ 2,3,5,10], the optimal param-\\neter settings of EAs can change during the optimization process.\\nMany empirical and theoretical results [ 3,5,14,21] have shown\\nthat the EAs with self-adaptive mutation can outperform the stan-\\ndard bit mutation for classical single-objective problems such as\\nOneMax ,LeadingOnes , and Jump . We expect that GSEMO can\\nbenefit from using self-adaptation of mutation. Moreover, we are\\ncurious whether the self-adaption methods can perform similarly\\nas in single-objective optimization.\\nFor self-adaption in MOEAs, the (1+(\\ud835\\udf06,\\ud835\\udf06))GSEMO was pro-\\nposed and shown in [ 6] improvements in running time compared\\nto the classic GSEMO. The algorithm uses the variators of mutation\\nand crossover with dynamic parameters. Recently, a self-adjusting\\nGSEMO [ 7], which increases mutation rate after \\ud835\\udc47iterations not\\nobtaining new non-dominated solutions, was tested for OneJumpZe-\\nroJump . In this section, we test three GSEMO variants using self-\\nadaptive mutation that has been studied in much single-objective\\nbenchmarking work [ 8]. The GSEMO variants sample an offspring\\npopulation of size \\ud835\\udf06for each generation, and the sampling distribu-\\ntions adjust based on the performance \\ud835\\udc52(\\ud835\\udc65)of the new solutions.\\nThe procedures of GSEMO variants are introduced below, and we\\nintroduce the design of \\ud835\\udc52(\\ud835\\udc65)and study its impact in the later sec-\\ntions.3.1.1 The two-rate GSEMO. The two-rate EA with self-adaptive\\nmutation rate was proposed and analyzed in [ 5] for OneMax . It\\nstarts with an initial mutation rate of \\ud835\\udc5f\\/\\ud835\\udc5b. In each generation, it\\nsamples half offspring using doubled mutation rate and samples\\nthe other half using a half mutation rate. Then the mutation rate\\nthat is used to create the best offspring in this generation has a\\nhigher probability of 3\\/4to be chosen for the next generation. In\\nthe two-rate GSEMO (see Algorithm 4), we follow the outline of\\nGSEMO and sample offspring populations using the same mutation\\nstrategy of the two-rate EA. The solutions\\u2019 performance \\ud835\\udc52(\\ud835\\udc66)(line\\n12) is evaluated using the three measures mentioned above.\\nAlgorithm 2: The two-rate GSEMO\\n1Input: Population size \\ud835\\udf06,\\ud835\\udc5finit;\\n2Initialization: Sample\\ud835\\udc65\\u2208{0,1}\\ud835\\udc5buniformly at random\\n(u.a.r.), and evaluate \\ud835\\udc53(\\ud835\\udc65);\\n3\\ud835\\udc5f\\u2190\\ud835\\udc5finit,\\ud835\\udc43\\u2190{\\ud835\\udc65};\\n4Optimization: while not stop condition do\\n5 fori = 1,. . . ,\\ud835\\udf06do\\n6 Select\\ud835\\udc65\\u2208\\ud835\\udc43u.a.r.;\\n7 if\\ud835\\udc56<\\u230a\\ud835\\udf06\\/2\\u230bthen\\n8 Sample\\u2113(\\ud835\\udc56)\\u223cBin>0(\\ud835\\udc5b,\\ud835\\udc5f\\/(2\\ud835\\udc5b))\\n9 else\\n10 Sample\\u2113(\\ud835\\udc56)\\u223cBin>0(\\ud835\\udc5b,(2\\ud835\\udc5f)\\/\\ud835\\udc5b),\\n11 Create\\ud835\\udc66(\\ud835\\udc56)\\u2190flip\\u2113(\\ud835\\udc56)(\\ud835\\udc65), and evaluate \\ud835\\udc52(\\ud835\\udc66(\\ud835\\udc56));\\n12\\ud835\\udc66(\\ud835\\udc56\\u2217)\\u2190arg max{\\ud835\\udc52(\\ud835\\udc66(1)),...,\\ud835\\udc52(\\ud835\\udc66(\\ud835\\udf06))}; 6 Select\\ud835\\udc65\\u2208\\ud835\\udc43u.a.r.;\\n7 if\\ud835\\udc56<\\u230a\\ud835\\udf06\\/2\\u230bthen\\n8 Sample\\u2113(\\ud835\\udc56)\\u223cBin>0(\\ud835\\udc5b,\\ud835\\udc5f\\/(2\\ud835\\udc5b))\\n9 else\\n10 Sample\\u2113(\\ud835\\udc56)\\u223cBin>0(\\ud835\\udc5b,(2\\ud835\\udc5f)\\/\\ud835\\udc5b),\\n11 Create\\ud835\\udc66(\\ud835\\udc56)\\u2190flip\\u2113(\\ud835\\udc56)(\\ud835\\udc65), and evaluate \\ud835\\udc52(\\ud835\\udc66(\\ud835\\udc56));\\n12\\ud835\\udc66(\\ud835\\udc56\\u2217)\\u2190arg max{\\ud835\\udc52(\\ud835\\udc66(1)),...,\\ud835\\udc52(\\ud835\\udc66(\\ud835\\udf06))};\\n13 if\\ud835\\udc56\\u2217<\\u230a\\ud835\\udf06\\/2\\u230bthen\\ud835\\udc60\\u21903\\/4else\\ud835\\udc60\\u21901\\/4;\\n14 Sample\\ud835\\udc5e\\u2208[0,1]u.a.r.;\\n15 if\\ud835\\udc5e\\u2264\\ud835\\udc60then\\ud835\\udc5f\\u2190max{\\ud835\\udc5f\\/2,1\\/2}else\\n\\ud835\\udc5f\\u2190min{2\\ud835\\udc5f,\\ud835\\udc5b\\/4};\\n16 for\\ud835\\udc56=1,...,\\ud835\\udf06 do\\n17 ifthere is no\\ud835\\udc67\\u2208\\ud835\\udc34such that\\ud835\\udc66(\\ud835\\udc56)\\u2aaf\\ud835\\udc67then\\n18 \\ud835\\udc34={\\ud835\\udc67\\u2208\\ud835\\udc34|\\ud835\\udc67\\u2aaf\\u0338\\ud835\\udc66(\\ud835\\udc56)}\\u222a{\\ud835\\udc66(\\ud835\\udc56)}\\n3.1.2 The Log-Normal GSEMO. The logNormal GSEMO applies\\nthe standard bit mutation and adjusts the mutation rate \\ud835\\udc5dusing a\\nlog-normal update rule [ 14]. While creating offspring populations,\\na new mutation rate \\ud835\\udc5d\\u2032is sampled as shown in line 7 of Algorithm 3.\\nThis strategy allows the mutation rate to increase and decrease with\\nidentical probabilities. However, the new mutation rate can be any\\nvalue around the mean of the sample distribution, which is different\\nfrom the two-rate strategy that allows only double and half values.\\nMoreover, it chooses the \\ud835\\udc5d\\u2032that is used to create the best solution\\nas the\\ud835\\udc5dfor the next generation.\\n3.1.3 The Variance Controlled GSEMO. The variance-controlled\\nGSEMO (var-ctrl) applies the normalized bit mutation [ 21] that sam-\\nples\\u2113from a normal distribution \\ud835\\udc41(\\ud835\\udc5f,\\ud835\\udf0e2)(line 6 in Algorithm 4).\\nSimilar to the log-normal GSEMO, it also uses a greedy strategy to\\nadjust\\ud835\\udc5f, which is replaced by the value of \\u2113that creates the best\\nsolution. An advantage of using normal distributions is that we can\\ncontrol not only the mean of sampled \\u2113but also its variance. In this GECCO \\u201923, July 15\\u201319, 2023, Lisbon, Portugal Furong Ye, Frank Neumann, Jacob de Nobel, Aneta Neumann, and Thomas B\\u00e4ck\\nAlgorithm 3: The log-normal GSEMO\\n1Input: Population size \\ud835\\udf06, mutation rate \\ud835\\udc5d;\\n2Initialization: Sample\\ud835\\udc65\\u2208{0,1}\\ud835\\udc5buniformly at random\\n(u.a.r.), and evaluate \\ud835\\udc53(\\ud835\\udc65);\\n3\\ud835\\udc43\\u2190{\\ud835\\udc65};\\n4Optimization: while not stop condition do\\n5 for\\ud835\\udc56=1,...,\\ud835\\udf06 do\\n6 Select\\ud835\\udc65\\u2208\\ud835\\udc43u.a.r.;\\n7\\ud835\\udc5d(\\ud835\\udc56)=\\u00001+1\\u2212\\ud835\\udc5d\\n\\ud835\\udc5d\\u00b7exp(0.22\\u00b7N(0,1))\\u0001\\u22121;\\n8 Sample\\u2113(\\ud835\\udc56)\\u223cBin>0(\\ud835\\udc5b,\\ud835\\udc5d(\\ud835\\udc56));\\n9 create\\ud835\\udc66(\\ud835\\udc56)\\u2190flip\\u2113(\\ud835\\udc56)(\\ud835\\udc65), and evaluate \\ud835\\udc53(\\ud835\\udc66(\\ud835\\udc56));\\n10\\ud835\\udc66(\\ud835\\udc56\\u2217)\\u2190arg max{\\ud835\\udc52(\\ud835\\udc66(1)),...,\\ud835\\udc52(\\ud835\\udc66(\\ud835\\udf06))};\\n11\\ud835\\udc5d\\u2190\\ud835\\udc5d(\\ud835\\udc56\\u2217);\\n12 for\\ud835\\udc56=1,...,\\ud835\\udf06 do\\n13 ifthere is no\\ud835\\udc67\\u2208\\ud835\\udc43such that\\ud835\\udc66(\\ud835\\udc56)\\u2aaf\\ud835\\udc67then\\n14 \\ud835\\udc43={\\ud835\\udc67\\u2208\\ud835\\udc43|\\ud835\\udc67\\u2aaf\\u0338\\ud835\\udc66(\\ud835\\udc56)}\\u222a{\\ud835\\udc66(\\ud835\\udc56)}\\nAlgorithm 4: The var-ctrl GSEMO\\n1Input: Population size \\ud835\\udf06,\\ud835\\udc5finitand a Factor \\ud835\\udc39;\\nInitialization: Sample\\ud835\\udc65\\u2208{0,1}\\ud835\\udc5buniformly at random\\n(u.a.r.), and evaluate \\ud835\\udc53(\\ud835\\udc65);\\n2\\ud835\\udc5f\\u2190\\ud835\\udc5finit;\\ud835\\udc50\\u21900;\\ud835\\udc43\\u2190{\\ud835\\udc65};\\n3Optimization: while not stop condition do\\n4 for\\ud835\\udc56=1,...,\\ud835\\udf06 do\\n5 Select\\ud835\\udc65\\u2208\\ud835\\udc43u.a.r.;\\n6 Sample\\u2113(\\ud835\\udc56)\\u223cmin{\\ud835\\udc41>0(\\ud835\\udc5f,\\ud835\\udc39\\ud835\\udc50\\ud835\\udc5f(1\\u2212\\ud835\\udc5f\\/\\ud835\\udc5b)),\\ud835\\udc5b}, create\\n\\ud835\\udc66(\\ud835\\udc56)\\u2190flip\\u2113(\\ud835\\udc56)(\\ud835\\udc65), and evaluate \\ud835\\udc53(\\ud835\\udc66(\\ud835\\udc56));\\n7\\ud835\\udc66(\\ud835\\udc56\\u2217)\\u2190arg max{\\ud835\\udc52(\\ud835\\udc66(1)),...,\\ud835\\udc52(\\ud835\\udc66(\\ud835\\udf06))};\\n8 if\\ud835\\udc5f=\\u2113(\\ud835\\udc56\\u2217)then\\ud835\\udc50\\u2190\\ud835\\udc50+1else\\ud835\\udc50\\u21900;\\n9\\ud835\\udc5f\\u2190\\u2113(\\ud835\\udc56\\u2217);\\n10 for\\ud835\\udc56=1,...,\\ud835\\udf06 do\\n11 ifthere is no\\ud835\\udc67\\u2208\\ud835\\udc43such that\\ud835\\udc66(\\ud835\\udc56)\\u2aaf\\ud835\\udc67then\\n12 \\ud835\\udc43={\\ud835\\udc67\\u2208\\ud835\\udc43|\\ud835\\udc67\\u2aaf\\u0338\\ud835\\udc66(\\ud835\\udc56)}\\u222a{\\ud835\\udc66(\\ud835\\udc56)}\\nwork, we follow the setting in [ 21] reducing the variance by a factor\\n\\ud835\\udc39=0.98if the value of \\ud835\\udc5fremains the same after a generation.\\n3.2 Using Multi-objective Metrics\\nRecall that the GSEMO variants rely on the progress in \\ud835\\udc52(\\ud835\\udc65)for\\nself-adaption. Many performance measures have been applied to\\nassess the performance of MOEAs, and those measures would be\\nstraightforward candidates for calculating \\ud835\\udc52(\\ud835\\udc65). Therefore, we ap-\\nply three multi-objective performance measures, which have been\\ncommonly considered in MOEAs, for the metric of the GSEMO vari-\\nants. In practice, given a population \\ud835\\udc43of the current non-dominated\\nsolutions, we measure the performance of a new solution \\ud835\\udc65by cal-\\nculating for the set \\ud835\\udc43\\u222a{\\ud835\\udc65}thenumber of solutions (NUM) in the\\nfull Pareto frontP, the hypervolume (HV) regarding the reference\\npoint(\\u22121,\\u22121), and the inverted generational distance (IGD) to P.For ease of reading, we use \\u201cmetric\\u201d referring to the progress re-\\nsulting from the newly sampled solutions, and \\u201cmeasure\\u201d referring\\nto the algorithms\\u2019 performance.\\nWe provide the definitions of the three metrics below:\\n\\u2022Number of Pareto solutions (NUM): It is, for the current non-\\ndominated set, the number of its solutions that belong to the\\nPareto frontP. NUM(\\ud835\\udc43)=|P\\u2229\\ud835\\udc43|.\\n\\u2022Hypervolume (HV) [ 23]: The hypervolume indicator \\ud835\\udc3c\\ud835\\udc3b(\\ud835\\udc43)\\nis the volume of the object space of the solutions that are\\ndominated by the solution set \\ud835\\udc43. Given a reference point\\n\\ud835\\udc5f\\u2208R\\ud835\\udc5a,\\ud835\\udc3c\\ud835\\udc3b(\\ud835\\udc43)=\\u039b(\\u00d0\\n\\ud835\\udc65\\u2208\\ud835\\udc43[\\ud835\\udc531(\\ud835\\udc65),\\ud835\\udc5f1]\\u00d7...\\u00d7[\\ud835\\udc53\\ud835\\udc5a(\\ud835\\udc65),\\ud835\\udc5f\\ud835\\udc5a]),\\nwhere \\u039b(\\ud835\\udc43)is the Lebesgue measure of \\ud835\\udc43and[\\ud835\\udc531(\\ud835\\udc65),\\ud835\\udc5f1]\\u00d7\\n...\\u00d7[\\ud835\\udc53\\ud835\\udc5a(\\ud835\\udc65),\\ud835\\udc5f\\ud835\\udc5a]is the orthotope with \\ud835\\udc53(\\ud835\\udc65)and\\ud835\\udc5fin opposite\\ncorners.\\n\\u2022Inverted generational distance (IGD) [ 24]: The generational\\ndistance (GD) measures how far the obtained non-dominated\\nset\\ud835\\udc43is from the full Pareto front. It is defined as GD(\\ud835\\udc43)=\\u221a\\ufe03\\u00cd\\ud835\\udc512\\n\\ud835\\udc56\\/|\\ud835\\udc43|, where\\ud835\\udc51\\ud835\\udc56is the Euclidean distance between the\\n\\ud835\\udc56-th solution in \\ud835\\udc43and its nearest solution in P. Following\\nthe suggestion in [ 24], we apply IGD using the obtained\\nnon-dominated set \\ud835\\udc43as the reference set and calculate the\\ndistances from each solution of the Pareto front Pto its\\nnearest neighbor in \\ud835\\udc43. In this way, the dynamic size of \\ud835\\udc43has\\nless influence on the value of IGD.\\nIt is worth mentioning that NUM and IGD are accuracy metrics\\nreferring to the convergence to the full Pareto front P, and HV and\\nIGD are diversity metrics referring to the range of search space that\\nis covered by the obtained non-dominated set \\ud835\\udc43. The value of HV\\nis based on a given reference set instead of the Parent front.\\nWe introduce in the following our first results for the 100di-\\nmensional OneMinMax ,COCZ , and LOTZ , and 50dimensional\\nOneJumpZeroJump . For reproducibility, we provide the codes and\\ndata in GitHub [20].\\n3.2.1 Results for OneMinMax .Table 1 lists the average running\\ntime to find the full Pareto set for the GSEMO variants, which use\\nmulti-objective metrics to guide self-adaptation. Compared to the 3.2.1 Results for OneMinMax .Table 1 lists the average running\\ntime to find the full Pareto set for the GSEMO variants, which use\\nmulti-objective metrics to guide self-adaptation. Compared to the\\nGSEMO using static mutation rate, we observe that the two-rate\\nGSEMO presents better results for OneMinMax . However, the log-\\nnormal and var-ctrl use a significantly large number of function\\nevaluations to converge to the full Pareto front. This performance\\nsurprisingly differs from the performance for OneMax since, as\\nshown in [ 21], the var-ctrl shows competitive results for OneMax .\\nTherefore, we plot in Figure 1 the average function evaluations to\\nfind each Pareto solution. Because all solutions to OneMinMax\\nare located at the Pareto front, this figure can represent the com-\\nplete convergence procedure. We observe that the self-adaptation\\nmethods outperform the static setting in the premature optimiza-\\ntion stage. However, the performance of log-normal and var-ctrl\\ndeteriorates later in the convergence process.\\n3.2.2 Results for COCZ .For another extension of OneMax , we\\nobserve that for COCZ , the two-rate GSEMO can again outperform\\nthe static GSEMO, and the log-normal GSEMO can obtain compa-\\nrable results when using HV as the metric. However, the var-ctrl\\nstrategy seems not work when using the three metrics. Accord-\\ning to Figure 7 (four sub-figures on the left), the static GSEMO What Performance Indicators to Use for Self-Adaptation in Multi-Objective Evolutionary Algorithms GECCO \\u201923, July 15\\u201319, 2023, Lisbon, Portugal\\ntwo-rate log-normal var-ctrl\\nOneMinMax HV 61 269 135 509 464 034\\nstatic =67 442 IGD 62 082 137 870 510 337\\nNUM 64 409 103 103 553 427\\nCOCZ HV 29 535 31 323 45866\\nstatic =31 369 IGD 28 146 65 870 121771\\nNUM 34 843 35 540 62521\\nLOTZ HV 377 091 348 008 905 752\\nstatic =351 671 IGD 368 877 677 082 2 172 211\\nNUM 350 914 1 059 784 2 556 265\\nOneJumpZeroJump HV 335 151 449 930 452 865\\nstatic =360 623 IGD 423 578 368 994 449 893\\nNUM 372 991 522 165 468 275\\nTable 1: Function evaluations to obtain the full Parent front\\nof the 100-dimensional OneMinMax ,COCZ , and LOTZ , and\\n50-dimensional OneJumpZeroJump . The results are average\\nof20runs.\\ud835\\udf06=10. The results of the GSEMO with static \\ud835\\udc5d=\\n1\\/\\ud835\\udc5bare listed in the first column. Bold values indicate the\\nbest result for each problem.\\nFigure 1: Convergence process of the GSEMO variants\\nfor the 100-dimensional OneMinMax . The\\ud835\\udc65-axis \\u201ctarget\\u201d\\nrefers to the objective of \\u201cOneMax\\u201d, i.e., the number of\\none bits.\\ud835\\udc66-axis plots the average function evaluation times\\nto find the corresponding target. The algorithms use \\u2019HV\\u2019,\\n\\u2019IGD\\u2019, and \\u2019NUM\\u2019 metrics.\\ncan traverse an ample search space faster than the three GSEMO\\nvariants.\\n3.2.3 Results for LOTZ .We obtain for LOTZ similar results to\\nthe other two problems. The two-rate GSEMO shows comparable\\nresults to the static GSEMO, and the log-normal GSEMO obtains\\nfavorable results only when using HV as the metric. Figure 2 plots\\nthe average hitting time for each solution located at the Parent\\nfront of LOTZ , and Figure 7 visualizes complete convergence pro-\\ncesses. We observe in Figure 2 that the static and two-rate GSEMOs\\nuse a similar amount of function evaluations to reach the Pareto\\nfront. However, the log-normal and var-ctrl GSEMOs obtain the\\nfirst Pareto solution slower than the other two algorithms, espe-\\ncially when using IGD and NUM. In addition, we observe that the\\nfirst obtained Parent solutions contain similar amounts ( \\u2248\\ud835\\udc5b\\/2) of\\nleading ones and trailing zeros.\\n3.2.4 Results for OneJumpZeroJump .Regarding the function eval-\\nuations to reach the full Pareto front of OneJumpZeroJump , the\\nperformances of the algorithms except for the var-ctrl GSEMO are\\nFigure 2: Convergence process of the GSEMO variants for\\nthe100-dimensional LOTZ . The\\ud835\\udc65-axis \\u201ctarget\\u201d refers to the\\nobjective of \\u201cLeadingOnes\\u201d. \\ud835\\udc66-axis plots the average func-\\ntion evaluation times to find the corresponding target. The\\nalgorithms use HV, IGD, and NUM metrics.\\nFigure 3: Convergence process of the GSEMO variants for\\nthe50-dimensional OneJumpZeroJump . The\\ud835\\udc65-axis \\u201ctarget\\u201d\\nrefers to the objective of \\u201cJump\\u201d function regarding the\\nnumber of one bits. \\ud835\\udc66-axis plots the average function evalua-\\ntion times to find the corresponding target. The algorithms\\nuse HV, IGD, and NUM metrics.\\nsimilar to each other. Due to that GSEMO requires a large mutation\\nrate and takes much time jumping to the Pareto solutions that are\\nlocated at the edge of search space, i.e., (\\ud835\\udc58,\\ud835\\udc5b+\\ud835\\udc58)and(\\ud835\\udc5b+\\ud835\\udc58,\\ud835\\udc58),\\nwe do not expect the methods tested in this paper can learn useful\\ninformation in the last step of searching, though some of them, e.g.,\\nlogNormal and var-ctrl, may obtain good performance by sampling\\nlarge mutation rate in the adaptation process. However, we can still\\nobserve in Figure 3 that the two-rate GSEMO outperforms the oth-\\ners for reaching the Pareto solutions {(\\ud835\\udc4e,2\\ud835\\udc58+\\ud835\\udc5b\\u22121)|\\ud835\\udc4e\\u2208[2\\ud835\\udc58,\\ud835\\udc5b]}\\nbeing located at the smoothy search space.\\nSo far, the performance of the self-adaptive GSEMO variants is un-\\nforeseen based on the performance improvements that self-adaptation\\nachieved in single-objective EAs [ 3,5,14,21]. However, we do not\\nobserve such improvements, except for the two-rate GSEMO, when\\nusing the multi-objective metrics. Therefore, we are investigating\\nthe impact of metrics for self-adaptation in the next section.\\n4 THE BEHAVIOR OF SELF-ADAPTATION using the multi-objective metrics. Therefore, we are investigating\\nthe impact of metrics for self-adaptation in the next section.\\n4 THE BEHAVIOR OF SELF-ADAPTATION\\nTo understand why the GSEMO variants perform differently using\\nthe three multi-objectives, we investigate the algorithms\\u2019 conver-\\ngence process along the search space and the dynamic evolutions\\nof mutation rates.\\n4.1 The Impact of Metrics\\nWe plot in Figure 4 how mutation rates self-adjust along the opti-\\nmization process of one run for OneMinMax andLOTZ . For ease of\\nvisualization, we plot the mutation rates (i.e., \\ud835\\udc5dfor the log-normal, GECCO \\u201923, July 15\\u201319, 2023, Lisbon, Portugal Furong Ye, Frank Neumann, Jacob de Nobel, Aneta Neumann, and Thomas B\\u00e4ck\\nFigure 4: The self-adaptive mutation rates of the GSEMO variants along the optimization process. The results are from the\\nGSEMO variants using six metrics for 100-dimensional OneMinMax (Top) and LOTZ (Bottom). The plotted values are the\\nexact values of one run for each algorithm. For ease of visualization, we plot the values at every five generations and cap the\\nmaximal generation by 10 000 .\\nand\\ud835\\udc5f\\/\\ud835\\udc5bfor the two-rate and log-normal GSEMO) at every five gen-\\nerations. The maximum generation is capped by 10 000 . Due to the\\ndifference in the final function evaluations used to obtain the full\\nPareto front, the lines terminate at different generations at the \\ud835\\udc65-\\naxis. The results of using HV, IGD, and NUM (the three sub-figures\\non the left) indicate considerable fluctuation in the self-adaptation\\nof mutation rates.\\nForOneMax , optimizers search towards increasing the num-\\nber of one bit, and the optimal mutation rates decrease while the\\nfitness value increases along the optimization process [ 1]. When\\napplying self-adaptation for OneMax and other pseudo-Boolean\\noptimization problems, EAs usually compare the parent and off-\\nspring\\u2019s fitness values to obtain useful information even for the\\nnon-improvement situation. However, when we consider the con-\\ntributions to the obtained non-dominated set for self-adaptation,\\nuseful information can be missing for the non-improvement situa-\\ntion. Therefore, we can observe in Figure 4 considerable fluctuations\\nin mutation rates during adaption. However, as presented in the\\nprevious section, the two-rate GSEMO can still outperform the one\\nusing a static mutation rate, indicating that static settings can not\\nbe the optimal solution.\\nA similar story holds for LOTZ . However, we observe different\\nbehavior of GSEMO using HV compared to the other two metrics.\\nFor example, if we describe solving OneMinMax as filling the full\\nPareto front, solving LOTZ needs to search and fill in the Pareto\\nfront. However, before obtaining a Pareto solution, the NUM metric\\ncan not provide useful information for GSEMO. Also, after obtaining\\na solution close to the Pareto front but far away from the other\\nobtained non-dominated solutions, the IGD value may remain the\\nsame even though the set of non-dominated solutions is moving\\ntowards the Pareto front. In contrast, HV can be improved anytime\\nwhen the current non-dominated set is updated, which frequently\\nhappens in the optimization process. Therefore, for LOTZ , Figure 4\\nplots an appropriate adaptation of mutation rate for the GSEMO\\nusing HV.\\nAlso, the mutual interface between multiple objectives can im-\\npact the adaptation. We only plot the function evaluations to find\\nthe solutions in the partial search space (\\ud835\\udc661>40,\\ud835\\udc662>40)ofCOCZ because the algorithms do not necessarily traverse all so-\\nlutions to obtain the Pareto front. When using the static \\ud835\\udc5d=1\\/\\ud835\\udc5b,\\nwe observe that the GSEMO searches towards the Pareto front\\nand skips many non-Pareto solutions. However, the self-adaptation\\nmethods search an ample space to obtain the full Pareto front.\\nMoreover, the log-normal and var-ctrl adjust the mutation rates\\nusing a log-normal and normal distribution, respectively. They both\\ngreedily choose the best one for the next generation. Therefore,\\nthey are easier to adjust towards high values when the algorithms\\nachieve progress by using high mutation rates, or no improvement\\nis obtained in a generation. However, the two-rate carries the half\\nor double strategy, so it learns only trends instead of explicit new\\nvalues. Therefore, the two-rate GSEMO does not use fluctuated\\nhigh mutation rates and results in better performance than log-\\nnormal and var-ctrl, regarding our experiments using the three\\nmulti-objective metrics.\\n4.2 Concerning Single Objective\\nSince we observe that the tested self-adaptation may fail when using normal and var-ctrl, regarding our experiments using the three\\nmulti-objective metrics.\\n4.2 Concerning Single Objective\\nSince we observe that the tested self-adaptation may fail when using\\nthe multi-objective metrics, we propose the GSEMO variants using\\nthe metrics referring to only one objective in this section. More\\nprecisely, we evaluate the \\ud835\\udc52(\\ud835\\udc65)of a solution \\ud835\\udc65with objective values\\n(\\ud835\\udc661,\\ud835\\udc662)by either(\\ud835\\udc661\\u2212\\ud835\\udc66\\u2217\\n1)or(\\ud835\\udc662\\u2212\\ud835\\udc66\\u2217\\n2), where\\ud835\\udc66\\u2217\\n1and\\ud835\\udc66\\u2217\\n2are the\\nbest corresponding objective values of the current non-dominated\\nsolution set. We sample a value \\ud835\\udc5f\\ud835\\udc4e\\ud835\\udc5b\\ud835\\udc51\\u2208[0,1]u.a.r. regularly. When\\n\\ud835\\udc5f\\ud835\\udc4e\\ud835\\udc5b\\ud835\\udc51 <0.5,\\ud835\\udc52(\\ud835\\udc65)=(\\ud835\\udc661\\u2212\\ud835\\udc66\\u2217\\n1); otherwise, \\ud835\\udc52(\\ud835\\udc65)=(\\ud835\\udc662\\u2212\\ud835\\udc66\\u2217\\n2). As\\ndiscussed in Section 3, multi-objective metrics (e.g., IGD and NUM)\\nmay not provide information on the convergence process in time.\\nFurthermore, the mutual interface between two objectives may lead\\nto misunderstanding the current optimization state. Therefore, we\\nexpect more helpful information concerning only one objective\\ncompared to using multi-objective metrics.\\nIn this section, We test three different sampling frequencies of\\n\\ud835\\udc5f\\ud835\\udc4e\\ud835\\udc5b\\ud835\\udc51 , and OneObj%50, OneObj%10, and OneObj denote the metrics\\nof sampling \\ud835\\udc5f\\ud835\\udc4e\\ud835\\udc5b\\ud835\\udc51 at every 50,10, and 1generation, respectively.\\nWe apply the same experimental settings described in Section 3,\\nwhere the metric of calculating \\ud835\\udc52(\\ud835\\udc65)is the only difference. What Performance Indicators to Use for Self-Adaptation in Multi-Objective Evolutionary Algorithms GECCO \\u201923, July 15\\u201319, 2023, Lisbon, Portugal\\nFigure 5: Convergence process of the GSEMO variants\\nfor the 100-dimensional OneMinMax . The\\ud835\\udc65-axis \\u201ctar-\\nget\\u201d refers to the objective of \\u201cOneMax\\u201d, i.e., the number\\nof one bits. \\ud835\\udc66-axis plots the average function evaluation\\ntimes to find the corresponding target. The algorithms use\\nOneObj %50, OneObj %10, and OneObj metrics.\\n4.2.1 Results. Compared to Table 1, we observe in Table 2 that\\nthe log-normal and var-ctrl GSEMO perform better when using\\nthe metric concerning only one objective. According to Figure 4,\\nthe fluctuation in mutation rates reduces significantly. Also, we\\nobserve in Figure 5 that the self-adaptation methods outperform\\nthe static GSEMO in the premature optimization stage, though we\\nstill need to investigate in the future how to adjust the mutation\\nrate to fill in the full Pareto front of OneMinMax faster.\\nHowever, we achieve significant improvement for COCZ and\\nLOTZ by concerning each objective alternatively. According to\\nTable 2, the log-normal and var-ctrl GSEMO outperform the static\\nsetting. In contrast, the performance of the two-rate GSEMO dete-\\nriorates a bit compared to using the multi-objective metrics. After\\nfinding the first Pareto solution, the var-ctrl GSEMO converges\\nto the full Pareto front faster than the other algorithms (Figure 6).\\nMoreover, as shown in Figure 4, the GSEMO variants do not choose\\nthose extremely large mutation rates for LOTZ when using the\\nOneObj metrics. This result indicates that we can avoid the dis-\\nturbance by concerning only one objective while performing self-\\nadaptation.\\nMoreover, the convergence process shows differences for COCZ\\nandLOTZ in Figure 7, comparing using the multi-objective metrics\\nand concerning progress in one objective. For COCZ , The two-\\nrate GSEMO works slower in premature convergence after using\\nOneObj, but the log-normal one moves faster. The latter results\\nin the best performance among the tested algorithms. For LOTZ ,\\nit is evident that all the self-adaptive algorithms converge faster\\n(plotted in darker colors) than the static one in the early stages\\nof optimization, starting from the point (0,0). Moreover, we can\\nobserve that the optimization process differs for each algorithm. We\\nsee orthogonal color lines representing function evaluations along\\ntwo objectives for the static, two-rate, and log-normal GSEMO.\\nHowever, the function evaluations to find each solution distribute\\nmore smoothly over the search space for the var-ctrl GSEMO. Recall\\nthat the var-ctrl using OneObj metrics performs the best among the\\ntested algorithms. This algorithm is designed for interpolating local\\nand global searches. After an identical mutation length has been\\nchosen for a few generations, the variance of the normal distribution\\nused to sample \\u2113will decrease so that the EA can perform a local\\nsearch. In Figure 4, we can observe that the same mutation rates\\nhave indeed been used for successive generations, which results in\\npromising performance.two-rate log-normal var-ctrl\\nOneMinMax OneObj 100 524 79 751 135 250\\nstatic =67 443 OneObj%10 103 315 74 726 132 700\\nOneObj%50 104 009 87 285 119 209\\nCOCZ OneObj 39 217 29 413 51 515\\nstatic =31 369 OneObj%10 37 860 26 585 53 530\\nOneObj%50 41 560 33 083 45 776\\nLOTZ OneObj 414 198 316 529 255 346\\nstatic =351 671 OneObj%10 394 350 325 271 272 238\\nOneObj%50 428 544 310 864 261 280\\nOneJumpZeroJump OneObj 456 250 329 995 386 858\\nstatic =360 623 OneObj%10 254 748 518 555 331 763\\nOneObj%50 375 875 324 141 453 305\\nTable 2: Function evaluations that the GSEMO variants us-\\ning OneObj metrics use to obtain the full Parent front of\\nthe100-dimensional OneMinMax ,COCZ , and LOTZ , and 50-\\ndimensional OneJumpZeroJump . The results are average of\\n20runs.\\ud835\\udf06=10. The results of the GSEMO with static \\ud835\\udc5d=1\\/\\ud835\\udc5b\\nare listed in the first column. Bold values indicate the best\\nresult for each problem. dimensional OneJumpZeroJump . The results are average of\\n20runs.\\ud835\\udf06=10. The results of the GSEMO with static \\ud835\\udc5d=1\\/\\ud835\\udc5b\\nare listed in the first column. Bold values indicate the best\\nresult for each problem.\\nFigure 6: Convergence process of the GSEMO variants for\\nthe100-dimensional LOTZ . The\\ud835\\udc65-axis \\u201ctarget\\u201d refers to the\\nobjective of \\u201cLeadingOnes\\u201d. \\ud835\\udc66-axis plots the average func-\\ntion evaluation times to find the corresponding target. The\\nalgorithms use OneObj %50, OneObj %10, and OneObj met-\\nrics.\\ud835\\udf06=10\\nForOneJumpZeroJump , the two-rate GSEMO using OneObj%10\\nis the best-performed in Table 2. Moreover, our results show that\\nthe log-normal GSEMO outperforms the others in the search space\\nof{(\\ud835\\udc4e,2\\ud835\\udc58+\\ud835\\udc5b\\u22121) |\\ud835\\udc4e\\u2208 [2\\ud835\\udc58,\\ud835\\udc5b]}. Regarding the result for the\\ntargets(\\ud835\\udc58,\\ud835\\udc5b+\\ud835\\udc58)and(\\ud835\\udc5b+\\ud835\\udc58,\\ud835\\udc58), the two-rate GSEMO using OneObj\\npresents the best average hitting time, followed by the log-normal\\none using OneObj%50 and var-ctrl one using OneObj%10. However,\\nthe results for those solutions on the edges of the search space also\\nshow considerable variances.\\n4.3 The Impact of Population Size\\nAt last, we would like a brief insight into the impact of the pop-\\nulation size, which refers to the offspring size here, for GSEMO\\nvariants. The impact from the population may exist for EAs or not,\\ndepending on the expected function evaluations required to achieve\\nprogress [ 9,11]. Research regarding adjusting population size for\\nMOEAs has been discussed recently in [ 6,19]. The population size GECCO \\u201923, July 15\\u201319, 2023, Lisbon, Portugal Furong Ye, Frank Neumann, Jacob de Nobel, Aneta Neumann, and Thomas B\\u00e4ck\\nFigure 7: Average function evaluations to find each solution in the search space of 100-dimensional COCZ (Top) and LOTZ\\n(Bottom). For COCZ ,\\ud835\\udc661indicates the objective of \\u201cOneMax\\u201d, and \\ud835\\udc662indicates the objective of \\u201cCount Ones Count Zeroes\\u201d for\\neach half bit-string. For LOTZ ,\\ud835\\udc661indicates the objective of \\u201cLeadingOnes\\u201d, and \\ud835\\udc662indicates the objective of \\u201cTrailing Zeros\\u201d.\\nValues are in log 10 scale.\\ud835\\udf06=10\\nFigure 8: The average function evaluations to find the full\\nPareto front of 100-dimensional OneMinMax (Top, normal-\\nized by\\ud835\\udc5b2log\\ud835\\udc5b),LOTZ (Middle, by \\ud835\\udc5b3), and COCZ (Bottom,\\nby\\ud835\\udc5b2log\\ud835\\udc5b), for the GSEMO variants. \\ud835\\udc65-axis plots population\\nsize\\ud835\\udf06. The GSEMO variants use the OneObj metric.\\nand mutation rate can be complementary to speed up EAs since they\\nboth shall work with the probability of achieving improvement, i.e.,\\nsuccess rate.\\nTo investigate the impact of population size on our GSEMO vari-\\nants, we plot in Figure 8 how their performance changes across\\ndifferent population size \\ud835\\udf06. Note that we plot the GSEMO vari-\\nants using the OneObj metric due to the promising performances.\\nThe static and log-normal GSEMO show stable performance for\\nthe3plotted problems across all 5tested dimensions. For LOTZ ,\\nthe two-rate and var-ctrl GSEMO\\u2019s performance does not change\\nmuch either. This behavior is consistent with the results of EAs\\nforLeadingOnes [9]. The var-ctrl GSEMO outperforms the others\\nfor all tested dimensions for LOTZ , and the static and log-normal\\nshow competitive results for OneMinMax andCOCZ . However, as\\nthe dimension increases, the var-ctrl shows potential performance\\nimprovements for OneMinMax andCOCZ .5 CONCLUSION\\nSelf-adaptation of mutation has succeeded in accelerating the con-\\nvergence speed in single-objective pseudo-Boolean optimization.\\nMoreover, benchmarking work and related collaboration in theoret-\\nical studies helped us understand the behavior of EAs in those prob-\\nlems. In this paper, we conduct detailed experimental investigations\\nof self-adaptation in multi-objective optimization by transmitting\\nthe existing self-adaptive mutation in single-objective optimization.\\nWe test three variants of GSEMO for OneMinMax ,LOTZ , and\\nOneJumpZeroJump . To transmit the single-objective techniques\\nto multi-objective problems, we consider four metrics to evaluate\\nthe progress during the optimization process and use these metrics\\nto guide self-adaptation. The experimental results show the poten-\\ntial benefits of self-adaptive mutation for GSEMO. Moreover, the\\nchoice of metrics significantly impacts the algorithms\\u2019 performance.\\nDue to the mutual interface between the progress referring to the\\ntwo objectives and the prolonged absence of progress updates, the\\nself-adaptive MOEAs may be misled and result in deteriorated per-\\nformance. However, by concerning the progress referring to a single\\nobjective as the metric, we obtain performance improvement for\\nGSEMO.\\nWe conclude that hypervolume can be a helpful metric guiding\\nthe self-adaptation, compared to the inverted generational distance\\nand the number of obtained Pareto solutions. In addition, concern-\\ning only one objective can accelerate MOEAs to move towards the\\nPareto front and lead to faster convergence.\\nMoreover, we study the impact of the population size on our\\nGSEMO variants. The self-adaptive algorithms can benefit from\\nlarge population sizes for OneMinMax andCOCZ , which suggests\\nfuture work researching self-adaptive techniques manipulating the\\ntwo parameters, population size and mutation rates, for MOEAs.\\nOverall, we hope the extensive experimental results in this paper\\nmotivate insights into the theoretical understanding of MOEAs,\\ne.g., dynamic optimal mutation rates and population sizes, and\\nalgorithm design for practical problems, e.g., real-time systems. What Performance Indicators to Use for Self-Adaptation in Multi-Objective Evolutionary Algorithms GECCO \\u201923, July 15\\u201319, 2023, Lisbon, Portugal\\nREFERENCES\\n[1]Buzdalov, M., and Doerr, C. Optimal mutation rates for the (1+ \\ud835\\udf06) EA on\\nonemax. In Proc. of Parallel Problem Solving from Nature (PPSN\\u201920) , Proceedings,\\nPart II (2020), T. B\\u00e4ck, M. Preuss, A. H. Deutz, H. Wang, C. Doerr, M. T. M.\\nEmmerich, and H. Trautmann, Eds., vol. 12270 of Lecture Notes in Computer\\nScience , Springer, pp. 574\\u2013587.\\n[2]Dang, N., and Doerr, C. Hyper-parameter tuning for the (1 + ( \\ud835\\udf06,\\ud835\\udf06)) GA. In\\nProc. Genetic and Evolutionary Computation Conference (GECCO\\u201919) (2019), ACM,\\npp. 889\\u2013897.\\n[3]Doerr, B., and Doerr, C. Optimal static and self-adjusting parameter choices\\nfor the (1+(\\ud835\\udf06,\\ud835\\udf06)) genetic algorithm. Algorithmica 80 , 5 (2018), 1658\\u20131709.\\n[4]Doerr, B., Gao, W., and Neumann, F. Runtime analysis of evolutionary diversity\\nmaximization for oneminmax. In Proc. of Genetic and Evolutionary Computation\\nConference (GECCO 16\\u2019) (2016), ACM, pp. 557\\u2013564.\\n[5]Doerr, B., Giessen, C., Witt, C., and Yang, J. The (1+\\ud835\\udf06) evolutionary algorithm\\nwith self-adjusting mutation rate. Algorithmica 81 , 2 (2019), 593\\u2013631.\\n[6]Doerr, B., Hadri, O. E., and Pinard, A. The (1 + (\\ud835\\udf06,\\ud835\\udf06)) global SEMO algorithm.\\nInProc. of Genetic and Evolutionary Computation Conference (GECCO\\u201922) (2022),\\nJ. E. Fieldsend and M. Wagner, Eds., ACM, pp. 520\\u2013528.\\n[7]Doerr, B., and Zheng, W. Theoretical analyses of multi-objective evolutionary\\nalgorithms on multi-modal objectives. In Proc. of AAAI Conference on Artificial\\nIntelligence (AAAI\\u201921) (2021), AAAI Press, pp. 12293\\u201312301.\\n[8]Doerr, C., Ye, F., Horesh, N., Wang, H., Shir, O. M., and B\\u00e4ck, T. Benchmarking\\ndiscrete optimization heuristics with iohprofiler. Applied Soft Computing 88 (2020),\\n106027.\\n[9]Doerr, C., Ye, F., van Rijn, S., Wang, H., and B\\u00e4ck, T. Towards a theory-guided\\nbenchmarking suite for discrete black-box optimization heuristics: profiling (1 +\\n\\ud835\\udf06) EA variants on onemax and leadingones. In Proc. of Genetic and Evolutionary\\nComputation Conference (GECCO 18\\u2019) (2018), H. E. Aguirre and K. Takadama,\\nEds., ACM, pp. 951\\u2013958.\\n[10] Eiben, A. E., Hinterding, R., and Michalewicz, Z. Parameter control in evolu-\\ntionary algorithms. IEEE Transactions on Evolutionary Computation 3 , 2 (1999),\\n124\\u2013141.\\n[11] Fajardo, M. A. H., and Sudholt, D. Self-adjusting population sizes for non-\\nelitist evolutionary algorithms: why success rates matter. In Proc. of Genetic\\nand Evolutionary Computation Conference (GECOO\\u201921) (2021), F. Chicano and\\nK. Krawiec, Eds., ACM, pp. 1151\\u20131159.[12] Giel, O. Expected runtimes of a simple multi-objective evolutionary algorithm.\\nInProc. of the IEEE Congress on Evolutionary Computation, (CEC\\u201903) (2003), IEEE,\\npp. 1918\\u20131925.\\n[13] Giel, O., and Lehre, P. K. On the effect of populations in evolutionary multi-\\nobjective optimisation. Evolutionary Computation 18 , 3 (2010), 335\\u2013356.\\n[14] Kruisselbrink, J. W., Li, R., Reehuis, E., Eggermont, J., and B\\u00e4ck, T. On\\nthe log-normal self-adaptation of the mutation rate in binary search spaces. In\\nProc. of Genetic and Evolutionary Computation Conference (GECCO 11\\u2019) (2011),\\nN. Krasnogor and P. L. Lanzi, Eds., ACM, pp. 893\\u2013900.\\n[15] Laumanns, M., Thiele, L., and Zitzler, E. Running time analysis of multiobjec-\\ntive evolutionary algorithms on pseudo-boolean functions. IEEE Transactions on\\nEvolutionary Computtation 8 , 2 (2004), 170\\u2013182.\\n[16] Nguyen, A. Q., Sutton, A. M., and Neumann, F. Population size matters:\\nRigorous runtime results for maximizing the hypervolume indicator. Theoretical\\nComputer Science 561 (2015), 24\\u201336.\\n[17] Osuna, E. C., Gao, W., Neumann, F., and Sudholt, D. Design and analysis of\\ndiversity-based parent selection schemes for speeding up evolutionary multi-\\nobjective optimisation. Theoretical Computer Science 832 (2020), 123\\u2013142.\\n[18] Pinto, E. C., and Doerr, C. Towards a more practice-aware runtime analysis of\\nevolutionary algorithms. CoRR abs\\/1812.00493 (2018). [18] Pinto, E. C., and Doerr, C. Towards a more practice-aware runtime analysis of\\nevolutionary algorithms. CoRR abs\\/1812.00493 (2018).\\n[19] Shi, F., Schirneck, M., Friedrich, T., K\\u00f6tzing, T., and Neumann, F. Reoptimiza-\\ntion time analysis of evolutionary algorithms on linear functions under dynamic\\nuniform constraints. Algorithmica 81 , 2 (2019), 828\\u2013857.\\n[20] Ye, F., and de Nobel, J. GSEMO. https:\\/\\/github.com\\/FurongYe\\/GSEMO, 2022.\\n[21] Ye, F., Doerr, C., and B\\u00e4ck, T. Interpolating local and global search by controlling\\nthe variance of standard bit mutation. In Proc. of IEEE Congress on Evolutionary\\nComputation (CEC\\u201919) (2019), IEEE, pp. 2292\\u20132299.\\n[22] Zheng, W., Liu, Y., and Doerr, B. A first mathematical runtime analysis of\\nthe non-dominated sorting genetic algorithm II (NSGA-II). In Proc. of AAAI\\nConference on Artificial Intelligence (AAAI,22) (2022), AAAI Press, pp. 10408\\u2013\\n10416.\\n[23] Zitzler, E., and Thiele, L. Multiobjective evolutionary algorithms: a com-\\nparative case study and the strength pareto approach. IEEE Transactions on\\nEvolutionary Computation 3 , 4 (1999), 257\\u2013271.\\n[24] Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C. M., and Da Fonseca, V. G.\\nPerformance assessment of multiobjective optimizers: An analysis and review.\\nIEEE Transactions on Evolutionary Computation 7 , 2 (2003), 117\\u2013132.\",\"4\":\" Using Affine Combinations of BBOB Problems for Performance\\nAssessment\\nDiederick Vermetten\\nLeiden Institute for Advanced\\nComputer Science\\nLeiden, The Netherlands\\nd.l.vermetten@liacs.leidenuniv.nlFurong Ye\\nLeiden Institute for Advanced\\nComputer Science\\nLeiden, The Netherlands\\nf.ye@liacs.leidenuniv.nlCarola Doerr\\nSorbonne Universit\\u00e9, CNRS, LIP6\\nParis, France\\nCarola.Doerr@lip6.fr\\nABSTRACT\\nBenchmarking plays a major role in the development and analysis\\nof optimization algorithms. As such, the way in which the used\\nbenchmark problems are defined significantly affects the insights\\nthat can be gained from any given benchmark study. One way\\nto easily extend the range of available benchmark functions is\\nthrough affine combinations between pairs of functions. From the\\nperspective of landscape analysis, these function combinations\\nsmoothly transition between the two base functions.\\nIn this work, we show how these affine function combinations\\ncan be used to analyze the behavior of optimization algorithms. In\\nparticular, we highlight that by varying the weighting between the\\ncombined problems, we can gain insights into the effects of added\\nglobal structure on the performance of optimization algorithms. By\\nanalyzing performance trajectories on more function combinations,\\nwe also show that aspects such as the scaling of objective functions\\nand placement of the optimum can greatly impact how these results\\nare interpreted.\\nKEYWORDS\\nBlack-box Optimization, Benchmarking, Performance Analysis\\n1 INTRODUCTION\\nBenchmarking is a key aspect in the development of optimization\\nalgorithms. Not only are benchmark problems used to compare the\\neffectiveness of different optimizers with regard to a standardized\\nset of problems, the analysis of algorithm behavior on these prob-\\nlems is often used to gain insight into the characteristics of the\\nalgorithm. Because of this, the design of benchmark problems has\\na major impact on the field of optimization as a whole [1].\\nOne of the most common benchmark suites in single-objective,\\ncontinuous, noiseless optimization is fittingly called Black Box Op-\\ntimization Benchmark (BBOB) [ 7]. This suite is part of the COCO\\nframework [ 6], which has seen significant adoption in the last\\ndecade. This suite consists of 24 problems, each defined to repre-\\nsent a set of global landscape properties. For each of these problems,\\nmany different instances can be created through a set of transfor-\\nmations, allowing researchers to test different invariances of their\\nalgorithm. Because of its popularity, studies into the specifics of\\nthe BBOB suite are numerous [13, 16, 17].\\nOne particularly popular method to investigate continuous opti-\\nmization problems is Exploratory Landscape Analysis (ELA) [ 15].\\nThis technique aims to characterize the low-level landscape proper-\\nties through a large set of features. Applying this to the BBOB suite\\nshows that instances of the 24 functions generally group together,\\nwith separation between functions being relatively robust [ 20].This observation raised the question of how the spaces between\\nproblems could be explored.\\nIn a recent study, affine combinations between pairs of BBOB\\nproblems were proposed and analyzed using ELA [ 4]. The resulting\\nanalysis shows that varying the weight of these combinations has a\\nrelatively smooth impact on the landscape features. As such, these\\nnew functions could potentially be used to study the transition\\nbetween different landscapes, which opens up a more in-depth\\nanalysis of the relation between landscapes and algorithm behavior.\\nTo investigate to what extent the affine function combinations\\ncan be used to study algorithmic behavior, we perform a bench-\\nmarking study through which we investigate the effect of the affine\\ncombinations on the performance of five numerical black-box opti-\\nmization algorithms. We make use of function combinations which\\ninclude a sphere model to show the impact of added global structure combinations on the performance of five numerical black-box opti-\\nmization algorithms. We make use of function combinations which\\ninclude a sphere model to show the impact of added global structure\\non the relative ranking between algorithms. Additionally, we show\\nthat by combining functions with different global properties we\\ndon\\u2019t always obtain smooth transitions in performance. We pro-\\nvide examples where the combination of two functions can either\\nbe significantly more challenging or slightly easier than the base\\nfunctions it consists of.\\n2 RELATED WORK\\n2.1 BBOB Problem Suite\\nWithin continuous optimization benchmarking, one of the most\\npopular suites of benchmarks is the BBOB family, which has been\\ndesigned as part of the COCO framework. The noiseless, single-\\nobjective suite consists of 24 problems, each of which can be in-\\nstantiated with a set of different transformations. These function\\ninstances aim to preserve the global function properties while vary-\\ning factors such as the location of the global optimum, such that an\\noptimizer can not directly exploit these aspects. However, the exact\\ninfluence these transformations have on the low-level landscape\\nproperties is not as straightforward, which can lead to noticeable\\ndifferences in algorithm behavior on different instances of the same\\nfunction [13].\\n2.2 Affine Function Combinations\\nWhile using function instances allows the BBOB suite to cover a\\nwider range of problem landscapes than the raw functions alone,\\nthere are limits to the types of landscapes which can be created in\\nthis way. Recently, it has been proposed to use affine combinations\\nbetween pairs of BBOB functions to generate new benchmark func-\\ntions [ 4]. These combinations have been shown to smoothly fill the\\nspace of low-level landscape properties, as measured through a set\\nof ELA features. These results have shown that even a relatively\\n1arXiv:2303.04573v1  [cs.NE]  8 Mar 2023 Diederick Vermetten, Furong Ye, and Carola Doerr\\nsimple function creation procedure has the potential to give us new\\ninsights into the way function landscapes work.\\n3 EXPERIMENTAL SETUP\\nIn this work, we make use of a slightly modified version of the\\naffine function combinations from [ 4]. In particular, we define the\\ncombination between two functions from the BBOB suite as follows:\\n\\ud835\\udc36(\\ud835\\udc391,\\ud835\\udc3c1,\\ud835\\udc392,\\ud835\\udc3c2,\\ud835\\udefc)(\\ud835\\udc65)=\\nexp\\u0010\\n\\ud835\\udefclog\\u0000\\ud835\\udc391(\\ud835\\udc65)\\u2212\\ud835\\udc391(\\ud835\\udc421)\\u0001+\\n(1\\u2212\\ud835\\udefc)log\\u0000\\ud835\\udc392(\\ud835\\udc65\\u2212\\ud835\\udc421+\\ud835\\udc422)\\u2212\\ud835\\udc392(\\ud835\\udc422)\\u0001\\u0011\\nWhere\\ud835\\udc391,\\ud835\\udc3c1,\\ud835\\udc392,\\ud835\\udc3c2are the two base functions and their instance\\nnumber, as defined in BBOB [7]. \\ud835\\udc421and\\ud835\\udc422represent the location\\nof the optimum of functions \\ud835\\udc391and\\ud835\\udc392respectively. The transforma-\\ntion to\\ud835\\udc65when evaluating \\ud835\\udc392is performed to make sure the location\\nof the optimum is at \\ud835\\udc421. As opposed to the original definition, we\\nsubtract the optimal values before aggregating, so we can take a\\nlogarithmic mean between the problems. This way, we can use\\nconsistent values for \\ud835\\udefcacross problems, without having to perform\\nthe entropy-based selection performed in [ 4]. It has the additional\\nbenefit of ensuring the objective value of the optimal solution is\\nalways 0, so the comparison of performance across instances and\\nacross problems is simplified. In Figure 1, we illustrate the change\\nin landscape for the combination of F21 and F1, for different values\\nof\\ud835\\udefc.\\nIn order to implement these function combinations, we make\\nuse of the IOHexperimenter [ 3] framework. We access the BBOB\\nproblems, combine them together as described, and wrap them into\\na new problem. This enables us to use any of the built-in logging\\nand tracking options of IOHexperimter. In particular, it allows us to\\nstore the performance data into a file-format which can be directly\\nprocessed into IOHanalyzer [25] for post-processing.\\nFor our algorithm portfolio, we make use of the Nevergrad tool-\\nbox, which provides a common interface to a wide range of opti-\\nmization algorithms [ 19]. In this study, we benchmark the following\\nalgorithms:\\n\\u2022Particle Swarm Optimization (PSO) [10]\\n\\u2022Constrained Optimization BY Linear Approximation\\n(Cobyla) [18]\\n\\u2022Differential Evolution (DE) [21]\\n\\u2022Estimation of Multivariate Normal Algorithm (EMNA) [ 12]\\n\\u2022Diagonal Covariance Matrix Adaptation Evolution Strategy\\n(dCMA-ES) [8]\\nFor each of these algorithms, we make use of the default parameters\\nas chosen in Nevergrad. Each run of the algorithm has a budget\\nof2 000\\ud835\\udc37, where\\ud835\\udc37is the dimension of the problem. We perform\\n5independent runs per instance. In the remainder of this paper,\\nwe set\\ud835\\udc3c2=1. As such, when discussing the instance of an affine\\nfunction combination \\ud835\\udc36(\\ud835\\udc391,\\ud835\\udc3c1,\\ud835\\udc392,\\ud835\\udc3c2,\\ud835\\udefc), we are referring to \\ud835\\udc3c1.\\nReproducibility To ensure reproducibility, we make all code\\nused in the creation of this paper available in a Zenodo reposi-\\ntory [ 24]. This repository contains the data generation code, raw\\ndata generated, and post-processing scripts used to create the re-\\nsults discussed in the following sections, following the recommen-\\ndations proposed in [ 14]. In addition to this, we also make availablea Figshare repository containing additional figures and animations\\nwhich could not be included in this paper [24].\\n4 PERFORMANCE COMPARISON FOR\\nAFFINE COMBINATIONS WITH F1\\nFor a first set of experiments, we make use of affine combinations\\nwhere we combine each function with F1: the sphere model. As can\\nbe seen in Figure 1, adding a sphere model to another function cre-\\nates an additional global structure that can guide the optimization\\ntoward the global optimum. As such, these kinds of combinations\\nmight allow us to investigate the influence of an added global struc-\\nture on the performance of optimization algorithms. While to some\\nextent this can already be investigated by comparing results on the\\nfunction groups of the original BBOB with different levels of global\\nstructure, the affine function combinations allow for a much more\\nfine-grained investigation. Since the landscape features of these\\ncombined functions seem to shift smoothly when varying \\ud835\\udefc, we structure, the affine function combinations allow for a much more\\nfine-grained investigation. Since the landscape features of these\\ncombined functions seem to shift smoothly when varying \\ud835\\udefc, we\\nmight assume similar behavior on algorithmic performance.\\nIn Figure 2, we show the performance of diagonal CMA-ES,\\nmeasured as the area under the Empirical Cumulative Distribution\\nFunction (ECDF) [ 5], for varying function combinations and \\ud835\\udefc\\nvalues. As is widely accepted for BBOB functions, we make use of 51\\ntargets logarithmically spaced between 102and10\\u22128to compute the\\nECDF. The resulting Area Under the Curve (AUC) is normalized, so\\nan algorithm which reaches all targets in the first evaluation would\\nhave an AUC of 1. The top of this figure, with \\ud835\\udefc=0, shows the\\nperformance on the sphere function, on which CMA-ES performs\\nvery well. There are however differences between the columns,\\nsince the location of the affine function combination is set to the\\noptimum of the second function.\\nIn Figure 2, we can see that the performance of CMA-ES does\\nindeed seem to move smoothly between the sphere and the function\\nwith which it is combined. It is however interesting to note the\\ndifferences in speed at which this transition occurs. While the final\\nperformance on e.g. functions 3 and 11 seems similar, the transition\\nspeed differs significantly. This seems to indicate that for F11, the\\naddition of some global structure has a relatively weak influence on\\nthe challenges of this landscape from the perspective of the CMA-ES,\\nwhile even small amounts of global structure significantly simplify\\nthe landscape of F3.\\nWe can perform a similar analysis on other optimization algo-\\nrithms. In Figure 3 and Figure 4, we show the same heatmap as\\nFigure 2, but for Differential Evolution and Cobyla respectively. It\\nis clear from these heatmaps that the performance of DE is more\\nvariable than that of CMA-ES, while Cobyla\\u2019s performance drops\\noff much more quickly. The overall trendlines for DE do seem to be\\nsomewhat similar to those seen for diagonal CMA-ES: the transition\\npoints between high and low AUC in Figure 3 are comparable to\\nthose seen in Figure 2. There are however still some differences in\\nbehavior, espcially relative to Cobyla. These differences then lead\\nto the question of whether there exist transition points in ranking\\nbetween algorithms as well. Specifically, if one algorithm performs\\nwell for\\ud835\\udefc=0but gets overtaken as \\ud835\\udefc\\u21921, exploring this change\\nin ranking would give further insight into the relative strengths\\nand weaknesses of the considered algorithms.\\n2 Using Affine Combinations of BBOB Problems for Performance Assessment\\n\\u22124\\u22122 024\\n0\\u22124\\u22122024\\n\\u22124\\u22122 024\\n0.25\\u22124\\u22122 024\\n0.5\\u22124\\u22122 024\\n0.75\\u22124\\u22122 024\\n1\\u22129.6\\u22128.0\\u22126.4\\u22124.8\\u22123.2\\u22121.60.01.63.2\\nFigure 1: Evolution of the landscape (log-scaled function-values) of the affine combination between F21 ( \\ud835\\udefc=1) and F1 (\\ud835\\udefc=0),\\ninstance 1 for both functions, for varying \\ud835\\udefc. The red circle highlights the location of the global optimum.\\n23456789101112131415161718192021222324\\nFunction ID0.00\\n0.05\\n0.10\\n0.15\\n0.20\\n0.25\\n0.30\\n0.35\\n0.40\\n0.45\\n0.50\\n0.55\\n0.60\\n0.65\\n0.70\\n0.75\\n0.80\\n0.85\\n0.90\\n0.95\\n1.00Alpha\\n0.00.20.40.60.81.0\\nFigure 2: Normalized area under the ECDF curve of Diago-\\nnal CMA-ES for each combination of the BBOB-function (x-\\naxis) with a sphere model, for given value of \\ud835\\udefc(y-axis). AUC\\nis calculated after 10 000 function evaluations, based on 50\\nruns on 10 instances.\\n23456789101112131415161718192021222324\\nFunction ID0.00\\n0.05\\n0.10\\n0.15\\n0.20\\n0.25\\n0.30\\n0.35\\n0.40\\n0.45\\n0.50\\n0.55\\n0.60\\n0.65\\n0.70\\n0.75\\n0.80\\n0.85\\n0.90\\n0.95\\n1.00Alpha\\n0.00.20.40.60.81.0\\nFigure 3: Normalized area under the ECDF curve of Differen-\\ntial Evolution for each combination of the BBOB-function\\n(x-axis) with a sphere model, for given value of \\ud835\\udefc(y-axis).\\nAUC is calculated after 10 000 function evaluations, based on\\n50 runs on 10 instances.\\n23456789101112131415161718192021222324\\nFunction ID0.00\\n0.05\\n0.10\\n0.15\\n0.20\\n0.25\\n0.30\\n0.35\\n0.40\\n0.45\\n0.50\\n0.55\\n0.60\\n0.65\\n0.70\\n0.75\\n0.80\\n0.85\\n0.90\\n0.95\\n1.00Alpha\\n0.00.20.40.60.81.0Figure 4: Normalized area under the ECDF curve of Cobyla\\nfor each combination of the BBOB-function (x-axis) with a\\nsphere model, for given value of \\ud835\\udefc(y-axis). AUC is calculated\\nafter 10 000 function evaluations, based on 50 runs on 10 in-\\nstances.\\nIn order to answer this question about the relative ranking of\\nalgorithms, we make use of the portfolio of 5 algorithms and rank\\nthem based on AUC on each affine function combination. We then\\nvisualize the top ranking algorithm on each setting in Figure 5.\\nImportant to note is that both PSO and EMNA never ranked first\\nfor the selected budget, and are thus not visible on the figure.\\nFrom Figure 5, we can clearly see that Cobyla deals well with\\nthe sphere model, managing to outperform the other algorithm\\nwhen the weighting of the sphere is relatively high. Then, after a\\ncertain threshold, the CMA-ES consistently outperforms the rest\\nof the portfolio. However, as \\ud835\\udefcincreases further, and the influence\\nof the sphere model diminishes, an interesting pattern seems to\\noccur. For several problems, there is a second transition point, to\\neither DE or Cobyla. For some functions, e.g. F3 and F4, one factor\\nwhich might explain this phenomenon is the strength of the local\\noptima increasing, making it harder for CMA-ES to explore the full\\nlandscape, while the uniform initialization of DE causes it to be\\nslightly less impacted.\\nIn order to better understand what the transitions in algorithm\\nranking look like, we can zoom in on one of the functions and\\n3 Diederick Vermetten, Furong Ye, and Carola Doerr\\n23456789101112131415161718192021222324\\nFunction ID0.00\\n0.05\\n0.10\\n0.15\\n0.20\\n0.25\\n0.30\\n0.35\\n0.40\\n0.45\\n0.50\\n0.55\\n0.60\\n0.65\\n0.70\\n0.75\\n0.80\\n0.85\\n0.90\\n0.95\\n1.00Alpha\\nDiagonalCMA DE RCobyla\\nFigure 5: Algorithm with the highest area under the ECDF-\\ncurve for each combination of the BBOB-function (x-axis)\\nwith a sphere model, for given value of \\ud835\\udefc(y-axis). AUC is\\ncalculated after 10 000 function evaluations, based on 50 runs\\non 10 instances. PSO and EMNA are not shown since they\\nnever ranked first.\\nplot the expected running time (ERT) for several values of \\ud835\\udefc. This\\nis done in Figure 6, where we look at the combination between\\nF10 and the sphere model. we clearly see that Cobyla is very ef-\\nfective at optimizing the sphere model, solving it almost an order\\nof magnitude faster than the second ranked algorithm, which is\\nDiagonalCMA. However, when \\ud835\\udefcincreases, Cobyla quickly starts\\nto fail, while DiagonalCMA still manages to solve most instances\\nat\\ud835\\udefc=0.25within similar amounts of evaluations. However, it is\\nclear from the bifurcation in the plot that on some instances, the\\nDiagonalCMA is no longer able to find the optimum within the\\nallocated budget. When \\ud835\\udefcincreases further, none of the instances\\nare able to be solved anymore by any of the three algorithms. When\\n\\ud835\\udefc\\u22650.75, we see that DE overtakes the other two, which explains\\nthe better ranking seen in Figure 5.\\n5 COMBINATIONS BETWEEN DIFFERENT\\nFUNCTION GROUPS\\nWhile combining functions with a sphere model can be viewed\\nas adding global structure to a problem, combinations between\\nother functions can provide interesting insights into the transition\\npoints between different types of problems. To illustrate the kinds\\nof insights that can be gained from these combinations, we select\\na subset of 5 functions and collect performance data on each com-\\nbination with the same 21 \\ud835\\udefcvalues (with both orderings of the\\nfunction). We show the performance in terms of normalized AUC\\nof diagonal CMA-ES on these function combinations in Figure 7.\\nNote that for \\ud835\\udefc=1, we are using the function specified in the\\ncolumn label, while for \\ud835\\udefc=0we have the function specified in the\\nrow label, but with the optimum of the column function.\\nFrom Figure 7, we can see that the transition of performance\\nbetween the two extreme \\ud835\\udefcvalues is mostly smooth. While there\\nare some rather quick changes, e.g. for the transition between\\nF2 and F11, these seem to be the exception rather than the rule.\\nParticularly interesting are the settings where the performance of\\naffine combinations between two functions proves to be much easieror harder than the functions which are being combined. This is the\\ncase e.g. for the combinations of F21 and F9. Of note in this function\\ncombination is the fact that its mirrored combination around the\\ndiagonal does not display similar behavior. In fact, Figure 7 in\\ngeneral is not fully symmetric around the diagonal.\\nWe might expect(\\ud835\\udc391,\\ud835\\udc392,\\ud835\\udefc)to be similar to(\\ud835\\udc392,\\ud835\\udc391,1\\u2212\\ud835\\udefc). How-\\never, the combination between F9 and F21 shows that this is not al-\\nways the case. Specifically, the AUC for the combination (\\ud835\\udc3921,\\ud835\\udc399,1)\\nis significantly worse than that of (\\ud835\\udc399,\\ud835\\udc3921,0), even though F21 does\\nnot contribute directly to the function value of the affine combi-\\nnation. The only way in which these two problems differ is in the\\nlocation of the optima. For F21, the default location of the optimum\\nis hard-coded to be at distance 1 from the optimum [ 13], which\\nis not the case for F9. Since the CMA-ES initializes its center of\\nmass in the origin of the space and uses a default initial stepsize\\nof0.3[19], it is able to find the optimum in the default setting,\\nwhile the translated version of the function becomes much more\\nchallenging. This highlights a potential issue with the traditional\\nanalysis of performance on BBOB problems: if we don\\u2019t take into\\naccount the built-in limitations on e.g. the location of the optimum\\nin our analysis, there is a risk of misinterpreting the results of a account the built-in limitations on e.g. the location of the optimum\\nin our analysis, there is a risk of misinterpreting the results of a\\nstructurally biased algorithm [ 23] and viewing it as optimal on this\\ntype of multimodal problem, while it is unable to solve a translated\\nversion of the same function.\\nTo see how much this initialization really impacts the differences\\nin performance, we perform an additional experiment with a dif-\\nferent version of CMA-ES. We opt to use the modular CMA-ES [ 2]\\nand set the initial stepsize to 0.2times the range of the domain, so\\n2in our case. The resulting performance is visualized in Figure 8.\\nIn this figure, it is clear that the overall performance of this set-\\nting of CMA-ES performs better overall, but of particular note is\\nthat the asymmetries have been somewhat reduced, although not\\ndisappeared entirely.\\nAdditionally, Figure 8 shows several interesting trends in per-\\nformance which were not present for the Diagonal CMA-ES. For\\nexample, the combinations between F2 and F9 show a large dip\\nin AUC near the center, even though both functions separately\\nseem relatively easy to solve for this version of CMA-ES. While the\\ndifferences between the two versions of CMA-ES are noticeable,\\nmany of the trends, e.g. decreased performance for combinations\\nbetween F11 and F16, are present to some extent in Figure 7 as well.\\nAs a final algorithm, we run DE on the same set of function\\ncombinations. The results are visualized in Figure 9. In this figure,\\nwe see that the overall performance of DE is indeed worse than\\nthe two versions of CMA-ES. It is worth noting that the amount of\\nasymmetry along the diagonal is lower than for the diagonal CMA-\\nES. This could be caused by the change in initialization (Gaussian\\nfor CMA-ES, uniform for DE) reducing the initial bias to the cen-\\nter of the space. Another factor to consider is the variance of the\\nperformance. For CMA-ES, performance can vary significantly as\\n\\ud835\\udefcchanges, while the changes in AUC seem to be much smaller for\\nDE.\\n4 Using Affine Combinations of BBOB Problems for Performance Assessment\\n10610210\\u2212210\\u22126\\n0100101102103104105ERTAlgorithm\\nDiagonalCMA\\nDifferentialEvolution\\nRCobyla\\n10610210\\u2212210\\u22126\\n0.25Algorithm\\nDiagonalCMA\\nDifferentialEvolution\\nRCobyla\\n10610210\\u2212210\\u22126\\n0.5Algorithm\\nDiagonalCMA\\nDifferentialEvolution\\nRCobyla\\n10610210\\u2212210\\u22126\\n0.75Algorithm\\nDiagonalCMA\\nDifferentialEvolution\\nRCobyla\\n10610210\\u2212210\\u22126\\n1Algorithm\\nDiagonalCMA\\nDifferentialEvolution\\nRCobyla\\nFigure 6: ERT per instance for three algorithms on the affine combinations between F10 ( \\ud835\\udefc=1) and F1 (\\ud835\\udefc=0), for selected\\nvalues of\\ud835\\udefc. Each dot corresponds to the ERT calculated based on 5 runs on 1 instance, for a total of 10 instances.\\n0.10.50.9F2\\n0.10.50.9F9\\n0.10.50.9F11\\n0.10.50.9F16\\n00.5 1\\nF20.10.50.9F21\\n00.5 1\\nF900.5 1\\nF1100.5 1\\nF1600.5 1\\nF21\\nFigure 7: Area under the ECDF-curve for Diagonal CMA-\\nES on each of the affine combinations between the selected\\nBBOB problems. Each facet corresponds to the combination\\nof the row and column function, with the x-axis indicating\\nthe used\\ud835\\udefc. AUC values are calculated based on 50 runs on 5\\ninstances, with a budget of 10 000 function evaluations.\\n6 ZOOMING INTO ONE FUNCTION\\nCOMBINATION\\nTo further analyze the impact of changing the weighting of the\\nfunction combinations, we can zoom in on one particular combi-\\nnation and study it in more detail. First, we gauge the impact of\\nusing different instances to measure performance. This is done by\\nconsidering the distribution of AUC values for a specific function\\ncombination, F2 to F16, in Figure 10, on a per-instance basis. From\\nthis figure, we see that in general, the distribution of AUC values\\nis rather stable. However, at the transition point for the CMA-ES\\nvariants, around \\ud835\\udefc\\u22480.8, we see a clear increase in variance. To\\ncheck whether this behavior also occurs for other function com-\\nbinations, we create the same visualization for the combination\\nof F21 and F9 in Figure 11. In this figure, we see a similar pattern\\n0.10.50.9F2\\n0.10.50.9F9\\n0.10.50.9F11\\n0.10.50.9F16\\n00.5 1\\nF20.10.50.9F21\\n00.5 1\\nF900.5 1\\nF1100.5 1\\nF1600.5 1\\nF21Figure 8: Area under the ECDF-curve for modular CMA-ES\\non each of the affine combinations between the selected\\nBBOB problems. Each facet corresponds to the combination\\nof the row and column function, with the x-axis indicating\\nthe used\\ud835\\udefc. AUC values are calculated based on 50 runs on 5\\ninstances, with a budget of 10 000 function evaluations.\\nfor the diagonal CMA-ES, where the distribution of AUC at high \\ud835\\udefc\\nranges from almost 0to almost 1.\\nThe variance observed in Figure 10 might indicate that, in order\\nto get a stable view of the exact behavior at this transition point,\\na wider variety of instances should be used to get a more robust\\nperformance estimate. However, when considering the extreme\\ndifferences in AUC observed in Figure 11, this variance invites\\na more detailed study into the interaction between the instance\\ngeneration process (e.g., the placement of the optimal solution) and\\nthe search behavior of the used algorithm.\\nNext to the instance generation process, another important fac-\\ntor to consider when analyzing the performance of optimization\\nalgorithms on these affine function combinations is the scaling of\\nthe objective values. While it is common practice to ignore the\\n5 Diederick Vermetten, Furong Ye, and Carola Doerr\\n0.10.50.9F2\\n0.10.50.9F9\\n0.10.50.9F11\\n0.10.50.9F16\\n00.5 1\\nF20.10.50.9F21\\n00.5 1\\nF900.5 1\\nF1100.5 1\\nF1600.5 1\\nF21\\nFigure 9: Area under the ECDF-curve for Differential Evolu-\\ntion on each of the affine combinations between the selected\\nBBOB problems. Each facet corresponds to the combination\\nof the row and column function, with the x-axis indicating\\nthe used\\ud835\\udefc. AUC values are calculated based on 50 runs on 5\\ninstances, with a budget of 10 000 function evaluations.\\n0.0 0.2 0.4 0.6 0.8 1.0\\nalpha0.00.20.40.60.8aucAlgorithm\\nDiagonalCMA\\nDifferentialEvolution\\nmodcma\\nFigure 10: Distribution of per-instance normalized AUC val-\\nues for the selected algorithm on the affine combination be-\\ntween F2 and F16. AUC values are calculated based on 50\\nruns on 5 instances, with a budget of 10 000 function evalua-\\ntions.\\nscaling, so the same target values (precision to the optimum) can be\\nused, for example to compute aggregated ECDF curves, the ways in\\nwhich different problems scale their objective values does influence\\nhow we should interpret their results. This becomes increasingly ob-\\nvious when considering the affine combinations of these problems.\\nIn Figure 12, we show the convergence plot of diagonal CMA-ES\\non the combination of F16 and F11. We clearly see from the left\\npart of this curve that the initial values found vary widely for dif-\\nferent combinations, ranging from 107when\\ud835\\udefc=0to102when\\n\\ud835\\udefc=1. However, the change in scale is not the only factor impacting\\n0.0 0.2 0.4 0.6 0.8 1.0\\nalpha0.00.20.40.60.81.0auc\\nAlgorithm\\nDiagonalCMA\\nDifferentialEvolution\\nmodcmaFigure 11: Distribution of per-instance normalized AUC val-\\nues for the selected algorithm on the affine combination be-\\ntween F21 and F9. AUC values are calculated based on 50\\nruns on 5 instances, with a budget of 10 000 function evalua-\\ntions.\\n12 5 102 5 1002 5 1e+32 5 1e+41e\\u221240.0111001e+41e+6\\n0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60\\n0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00Function EvaluationsBest-so-far f(x)\\nFigure 12: Evolution of geometric mean function value\\nfound by modular CMA-ES for the affine combination of F16\\nand F11, instance 1 for both functions. Each line corresponds\\nto 50 runs with the specified \\ud835\\udefc.\\nthe performance. The shape of the curve changes noticeably after\\nthe initialization, which matches the change in AUC observed in\\nFigure 7.\\nTo investigate the reason for this change in behavior, we can\\nstudy the optimization trajectory of diagonal CMA-ES on these\\nfunctions. Since this is not feasible to visualize in the original 5-\\ndimensional space, we repeat the data collection on the 2-dimensional\\nversion of these functions. In Figure 13, we show the landscapes\\nof the affine combinations between F11 and F16 for several values\\nof\\ud835\\udefc. We highlight the best point found by the diagonal CMA-ES\\nin each of its 50 runs on this instance. This plot clearly shows the\\ndifferences in scale between the original problems. In addition, we\\nsee that as\\ud835\\udefcgets closer to 1, the algorithm gets stuck in the lo-\\ncal optima less often. The global structure added by F11 is strong\\nenough to guide the CMA-ES to the area containing the global\\noptimum. However, when the influence of F11 becomes too large,\\nthe difficulties of finding the correct search direction have a strong\\nimpact on the convergence behavior. As such, values of \\ud835\\udefccloser\\n6 Using Affine Combinations of BBOB Problems for Performance Assessment\\n\\u22124\\u22122 024\\n0\\u22124\\u22122024\\n\\u22124\\u22122 024\\n0.25\\u22124\\u22122 024\\n0.5\\u22124\\u22122 024\\n0.75\\u22124\\u22122024\\n1024681012141618\\nFigure 13: Evolution of the landscape (log-scaled function-values) of the affine combination between F11 ( \\ud835\\udefc=1) and F16\\n(\\ud835\\udefc=0), instance 1 for both functions, for varying \\ud835\\udefc. The red circle highlights the location of the global optimum. The crosses\\ncorrespond to the best point found in each of 50 runs of the modular CMA-ES.\\n\\u22124\\u22122 024\\n0\\u22124\\u22122024\\n\\u22124\\u22122 024\\n0.25\\u22124\\u22122 024\\n0.5\\u22124\\u22122 024\\n0.75\\u22124\\u22122 024\\n1\\u22129.6\\u22128.0\\u22126.4\\u22124.8\\u22123.2\\u22121.60.01.63.2\\nFigure 14: Evolution of the landscape (log-scaled function-values) of the affine combination between F21 ( \\ud835\\udefc=1) and F9 (\\ud835\\udefc=\\n0), instance 1 for both functions, for varying \\ud835\\udefc. The red circle highlights the location of the global optimum. The crosses\\ncorrespond to the best point found in each of 50 runs of Diagonal CMA-ES.\\nto0.5seem to provide a mix of the multimodality of F16 and the\\nchallenges of F11, which makes it a challenging problem to solve\\nfor the CMA-ES.\\nWhile the combination between F11 and F16 seems to create\\nfunctions that are more challenging, Figure 7 shows that there are\\nfunction combinations where the opposite is true. The combina-\\ntion between F9 and F21 displays interesting behavior. While the\\nway of performing initialization might explain the asymmetry be-\\ntween(\\ud835\\udc399,\\ud835\\udc3921,0)and(\\ud835\\udc3921,\\ud835\\udc399,1), it does not explain the increase\\nin AUC for \\ud835\\udefcclose to 0.5. We visualize the change in landscape,\\nand corresponding solutions found by the diagonal CMA-ES, in\\nFigure 14. In this figure, we see that when \\ud835\\udefc=0, the CMA-ES finds\\nsolutions on the ridge of the function, but most of the runs don\\u2019t\\nreach the optimum within the given budget. This indicates that the\\ncharacteristic difficulty of F9, the algorithm having to consistently\\nadapt its search direction [ 7], hinders the convergence of the used\\ndiagonal CMA-ES. However, as \\ud835\\udefcincreases, the structure of F21\\ngets added, which increases the ways in which the algorithm can\\napproach the optimum value. For \\ud835\\udefc=1, the multimodality from\\nF21 completely takes over, trapping some runs in local optima, thus\\ndecreasing the performance of the algorithm. This showcases thatcombining these two functions in this way creates a function where\\nthe original difficulties of both are combined in a way that negates\\nboth of them, which is then exploited by the CMA-ES.\\n7 CONCLUSIONS AND FUTURE WORK\\nAffine combinations of BBOB problems offer a new way to inves-\\ntigate the behavior of optimization algorithms. We have shown\\nhow combinations of arbitrary functions with a sphere model can\\nbe used to identify the impact of added global structure on the\\nperformance of a set of algorithms. In addition, combinations be-\\ntween functions with different high-level characteristics allowed\\nus to observe transitions between different optimization challenges.\\nWhile this investigation is not exhaustive, it highlights the potential\\nbenefit of utilizing these new function combinations for gaining an\\nunderstanding of the behavior of optimization algorithms.\\nHowever, these benefits in terms of analysis options also come\\nwith several challenges which have to be considered. We identified\\nthe following aspects:\\nScaling. As identified when these combinations were proposed [ 4],\\nthe differences in scale between two problems can be significant.\\n7 Diederick Vermetten, Furong Ye, and Carola Doerr\\nWhile we aimed to reduce this impact by considering a logarith-\\nmically scaled weighting, it is clear from our experiments that the\\nscale still plays a large role in the way we interpret the performance.\\nFinding ways to combine the landscapes of two functions while\\nmaintaining a consistent range of function values is still an open\\nquestion.\\nInstances. The BBOB suite is built on the idea that each func-\\ntion can be instantiated in many ways. This is achieved through\\nseveral transformations, the most common of which is moving\\nthe optimum to a different location in the domain. The results we\\npresent show that the way in which these optimal locations are\\nchosen can have a large impact on the performance of optimization\\nalgorithms. Since the optima are not distributed uniformly in the\\ndomain, some functions have different kinds of bias, which can\\nbe exploited by an algorithm. The question on how to fairly con-\\nsider different instance generation mechanisms when making use\\nof function combination is thus highly interlinked with questions\\nabout how well performance observed on a set of BBOB instances\\ngeneralizes.\\nEven with these challenges in mind, there are many potential\\nuse cases for these affine function combinations. One aspect in\\nwhich they can prove useful is in the training of algorithm selection\\nmodels [ 11], as they can significantly increase the size and variety of\\ntraining data, which is an important consideration towards testing\\ngeneralizability.\\nOne final aspect in which the benchmark data on these function\\ncombinations can be further utilized is by linking it back to the\\nexploratory landscape analysis which inspired their creation. Since\\nthe combinations can smoothly fill the landscape feature space,\\nthis can be combined with algorithm performance to get a more\\nfine-grained view of the way in which the landscape interacts with\\ndifferent algorithms [9, 22].\\nACKNOWLEDGMENTS\\nOur work is financially supported by ANR-22-ERCS-0003-01 project\\nVARIATION and by the CNRS INS2I project IOHprofiler. This work\\nwas performed using the ALICE compute resources provided by\\nLeiden University.\\nREFERENCES\\n[1]Thomas Bartz-Beielstein, Carola Doerr, Jakob Bossek, Sowmya Chandrasekaran,\\nTome Eftimov, Andreas Fischbach, Pascal Kerschke, Manuel L\\u00f3pez-Ib\\u00e1\\u00f1ez, Kather-\\nine M. Malan, Jason H. Moore, Boris Naujoks, Patryk Orzechowski, Vanessa Volz,\\nMarkus Wagner, and Thomas Weise. 2020. Benchmarking in Optimization:\\nBest Practice and Open Issues. CoRR abs\\/2007.03488 (2020). arXiv:2007.03488\\nhttps:\\/\\/arxiv.org\\/abs\\/2007.03488\\n[2]Jacob de Nobel, Diederick Vermetten, Hao Wang, Carola Doerr, and Thomas B\\u00e4ck.\\n2021. Tuning as a means of assessing the benefits of new ideas in interplay with\\nexisting algorithmic modules. In Proc. of Genetic and Evolutionary Computation\\nConference (GECCO\\u201921, Companion material) , Krzysztof Krawiec (Ed.). ACM,\\n1375\\u20131384. https:\\/\\/doi.org\\/10.1145\\/3449726.3463167\\n[3]Jacob de Nobel, Furong Ye, Diederick Vermetten, Hao Wang, Carola Doerr, and\\nThomas B\\u00e4ck. 2021. IOHexperimenter: Benchmarking Platform for Iterative\\nOptimization Heuristics. CoRR abs\\/2111.04077 (2021). arXiv:2111.04077 https:\\n\\/\\/arxiv.org\\/abs\\/2111.04077\\n[4]Konstantin Dietrich and Olaf Mersmann. 2022. Increasing the Diversity of\\nBenchmark Function Sets Through Affine Recombination. In Parallel Problem\\nSolving from Nature - PPSN XVII - 17th International Conference, PPSN 2022,\\nDortmund, Germany, September 10-14, 2022, Proceedings, Part I (Lecture Notes in\\nComputer Science, Vol. 13398) , G\\u00fcnter Rudolph, Anna V. Kononova, Hern\\u00e1n E.\\nAguirre, Pascal Kerschke, Gabriela Ochoa, and Tea Tusar (Eds.). Springer, 590\\u2013602.\\nhttps:\\/\\/doi.org\\/10.1007\\/978-3-031-14714-2_41[5]Nikolaus Hansen, Anne Auger, Dimo Brockhoff, and Tea Tu\\u0161ar. 2022. Any-\\ntime Performance Assessment in Blackbox Optimization Benchmarking. IEEE\\nTransactions on Evolutionary Computation 26, 6 (2022), 1293\\u20131305. time Performance Assessment in Blackbox Optimization Benchmarking. IEEE\\nTransactions on Evolutionary Computation 26, 6 (2022), 1293\\u20131305.\\n[6]Nikolaus Hansen, Anne Auger, Raymond Ros, Olaf Mersmann, Tea Tu\\u0161ar, and\\nDimo Brockhoff. 2021. COCO: A platform for comparing continuous optimizers\\nin a black-box setting. Optimization Methods and Software 36, 1 (2021), 114\\u2013144.\\n[7]Nikolaus Hansen, Steffen Finck, Raymond Ros, and Anne Auger. 2009. Real-\\nParameter Black-Box Optimization Benchmarking 2009: Noiseless Functions Defi-\\nnitions . Technical Report RR-6829. INRIA. https:\\/\\/hal.inria.fr\\/inria-00362633\\/\\ndocument\\n[8]Nikolaus Hansen and Andreas Ostermeier. 2001. Completely Derandomized\\nSelf-Adaptation in Evolution Strategies. Evolutionary Computation 9, 2 (2001),\\n159\\u2013195. https:\\/\\/doi.org\\/10.1162\\/106365601750190398\\n[9]Anja Jankovic and Carola Doerr. 2020. Landscape-aware fixed-budget perfor-\\nmance regression and algorithm selection for modular CMA-ES variants. In\\nProceedings of the 2020 Genetic and Evolutionary Computation Conference . ACM,\\n841\\u2013849.\\n[10] James Kennedy and Russell Eberhart. 1995. Particle swarm optimization. In\\nProceedings of International Conference on Neural Networks (ICNN\\u201995), Perth, WA,\\nAustralia, November 27 - December 1, 1995 . IEEE, 1942\\u20131948. https:\\/\\/doi.org\\/10.\\n1109\\/ICNN.1995.488968\\n[11] Pascal Kerschke, Holger H. Hoos, Frank Neumann, and Heike Trautmann. 2019.\\nAutomated Algorithm Selection: Survey and Perspectives. Evolutionary Compu-\\ntation 27, 1 (2019), 3\\u201345. https:\\/\\/doi.org\\/10.1162\\/evco_a_00242\\n[12] Pedro Larra\\u00f1aga and Jose A Lozano. 2001. Estimation of distribution algorithms:\\nA new tool for evolutionary computation . Vol. 2. Springer Science & Business\\nMedia.\\n[13] Fu Xing Long, Diederick Vermetten, Bas van Stein, and Anna V. Kononova. 2022.\\nBBOB Instance Analysis: Landscape Properties and Algorithm Performance\\nacross Problem Instances. CoRR abs\\/2211.16318 (2022). https:\\/\\/doi.org\\/10.48550\\/\\narXiv.2211.16318 arXiv:2211.16318\\n[14] Manuel L\\u00f3pez-Ib\\u00e1\\u00f1ez, Juergen Branke, and Lu\\u00eds Paquete. 2021. Reproducibility\\nin evolutionary computation. ACM Transactions on Evolutionary Learning and\\nOptimization 1, 4 (2021), 1\\u201321.\\n[15] Olaf Mersmann, Bernd Bischl, Heike Trautmann, Mike Preuss, Claus Weihs, and\\nG\\u00fcnter Rudolph. 2011. Exploratory landscape analysis. In Proc. of Genetic and\\nEvolutionary Computation (GECCO\\u201911) . ACM, 829\\u2013836.\\n[16] Mario Andr\\u00e9s Mu\\u00f1oz, Michael Kirley, and Kate Smith-Miles. 2022. Analyzing\\nrandomness effects on the reliability of exploratory landscape analysis. Natural\\nComputing 21, 2 (2022), 131\\u2013154.\\n[17] Mario A Mu\\u00f1oz, Yuan Sun, Michael Kirley, and Saman K Halgamuge. 2015.\\nAlgorithm selection for black-box continuous optimization problems: A survey\\non methods and challenges. Information Sciences 317 (2015), 224\\u2013245.\\n[18] Michael JD Powell. 1994. A direct search optimization method that models the\\nobjective and constraint functions by linear interpolation . Springer.\\n[19] J\\u00e9r\\u00e9my Rapin and Olivier Teytaud. 2018. Nevergrad: A gradient-free optimization\\nplatform. https:\\/\\/GitHub.com\\/FacebookResearch\\/Nevergrad.\\n[20] Quentin Renau, Johann Dr\\u00e9o, Carola Doerr, and Benjamin Doerr. 2021. To-\\nwards explainable exploratory landscape analysis: extreme feature selection for\\nclassifying BBOB functions. In Applications of Evolutionary Computation: 24th\\nInternational Conference, EvoApplications 2021, Held as Part of EvoStar 2021, Virtual\\nEvent, April 7\\u20139, 2021, Proceedings 24 . Springer, 17\\u201333.\\n[21] Rainer Storn and Kenneth Price. 1997. Differential evolution-a simple and effi-\\ncient heuristic for global optimization over continuous spaces. Journal of global\\noptimization 11, 4 (1997), 341.\\n[22] Risto Trajanov, Stefan Dimeski, Martin Popovski, Peter Koro\\u0161ec, and Tome Efti-\\nmov. 2021. Explainable landscape-aware optimization performance prediction.\\nIn2021 IEEE Symposium Series on Computational Intelligence (SSCI) . IEEE, 01\\u201308.\\n[23] Diederick Vermetten, Bas van Stein, Fabio Caraffini, Leandro L. Minku, and In2021 IEEE Symposium Series on Computational Intelligence (SSCI) . IEEE, 01\\u201308.\\n[23] Diederick Vermetten, Bas van Stein, Fabio Caraffini, Leandro L. Minku, and\\nAnna V. Kononova. 2022. BIAS: A Toolbox for Benchmarking Structural Bias\\nin the Continuous Domain. IEEE Trans. Evol. Comput. 26, 6 (2022), 1380\\u20131393.\\nhttps:\\/\\/doi.org\\/10.1109\\/TEVC.2022.3189848\\n[24] Diederick Vermetten, Furong Ye, and Carola Doerr. 2023. Reproducibility files\\nand additional figures. Code and data repository: https:\\/\\/doi.org\\/10.5281\\/zenodo.\\n7629706 Figure repository: https:\\/\\/figshare.com\\/s\\/68587b6a82d9c6e5eccf.\\n[25] Hao Wang, Diederick Vermetten, Furong Ye, Carola Doerr, and Thomas B\\u00e4ck.\\n2022. IOHanalyzer: Detailed Performance Analysis for Iterative Optimization\\nHeuristic. ACM Trans. Evol. Learn. Optim. 2, 1 (2022), 3:1\\u20133:29. https:\\/\\/doi.org\\/\\n10.1145\\/3510426 IOHanalyzer is available at CRAN, on GitHub, and as web-based\\nGUI, see https:\\/\\/iohprofiler.github.io\\/IOHanalyzer\\/ for links.\\n8\",\"5\":\" arXiv:2303.04347v1  [cs.NE]  8 Mar 2023Published as a conference paper at ICLR 2022\\nOPTIMAL ANN-SNN C ONVERSION FOR HIGH-\\nACCURACY AND ULTRA -LOW -LATENCY SPIKING\\nNEURAL NETWORKS\\nTong Bu1, Wei Fang1, Jianhao Ding1, PengLin Dai2, Zhaofei Yu1 *, Tiejun Huang1\\n1Peking University,2Southwest Jiaotong University\\n*Corresponding author: yuzf12@pku.edu.cn\\nABSTRACT\\nSpiking Neural Networks (SNNs) have gained great attractio n due to their dis-\\ntinctive properties of low power consumption and fast infer ence on neuromorphic\\nhardware. As the most effective method to get deep SNNs, ANN- SNN conversion\\nhas achieved comparable performance as ANNs on large-scale datasets. Despite\\nthis, it requires long time-steps to match the \\ufb01ring rates of SNNs to the activation\\nof ANNs. As a result, the converted SNN suffers severe perfor mance degradation\\nproblems with short time-steps, which hamper the practical application of SNNs.\\nIn this paper, we theoretically analyze ANN-SNN conversion error and derive the\\nestimated activation function of SNNs. Then we propose the q uantization clip-\\n\\ufb02oor-shift activation function to replace the ReLU activat ion function in source\\nANNs, which can better approximate the activation function of SNNs. We prove\\nthat the expected conversion error between SNNs and ANNs is z ero, enabling us\\nto achieve high-accuracy and ultra-low-latency SNNs. We ev aluate our method\\non CIFAR-10\\/100 and ImageNet datasets, and show that it outp erforms the state-\\nof-the-art ANN-SNN and directly trained SNNs in both accura cy and time-steps.\\nTo the best of our knowledge, this is the \\ufb01rst time to explore h igh-performance\\nANN-SNN conversion with ultra-low latency (4 time-steps). Code is available at\\nhttps:\\/\\/github.com\\/putshua\\/SNN conversion QCFS\\n1 I NTRODUCTION\\nSpiking neural networks (SNNs) are biologically plausible neural networks based on the dynamic\\ncharacteristic of biological neurons (McCulloch & Pitts, 1 943; Izhikevich, 2003). As the third gener-\\nation of arti\\ufb01cial neural networks (Maass, 1997), SNNs have attracted great attention due to their dis-\\ntinctive properties over deep analog neural networks (ANNs ) (Roy et al., 2019). Each neuron trans-\\nmits discrete spikes to convey information when exceeding a threshold. For most SNNs, the spiking\\nneurons will accumulate the current of the last layer as the o utput within Tinference time steps.\\nThe binarized activation has rendered dedicated hardware o f neuromorphic computing (Pei et al.,\\n2019; DeBole et al., 2019; Davies et al., 2018). This kind of h ardware has excellent advantages in\\ntemporal resolution and energy budget. Existing work has sh own the potential of tremendous energy\\nsaving with considerably fast inference (St\\u00a8 ockl & Maass, 2 021).\\nIn addition to ef\\ufb01ciency advantages, the learning algorith m of SNNs has been improved by leaps\\nand bounds in recent years. The performance of SNNs trained b y backpropagation through time\\nand ANN-SNN conversion techniques has gradually been compa rable to ANNs on large-scale\\ndatasets (Fang et al., 2021; Rueckauer et al., 2017). Both te chniques bene\\ufb01t from the setting of\\nSNN inference time. Setting longer time-steps in backpropa gation can make the gradient of surro-\\ngate functions more reliable (Wu et al., 2018; Neftci et al., 2019; Zenke & V ogels, 2021). However,\\nthe price is enormous resource consumption during training . Existing platforms such as TensorFlow\\nand PyTorch based on CUDA have limited optimization for SNN t raining. In contrast, ANN-SNN\\nconversion usually depends on a longer inference time to get comparable accuracy as the original\\nANN (Sengupta et al., 2019) because it is based on the equival ence of ReLU activation and integrate-\\nand-\\ufb01re model\\u2019s \\ufb01ring rate (Cao et al., 2015). Although long er inference time can further reduce the\\nconversion error, it also hampers the practical applicatio n of SNNs on neuromorphic chips.\\n1 Published as a conference paper at ICLR 2022\\nThe dilemma of ANN-SNN conversion is that there exists a rema ining potential in the conver-\\nsion theory, which is hard to be eliminated in a few time steps (Rueckauer et al., 2016). Although\\nmany methods have been proposed to improve the conversion ac curacy, such as weight normaliza-\\ntion (Diehl et al., 2015), threshold rescaling (Sengupta et al., 2019), soft-reset (Han & Roy, 2020)\\nand threshold shift (Deng & Gu, 2020), tens to hundreds of tim e-steps in the baseline works are still\\nunbearable. To obtain high-performance SNNs with ultra-lo w latency (e.g., 4 time-steps), we list the\\ncritical errors in ANN-SNN conversion and provide solution s for each error. Our main contributions\\nare summarized as follows:\\n\\u2022 We go deeper into the errors in the ANN-SNN conversion and as cribe them to clipping\\nerror, quantization error, and unevenness error. We \\ufb01nd tha t unevenness error, which is\\ncaused by the changes in the timing of arrival spikes and has b een neglected in previous\\nworks, can induce more spikes or fewer spikes as expected.\\n\\u2022 We propose the quantization clip-\\ufb02oor-shift activation f unction to replace the ReLU activa-\\ntion function in source ANNs, which better approximates the activation function of SNNs.\\nWe prove that the expected conversion error between SNNs and ANNs is zero, indicating\\nthat we can achieve high-performance converted SNN at ultra -low time-steps.\\n\\u2022 We evaluate our method on CIFAR-10, CIFAR-100, and ImageNe t datasets. Compared\\nwith both ANN-SNN conversion and backpropagation training methods, the proposed\\nmethod exceeds state-of-the-art accuracy with fewer time- steps. For example, we reach\\ntop-1 accuracy 91.18% on CIFAR-10 with unprecedented 2 time -steps.\\n2 PRELIMINARIES\\nIn this section, we \\ufb01rst brie\\ufb02y review the neuron models for S NNs and ANNs. Then we introduce\\nthe basic framework for ANN-SNN conversion.\\nNeuron model for ANNs. For ANNs, the computations of analog neurons can be simpli\\ufb01e d as the\\ncombination of a linear transformation and a non-linear map ping:\\nal=h(Wlal\\u22121), l= 1,2,...,M (1)\\nwhere the vector aldenotes the output of all neurons in l-th layer, Wldenotes the weight matrix\\nbetween layer land layer l\\u22121, andh(\\u00b7)is the ReLU activation function.\\nNeuron model for SNNs. Similar to the previous works (Cao et al., 2015; Diehl et al., 2015;\\nHan et al., 2020), we consider the Integrate-and-Fire (IF) m odel for SNNs. If the IF neurons in\\nl-th layer receive the input xl\\u22121(t)from last layer, the temporal potential of the IF neurons can be\\nde\\ufb01ned as:\\nml(t) =vl(t\\u22121)+Wlxl\\u22121(t), (2)\\nwhereml(t)andvl(t)represent the membrane potential before and after the trigg er of a spike\\nat time-step t.Wldenote the weight in l-th layer. As soon as any element ml\\ni(t)ofml(t)ex-\\nceeds the \\ufb01ring threshold \\u03b8l, the neuron will elicit a spike and update the membrane poten tialvl\\ni(t).\\nTo avoid information loss, we use the \\u201creset-by-subtractio n\\u201d mechanism (Rueckauer et al., 2017;\\nHan et al., 2020) instead of the \\u201creset-to-zero\\u201d mechanism, which means the membrane potential\\nvl\\ni(t)is subtracted by the threshold value \\u03b8lif the neuron \\ufb01res. Based on the threshold-triggered\\n\\ufb01ring mechanism and the \\u201creset-by-subtraction\\u201d of the memb rane potential after \\ufb01ring discussed\\nabove, we can write the uplate rule of membrane potential as:\\nsl(t) =H(ml(t)\\u2212\\u03b8l), (3)\\nvl(t) =ml(t)\\u2212sl(t)\\u03b8l. (4)\\nHeresl(t)refers to the output spikes of all neurons in layer lat timet, the element of which equals\\n1 if there is a spike and 0 otherwise. H(\\u00b7)is the Heaviside step function. \\u03b8lis the vector of the\\n\\ufb01ring threshold \\u03b8l. Similar to Deng & Gu (2020), we suppose that the postsynapti c neuron in l-th\\nlayer receives unweighted postsynaptic potential \\u03b8lif the presynaptic neuron in l\\u22121-th layer \\ufb01res\\na spike, that is:\\nxl(t) =sl(t)\\u03b8l. (5)\\n2 Published as a conference paper at ICLR 2022\\nTable 1: Summary of notations in this paper\\nSymbol De\\ufb01nition Symbol De\\ufb01nition\\nl Layer index xl(t) Unweighted PSP1\\ni Neuron index sl(t) Output spikes\\nWlWeight \\u03c6l(T) Average unweigthed PSP before time T\\nalANN activation values zlWeighted input from l\\u22121layer\\nt Time-steps h(\\u00b7) ReLU function\\nT Total time-step H(\\u00b7) Heaviside step function\\n\\u03b8lThreshold L Quantization step for ANN\\n\\u03bblTrainable threshold in ANN ErrlConversion Error\\nml(t) Potential before \\ufb01ring \\/tildewidestErrlEstimated conversion Error\\nvl(t) Potential after \\ufb01ring \\u03d5 Shift of quantization clip-\\ufb02oor function\\n1Postsynaptic potential\\nANN-SNN conversion. The key idea of ANN-SNN conversion is to map the activation va lue of an\\nanalog neuron in ANN to the \\ufb01ring rate (or average postsynapt ic potential) of a spiking neuron in\\nSNN. Speci\\ufb01cally, we can get the potential update equation b y combining Equation 2 \\u2013 Equation 4:\\nvl(t)\\u2212vl(t\\u22121) =Wlxl\\u22121(t)\\u2212sl(t)\\u03b8l. (6)\\nEquation 6 describes the basic function of spiking neurons u sed in ANN-SNN conversion. By\\nsumming Equation 6 from time 1toTand dividing Ton both sides, we have:\\nvl(T)\\u2212vl(0)\\nT=Wl\\/summationtextT\\ni=1xl\\u22121(i)\\nT\\u2212\\/summationtextT\\ni=1sl(i)\\u03b8l\\nT. (7)\\nIf we use \\u03c6l\\u22121(T) =\\/summationtextT\\ni=1xl\\u22121(i)\\nTto denote the average postsynaptic potential during the per iod\\nfrom 0 to Tand substitute Equation 5 into Equation 7, then we get:\\n\\u03c6l(T) =Wl\\u03c6l\\u22121(T)\\u2212vl(T)\\u2212vl(0)\\nT. (8)\\nEquation 8 describes the relationship of the average postsy naptic potential of neurons in adjacent\\nlayers. Note that \\u03c6l(T)\\/greaterorequalslant0. If we set the initial potential vl(0)to zero and neglect the remaining\\ntermvl(T)\\nTwhen the simulation time-steps Tis long enough, the converted SNN has nearly the\\nsame activation function as source ANN (Equation 1). Howeve r, highTwould cause long inference\\nlatency that hampers the practical application of SNNs. The refore, this paper aims to implement\\nhigh-performance ANN-SNN conversion with extremely low la tency.\\n3 CONVERSION ERROR ANALYSIS\\nIn this section, we will analyze the conversion error betwee n the source ANN and the converted\\nSNN in each layer in detail. In the following, we assume that b oth ANN and SNN receive the same\\ninput from the layer l\\u22121, that is, al\\u22121=\\u03c6l\\u22121(T), and then analyze the error in layer l. For\\nsimplicity, we use zl=Wl\\u03c6l\\u22121(T) =Wlal\\u22121to substitute the weighted input from layer l\\u22121\\nfor both ANN and SNN. The absolute conversion error is exactl y the outputs from converted SNN\\nsubtract the outputs from ANN:\\nErrl=\\u03c6l(T)\\u2212al=zl\\u2212vl(T)\\u2212vl(0)\\nT\\u2212h(zl), (9)\\nwhereh(zl) =ReLU(zl). It can be found from Equation 9 that the conversion error is n onzero if\\nvl(T)\\u2212vl(0)\\/negationslash= 0andzl>0. In fact, the conversion error is caused by three factors.\\nClipping error. The output \\u03c6l(T)of SNNs is in the range of [0,\\u03b8l]as\\u03c6l(T) =\\/summationtextT\\ni=1xl(i)\\nT=\\n\\/summationtextT\\ni=1sl(i)\\nT\\u03b8l(see Equation 5). However, the output alof ANNs is in a much lager range of [0,al\\nmax],\\nwhereal\\nmax denotes the maximum value of al. As illustrated in Figure 1a, alcan be mapped to\\n\\u03c6l(T)by the following equation:\\n\\u03c6l(T) =clip\\/parenleftbigg\\u03b8l\\nT\\/floorleftbiggalT\\n\\u03bbl\\/floorrightbigg\\n,0,\\u03b8l\\/parenrightbigg\\n. (10)\\n3 Published as a conference paper at ICLR 2022\\n\\u000f\\u000f\\u000f \\nclipped\\n\\u000f\\u000f\\u000f\\n(a) Clipping error\\/uni00000013 \\/uni00000014 \\/uni00000015 \\/uni00000016 \\/uni00000017 \\/uni00000018\\n\\/uni00000037\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053Sl\\u2212 1\\n2 Sl\\u2212 1\\n1\\/uni00000010\\/uni00000017\\/uni00000010\\/uni00000015\\/uni00000013\\/uni00000015\\/uni00000017\\/uni00000033\\/uni00000052\\/uni00000057\\/uni00000048\\/uni00000051\\/uni00000057\\/uni0000004c\\/uni00000044\\/uni0000004fSl\\n1\\n(b) Even spikes\\/uni00000013 \\/uni00000014 \\/uni00000015 \\/uni00000016 \\/uni00000017 \\/uni00000018\\n\\/uni00000037\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053Sl\\u2212 1\\n2 Sl\\u2212 1\\n1\\/uni00000010\\/uni00000017\\/uni00000010\\/uni00000015\\/uni00000013\\/uni00000015\\/uni00000017\\/uni00000033\\/uni00000052\\/uni00000057\\/uni00000048\\/uni00000051\\/uni00000057\\/uni0000004c\\/uni00000044\\/uni0000004fSl\\n1\\n(c) More spikes\\/uni00000013 \\/uni00000014 \\/uni00000015 \\/uni00000016 \\/uni00000017 \\/uni00000018\\n\\/uni00000037\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053Sl\\u2212 1\\n2 Sl\\u2212 1\\n1\\/uni00000010\\/uni00000017\\/uni00000010\\/uni00000015\\/uni00000013\\/uni00000015\\/uni00000017\\/uni00000033\\/uni00000052\\/uni00000057\\/uni00000048\\/uni00000051\\/uni00000057\\/uni0000004c\\/uni00000044\\/uni0000004fSl\\n1\\n(d) Fewer spikes\\nFigure 1: Conversion error between source ANN and converted SNN.sl\\u22121\\n1andsl\\u22121\\n2denote the\\noutput spikes of two neurons in layer l\\u22121, andsl\\n1denotes the output spikes of a neuron in layer l.\\nHere the clip function sets the upper bound \\u03b8land the lower bound 0.\\u230a\\u00b7\\u230bdenotes the \\ufb02oor func-\\ntion.\\u03bblrepresents the actual maximum value of output almapped to the maximum value \\u03b8lof\\n\\u03c6l(T). Considering that nearly 99.9% activations of alin ANN are in the range of [0,al\\nmax\\n3],\\nRueckauer et al. (2016) suggested to choose \\u03bblaccording to 99.9% activations. The activations\\nbetween\\u03bblandal\\nmaxin ANN are mapped to the same value \\u03b8lin SNN, which will cause conversion\\nerror called clipping error.\\nQuantization error (\\ufb02ooring error). The output spikes sl(t)are discrete events, thus \\u03c6l(T)are\\ndiscrete with quantization resolution\\u03b8l\\nT(see Equation 10). When mapping alto\\u03c6l(T), there exists\\nunavoidable quantization error. For example, as illustrat ed in Figure 1a, the activations of ANN in\\nthe range of [\\u03bbl\\nT,2\\u03bbl\\nT)are mapped to the same value\\u03b8l\\nTof SNN.\\nUnevenness error. Unevenness error is caused by the unevenness of input spikes . If the timing of\\narrival spikes changes, the output \\ufb01ring rates may change, w hich causes conversion error. There are\\ntwo situations: more spikes as expected or fewer spikes as ex pected. To see this, in source ANN,\\nwe suppose that two analog neurons in layer l\\u22121are connected to an analog neuron in layer l\\nwith weights 2 and -2, and the output vector al\\u22121of neurons in layer l\\u22121is[0.6,0.4]. Besides, in\\nconverted SNN, we suppose that the two spiking neurons in lay erl\\u22121\\ufb01re 3 spikes and 2 spikes in\\n5 time-steps (T=5), respectively, and the threshold \\u03b8l\\u22121= 1. Thus,\\u03c6l\\u22121(T) =\\/summationtextT\\ni=1sl\\u22121(i)\\nT\\u03b8l\\u22121=\\n[0.6,0.4]. Even though \\u03c6l\\u22121(T) =al\\u22121and the weights are same for the ANN and SNN, \\u03c6l(T)\\ncan be different from alif the timing of arrival spikes changes. According to Equati on 1, the ANN\\noutputal=Wlal\\u22121= [2,\\u22122][0.6,0.4]T= 0.4. As for SNN, supposing that the threshold \\u03b8l= 1,\\nthere are three possible output \\ufb01ring rates, which are illus trated in Figure 1 (b)-(d). If the two\\npresynaptic neurons \\ufb01res at t= 1,3,5andt= 2,4(red bars) respectively with weights 2 and -2, the\\npostsynaptic neuron will \\ufb01re two spikes at t= 1,3(red bars), and \\u03c6l(T) =\\/summationtextT\\ni=1sl(i)\\nT\\u03b8l= 0.4 =al.\\nHowever, if the presynaptic neurons \\ufb01res at t= 1,2,3andt= 4,5, respectively, the postsynaptic\\nneuron will \\ufb01re four spikes at t= 1,2,3,4, and\\u03c6l(T) = 0.8>al. If the presynaptic neurons \\ufb01res\\natt= 3,4,5andt= 1,2, respectively, the postsynaptic neuron will \\ufb01re only one sp ikes att= 5,\\nand\\u03c6l(T) = 0.2<al. att= 3,4,5andt= 1,2, respectively, the postsynaptic neuron will \\ufb01re only one sp ikes att= 5,\\nand\\u03c6l(T) = 0.2<al.\\nNote that the clipping error and quantization error have bee n proposed in Li et al. (2021). There exist\\ninterdependence between the above three kinds of errors. Sp eci\\ufb01cally, the unevenness error will\\ndegenerate to the quantization error if vl(T)is in the range of [0,\\u03b8l]. Assuming that the potential\\nvl(T)falls into [0,\\u03b8l]will enable us to estimate the activation function of SNNs ig noring the effect\\nof unevenness error. Therefore, an estimation of the output value\\u03c6l(T)in a converted SNN can be\\nformulated with the combination of clip function and \\ufb02oor fu nction, that is:\\n\\u03c6l(T)\\u2248\\u03b8lclip\\/parenleftbigg1\\nT\\/floorleftbiggzlT+vl(0)\\n\\u03b8l\\/floorrightbigg\\n,0,1\\/parenrightbigg\\n. (11)\\nThe detailed derivation is in the Appendix. With the help of t his estimation for the SNN output, the\\nestimated conversion error \\/tildewidestErrlcan be derived from Equation 9:\\n\\/tildewidestErrl=\\u03b8lclip\\/parenleftbigg1\\nT\\/floorleftbiggzlT+vl(0)\\n\\u03b8l\\/floorrightbigg\\n,0,1\\/parenrightbigg\\n\\u2212h(zl)\\u2248Errl. (12)\\n4 Published as a conference paper at ICLR 2022\\n\\/uni00000013\\/uni00000011\\/uni00000015\\/uni00000018 \\u03bbl\\n\\/uni00000013\\/uni00000011\\/uni00000018 \\u03bbl\\n\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018 \\u03bbl\\n\\u03bbl\\nzl\\u2212 0 .125 \\u03b8l\\/uni00000013\\/uni000000030 .125 \\u03b8l\\/tildewidest\\nErrl\\/uni00000013\\/uni00000013\\/uni00000011\\/uni00000015\\/uni00000018 \\u03b8l\\/uni00000013\\/uni00000011\\/uni00000018 \\u03b8l\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018 \\u03b8l\\u03b8l\\u03c6l\\n( T ) \\/uni00000003\\/uni0000003f\\/uni00000003 al\\n(a)L=T= 4\\/uni00000013\\/uni00000011\\/uni00000015\\/uni00000018 \\u03bbl\\n\\/uni00000013\\/uni00000011\\/uni00000018 \\u03bbl\\n\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018 \\u03bbl\\n\\u03bbl\\nzl\\u2212 0 .125 \\u03b8l\\/uni00000013\\/uni000000030 .125 \\u03b8l\\/tildewidest\\nErrl\\/uni00000013\\/uni00000013\\/uni00000011\\/uni00000015\\/uni00000018 \\u03b8l\\/uni00000013\\/uni00000011\\/uni00000018 \\u03b8l\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018 \\u03b8l\\u03b8l\\u03c6l\\n( T ) \\/uni00000003\\/uni0000003f\\/uni00000003 al\\n(b)L= 4,T= 8\\/uni00000013\\/uni00000011\\/uni00000015\\/uni00000018 \\u03bbl\\n\\/uni00000013\\/uni00000011\\/uni00000018 \\u03bbl\\n\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018 \\u03bbl\\n\\u03bbl\\nzl\\u2212 0 .125 \\u03b8l\\/uni00000013\\/uni000000030 .125 \\u03b8l\\/tildewidest\\nErrl\\/uni00000013\\/uni00000013\\/uni00000011\\/uni00000015\\/uni00000018 \\u03b8l\\/uni00000013\\/uni00000011\\/uni00000018 \\u03b8l\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018 \\u03b8l\\u03b8l\\u03c6l\\n( T ) \\/uni00000003\\/uni0000003f\\/uni00000003 al\\n(c)L= 4,T= 8,\\u03d5=0.5\\nFigure 2: Comparison of SNN output \\u03c6l(T)and ANN output alwith same input zl\\n4 O PTIMAL ANN-SNN CONVERSION\\n4.1 QUANTIZATION CLIP -FLOOR ACTIVATION FUNCTION\\nAccording to the conversion error of Equation 12, it is natur al to think that if the commonly used\\nReLU activation function h(zl)is substituted by a clip-\\ufb02oor function with a given quantiza tion\\nstepsL(similar to Equation 11), the conversion error at time-step sT=Lwill be eliminated. Thus\\nthe performance degradation problem at low latency will be s olved. As shown in Equation 13, we\\nproposed the quantization clip-\\ufb02oor activation function t o train ANNs.\\nal=\\u00afh(zl) =\\u03bblclip\\/parenleftbigg1\\nL\\/floorleftbiggzlL\\n\\u03bbl\\/floorrightbigg\\n,0,1\\/parenrightbigg\\n, (13)\\nwhere the hyperparameter Ldenotes quantization steps of ANNs, the trainable \\u03bbldecides the\\nmaximum value of alin ANNs mapped to the maximum of \\u03c6l(T)in SNNs. Note that zl=\\nWl\\u03c6l\\u22121(T) =Wlal\\u22121. With this new activation function, we can prove that the est imated con-\\nversion error between SNNs and ANNs is zero, and we have the fo llowing Theorem.\\nTheorem 1. An ANN with activation function (13) is converted to an SNN wi th the same weights. If\\nT=L,\\u03b8l=\\u03bbl, andvl(0) =0, then:\\n\\/tildewidestErrl=\\u03c6l(T)\\u2212al=0. (14)\\nProof. According to Equation 12, and the conditions T=L,\\u03b8l=\\u03bbl,vl(0) =0, we have \\/tildewidestErrl=\\n\\u03c6l(T)\\u2212al=\\u03b8lclip\\/parenleftBig\\n1\\nT\\/floorleftBig\\nzlT+vl(0)\\n\\u03b8l\\/floorrightBig\\n,0,1\\/parenrightBig\\n\\u2212\\u03bblclip\\/parenleftBig\\n1\\nL\\/floorleftBig\\nzlL\\n\\u03bbl\\/floorrightBig\\n,0,1\\/parenrightBig\\n= 0.\\nTheorem 1 implies that if the time-steps Tof the converted SNN is the same as the quantization\\nstepsLof the source ANN, the conversion error will be zero. An examp le is illustrated in Figure 2a,\\nwhereT=L= 4,\\u03b8l=\\u03bbl. The red curve presents the estimated output \\u03c6l(T)of the converted\\nSNNs with respective to different input zl, while the green curve represents the out alof the source\\nANN with respective to different input zl. As the two curve are the same, the estimated conversion\\nerror\\/tildewidestErrlis zero. Nevertheless, in practical application, we focus o n the performance of SNNs\\nat different time-steps. There is no guarantee that the conv ersion error is zero when Tis not equal\\ntoL. As illustrated in Figure 2b, where L= 4 andL= 8, we can \\ufb01nd the conversion error is\\ngreater than zero for some zl. This error will transmit layer-by-layer and eventually de grading the\\naccuracy of the converted SNN. One way to solve this problem i s to train multiple source ANNs\\nwith different quantization steps, then convert them to SNN s with different time-steps, but it comes\\nat a considerable cost. In the next section, we propose the qu antization clip-\\ufb02oor activation function at a considerable cost. In the next section, we propose the qu antization clip-\\ufb02oor activation function\\nwith a shift term to solve this problem. Such an approach can a chieve high accuracy for different\\ntime-steps, without extra computation cost.\\n4.2 QUANTIZATION CLIP -FLOOR -SHIFT ACTIVATION FUNCTION\\nWe propose the quantization clip-\\ufb02oor-shift activation fu nction to train ANNs.\\nal=\\/hatwideh(zl) =\\u03bblclip\\/parenleftbigg1\\nL\\/floorleftbiggzlL\\n\\u03bbl+\\u03d5\\/floorrightbigg\\n,0,1\\/parenrightbigg\\n. (15)\\n5 Published as a conference paper at ICLR 2022\\nCompared with Equation 13, there exists a hyperparameter ve ctor\\u03d5that controls the shift of the\\nactivation function. When L\\/negationslash=T, we cannot guarantee the conversion error is 0. However, we\\ncan estimate the expectation of conversion error. Similar t o (Deng & Gu, 2020), we assume that\\nzl\\niis uniformly distributed within intervals [(t\\u22121)\\u03bbl\\/T,(t)\\u03bbl\\/T]and[(l\\u22121)\\u03bbl\\/L,(l)\\u03bbl\\/L]for\\nt= 1,2,...,T andL= 1,2,...,L , we have the following Theorem.\\nTheorem 2. An ANN with activation function (15) is converted to an SNN wi th the same weights.\\nIf\\u03b8l=\\u03bbl,vl(0) =\\u03b8l\\u03d5, then for arbitrary TandL, the expectation of conversion error reaches 0\\nwhen the shift term \\u03d5in source ANN is1\\n2.\\n\\u2200T,LEz\\/parenleftBig\\n\\/tildewidestErrl\\/parenrightBig\\/vextendsingle\\/vextendsingle\\/vextendsingle\\n\\u03d5=1\\n2=0. (16)\\nThe proof is in the Appendix. Theorem 2 indicates that the shi ft term1\\n2is able to optimize the\\nexpectation of conversion error. By comparing Figure 2b and Figure 2c, we can \\ufb01nd that when the\\nshift term \\u03d5= 0.5is added, the mean conversion error reaches zero, even thoug hL\\/negationslash=T. These\\nresults indicate we can achieve high-performance converte d SNN at ultra-low time-steps.\\nLis the only undetermined hyperparameter of the quantizatio n clip-\\ufb02oor-shift activation. When\\nT=L, the conversion error reaches zero. So we naturally think th at the parameter Lshould be set\\nas small as possible to get better performance at low time-st eps. However, a too low quantization\\nof the activation function will decrease the model capacity and further lead to accuracy loss when\\nthe time-steps is relatively large. Choosing the proper Lis a trade-off between the accuracy at low\\nlatency and the best accuracy of SNNs. We will further analyz e the effects of quantization steps L\\nin the experiment section.\\n4.3 ALGORITHM FOR TRAINING QUANTIZATION CLIP -FLOOR -SHIFT ACTIVATION FUNCTION\\nTraining an ANN with quantization clip-\\ufb02oor-shift activat ion instead of ReLU is also a tough prob-\\nlem. To direct train the ANN, we use the straight-through est imator (Bengio et al., 2013) for the\\nderivative of the \\ufb02oor function, that isd\\u230ax\\u230b\\ndx= 1. The overall derivation rule is given in Equation 17.\\n\\u2202\\/hatwidehi(zl)\\n\\u2202zl\\ni=\\/braceleftBigg\\n1,\\u2212\\u03bbl\\n2L< zl\\ni< \\u03bbl\\u2212\\u03bbl\\n2L\\n0,otherwise,\\u2202\\/hatwidehi(zl)\\n\\u2202\\u03bbl=\\uf8f1\\n\\uf8f4\\uf8f2\\n\\uf8f4\\uf8f3\\/hatwidehi(zl)\\u2212zl\\ni\\n\\u03bbl,\\u2212\\u03bbl\\n2L\\/lessorequalslantzl\\ni< \\u03bbl\\u2212\\u03bbl\\n2L\\n0, zl\\ni<\\u2212\\u03bbl\\n2L\\n1, zl\\ni\\/greaterorequalslant\\u03bbl\\u2212\\u03bbl\\n2L(17)\\nHerezl\\niis the i-th element of zl. Then we can train the ANN with quantization clip-\\ufb02oor-shif t\\nactivation using Stochastic Gradient Descent algorithm (B ottou, 2012).\\n5 R ELATED WORK\\nThe study of ANN-SNN conversion is \\ufb01rst launched by Cao et al. (2015). Then Diehl et al. (2015)\\nconverted a three-layer CNN to an SNN using data-based and mo del-based normalization. To ob-\\ntain high-performance SNNs for complex datasets and deeper networks, Rueckauer et al. (2016)\\nand Sengupta et al. (2019) proposed more accurate scaling me thods to normalize weights and scale\\nthresholds respectively, which were later proved to be equi valent (Ding et al., 2021). Neverthe-\\nless, the converted deep SNN requires hundreds of time steps to get accurate results due to the\\nconversion error analyzed in Sec. 3. To address the potentia l information loss, Rueckauer et al.\\n(2016) and Han et al. (2020) suggested using \\u201creset-by-subt raction\\u201d neurons rather than \\u201creset-\\nto-zero\\u201d neurons. Recently, many methods have been propose d to eliminate the conversion\\nerror. Rueckauer et al. (2016) recommended 99.9% percentil e of activations as scale factors,\\nand Ho & Chang (2020) added the trainable clipping layer. Bes ides, Han et al. (2020) rescaled\\nthe SNN thresholds to avoid the improper activation of spiki ng neurons. Massa et al. (2020) and\\nSingh et al. (2021) evaluated the performance of converted S NNs on the Loihi Neuromorphic Pro-\\ncessor. Our work share similarity with Deng & Gu (2020); Li et al. (2021), which also shed light Singh et al. (2021) evaluated the performance of converted S NNs on the Loihi Neuromorphic Pro-\\ncessor. Our work share similarity with Deng & Gu (2020); Li et al. (2021), which also shed light\\non the conversion error. Deng & Gu (2020) minimized the layer -wise error by introducing ex-\\ntra bias in addition to the converted SNN biases. Li et al. (20 21) further proposed calibration for\\nweights and biases using quantized \\ufb01ne-tuning. They got goo d results with 16 and 32 time-steps\\n6 Published as a conference paper at ICLR 2022\\n\\/uni00000015 \\/uni00000017 \\/uni0000001b \\/uni00000014\\/uni00000019 \\/uni00000016\\/uni00000015\\n\\/uni00000034\\/uni00000058\\/uni00000044\\/uni00000051\\/uni00000057\\/uni0000004c\\/uni0000005d\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000003\\/uni0000002f\\/uni00000013\\/uni00000011\\/uni0000001b\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001c\\/uni00000013\\/uni00000011\\/uni0000001c\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c\\n\\/uni0000000b\\/uni00000044\\/uni0000000c\\/uni00000003\\/uni00000039\\/uni0000002a\\/uni0000002a\\/uni00000010\\/uni00000014\\/uni00000019\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni0000005a\\/uni00000012\\/uni00000003\\/uni00000056\\/uni0000004b\\/uni0000004c\\/uni00000049\\/uni00000057\\n\\/uni0000005a\\/uni00000012\\/uni00000052\\/uni00000003\\/uni00000056\\/uni0000004b\\/uni0000004c\\/uni00000049\\/uni00000057\\n\\/uni00000015 \\/uni00000017 \\/uni0000001b \\/uni00000014\\/uni00000019 \\/uni00000016\\/uni00000015\\n\\/uni00000034\\/uni00000058\\/uni00000044\\/uni00000051\\/uni00000057\\/uni0000004c\\/uni0000005d\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000003\\/uni0000002f\\/uni00000013\\/uni00000011\\/uni0000001b\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001c\\/uni00000013\\/uni00000011\\/uni0000001c\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c\\n\\/uni0000000b\\/uni00000045\\/uni0000000c\\/uni00000003\\/uni00000035\\/uni00000048\\/uni00000056\\/uni00000031\\/uni00000048\\/uni00000057\\/uni00000010\\/uni00000015\\/uni00000013\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni00000015 \\/uni00000017 \\/uni0000001b \\/uni00000014\\/uni00000019 \\/uni00000016\\/uni00000015\\n\\/uni00000034\\/uni00000058\\/uni00000044\\/uni00000051\\/uni00000057\\/uni0000004c\\/uni0000005d\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000003\\/uni0000002f\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c\\n\\/uni0000000b\\/uni00000046\\/uni0000000c\\/uni00000003\\/uni00000039\\/uni0000002a\\/uni0000002a\\/uni00000010\\/uni00000014\\/uni00000019\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni00000013\\/uni00000015 \\/uni00000017 \\/uni0000001b \\/uni00000014\\/uni00000019 \\/uni00000016\\/uni00000015\\n\\/uni00000034\\/uni00000058\\/uni00000044\\/uni00000051\\/uni00000057\\/uni0000004c\\/uni0000005d\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000003\\/uni0000002f\\/uni00000013\\/uni00000011\\/uni00000018\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000019\\/uni00000013\\/uni00000011\\/uni00000019\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c\\n\\/uni0000000b\\/uni00000047\\/uni0000000c\\/uni00000003\\/uni00000035\\/uni00000048\\/uni00000056\\/uni00000031\\/uni00000048\\/uni00000057\\/uni00000010\\/uni00000015\\/uni00000013\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni00000013\\nFigure 3: Compare ANNs accuracy.\\nwithout trails for more extreme time-steps. In comparison, our work aims to \\ufb01t ANN into SNN\\nwith techniques eliminating the mentioned conversion erro r. The end-to-end training of quantiza-\\ntion layers is implemented to get better overall performanc e. Our shift correction can lead to a\\nsingle SNN which performs well at both ultra-low and large ti me-steps. Maintaining SNN per- tion layers is implemented to get better overall performanc e. Our shift correction can lead to a\\nsingle SNN which performs well at both ultra-low and large ti me-steps. Maintaining SNN per-\\nformance within extremely few time-steps is dif\\ufb01cult even f or supervised learning methods like\\nbackpropagation through time (BPTT). BPTT usually require s fewer time-steps because of thor-\\nough training, yet at the cost of heavy GPU computation (Wu et al., 2018; 2019; Lee et al., 2016;\\nNeftci et al., 2019; Lee et al., 2020; Zenke & V ogels, 2021). T he timing-based backpropagation\\nmethods (Bohte et al., 2002; Tavanaei et al., 2019; Kim et al. , 2020) could train SNNs over a very\\nshort temporal window, e.g. over 5-10 time-steps. However, they are usually limited to sim-\\nple datasets like MNIST (Kheradpisheh & Masquelier, 2020) a nd CIFAR10 (Zhang & Li, 2020).\\nRathi et al. (2019) shortened simulation steps by initializ ing SNN with conversion method and then\\ntuning SNN with STDP. In this paper, the proposed method achi eves high-performance SNNs with\\nultra-low latency (4 time-steps).\\n6 E XPERIMENTS\\nIn this section, we validate the effectiveness of our method and compare our method with other\\nstate-of-the-art approaches for image classi\\ufb01cation task s on CIFAR-10 (LeCun et al., 1998), CIFAR-\\n100 (Krizhevsky et al., 2009), and ImageNet datasets (Deng e t al., 2009). Similar to previous works,\\nwe utilize VGG-16 (Simonyan & Zisserman, 2014), ResNet-18 ( He et al., 2016), and ResNet-20\\nnetwork structures for source ANNs. We compare our method wi th the state-of-the-art ANN-\\nSNN conversion methods, including Hybrid-Conversion (HC) from Rathi et al. (2019), RMP from\\nHan et al. (2020), TSC from Han & Roy (2020), RNL from Ding et al . (2021), ReLUThreshold-\\nShift (RTS) from Deng & Gu (2020), and SNN Conversion with Adv anced Pipeline (SNNC-AP)\\nfrom Li et al. (2021). Comparison with different SNN trainin g methods is also included to mani-\\nfest the superiority of low latency inference, including Hy bridConversion-STDB (HC-STDB) from\\nRathi et al. (2019), STBP from Wu et al. (2018), DirectTraini ng (DT) from Wu et al. (2019), and\\nTSSL from Zhang & Li (2020). The details of the proposed ANN-S NN algorithm and training con-\\n\\ufb01gurations are provided in the Appendix.\\n6.1 T EST ACCURACY OF ANN WITH QUANTIZATION CLIP -FLOOR -SHIFT ACTIVATION\\nWe \\ufb01rst compare the performance of ANNs with quantization cl ip-\\ufb02oor activation (green curve),\\nANNs with quantization clip-\\ufb02oor-shift activation (blue c urve), and original ANNs with ReLU acti-\\nvation (black dotted line). Figure 3(a)-(d) report the resu lts about VGG-16 on CIFAR-10, ResNet-20\\non CIFAR-10, VGG-16 on CIFAR-100 and ResNet-20 on CIFAR-100 . The performance of ANNs\\nwith quantization clip-\\ufb02oor-shift activation is better th an ANNs with quantization clip-\\ufb02oor activa-\\ntion. These two ANNs can achieve the same performance as orig inal ANNs with ReLU activation\\nwhenL >4. These results demonstrate that our quantization clip-\\ufb02oo r-shift activation function\\nhardly affects the performance of ANN.\\n6.2 C OMPARISON WITH THE STATE -OF-THE-ART\\nTable 2 compares our method with the state-of-the-art ANN-S NN conversion methods on CIFAR-\\n10. As for low latency inference (T \\u226464), our model outperforms all the other methods with the\\nsame time-step setting. For T = 32 , the accuracy of our method is slightly better than that of AN N\\n(95.54% vs. 95.52%), whereas RMP, RTS, RNL, and SNNC-AP meth ods have accuracy loss of\\n33.3%, 19.48%, 7.42%, and 2.01%. Moreover, we achieve an acc uracy of 93.96% using only 4\\ntime-steps, which is 8 times faster than SNNC-AP that takes 3 2 time-steps. For ResNet-20, we\\nachieve an accuracy of 83.75% with 4 time-steps. Notably, ou r ultra-low latency performance is\\n7 Published as a conference paper at ICLR 2022\\n\\/uni00000014\\/uni00000015\\/uni00000017\\/uni0000001b\\/uni00000014\\/uni00000019\\/uni00000016\\/uni00000015\\/uni00000019\\/uni00000017\\/uni00000014\\/uni00000015\\/uni0000001b\\n\\/uni00000036\\/uni0000004c\\/uni00000050\\/uni00000058\\/uni0000004f\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000057\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000056\\/uni00000013\\/uni00000011\\/uni00000014\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000016\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000018\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001c\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c\\n\\/uni0000000b\\/uni00000044\\/uni0000000c\\/uni00000003\\/uni00000039\\/uni0000002a\\/uni0000002a\\/uni00000010\\/uni00000014\\/uni00000019\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni0000005a\\/uni00000012\\/uni00000003\\/uni00000056\\/uni0000004b\\/uni0000004c\\/uni00000049\\/uni00000057\\n\\/uni0000005a\\/uni00000012\\/uni00000052\\/uni00000003\\/uni00000056\\/uni0000004b\\/uni0000004c\\/uni00000049\\/uni00000057\\n\\/uni00000014 \\/uni00000015 \\/uni00000017 \\/uni0000001b \\/uni00000014\\/uni00000019 \\/uni00000016\\/uni00000015 \\/uni00000019\\/uni00000017 \\/uni00000014\\/uni00000015\\/uni0000001b\\n\\/uni00000036\\/uni0000004c\\/uni00000050\\/uni00000058\\/uni0000004f\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000057\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000056\\/uni00000013\\/uni00000011\\/uni00000014\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000016\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000018\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001c\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c\\n\\/uni0000000b\\/uni00000045\\/uni0000000c\\/uni00000003\\/uni00000035\\/uni00000048\\/uni00000056\\/uni00000031\\/uni00000048\\/uni00000057\\/uni00000010\\/uni00000015\\/uni00000013\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni00000014\\/uni00000015\\/uni00000017\\/uni0000001b\\/uni00000014\\/uni00000019\\/uni00000016\\/uni00000015\\/uni00000019\\/uni00000017\\/uni00000014\\/uni00000015\\/uni0000001b\\n\\/uni00000036\\/uni0000004c\\/uni00000050\\/uni00000058\\/uni0000004f\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000057\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000056\\/uni00000013\\/uni00000011\\/uni00000014\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000016\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000018\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c\\n\\/uni0000000b\\/uni00000046\\/uni0000000c\\/uni00000003\\/uni00000039\\/uni0000002a\\/uni0000002a\\/uni00000010\\/uni00000014\\/uni00000019\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni00000013\\/uni00000014\\/uni00000015\\/uni00000017\\/uni0000001b\\/uni00000014\\/uni00000019\\/uni00000016\\/uni00000015\\/uni00000019\\/uni00000017\\/uni00000014\\/uni00000015\\/uni0000001b\\n\\/uni00000036\\/uni0000004c\\/uni00000050\\/uni00000058\\/uni0000004f\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000057\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000056\\/uni00000013\\/uni00000011\\/uni00000014\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000016\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000018\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c \\/uni0000000b\\/uni00000047\\/uni0000000c\\/uni00000003\\/uni00000035\\/uni00000048\\/uni00000056\\/uni00000031\\/uni00000048\\/uni00000057\\/uni00000010\\/uni00000015\\/uni00000013\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni00000013\\nFigure 4: Compare quantization clip-\\ufb02oor activation with\\/ without shift term\\nTable 2: Comparison between the proposed method and previou s works on CIFAR-10 dataset.\\nArchitecture Method ANN T=2 T=4 T=8 T=16 T=32 T=64 T \\u2265512\\nVGG-16RMP 93.63% - - - - 60.30% 90.35% 93.63%\\nTSC 93.63% - - - - - 92.79% 93.63%\\nRTS 95.72% - - - - 76.24% 90.64% 95.73%\\nRNL 92.82% - - - 57.90% 85.40% 91.15% 92.95%\\nSNNC-AP 95.72% - - - - 93.71% 95.14% 95.79%\\nOurs 95.52% 91.18% 93.96% 94.95% 95.40% 95.54% 95.55% 95.59%\\nResNet-20RMP 91.47% - - - - - - 91.36%\\nTSC 91.47% - - - - - 69.38% 91.42%\\nOurs 91.77% 73.20% 83.75% 89.55% 91.62% 92.24% 92.35% 92.41%\\nResNet-18RTS195.46% - - - - 84.06% 92.48% 94.42%\\nSNNC-AP195.46% - - - - 94.78% 95.30% 95.45%\\nOurs 96.04% 75.44% 90.43% 94.82% 95.92% 96.08% 96.06% 96.06%\\n1RTS and SNNC-AP use altered ResNet-18, while ours use standa rd ResNet-18.\\ncomparable with other state-of-the-art supervised traini ng methods, which is shown in Table S3 of\\nthe Appendix.\\nWe further test the performance of our method on the large-sc ale dataset. Table 3 reports the results\\non ImageNet, our method also outperforms the others both in t erms of high accuracy and ultra-low\\nlatency. For ResNet-34, the accuracy of the proposed method is 4.83% higher than SNNC-AP and\\n69.28% higher than RTS when T= 32 . When the time-steps is 16, we can still achieve an accuracy\\nof 59.35%. For VGG-16, the accuracy of the proposed method is 4.83% higher than SNNC-AP\\nand 68.356% higher than RTS when T= 32 . When the time-steps is 16, we can still achieve an\\naccuracy of 50.97%. These results demonstrate that our meth od outperforms the previous conversion\\nmethods. More experimental results on CIFAR-100 is in Table S4 of the Appendix.\\n6.3 C OMPARISON OF QUANTIZATION CLIP -FLOOR AND QUANTIZATION CLIP -FLOOR -SHIFT\\nHere we further compare the performance of SNNs converted fr om ANNs with quantization clip-\\n\\ufb02oor activation and ANN with quantization clip-\\ufb02oor-shift activation. In Sec. 4, we prove that\\nthe expectation of the conversion error reaches 0 with quant ization clip-\\ufb02oor-shift activation, no\\nmatter whether TandLare the same or not. To verify these, we set Lto 4 and train ANNs with\\nquantization clip-\\ufb02oor activation and quantization clip- \\ufb02oor-shift activation, respectively. Figure 4\\nshows how the accuracy of converted SNNs changes with respec t to the time-steps T. The accuracy\\nof the converted SNN (green curve) from ANN with quantizatio n clip-\\ufb02oor activation (green dotted\\nline) \\ufb01rst increases and then decreases rapidly with the inc rease of time-steps, because we cannot\\nguarantee that the conversion error is zero when Tis not equal to L. The best performance is still\\nlower than source ANN (green dotted line). In contrast, the a ccuracy of the converted SNN from\\nANN with quantization clip-\\ufb02oor-shift activation (blue cu rve) increases with the increase of T. It\\ngets the same accuracy as source ANN (blue dotted line) when t he time-steps is larger than 16.\\n6.4 E FFECT OF QUANTIZATION STEPS L\\nIn our method, the quantization steps Lis a hyperparameter, which affects the accuracy of the con-\\nverted SNN. To analyze the effect of Land better determine the optimal value, we train VGG-\\n16\\/ResNet-20 networks with quantization clip-\\ufb02oor-shift activation using different quantization\\nsteps L, including 2,4,8,16 and 32, and then converted them t o SNNs. The experimental results\\n8 Published as a conference paper at ICLR 2022\\n\\/uni00000014\\/uni00000015\\/uni00000017\\/uni0000001b\\/uni00000014\\/uni00000019\\/uni00000016\\/uni00000015\\/uni00000019\\/uni00000017\\/uni00000014\\/uni00000015\\/uni0000001b\\n\\/uni00000036\\/uni0000004c\\/uni00000050\\/uni00000058\\/uni0000004f\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000057\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000056\\/uni00000013\\/uni00000011\\/uni00000014\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000016\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000018\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001c\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c\\n\\/uni0000000b\\/uni00000044\\/uni0000000c\\/uni00000003\\/uni00000039\\/uni0000002a\\/uni0000002a\\/uni00000010\\/uni00000014\\/uni00000019\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni0000002f\\/uni00000020\\/uni00000015\\n\\/uni0000002f\\/uni00000020\\/uni00000017\\n\\/uni0000002f\\/uni00000020\\/uni0000001b\\n\\/uni0000002f\\/uni00000020\\/uni00000014\\/uni00000019\\n\\/uni0000002f\\/uni00000020\\/uni00000016\\/uni00000015\\n\\/uni00000014 \\/uni00000015 \\/uni00000017 \\/uni0000001b \\/uni00000014\\/uni00000019 \\/uni00000016\\/uni00000015 \\/uni00000019\\/uni00000017 \\/uni00000014\\/uni00000015\\/uni0000001b\\n\\/uni00000036\\/uni0000004c\\/uni00000050\\/uni00000058\\/uni0000004f\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000057\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000056\\/uni00000013\\/uni00000011\\/uni00000014\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000016\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000018\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001c\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c\\n\\/uni0000000b\\/uni00000045\\/uni0000000c\\/uni00000003\\/uni00000035\\/uni00000048\\/uni00000056\\/uni00000031\\/uni00000048\\/uni00000057\\/uni00000010\\/uni00000015\\/uni00000013\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni00000014\\/uni00000015\\/uni00000017\\/uni0000001b\\/uni00000014\\/uni00000019\\/uni00000016\\/uni00000015\\/uni00000019\\/uni00000017\\/uni00000014\\/uni00000015\\/uni0000001b\\n\\/uni00000036\\/uni0000004c\\/uni00000050\\/uni00000058\\/uni0000004f\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000057\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000056\\/uni00000013\\/uni00000011\\/uni00000014\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000016\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000018\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c\\n\\/uni0000000b\\/uni00000046\\/uni0000000c\\/uni00000003\\/uni00000039\\/uni0000002a\\/uni0000002a\\/uni00000010\\/uni00000014\\/uni00000019\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni00000013\\/uni00000014\\/uni00000015\\/uni00000017\\/uni0000001b\\/uni00000014\\/uni00000019\\/uni00000016\\/uni00000015\\/uni00000019\\/uni00000017\\/uni00000014\\/uni00000015\\/uni0000001b\\n\\/uni00000036\\/uni0000004c\\/uni00000050\\/uni00000058\\/uni0000004f\\/uni00000044\\/uni00000057\\/uni0000004c\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000057\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053\\/uni00000056\\/uni00000013\\/uni00000011\\/uni00000014\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000016\\/uni00000018\\/uni00000013\\/uni00000011\\/uni00000018\\/uni00000018\\/uni00000013\\/uni00000011\\/uni0000001a\\/uni00000018\\/uni00000024\\/uni00000046\\/uni00000046\\/uni00000058\\/uni00000055\\/uni00000044\\/uni00000046\\/uni0000005c \\/uni0000000b\\/uni00000047\\/uni0000000c\\/uni00000003\\/uni00000035\\/uni00000048\\/uni00000056\\/uni00000031\\/uni00000048\\/uni00000057\\/uni00000010\\/uni00000015\\/uni00000013\\/uni00000003\\/uni00000052\\/uni00000051\\/uni00000003\\/uni00000026\\/uni0000002c\\/uni00000029\\/uni00000024\\/uni00000035\\/uni00000010\\/uni00000014\\/uni00000013\\/uni00000013\\nFigure 5: In\\ufb02uence of different quantization steps\\nTable 3: Comparison between the proposed method and previou s works on ImageNet dataset.\\nArchitecture Method ANN T=16 T=32 T=64 T=128 T=256 T \\u22651024\\nResNet-34RMP 70.64% - - - - - 65.47%\\nTSC 70.64% - - - - 61.48% 65.10%\\nRTS 75.66% - 0.09% 0.12% 3.19% 47.11% 75.08%\\nSNNC-AP 75.66% - 64.54% 71.12% 73.45% 74.61% 75.45%\\nOurs 74.32% 59.35% 69.37% 72.35% 73.15% 73.37% 73.39%\\nVGG-16RMP 73.49% - - - - 48.32% 73.09%\\nTSC 73.49% - - - - 69.71% 73.46%\\nRTS 75.36% - 0.114% 0.118% 0.122% 1.81% 73.88%\\nSNNC-AP 75.36% - 63.64% 70.69% 73.32% 74.23% 75.32%\\nOurs 74.29% 50.97% 68.47% 72.85% 73.97% 74.22% 74.32%\\non CIFAR-10\\/100 dataset are shown in Table S2 and Figure 5, wh ere the black dotted line denotes\\nthe ANN accuracy and the colored curves represent the accura cy of the converted SNN. In order to\\nbalance the trade-off between low latency and high accuracy , we evaluate the performance of con-\\nverted SNN mainly in two aspects. First, we focus on the SNN ac curacy at ultra-low latency (within\\n4 time-steps). Second, we consider the best accuracy of SNN. It is obvious to \\ufb01nd that the SNN\\naccuracy at ultra-low latency decreases as Lincreases. However, a too small Lwill decrease the\\nmodel capacity and further lead to accuracy loss. When L= 2, there exists a clear gap between the\\nbest accuracy of SNN and source ANN. The best accuracy of SNN a pproaches source ANN when\\nL >4. In conclusion, the setting of parameter Lmainly depends on the aims for low latency or best\\naccuracy. The recommend quantization step Lis 4 or 8, which leads to high-performance converted\\nSNN at both small time-steps and very large time-steps.\\n7 D ISCUSSION AND CONCLUSION\\nIn this paper, we present ANN-SNN conversion method, enabli ng high-accuracy and ultra-low-\\nlatency deep SNNs. We propose the quantization clip-\\ufb02oor-s hift activation to replace ReLU activa-\\ntion, which hardly affects the performance of ANNs and is clo ser to SNNs activation. Furthermore,\\nwe prove that the expected conversion error is zero, no matte r whether the time-steps of SNNs and\\nthe quantization steps of ANNs is the same or not. We achieve s tate-of-the-art accuracy with fewer\\ntime-steps on CIFAR-10, CIFAR-100, and ImageNet datasets. Our results can bene\\ufb01t the imple-\\nmentations on neuromorphic hardware and pave the way for the large-scale application of SNNs.\\nDifferent from the work of Deng & Gu (2020), which adds the bia s of the converted SNNs to shift\\nthe theoretical ANN-SNN curve to minimize the quantization error, we add the shift term in the\\nquantization clip-\\ufb02oor activation function, and use this q uantization clip-\\ufb02oor-shift function to train\\nthe source ANN. We show that the shift term can overcome the pe rformance degradation problem\\nwhen the time-steps and the quantization steps are not match ed. Due to the unevenness error, there\\nstill exists a gap between ANN accuracy and SNN accuracy, eve n whenL=T. Moreover, it is hard\\nto achieve high-performance ANN-SNN conversion when the ti me-steps T= 1. All these problems\\ndeserve further research. One advantage of conversion-bas ed methods is that they can reduce the\\noverall computing cost while maintaining comparable perfo rmance as source ANN. Combining the\\nconversion-based methods and model compression may help si gni\\ufb01cantly reduce the neuron activity\\n9 Published as a conference paper at ICLR 2022\\nand thus reduce energy consumptions without suffering from accuracy loss (Kundu et al., 2021;\\nRathi & Roy, 2021), which is a promising direction.\\n10 Published as a conference paper at ICLR 2022\\nACKNOWLEDGEMENT\\nThis work was supported by the National Natural Science Foun dation of China under contracts\\nNo.62176003 and No.62088102.\\nREFERENCES\\nYoshua Bengio, Nicholas L\\u00b4 eonard, and Aaron Courville. Est imating or propagating gradients\\nthrough stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432 , 2013.\\nSander M Bohte, Joost N Kok, and Han La Poutre. Error-backpro pagation in temporally encoded\\nnetworks of spiking neurons. Neurocomputing , 48(1-4):17\\u201337, 2002.\\nL\\u00b4 eon Bottou. Stochastic gradient descent tricks. In Neural networks: Tricks of the trade , pp. 421\\u2013\\n436. Springer, 2012.\\nYongqiang Cao, Yang Chen, and Deepak Khosla. Spiking deep co nvolutional neural networks for\\nenergy-ef\\ufb01cient object recognition. International Journal of Computer Vision , 113(1):54\\u201366,\\n2015.\\nEkin D Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc V Le. Autoaugment:\\nLearning augmentation strategies from data. In IEEE Conference on Computer Vision and Pattern\\nRecognition , pp. 113\\u2013123, 2019.\\nMike Davies, Narayan Srinivasa, Tsung-Han Lin, Gautham Chi nya, Yongqiang Cao, Sri Harsha\\nChoday, Georgios Dimou, Prasad Joshi, Nabil Imam, Shweta Ja in, et al. Loihi: A neuromorphic\\nmanycore processor with on-chip learning. IEEE Micro , 38(1):82\\u201399, 2018.\\nMichael V DeBole, Brian Taba, Arnon Amir, Filipp Akopyan, Al exander Andreopoulos, William P\\nRisk, Jeff Kusnitz, Carlos Ortega Otero, Tapan K Nayak, Rath inakumar Appuswamy, et al.\\nTrueNorth: Accelerating from zero to 64 million neurons in 1 0 years. Computer , 52(5):20\\u201329,\\n2019.\\nJia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li F ei-Fei. Imagenet: A large-scale\\nhierarchical image database. In IEEE Conference on Computer Vision and Pattern Recognition ,\\npp. 248\\u2013255. Ieee, 2009.\\nShikuang Deng and Shi Gu. Optimal conversion of conventiona l arti\\ufb01cial neural networks to spiking\\nneural networks. In International Conference on Learning Representations , 2020.\\nTerrance DeVries and Graham W Taylor. Improved regularizat ion of convolutional neural networks\\nwith cutout. arXiv preprint arXiv:1708.04552 , 2017.\\nPeter U Diehl, Daniel Neil, Jonathan Binas, Matthew Cook, Sh ih-Chii Liu, and Michael Pfeiffer.\\nFast-classifying, high-accuracy spiking deep networks th rough weight and threshold balancing.\\nInInternational Joint Conference on Neural Networks , pp. 1\\u20138, 2015.\\nJianhao Ding, Zhaofei Yu, Yonghong Tian, and Tiejun Huang. O ptimal ann-snn conversion for fast\\nand accurate inference in deep spiking neural networks. In International Joint Conference on\\nArti\\ufb01cial Intelligence , pp. 2328\\u20132336, 2021.\\nWei Fang, Zhaofei Yu, Yanqi Chen, Tiejun Huang, Timoth\\u00b4 ee Ma squelier, and Yonghong Tian. Deep\\nresidual learning in spiking neural networks. arXiv preprint arXiv:2102.04159 , 2021.\\nBing Han and Kaushik Roy. Deep spiking neural network: Energ y ef\\ufb01ciency through time based\\ncoding. In European Conference on Computer Vision , pp. 388\\u2013404, 2020.\\nBing Han, Gopalakrishnan Srinivasan, and Kaushik Roy. RMP- SNN: Residual membrane poten-\\ntial neuron for enabling deeper high-accuracy and low-late ncy spiking neural network. In IEEE\\nConference on Computer Vision and Pattern Recognition , pp. 13558\\u201313567, 2020.\\nKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep r esidual learning for image recog-\\nnition. In IEEE conference on Computer Vision and Pattern Recognition , pp. 770\\u2013778, 2016.\\n11 Published as a conference paper at ICLR 2022\\nNguyen-Dong Ho and Ik-Joon Chang. Tcl: an ann-to-snn conver sion with trainable clipping layers.\\narXiv preprint arXiv:2008.04509 , 2020.\\nEugene M Izhikevich. Simple model of spiking neurons. IEEE Transactions on neural networks ,\\n14(6):1569\\u20131572, 2003.\\nSaeed Reza Kheradpisheh and Timoth\\u00b4 ee Masquelier. Tempora l backpropagation for spiking neural\\nnetworks with one spike per neuron. International Journal of Neural Systems , 30(06):2050027,\\n2020.\\nJinseok Kim, Kyungsu Kim, and Jae-Joon Kim. Unifying activa tion- and timing-based learning\\nrules for spiking neural networks. In Advances in Neural Information Processing Systems , pp.\\n19534\\u201319544, 2020.\\nAlex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images.\\n2009.\\nSouvik Kundu, Gourav Datta, Massoud Pedram, and Peter A Beer el. Spike-thrift: Towards energy-\\nef\\ufb01cient deep spiking neural networks by limiting spiking a ctivity via attention-guided compres-\\nsion. In Proceedings of the IEEE\\/CVF Winter Conference on Applicati ons of Computer Vision\\n(WACV) , pp. 3953\\u20133962, 2021.\\nYann LeCun, L\\u00b4 eon Bottou, Yoshua Bengio, and Patrick Haffne r. Gradient-based learning applied to\\ndocument recognition. Proceedings of the IEEE , 86(11):2278\\u20132324, 1998.\\nChankyu Lee, Syed Shakib Sarwar, Priyadarshini Panda, Gopa lakrishnan Srinivasan, and Kaushik\\nRoy. Enabling spike-based backpropagation for training de ep neural network architectures. Fron-\\ntiers in Neuroscience , 14, 2020.\\nJun Haeng Lee, Tobi Delbruck, and Michael Pfeiffer. Trainin g deep spiking neural networks using\\nbackpropagation. Frontiers in Neuroscience , 10:508, 2016.\\nYuhang Li, Shikuang Deng, Xin Dong, Ruihao Gong, and Shi Gu. A free lunch from ann: Towards\\nef\\ufb01cient, accurate spiking neural networks calibration. I nInternational Conference on Machine\\nLearning , pp. 6316\\u20136325, 2021.\\nIlya Loshchilov and Frank Hutter. Sgdr: Stochastic gradien t descent with warm restarts. In Interna-\\ntional Conference on Learning Representations , 2016.\\nWolfgang Maass. Networks of spiking neurons: the third gene ration of neural network models.\\nNeural Networks , 10(9):1659\\u20131671, 1997.\\nRiccardo Massa, Alberto Marchisio, Maurizio Martina, and M uhammad Sha\\ufb01que. An ef\\ufb01cient\\nspiking neural network for recognizing gestures with a DVS c amera on the Loihi neuromorphic\\nprocessor. In International Joint Conference on Neural Networks , pp. 1\\u20139, 2020.\\nWarren S McCulloch and Walter Pitts. A logical calculus of th e ideas immanent in nervous activity.\\nThe Bulletin of Mathematical Biophysics , 5(4):115\\u2013133, 1943.\\nPaul A Merolla, John V Arthur, Rodrigo Alvarez-Icaza, Andre w S Cassidy, Jun Sawada, Filipp\\nAkopyan, Bryan L Jackson, Nabil Imam, Chen Guo, Yutaka Nakam ura, et al. A million spiking-\\nneuron integrated circuit with a scalable communication ne twork and interface. Science , 345\\n(6197):668\\u2013673, 2014.\\nEmre O Neftci, Hesham Mostafa, and Friedemann Zenke. Surrog ate gradient learning in spiking\\nneural networks: Bringing the power of gradient-based opti mization to spiking neural networks.\\nIEEE Signal Processing Magazine , 36(6):51\\u201363, 2019.\\nJing Pei, Lei Deng, Sen Song, Mingguo Zhao, Youhui Zhang, Shu ang Wu, Guanrui Wang, Zhe\\nZou, Zhenzhi Wu, Wei He, et al. Towards arti\\ufb01cial general int elligence with hybrid tianjic chip\\narchitecture. Nature , 572(7767):106\\u2013111, 2019.\\nNing Qiao, Hesham Mostafa, Federico Corradi, Marc Osswald, Fabio Stefanini, Dora Sumislawska,\\nand Giacomo Indiveri. A recon\\ufb01gurable on-line learning spi king neuromorphic processor com-\\nprising 256 neurons and 128K synapses. Frontiers in neuroscience , 9:141, 2015.\\n12 Published as a conference paper at ICLR 2022\\nNitin Rathi and Kaushik Roy. Diet-snn: A low-latency spikin g neural network with direct input\\nencoding and leakage and threshold optimization. IEEE Transactions on Neural Networks and\\nLearning Systems , 2021.\\nNitin Rathi, Gopalakrishnan Srinivasan, Priyadarshini Pa nda, and Kaushik Roy. Enabling deep\\nspiking neural networks with hybrid conversion and spike ti ming dependent backpropagation. In\\nInternational Conference on Learning Representations , 2019.\\nKaushik Roy, Akhilesh Jaiswal, and Priyadarshini Panda. To wards spike-based machine intelligence\\nwith neuromorphic computing. Nature , 575(7784):607\\u2013617, 2019.\\nBodo Rueckauer, Iulia-Alexandra Lungu, Yuhuang Hu, and Mic hael Pfeiffer. Theory and\\ntools for the conversion of analog to spiking convolutional neural networks. arXiv preprint\\narXiv:1612.04052 , 2016.\\nBodo Rueckauer, Iulia-Alexandra Lungu, Yuhuang Hu, Michae l Pfeiffer, and Shih-Chii Liu. Con-\\nversion of continuous-valued deep networks to ef\\ufb01cient eve nt-driven networks for image classi\\ufb01-\\ncation. Frontiers in Neuroscience , 11:682, 2017.\\nOlga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjee v Satheesh, Sean Ma, Zhiheng\\nHuang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei.\\nImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision\\n(IJCV) , 115(3):211\\u2013252, 2015. doi: 10.1007\\/s11263-015-0816-y.\\nAbhronil Sengupta, Yuting Ye, Robert Wang, Chiao Liu, and Ka ushik Roy. Going deeper in spiking\\nneural networks: VGG and residual architectures. Frontiers in Neuroscience , 13:95, 2019.\\nKaren Simonyan and Andrew Zisserman. Very deep convolution al networks for large-scale image\\nrecognition. arXiv preprint arXiv:1409.1556 , 2014.\\nSonali Singh, Anup Sarma, Sen Lu, Abhronil Sengupta, Vijayk rishnan Narayanan, and Chita R\\nDas. Gesture-snn: Co-optimizing accuracy, latency and ene rgy of snns for neuromorphic vision\\nsensors. In IEEE\\/ACM International Symposium on Low Power Electronics and Design , pp. 1\\u20136,\\n2021.\\nChristoph St\\u00a8 ockl and Wolfgang Maass. Optimized spiking ne urons can classify images with high\\naccuracy through temporal coding with two spikes. Nature Machine Intelligence , 3(3):230\\u2013238,\\n2021.\\nAmirhossein Tavanaei, Masoud Ghodrati, Saeed Reza Kheradp isheh, Timoth\\u00b4 ee Masquelier, and\\nAnthony Maida. Deep learning in spiking neural networks. Neural Networks , 111:47\\u201363, 2019.\\nYujie Wu, Lei Deng, Guoqi Li, Jun Zhu, and Luping Shi. Spatio- temporal backpropagation for\\ntraining high-performance spiking neural networks. Frontiers in Neuroscience , 12:331, 2018.\\nYujie Wu, Lei Deng, Guoqi Li, Jun Zhu, Yuan Xie, and Luping Shi . Direct training for spiking\\nneural networks: Faster, larger, better. In AAAI Conference on Arti\\ufb01cial Intelligence , pp. 1311\\u2013\\n1318, 2019.\\nFriedemann Zenke and Tim P V ogels. The remarkable robustnes s of surrogate gradient learning\\nfor instilling complex function in spiking neural networks .Neural Computation , 33(4):899\\u2013925,\\n2021.\\nWenrui Zhang and Peng Li. Temporal spike sequence learning v ia backpropagation for deep spiking\\nneural networks. In Advances in Neural Information Processing Systems , pp. 12022\\u201312033, 2020.\\n13 Published as a conference paper at ICLR 2022\\nA A PPENDIX\\nA.1 NETWORK STRUCTURE AND TRAINING CONFIGURATIONS\\nBefore training ANNs, we \\ufb01rst replace max-pooling with aver age-pooling and then replace the\\nReLU activation with the proposed quantization clip-\\ufb02oor- shift activation (Equation 15). After\\ntraining, we copy all weights from the source ANN to the conve rted SNN, and set the threshold\\n\\u03b8lin each layer of the converted SNN equal to the maximum activa tion value \\u03bblof the source ANN\\nin the same layer. Besides, we set the initial membrane poten tialvl(0)in converted SNN as \\u03b8l\\/2to\\nmatch the optimal shift \\u03d5=1\\n2of quantization clip-\\ufb02oor-shift activation in the source A NN.\\nDespite the common data normalization, we use some data pre- processing techniques. For CIFAR\\ndatasets, we resize the images into 32\\u00d732, and for ImageNet dataset, we resize the image into\\n224\\u00d7224. Besides, we use random crop images, Cutout (DeVries & Taylo r, 2017) and AutoAug-\\nment (Cubuk et al., 2019) for all datasets.\\nWe use the Stochastic Gradient Descent optimizer (Bottou, 2 012) with a momentum parameter of\\n0.9. The initial learning rate is set to 0.1 for CIFAR-10 and I mageNet, and 0.02 for CIFAR-100. A\\ncosine decay scheduler (Loshchilov & Hutter, 2016) is used t o adjust the learning rate. We apply a\\n5\\u00d710\\u22124weight decay for CIFAR datasets while applying a 1\\u00d710\\u22124weight decay for ImageNet.\\nWe train all models for 300 epochs. The quantization steps Lis set to 4 when training all the\\nnetworks on CIFAR-10, and VGG-16, ResNet-18 on CIFAR-100 da taset. When training ResNet-20\\non CIFAR-100, the parameter Lis set to 8. When training ResNet-34 and VGG-16 on ImageNet,\\nthe parameter Lis set to 8, 16, respectively. We use constant input when eval uating the converted\\nSNNs.\\nA.2 I NTRODUCTION OF DATASETS\\nCIFAR-10. The CIFAR-10 dataset (Krizhevsky et al., 2009) consists of 6 000032\\u00d732images in\\n10 classes. There are 50000 training images and 10000 test im ages.\\nCIFAR-100. The CIFAR-100 dataset (Krizhevsky et al., 2009) consists of 6000032\\u00d732images in\\n100 classes. There are 50000 training images and 10000 test i mages.\\nImageNet. We use the ILSVRC 2012 dataset (Russakovsky et al., 2015), wh ich consists 1,281,167\\ntraining images and 50000 testing images.\\nA.3 D ERIVATION OF EQUATION 12AND PROOF OF THEOREM 2\\nDerivation of Equation 11\\nSimilar to ,, We de\\ufb01ne\\nul(t) =Wlxl\\u22121(t). (S1)\\nWe useul\\ni(t)andzl\\nito denote the i-th element in vector ul(t)andzl, respectively. To derive\\nEquation 11, some extra assumptions on the relationship bet ween ANN activation value and SNN\\npostsynaptic potentials are needed, which are showed in Equ ation S2.\\n\\uf8f1\\n\\uf8f2\\n\\uf8f3ifzl\\ni<0,then\\u2200t ul\\ni(t)<0,\\nif 0\\/lessorequalslantzl\\ni\\/lessorequalslant\\u03b8l,then\\u2200t0\\/lessorequalslantul\\ni(t)\\/lessorequalslant\\u03b8l,\\nifzl\\ni> \\u03b8l,then\\u2200t ul\\ni(t)> \\u03b8l.(S2)\\nWith the assumption above, we can discuss the \\ufb01ring behavior of the neurons in each time-step.\\nWhenzl\\ni<0orzl\\ni> \\u03b8l, the neuron will never \\ufb01re or \\ufb01re all the time-steps, which me ans\\u03c6l\\ni(T) = 0\\nor\\u03c6l\\ni(T) =\\u03b8l. In this situation, we can use a clip function to denote \\u03c6l\\ni(T).\\n\\u03c6l\\ni(T) = clip(zl\\ni,0,\\u03b8l). (S3)\\n14 Published as a conference paper at ICLR 2022\\nWhen0< zl\\ni< \\u03b8l, every input from the presynaptic neuron in SNNs falls into [0,\\u03b8l], then we have\\n\\u2200t, vl\\ni(t)\\u2208[0,\\u03b8]. We can rewrite Equation 8 into the following equation.\\n\\u03c6l\\ni(T)T\\n\\u03b8l=zl\\niT+vl\\ni(0)\\n\\u03b8l\\u2212vl\\ni(T)\\n\\u03b8l. (S4)\\nConsidering that\\u03c6l\\ni(T)T\\n\\u03b8l=\\/summationtextT\\nt=1sl\\ni(t)\\u2208Nand0<vl\\ni(T)\\n\\u03b8l<1, Equation S4 is changed to:\\n\\u03c6l\\ni(T) =\\u03b8l\\nT\\/floorleftbiggzl\\niT+vl\\ni(0)\\n\\u03b8l\\/floorrightbigg\\n. (S5)\\nWe combine these two situations (Equation S3 and Equation S4 ), and we have:\\n\\u03c6l(T) =\\u03b8lclip\\/parenleftbigg1\\nT\\/floorleftbiggzlT+vl(0)\\n\\u03b8l\\/floorrightbigg\\n,0,1\\/parenrightbigg\\n. (S6)\\nProof of Theorem 2\\nBefore prove Theorem 2, we \\ufb01rst introduce Lemma 1.\\nLemma 1. If random variable x\\u2208[0,\\u03b8]is uniformly distributed in every small interval [mt,mt+1]\\nwith the probability density function pt(t= 0,1,...,T ), wherem0= 0,mT+1=\\u03b8,mt=(t\\u22121\\n2)\\u03b8\\nTfort= 1,2,...,T ,p0=pT, we can conclude that\\nEx\\/parenleftbigg\\nx\\u2212\\u03b8\\nT\\/floorleftbiggTx\\n\\u03b8+1\\n2\\/floorrightbigg\\/parenrightbigg\\n= 0. (S7)\\nProof.\\nEx\\/parenleftbigg\\nx\\u2212\\u03b8\\nT\\/floorleftbiggTx\\n\\u03b8+1\\n2\\/floorrightbigg\\/parenrightbigg\\n=\\/integraldisplay\\u03b8\\/2T\\n0p0\\/parenleftbigg\\nx\\u2212\\u03b8\\nT\\/floorleftbiggxT\\n\\u03b8+1\\n2\\/floorrightbigg\\/parenrightbigg\\ndx\\n+T\\u22121\\/summationdisplay\\nt=1\\/integraldisplay(2t+1)\\u03b8\\/2T\\n(2t\\u22121)\\u03b8\\/2Tpt\\/parenleftbigg\\nx\\u2212\\u03b8\\nT\\/floorleftbiggxT\\n\\u03b8+1\\n2\\/floorrightbigg\\/parenrightbigg\\ndx\\n+\\/integraldisplay\\u03b8\\n(2T\\u22121)\\u03b8\\/2TpT\\/parenleftbigg\\nx\\u2212\\u03b8\\nT\\/floorleftbiggxT\\n\\u03b8+1\\n2\\/floorrightbigg\\/parenrightbigg\\ndx\\n=p0\\/integraldisplay\\u03b8\\/2T\\n0xdx+T\\u22121\\/summationdisplay\\nt=1pt\\/integraldisplay(2t+1)\\u03b8\\/2T\\n(2t\\u22121)\\u03b8\\/2T(x\\u2212t\\u03b8\\nT) dx+pT\\/integraldisplay\\u03b8\\n(2T\\u22121)\\u03b8\\/2T(x\\u2212\\u03b8) dx\\n=p0\\u03b82\\n8T2+0\\u2212pT\\u03b82\\n8T2= (p0\\u2212pT)\\u03b82\\n8T2= 0. (S8)\\nTheorem 2. An ANN with activation function (15) is converted to an SNN wi th the same weights.\\nIf\\u03b8l=\\u03bbl,vl(0) =\\u03b8l\\u03d5, then for arbitrary TandL, the expectation of conversion error reaches 0\\nwhen the shift term \\u03d5in source ANN is1\\n2.\\n\\u2200T,LEz\\/parenleftBig\\n\\/tildewidestErrl\\/parenrightBig\\/vextendsingle\\/vextendsingle\\/vextendsingle\\n\\u03d5=1\\n2=0. (S9)\\nProof.\\nEz\\/parenleftBig\\n\\/tildewidestErrl\\/parenrightBig\\/vextendsingle\\/vextendsingle\\/vextendsingle\\n\\u03d5=1\\n2=Ez\\/parenleftbigg\\u03b8l\\nT\\/floorleftbiggzlT+vl(0)\\n\\u03b8l\\/floorrightbigg\\n\\u2212\\u03bbl\\nL\\/floorleftbiggzlL\\n\\u03bb+\\u03d5\\/floorrightbigg\\/parenrightbigg\\n. (S10)\\n15 Published as a conference paper at ICLR 2022\\n\\/uni00000013 \\/uni00000014 \\/uni00000015 \\/uni00000016 \\/uni00000017 \\/uni00000018\\n\\/uni00000037\\/uni0000004c\\/uni00000050\\/uni00000048\\/uni00000010\\/uni00000056\\/uni00000057\\/uni00000048\\/uni00000053Sl\\u2212 1\\n2 Sl\\u2212 1\\n1\\/uni00000010\\/uni00000017\\/uni00000010\\/uni00000015\\/uni00000013\\/uni00000015\\/uni00000017\\/uni00000033\\/uni00000052\\/uni00000057\\/uni00000048\\/uni00000051\\/uni00000057\\/uni0000004c\\/uni00000044\\/uni0000004fSl\\n1\\nFigure S1: More spikes than expected exists for the method of setting the maximum activation.\\nAs every element in vector zis identical, we only need to consider one element.\\nEzi\\/parenleftbigg\\u03b8l\\nT\\/floorleftbiggzl\\niT+vl\\ni(0)\\n\\u03b8l\\/floorrightbigg\\n\\u2212\\u03bbl\\nL\\/floorleftbiggzl\\niL\\n\\u03bb+\\u03d5i\\/floorrightbigg\\/parenrightbigg\\n=Ezi\\/parenleftbigg\\u03b8l\\nT\\/floorleftbiggzl\\niT+vl\\ni(0)\\n\\u03b8l\\/floorrightbigg\\n\\u2212zl\\ni\\/parenrightbigg\\n+Ezi\\/parenleftbigg\\nzl\\ni\\u2212\\u03bbl\\nL\\/floorleftbiggzl\\niL\\n\\u03bb+\\u03d5i\\/floorrightbigg\\/parenrightbigg\\n. (S11)\\nAccording to Lemma 1, we have\\nEzi\\/parenleftbigg\\u03b8l\\nT\\/floorleftbiggzl\\niT+vl\\ni(0)\\n\\u03b8l\\/floorrightbigg\\n\\u2212zl\\ni\\/parenrightbigg\\/vextendsingle\\/vextendsingle\\/vextendsingle\\/vextendsingle\\nvl\\ni(0)=1\\/2= 0, (S12)\\nEzi\\/parenleftbigg\\nzl\\ni\\u2212\\u03bbl\\nL\\/floorleftbiggzl\\niL\\n\\u03bb+\\u03d5i\\/floorrightbigg\\/parenrightbigg\\/vextendsingle\\/vextendsingle\\/vextendsingle\\/vextendsingle\\n\\u03d5=1\\/2= 0. (S13)\\nThus the sum of both terms also equals zero.\\nA.4 C OMPARISON OF THE METHODS WITH OR WITHOUT DYNAMIC THRESHOLD O N THE\\nCIFAR-100 DATASET\\nIn this paper we use a training parameter \\u03bblto decide the maximum value of ANN activation.\\nThe previous works suggested to set the maximum value of ANN a ctivation after training as the\\nthreshold. If we set \\u03b8l= max s\\u2208{0,1}n\\/parenleftbig\\nmax(\\u03b8l\\u22121Wls)\\/parenrightbig\\n, the situation of fewer spikes as expected\\nnever happens, as we can prove that vl(T)< \\u03b8l(see Theorem 3). Despite this, there still exists\\nthe situation of more spikes as expected. An example is given in Figure S1. Here we consider\\nthe same example as in Figure 1. In source ANN, we suppose that two analog neurons in layer\\nl\\u22121are connected to an analog neuron in layer lwith weights 2 and -2, and the output vector\\nal\\u22121of neurons in layer l\\u22121is[0.6,0.4]. Besides, in converted SNN, we suppose that the two\\nspiking neurons in layer l\\u22121\\ufb01re 3 spikes and 2 spikes in 5 time-steps (T=5), respectively , and the\\nthreshold \\u03b8l\\u22121= 1. Thus,\\u03c6l\\u22121(T) =\\/summationtextT\\ni=1sl\\u22121(i)\\nT\\u03b8l\\u22121= [0.6,0.4]. According to Equation 1,\\nthe ANN output al=Wlal\\u22121= [2,\\u22122][0.6,0.4]T= 0.4. As for SNN, we suppose that the\\npresynaptic neurons \\ufb01res at t= 1,2,3andt= 4,5, respectively. Even through we set the threshold\\n\\u03b8l= 1to the maximum activation 2, the postsynaptic neuron will \\ufb01r e three spikes at t= 1,2,3, and\\n\\u03c6l(T) = 0.6>al.\\nBesides, setting maxs\\u2208{0,1}n\\/parenleftbig\\nmax(\\u03b8l\\u22121Wls)\\/parenrightbig\\nas the threshold brings two other problems. First,\\nthe spiking neurons will take a long time to \\ufb01re spikes becaus e of the large value of the threshold,\\nwhich makes it hard to maintain SNN performance within a few t ime-steps. Second, the quantization\\nerror will be large as it is proportional to the threshold. If the conversion error is not zero for\\none layer, it will propagate layer by layer and will be magni\\ufb01 ed by larger quantization errors. We\\ncompare our method and the method of setting the maximum acti vation on the CIFAR-100 dataset.\\nThe results are reported in Table S1, where DT represents the dynamic threshold in our method. The\\nresults show that our method can achieve better performance .\\n16 Published as a conference paper at ICLR 2022\\nTable S1: Comparison between our method and the method of set ting the maximum activation.\\nDT1w\\/o shift T=4 T=8 T=16 T=32 T=64 T=128 T=256 T \\u2265512\\nVGG-16 on CIFAR-100 with L=4\\n\\/check \\/check 69.62% 73.96% 76.24% 77.01% 77.10% 77.05% 77.08% 77.08%\\n\\/check\\u00d7 21.57% 41.13% 58.92% 65.38% 64.19% 58.60% 52.99% 49.41%\\n\\u00d7\\/check 1.00% 0.96% 1.00% 1.10% 2.41% 13.76% 51.70% 77.10%\\n\\u00d7 \\u00d7 1.00% 1.00% 0.90% 1.00% 1.01% 2.01% 19.59% 70.86%\\n1Dynamic threshold.\\nTheorem 3. If the threshold is set to the maximum value of ANN activation , that is \\u03b8l=\\nmaxs\\u2208{0,1}n\\/parenleftbig\\nmax(\\u03b8l\\u22121Wls)\\/parenrightbig\\n,andvl\\ni(0)< \\u03b8l.Then at any time-step, the membrane potential\\nof each neuron after spike vl\\ni(t)will be less than \\u03b8l, whereirepresents the index of each neuron.\\nProof. We prove it by induction. For t= 0, it is easy to see vl\\ni(0)< \\u03b8l. Fort >0, we suppose\\nthatvl\\ni(t\\u22121)< \\u03b8l. Since we have set the threshold to the maximum possible inpu t, andxl\\u22121\\ni(t)\\nrepresents the input from layer l\\u22121to thei-th neuron in layer l,xl\\u22121\\ni(t)will be no larger than \\u03b8l\\nfor arbitrary t. Thus we have\\nml\\ni(t) =vl\\ni(t\\u22121)+xl\\u22121\\ni(t)< \\u03b8l+\\u03b8l= 2\\u03b8l, (S14)\\nsl\\ni(t) =H(ml\\ni(t)\\u2212\\u03b8l), (S15)\\nvl\\ni(t) =ml\\ni(t)\\u2212sl\\ni(t)\\u03b8l. (S16)\\nIf\\u03b8l\\/lessorequalslantml\\ni(t)<2\\u03b8l, then we have vl\\ni(t) =ml\\ni(t)\\u2212\\u03b8l< \\u03b8l. Ifml\\ni(t)< \\u03b8l, thenvl\\ni(t) =ml\\ni(t)< \\u03b8l.\\nBy mathematical induction, vl\\ni(t)< \\u03b8lholds for any t\\/greaterorequalslant0.\\nA.5 E FFECT OF QUANTIZATION STEPS L\\nTable S2 reports the performance of converted SNNs with diff erent quantization steps Land differ-\\nent time-steps T. For VGG-16 and quantization steps L= 2, we achieve an accuracy of 86.53% on\\nCIFAR-10 dataset and an accuracy of 61.41% on CIFAR-100 data set with 1 time-steps. When the\\nquantization steps L= 1, we cannot train the source ANN.\\nA.6 C OMPARISON WITH STATE -OF-THE-ART SUPERVISED TRAINING METHODS ON\\nCIFAR-10 DATASET\\nNotably, our ultra-low latency performance is comparable w ith other state-of-the-art supervised\\ntraining methods. Table S3 reports the results of hybrid tra ining and backpropagation methods\\non CIFAR-10. The backpropagation methods require suf\\ufb01cien t time-steps to convey discriminate\\ninformation. Thus, the list methods need at least 5 time-ste ps to achieve\\u223c91% accuracy. On the\\ncontrary, our method can achieve 94.73% accuracy with 4 time -steps. Besides, the hybrid training\\nmethod requires 200 time-steps to obtain 92.02% accuracy be cause of further training with STDB,\\nwhereas our method achieves 93.96% accuracy with 4 time-ste ps.\\nA.7 C OMPARISON ON CIFAR-100 DATASET\\nTable S4 reports the results on CIFAR-100, our method also ou tperforms the others both in terms of\\nhigh accuracy and ultra-low latency. For VGG-16, the accura cy of the proposed method is 3.46%\\nhigher than SNNC-AP and 69.37% higher than RTS when T= 32 . When the time-steps is only 4,\\nwe can still achieve an accuracy of 69.62%. These results dem onstrate that our method outperforms\\nthe previous conversion methods.\\n17 Published as a conference paper at ICLR 2022\\nTable S2: In\\ufb02uence of different quantization steps.\\nquantization\\nsteps T=1 T=2 T=4 T=8 T=16 T=32 T=64 T=128\\nVGG-16 on CIFAR-10\\nL=2 86.53% 91.98% 93.00% 93.95% 94.18% 94.22% 94.18% 94.14%\\nL=4 88.41% 91.18% 93.96% 94.95% 95.40% 95.54% 95.55% 95.59%\\nL=8 62.89% 83.93% 91.77% 94.45% 95.22% 95.56% 95.74% 95.79%\\nL=16 61.48% 76.76% 89.61% 93.03% 93.95% 94.24% 94.25% 94.22 %\\nL=32 13.05% 73.33% 89.67% 94.13% 95.31% 95.66% 95.73% 95.77 %\\nResNet-20 on CIFAR-10\\nL=2 77.54% 82.12% 85.77% 88.04% 88.64% 88.79% 88.85% 88.76%\\nL=4 62.43% 73.2% 83.75% 89.55% 91.62% 92.24% 92.35% 92.35%\\nL=8 46.19% 58.67% 75.70% 87.79% 92.14% 93.04% 93.34% 93.24%\\nL=16 30.96% 39.87% 57.04% 79.5% 90.87% 93.25% 93.44% 93.48%\\nL=32 22.15% 27.83% 43.56% 70.15% 88.81% 92.97% 93.48% 93.48 %\\nVGG-16 on CIFAR-100\\nL=2 61.41% 64.96% 68.0% 70.72% 71.87% 72.28% 72.35% 72.4%\\nL=4 57.5% 63.79% 69.62% 73.96% 76.24% 77.01% 77.1% 77.05%\\nL=8 44.98% 52.46% 62.09% 70.71% 74.83% 76.41% 76.73% 76.73%\\nL=16 33.12% 41.71% 53.38% 65.76% 72.80% 75.6% 76.37% 76.36%\\nL=32 15.18% 21.41% 32.21% 50.46% 67.32% 74.6% 76.18% 76.24%\\nResNet-20 on CIFAR-100\\nL=2 38.65% 47.35% 55.23% 59.69% 61.29% 61.5% 61.03% 60.81%\\nL=4 25.62% 36.33% 51.55% 63.14% 66.70% 67.47% 67.47% 67.41%\\nL=8 13.19% 19.96% 34.14% 55.37% 67.33% 69.82% 70.49% 70.55%\\nL=16 6.09% 9.25% 17.48% 38.22% 60.92% 68.70% 70.15% 70.20%\\nL=32 5.44% 7.41% 13.36% 31.66% 58.68% 68.12% 70.12% 70.27%\\nA.8 E NERGY CONSUMPTION ANALYSIS\\nWe evaluate the energy consumption of our method and the comp ared methods (Li et al., 2021;\\nDeng & Gu, 2020) on CIFAR-100 datasets. Here we use the same ne twork structure of VGG-16.\\nFollowing the analysis in Merolla et al. (2014), we use synap tic operation (SOP) for SNN to rep-\\nresent the required basic operation numbers to classify one image. We utilize 77fJ\\/SOP for SNN\\nand 12.5pJ\\/FLOP for ANN as the power consumption baseline, w hich is reported from the ROLLS\\nneuromorphic processor (Qiao et al., 2015). Note that we do n ot consider the memory access energy\\nin our study because it depends on the hardware. As shown in Ta ble S5, when the time-steps is the\\nsame, the energy consumption of our method is about two times of SNNC-AP. However, to achieve\\nthe same accuracy of 73.55%, our method requires less energy consumption.\\nA.9 PSEUDO -CODE FOR OVERALL CONVERSION ALGORITHM\\nIn this section, we summarize the entire conversion process in Algorithm 1, including training ANNs\\nfrom scratch and converting ANNs to SNNs. The QCFS in the pseu do-code represents the proposed\\nquantization clip-\\ufb02oor-shift function.\\n18 Published as a conference paper at ICLR 2022\\nTable S3: Compare with state-of-the-art supervised traini ng methods on CIFAR-10 dataset\\nModel Method Architecture SNN Accuracy Timesteps\\nCIFAR-10\\nHC Hybrid VGG-16 92.02 200\\nSTBP Backprop CIFARNet 90.53 12\\nDT Backprop CIFARNet 90.98 8\\nTSSL Backprop CIFARNet 91.41 5\\nDThIR1ANN-SNN cNet 77.10 256\\nOurs ANN-SNN VGG-16 93.96 4\\nOurs ANN-SNN CIFARNet294.73 4\\n1Implemented on Loihi neuromorphic processor\\n2For CIFARNet, we use the same architecture as Wu et al. (2018) .\\nTable S4: Comparison between the proposed method and previo us works on CIFAR-100 dataset.\\nArchitecture Method ANN T=2 T=4 T=8 T=16 T=32 T=64 T \\u2265512\\nVGG-16RMP 71.22% - - - - - - 70.93%\\nTSC 71.22% - - - - - - 70.97%\\nRTS 77.89% - - - - 7.64% 21.84% 77.71%\\nSNNC-AP 77.89% - - - - 73.55% 76.64% 77.87%\\nOurs 76.28% 63.79% 69.62% 73.96% 76.24% 77.01% 77.10% 77.08%\\nResNet-20RMP 68.72% - - - - 27.64% 46.91% 67.82%\\nTSC 68.72% - - - - - - 68.18%\\nOurs 69.94% 19.96% 34.14% 55.37% 67.33% 69.82% 70.49% 70.50%\\nResNet-18RTS 77.16% - - - - 51.27% 70.12% 77.19%\\nSNNC-AP 77.16% - - - - 76.32% 77.29% 77.25%\\nOurs 78.80% 70.79% 75.67% 78.48% 79.48% 79.62% 79.54% 79.61%\\n1RTS and SNNC-AP use altered ResNet-18, while ours use standa rd ResNet-18.\\nTable S5: Comparison of the energy consumption with previou s works\\nMethod ANN T=2 T=4 T=8 T=16 T=32 T=64\\nRTSAccuracy 77.89% - - - - 7.64% 21.84%\\nOP (GFLOP\\/GSOP) 0.628 - - - - 0.508 0.681\\nEnergy (mJ) 7.85 - - - - 0.039 0.052\\nSNNC-APAccuracy 77.89% - - - - 73.55% 76.64%\\nOP (GFLOP\\/GSOP) 0.628 - - - - 0.857 1.22\\nEnergy (mJ) 7.85 - - - - 0.660 0.094\\nOursAccuracy 76.28% 63.79% 69.62% 73.96% 76.24% 77.01% 77.10%\\nOP (GFLOP\\/GSOP) 0.628 0.094 0.185 0.364 0.724 1.444 2.884\\nEnergy (mJ) 7.85 0.007 0.014 0.028 0.056 0.111 0.222\\n19 Published as a conference paper at ICLR 2022\\nAlgorithm 1 Algorithm for ANN-SNN conversion.\\nInput : ANN model MANN(x;W)with initial weight W; Dataset D; Quantization step L; Initial\\ndynamic thresholds \\u03bb; Learning rate \\u01eb.\\nOutput :MSNN(x;\\u02c6W)\\n1:forl= 1toMANN.layers do\\n2: ifis ReLU activation then\\n3: Replace ReLU (x)by QCFS (x;L,\\u03bbl)\\n4: end if\\n5: ifis MaxPooling layer then\\n6: Replace MaxPooling layer by AvgPooling layer\\n7: end if\\n8:end for\\n9:fore= 1to epochs do\\n10: forlength of Dataset Ddo\\n11: Sample minibatch (x0,y)fromD\\n12: forl= 1toMANN.layers do\\n13: xl=QCFS(Wlxl\\u22121;L,\\u03bbl)\\n14: end for\\n15: Loss = CrossEntropy (xl,y)\\n16: forl= 1toMANN.layers do\\n17: Wl\\u2190Wl\\u2212\\u01eb\\u2202Loss\\n\\u2202Wl\\n18: \\u03bbl\\u2190\\u03bbl\\u2212\\u01eb\\u2202Loss\\n\\u2202\\u03bbl\\n19: end for\\n20: end for\\n21:end for\\n22:forl= 1toMANN.layers do\\n23:MSNN.\\u02c6Wl\\u2190MANN.Wl\\n24:MSNN.\\u03b8l\\u2190MANN.\\u03bbl\\n25:MSNN.vl(0)\\u2190MSNN.\\u03b8l\\/2\\n26:end for\\n27:returnMSNN\\n20\",\"6\":\" Evolutionary Reinforcement Learning: A Survey\\nHui Bai1, Ran Cheng1, and Yaochu Jin2,3\\n1Department of Computer Science and Engineering, Southern University of Science\\nand Technology, Shenzhen, China.\\n2Faculty of Technology, Bielefeld University, 33615 Bielefeld, Germany.\\n3Department of Computer Science, University of Surrey, Guildford, Surrey GU2\\n7XH, U.K.\\nAbstract\\nReinforcement learning (RL) is a machine learning approach that trains agents to maximize\\ncumulative rewards through interactions with environments. The integration of RL with deep\\nlearning has recently resulted in impressive achievements in a wide range of challenging tasks,\\nincluding board games, arcade games, and robot control. Despite these successes, there remain\\nseveral crucial challenges, including brittle convergence properties caused by sensitive hyperpa-\\nrameters, di\\u000eculties in temporal credit assignment with long time horizons and sparse rewards,\\na lack of diverse exploration, especially in continuous search space scenarios, di\\u000eculties in credit\\nassignment in multi-agent reinforcement learning, and con\\ricting objectives for rewards. Evolu-\\ntionary computation (EC), which maintains a population of learning agents, has demonstrated\\npromising performance in addressing these limitations. This article presents a comprehensive\\nsurvey of state-of-the-art methods for integrating EC into RL, referred to as evolutionary rein-\\nforcement learning (EvoRL). We categorize EvoRL methods according to key research \\felds in\\nRL, including hyperparameter optimization, policy search, exploration, reward shaping, meta-\\nRL, and multi-objective RL. We then discuss future research directions in terms of e\\u000ecient\\nmethods, benchmarks, and scalable platforms. This survey serves as a resource for researchers\\nand practitioners interested in the \\feld of EvoRL, highlighting the important challenges and\\nopportunities for future research. With the help of this survey, researchers and practitioners\\ncan develop more e\\u000ecient methods and tailored benchmarks for EvoRL, further advancing this\\npromising cross-disciplinary research \\feld.\\n1 Introduction\\nReinforcement learning (RL) has achieved remarkable success in recent years, particularly with the\\nintegration of deep learning (DL), in solving complex sequential decision-making problems [1, 2].\\nDespite these advancements, RL still faces several challenges, such as sensitivity to hyperparameters\\n[3], di\\u000eculties in credit assignment in tasks with long time horizons, sparse rewards, and multiple\\n1arXiv:2303.04150v1  [cs.NE]  7 Mar 2023 agents [4, 5], limited diverse exploration in tasks with deceptive rewards or continuous state and\\naction spaces [6], and con\\ricting objectives for rewards [7].\\nTo address these challenges, the \\feld of evolutionary reinforcement learning (EvoRL) has emerged\\nby integrating RL with evolutionary computation (EC) [8, 9]. EvoRL involves maintaining a popula-\\ntion of agents, which o\\u000bers several bene\\fts, such as provision of redundant information for improved\\nrobustness [9], enabling diverse exploration [10], ability to evaluate agents using an episodic \\ftness\\nmetric [9], and the ease of generating trade-o\\u000b solutions through multi-objective EC algorithms [11].\\nEvoRL has a rich history, dating back to early work in neuroevolution, which used EC algorithms\\nto generate the weights and\\/or topology of arti\\fcial neural networks (ANNs) for agent policies\\n[12, 13]. Since the proposal of OpenAI ES [8], EvoRL has gained increasing attention in both EC\\nand RL communities. While there have been some surveys focusing on various aspects of EvoRL,\\nsuch as neuroevolution [14], multi-objective RL [7, 15], automated RL [16], and derivative-free RL\\n[17], they either focus on a narrow research \\feld within RL or lack a comprehensive overview of EC\\nmethods as applied to RL.\\nTo bridge the gap between EC and RL communities, this article provides a comprehensive survey\\nof EvoRL, elaborating on six key research \\felds of RL, as shown in Figure 1. The EvoRL methods\\nare introduced and discussed separately for each \\feld, focusing on their advantages and limitations.\\nFinally, the article discusses potential improvement approaches for future research, including e\\u000ecient\\nmethods in terms of EvoRL processes, tailored benchmarks, and scalable platforms.\\n2 Background\\n2.1 Reinforcement Learning\\nReinforcement learning (RL) is a powerful tool for decision-making in complex and stochastic en-\\nvironments. In RL, an agent interacts with its environment by taking a sequence of actions and\\nreceiving a sequence of rewards over time. The objective of the agent is to maximize the expected\\ncumulative reward. This problem can be modeled as a Markov Decision Process (MDP), which is\\nde\\fned as< S;A;T;R;\\u001a 0;\\r > , with a state space S, an action space A, a stochastic transition\\nfunctionT:S\\u0002A!P(S) that represents the probability distribution over possible next states, a\\nreward function R:S\\u0002A!R, an initial state distribution \\u001a0:S!R2[0;1], and a discount factor\\n\\r2[0;1).\\nThe agent's behavior is determined by its policy, which is denoted by \\u0019\\u0012:S!P(A), withP(A)\\nbeing the set of probability measures on Aand\\u00122Rnbeing a vector of nparameters. The agent\\nupdates its policy over time to maximize the expected cumulative discounted reward, as given by\\nJ(\\u0019) =E\\u001a0;\\u0019;T\\\"1X\\nt=0\\rtrt#\\n; (1)\\nwheres0\\u0018\\u001a0(s0),at\\u0018\\u0019(st),st+1\\u0018T(\\u0001jst;at), andrt=R(st;at).\\nRL algorithms can be divided into two categories: model-based and model-free. While model-\\nbased algorithms establish a complete MDP by estimating the transition function and reward func-\\n2 Darwinian evolutionary approaches\\nLamarckian evolutionary approaches\\nEvolution strategies based methods\\nGenetic algorithms based methodsHybrid approaches\\nEvolutionary \\nreinforcement learningHyperparameter\\n optimization\\nPolicy search\\nExploration\\nReward shaping\\nMeta-RL\\nMulti-objective RLGenetic programming based methods\\nDiversity encouragement methods\\nEvolution of reward functions\\nHyperparameter optimization methodsEvolution-guided exploration methods\\nParameters initialization\\nLoss learning\\nEnvironment synthesis\\nAlgorithms generation\\nMulti-objective evolutionary algorithms\\nMulti-objectivizationPopulation based training\\nNatural evolution strategies\\nCanonical evolution strategies\\nCMA-ES\\nAlgorithmic frameworks\\nIndirect encoding\\nVariation operators\\nRepresentations\\nEvolution of direct controllers\\nSymbolic regression\\nFeature Discovery\\nNovelty search\\nQuality diversity\\nSurprise search\\nEvolvability search\\nEvolvability search\\nUse of diverse experiences\\nUse of gradient informationFigure 1: Key research \\felds of evolutionary reinforcement learning. Hyperparameter optimization\\nis a universal method for algorithms in the other \\fve research \\felds to realize end-to-end learning\\nand improve performance simultaneously. Policy search seeks to identify a policy that maximizes\\nthe cumulative reward for a given task. Exploration encourages agents to explore more states and\\nactions and trains robust agents to better respond to dynamic changes in environments. Reward\\nshaping is aimed at enhancing the original reward with additional shaping rewards for tasks with\\nsparse rewards. Meta-RL seeks to develop a general-purpose learning algorithm that can adapt\\nto di\\u000berent tasks. Multi-objective RL aims to obtain trade-o\\u000b agents in tasks with a number of\\ncon\\ricting objectives.\\n3 tion, by contrast, model-free algorithms are data-driven and optimize the policy by using a large\\nnumber of samples, without the need to know the transition function and reward function. Due to\\nthe di\\u000eculties of establishing a complete MDP and the success of neural networks (NNs) in repre-\\nsenting policies, model-free RL has become the main focus of research in recent years [18]. In this\\nsurvey, we focus on model-free RL methods.\\nMore speci\\fcally, model-free RL methods can further be divided into two categories: policy-based\\nand value-based methods. In policy-based methods, the parameters \\u0012of the policy are adjusted in\\nthe direction of the performance gradient, according to the Policy Gradients Theorem [1]. Some\\nof the state-of-the-art policy-based algorithms include TRPO [19], PPO [20], A3C [21], DDPG\\n[22], TD3 [23], and SAC [24]. In value-based methods, a parameterized Q-function is optimized to\\nestimate the value of a state-action pair. One of the state-of-the-art value-based methods is DQN\\n[25], which updates the parameters of the Q-function by minimizing the Temporal Di\\u000berence (TD)\\nloss using a batch of samples. Techniques such as experience replay [26] and double Q-network [27]\\nhave been proposed to improve the sample e\\u000eciency and exploration of DQN.\\n2.2 Evolutionary Computation\\nEvolutionary computation (EC) refer to a family of stochastic search algorithms that have been\\ndeveloped based on the principle of natural evolution. The primary objective of EC is to approximate\\nglobal optima of optimization problems by iteratively performing a range of mechanisms, such as\\nvariation (i.e., crossover and mutation), evaluation, and selection. Among various EC paradigms, the\\nEvolution Strategies (ESs) [28] are the mostly adopted in EvoRL, together with the classic Genetic\\nAlgorithm (GAs) [29] and the Genetic Programming (GP) [30].\\nESs primarily tackle continuous black-box optimization problems where the search space lies\\nwithin the continuous domain. Therefore, it is predominantly applied to weight optimization of pol-\\nicy search in RL. ESs for RL are typically categorized into three main classes, which are Canonical\\nES [31], Covariance Matrix Adaptation ES (CMA-ES) [32], and Natural ES (NES) [33]. Canonical\\nES is aimed at obtaining \\fnal solutions with high \\ftness values while using a small number of \\ftness\\nevaluations through conducting an iterative process involving variation, evaluation, and selection.\\nHere, we illustrate the canonical ( \\u0016,\\u0015)-ES algorithm. Following the initialization of policy param-\\netersx2Rnand a set of hyperparameters, the algorithm generates \\u0015o\\u000bspringx1;:::;x\\u0015from a\\nsearch distribution with mean xand variance \\u001b2C. Subsequently, all o\\u000bspring are evaluated using\\na \\ftness evaluation function. Following this, a new population mean is generated by moving the old\\npopulation mean towards the best \\u0016o\\u000bspring. Then, \\u001bis optionally updated, and an adaptive \\u001b\\ncan improve ESs performance. CMA-ES shares the same procedure with Canonical ES but is more\\ne\\u000bective since its mutation step size \\u001band covariance matrix Care updated adaptively, enabling\\nthe capture of the anisotropy properties of general optimization problems. NES also shares the same\\niteration with Canonical ES. However, it updates a search distribution iteratively by estimating a\\nsearch gradient (i.e., the second-order gradient) on the distribution parameters towards higher ex-\\npected \\ftness values. The distribution parameters are updated through estimating a natural gradient\\nthat can \\fnd a parameterization-independent ascent direction compared to a plain search gradient\\n4 EnvironmentGiven current state \\ud835\\udc60\\u2208\\ud835\\udc46\\nTake action \\ud835\\udc4e\\u2208\\ud835\\udc34\\nGet reward \\ud835\\udc5f\\nNext state \\ud835\\udc60\\u2032\\u2208\\ud835\\udc46Agent Policy\\nOffspring \\ngenerationMating selectionPopulation\\nEnvironmental selectionEvaluationFigure 2: A simple and general framework of EvoRL. The framework consists of two loops: the\\nouter loop shows the evolution process of EC, while the inner loop illustrates the agent-environment\\ninteraction process in RL. Initially, a population of parent candidate solutions is randomly initialized,\\nand then the o\\u000bspring candidate solutions are generated from the parents via variation. Each\\no\\u000bspring is evaluated using an RL task to obtain its \\ftness value, and a new population is selected\\nfor the next iteration by combining all parents and o\\u000bspring.\\n[34]. The natural gradient er\\u0012Jis formulated as F\\u00001r\\u0012J, where Fis the Fisher information matrix\\n(FIM) of the parametric family of the search distribution, and r\\u0012Jis the estimated search gradient\\nof expected \\ftness by Monte Carlo estimation. The Fisher information matrix implies the degree of\\ncertainty of updating \\u0012, which has the e\\u000bect of punishing the natural gradient with high variance\\nand boosting the natural gradient with low variance.\\nGAs, as the most classic EC paradigm, are also commonly adopted in EvoRL. GAs follow\\nthe work\\row where a population of candidate solutions is iteratively improved through selection,\\ncrossover, and mutation. The encoding or representation of the search space in GAs can be tailored\\nto the speci\\fc problem at hand, allowing for binary encoding and discrete encoding for combinatorial\\noptimization problems, and real encoding for numerical optimization problems. This versatility in\\nencoding types makes GAs a widely applicable method for solving various problems in RL [35, 36].\\nGP is a distinctive EC paradigm that is di\\u000berent from ESs or GAs which mainly solve numerical\\noptimization problems. In GP, the search space is composed of a set of programs that are represented\\nusing various encoding methods, such as abstract syntax tree, executable graph (e.g., Cartesian GP\\nand tangled program graphs), \\fnite-state machine, and context-free grammar, among others [28].\\nThe \\ftness of a program is evaluated by executing it to observe its behavior, and the programs can\\nbe viewed as data when crossover and mutation operations are performed on them. In contrast, the\\ndata are interpreted as programs when they are executed. GP has several advantages over other EC\\nparadigms, including the ability to handle complex problems that require a programmatic solution,\\nsuch as symbolic regression and control problems, among others [37, 38, 39].\\n5 2.3 Discussion\\nEC algorithms have been recognized as competitive tools to handle complex optimization problems\\nthat exhibit non-convex, non-di\\u000berentiable, non-smooth, and multi-modal properties [40]. In RL, EC\\nis particularly useful for complex problems with numerous local optima and is suitable for problems\\nwithout gradient information. Furthermore, it is applicable for problems without explicit objective\\nfunctions by novelty search [41]. The population-based search strategy of EC makes it robust to\\ndynamic changes that are commonly found in real-world applications of RL, such as sim-to-real\\ntransfer in robot control [42]. A simple and general framework that combines EC and RL is shown\\nin Figure 2.\\nSpeci\\fcally, EC has been introduced into RL in six major key research \\felds of RL, including hy-\\nperparameter optimization ,policy search ,exploration ,reward shaping ,meta-RL , and multi-objective\\nRL, as presented in Figure 1. These six key research \\felds will be introduced in detail in the following\\nsections, respectively.\\n3 Evolutionary Computation in Hyperparameter Optimiza-\\ntion\\nFinding the optimal hyperparameter con\\fgurations for reinforcement learning (RL) can be a chal-\\nlenging task due to the large number of hyperparameters involved, including those related to RL\\nalgorithms (e.g., the learning rate \\u000band discount factor of future rewards \\r) and the neural network\\narchitectures of policies (e.g., the number and size of layers). To overcome this challenge, researchers\\nhave introduced hyperparameter optimization (HPO) to automatically set con\\fgurations for optimal\\nperformance. HPO has been shown to improve the performance and robustness of RL algorithms\\n[43, 44].\\nHowever, HPO for RL faces several challenges. First, performance evaluation can be extremely\\nexpensive for complex tasks. Second, the search space of hyperparameters can be complex, involving\\nmixed encoding, high dimensionality, and non-convexity. Third, there may be two or more objectives\\nthat need to be traded o\\u000b. To address these challenges, there are several major classes of HPO\\nmethods, including random search [45], Bayesian optimization [46], gradient-based methods [47],\\nand evolutionary computation (EC) methods [43]. Among these, EC methods can simultaneously\\nmeet the challenges of HPO, owing to their high degree of parallelism, gradient-free properties, and\\nability to obtain a set of trade-o\\u000b optimal solutions.\\nEC-based HPO methods can be mainly classi\\fed into three categories: Darwinian evolution-\\nary methods ,Lamarckian evolutionary methods , and hybrid methods . In Darwinian evolutionary\\nmethods, parameters are initialized while hyperparameters are evolved. In contrast, in Lamarckian\\nevolutionary methods, parameters are inherited while hyperparameters are evolved. Additionally,\\nhybrid methods further combine the former two methods and gradient-based methods.\\n6 3.1 Darwinian Evolutionary Methods\\nIn Darwinian evolutionary methods, the parameters are randomly initialized while hyperparameters\\nare evolved using genetic algorithms (GAs) [48]. For instance, Eriksson et al. applied GA to evolve\\ntwo hyperparameters, the learning rate \\u000band the temperature \\u001c, which control the trade-o\\u000b between\\nexploration and exploitation in softmax action selection, for Sarsa( \\u0015) in food capture tasks. Elfwing\\net al. [49] also applied GA to evolve hyperparameters and weights in potential-based reward shaping\\nfor Sarsa(\\u0015) in the same tasks. These methods integrate learning and evolution to e\\u000bectively improve\\nthe performance of RL algorithms and obtain sim-to-real robust policies. Nonetheless, the Darwinian\\nevolutionary methods are ine\\u000ecient since parameters are reinitialized in each generation, causing a\\nloss of knowledge already acquired during previous generations.\\n3.2 Lamarckian Evolutionary Methods\\nIn Lamarckian evolutionary methods, parameters are inherited while hyperparameters are evolved,\\nmeaning that hyperparameters are adapted to the current learning process to make agents learn\\nmore e\\u000eciently. A state-of-the-art asynchronous parallel evolutionary method called population-\\nbased training (PBT) has recently been proposed to improve the e\\u000eciency of HPO [43]. In the\\nasynchronous PBT, only one ready individual is compared with a randomly selected individual from\\nthe remaining population in each generation, and then the worse individual copies the parameters\\nand hyperparameters of the better individual and adds noise to its hyperparameters. PBT has\\nsuccessfully trained a series of RL agents in a number of complex tasks, such as the 3D multi-player\\n\\frst-person video game, DMLab, the MuJoCo multi-agent soccer game, and ELF OpenGo, achieving\\nnew state-of-the-art performance of RL algorithms [50, 51, 52, 53]. The PBT-style evolution is\\nquite similar to the steady-state EC (i.e., a new individual is inserted into the population in each\\ngeneration) that is believed to be e\\u000bective for non-stationary\\/dynamic environments [54]. However,\\nPBT does not consider the diversity of the population. That is, PBT prefers the higher-performing\\ncon\\fgurations, but it may lose the individuals that are \\\\late bloomers\\\". Therefore, the faster\\nimprovement rate PBT (FIRE PBT) has been proposed based on the assumption that when two\\nNNs have similar performance and hyperparameters, the NN with a faster rate of improvement\\nwill bring about a better \\fnal performance [55]. FIRE PBT derives a \\ftness metric based on\\nthe assumption and introduces subpopulations to increase diversity. Further, a sample-e\\u000ecient\\nautomated RL framework (SEARL) has been proposed for o\\u000b-policy RL algorithms, which follows\\nthe PBT-style to evolve dynamic con\\fgurations of hyperparameters and shares experiences across\\nthe population with di\\u000berent con\\fgurations [44].\\n3.3 Hybrid Methods\\nHybrid methods combine the above two evolutionary methods and other gradient-based methods to\\nimprove training e\\u000eciency. For example, in the work of [56], a hybrid method combines Darwinian\\nand Lamarckian evolutionary methods, following the mainstream of Lamarckian evolution while con-\\nducting multiple random restarts of parameters in the evolutionary process to escape local optima.\\n7 In addition, an evolutionary stochastic gradient descent framework is proposed, aiming at combining\\nthe merits of stochastic gradient descent and evolutionary computation [57]. In the framework, a\\nset of neural network weights with distinct hyperparameters are optimized independently by vari-\\nous stochastic gradient descent variants, and then their information is exchanged by evolutionary\\ncomputation. However, the initial hyperparameters are preset by humans, which still involves a\\ncertain amount of human knowledge. Moreover, [58] has proposed a collection of benchmarks de-\\nrived from hyperparameter optimization to verify the performance of quality diversity methods. In\\nother words, hyperparameter optimization methods can be resolved by quality diversity methods to\\nintroduce diversity by niches.\\n3.4 Discussion\\nHPO is crucial for achieving state-of-the-art performance in RL, and EC methods have shown\\ngreat potential in automating this process. However, the current literature on HPO still faces\\nseveral challenges, such as the lack of comprehensive performance metrics and the need for e\\u000ecient\\nconvergence speed. Additionally, selecting hyperparameters from a large number of options is a\\ncombinatorial optimization problem that brings new challenges to EC methods.\\nTo address these challenges, future research should focus on developing comprehensive evaluation\\nmetrics that consider both e\\u000bectiveness and e\\u000eciency. This can be achieved by benchmarking\\ndi\\u000berent HPO methods on a wide range of tasks and considering factors such as training time,\\nconvergence speed, and performance. Additionally, future research should explore new EC methods\\nthat can e\\u000eciently search high-dimensional and combinatorial search spaces for hyperparameter\\noptimization. Furthermore, it is essential to investigate how to combine di\\u000berent HPO methods\\nto improve the overall e\\u000eciency and e\\u000bectiveness of the optimization process. By addressing these\\nchallenges, EC-based HPO can further accelerate the development and deployment of RL algorithms\\nin real-world scenarios.\\n4 Evolutionary Computation in Policy Search\\nIn the context of RL, policy search seeks to identify a policy that maximizes the cumulative reward\\nfor a given task. The incorporation of neural networks as function approximators for policies has been\\nfacilitated by the surge of deep learning, despite the vast search space of states and actions. Stochas-\\ntic gradient descent (SGD) methods are widely used for training neural network weights in deep RL.\\nAlternatively, neuroevolution has emerged as an alternative approach, leveraging gradient-free EC\\nmethods for policy search, which can optimize neural network weights, architectures, hyperparame-\\nters, building blocks, and even learning rules [59].\\nEarly work in neuroevolution focused primarily on the evolution of the weights of small and\\n\\fxed-architecture neural networks. Recent advancements, however, have demonstrated the promise\\nof evolving the architecture together with the weights of neural networks for complex RL tasks\\n[13]. Moreover, a new perspective on policy search has been established by ignoring the weights and\\nconducting only architecture search [60]. This section will review EC techniques such as evolutionary\\n8 strategies (ESs) ,genetic algorithms (GAs) , and genetic programming (GP) for policy search in RL.\\n4.1 ES based Methods\\nThe subsection will review three popular ESs used in RL tasks, namely Canonical ES, Natural\\nEvolution Strategies (NES), and Covariance Matrix Adaptation ES (CMA-ES).\\n4.1.1 Canonical ES based Methods\\nThe high-parallel framework of OpenAI ES has led to the successful application of a simpli\\fed\\ncanonical ES to Atari games [61]. Prior to this, the canonical ES had not been applied much to RL\\ntasks, since it performs poorly on high-dimensional tasks. Although the canonical ES can achieve\\nsimilar performance to OpenAI ES on several Atari games with discrete state and action spaces, its\\nperformance on continuous state and action spaces (where EC is more preferable [62]) has not been\\ninvestigated.\\nAs ESs treat RL tasks as black-box optimization problems directly instead of taking advantage of\\ntheir intrinsic MDP structures, it may have a large variance in multiple runs. To address this issue,\\nseveral variance reduction techniques have been introduced. On the one hand, various ESs gradient\\nestimators using Monte Carlo techniques have been proposed, such as the antithetic ES gradient\\nestimator in OpenAI ES [8] and the forward \\fnite-di\\u000berence ES gradient estimator in the structured\\nES [63]. Speci\\fcally, the structured ES performed well on most of the MuJoCo tasks using less than\\n300 policy parameters. Furthermore, the adaptive ES-active subspace method further combines\\nstructured ES with the techniques from active subspaces to learn the changing dimensionality of the\\ngradient space, which achieved competitive performance compared to PPO, TRPO, and several ES\\nvariants on a subset of MuJoCo tasks [64]. On the other hand, structured methods leveraging the\\nunderlying MDP structures have been developed. Speci\\fcally, the control variate, also known as the\\nadvantages function in RL [21], has been introduced into ESs to reduce the variance of Monte Carlo\\ngradient estimation [65].\\nSince applying ESs to large-scale RL tasks is low-e\\u000ecient, a number of works have been proposed\\nto improve sample e\\u000eciency in two ways: sampling from diverse search directions and making full\\nuse of previous samples. In the \\frst way, the Gaussian orthogonal exploration searches a number of\\ndiverse directions [63]. Based on this method, the Guided ES further combines ES with surrogate\\ngradients (correlated with the true gradients) [66]. Then, the self-guided ES trades o\\u000b exploitation\\nin the gradient subspace and exploration in its orthogonal complement subspace, which has obtained\\nhigher returns and faster convergence speed than PPO, TRPO, and Guided ES on several MuJoCo\\ntasks [67]. In the second way, the trust region ES (TRES) approximately optimizes a surrogate\\nobjective by reusing the data sampled from the old policy parameters instead of sampling from new\\nparameters, which has achieved faster convergence speed than PPO and TRPO on several MuJoCo\\ntasks [68].\\n9 4.1.2 NES based Methods\\nThe original NES is known to be limited in scalability for high dimensional problems due to the time-\\nconsuming calculation of Fisher information matrix (FIM) [33]. To improve e\\u000eciency and robustness,\\nthe exact NES computes the inverse of the exact FIM instead of the empirical FIM and has shown\\ncompetitive performance on the double pole balancing task using an NN with 21 weights [69].\\nHowever, NES was not successful for large-scale tasks until the development of OpenAI ES [8], which\\nis a simpli\\fed variant of NES with an isotropic multivariate Gaussian and \\fxed variances \\u0006. OpenAI\\nES is closely related to the policy-based RL algorithm PEPG in the theoretical relation [70]. To\\nreduce variance and realize high-parallel ability, OpenAI ES has introduced several techniques such as\\nmirrored sampling of the perturbation, rank-based \\ftness shaping, virtual batch normalization, and\\nthe random seeds sharing strategy. As a result, OpenAI ES has achieved competitive performance on\\nMuJoCo and Atari games with over a million policy parameters by using thousands of CPU workers,\\ndemonstrating the advantages of ESs as a black-box optimization algorithm for complex RL tasks.\\nMoreover, OpenAI ES has been shown to be close to SGD with a large number of o\\u000bspring [71] and\\nresembles traditional \\fnite-di\\u000berence approximators [72].\\nSeveral diversity encouragement methods from EC such as novelty search (NS) and quality\\ndiversity (QD) have been introduced to enhance the exploration of OpenAI ES, encouraging agents\\nto exhibit diverse behaviors [6]. Hybrid algorithms NS-ES and NSR-ES can solve tasks with noisy\\nand deceptive rewards. Furthermore, progressive episode lengths (PEL) have been proposed to\\nimprove the learning e\\u000eciency in evaluating \\ftness values of samples [73]. PEL enables agents\\nto learn from simple tasks to complex tasks by dividing the time budget and episode lengths into\\nincreasing numbers of fragments concurrently.\\n4.1.3 CMA-ES based Methods\\nThe application of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) to RL was\\n\\frst proposed by Igel in 2003, who demonstrated that CMA-ES can achieve faster convergence than\\nseveral state-of-the-art GAs based neuroevolution algorithms on double pole balancing tasks using a\\nsingle hidden layer policy [74]. CMA-ES typically uses rank-based \\ftness shaping instead of absolute\\n\\ftness values to reduce its susceptibility to noise. However, the accuracy of rank still plays a critical\\nrole in performance. To address this, Heidrich-Meisner augmented CMA-ES with Hoe\\u000bding- and\\nBernstein-based racing algorithms to obtain a reliable rank, which led to faster convergence and\\nmore robust hyperparameter selection on single and double pole balancing tasks than several GAs\\nbased neuroevolutionary algorithms [75, 76].\\nRecently, Chen proposed a restart-based rank-1 ES (R-R1-ES), a simpli\\fed CMA-ES, to play\\nAtari games using a two-hidden-layer neural network, which is a groundbreaking work applying\\ne\\u000ecient CMA-ES to complex RL tasks [77]. R-R1-ES integrates a Gaussian-distributed model with\\ntwo mechanisms, including the adaptation of the number of parents and a restart procedure, and\\nhas achieved higher scores than OpenAI ES, canonical ES, NS-ES, and NSR-ES on a subset of Atari\\ngames. Despite the development of several high-e\\u000ecient CMA-ES variants, such as R1\\/Rm-ES [78],\\nLM-MA-ES [79], and fast CMA-ES [80] to deal with the time-consuming adaptation of the full\\n10 covariance matrix for large scale optimization problems with up to 10,000 dimensions, OpenAI ES\\nhas been veri\\fed on millions of policy parameters. Therefore, their potential for complex RL tasks\\nis yet to be studied.\\n4.2 GA based Methods\\nGAs have been widely adopted to optimize the weights and architectures for policy search in RL,\\nowing to their diverse encoding types. The GAs based methods have primarily focused on three\\nresearch topics: algorithmic frameworks, indirect encoding, and variation operators.\\n4.2.1 Algorithmic Frameworks\\nPure GAs based Frameworks In the 1990s, several studies utilized GAs to optimize policy\\nweights for pole balancing problems [81, 12]. Among these works, GENITOR, which represented\\nweight with real values instead of binary strings, improved the precision and e\\u000eciency of the search\\nspace. Subsequently, a range of works has been developed, with the most popular one being NEAT,\\nwhich can obtain a minimal neural network by adding nodes and connections from a smallest neural\\nnetwork without hidden nodes [13]. NEAT has several highlights, including genetic encoding that\\naligns corresponding genes easily when mating two genomes, historical markings that enable tracking\\nand matching of genes during crossover, and speciation within smaller niches to protect topological\\ninnovations. NEAT has since been improved and tailored to various tasks, such as evolving dynamic\\npolicies to adapt to environmental changes in a dangerous foraging task [82], evolving complex\\npolicy architectures in robot competition and coevolution tasks [83], video games [84], and strategic\\ndecision-making problems [85].\\nDespite the e\\u000bectiveness of NEAT, it does not fully optimize the weights under a potential\\narchitecture. Hence, ENAT was developed based on NEAT, which adopts the idea of incremental\\ngrowth from a minimal structure [86]. However, ENAT applies CMA-ES to fully optimize the weights\\nand introduces a compact genetic encoding to encode a tree-based program in a linear genome. As\\na result, ENAT can \\fnd better weights than NEAT with the same network size on a robot arm\\ncontrol task. To reduce the side e\\u000bect of topology change, CMA-TWEANN replaces the mutation\\nby random weights in NEAT with a seamless topology mutation by zero weights and applies CMA-\\nES to optimize weights [87]. However, whether weight optimization is more important than topology\\noptimization is still debatable. The weight-agnostic search method answers this problem by only\\nsearching the topology by NEAT without training of weights. This method has found policies with\\nminimal architectures in continuous control tasks [60].\\nIn the highly parallel framework of OpenAI ES, a simple genetic algorithm has been used to\\noptimize large-scale policies with millions of weights in Atari games and MuJoCo [88]. In a simi-\\nlar vein, a massively parallel method has been applied to search recurrent neural network (RNN)\\narchitectures using only mutation. This method has achieved high performance with orders of mag-\\nnitude fewer parameters than several state-of-the-art RL methods in MuJoCo tasks [89]. While\\nthese methods require huge computational resources, the training of large-scale policies using EC\\nmethods is still very low-e\\u000ecient. To improve the e\\u000eciency, a hybrid agent model consisting of a\\n11 large world model and a small controller model has been proposed to tackle complex RL tasks [90].\\nThe world model extracts low-dimensional features from real-world observations and predicts future\\nstates based on historical information. The controller model, such as a single-layer linear neural\\nnetwork, determines the actions to take by receiving current and predicted features to maximize the\\nexpected cumulative reward. The controller model has been evolved by EC methods in vision-based\\ngame tasks [91, 92]. Moreover, an end-to-end training of the whole agent model using GAs has\\nshown comparable performance in car racing tasks [93].\\nFrameworks Hybridizing GAs and RL A combination of GAs based methods (i.e., NEAT)\\nand TD methods (i.e., Q-learning) has led to the development of two methods, namely Lamarckian\\nNEAT+Q and Darwinian NEAT+Q. In these methods, NEAT is used to optimize the architectures\\nand initial weights of Q networks, while the policy weights are updated using backpropagation [94].\\nTo balance the exploration and exploitation of the EC method, \\\"-greedy selection and softmax selec-\\ntion in RL have been incorporated into NEAT. However, the high sample complexity of NEAT+Q,\\nresulting from the fully training of each candidate policy in highly stochastic domains, has prompted\\nthe proposal of an e\\u000ecient NEAT+Q method that reuses previous samples to train a population\\nof candidate policies [95]. A comparison study has suggested that EC methods are more e\\u000bective\\nwhen \\ftness can be rapidly evaluated in deterministic domains, whereas TD methods exhibit an\\nadvantage in fully observable but non-deterministic domains [96].\\nCooperative Coevolution In the \\feld of neural network optimization, the search space can\\nbecome excessively large when the numbers of the neural network input, output, architecture, and\\nweight are large. To overcome this challenge, Cooperative Coevolution (CC) based GAs can be\\nused, which decompose the problem into smaller components to reduce complexity and enable more\\ne\\u000ecient resolution through CC methods [97]. In a CC algorithm, each individual represents a partial\\nsolution, or a component of a complete solution, which is resolved by a species, or a set of individuals,\\nindependently. The individuals are evaluated based on their contributions to the complete solution.\\nThis approach introduces diversity and robustness in the maintenance of various components and\\nenables parallel search to improve training e\\u000eciency. The search granularity in the decomposition\\nmethods of neural networks indirectly in\\ruences search performance and e\\u000eciency. In neuron level\\nCC methods, weights connected to a neuron are grouped into a component. SANE is an earlier CC\\nalgorithm that evolves a population of hidden neurons for a neural network with a \\fxed architecture,\\nand has shown better performance than Q-learning and GENITOR on pole balancing tasks [98]. ESP\\nimproves the e\\u000eciency of SANE and supports the evolution of Recurrent Neural Networks (RNN)\\nby allocating a species for a hidden neuron and conducting variation within species [99]. NSP further\\ngroups the weights connected to a hidden neuron into \\fner granularity [100]. In synapse-level CC\\nmethods, each weight is considered a component. Based on ESP, CoSyNE groups each weight into\\na component and has shown better e\\u000eciency than SANE, NEAT, and ESP on pole balancing tasks\\n[101]. However, CoSyNE may not be applicable for large-scale neural networks since it cannot fully\\nexploit the weights to avoid inaccurate evaluation, which is a problem in neuron-level methods.\\nOther state-of-the-art decomposition methods include COVNET [102], Modular NEAT [103], and\\n12 CCNCS [104].\\n4.2.2 Indirect Encoding\\nResearch on indirect encoding has been promoted for two reasons. Firstly, direct encoding has\\nlimitations in scaling up to large-scale NN scenarios. Secondly, in biological genetic encoding, phe-\\nnotypes typically contain more genetic components than genotypes, and the mapping of phenotypes\\nto genotypes is indirect.\\nArti\\fcial embryogeny, which includes cellular encoding [105] and generative encoding [106], uti-\\nlizes a developmental phase by reusing genes or rules to evolve arti\\fcial systems from a small starting\\npoint [107]. In addition to repetition by reuse, the properties of physical space, such as symmetry\\nand locality, have motivated the design of encoding that seeks to discover regularity by local con-\\nnectivity. Compositional pattern producing networks (CPPNs) capture structural relationships and\\nare encoded by a composition of functions organized in the form of neural networks [108]. CPPNs\\ncan be trained in the same way as neural networks, and HyperNEAT is tailored to training CPPNs,\\nwhich can evolve increasingly complex expression patterns to capture the complete regularities of\\nproblem structures [109, 110]. The ability to learn from geometric regularity has enabled Hyper-\\nNEAT to be successfully applied to complex tasks such as checkers [111], Go [35], and Atari games\\n[112]. Additionally, several HyperNEAT variants have been proposed, such as adaptive HyperNEAT\\n[113] and ES-HyperNEAT [114].\\nBy modeling modularity as an optimization objective, NSGA-II (a multi-objective evolutionary\\nalgorithm) [115] has been applied to evolve CPPNs. This method performs better than HyperNEAT\\nby generating lower modularity of genotypes and phenotypes in a robotics task [116]. However,\\nCPPNs may lose continuity when mapping genotypes to phenotypes (i.e., a small change in the\\ngenotype may lead to a large change in the phenotype). Therefore, compressed encoding uses\\ndiscrete cosine transform (DCT) to reduce the dimensionality of the search space by exploiting the\\nspatial regularities of the weight matrix and obtains large-scale neural networks on a vision-input\\ncar driving task [117].\\nFurthermore, several works have combined indirect encoding and direct encoding to discover\\nregularities by indirect encoding and compensate for irregularities by direct encoding [118, 119, 120].\\n4.2.3 Variation Operators\\nVariation operators aim to preserve the characteristics of parents while introducing diversity, and\\nthey typically include crossover and mutation. Crossover combines the properties of more than one\\nparent, while mutation inherits the properties from one parent to a large extent. In earlier binary\\nencoding, the single-point crossover and \\rip mutation are widely used operators [28]. Since the\\nlength of the binary string determines the representation precision of a real number and in\\ruences the\\nvariation granularity, real encoding has been proposed to represent a weight as a real number directly.\\nAccordingly, the simulated binary crossover and the polynomial mutation have been proposed for\\ncontinuous search space [121]. Gaussian mutation is also widely used by adding a random value\\nfrom a Gaussian distribution to a real-encoded weight [88].\\n13 However, since NNs are sensitive to small modi\\fcations of weights, the above variation operators\\ntypically cause catastrophic forgetting of the characteristics of parents. Hence, imitation learning\\nor network distillation has been applied to variation operators for NNs, such as the state-space\\ncrossover [122], the Q-\\fltered distillation crossover [123], as well as the distilled topology mutation\\n[124]. Both crossover methods apply imitation learning to distill better behaviors of parents into\\no\\u000bspring. The mutation method \\frst generates an o\\u000bspring by augmenting topology and then\\npretrains the o\\u000bspring by distilling the behavior of its parent as a necessary initialization.\\nFurthermore, [125] proposes a family of safe mutation to deal with the catastrophic forgetting\\nproblem, where the mutation degree of each weight is scaled by the sensitivity of the weight to the\\nNN outputs, and thus an o\\u000bspring will not diverge too much from its parent. In contrast, [126]\\nproposes a di\\u000berent concept of safe mutation for safe exploration, which uses visited unsafe states\\nto explore safer actions.\\nApart from the catastrophic forgetting problem, the permutation problem (i.e., the same solution\\ncan be represented by di\\u000berent NNs) can lead to low-e\\u000ecient search when encoding schemes or vari-\\nation operators are not designed sophisticated [127]. Hence, two types of approaches are developed.\\nOne approach tailors encoding methods for evolving NN architectures. For example, NEAT designs\\na genetic encoding to track parent solutions by using an innovation number [13]. Another approach\\naligns neurons from two NNs by analyzing their functional correlations before crossover and provides\\na general method for di\\u000berent encoding [127].\\n4.3 Genetic Programming based Methods\\nGenetic programming (GP) is a popular method for solving complex RL tasks since a program can\\nemulate any model of computation given su\\u000ecient time and search space [128]. Three commonly\\nused representations in GP are the abstract syntax tree, Cartesian GP (CGP), and tangled program\\ngraphs (TPG). These representations are applied to various tasks such as evolving direct controllers,\\nsymbolic regression, and feature discovery for RL tasks.\\n4.3.1 Representations\\nThe syntax tree is a commonly used representation in GP, where programs are directly encoded as\\ngenomes. As shown in Figure 3a, nodes can represent functions, operations, variables, and constants,\\nand the tree outputs a unique program through tree traversal algorithms [28]. Typically, the tree\\ngrows randomly by crossover and mutation from a null node, but it can su\\u000ber from the bloat issue,\\nwhere programs grow in size without showing obvious \\ftness improvement over time if depth limiting\\nis not enforced.\\nTo address the bloat issue, Cartesian GP (CGP) uses an integer-array genome with a \\fxed length\\nto encode an executable tree graph [129], as depicted in Figure 3b. All nodes are placed in a grid\\nmanner and are indexed sequentially. The nodes in the \\frst column are input nodes. The integer-\\narray genome is divided into blocks, with each block consisting of three integer genes representing\\na single non-input node, where the third integer speci\\fes a function, and the former two integers\\nspecify the indices of its input nodes. The data \\row is from left to right in the graph, and the graph\\n14 +\\n*\\nx y 2sin\\nOutput programs:\\nxy+sin(2)(a) Syntax Tree\\nyx\\nz*\\n\\/+\\n++-0\\n1\\n20\\n1\\n1\\n2\\n0\\n13\\n4\\n56\\n7\\n83\\n4\\n3\\n4\\n4\\n5[012 120 013 341 340 450]\\n[01* 12+ 01\\/ 34 -34+ 45+]\\nOutput programs:\\nNode 6: xy \\u2013(y + z )\\nNode 7: xy + y + z\\nNode 8: xy + y \\/ z (b) Cartesian GP\\n{ }{ }{ }{ }{ }\\n{ }\\n{ }{ }\\n{ }\\n{ }{ }\\n{ }{ }{ }{ }\\n{ }\\nN A { }\\n{ }{ }\\n{ } { }{ }\\nTeam\\nProgram\\nAtomic ActionRoot Team (c) Tangled Program Graph\\nFigure 3: Three illustrative examples for the representations of GP in RL: syntax tree, Cartesian\\nGP (CGP), and tangled program graphs (TGP), respectively.\\ncan have multiple output programs. Finally, users can specify only one output among the multiple\\noutputs.\\nTangled Program Graphs (TPG) is a framework for organizing multiple programs into a structure\\nwith high modularity [130]. TPG evolves two populations: a Node (i.e., Team) population and a\\nProgram population, where the Node population constructs a good organization of multiple teams\\nof programs from the Program population, and the Program population discovers programs that\\noutput useful Atomic Actions. As shown in Figure 3c, the black point represents the root node that\\nreceives state inputs. Evaluation of an agent starts at the root node and reaches an Atomic Action\\nthrough a path in the direction of the arrow. This process only executes a fraction of the programs\\nfor a task, making TPG more e\\u000ecient than other neuroevolutionary algorithms that require covering\\nall topology for a decision.\\n4.3.2 Evolution of Direct Controllers\\nThe syntax tree directly corresponds to the parse tree created by compilers, and has been applied\\nto evolve controllers for robot control tasks [131], bipedal locomotion tasks [132], acrobat tasks,\\nand helicopter hovering tasks [133] through syntax tree-based GP. Integration of the tree-based GP\\nand RL has been tailored for real robots such that a precise simulator is not required in complex\\nrobot control tasks [134]. This method executes GP in a simpli\\fed simulator to generate simple\\ncontrollers, which are then adapted to a particular real robot by RL. It has outperformed Q-learning\\non two complex robot tasks. In addition, cellular encoding and the syntax tree have been integrated\\nto evolve NNs with particular structures [135]. Furthermore, a hybrid of NEAT and GP, known\\nas HyperGP, has been applied to evolve weights of CPPN, and showed similar performance to\\nHyperNEAT with signi\\fcantly fewer evaluations [37].\\n15 Both Cartesian Genetic Programming (CGP) and neural networks (NNs) can be viewed as ex-\\necutable graphs, which has motivated researchers to extend the \\rexible CGP representation to the\\nevolution of NNs. The CGP-based NN has been proposed to evolve both topology and weights for\\nNNs with feed-forward or recurrent architectures, outperforming NEAT, ESP, and CoSyNE in pole\\nbalancing tasks [136]. Additionally, CGP has been applied to evolving the parameters of trans-\\nfer functions such as the Gaussian function and the logistic sigmoid function, which improved the\\nperformance of NNs on a ball throwing task [137]. Furthermore, CGP has been applied to evolv-\\ning game-playing agents directly from high-dimensional pixel inputs of Atari games [138], and the\\nevolved programs are easier to understand than NNs.\\nTPG is tailored for visual RL tasks with high-dimensional pixel state inputs from environments,\\nand has been applied to 20 Challenging Atari games, where TPG exceeds DQN in 15 of the 20\\ngames, and further exceeds the human level in 7 of the 15 games [139]. Surprisingly, TPG requires\\nsigni\\fcantly lower computational resources and is without specialized hardware such as GPUs. Be-\\nsides, by interacting with environments, TPG introduces emergent modularity and thereby leads to\\ntask decomposition. Due to these advantages, TPG is capable of generating multi-task policies in\\nAtari games [140, 141], and has been successfully applied to the partially-observable ViZDoom [142]\\nand Dota2 [143].\\n4.3.3 Symbolic Regression\\nInterpretable RL is of high interest to academics and industrial areas, and interpretable controllers\\nare more likely to be employed especially in industrial systems. The symbolic regression of GP\\nis an e\\u000bective approach to interpretable RL by \\ftting the policy or value functions in a human-\\nunderstandable way. The syntax tree is naturally the appropriate representation due to its high in-\\nterpretability and understandability for humans. For example, the value function discovery method\\nhas been proposed to discover algebraic expressions of an obtained V-function by minimizing a sim-\\nulation error between the expressions and sampled V-function data [144]; the genetic programming\\nfor RL method has been proposed to generate simple algebraic policies by the data sampled from\\nthe world models [145].\\nIn addition to interpretability, symbolic regression can generate smoother and more adaptive\\nsymbolic approximators than numerical approximators, such as neural networks (NNs). For example,\\na variant of the single-node GP has been applied to evolve a smooth proxy of the V-function by\\nmaximizing the number of correct choices of actions for sampled training states [146]. In addition,\\nthe single-node GP has constructed a symbolic process model for model-based RL methods to reduce\\nthe number of training data and adapt to the dynamic system in real-time robot control tasks. This\\nmethod has advantages over NNs in that it requires no hyperparameter tuning and can generate\\nsmooth V-functions [147].\\n4.3.4 Feature Discovery\\nFeature discovery is the process of transforming input data into a form that can be more easily\\nprocessed by RL agents. Unlike most feature extraction methods in machine learning, where useful\\n16 features are extracted from inputs, feature discovery in RL may add features to aid in better learning\\nby agents. For example, in pole balancing tasks, the optimal policy can be learned faster by adding\\ntwo angle features of the pole.\\nIn the works of [148, 149], each individual encodes programs with multiple S-expressions (feature-\\nfunctions), where each S-expression corresponds to a unique feature. In [149], the number of useful\\nfeatures is \\fxed, while in [148], the number of useful features can vary within a range since the prior\\nis unknown in advance. The obtained features are human-readable, allowing for \\fne-tuning and\\nknowledge transfer during the process of feature discovery.\\n4.4 Discussion\\nIn general, EC based methods have shown great potential in solving complex RL tasks. Each method\\nhas its unique advantages and disadvantages, making them suitable for di\\u000berent situations. ESs are\\nsimple and e\\u000ecient but can be limited by the high-dimensional search space and \\ftness noise.\\nNES improves upon ES by using the reward gradient of all o\\u000bspring, but there is still room for\\nbetter exploitation of low-\\ftness o\\u000bspring. GA-based methods can employ safe mutation methods\\nto broaden their applicability. GP-based methods, such as CGP can evolve policies with better\\ninterpretability, and TPG is a unique method that is speci\\fcally tailored to visual RL tasks and\\ncan solve challenging games with high-dimensional pixel inputs while using fewer computational\\nresources than other EC based methods.\\nIt is also worth noting that there are still areas for improvement in EC-based RL methods. The\\nhighly-parallel framework of OpenAI ES requires a large number of CPU resources, which is low-\\ne\\u000ecient for large-scale image inputs, and allocating more resources to promising samples may enable\\nbetter performance within limited computational resources. Furthermore, ENAS has shown great\\npotential in automatically designing the architecture of deep NNs for image classi\\fcation, but more\\nresearch is needed to explore its applicability to policy search of complex RL tasks. Overall, as EC\\nbased RL methods continue evolving and improving, their potential in solving complex RL tasks\\nmakes them an exciting area of research.\\n5 Evolutionary Computation in Exploration\\nIn RL, agents must interact with environments by taking actions and observing environmental\\nstates to collect trajectories to improve their behaviors. The learning e\\u000eciency of agents relies on\\nthe data that they gather. However, if an agent only visits a small portion of its environment, its\\nknowledge will be limited, leading to suboptimal decision-making. Therefore, diverse exploration of\\nthe environment is desired. Agents typically explore environments by adding noise to the action space\\nor to the parameter space of their policies. In state-of-the-art methods, the \\u000f-greedy exploration\\nmethod encourages agents to take di\\u000berent feasible actions instead of the current optimal action for\\na state with a certain probability [1], while the parameter space noise method adds Gaussian noise\\nto the policy weights to change the original output actions [150].\\nTo achieve e\\u000ecient exploration, four key challenges must be addressed [151]. First, the state-\\n17 action space is often large, making it challenging for the agent to access the e\\u000bective space. Second,\\nthe environment returns sparse and delayed rewards, meaning that agents cannot receive timely\\nand informative feedback on their behaviors. Third, real-world environments often contain highly-\\nrandom and unpredictable things (i.e., white-noise problems), making it challenging for the agent to\\ndistinguish important information from unimportant information, leading to unstable and ine\\u000ecient\\nexploration. Fourth, exploration in multi-agent RL is more challenging since the state-action space\\nincreases exponentially, and agents must explore coordinately to achieve local and global exploration\\ntrade-o\\u000bs.\\nTo deal with these challenges, EC methods for RL enable extreme exploration, competition and\\ncooperation, and massive parallelization by maintaining a set of diverse agents during search. In\\nthe implementation, EC methods introduce exploration for RL from two perspectives: diversity\\nencouragement methods , especially for neuroevolution, and evolution-guided exploration methods for\\ntraditional RL algorithms.\\n5.1 Diversity Encouragement Methods\\nIn diversity encouragement methods for neuroevolution, EC as a policy search method directly\\nevolves a set of diverse agents by encouraging diversity in the parameter space or in the behav-\\nior space. Most initial work focuses on the parameter space with the purpose of avoiding local\\noptima. The widely applied diversity maintenance techniques have speciation (i.e., niching) and\\n\\ftness sharing [13]. Speciation divides a population into a number of species according to their\\ngenetic similarity, and \\ftness sharing enables individuals with similar genomes to share their \\ftness\\nso that innovation can be protected in their own species. However, diversity in the parameter space\\ncannot ensure diversity in the behavior space, since there are in\\fnite ways for NN weight settings to\\nproduce the same outputs of behaviors. Therefore, a number of recent approaches from or inspired\\nby the diversity maintenance techniques of EC have been introduced into RL to directly reward\\ndiverse behaviors or novel states, such as novelty search [41], quality diversity [152], surprise search\\n[153], evolvability search [154], and curiosity search [155].\\n5.1.1 Novelty Search\\nNovelty search (NS) abandons the \\ftness objective while rewarding novel behaviors that are di\\u000berent\\nfrom previous behaviors. Behavior characterizations (BCs) should be \\frst designed to map the high-\\ndimensional search space into a lower-dimensional behavior space. Then, to measure the novelty of a\\nnewly generated individual, a novel metric is de\\fned as the task-speci\\fc distance between behaviors.\\nAfter that, NS can be easily integrated into EC algorithms with little change of replacing the \\ftness\\nobjective with the novel metric. NS has been applied to NEAT and outperformed \\ftness-based\\nmethods on the deceptive T-Maze and biped walking tasks [156, 41]. Empirical studies demonstrated\\nthat NS can bring unique advantages over \\ftness-based EC methods in overriding the deceptiveness\\nof most \\ftness functions and making the evolutionary process more open-ended.\\nHowever, when faced with a task with a large state-action space, pursuing novelty alone does not\\nperform better than \\ftness-based methods [157]. Thus, in high-dimensional evolutionary robotics\\n18 tasks, NS is used to augment \\ftness-based EC methods as a diversity maintenance technique or\\nserved as the second objective to be optimized simultaneously with the \\ftness objective [158]. NS is\\naugmented with local competition (NSLC) to create a set of diverse locomotion evolutionary robotics\\n[159]. Additionally, NS is combined with OpenAI ES in three ways. The \\frst version, NS-ES, replaces\\nthe gradients of expected rewards with the gradients of expected novelty, and the other two versions,\\nNSR-ES and NSRA-ES, trade o\\u000b the gradients of expected rewards and novelty [6]. Empirical studies\\nhave shown that the three versions performed better than OpenAI ES on Humanoid Locomotion\\nand Atari tasks with deceptive traps. NS is also combined with sub-population to promote directed\\nexploration in the population-guided NS method [160]. In general, all the above empirical studies\\nhave demonstrated that increasing behavioral diversity makes problems more easily resolved.\\n5.1.2 Quality Diversity\\nThe pursuit of both \\ftness and novelty has led to the development of quality diversity (QD) algo-\\nrithms, which aim to \\fnd a large set of both diverse and high-performing solutions in a single run.\\nThe set of solutions aims to cover as many solution types, or behavioral characterizations (BCs),\\nas possible and \\fnd the best solution for each type. Two main state-of-the-art QD algorithms are\\nnovelty search with local competition (NSLC) and the multi-dimensional archive of phenotypic elites\\n(MAP-Elites) [161]. A comparative study of the two algorithms in a set of maze tasks has revealed\\nthat the selection of BCs is a crucial and challenging issue, since it is task-dependent and should\\nalign with quality, otherwise it can change the di\\u000eculty of \\fnding a good solution [152]. Therefore,\\na number of automated BCs methods have been proposed to improve exploration e\\u000eciency, such\\nas using dimensionality reduction methods to autonomously learn BCs [162, 163], or mapping the\\nhigh-dimensional parameter space into a low-dimensional manifold in which a high-density of good\\npolicies is located [164].\\nMoreover, exploration e\\u000eciency can be improved from other aspects. Several e\\u000ecient behavioral\\ndiversity measurement methods have been proposed to measure the diversity of the entire popula-\\ntion by determinants of behavioral embedding of policies [165] or use a string edit metric to measure\\nbehavioral distance [166]. To improve evaluation e\\u000eciency, the quality and novelty of new candi-\\ndate solutions are predicted by a neural network (NN) in open-ended robot object manipulation\\ntasks [167]. To improve sample e\\u000eciency, a few-shot quality-diversity optimization method learns\\na population of prior policies for the initialization of QD [168]. To improve selection e\\u000eciency, an\\nevolutionary diversity optimization algorithm with clustering-based selection selects a high-quality\\npolicy in each cluster for reproduction [169].\\nIn addition, QD o\\u000bers the potential for open-ended innovation that RL agents can generate and\\nlearn their own never-ending curriculum without human intervention. The paired open-ended trail-\\nblazer (POET) algorithm has been proposed to generate increasingly complex environments and\\noptimize their solutions concurrently by combining the methods of NS, MAP-Elites, and minimal\\ncriterion coevolution [170, 171]. The empirical studies on 2-D bipedal-walking obstacle-course tasks\\nhave demonstrated that solutions found by POET for challenging environments cannot be found\\nby directly learning from scratch for the same environments. Further, a sample-e\\u000ecient QD envi-\\n19 ronment generation algorithm is proposed to apply a deep surrogate model to predict behaviors of\\nagents in new environments [172]. Surprisingly, the open-ended coevolution of environments and\\nsolutions has provided novel ideas for addressing complex tasks.\\n5.1.3 MAP-Elites\\nThe focus of QD algorithms is mainly on MAP-Elites, which divides the behavior space of BCs\\ninto discrete bins according to the number of discretizations required for each dimension. Each bin\\nrecords the best-found solution, and only one solution replaces a previous one if it outperforms the\\nprevious one both in terms of quality and diversity. MAP-Elites has been applied to generate elites of\\ndiverse behaviors (e.g., walking strategies) to help a robot adapt quickly to various kinds of damages\\n[10], and has achieved better performance and robustness than PPO for simulated hexapod robot\\ntasks [173]. However, MAP-Elites su\\u000bers from a scaling-up limitation in that the dimensionality of\\nBCs must be low since the number of discrete bins increases exponentially with the dimensionality\\nof BCs. To address this limitation, CVT-MAP-Elites uses centroidal Voronoi tessellation instead of\\ngrid-shaped bins to divide the behavior space into a desired number of regions [174]. In addition,\\nseveral improvement methods have been proposed to scale up MAP-Elites to high-dimensional tasks.\\nThese include biased cell sampling [175] and gradient-based mutation operators [176] for e\\u000ecient\\nreproduction, approximated gradient [177], policy gradient assisted MAP-Elites [178], and deep\\nsurrogate-assisted MAP-Elites [179] for the acceleration of optimization.\\nWhen facing hard-exploration tasks with sparse and deceptive rewards, RL algorithms, even with\\nintrinsic motivation, perform poorly due to two challenges: detachment and derailment. Contem-\\nporary RL algorithms do not remember well-explored states (detachment), and random exploration\\nmay not lead back to well-explored states (derailment). Hence, [4, 180] proposed Go-Explore, a\\nfamily of QD algorithms based on MAP-Elites. Go-Explore follows the key ideas of remembering\\nstates, returning to them (GO), and exploring from them (Explore). Go-Explore has greatly sur-\\npassed state-of-the-art RL algorithms on two challenging games such as Montezuma's Revenge and\\nPitfall.\\n5.1.4 Surprise Search\\nSurprise search is a new method of evolutionary divergent search that rewards deviation from the\\nexpected solution [153, 181]. In contrast, novelty search rewards deviation from the prior solutions.\\nSurprise search models the prediction of expected behavior and derivation. The expectation is based\\non the reasoning about past information, and thus surprise search can be viewed as a temporal\\nnovelty process. Both surprise search and novelty search are divergent search variants of QD, and\\ntheir combination along with local competition has led to comparable \\ftness, higher e\\u000eciency,\\nand better robustness (i.e., exploration and behavioral diversity) than novelty search on 60 highly\\ndeceptive maze navigation tasks [181].\\n20 5.1.5 Evolvability Search\\nEvolvability search is a new class of EC algorithms where the \\ftness function is a direct measure\\nof the evolvability of an individual [154]. Evolvability is the potential for the future evolution of\\nan individual. Evolvability search calculates the behavioral diversity of immediate o\\u000bspring of an\\nindividual to estimate its future potential for diversity and then directly selects individuals with\\nbetter potential diversity to enter into the next environmental selection. Encouraging behavioral\\ndiversity increases the adaptive ability of a lineage. Though resembling diversity-seeking methods\\nsuch as novelty search, evolvability search outperforms novelty search on maze navigation and biped\\nlocomotion tasks [154]. However, evolvability search is computationally expensive due to its \\ftness\\nevaluation process.\\n5.1.6 Curiosity Search\\nCuriosity search is a class of intrinsic motivated methods that learn intrinsic reward signals to\\nenable agents to explore their environments. Exploration by intrinsic curiosity is a widely used\\nmethod in RL algorithms, where intrinsic curiosity is used to complement the extrinsic rewards [182],\\npredict the sequences of future actions or states [155], and achieve self-generated goals [183, 184].\\nGoal exploration processes (GEP) from EC methods explore robustly, designing a set of behavioral\\nfeatures (i.e., goals) based on the outcome trajectories of policies, and then exploit around these\\ngenerated goals by directed behavioral diversity without being aware of external rewards. GEP has\\nbeen integrated with o\\u000b-policy RL methods to exploit policy parameters, and the diverse samples\\ngenerated by GEP can be inserted into the replay bu\\u000ber of deep deterministic policy gradient\\n(DDPG) for training [184]. The intrinsically motivated GEP method integrates curiosity search\\nand GEP to discover and acquire skills by self-generation, self-selection, self-ordering, and self-\\nexperimentation of learning goals [183]. Additionally, [185] rewards intra-life novelty to encourage\\nagents to explore new states within their lifetime. This method discretizes the pixel space into\\ncuriosity grids and rewards agents for visiting new locations on the grids. In contrast to the across-\\ntraining novelty of novelty search, curiosity search can revisit previously visited potential states.\\n5.2 Evolution-guided Exploration Methods\\nIn evolution-guided exploration methods, the Evolutionary Reinforcement Learning (ERL) method\\nwas the pioneer [9]. Since then, a number of works sharing the framework of ERL have been\\nproposed. In these methods, EC introduces exploration in mainly two ways: EC agents generate\\ndiverse experiences stored in the replay bu\\u000ber for the training of o\\u000b-policy RL methods (e.g., DDPG,\\nTD3, SAC), or RL agents are directly updated using the gradient information of EC agents.\\n5.2.1 Use of Diverse Experiences\\nERL is a basic framework for evolution-guided exploration methods that has outperformed PPO,\\nDDPG, and GA on MuJoCo continuous control benchmarks [9]. In this method, EC employs a\\npopulation of agents to explore the parameter space to generate diverse experiences for the training\\n21 of o\\u000b-policy RL agents, and periodically copies the RL agent into the EC population to inject\\ngradient information into evolution. As a result, ERL is able to deal with the challenges of sparse\\nrewards, ine\\u000bective exploration, and brittle convergence properties. EC is indi\\u000berent to the reward\\nsparsity by using an episodic \\ftness metric and enables diverse exploration and introduces redundant\\ninformation with a population of actors. The periodic injection of gradient information of RL into\\nEC deals with the ine\\u000ecient exploration issues of EC.\\nA more general framework, Collaborative ERL (CERL), maintains a population of TD3 agents\\nto optimize over di\\u000berent hyperparameters (e.g., discount rate \\r) and applies a resource manager\\nto allocate computational resources to the agents adaptively according to their cumulative returns\\n[3]. CERL has outperformed ERL and TD3 in Humanoid and Swimmer tasks, where state-of-the-\\nart RL algorithms are highly sensitive to their hyperparameters. The competitive and cooperative\\nheterogeneous DRL (C2HRL) leverages the advantages of both gradient-based and gradient-free\\nagents and introduces two agent management mechanisms to compete for computational resources\\nand share exploration experiences [186]. C2HRL has shown faster convergence than CERL on\\nMuJoCo tasks.\\nTo generate legal o\\u000bspring, more sophisticated variation operators such as safe mutation [125,\\n126] and Q-\\fltered distillation crossovers [122] have been introduced into the proximal distilled\\nERL algorithm [123] and the safe-oriented search method [126]. Additionally, ERL is augmented\\nwith imitation learning, where RL agents learn from the experiences sampled by high-\\ftness EC\\nindividuals, and low-\\ftness EC individuals learn from RL agents by imitating behavior patterns\\n[187]. This method has outperformed DDPG and ERL on four MuJoCo tasks.\\nHowever, the EC part of ERL applies undirected exploration by adding noise to the parameters.\\nHence, directed exploration methods such as NS, QD, and curiosity search have been applied to cover\\nthe state-action space more uniformly and e\\u000eciently. For example, GEP-PG applies curiosity search\\nto generate diverse targeted samples for the training of DDPG [184]. In addition to adding noise\\nin the parameter space, EC enables the introduction of action noise for o\\u000b-policy RL algorithms.\\nThe evolutionary action selection-TD3 (EASTD3) uses samples generated by RL agents to form an\\nEC population, applies particle swarm optimization (PSO) to evolve continuous action values, and\\n\\fnally uses the best actions to guide action selection for RL agents [188]. EASTD3 has shown better\\nperformance than ERL, PDERL, CERL, and TD3 on MuJoCo tasks.\\nOther exploration methods in robotics include evolving a foot trajectory generator to provide\\ndiversi\\fed motion priors to guide policy learning [189] and augmenting NS with multiple behavior\\nspaces to deal with the challenge of automated data collection in robotic grasping tasks [190].\\n5.2.2 Use of Gradient Information\\nIn addition to the combinations of GAs and o\\u000b-policy RL algorithms in the ERL variants, the\\ncross-entropy method (CEM) has been combined with TD3 or DDPG to create CEM-RL [191]. In\\nCEM-RL, the gradient information of the RL agent is directly applied to half of the CEM agents at\\neach iteration to increase training e\\u000eciency. Asynchronous ES-RL, based on CEM-RL and OpenAI\\nES, has been developed by [192] to integrate ES with o\\u000b-policy RL methods and improve time\\n22 e\\u000eciency and performance over ERL and CEM-RL. Since ESs are similar to gradient-based RL\\nmethods, they are naturally applied to the EC loop of ERL to share gradient information with\\nRL methods. The combination of ES and SAC, called ESAC, enables e\\u000bective exploration in the\\nparameter space [193]. ESAC has obtained improved performance over SAC, TD3, PPO, and ES\\non many MuJoCo and DeepMind control suite locomotion tasks.\\nERL and its variants have been applied only to the o\\u000b-policy actor-critic methods. Therefore,\\nSupe-RL has been proposed by [194] to generalize to any RL methods using soft updates for policy\\nevolution. Supe-RL generates a set of children by adding Gaussian mutation to the policy weights,\\nand then soft updates the best individual from the children periodically or keeps the weights the\\nsame to avoid detrimental behaviors. Supe-RL has outperformed ERL and PPO on several MuJoCo\\ntasks. Additionally, [195] has proposed the gradient-evolutionary algorithm with temporal logic for\\non-policy methods.\\n5.3 Discussion\\nWhile QD algorithms have shown promise in encouraging diversity in neuroevolution and potentially\\nrealizing the third pillar of AI-generating algorithms (AI-GAs) [196], their e\\u000bectiveness relies heavily\\non the selection of appropriate behavioral characteristics (BCs). Ensuring that the extraction of BCs\\naligns with quality is crucial to the success of QD, as otherwise, it may perform worse than other\\nmethods like NS. This highlights the importance of careful consideration and experimentation when\\nselecting BCs for QD algorithms.\\nAnother important consideration when using evolution-guided exploration methods in o\\u000b-policy\\nRL algorithms is the alignment between the EC population and the RL agents. If the EC population\\nis too di\\u000berent from the RL agents, the experiences or gradients generated by EC may not be e\\u000ecient\\nfor updating the RL agents. This underscores the need to carefully design and tune EC algorithms\\nto ensure that they are suitable for the speci\\fc RL tasks at hand. Additionally, it may be bene\\fcial\\nto explore hybrid methods that combine EC and gradient-based methods to leverage the strengths\\nof both methods.\\n6 Evolutionary Computation in Reward Shaping\\nThe reward signal is crucial in re\\recting the task objective in RL. However, in many scenarios, the\\nreward is sparse, making it di\\u000ecult for agents to learn useful information. To tackle this issue,\\nreward shaping has been introduced, which enhances the original reward with additional shaping\\nrewards. These subrewards provide feedback about the progress of the task, adjust the importance\\nof di\\u000berent aspects of the task, or learn proxy rewards [197]. Empirical studies have shown that\\nreward shaping can reduce the amount of exploration and accelerate convergence [198]. Despite\\nthese bene\\fts, reward shaping still faces several challenges. Firstly, it can alter the problem itself.\\nSecondly, designing appropriate subrewards requires expert domain knowledge. Thirdly, achieving\\na balance between multiple subrewards requires careful manual tuning. Finally, credit assignment\\nis a di\\u000ecult problem in multi-agent reinforcement learning. EC has been applied to deal with\\n23 these challenges by the evolution of reward functions andhyperparameters for both single-agent and\\nmulti-agent RL.\\n6.1 Evolution of Reward Functions\\nPotential-based reward shaping, which originated from the potential energy, was proposed in the\\n1990s to deal with the challenge of changes in the problem itself [199]. The method derives a\\npotential-based shaping function that can guarantee its consistency with the optimal policy. How-\\never, designing the potential function is still an open question, requiring extensive manual search to\\nproduce acceptable results. Therefore, the reward network in [200] employs potential-based reward\\nshaping, representing the potential function as a neural network whose weights are optimized by\\nnatural evolution strategies (NES) in a highly parallel way, as in OpenAI ES. The reward network,\\nalong with the proposed synthetic environments (both trained by NES), is robust to hyperparameter\\nvariation and can be transferred to unseen agents.\\nIn contrast, a general computational reward framework from the evolutionary perspective achieves\\nthe optimal reward function by maximizing the expected \\ftness over the distribution of environ-\\nments [201]. Experiments in a hungry-thirsty task have shown that the optimal reward function can\\ncapture both physical regularities across environments and speci\\fc properties of agent-environment\\ninteractions. Moreover, maximizing the expected \\ftness can lead to the emergence of interesting\\nreward functions such as intrinsic and extrinsic rewards. As a result, the framework avoids the issues\\nof designing task-dependent subrewards. However, the automated quasi-exhaustive search used for\\n\\fnding good reward functions is quite time-consuming. Hence, PushGP has been applied to \\fnd\\nreward functions e\\u000eciently, which can discover common features of environments and reduce the\\nexpensive costs for sim-to-real transfer in robot control tasks [202]. Other EC methods to obtain\\nintrinsic rewards include hyperparameter tuning for reward shaping [203], symbolic reward search\\n[204], and exploration methods such as novelty search [205] and curiosity search [155] (demonstrated\\nin Section 5).\\nIn a multi-agent Cyber Rodent robots task, the parameters of an intrinsic reward function are\\nevolved to dynamically control the exploration for multiple hand-coded extrinsic rewards [203]. In\\nseveral MuJoCo and Atari tasks, dense intrinsic reward functions are evolved directly by symbolic\\nregression for the purpose of obtaining interpretable low-dimensional reward functions [204]. In a\\ngrounded communication environment, the multi-agent evolutionary reinforcement learning (MERL)\\nmethod maintains a population of evolving teams by neuroevolutionary algorithms to achieve sparse\\nteam-based rewards and optimizes agent-speci\\fc policies by gradient-based methods to achieve a\\nlocal reward for each agent [206]. MERL has outperformed the state-of-the-art MARL algorithm\\nMADDPG [207]. In tasks with both spatial and temporal constraints (i.e., a team of agents should\\ncomplete a task in temporal and spatial order), the multi-agent evolution via dynamic skill selection\\n(MAEDyS) method \\frst decomposes a task into subcomponents with local rewards and then applies\\na coevolutionary method to optimize multiple local rewards and a global reward [208].\\n24 6.2 Hyperparameter Optimization Methods\\nThe issue of manual tuning can be addressed through hyperparameter optimization (HPO) methods.\\nFor example, the AutoRL method applies large-scale HPO to the parameterized reward function and\\nthe neural network (NN) architecture of the policy [209]. The method \\frst learns the reward function\\nusing evolutionary strategies with \\fxed NN architectures of the actor and critic, and then learns\\nNN architectures by only adjusting the number of neurons in each layer with the previously learned\\nreward. Although it can \\fnd robust policies for point-to-point robot navigation tasks, it is still\\nrelatively ine\\u000ecient. Besides, AutoRL has been applied to several MuJoCo tasks and outperformed\\nstate-of-the-art algorithms such as SAC and PPO [197]. In addition, in [49], a number of weights in\\nthe hand-designed potential-based reward shaping is evolved together with other hyperparameters\\nto control the trade-o\\u000b between exploration and exploitation in a robot foraging task.\\nThe credit assignment issue in tasks with sparse rewards is quite challenging, as estimating\\nthe contribution of an agent is particularly hard, making it di\\u000ecult to optimize team rewards. A\\nnumber of works have dealt with this issue using HPO methods. The state-of-the-art population-\\nbased training (PBT) has been used to automatically learn dense internal rewards for the popular\\n3D multiplayer \\frst-person video game [50], and optimize the relative importance between a set\\nof dense shaping rewards along with their discount rates automatically for the continuous multi-\\nagent MuJoCo soccer game [52]. Both methods aim to align the myopic shaping rewards with the\\nsparse long-horizon team rewards and generate cooperative behaviors. In addition, [210] deals with\\nintertemporal social dilemmas by trading o\\u000b collective welfare and individual utility. Speci\\fcally,\\na shared intrinsic reward network takes features input from all agents, while each agent trains a\\ndistinct policy network in each episode. Then, PBT is applied to optimize the weights of the reward\\nnetwork and other hyperparameters to evolve altruistic behavior.\\n6.3 Discussion\\nThe use of EC for reward shaping has shown promising results in addressing the challenges of sparse\\nrewards in RL. Potential-based reward shaping, a method originating from potential energy, can\\nguarantee consistency with the optimal policy. A computational reward framework can achieve\\noptimal reward functions through a quasi-exhaustive search, and PushGP can e\\u000eciently \\fnd reward\\nfunctions that discover common features of environments. Hyperparameter optimization methods,\\nsuch as AutoRL, can be used for parameterized reward functions and neural network architecture\\ntuning. In multi-gent RL, PBT has been used to automatically learn dense internal rewards and\\noptimize the relative importance between shaping rewards to generate cooperative behaviors.\\nAlthough EC methods o\\u000ber an automated and e\\u000ecient approach to reward shaping in RL, relying\\nsolely on EC methods for reward shaping can be ine\\u000ecient. To mitigate this, incorporating prior\\nknowledge can be bene\\fcial. Intrinsic rewards can guide agents to explore interesting parts of the\\nstate space, while extrinsic rewards de\\fne the task's objective. EC methods can generate intrinsic\\nrewards and dynamically tune their weights to address the issue of sparse rewards.\\n25 7 Evolutionary Computation in Meta-RL\\nRL has proven successful in tackling complex tasks, but it often requires a large number of samples\\nto learn from scratch for each task. Moreover, the choice of a pre-speci\\fed RL algorithm can impact\\nthe performance, such as cumulative rewards or sample e\\u000eciency. To address these challenges,\\nmeta-RL seeks to develop a general-purpose learning algorithm that can adapt to di\\u000berent tasks. In\\nother words, it aims to leverage knowledge from previous tasks to facilitate fast learning in new ones.\\nMeta-RL can handle various scenarios, including learning in similar environments from a single task\\nor substantially distinct environments from multiple tasks.\\nFrom the perspective of optimization-based methods, meta-RL can be formulated as a bi-level\\noptimization problem, where the inner level involves learning an agent using standard RL techniques,\\nwhile the outer level optimizes RL con\\fgurations such as policy update rules, hyperparameters, and\\nreward formulation to achieve a meta-objective. Gradient-based methods [211], RL methods [212],\\nand EC methods [213] can be used to optimize either level. EC methods are particularly promising\\nfor meta-RL since they are applicable for non-di\\u000berentiable meta-objectives and can avoid the high\\ncomputational overhead of high-order gradients. Speci\\fcally, EC methods have been introduced in\\nvarious aspects of meta-RL, such as parameter initialization ,loss learning ,environment synthesis ,\\nand algorithm generation .\\n7.1 Parameter Initialization\\nParameter initialization is a critical aspect of meta-RL that aims to \\fnd a suitable policy initializa-\\ntion such that good general performance can be achieved with only a few gradient steps over di\\u000berent\\nenvironments sampled from a distribution of tasks. The model-agnostic meta-learning (MAML) is\\na state-of-the-art method that formulates the learning of an easily-adaptable policy as an optimiza-\\ntion problem with a meta-objective of minimizing the loss of a small number of gradient steps on\\na new task [211]. MAML has no requirement for the model representation and has been applied\\nsuccessfully to various tasks, including regression, classi\\fcation, and RL. However, estimating the\\nsecond derivatives of the reward function is challenging using backpropagation, and policy gradient\\nmethods have inherently high variance. To overcome these limitations, ES-MAML integrates MAML\\ninto the ES framework, which avoids calculating the second derivatives by Gaussian smoothing of\\nthe MAML reward [214]. Additionally, ES-MAML simultaneously optimizes hyperparameters and\\ninitial parameters, leading to better performance and exploration on tasks with sparse rewards.\\nAlternatively, the Baldwin evolutionary methods, which involve reinitializing parameters and hy-\\nperparameters, have been used for meta-learning when MAML is not applicable [215]. However,\\nit is worth noting that meta-learning parameter initialization is limited to a single task or similar\\ndistribution of tasks.\\n7.2 Loss Learning\\nMeta-learning the loss has demonstrated generalization ability across substantially di\\u000berent tasks,\\nincluding out-of-distribution tasks. Evolved Policy Gradient (EPG) is a representative work that\\n26 meta-learns a di\\u000berentiable loss function parameterized by temporal experiences [213]. EPG has two\\noptimization loops: an inner loop learns a policy to minimize the loss, and an outer loop learns the\\nloss such that an agent trained by the loss can achieve high expected returns in a task distribution.\\nThe parameters of the loss function, represented by a neural network, are optimized by OpenAI ES,\\nwhere a population of workers runs in parallel to obtain the update gradients of the loss. EPG has\\ndemonstrated better generalization than MAML on several MuJoCo tasks, although its e\\u000bectiveness\\nis limited to a small family of tasks at a time.\\nSeveral works have meta-learned interpretable loss functions by symbolic regression of GP. For\\nexample, Co-Reyes et al. [216] generate symbolic loss functions for general policy update rules\\nby regularized evolution and directly evolve a population of RL algorithms. Regularized evolution\\nremoves the oldest solutions to prevent over\\ftting to training noise, rather than removing the worst\\nsolutions from the whole population, such that algorithms retrained well are more likely to remain in\\nthe population. Experiments on complex tasks demonstrate that, when learning from scratch, this\\nmethod can rediscover DQN, and when inserting the existing algorithm DQN into the population\\nas bootstrap, the method can further improve generalization of DQN.\\nAdditionally, to simultaneously meet the requirements of performance, generalizability, and sta-\\nbility, MetaPG formulates the requirements as three meta-objectives and applies NSGA-II to discover\\nnew RL algorithms by designing directed acyclic graphs, a symbolic representation [217]. Experi-\\nments on three continuous control tasks show that MetaPG has improved both the performance and\\ngeneralizability of SAC by using a graph-based implementation of SAC to initialize the population\\n[24].\\n7.3 Environment Synthesis\\nEnvironment synthesis aims to generate synthetic data that can enhance the training e\\u000eciency of\\nRL models. In the world models approach, for instance, the MDN-RNN model receives compressed\\ninput from the agent's visual perception and predicts future states to improve training e\\u000eciency\\n[90]. However, in real-world applications, it is essential to consider both states and rewards. There-\\nfore, [200] proposes a method that simultaneously learns synthetic environments (SEs) and reward\\nnetworks (RNs) to generate synthetic MDPs that mimic real-world environments. The method rep-\\nresents SEs and RNs as NN proxies and applies a bi-level optimization method to optimize them for\\nbetter performance on real environments. The inner optimization trains RL agents on the proxies,\\nwhile the outer loop evolves parameters of SEs and RNs using NES to maximize performance on\\nreal environments. This method allows training competitive agents with fewer interactions with\\nreal environments on MuJoCo tasks. In [50], the population based training (PBT) [43] has been\\napplied to meta-learn the internal rewards and hyperparameters of the RL algorithm simultaneously.\\nThis method can be viewed as a two-tier optimization problem where the inner tier maximizes the\\nexpected discounted internal rewards, and the outer tier maximizes a meta-reward based on the\\ninternal rewards and hyperparameters. The joint optimization of policy and the RL process itself\\nenables the training of better agents in large-scale complex tasks.\\n27 7.4 Algorithms Generation\\nWhen rewards from extrinsic environments are sparse, curiosity can provide intrinsic motivation for\\nagents to explore environments. In [218], learning how to explore is formulated as a meta-learning\\nproblem of generating curious behaviors, where e\\u000bective curiosity algorithms can be generated by\\nlearning proxy rewards for exploration. The meta-learning method has two optimization loops: the\\nouter loop evolves a population of curiosity algorithms (represented as programs) by GP to learn\\nintrinsic reward signals dynamically, and the inner loop performs a standard RL pipeline using the\\nlearned reward signals. The method designs a domain-speci\\fc language, represented as directed\\nacyclic graphs, to generate programs that include building blocks such as NNs, objective functions,\\nensembles, bu\\u000bers, and other regressors as polymorphic data types. Empirical studies have shown\\nthat the method can discover algorithms similar to novelty search and generalize across a much\\nbroader distribution of environments.\\n7.5 Discussion\\nMeta-RL has undoubtedly made signi\\fcant progress in enabling RL to learn new tasks e\\u000eciently\\nand generalize across di\\u000berent tasks. However, the number and complexity of tasks that can be\\nsolved using meta-RL are still limited, particularly in real-world applications. Furthermore, the\\ncomputational cost of meta-RL is high due to the concurrent optimization of two loops and training\\nover a large number of tasks. Therefore, exploiting the model-agnostic and highly-parallel properties\\nof EC is a promising direction to unlock the full potential of meta-RL. EC methods can avoid the high\\ncomputational overhead of high-order gradients and can handle non-di\\u000berentiable meta-objectives,\\nwhich are challenging for gradient-based methods. The use of EC methods in various aspects of meta-\\nRL, such as parameter initialization, loss learning, environment synthesis, and algorithm generation,\\nhas demonstrated promising results. With further research and development, EC-based meta-RL\\nhas the potential to enable RL to learn and generalize across more complex tasks and to be more\\ncomputationally e\\u000ecient in real-world scenarios.\\nAlthough EC methods have shown great potential in meta-RL, there are still several challenges\\nto overcome. One challenge is the high dimensionality of the search space, which can be very large,\\nespecially when dealing with complex environments or when optimizing both policy and hyperpa-\\nrameters. This can lead to slow convergence and high computational costs. Another challenge is\\n\\fnding a balance between exploration and exploitation. EC methods often rely on some form of\\nexploration to \\fnd good solutions, but too much exploration can result in wasted computational re-\\nsources and slow progress. To address these challenges, future research in EC-based meta-RL could\\nfocus on developing more e\\u000ecient search algorithms, reducing the dimensionality of the search space,\\nand designing exploration strategies that balance exploration and exploitation more e\\u000bectively.\\n8 Evolutionary Computation in Multi-objective RL\\nWhile the aforementioned methods focus on single-objective RL, many real-world problems have\\nmultiple con\\ricting objectives. For example, an agent may need to grasp an object while minimizing\\n28 its energy consumption. These problems can be formulated as multi-objective MDPs (MOMDPs),\\nwhere the reward function R= [r1;:::;rm]Trepresents a vector of mrewards, and the discount\\nfactor is a vector \\r= [\\r1;::;\\rm]T. Multi-objective RL (MORL) learns a policy \\u0019\\u0012to simultaneously\\noptimize the multiple objectives J(\\u0019) = [J1(\\u0019);:::;Jm(\\u0019)]T, where each objective Ji(\\u0019) is associated\\nwith a dimension of the reward vector:\\nJi(\\u0019) =E\\u001a0;\\u0019;T\\\"1X\\nt=0\\rt\\nirt\\ni#\\n; (2)\\nTherefore, MORL is intrinsically a multi-objective optimization problem, where there is no single\\noptimal solution that can satisfy multiple con\\ricting objectives. Instead, a set of trade-o\\u000b solutions\\nnamed Pareto optimality is desired. In this context, Pareto optimality de\\fnes a solution that is\\nnot dominated by any other solution in the objective space. The set of such solutions is called the\\nPareto set, and the mapping of the Pareto set in the objective space is referred to as the Pareto front.\\nTo \\fnd a set of solutions approximating the Pareto front, multi-objective evolutionary algorithms\\n(MOEAs) are e\\u000bective tools to obtain optimal solutions in a single run [219]. The solution qualities\\nare often justi\\fed based on two aspects: convergence toward the true Pareto front and diversity along\\nthe Pareto front [220]. The hypervolume (HV) metric is often employed to measure convergence and\\ndiversity concurrently, which calculates the size of the hypervolume enclosed by the solution set and\\na reference point (e.g., a vector of nadir points of all obtained solutions in each dimension) [221].\\nThus, a larger HV metric value indicates a closer approximation of the Pareto front of solutions.\\nOther metrics employed in MORL include the (inverted) generational distance [222], the generalized\\nspread indicator [223], the cardinality indicator [224], and sparsity metrics [225].\\nAccording to the number of policies obtained at the end of optimization, MORL methods can be\\nroughly divided into two categories: single-policy methods and multi-policy methods [15]. Single-\\npolicy methods aim to learn a unique optimal policy each time. To achieve this, multiple rewards are\\ntransformed into a scalar reward by using a scalarization function, and then the scalar-rewarded task\\nis learned by general RL methods [226]. However, single-policy methods have several drawbacks,\\nincluding the requirement for domain knowledge, low e\\u000eciency, and sub-optimality due to preference\\n[15]. In contrast, multi-policy methods aim to \\fnd a set of diverse policies that approximate the\\nPareto front. These methods learn a set of policies that provide users with multiple options to\\nchoose from. One class of multi-policy methods focuses on how to apply the weights of objectives to\\nguide the optimization of policies [227]. However, scalarizing for all weights and approximating the\\nwhole Pareto front are both challenging. In contrast, another class of multi-policy methods applies\\nMOEAs to obtain a set of optimal solutions without setting weights. Further, multi-objectivization\\nis used to transfer single-objective problems into multi-objective problems to make problems more\\neasily to be resolved.\\n8.1 Multi-objective Evolutionary Algorithms\\nTo deal with the existing issues of MORL, EC methods are e\\u000bective tools to obtain a set of trade-\\no\\u000b solutions that can not only converge toward the Pareto front but also distribute well. In [11],\\n29 NSGA-II or MCMA (a multi-objective variant of CMA-ES) augmented with a local search method\\nhas been applied to search the graph-represented policies in robots load tasks with two, three, and\\n\\fve objectives. In addition, evaluation metrics have been applied to select solutions with good\\nperformance to enter into the next generation. The metrics-based MORL method applies HV and\\nsparsity metrics to select the best policy-weight pairs [225]. With a prediction model to predict policy\\nimprovements and an intra-family interpolation method to construct continuous Pareto fronts, the\\nmethod has achieved better HV and sparsity metrics over MOEA\\/D and a meta-learning method\\non a set of multi-objective MuJoCo tasks. In addition, the ideas of selecting good performance\\nby metrics have been borrowed into the action selection mechanism in MORL. The hypervolume-\\nbased MORL algorithm selects the action with the largest HV contribution into the next generation,\\nand this method has outperformed a linear scalarization-based action selection mechanism on two-\\nobjective deep sea treasure and three-objective mountain car benchmark tasks [228]. Besides, HV\\nhas been applied to select actions in an interactive way [229]. The Pareto-Q-learning algorithm used\\nthree evaluation methods, i.e., Pareto dominance relation, HV metric, and the cardinality metric,\\nto select the most potential actions in the Q-learning process [230].\\n8.2 Multi-objectivization\\nMulti-objectivization is a technique that transforms a single-objective problem into a multi-objective\\nproblem by decomposing a single objective or adding extra objectives [231]. The motivation behind\\nthis approach is to make the problem easier to solve by introducing more exploration through prior\\nknowledge. In the context of RL, multi-objectivization is typically used as a reward shaping method\\nthat focuses on not only completing tasks but also on encouraging behavior diversity among learned\\nagents. Recently, the evolutionary multi-objective game AI (EMOGI) framework was proposed as\\na method for generating behavior-diverse game agents [232]. EMOGI \\frst tailors a reward function\\nwith multiple objectives consisting of a performance objective and a number of behavior objectives,\\nand then applies NSGA-II to evolve a set of trade-o\\u000b policies. Further studies on EMOGI have\\nshown that introducing behavior objectives alone, without the performance objective, is able to\\ngenerate a set of behavior-diverse agents [233].\\n8.3 Discussion\\nEC has a wide range of MOEAs such as Pareto-based, decomposition-based, and indicator-based\\nmethods, but MORL has mainly used NSGA-II and the HV metric. The lack of diversity in MOEAs\\nfor MORL has limited the development of e\\u000ecient and e\\u000bective algorithms for multi-objective RL\\nproblems. One possible solution to address this issue is to introduce user preference-based MORL\\nmethods. Such methods can take inspiration from preference-based MOEAs and enable users to\\nspecify their preferences for di\\u000berent objectives. Furthermore, to advance the \\feld of MORL, re-\\nsearchers need to design and study tasks with more than three objectives. This would allow the\\nuse of many-objective evolutionary algorithms (MaOEAs) [234], which have the potential to provide\\nbetter solutions for such problems.\\nWhile EC has proved to be a valuable tool for MORL, there is still a need to explore more\\n30 Application scenarios\\nIndustry Healthcare Finance Robotics Games Scheduling\\nEvoRL platforms EvoRL benchmarks\\nLamarckian, EvoJAX, Evosax, EvoX,\\u2026 OpenAI Gym, ViZDoom, Maze navigation,\\u2026\\nEvoRL processes\\nEncoding\\nSamplingSearch \\noperatorsAlgorithmic \\nframeworksEvaluation\\nFigure 4: Overview of future directions from four \\felds of EvoRL: processes, benchmarks, platforms,\\nand application scenarios.\\nMOEAs and essential ideas to develop e\\u000ecient and e\\u000bective algorithms for multi-objective RL prob-\\nlems. Preference-based MORL methods and MaOEAs can o\\u000ber new opportunities to overcome the\\nlimitations of existing approaches and advance the \\feld further. Future research e\\u000borts should fo-\\ncus on designing and studying more complex tasks with multiple objectives to better evaluate the\\nperformance of MORL algorithms and compare them with existing state-of-the-art methods.\\n9 Future Research Directions\\nAlthough EvoRL has been successfully applied to large-scale complex RL tasks, even with sparse\\nand deceptive rewards, it is still computationally expensive. A number of e\\u000ecient methods in terms\\nof EvoRL processes including encoding, sampling, search operators, algorithmic framework, and\\nevaluation, as well as benchmarks, platforms, and applications, are desirable research directions, as\\noverviewed in Figure 4.\\n9.1 Encodings\\nE\\u000ecient encodings of decision variables are essential for the optimization of EC algorithms. While\\nreal-encoding is the most widely used representation in most research \\felds of RL, it is not the most\\ne\\u000ecient representation for policy search. Indirect encodings, such as CPPN, can be easily evolved\\nto capture structural relationships that exist in the human body structure. Additionally, TPG for\\nGP is a highly compressed representation that organizes multiple execution programs into modular\\n31 structures. Despite these indirect encodings, the development of e\\u000ecient encoding schemes is still\\nlimited. This includes the development of human-understandable encodings and e\\u000ecient encodings\\napplied to ESs. Furthermore, it is desirable to apply existing encodings to large-scale complex tasks\\nto investigate their scalability and generality.\\nE\\u000ecient encodings have the potential to improve the scalability and generality of EC algorithms.\\nIt is crucial to develop e\\u000ecient encoding schemes that can handle large-scale complex tasks to\\nreduce computational costs. Furthermore, there is a need for human-understandable encodings\\nthat can increase the interpretability of evolved policies. The development of e\\u000ecient encoding\\nschemes can also bene\\ft from incorporating insights from other research \\felds, such as information\\ntheory and statistical learning theory. E\\u000ecient encoding schemes can also be combined with other\\ne\\u000ecient methods, such as e\\u000ecient sampling and e\\u000ecient sample utilization, to further improve the\\nperformance of EC algorithms. Overall, the development of e\\u000ecient representations is an important\\nresearch direction for improving the scalability and generality of EC algorithms.\\n9.2 Sampling Methods\\nThe fundamental process of EC is to continuously sample solutions in search of the optimal direction.\\nTherefore, e\\u000ecient sampling can signi\\fcantly reduce time and computational overhead. In ESs based\\nmethods for policy search, e\\u000ecient sampling can decrease variance and accelerate convergence, such\\nas mirrored sampling in OpenAI ES [8]. E\\u000ecient sampling from the decision space is advantageous\\nfor ESs, but it has not been thoroughly investigated in GAs or GP based methods.\\nOn the other hand, in GAs based methods, sampling is often conducted based on the performance\\nin the objective space (i.e., behavior space). For instance, in exploration, new individuals are\\nfrequently generated in sparsely populated areas to help \\fnd agents with diverse behaviors as quickly\\nas possible. However, designing e\\u000ecient sampling methods remains a challenging issue for scaling\\nup algorithms to tasks with large-scale search spaces, sparse and deceptive rewards, and complex\\nand varied landscapes during training.\\n9.3 Sample Utilization\\nIn o\\u000b-policy RL algorithms such as DQN, samples are stored in a replay bu\\u000ber for multiple epochs of\\nupdates to improve sample utilization and break sample correlations. On-policy RL algorithms such\\nas PPO use importance sampling to enable sample reuse by correcting for the discrepancy between\\nold and new policies. Other RL techniques such as GAE and V-trace allow the reuse of old samples\\nto further reduce variance in updates [235].\\nIn ESs (similar to policy-gradient RL methods), techniques such as importance sampling have\\nbeen introduced to enable sample reuse [68]. However, importance sampling alone is not su\\u000ecient\\nto compete with RL methods in terms of sample e\\u000eciency. Hence, more e\\u000ecient techniques need\\nto be introduced into ESs. Furthermore, it remains to be investigated whether these techniques are\\nfeasible in GAs.\\n32 9.4 Search Operators\\nAlthough there have been a number of variational operators tailored for the search of NNs, such as\\nsafe mutation [125] and distillation crossover [123], the related research is still limited due to the\\nintrinsic features of various encodings. Besides topology distillation and gradient sensitive variation\\nthrough imitation learning and distillation, more methods or techniques can be employed to improve\\nthe e\\u000eciency of variational operators, such as transfer learning and surrogate gradient methods.\\nIn addition, the operators need to be customized according to the encoding method and can\\nbe controlled to generate safe o\\u000bspring. To do this, the use of surrogate models or other auxiliary\\ntools can help identify the most promising o\\u000bspring to preserve during the selection process, while\\ndiscarding less promising ones. Finally, more search operators can be borrowed from EC (e.g., DE\\n[236], PSO [237], and CSO [238]) to investigate their e\\u000bectiveness for RL tasks. The development\\nof e\\u000ecient search operators can signi\\fcantly reduce the search time and improve the e\\u000bectiveness of\\nEvoRL algorithms.\\n9.5 Algorithmic Frameworks\\nSeveral EvoRL frameworks have been developed, such as NEAT, OpenAI ES, ERL, and PBT. How-\\never, there are still several issues that require further research. In policy search, optimizing a policy\\nwith a large number of parameters can be challenging. If the policy can be divided into components\\nat a \\fner granularity, various functional modules can be automatically discovered using cooperative\\ncooperation methods. Moreover, this divide-and-conquer method enables parallel computation of\\ndi\\u000berent components, thereby further reducing computational resources.\\nResearch on evolution-guided exploration methods has mainly focused on o\\u000b-policy RL algo-\\nrithms since experiences of EC are more easily used than gradient information to ensure RL algo-\\nrithms do not degenerate. If gradient information of EC can be introduced, the training e\\u000eciency\\nof RL algorithms can be improved. Additionally, more e\\u000bort is needed to develop frameworks for\\non-policy RL algorithms. Although PBT is e\\u000bective, selecting optimized hyperparameters from a\\nlarge number of hyperparameters is challenging. This issue can be formulated as a combinatorial op-\\ntimization problem, and then EC can be applied to the whole framework of hyperparameter selection\\nand optimization to realize end-to-end automated HPO.\\n9.6 Evaluation Methods\\nE\\u000ecient evaluation methods are crucial to reduce the computational burden of evaluating the \\ft-\\nness of each newly-generated agent in EvoRL. Surrogate-assisted methods have been introduced in\\nEvoRL to predict \\ftness values [239, 240]. However, accurately modeling the relationship between\\npolicy parameters\\/behaviors and performance remains a challenging issue, especially in tasks with\\na large number of parameters and multiple performance indicators. In addition, ranking agents\\nwithout \\ftness estimation is a feasible method that has been explored, such as rank-based \\ftness\\nshaping which introduced racing methods into ESs to judge the relative performance of agents [75].\\nThese existing attempts have shown that accurate \\ftness values are not always necessary for e\\u000e-\\n33 cient evaluation. A partial order relation among agents may be su\\u000ecient, further facilitating the\\nparallelism of algorithms.\\nE\\u000ecient evaluation methods are not only essential for speeding up the evaluation process but\\nalso for improving the accuracy and robustness of evaluation results. In addition to surrogate-\\nassisted methods and rank-based \\ftness shaping, other techniques such as transfer learning, meta-\\nlearning, and Bayesian optimization can be explored to reduce the number of interactions with\\nthe environment and improve the e\\u000eciency of evaluation. These techniques can also be used to\\nimprove the generalization capability of the agents, making them more adaptable to new tasks and\\nenvironments. E\\u000ecient evaluation methods can signi\\fcantly improve the scalability and generality\\nof EvoRL algorithms, enabling them to tackle more complex and challenging tasks in a more e\\u000ecient\\nmanner.\\n9.7 Benchmarks for EvoRL\\nThe earlier tasks resolved by EvoRL are simple, such as pole balancing tasks and mountain car tasks\\n[74]. Then, EvoRL has been applied to various mobile robotics tasks such as maze navigation and\\nrobot arm control, as well as sim-to-real robotic tasks. These works have contributed to the formation\\nof the research \\feld of evolutionary robotics [158]. Recently, bene\\fting from the uni\\fed platform\\nand agent-environment interaction interface of integrated OpenAI Gym [241], games have become\\nexcellent benchmarks for evaluating EvoRL compared to existing testbeds. Especially, EvoRL has\\nbeen widely applied to tasks with continuous state-action space (e.g., robot control in MuJoCo),\\nand has shown promise for learning agents in large-scale visual-input games (e.g., ViZDoom [142]\\nand Dota2 [143]).\\nHowever, in empirical studies, EvoRL methods have typically been compared with RL methods\\nto observe their performance improvement, rather than compared with other EvoRL methods. The\\nreason is that existing RL benchmarks are not su\\u000ecient to investigate the properties of various\\nEvoRL methods. On the other hand, multi-objective EvoRL methods and quality diversity EvoRL\\nmethods do not have tailored benchmarks. Therefore, developing tailored EvoRL benchmarks is\\nmuch needed. While designing EvoRL benchmarks is not easy, a quick way is to modify existing RL\\nbenchmarks, such as the modi\\fed multi-objective MuJoCo tasks [7] to verify multi-objective EvoRL\\nmethods and the 2D bipedal walking adapted tasks to verify the open-endedness of EvoRL methods\\n[171].\\n9.8 Scalable Platforms\\nRecently, several scalable platforms for EvoRL have been developed, such as the Lamarckian plat-\\nform, an open-source high-performance platform that can scale up to thousands of CPU cores and\\nhas been veri\\fed to perform well in large-scale commercial games [242]. Another platform is the\\nparallel evolutionary and reinforcement learning library (PEARL), although it does not show highly\\nscalable abilities [243]. With the fast computational ability of GPUs, the JAX library has been re-\\nleased, which o\\u000bers a NumPy-like API for GPU-accelerated numerical calculations. Based on JAX,\\nthe platforms for neuroevolution such as EvoJAX [244] and evosax [245], the platform EvoX [246]\\n34 for general EC algorithms, and the platform QDax [247] for QD algorithms have been developed.\\nThese platforms have shown the ability to \\fnd solutions in Atari or MuJoCo tasks with signi\\fcantly\\nless time than using CPUs.\\nIn addition, [248] has proposed a platform for developing co-evolution algorithms for co-optimizing\\nthe design and control of robots. Despite these advancements, the research on e\\u000ecient and scalable\\nplatforms for EvoRL is still limited, as they may not be user-friendly or focus on limited EvoRL\\nalgorithms. Further research is needed to develop more e\\u000ecient and scalable platforms that can\\nhandle large-scale complex tasks and integrate with various EvoRL algorithms.\\n10 Conclusion\\nThe article has presented a comprehensive survey of EvoRL, mainly focusing on its methodologies\\nand future directions. Firstly, the article has introduced EvoRL methods by classifying them ac-\\ncording to six key research \\felds of RL: hyperparameter optimization, policy search, exploration,\\nreward shaping, meta-RL, and multi-objective RL. For each \\feld, the applied EC methods (ESs,\\nGAs, and GP) are elaborated and their main advantages and disadvantages are discussed. Secondly,\\nthe article has discussed several future directions for e\\u000ecient methods in EvoRL processes, as well\\nas tailored EvoRL benchmarks and platforms. By discussing these future directions, the article has\\nprovided guidance for researchers and practitioners interested in the \\feld of EvoRL, and promotes\\nthe further development of this cross-disciplinary research \\feld. Overall, this survey is a resource\\nfor anyone interested in learning about EvoRL and its potential applications in RL, and provides\\ninsights into the future direction of this rapidly growing \\feld.\\nReferences\\n[1] R. S. Sutton and A. G. Barto, Reinforcement learning: An introduction . MIT press, 2018.\\n[2] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves,\\nM. Riedmiller, A. K. Fidjeland, G. Ostrovski et al. , \\\\Human-level control through deep rein-\\nforcement learning,\\\" Nature , vol. 518, no. 7540, pp. 529{533, 2015.\\n[3] S. Khadka, S. Majumdar, T. Nassar, Z. Dwiel, E. Tumer, S. Miret, Y. Liu, and K. Tumer,\\n\\\\Collaborative evolutionary reinforcement learning,\\\" International Conference on Machine\\nLearning , 2019.\\n[4] A. Eco\\u000bet, J. Huizinga, J. Lehman, K. O. Stanley, and J. Clune, \\\\Go-explore: A new approach\\nfor hard-exploration problems,\\\" arXiv preprint arXiv:1901.10995 , 2019.\\n[5] Q. Long, Z. Zhou, A. Gupta, F. Fang, Y. Wu, and X. Wang, \\\\Evolutionary population curricu-\\nlum for scaling multi-agent reinforcement learning,\\\" in International Conference on Learning\\nRepresentations , 2020.\\n35 [6] E. Conti, V. Madhavan, F. Petroski Such, J. Lehman, K. Stanley, and J. Clune, \\\\Improving\\nexploration in evolution strategies for deep reinforcement learning via a population of novelty-\\nseeking agents,\\\" Advances in Neural Information Processing Systems , vol. 31, 2018.\\n[7] D. M. Roijers, P. Vamplew, S. Whiteson, and R. Dazeley, \\\\A survey of multi-objective sequen-\\ntial decision-making,\\\" Journal of Arti\\fcial Intelligence Research , vol. 48, pp. 67{113, 2013.\\n[8] T. Salimans, J. Ho, X. Chen, S. Sidor, and I. Sutskever, \\\\Evolution strategies as a scalable\\nalternative to reinforcement learning,\\\" arXiv preprint arXiv:1703.03864 , 2017.\\n[9] S. Khadka and K. Tumer, \\\\Evolution-guided policy gradient in reinforcement learning,\\\" in\\nInternational Conference on Neural Information Processing Systems , 2018.\\n[10] A. Cully, J. Clune, D. Tarapore, and J.-B. Mouret, \\\\Robots that can adapt like animals,\\\"\\nNature , vol. 521, no. 7553, p. 503, 2015.\\n[11] H. Soh and Y. Demiris, \\\\Evolving policies for multi-reward partially observable markov deci-\\nsion processes (mr-pomdps),\\\" in Proceedings of the 13th Annual Conference on Genetic and\\nEvolutionary Computation , 2011, pp. 713{720.\\n[12] D. Whitley, S. Dominic, R. Das, and C. W. Anderson, \\\\Genetic reinforcement learning for\\nneurocontrol problems,\\\" Machine Learning , vol. 13, no. 2, pp. 259{284, 1993.\\n[13] K. O. Stanley and R. Miikkulainen, \\\\Evolving neural networks through augmenting topolo-\\ngies,\\\" Evolutionary Computation , vol. 10, no. 2, pp. 99{127, 2002.\\n[14] O. Sigaud, \\\\Combining evolution and deep reinforcement learning for policy search: A survey,\\\"\\narXiv preprint arXiv:2203.14009 , 2022.\\n[15] C. Liu, X. Xu, and D. Hu, \\\\Multiobjective reinforcement learning: A comprehensive overview,\\\"\\nIEEE Transactions on Systems, Man, and Cybernetics: Systems , vol. 45, no. 3, pp. 385{398,\\n2014.\\n[16] J. Parker-Holder, R. Rajan, X. Song, A. Biedenkapp, Y. Miao, T. Eimer, B. Zhang, V. Nguyen,\\nR. Calandra, A. Faust et al. , \\\\Automated reinforcement learning (autorl): A survey and open\\nproblems,\\\" arXiv preprint arXiv:2201.03916 , 2022.\\n[17] H. Qian and Y. Yu, \\\\Derivative-free reinforcement learning: A review,\\\" Frontiers of Computer\\nScience , 2021.\\n[18] Y. Li, \\\\Deep reinforcement learning: An overview,\\\" arXiv preprint arXiv:1701.07274 , 2018.\\n[19] J. Schulman, S. Levine, P. Abbeel, M. Jordan, and P. Moritz, \\\\Trust region policy optimiza-\\ntion,\\\" in International Conference on Machine Learning . PMLR, 2015, pp. 1889{1897.\\n[20] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov, \\\\Proximal policy optimiza-\\ntion algorithms,\\\" arXiv preprint arXiv:1707.06347 , 2017.\\n36 [21] V. Mnih, A. P. Badia, M. Mirza, A. Graves, T. Lillicrap, T. Harley, D. Silver, and\\nK. Kavukcuoglu, \\\\Asynchronous methods for deep reinforcement learning,\\\" in International\\nConference on Machine Learning , 2016.\\n[22] T. P. Lillicrap, J. J. Hunt, A. Pritzel, N. Heess, T. Erez, Y. Tassa, D. Silver, and D. Wier-\\nstra, \\\\Continuous control with deep reinforcement learning,\\\" in International Conference on\\nLearning Representations , 2016.\\n[23] S. Fujimoto, H. Hoof, and D. Meger, \\\\Addressing function approximation error in actor-critic\\nmethods,\\\" in International Conference on Machine Learning . PMLR, 2018, pp. 1587{1596.\\n[24] T. Haarnoja, A. Zhou, P. Abbeel, and S. Levine, \\\\Soft actor-critic: O\\u000b-policy maximum\\nentropy deep reinforcement learning with a stochastic actor,\\\" in International Conference on\\nMachine Learning . PMLR, 2018, pp. 1861{1870.\\n[25] V. Mnih, K. Kavukcuoglu, D. Silver, A. Graves, I. Antonoglou, D. Wierstra, and M. Riedmiller,\\n\\\\Playing atari with deep reinforcement learning,\\\" arXiv preprint arXiv:1312.5602 , 2013.\\n[26] M. Hessel, J. Modayil, H. Van Hasselt, T. Schaul, G. Ostrovski, W. Dabney, D. Horgan,\\nB. Piot, M. Azar, and D. Silver, \\\\Rainbow: Combining improvements in deep reinforcement\\nlearning,\\\" in Thirty-second AAAI Conference on Arti\\fcial Intelligence , 2018.\\n[27] H. Van Hasselt, A. Guez, and D. Silver, \\\\Deep reinforcement learning with double q-learning,\\\"\\ninProceedings of the AAAI Conference on Arti\\fcial Intelligence , vol. 30, no. 1, 2016.\\n[28] N. Hansen, D. V. Arnold, and A. Auger, \\\\Evolution strategies,\\\" in Springer handbook of\\ncomputational intelligence . Springer, 2015, pp. 871{898.\\n[29] D. Whitley, \\\\A genetic algorithm tutorial,\\\" Statistics and Computing , vol. 4, no. 2, pp. 65{85,\\n1994.\\n[30] E. K. Burke, S. Gustafson, and G. Kendall, \\\\Diversity in genetic programming: an analysis\\nof measures and correlation with \\ftness,\\\" IEEE Transactions on Evolutionary Computation ,\\nvol. 8, no. 1, pp. 47{62, 2004.\\n[31] G. Rudolph, Convergence properties of evolutionary algorithms . Verlag Dr. Kova\\u0014 c, 1997.\\n[32] N. Hansen, S. D. M\u007f uller, and P. Koumoutsakos, \\\\Reducing the time complexity of the de-\\nrandomized evolution strategy with covariance matrix adaptation (CMA-ES),\\\" Evolutionary\\nComputation , vol. 11, no. 1, pp. 1{18, 2003.\\n[33] D. Wierstra, T. Schaul, T. Glasmachers, Y. Sun, J. Peters, and J. Schmidhuber, \\\\Natural\\nevolution strategies,\\\" The Journal of Machine Learning Research , vol. 15, no. 1, pp. 949{980,\\n2014.\\n[34] S. Amari and S. C. Douglas, \\\\Why natural gradient?\\\" in IEEE International Conference on\\nAcoustics , 1998.\\n37 [35] J. Gauci and K. O. Stanley, \\\\Indirect encoding of neural networks for scalable go,\\\" in Inter-\\nnational Conference on Parallel Problem Solving from Nature . Springer, 2010, pp. 354{363.\\n[36] S. Risi and J. Togelius, \\\\Neuroevolution in games: State of the art and open challenges,\\\"\\nIEEE Transactions on Computational Intelligence and AI in Games , vol. PP, no. 99, 2015.\\n[37] Z. Buk, J. Koutn\\u0013 \\u0010k, and M. \\u0014Snorek, \\\\Neat in hyperneat substituted with genetic program-\\nming,\\\" in International Conference on Adaptive and Natural Computing Algorithms . Springer,\\n2009, pp. 243{252.\\n[38] A. Moraglio, C. Di Chio, J. Togelius, and R. Poli, \\\\Geometric particle swarm optimization,\\\"\\nJournal of Arti\\fcial Evolution and Applications , 2008.\\n[39] R. I. McKay, N. X. Hoai, P. A. Whigham, Y. Shan, and M. O'neill, \\\\Grammar-based genetic\\nprogramming: A survey,\\\" Genetic Programming and Evolvable Machines , vol. 11, no. 3, pp.\\n365{396, 2010.\\n[40] K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms , 1st ed., ser. Wiley-\\nInterscience series in systems and optimization. Chichester, New York: John Wiley & Sons,\\n2001.\\n[41] J. Lehman and K. O. Stanley, \\\\Abandoning objectives: Evolution through the search for\\nnovelty alone,\\\" Evolutionary Computation , vol. 19, no. 2, pp. 189{223, 2011.\\n[42] W. Zhao, J. P. Queralta, and T. Westerlund, \\\\Sim-to-real transfer in deep reinforcement learn-\\ning for robotics: A survey,\\\" in 2020 IEEE Symposium Series on Computational Intelligence ,\\n2020, pp. 737{744.\\n[43] M. Jaderberg, V. Dalibard, S. Osindero, W. M. Czarnecki, J. Donahue, A. Razavi, O. Vinyals,\\nT. Green, I. Dunning, K. Simonyan et al. , \\\\Population based training of neural networks,\\\"\\narXiv preprint arXiv:1711.09846 , 2017.\\n[44] J. K. Franke, G. K\u007f ohler, A. Biedenkapp, and F. Hutter, \\\\Sample-e\\u000ecient automated deep\\nreinforcement learning,\\\" arXiv preprint arXiv:2009.01555 , 2020.\\n[45] J. Bergstra and Y. Bengio, \\\\Random search for hyper-parameter optimization.\\\" Journal of\\nmachine learning research , vol. 13, no. 2, 2012.\\n[46] J. Snoek, H. Larochelle, and R. P. Adams, \\\\Practical bayesian optimization of machine learning\\nalgorithms,\\\" Advances in Neural Information Processing Systems , vol. 25, 2012.\\n[47] T. Zahavy, Z. Xu, V. Veeriah, M. Hessel, J. Oh, H. P. van Hasselt, D. Silver, and S. Singh,\\n\\\\A self-tuning actor-critic algorithm,\\\" Advances in Neural Information Processing Systems ,\\nvol. 33, pp. 20 913{20 924, 2020.\\n[48] A. Eriksson, G. Capi, and K. Doya, \\\\Evolution of meta-parameters in reinforcement learning\\nalgorithm,\\\" in Proceedings 2003 IEEE\\/RSJ International Conference on Intelligent Robots and\\nSystems , vol. 1. IEEE, 2003, pp. 412{417.\\n38 [49] S. Elfwing, E. Uchibe, K. Doya, and H. I. Christensen, \\\\Co-evolution of shaping rewards and\\nmeta-parameters in reinforcement learning,\\\" Adaptive Behavior , vol. 16, no. 6, pp. 400{412,\\n2008.\\n[50] M. Jaderberg, W. M. Czarnecki, I. Dunning, L. Marris, G. Lever, A. G. Castaneda, C. Beat-\\ntie, N. C. Rabinowitz, A. S. Morcos, A. Ruderman et al. , \\\\Human-level performance in 3D\\nmultiplayer games with population-based reinforcement learning,\\\" Science , vol. 364, no. 6443,\\npp. 859{865, 2019.\\n[51] S. Schmitt, J. J. Hudson, A. Zidek, S. Osindero, C. Doersch, W. M. Czarnecki, J. Z. Leibo,\\nH. Kuttler, A. Zisserman, K. Simonyan et al. , \\\\Kickstarting deep reinforcement learning,\\\"\\narXiv preprint arXiv:1803.03835 , 2018.\\n[52] S. Liu, G. Lever, J. Merel, S. Tunyasuvunakool, N. Heess, and T. Graepel, \\\\Emergent co-\\nordination through competition,\\\" in International Conference on Learning Representations ,\\n2019.\\n[53] T. R. Wu, T. H. Wei, and I. C. Wu, \\\\Accelerating and improving alphazero using population\\nbased training,\\\" in Proceedings of the AAAI Conference on Arti\\fcial Intelligence , 2020.\\n[54] F. Vavak and T. C. Fogarty, \\\\Comparison of steady state and generational genetic algorithms\\nfor use in nonstationary environments,\\\" in Proceedings of IEEE International Conference on\\nEvolutionary Computation . IEEE, 1996, pp. 192{195.\\n[55] V. Dalibard and M. Jaderberg, \\\\Faster improvement rate population based training,\\\" arXiv\\npreprint arXiv:2109.13800 , 2021.\\n[56] F. C. Fernandez and W. Caarls, \\\\Parameters tuning and optimization for reinforcement learn-\\ning algorithms using evolutionary computing,\\\" in 2018 International Conference on Informa-\\ntion Systems and Computer Science . IEEE, 2018, pp. 301{305.\\n[57] X. Cui, W. Zhang, Z. T\u007f uske, and M. Picheny, \\\\Evolutionary stochastic gradient descent for\\noptimization of deep neural networks,\\\" in Advances in Neural Information Processing Systems ,\\n2018.\\n[58] L. Schneider, F. P\\fsterer, J. Thomas, and B. Bischl, \\\\A collection of quality diversity opti-\\nmization problems derived from hyperparameter optimization of machine learning models,\\\" in\\nProceedings of the Genetic and Evolutionary Computation Conference , 2022.\\n[59] K. O. Stanley, J. Clune, J. Lehman, and R. Miikkulainen, \\\\Designing neural networks through\\nneuroevolution,\\\" Nature Machine Intelligence , vol. 1, no. 1, pp. 24{35, 2019.\\n[60] A. Gaier and D. Ha, \\\\Weight agnostic neural networks,\\\" Advances in Neural Information\\nProcessing Systems , 2019.\\n[61] P. Chrabaszcz, I. Loshchilov, and F. Hutter, \\\\Back to basics: Benchmarking canonical evolu-\\ntion strategies for playing atari,\\\" in International Joint Conference on Arti\\fcial Intelligence ,\\n2018.\\n39 [62] S. Whiteson, Evolutionary computation for reinforcement learning . Springer Berlin Heidel-\\nberg, 2012, pp. 325{355.\\n[63] K. Choromanski, M. Rowland, V. Sindhwani, R. Turner, and A. Weller, \\\\Structured evolution\\nwith compact architectures for scalable policy optimization,\\\" in International Conference on\\nMachine Learning . PMLR, 2018, pp. 970{978.\\n[64] K. M. Choromanski, A. Pacchiano, J. Parker-Holder, Y. Tang, and V. Sindhwani, \\\\From\\ncomplexity to simplicity: Adaptive es-active subspaces for blackbox optimization,\\\" Advances\\nin Neural Information Processing Systems , vol. 32, 2019.\\n[65] Y. Tang, K. Choromanski, and A. Kucukelbir, \\\\Variance reduction for evolution strategies\\nvia structured control variates,\\\" in International Conference on Arti\\fcial Intelligence and\\nStatistics . PMLR, 2020, pp. 646{656.\\n[66] N. Maheswaranathan, L. Metz, G. Tucker, D. Choi, and J. Sohl-Dickstein, \\\\Guided evolu-\\ntionary strategies: augmenting random search with surrogate gradients,\\\" in Proceedings of the\\n36th International Conference on Machine Learning . PMLR, 2019, pp. 4264{4273.\\n[67] F.-Y. Liu, Z.-N. Li, and C. Qian, \\\\Self-guided evolution strategies with historical estimated\\ngradients,\\\" in International Joint Conference on Artifcial Intelligence , 2020, pp. 1474{1480.\\n[68] G. Liu, L. Zhao, F. Yang, J. Bian, T. Qin, N. Yu, and T.-Y. Liu, \\\\Trust region evolution\\nstrategies,\\\" in Proceedings of the AAAI Conference on Arti\\fcial Intelligence , vol. 33, no. 01,\\n2019, pp. 4352{4359.\\n[69] S. Yi, D. Wierstra, T. Schaul, and J. Schmidhuber, \\\\Stochastic search using the natural\\ngradient,\\\" in International Conference on Machine Learning , 2009.\\n[70] F. Sehnke, C. Osendorfer, T. R\u007f uckstiess, A. Graves, J. Peters, and J. Schmidhuber,\\n\\\\Parameter-exploring policy gradients,\\\" Neural Networks , vol. 23, no. 4, pp. 551{559, 2010.\\n[71] X. Zhang, J. Clune, and K. O. Stanley, \\\\On the relationship between the openai evolution\\nstrategy and stochastic gradient descent,\\\" arXiv preprint arXiv:1712.06564 , 2017.\\n[72] J. Lehman, J. Chen, J. Clune, and K. O. Stanley, \\\\Es is more than just a traditional \\fnite-\\ndi\\u000berence approximator,\\\" in Proceedings of the Genetic and Evolutionary Computation Con-\\nference , 2018, pp. 450{457.\\n[73] L. Fuks, N. H. Awad, F. Hutter, and M. Lindauer, \\\\An evolution strategy with progressive\\nepisode lengths for playing games.\\\" in International Joint Conferences on Arti\\fcial Intelli-\\ngence , 2019, pp. 1234{1240.\\n[74] C. Igel, \\\\Neuroevolution for reinforcement learning using evolution strategies,\\\" in The Congress\\non Evolutionary Computation , vol. 4. IEEE, 2003, pp. 2588{2595.\\n40 [75] V. Heidrich-Meisner and C. Igel, \\\\Hoe\\u000bding and bernstein races for selecting policies in evo-\\nlutionary direct policy search,\\\" in International Conference on Machine Learning , 2009, pp.\\n401{408.\\n[76] ||, \\\\Neuroevolution strategies for episodic reinforcement learning,\\\" Journal of Algorithms ,\\nvol. 64, no. 4, pp. 152{168, 2009.\\n[77] Z. Chen, Y. Zhou, X. He, and S. Jiang, \\\\A restart-based rank-1 evolution strategy for rein-\\nforcement learning,\\\" in International Joint Conferences on Arti\\fcial Intelligence , 2019, pp.\\n2130{2136.\\n[78] Z. Li and Q. Zhang, \\\\A simple yet e\\u000ecient evolution strategy for large-scale black-box op-\\ntimization,\\\" IEEE Transactions on Evolutionary Computation , vol. 22, no. 5, pp. 637{646,\\n2017.\\n[79] I. Loshchilov, T. Glasmachers, and H.-G. Beyer, \\\\Large scale black-box optimization\\nby limited-memory matrix adaptation,\\\" IEEE Transactions on Evolutionary Computation ,\\nvol. 23, no. 2, pp. 353{358, 2018.\\n[80] Z. Li, Q. Zhang, X. Lin, and H.-L. Zhen, \\\\Fast covariance matrix adaptation for large-scale\\nblack-box optimization,\\\" IEEE Transactions on Cybernetics , vol. 50, no. 5, pp. 2073{2083,\\n2020.\\n[81] A. P. Wieland, \\\\Evolving controls for unstable systems,\\\" in Connectionist Models . Elsevier,\\n1991, pp. 91{102.\\n[82] K. O. Stanley, B. D. Bryant, and R. Miikkulainen, \\\\Evolving adaptive neural networks with\\nand without adaptive synapses,\\\" in The 2003 Congress on Evolutionary Computation , vol. 4.\\nIEEE, 2003, pp. 2557{2564.\\n[83] K. O. Stanley and R. Miikkulainen, \\\\Competitive coevolution through evolutionary complex-\\ni\\fcation,\\\" Journal of Arti\\fcial Intelligence Research , vol. 21, pp. 63{100, 2004.\\n[84] K. O. Stanley, B. D. Bryant, and R. Miikkulainen, \\\\Evolving neural network agents in the\\nnero video game,\\\" Proceedings of the IEEE , pp. 182{189, 2005.\\n[85] N. Kohl and R. Miikkulainen, \\\\Evolving neural networks for strategic decision-making prob-\\nlems,\\\" Neural Networks , vol. 22, no. 3, pp. 326{337, 2009.\\n[86] Y. Kassahun and G. Sommer, \\\\E\\u000ecient reinforcement learning through evolutionary acqui-\\nsition of neural topologies,\\\" in Proceedings of The European Symposium on Arti\\fcial Neural\\nNetworks , 2005, pp. 259{266.\\n[87] H. Moriguchi and S. Honiden, \\\\Cma-tweann: E\\u000ecient optimization of neural networks via\\nself-adaptation and seamless augmentation,\\\" in Proceedings of the 14th Annual Conference on\\nGenetic and Evolutionary Computation , 2012, pp. 903{910.\\n41 [88] F. P. Such, V. Madhavan, E. Conti, J. Lehman, K. O. Stanley, and J. Clune, \\\\Deep neuroevo-\\nlution: Genetic algorithms are a competitive alternative for training deep neural networks for\\nreinforcement learning,\\\" arXiv preprint arXiv:1712.06567 , 2017.\\n[89] M. Le Clei and P. Bellec, \\\\Neuroevolution of recurrent architectures on control tasks,\\\" in\\nInternational Conference on Learning Representations Workshop on Agent Learning in Open-\\nEndedness , 2022.\\n[90] D. Ha and J. Schmidhuber, \\\\Recurrent world models facilitate policy evolution,\\\" Advances in\\nNeural Information Processing Systems , vol. 31, 2018.\\n[91] J. Koutn\\u0013 \\u0010k, J. Schmidhuber, and F. Gomez, \\\\Evolving deep unsupervised convolutional net-\\nworks for vision-based reinforcement learning,\\\" in Proceedings of the 2014 Annual Conference\\non Genetic and Evolutionary Computation , 2014, pp. 541{548.\\n[92] S. Alvernaz and J. Togelius, \\\\Autoencoder-augmented neuroevolution for visual doom play-\\ning,\\\" in 2017 IEEE Conference on Computational Intelligence and Games . IEEE, 2017, pp.\\n1{8.\\n[93] S. Risi and K. O. Stanley, \\\\Deep neuroevolution of recurrent and discrete world models,\\\" in\\nProceedings of the Genetic and Evolutionary Computation Conference , 2019, pp. 456{462.\\n[94] S. Whiteson and P. Stone, \\\\Evolutionary function approximation for reinforcement learning,\\\"\\nJournal of Machine Learning Research , vol. 7, 2006.\\n[95] ||, \\\\Sample-e\\u000ecient evolutionary function approximation for reinforcement learning,\\\" in\\nProceedings of the National Conference on Arti\\fcial Intelligence , vol. 21, no. 1, 2006, p. 518.\\n[96] S. Whiteson, M. E. Taylor, and P. Stone, \\\\Critical factors in the empirical performance of\\ntemporal di\\u000berence and evolutionary methods for reinforcement learning,\\\" Autonomous Agents\\nand Multi-Agent Systems , vol. 21, no. 1, pp. 1{35, 2010.\\n[97] M. A. Potter and K. A. D. Jong, \\\\Cooperative coevolution: An architecture for evolving\\ncoadapted subcomponents,\\\" Evolutionary Computation , vol. 8, no. 1, pp. 1{29, 2000.\\n[98] D. E. Moriarty and R. Mikkulainen, \\\\E\\u000ecient reinforcement learning through symbiotic evo-\\nlution,\\\" Machine Learning , vol. 22, no. 1, pp. 11{32, 1996.\\n[99] F. Gomez and R. Miikulainen, \\\\Solving non-markovian tasks with neuroevolution,\\\" in Pro-\\nceeding of the Sixteenth International Joint Conference on Arti\\fcial Intelligence , 1999, pp.\\n1356{1361.\\n[100] R. Chandra, M. Frean, M. Zhang, and C. W. Omlin, \\\\Encoding subcomponents in cooperative\\nco-evolutionary recurrent neural networks,\\\" Neurocomputing , vol. 74, no. 17, pp. 3223{3234,\\n2011.\\n42 [101] F. Gomez, J. Schmidhuber, R. Miikkulainen, and M. Mitchell, \\\\Accelerated neural evolution\\nthrough cooperatively coevolved synapses,\\\" Journal of Machine Learning Research , vol. 9,\\nno. 5, 2008.\\n[102] N. Garc\\u0013 \\u0010a-Pedrajas, C. Herv\\u0013 as-Mart\\u0013 \\u0010nez, and J. Mu~ noz-P\\u0013 erez, \\\\Covnet: A cooperative co-\\nevolutionary model for evolving arti\\fcial neural networks,\\\" IEEE Transactions on Neural\\nNetworks , vol. 14, no. 3, pp. 575{596, 2003.\\n[103] J. Reisinger, K. O. Stanley, and R. Miikkulainen, \\\\Evolving reusable neural modules,\\\" in\\nGenetic and Evolutionary Computation Conference . Springer, 2004, pp. 69{81.\\n[104] P. Yang, H. Zhang, Y. Yu, M. Li, and K. Tang, \\\\Evolutionary reinforcement learning via coop-\\nerative coevolutionary negatively correlated search,\\\" Swarm and Evolutionary Computation ,\\nvol. 68, p. 100974, 2022.\\n[105] F. Gruau, \\\\Automatic de\\fnition of modular neural networks,\\\" Adaptive Behavior , vol. 3, no. 2,\\npp. 151{183, 1994.\\n[106] G. S. Hornby and J. B. Pollack, \\\\Creating high-level components with a generative represen-\\ntation for body-brain evolution,\\\" Arti\\fcial Life , vol. 8, no. 3, pp. 223{246, 2002.\\n[107] K. O. Stanley and R. Miikkulainen, \\\\A taxonomy for arti\\fcial embryogeny,\\\" Arti\\fcial Life ,\\nvol. 9, no. 2, pp. 93{130, 2003.\\n[108] K. O. Stanley, \\\\Compositional pattern producing networks: A novel abstraction of develop-\\nment,\\\" Genetic Programming and Evolvable Machines , vol. 8, no. 2, pp. 131{162, 2007.\\n[109] K. O. Stanley, D. B. D'Ambrosio, and J. Gauci, \\\\A hypercube-based encoding for evolving\\nlarge-scale neural networks,\\\" Arti\\fcial Life , vol. 15, no. 2, pp. 185{212, 2009.\\n[110] J. Clune, K. O. Stanley, R. T. Pennock, and C. Ofria, \\\\On the performance of indirect encoding\\nacross the continuum of regularity,\\\" IEEE Transactions on Evolutionary Computation , vol. 15,\\nno. 3, pp. 346{367, 2011.\\n[111] J. Gauci and K. O. Stanley, \\\\A case study on the critical role of geometric regularity in machine\\nlearning,\\\" in Proceedings of the 23rd National Conference on Arti\\fcial Intelligence . AAAI\\nPress, 2008, pp. 628 { 633.\\n[112] M. Hausknecht, J. Lehman, R. Miikkulainen, and P. Stone, \\\\A neuroevolution approach to\\ngeneral atari game playing,\\\" IEEE Transactions on Computational Intelligence and AI in\\nGames , vol. 6, no. 4, pp. 355{366, 2014.\\n[113] S. Risi and K. O. Stanley, \\\\Indirectly encoding neural plasticity as a pattern of local rules,\\\" in\\nInternational Conference on Simulation of Adaptive Behavior . Springer, 2010, pp. 533{543.\\n[114] ||, \\\\An enhanced hypercube-based encoding for evolving the placement, density, and con-\\nnectivity of neurons,\\\" Arti\\fcial Life , vol. 18, no. 4, pp. 331{363, 2012.\\n43 [115] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, \\\\A fast and elitist multiobjective genetic\\nalgorithm: NSGA-II,\\\" IEEE Transactions on Evolutionary Computation , vol. 6, no. 2, pp.\\n182{197, 2002.\\n[116] J. Huizinga, J.-B. Mouret, and J. Clune, \\\\Does aligning phenotypic and genotypic modularity\\nimprove the evolution of neural networks?\\\" in Proceedings of the Genetic and Evolutionary\\nComputation Conference 2016 , 2016, pp. 125{132.\\n[117] J. Koutn\\u0013 \\u0010k, G. Cuccu, J. Schmidhuber, and F. Gomez, \\\\Evolving large-scale neural networks\\nfor vision-based reinforcement learning,\\\" in Proceedings of the 15th Annual Conference on\\nGenetic and Evolutionary Computation , 2013.\\n[118] J. Clune, B. E. Beckmann, R. T. Pennock, and C. Ofria, \\\\Hybrid: A hybridization of indirect\\nand direct encodings for evolutionary computation,\\\" in European Conference on Arti\\fcial Life .\\nSpringer, 2009, pp. 134{141.\\n[119] G.-A. Vargas-H\\u0013 akim, E. Mezura-Montes, and H.-G. Acosta-Mesa, \\\\Hybrid encodings for neu-\\nroevolution of convolutional neural networks: A case study,\\\" in Proceedings of the Genetic and\\nEvolutionary Computation Conference Companion , 2021, pp. 1762{1770.\\n[120] J. Schrum, B. Capps, K. Steckel, V. Volz, and S. Risi, \\\\Hybrid encoding for generating large\\nscale game llevel patterns with local variations,\\\" IEEE Transactions on Games , 2022.\\n[121] K. Deb and A. Kumar, \\\\Real-coded genetic algorithms with simulated binary crossover: Stud-\\nies on multimodal and multiobjective problems,\\\" Complex Systems , vol. 9, no. 6, pp. 431{454,\\n1995.\\n[122] T. Gangwani and J. Peng, \\\\Genetic policy optimization,\\\" in International Conference on\\nLearning Representations , 2018.\\n[123] C. Bodnar, B. Day, and P. Li\\u0013 o, \\\\Proximal distilled evolutionary reinforcement learning,\\\"\\nProceedings of the AAAI Conference on Arti\\fcial Intelligence , vol. 34, no. 04, pp. 3283{3290,\\n2020.\\n[124] J. K. Franke, G. K\u007f ohler, N. Awad, and F. Hutter, \\\\Neural architecture evolution in deep\\nreinforcement learning for continuous control,\\\" arXiv preprint arXiv:1910.12824 , 2019.\\n[125] L. J, C. J, C. J, and S. KO, \\\\Safe mutations for deep and recurrent neural networks through\\noutput gradients,\\\" in Proceedings of the Genetic and Evolutionary Computation Conference ,\\n2018.\\n[126] E. Marchesini, D. Corsi, and A. Farinelli, \\\\Exploring safer behaviors for deep reinforcement\\nlearning,\\\" in Proceedings of the AAAI Conference on Arti\\fcial Intelligence , vol. 36, no. 7,\\n2022, pp. 7701{7709.\\n[127] T. Uriot and D. Izzo, \\\\Safe crossover of neural networks through neuron alignment,\\\" in Pro-\\nceedings of the 2020 Genetic and Evolutionary Computation Conference , 2020, pp. 435{443.\\n44 [128] J. R. Woodward, \\\\Evolving turing complete representations,\\\" in The Congress on Evolutionary\\nComputation , vol. 2. IEEE, 2003, pp. 830{837.\\n[129] J. F. Miller, \\\\Cartesian genetic programming,\\\" in Cartesian Genetic Programming . Springer,\\n2011, pp. 17{34.\\n[130] S. Kelly, R. J. Smith, and M. I. Heywood, \\\\Emergent policy discovery for visual reinforcement\\nlearning through tangled program graphs: A tutorial,\\\" Genetic Programming Theory and\\nPractice XVI , pp. 37{57, 2019.\\n[131] J. R. Koza and J. P. Rice, \\\\Automatic programming of robots using genetic programming,\\\" in\\nProceedings of the Tenth National Conference on Arti\\fcial Intelligence . AAAI Press, 1992,\\npp. 194{207.\\n[132] S. Ok, K. Miyashita, and K. Hase, \\\\Evolving bipedal locomotion with genetic programming-a\\npreliminary report,\\\" in Proceedings of the 2001 Congress on Evolutionary Computation , vol. 2.\\nIEEE, 2001, pp. 1025{1032.\\n[133] D. C. Dracopoulos, D. E\\u000braimidis, and B. D. Nichols, \\\\Genetic programming as a solver to\\nchallenging reinforcement learning problems,\\\" International Journal of Computer Research ,\\nvol. 20, no. 3, p. 351, 2013.\\n[134] S. Kamio and H. Iba, \\\\Adaptation technique for integrating genetic programming and rein-\\nforcement learning for real robots,\\\" IEEE Transactions on Evolutionary Computation , vol. 9,\\nno. 3, pp. 318{333, 2005.\\n[135] F. Gruau, D. Whitley, and L. Pyeatt, \\\\A comparison between cellular encoding and direct\\nencoding for genetic neural networks,\\\" in Proceedings of the 1st Annual Conference on Genetic\\nProgramming , 1996, pp. 81{89.\\n[136] M. M. Khan, A. M. Ahmad, G. M. Khan, and J. F. Miller, \\\\Fast learning neural networks\\nusing cartesian genetic programming,\\\" Neurocomputing , vol. 121, pp. 274{289, 2013.\\n[137] A. J. Turner and J. F. Miller, \\\\Neuroevolution: Evolving heterogeneous arti\\fcial neural net-\\nworks,\\\" Evolutionary Intelligence , vol. 7, no. 3, pp. 135{154, 2014.\\n[138] D. G. Wilson, S. Cussat-Blanc, H. Luga, and J. F. Miller, \\\\Evolving simple programs for\\nplaying atari games,\\\" in Proceedings of the Genetic and Evolutionary Computation Conference ,\\n2018, pp. 229{236.\\n[139] S. Kelly and M. I. Heywood, \\\\Emergent tangled graph representations for atari game playing\\nagents,\\\" in European Conference on Genetic Programming . Springer, 2017, pp. 64{79.\\n[140] ||, \\\\Emergent tangled program graphs in multi-task learning,\\\" in International Joint Con-\\nference on Artifcial Intelligence , 2018, pp. 5294{5298.\\n45 [141] S. Kelly, T. Voegerl, W. Banzhaf, and C. Gondro, \\\\Evolving hierarchical memory-prediction\\nmachines in multi-task reinforcement learning,\\\" Genetic Programming and Evolvable Ma-\\nchines , vol. 22, no. 4, pp. 573{605, 2021.\\n[142] R. J. Smith and M. I. Heywood, \\\\A model of external memory for navigation in partially\\nobservable visual reinforcement learning tasks,\\\" in European Conference on Genetic Program-\\nming . Springer, 2019, pp. 162{177.\\n[143] ||, \\\\Evolving dota 2 shadow \\fend bots using genetic programming with external memory,\\\"\\ninProceedings of the Genetic and Evolutionary Computation Conference , 2019, pp. 179{187.\\n[144] M. Onderwater, S. Bhulai, and R. van der Mei, \\\\Value function discovery in markov decision\\nprocesses with evolutionary algorithms,\\\" IEEE Transactions on Systems, Man, and Cybernet-\\nics: Systems , vol. 46, no. 9, pp. 1190{1201, 2015.\\n[145] D. Hein, S. Udluft, and T. A. Runkler, \\\\Interpretable policies for reinforcement learning by\\ngenetic programming,\\\" Engineering Applications of Arti\\fcial Intelligence , vol. 76, pp. 158{169,\\n2018.\\n[146] E. Alibekov, J. Kubal\\u0013 \\u0010k, and R. Babu\\u0014 ska, \\\\Symbolic method for deriving policy in reinforce-\\nment learning,\\\" in 2016 IEEE 55th Conference on Decision and Control . IEEE, 2016, pp.\\n2789{2795.\\n[147] E. Derner, J. Kubal\\u0013 \\u0010k, and R. Babu\\u0014 ska, \\\\Data-driven construction of symbolic process models\\nfor reinforcement learning,\\\" in 2018 IEEE International Conference on Robotics and Automa-\\ntion, 2018, pp. 5105{5112.\\n[148] S. Girgin and P. Preux, \\\\Feature discovery in reinforcement learning using genetic program-\\nming,\\\" in European Conference on Genetic Programming . Springer, 2008, pp. 218{229.\\n[149] K. Krawiec, \\\\Genetic programming-based construction of features for machine learning and\\nknowledge discovery tasks,\\\" Genetic Programming and Evolvable Machines , vol. 3, no. 4, pp.\\n329{343, 2002.\\n[150] M. Plappert, R. Houthooft, P. Dhariwal, S. Sidor, R. Y. Chen, X. Chen, T. Asfour, P. Abbeel,\\nand M. Andrychowicz, \\\\Parameter space noise for eeploration,\\\" International Conference on\\nLearning Representations, 2018.\\n[151] T. Yang, H. Tang, C. Bai, J. Liu, J. Hao, Z. Meng, P. Liu, and Z. Wang, \\\\Exploration in deep\\nreinforcement learning: A comprehensive survey,\\\" arXiv preprint arXiv:2109.06668 , 2021.\\n[152] J. K. Pugh, L. B. Soros, and K. O. Stanley, \\\\Quality diversity: A new frontier for evolutionary\\ncomputation,\\\" Frontiers in Robotics and AI , 2016.\\n[153] D. Gravina, A. Liapis, and G. Yannakakis, \\\\Surprise search: Beyond objectives and novelty,\\\"\\ninProceedings of the Genetic and Evolutionary Computation Conference 2016 , 2016, pp. 677{\\n684.\\n46 [154] H. Mengistu, J. Lehman, and J. Clune, \\\\Evolvability search: Directly selecting for evolvability\\nin order to study and produce it,\\\" in Proceedings of the Genetic and Evolutionary Computation\\nConference 2016 , 2016, pp. 141{148.\\n[155] D. Pathak, P. Agrawal, A. A. Efros, and T. Darrell, \\\\Curiosity-driven exploration by self-\\nsupervised prediction,\\\" in International Conference on Machine Learning . PMLR, 2017, pp.\\n2778{2787.\\n[156] S. Risi, S. D. Vanderbleek, C. E. Hughes, and K. O. Stanley, \\\\How novelty search escapes the\\ndeceptive trap of learning to learn,\\\" in Proceedings of the 11th Annual Conference on Genetic\\nand Evolutionary Computation , 2009, pp. 153{160.\\n[157] G. Cuccu and F. Gomez, \\\\When novelty is not enough,\\\" in European Conference on the\\nApplications of Evolutionary Computation . Springer, 2011, pp. 234{243.\\n[158] J.-B. Mouret and S. Doncieux, \\\\Encouraging behavioral diversity in evolutionary robotics: An\\nempirical study,\\\" Evolutionary Computation , vol. 20, no. 1, pp. 91{133, 2012.\\n[159] J. Lehman and K. O. Stanley, \\\\Evolving a diversity of virtual creatures through novelty\\nsearch and local competition,\\\" in Proceedings of the 13th Annual Conference on Genetic and\\nEvolutionary Computation , 2011.\\n[160] Q. Liu, Y. Wang, and X. Liu, \\\\Pns: Population-guided novelty search for reinforcement learn-\\ning in hard exploration environments,\\\" in 2021 IEEE\\/RSJ International Conference on Intel-\\nligent Robots and Systems , 2021.\\n[161] J.-B. Mouret and J. Clune, \\\\Illuminating search spaces by mapping elites,\\\" arXiv preprint\\narXiv:1504.04909 , 2015.\\n[162] A. Cully, \\\\Autonomous skill discovery with quality-diversity and unsupervised descriptors,\\\"\\ninProceedings of the Genetic and Evolutionary Computation Conference , 2019, pp. 81{89.\\n[163] R. Y. Tao, V. Fran\\u0018 cois-Lavet, and J. Pineau, \\\\Novelty search in representational space for\\nsample e\\u000ecient exploration,\\\" Advances in Neural Information Processing Systems , vol. 33, pp.\\n8114{8126, 2020.\\n[164] N. Rakicevic, A. Cully, and P. Kormushev, \\\\Policy manifold search: Exploring the mani-\\nfold hypothesis for diversity-based neuroevolution,\\\" in Genetic and Evolutionary Computation\\nConference , 2021.\\n[165] J. Parker-Holder, A. Pacchiano, K. Choromanski, and S. Roberts, \\\\E\\u000bective diversity in\\npopulation-based reinforcement learning,\\\" arXiv preprint arXiv:2002.00632 , 2020.\\n[166] E. C. Jackson and M. Daley, \\\\Novelty search for deep reinforcement learning policy network\\nweights by action sequence edit metric distance,\\\" in Proceedings of the Genetic and Evolution-\\nary Computation Conference Companion , 2019, pp. 173{174.\\n47 [167] L. Keller, D. Tanneberg, S. Stark, and J. Peters, \\\\Model-based quality-diversity search for\\ne\\u000ecient robot learning,\\\" in 2020 IEEE\\/RSJ International Conference on Intelligent Robots\\nand Systems . IEEE, 2020, pp. 9675{9680.\\n[168] A. Salehi, A. Coninx, and S. Doncieux, \\\\Few-shot quality-diversity optimization,\\\" IEEE\\nRobotics and Automation Letters , vol. 7, no. 2, pp. 4424{4431, 2022.\\n[169] Y. Wang, K. Xue, and C. Qian, \\\\Evolutionary diversity optimization with clustering-based\\nselection for reinforcement learning,\\\" in International Conference on Learning Representations ,\\n2022.\\n[170] R. Wang, J. Lehman, J. Clune, and K. O. Stanley, \\\\Poet: Open-ended coevolution of en-\\nvironments and their optimized solutions,\\\" in Proceedings of the Genetic and Evolutionary\\nComputation Conference , 2019, pp. 142{151.\\n[171] R. Wang, J. Lehman, A. Rawal, J. Zhi, Y. Li, J. Clune, and K. Stanley, \\\\Enhanced POET:\\nOpen-ended reinforcement learning through unbounded invention of learning challenges and\\ntheir solutions,\\\" in International Conference on Machine Learning . PMLR, 2020.\\n[172] V. Bhatt, B. Tjanaka, M. C. Fontaine, and S. Nikolaidis, \\\\Deep surrogate assisted generation\\nof environments,\\\" arXiv preprint arXiv:2206.04199 , 2022.\\n[173] S. Brych and A. Cully, \\\\Competitiveness of map-elites against proximal policy optimization\\non locomotion tasks in deterministic simulations,\\\" arXiv preprint arXiv:2009.08438 , 2020.\\n[174] V. Vassiliades, K. Chatzilygeroudis, and J.-B. Mouret, \\\\Using centroidal voronoi tessellations\\nto scale up the multidimensional archive of phenotypic elites algorithm,\\\" IEEE Transactions\\non Evolutionary Computation , vol. 22, no. 4, pp. 623{630, 2017.\\n[175] C. Colas, J. Huizinga, V. Madhavan, and J. Clune, \\\\Scaling map-elites to deep neuroevolu-\\ntion,\\\" arXiv preprint arXiv:2003.01825 , 2020.\\n[176] T. Pierrot, V. Mac\\u0013 e, F. Chalumeau, A. Flajolet, G. Cideron, K. Beguir, A. Cully, O. Sigaud,\\nand N. Perrin-Gilbert, \\\\Diversity policy gradient for sample e\\u000ecient quality-diversity opti-\\nmization,\\\" in ICLR Workshop on Agent Learning in Open-Endedness , 2022.\\n[177] B. Tjanaka, M. C. Fontaine, J. Togelius, and S. Nikolaidis, \\\\Di\\u000berentiable quality diversity for\\nreinforcement learning by approximating gradients,\\\" in International Conference on Learning\\nRepresentations Workshop on Agent Learning in Open-Endedness , 2022.\\n[178] O. Nilsson and A. Cully, \\\\Policy gradient assisted map-elites,\\\" in Genetic and Evolutionary\\nComputation Conference , 2021.\\n[179] Y. Zhang, M. C. Fontaine, A. K. Hoover, and S. Nikolaidis, \\\\Dsa-me: Deep surrogate assisted\\nmap-elites,\\\" in International Conference on Learning Representations Workshop on Agent\\nLearning in Open-Endedness , 2022.\\n48 [180] A. Eco\\u000bet, J. Huizinga, J. Lehman, K. O. Stanley, and J. Clune, \\\\First return, then explore,\\\"\\nNature , vol. 590, no. 7847, pp. 580{586, 2021.\\n[181] D. Gravina, A. Liapis, and G. N. Yannakakis, \\\\Quality diversity through surprise,\\\" IEEE\\nTransactions on Evolutionary Computation , vol. 23, no. 4, pp. 603{616, 2018.\\n[182] M. Bellemare, S. Srinivasan, G. Ostrovski, T. Schaul, D. Saxton, and R. Munos, \\\\Unifying\\ncount-based exploration and intrinsic motivation,\\\" Advances in neural information processing\\nsystems , vol. 29, pp. 1471{1479, 2016.\\n[183] S. Forestier, R. Portelas, Y. Mollard, and P.-Y. Oudeyer, \\\\Intrinsically motivated goal ex-\\nploration processes with automatic curriculum learning,\\\" arXiv preprint arXiv:1708.02190 ,\\n2017.\\n[184] C. Colas, O. Sigaud, and P.-Y. Oudeyer, \\\\GEP-PG: Decoupling exploration and exploitation\\nin deep reinforcement learning algorithms,\\\" in International Conference on Machine Learning .\\nPMLR, 2018, pp. 1039{1048.\\n[185] C. Stanton and J. Clune, \\\\Deep curiosity search: Intra-life exploration improves performance\\non challenging deep reinforcement learning problems,\\\" arXiv preprint arXiv:1806.00553 , 2018.\\n[186] H. Zheng, J. Jiang, P. Wei, G. Long, and C. Zhang, \\\\Competitive and cooperative heteroge-\\nneous deep reinforcement learning,\\\" in Proceedings of the International Joint Conference on\\nAutonomous Agents and Multiagent Systems , 2020.\\n[187] S. L\u007f u, S. Han, W. Zhou, and J. Zhang, \\\\Recruitment-imitation mechanism for evolutionary\\nreinforcement learning,\\\" Information Sciences , vol. 553, pp. 172{188, 2021.\\n[188] Y. Ma, T. Liu, B. Wei, Y. Liu, K. Xu, and W. Li, \\\\Evolutionary action selection for gradient-\\nbased policy learning,\\\" arXiv preprint arXiv:2201.04286 , 2022.\\n[189] H. Shi, B. Zhou, H. Zeng, F. Wang, Y. Dong, J. Li, K. Wang, H. Tian, and M. Q.-H.\\nMeng, \\\\Reinforcement learning with evolutionary trajectory generator: A general approach for\\nquadrupedal locomotion,\\\" IEEE Robotics and Automation Letters , vol. 7, no. 2, pp. 3085{3092,\\n2022.\\n[190] A. Morel, Y. Kunimoto, A. Coninx, and S. Doncieux, \\\\Automatic acquisition of a repertoire\\nof diverse grasping trajectories through behavior shaping and novelty search,\\\" arXiv preprint\\narXiv:2205.08189 , 2022.\\n[191] A. Pourchot and O. Sigaud, \\\\Cem-rl: Combining evolutionary and gradient-based methods\\nfor policy search,\\\" International Conference on Learning Representations , 2019.\\n[192] K. Lee, B.-U. Lee, U. Shin, and I. S. Kweon, \\\\An e\\u000ecient asynchronous method for integrating\\nevolutionary and gradient-based policy search,\\\" Advances in Neural Information Processing\\nSystems , vol. 33, pp. 10 124{10 135, 2020.\\n49 [193] K. Suri, \\\\O\\u000b-policy evolutionary reinforcement learning with maximum mutations,\\\" in Pro-\\nceedings of the 21st International Conference on Autonomous Agents and Multiagent Systems ,\\n2022.\\n[194] E. Marchesini, D. Corsi, and A. Farinelli, \\\\Genetic soft updates for policy evolution in deep\\nreinforcement learning,\\\" in International Conference on Learning Representations , 2020.\\n[195] S. Zhu, F. Belardinelli, and B. G. Le\\u0013 on, \\\\Evolutionary reinforcement learning for sparse re-\\nwards,\\\" in Proceedings of the Genetic and Evolutionary Computation Conference , 2021, pp.\\n1508{1512.\\n[196] J. Clune, \\\\Ai-gas: Ai-generating algorithms, an alternate paradigm for producing general\\narti\\fcial intelligence,\\\" arXiv preprint arXiv:1905.10985 , 2019.\\n[197] A. Faust, A. Francis, and D. Mehta, \\\\Evolving rewards to automate reinforcement learning,\\\"\\narXiv preprint arXiv:1905.07628 , 2019.\\n[198] A. Laud and G. DeJong, \\\\The in\\ruence of reward on the speed of reinforcement learning: An\\nanalysis of shaping,\\\" in Proceedings of the 20th International Conference on Machine Learning ,\\n2003, pp. 440{447.\\n[199] A. Y. Ng, D. Harada, and S. Russell, \\\\Policy invariance under reward transformations: Theory\\nand application to reward shaping,\\\" in International Conference on Machine Learning , vol. 99,\\n1999, pp. 278{287.\\n[200] F. Ferreira, T. Nierho\\u000b, A. Saelinger, and F. Hutter, \\\\Learning synthetic environments and\\nreward networks for reinforcement learning,\\\" International Conference on Learning Represen-\\ntations , 2022.\\n[201] S. Singh, R. L. Lewis, A. G. Barto, and J. Sorg, \\\\Intrinsically motivated reinforcement learning:\\nAn evolutionary perspective,\\\" IEEE Transactions on Autonomous Mental Development , vol. 2,\\nno. 2, pp. 70{82, 2010.\\n[202] S. Niekum, A. G. Barto, and L. Spector, \\\\Genetic programming for reward function search,\\\"\\nIEEE Transactions on Autonomous Mental Development , vol. 2, no. 2, pp. 83{90, 2010.\\n[203] E. Uchibe and K. Doya, \\\\Finding intrinsic rewards by embodied evolution and constrained\\nreinforcement learning,\\\" Neural Networks , vol. 21, no. 10, pp. 1447{1455, 2008.\\n[204] H. U. Sheikh, S. Khadka, S. Miret, S. Majumdar, and M. Phielipp, \\\\Learning intrinsic symbolic\\nrewards in reinforcement learning,\\\" in International Joint Conference on Neural Networks .\\nIEEE, 2022, pp. 1{8.\\n[205] G. Paolo, A. Coninx, S. Doncieux, and A. La\\raqui\\u0012 ere, \\\\Sparse reward exploration via novelty\\nsearch and emitters,\\\" in Proceedings of the Genetic and Evolutionary Computation Conference ,\\n2021, pp. 154{162.\\n50 [206] S. Majumdar, S. Khadka, S. Miret, S. Mcaleer, and K. Tumer, \\\\Evolutionary reinforcement\\nlearning for sample-e\\u000ecient multiagent coordination,\\\" in International Conference on Machine\\nLearning , 2020.\\n[207] R. Lowe, Y. Wu, A. Tamar, J. Harb, P. Abbeel, and I. Mordatch, \\\\Multi-agent actor-critic for\\nmixed cooperative-competitive environments,\\\" arXiv preprint arXiv:1706.02275 , 2017.\\n[208] E. Sachdeva, S. Khadka, S. Majumdar, and K. Tumer, \\\\Maedys: Multiagent evolution via\\ndynamic skill selection,\\\" in Proceedings of the Genetic and Evolutionary Computation Confer-\\nence, 2021, pp. 163{171.\\n[209] H.-T. L. Chiang, A. Faust, M. Fiser, and A. Francis, \\\\Learning navigation behaviors end-\\nto-end with autorl,\\\" IEEE Robotics and Automation Letters , vol. 4, no. 2, pp. 2007{2014,\\n2019.\\n[210] J. X. Wang, E. Hughes, C. Fernando, W. M. Czarnecki, E. A. Du\\u0013 e~ nez-Guzm\\u0013 an, and J. Z. Leibo,\\n\\\\Evolving intrinsic motivations for altruistic behavior,\\\" arXiv preprint arXiv:1811.05931 ,\\n2018.\\n[211] C. Finn, P. Abbeel, and S. Levine, \\\\Model-agnostic meta-learning for fast adaptation of deep\\nnetworks,\\\" in International Conference on Machine Learning . JMLR. org, 2017, pp. 1126{\\n1135.\\n[212] Y. Duan, J. Schulman, X. Chen, P. L. Bartlett, I. Sutskever, and P. Abbeel, \\\\Rl2: Fast rein-\\nforcement learning via slow reinforcement learning,\\\" arXiv preprint arXiv:1611.02779 , 2016.\\n[213] R. Houthooft, Y. Chen, P. Isola, B. Stadie, F. Wolski, O. Jonathan Ho, and P. Abbeel, \\\\Evolved\\npolicy gradients,\\\" Advances in Neural Information Processing Systems , vol. 31, 2018.\\n[214] X. Song, W. Gao, Y. Yang, K. Choromanski, A. Pacchiano, and Y. Tang, \\\\Es-maml: Simple\\nhessian-free meta learning,\\\" arXiv preprint arXiv:1910.01215 , 2019.\\n[215] C. Fernando, J. Sygnowski, S. Osindero, J. Wang, T. Schaul, D. Teplyashin, P. Sprechmann,\\nA. Pritzel, and A. Rusu, \\\\Meta-learning by the baldwin e\\u000bect,\\\" in Proceedings of the Genetic\\nand Evolutionary Computation Conference Companion , 2018, pp. 1313{1320.\\n[216] J. D. Co-Reyes, Y. Miao, D. Peng, E. Real, Q. V. Le, S. Levine, H. Lee, and A. Faust, \\\\Evolving\\nreinforcement learning algorithms,\\\" in International Conference on Learning Representations ,\\n2021.\\n[217] J. J. Garau-Luis, Y. Miao, J. D. Co-Reyes, A. Parisi, J. Tan, E. Real, and A. Faust, \\\\Multi-\\nobjective evolution for generalizable policy gradient algorithms,\\\" International Conference on\\nLearning Representations , 2022.\\n[218] F. Alet, M. F. Schneider, T. Lozano-Perez, and L. P. Kaelbling, \\\\Meta-learning curiosity\\nalgorithms,\\\" in International Conference on Learning Representations , 2020.\\n51 [219] C. A. Coello Coello, S. Gonz\\u0013 alez Brambila, J. Figueroa Gamboa, M. G. Castillo Tapia, and\\nR. Hern\\u0013 andez G\\u0013 omez, \\\\Evolutionary multiobjective optimization: open research areas and\\nsome challenges lying ahead,\\\" Complex & Intelligent Systems , vol. 6, pp. 221{236, 2020.\\n[220] K. Van Mo\\u000baert, M. M. Drugan, and A. Now\\u0013 e, \\\\Scalarized multi-objective reinforcement learn-\\ning: Novel design techniques,\\\" in 2013 IEEE Symposium on Adaptive Dynamic Programming\\nand Reinforcement Learning . IEEE, 2013, pp. 191{199.\\n[221] J. M. Bader, Hypervolume-based search for multiobjective optimization: theory and methods .\\nJohannes Bader, 2010, no. 112.\\n[222] E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, and V. G. da Fonseca, \\\\Performance\\nassessment of multiobjective optimizers: An analysis and review,\\\" IEEE Transactions on\\nEvolutionary Computation , vol. 7, no. 2, pp. 117{132, 2003.\\n[223] C. M. Fonseca and P. J. Fleming, \\\\An overview of evolutionary algorithms in multiobjective\\noptimization,\\\" Evolutionary Computation , vol. 3, no. 1, pp. 1{16, 1995.\\n[224] N. Beume, C. M. Fonseca, M. Lopez-Ibanez, L. Paquete, and J. Vahrenhold, \\\\On the com-\\nplexity of computing the hypervolume indicator,\\\" IEEE Transactions on Evolutionary Com-\\nputation , vol. 13, no. 5, pp. 1075{1082, October 2009.\\n[225] J. Xu, Y. Tian, P. Ma, D. Rus, S. Sueda, and W. Matusik, \\\\Prediction-guided multi-objective\\nreinforcement learning for continuous robot control,\\\" in International Conference on Machine\\nLearning , 2020.\\n[226] E. A. Feinberg and A. Shwartz, \\\\Constrained markov decision models with weighted discounted\\nrewards,\\\" Mathematics of Operations Research , vol. 20, no. 2, pp. 302{320, 1995.\\n[227] A. Abels, D. Roijers, T. Lenaerts, A. Now\\u0013 e, and D. Steckelmacher, \\\\Dynamic weights in multi-\\nobjective deep reinforcement learning,\\\" in International Conference on Machine Learning .\\nPMLR, 2019, pp. 11{20.\\n[228] K. V. Mo\\u000baert, M. M. Drugan, and A. Now\\u0013 e, \\\\Hypervolume-based multi-objective reinforce-\\nment learning,\\\" in International Conference on Evolutionary Multi-Criterion Optimization .\\nSpringer, 2013, pp. 352{366.\\n[229] H. Yamamoto, T. Hayashida, I. Nishizaki, and S. Sekizaki, \\\\Hypervolume-based multi-\\nobjective reinforcement learning: Interactive approach,\\\" Advances in Science, Technology and\\nEngineering Systems Journal , vol. 4, 2019.\\n[230] K. Van Mo\\u000baert and A. Now\\u0013 e, \\\\Multi-objective reinforcement learning using sets of pareto\\ndominating policies,\\\" The Journal of Machine Learning Research , vol. 15, no. 1, pp. 3483{3512,\\n2014.\\n[231] T. Brys, A. Harutyunyan, P. Vrancx, M. E. Taylor, D. Kudenko, and A. Now\\u0013 e, \\\\Multi-\\nobjectivization of reinforcement learning problems by reward shaping,\\\" in 2014 international\\njoint conference on neural networks . IEEE, 2014, pp. 2315{2322.\\n52 [232] R. Shen, Y. Zheng, J. Hao, Z. Meng, Y. Chen, C. Fan, and Y. Liu, \\\\Generating behavior-diverse\\ngame AIs with evolutionary multi-objective deep reinforcement learning,\\\" in International\\nJoint Conference on Arti\\fcial Intelligence , 2020.\\n[233] V. Villin, N. Masuyama, and Y. Nojima, \\\\E\\u000bects of di\\u000berent optimization formulations in evo-\\nlutionary reinforcement learning on diverse behavior generation,\\\" in IEEE Symposium Series\\non Computational Intelligence , 2021.\\n[234] B. Li, J. Li, K. Tang, and X. Yao, \\\\Many-objective evolutionary algorithms: A survey,\\\" Acm\\nComputing Surveys , vol. 48, no. (1), pp. 1{35., 2015.\\n[235] S. Han and Y. Sung, \\\\Dimension-wise importance sampling weight clipping for sample-e\\u000ecient\\nreinforcement learning,\\\" in International Conference on Machine Learning , 2019.\\n[236] R. Storn and K. Price, \\\\Di\\u000berential evolution { a simple and e\\u000ecient heuristic for global\\noptimization over continuous spaces,\\\" Journal of Global Optimization , vol. 11, no. 4, pp. 341{\\n359, December 1997.\\n[237] J. Kennedy and R. Eberhart, \\\\Particle swarm optimization,\\\" in Proceedings of International\\nConference on Neural Networks , vol. 4. IEEE, 1995, pp. 1942{1948.\\n[238] R. Cheng and Y. Jin, \\\\A competitive swarm optimizer for large scale optimization,\\\" IEEE\\nTransactions on Cybernetics , vol. 45, no. 2, pp. 191{204, 2014.\\n[239] J. Stork, M. Zae\\u000berer, N. Eisler, P. Tichelmann, T. Bartz-Beielstein, and A. Eiben, \\\\Behavior-\\nbased neuroevolutionary training in reinforcement learning,\\\" in Proceedings of the Genetic and\\nEvolutionary Computation Conference , 2021, pp. 1753{1761.\\n[240] Y. Wang, T. Zhang, Y. Chang, B. Liang, X. Wang, and B. Yuan, \\\\A surrogate-assisted\\ncontroller for expensive evolutionary reinforcement learning,\\\" Information Sciences , vol. 616,\\npp. 539{557, 2022.\\n[241] G. Brockman, V. Cheung, L. Pettersson, J. Schneider, J. Schulman, J. Tang, and W. Zaremba,\\n\\\\OpenAI Gym,\\\" arXiv preprint arXiv:1606.01540 , 2016.\\n[242] H. Bai, R. Shen, Y. Lin, B. Xu, and R. Cheng, \\\\Lamarckian platform: Pushing the bound-\\naries of evolutionary reinforcement learning towards asynchronous commercial games,\\\" IEEE\\nTransactions on Games , pp. 1{14, 2022.\\n[243] R. Tangri, D. P. Mandic, and A. G. Constantinides, \\\\Pearl: Parallel evolutionary and rein-\\nforcement learning library,\\\" arXiv preprint arXiv:2201.09568 , 2022.\\n[244] Y. Tang, Y. Tian, and D. Ha, \\\\Evojax: Hardware-accelerated neuroevolution,\\\" 2022.\\n[245] R. T. Lange, \\\\Evosax: Jax-based evolution strategies,\\\" 2022.\\n[246] B. Huang, R. Cheng, Y. Jin, and K. C. Tan, \\\\Evox: A distributed gpu-accelerated library\\ntowards scalable evolutionary computation,\\\" arXiv preprint arXiv:2301.12457 , 2023.\\n53 [247] B. Lim, M. Allard, L. Grillotti, and A. Cully, \\\\Accelerated quality-diversity for robotics\\nthrough massive parallelism,\\\" in ICLR Workshop on Agent Learning in Open-Endedness , 2022.\\n[248] J. Bhatia, H. Jackson, Y. Tian, J. Xu, and W. Matusik, \\\\Evolution gym: A large-scale\\nbenchmark for evolving soft robots,\\\" Advances in Neural Information Processing Systems ,\\nvol. 34, pp. 2201{2214, 2021.\\n54\",\"7\":\" RADAM: T EXTURE RECOGNITION THROUGH RANDOMIZED\\nAGGREGATED ENCODING OF DEEPACTIVATION MAPS\\nLeonardo Scabini1,2, Kallil M. Zielinski1, Lucas C. Ribas3, Wesley N. Gon\\u00e7alves4, Bernard De Baets2, and Odemir M.\\nBruno1\\n1S\\u00e3o Carlos Institute of Physics, University of S\\u00e3o Paulo, postal code 13560-970, S\\u00e3o Carlos - SP, Brazil\\n2KERMIT, Department of Data Analysis and Mathematical Modelling, Ghent University, Coupure links 653, postal code 9000,\\nGhent, Belgium\\n3Institute of Biosciences, Humanities and Exact Sciences, S\\u00e3o Paulo State University, postal code 15054-000, S\\u00e3o Jos\\u00e9 do Rio Preto -\\nSP, Brazil\\n4Faculty of Computing, Federal University of Mato Grosso do Sul, postal code 79070-900, Campo Grande - MS, Brazil\\nABSTRACT\\nTexture analysis is a classical yet challenging task in computer vision for which deep neural networks\\nare actively being applied. Most approaches are based on building feature aggregation modules\\naround a pre-trained backbone and then \\ufb01ne-tuning the new architecture on speci\\ufb01c texture recogni-\\ntion tasks. Here we propose a new method named Random encoding of Aggregated DeepActivation\\nMaps (RADAM) which extracts rich texture representations without ever changing the backbone.\\nThe technique consists of encoding the output at different depths of a pre-trained deep convolutional\\nnetwork using a Randomized Autoencoder (RAE). The RAE is trained locally to each image using a\\nclosed-form solution, and its decoder weights are used to compose a 1-dimensional texture represen-\\ntation that is fed into a linear SVM. This means that no \\ufb01ne-tuning or backpropagation is needed. We\\nexplore RADAM on several texture benchmarks and achieve state-of-the-art results with different\\ncomputational budgets. Our results suggest that pre-trained backbones may not require additional\\n\\ufb01ne-tuning for texture recognition if their learned representations are better encoded.\\n1 Introduction\\nFor several decades, texture has been studied in Computer Vision as a fundamental visual cue for image recognition in\\nseveral applications. Despite lacking a widely accepted theoretical de\\ufb01nition, we all have developed an intuition for\\ntextures by analyzing the world around us from material surfaces in our daily life, through microscopic images, and\\neven through macroscopic images from telescopes and remote sensing. In digital images, one abstract de\\ufb01nition is\\nthat texture elements emerge from the local intensity constancy and\\/or variations of pixels producing spatial patterns\\nroughly independently at different scales [39].\\nThe classical approaches to texture recognition focus on the mathematical description of the textural patterns, considering\\nproperties such as statistics [10, 15, 24], frequency [1, 12], complexity\\/fractality [2, 38], and others [52]. Many such\\naspects of texture are challenging to model even in controlled imaging scenarios. Moreover, the wild nature of digital\\nimages also results in additional variability, making the task even more complex in real-world applications.\\nRecently, the power of deep neural networks has been extended to texture analysis by taking advantage of models\\npre-trained on big natural image datasets [4 \\u20136, 22, 46 \\u201351]. These transfer-learning approaches combine the general\\nvision capabilities of pre-trained models with dedicated techniques to capture additional texture information, achieving\\nstate-of-the-art performance on several texture recognition tasks. Therefore, most of the recent works on deep texture\\nrecognition propose to build new modules around a pre-trained deep network (backbone) and to retrain the new\\narchitecture for a speci\\ufb01c texture analysis task. However, even if the new modules are relatively cheap in terms of\\ncomputational complexity, resulting in good inference ef\\ufb01ciency, the retraining of the backbone itself is usually costly.\\nGoing in a different direction, Randomized Neural Networks [13,25,26,40] proposes a closed-form solution for training Going in a different direction, Randomized Neural Networks [13,25,26,40] proposes a closed-form solution for training\\nneural networks, instead of the common backpropagation, with various potential applications. For instance, the training\\ntime of randomization-based models was analyzed [17] on datasets such as MNIST, resulting in gains up to 150 times.\\nThese gains can be expressive when hundreds of thousands of images are used to train a model.arXiv:2303.04554v1  [cs.CV]  8 Mar 2023 In this work, we propose a new module for texture feature extraction from pre-trained deep convolutional neural\\nnetworks (DCNNs). The method, called Random encoding of Aggregated DeepActivation Maps (RADAM), goes\\nin a different direction than recent literature on deep texture recognition. Instead of increasing the complexity of\\nthe backbone and then retraining everything, we propose a simple codi\\ufb01cation of the backbone features using a new\\nrandomized module. The method is based on aggregating deep activation maps from different depths of a pre-trained\\nconvolutional network, and then training Randomized Autoencoders (RAEs) in a pixel-wise fashion for each image,\\nusing a closed-form solution. This module outputs the decoder weights from the learned RAEs, which are used as a\\n1-dimensional feature representation of the input image. This approach is simple and does not require hyperparameter\\ntuning or backpropagation training. Instead, we propose to attach a linear SVM at the top of our features, which can be\\nsimply used with standard parameters. Our code is open and is available in a public repository1. In summary, our main\\ncontributions are:\\n(i)We propose the RADAM texture feature encoding technique applied over a pre-trained DCNN backbone and\\ncoupled with a simple linear SVM. The model achieves impressive classi\\ufb01cation performance without needing\\nto \\ufb01ne-tune the backbone, in contrast to what has been proposed in previous works.\\n(ii)Bigger backbones and better pre-training improve the performance of RADAM considerably, suggesting that\\nour approach scales well.\\n2 Background\\nWe start by conducting a literature review on texture analysis with deep learning and randomized neural networks. The\\nmethods covered here are also considered for comparison in our experiments.\\n2.1 Texture Analysis with Deep Neural Networks\\nIn this work, we focus on transfer-learning-based texture analysis by taking advantage of pre-trained deep neural\\nnetworks. For a more comprehensive review of different approaches to texture analysis, the reader may consult [19].\\nThere have been numerous studies involving deep learning for texture recognition, and here we review them according\\nto two approaches: feature extraction or end-to-end \\ufb01ne-tuning. Some studies explore CNNs only for texture feature\\nextraction and use a dedicated classi\\ufb01er apart from the model architecture. Cimpoi et al. [6] was one of the \\ufb01rst works\\non the subject, where the authors compare the ef\\ufb01ciency of two different CNN architectures for feature extraction:\\nFC-CNN, which uses a fully connected (FC) layer, and FV-CNN, which uses a Fisher vector (FV) [5] as a pooling\\nmethod. They demonstrated that, in general, FC features are not that ef\\ufb01cient because their output is highly correlated\\nwith the spatial order of the pixels. Later on, Condori and Bruno [22] developed a model, called RankGP-3M-CNN,\\nwhich performs multi-layer feature aggregation employing Global Average Pooling (GAP) to extract the feature vectors\\nof activation maps at different depths of three combined CNNs (VGG-19, Inception-V3, and ResNet50). They propose\\na ranking technique to select the best activation maps given a training dataset, achieving promising results in some\\ncases but at the cost of increased computational load, since three backbones are needed. Lyra et al. [21] also proposes\\nfeature aggregation from multiple convolutional layers, but pooling is performed using an FV-based approach.\\nNumerous studies propose end-to-end architectures that enable \\ufb01ne-tuning of the backbone for texture recognition.\\nZhang et al. [51] proposed an orderless encoding layer on top of a DCNN, called Deep Texture Encoding Network\\n(Deep-TEN), which allows images of arbitrary size. Xue et al. [46] introduces a Deep Encoding Pooling Network\\n(DEPNet), which combines features from the texture encoding layer from Deep-TEN and a global average pooling (DEPNet), which combines features from the texture encoding layer from Deep-TEN and a global average pooling\\n(GAP) to explore both the local appearance and global context of the images. These features are further processed by\\na bilinear pooling layer [18]. In another work, Xue et al. [47] also combined features from differential images with\\nthe features of DEPNet into a new architecture. Using a different approach, Zhai et al. [50] proposed the Multiple-\\nAttribute-Perceived Network (MAP-Net), which incorporated visual texture attributes in a multi-branch architecture that\\naggregates features of different layers. Later on [49], they explored the spatial dependency among texture primitives for\\ncapturing structural information of the images by using a model called Deep Structure-Revealed Network (DSRNet).\\nChen et al. [4] introduced the Cross-Layer Aggregation of a Statistical Self-similarity Network (CLASSNet). This CNN\\nfeature aggregation module uses a differential box-counting pooling layer that characterizes the statistical self-similarity\\nof texture images. More recently, Yang et al. [48] proposed DFAEN (Double-order Knowledge Fusion and Attentional\\nEncoding Network), which takes advantage of attention mechanisms to aggregate \\ufb01rst- and second-order information\\nfor encoding texture features. Fine-tuning is employed in these methods to adapt the backbone to the new architecture\\nalong with the new classi\\ufb01cation head.\\n1https:\\/\\/github.com\\/scabini\\/RADAM\\n2 As an alternative to CNNs, Vision Transformers (ViTs) [8] are emerging in the visual recognition literature. Some works\\nhave brie\\ufb02y explored their potential for texture analysis through the Describable Textures Dataset (DTD) achieving state-\\nof-the-art results. Firstly, ViTs achieve competitive results compared to CNNs, but the lack of the typical convolutional\\ninductive bias usually results in the need for more training data. To overcome this issue, a promising alternative is\\nto use attention mechanisms to learn directly from text descriptions about images, e.g.using Contrastive Language\\nImage Pre-training (CLIP) [31]. There have also been proposed bigger datasets for the pre-training of ViTs, such as\\nBamboo [53], showing that these models scale well. Another approach is to optimize the construction of multitask\\nlarge-scale ViTs such as proposed by Gesmundo [9] with the \\u00162Net+ method.\\n2.2 Randomized Neural Networks for Texture Analysis\\nA Randomized Neural Network [13, 25, 26, 40], in its simplest form, is a single-hidden-layer feed-forward neural\\nnetwork whose input weights are random, while the weights of the output layer are learned by a closed-form solution,\\nin contrast to gradient-descent-based learning. Recently, several works have investigated RNNs to learn texture features\\nfor image analysis. S\\u00e1 Junior et al. [35] used small local regions of one image as inputs to an RNN, and the central\\npixel of the region as the target. The trained weights of the output layer for each image are then used as a texture\\nrepresentation. Ribas et al. [33] improved the previous approach with the incorporation of graph theory to model the\\ntexture image. Other works [16, 32] have also extended these concepts to video texture analysis (dynamic texture).\\nThe training of 1-layer RNNs as employed in previous works is a least-squares solution at the output layer. First,\\nconsiderX2Rn\\u0002zas the input matrix with ntraining samples and zfeatures, and g=\\u001e(XW)as the forward pass of\\nthe hidden layer with a sigmoid nonlinearity, where W2Rz\\u0002qrepresents the random input weights for qneurons.\\nGiven the desired output labels Y, the output weights fare obtained as the least-squares solution of a system of linear\\nequations:\\nf=YgT(ggT)\\u00001; (1)\\nwheregT(ggT)\\u00001is the Moore\\u2013Penrose pseudo-inverse [23, 29] of matrix g.\\nAn important aspect of RNNs is the generation of random weights for the \\ufb01rst layer. Evidence suggests that this choice\\nhas little impact once the weights are \\ufb01xed. In this sense, a common trend among previous works is the use of the Linear\\nCongruential Generator (LCG), a simple pseudo-random number generator in the form of xk+1= (axk+b) modc.\\nThe RNN can be used as a randomized autoencoder (RAE) [17] by considering the input feature matrix Xas the target\\noutputY=X. In this sense, the model is composed of a random encoder and a least-squares-based decoder that can\\nmap the input data. Kasun et al. [17] also suggests the use of random orthogonal weights [37] for the initialization of\\nthe encoder. In this way, the weight matrix fwill represent the transformation of the projected random space back into\\nthe input data X(output).\\n3 RADAM for Texture Feature Encoding\\nThe main idea of the proposed RADAM method is to use multi-depth feature aggregation and randomized pixel-wise\\nencoding to compose a single feature vector, given an input image processed by the backbone. First of all, consider\\nan input image I2Rw0\\u0002h0\\u00023fed into a backbone B= (d1;:::;d n), consisting of nblocks of convolutional layers.\\nAn activation map, i.e., the output of any convolutional block given I, is a 3-dimensional tensor (ignoring the batch\\ndimension, for simplicity) Xi2Rwi\\u0002hi\\u0002zi. The process of feature aggregation consists of combining the outputs\\nof different activation maps at different depths. To that end, we divide the backbone into a \\ufb01xed number of blocks\\naccording to different depths. This division is made to keep a \\ufb01xed number of blocks for feature extraction, regardless according to different depths. This division is made to keep a \\ufb01xed number of blocks for feature extraction, regardless\\nof the total depth of the backbone architecture.\\n3.1 Pre-trained Deep Convolutional Networks: Backbone selection\\nMost previous works on texture analysis consider pre-trained ResNets [11] (18 or 50) as backbones. Here, we consider\\nthe output of \\ufb01ve blocks of layers according to the ResNet architecture, meaning that \\ufb01ve activation maps are considered\\nfor feature aggregation. Additionally, we consider the ConvNeXt architecture [20], a more recent method with promising\\nresults in image recognition. For this backbone, we consider the activation maps from the four blocks of layers according\\nto the architecture described in the original work. More speci\\ufb01cally, the following ConvNeXt con\\ufb01gurations are used,\\nwith their corresponding number of channels ( zi) of each block:\\n\\u2022 ConvNeXt-nano2:zi= (80;160;320;640) .\\n2This variant was not presented in the original paper, but is available at https:\\/\\/github.com\\/rwightman\\/\\npytorch-image-models\\/blob\\/main\\/timm\\/models\\/ConvNeXt.py\\n3 \\u2022 ConvNeXt-T: zi= (96;192;384;768) .\\n\\u2022 ConvNeXt-B: zi= (128;256;512;1024)\\n\\u2022 ConvNeXt-L: zi= (192;384;768;1536) .\\n\\u2022 ConvNeXt-XL: zi= (256;512;1024;2048) .\\nAs a lightweight alternative, MobileNet V2 [36] is also compared within our experiments (with \\ufb01ve blocks\\nof layers) using either the original architecture or a 1.4 width multiplier, i.e., zi= (16;24;32;96;320) and\\nzi= (24;32;48;136;448) , respectively.\\n3.2 Deep Activation Map Preparation\\nGiven each deep activation map Xi, we apply a depth-wise lp-normalization ( p= 2, i.e., Euclidean norm)\\nXi(:;:;j) =Xi(:;:;j)\\nmax(jjXi(:;:;j)jj2); (2)\\nwhereXi(:;:;j)represents the 2-dimensional activation map at each channel j2ziwith spatial sizes (wi;hi).\\nFor feature aggregation, we propose to concatenate the activation maps along the third dimension ( zi). However, each\\nmapXiinitially has a different spatial dimension wiandhi. To overcome this, we simply resize all activation maps\\nwith bilinear interpolation using the spatial dimensions of Xn\\n2,(wn\\n2;hn\\n2), as the target sizes. In other words, we\\nconsider the spatial dimensions at the middle of the backbone as our anchor size, meaning that some activation maps\\nwill require upscaling (if i>n\\n2) and others downscaling (if i<n\\n2). Naturally, the information from activation maps at\\nhigher depths receives higher priority considering that upscaling preserves more information than downscaling. These\\nassumptions consider the most common structure of convolutional architectures where the spatial size decreases with\\nlayer depth. Nonetheless, the idea is to keep all activation maps with a \\ufb01xed spatial dimension. From now on, we will\\nrefer to spatial dimensions of all Xiasw=wn\\n2andh=hn\\n2. For an input size of 224x224, this results in w=h= 28\\nfor the backbones explored in this work. The concatenation of activation maps is then performed as\\nX0= [X1;:::;Xn]2Rw\\u0002h\\u0002z!X02Rwh\\u0002z; (3)\\nwhere [:;:]denotes the concatenation along the third dimension, and z=P\\niziis the resulting number of channels\\nafter concatenation. Considering common convolutional architectures where zi<zi+1, activation maps from higher\\ndepths have a higher in\\ufb02uence on the overall zfeatures. Additionally, the 2-dimensional activation map at each channel\\nziis \\ufb02attened, resulting in the reshaped 2D representation X0with sizeswh-by-z, which we refer to as an aggregated\\nactivation map. These steps are illustrated in Fig. 1(a), which reports the overall structure of the proposed method.\\n X1 \\n(w,h,z1) \\n Xn \\n(w,h,zn) L2\\nL2X'  \\n(wh, z)RAE 1\\nRAE mbackboned1\\ndnconcat +\\nreshape\\ntexture descriptor  \\n(z features)depth-wise\\nnormalization\\noutputinput\\naggregative\\nactivation mappositional\\nencodingsoup of RAEs\\nL2\\n(a) The proposed feature encoding module (RADAM).\\nRAE kEncoder gk: linear layer Wk \\n(random weights)\\nDecoder fk :least squares\\nsolution with X'Sigmoid activation(wh,z) X'\\noutput: decoder weights (b) Randomized Auto-encoder (RAE).\\nFigure 1: Illustration of the proposed RADAM architecture (a), given an input image to a \\ufb01nal descriptor. The RAE is\\nshown in detail (b), which is a simple 1-layer auto-encoder solved through least-squares, where we use the decoder\\nweights as descriptors (summed for mRAEs).\\n4 3.3 Pixel-wise Randomized Encoding\\nThe aggregated activation map of a single image is used to train an RAE considering each spatial point, or pixel (row of\\nX0), as a sample and each channel (column of X0) as a feature. In this sense, the method also works with arbitrary\\ninput sizes (if accepted by the backbone) since the spatial dimensions only affect the number of training samples for\\nthe RAE. Intuitively, larger input sizes would improve the RAE training, but would also increase the backbone cost\\nsigni\\ufb01cantly. Therefore, in this work, we consider only a constant input size of 224x224 (forced resizing), since this is\\nthe most common con\\ufb01guration of various backbones. Moreover, considering that the spatial organization of the pixels\\nis lost due to the \\ufb02attening procedure of X0, we add a positional encoding composed of sine and cosine functions of\\ndifferent frequencies with dimension zas proposed in [42], extended for 2 spatial dimensions as in [43]:\\nPE(x;y;2i) = sin(x\\n100004i=z);\\nPE(x;y;2i+ 1) = cos(x\\n100004i=z);\\nPE(x;y;2j+z=2) = sin(y\\n100004j=z);\\nPE(x;y;2j+ 1 +z=2) = cos(y\\n100004j=z);\\nwherex2wandy2h, which is then added to the aggregated activation map via element-wise sum:\\nX0=X0\\bPE: (4)\\nAfter summing the positional encoding to X0, the \\ufb01rst step of the RAE is to project the inputs using a random\\nfully-connected layer with weights Wk2Rz\\u0002q, followed by a sigmoid nonlinearity. The weights are generated\\nusing the LCG for simplicity and better replicability, followed by standardization (zero centered, unity variance) and\\northogonalization [37]. These con\\ufb01gurations were chosen according to previous works [17, 33]. As for the LCG\\nparameters, we use a= 75 ,b= 74 , andc= 216+1, starting with x= 0, which is a classical con\\ufb01guration according to\\nthe ZX81 computer from 1981. Here kworks like a seed for random sampling, denoting a starting index inside the LCG\\nspace generated with the given con\\ufb01guration. More details on LCG weights are given in the Supplementary Material,\\nsuch as an ablation on the impacts of different LCG con\\ufb01gurations. The forward pass of the encoder gk2Rwh\\u0002qfor\\nall samples is then obtained as\\ngk=\\u001e(X0Wk); (5)\\nand the decoder weights fk2Rz\\u0002qare obtained as the least-squares solution described in Eq. (1), changing the target\\nYtoX0:\\nfk=X0gT\\nk(gkgT\\nk)\\u00001: (6)\\nThe main idea of employing an individual randomized neural network for each image is to use the output weights\\nthemselves as a representation. In the case of RAEs, the output layer has the same dimension as the input layer.\\nTherefore, a single hidden neuron ( q= 1) is considered to maintain the dimensionality. In this sense, the resulting\\ndecoder weights are represented by\\nfk= (\\u00171;:::;\\u0017 z); (7)\\nwhere\\u0017irepresents the connection weight between the single hidden neuron and the output i, corresponding to feature\\ni2z.\\nA single-neuron RAE may be limited in encoding enough information contained in the deep activation maps. Therefore,\\nwe propose an ensemble of models or, as recently introduced [45], a model \\u201csoup\\u201d, which is achieved by combining the\\nweights ofmparallel models. Here, each model is an RAE with a different random encoder (using a different LCG\\nseed), and the combination is performed by summing the decoder weights\\n'm= (mX\\nk=1fk(\\u00171);:::;mX\\nk=1fk(\\u0017z)): (8)\\nIt is important to note that the encoders gkof each of the mRAEs have a different random weight initialization.\\nThis is achieved by creating an LCG sequence of size mzso that we have zweights for each of the mRAEs. The\\nstructure of the RAE is illustrated in Fig. 1(b), and following the whole RADAM pipeline shown in Fig. 1(a), a texture\\nrepresentation, or feature vector 'm, is obtained for the input image. The code for all these steps is available in the\\nSupplementary Material and in our online repository1. The feature vectors 'mare then used to train a linear classi\\ufb01er\\nfor a given texture recognition task (more details on the classi\\ufb01er can be consulted in Sec. 4.2.1).\\n5 4 Experiments and Results\\n4.1 Setup\\nOur model is implemented using PyTorch [27] (except for the classi\\ufb01cation step), making it easier to couple RADAM\\nwith several methods implemented in this library. The classi\\ufb01cation step is performed using Scikit-learn [28]. We\\nmeasure our results by the average classi\\ufb01cation accuracy and corresponding standard deviation, when applicable\\n(depending on the dataset). For the backbones, we consider the PyTorch Image Models library [44] (version 0.6.7),\\nwhich contains several pre-trained computer vision methods. In the Supplementary Material, we present the main code\\nfor RADAM, and the complete implementation including scripts for experimentation can be consulted in our GitHub\\nrepository1.\\nSeven texture datasets are used for evaluation purposes in this paper, of which the following two variants of the Outex\\ndataset [14] were used for analyzing the RADAM method alone:\\n\\u2022Outex10 : Composed of 4320 grayscale images in 24 different texture classes. This dataset focuses on rotation\\ninvariance;\\n\\u2022Outex13 : This suite holds 1360 RGB images in 68 texture classes, and evaluates color texture recognition.\\nThe following \\ufb01ve datasets are used for comparisons with other methods:\\n\\u2022Describable Texture Dataset (DTD) [5]: Composed of 5640 images in 47 different texture classes, evaluated\\nby the 10 provided splits for training, validation, and test;\\n\\u2022Flickr Material Dataset (FMD) [41]: Holds 1000 images representing 10 material categories, and validation is\\ndone through 10 repetitions of 10-fold cross-validation;\\n\\u2022KTH-TIPS2-b [3]: Contains 4752 images of 11 different materials. This dataset has a \\ufb01xed set of 4 splits for\\n4-fold cross-validation;\\n\\u2022Ground Terrain in Outdoor Scenes (GTOS) [47]: This dataset represents 34105 images divided into 40 outdoor\\nground materials classes. There is also a \\ufb01xed set of 5 train\\/test splits;\\n\\u2022GTOS-Mobile [46]: Consists of 100011 images captured from a mobile phone of 31 different outdoor ground\\nmaterials, and contains a single train\\/test split.\\n4.2 Analysis of RADAM properties\\nOur \\ufb01rst experimental evaluation concerns aspects of the proposed RADAM method. In the Supplementary Material, we\\nshow an additional analysis of the impacts caused by different random weights (LCG con\\ufb01gurations) and concluded that\\nthey are minimal, corroborating previous works. In the following, we evaluate and discuss other aspects of RADAM.\\n4.2.1 Positional encoding and different classi\\ufb01ers\\nWe evaluate two design choices for the RADAM pipeline, the use of positional encoding and the classi\\ufb01er. The\\nmethod is compared with or without encoding under two different classi\\ufb01ers: Linear Discriminant Analysis (LDA) [34]\\nand Support Vector Machines (SVM) [30]. For LDA the least-squares solution with automatic shrinkage using the\\nLedoit\\u2013Wolf lemma is used, and a linear kernel with C= 1is considered for SVM. Since the evaluation of positional\\nencoding concerns the spatial properties of texture, we consider the Outex10 benchmark, which focuses on rotation\\ninvariance. As the results in Tab. 1 demonstrate, positional encoding improves or maintains performance in all cases,\\nespecially under the LDA classi\\ufb01er. On the other hand, SVM provides the best results in all cases while gaining less\\nimprovement from positional encoding. Nevertheless, we keep the positional encoding in our architecture since the\\nadditional cost is negligible compared to the potential gains. The SVM is also used as the classi\\ufb01er for all the following\\nexperiments.\\n4.2.2 Soup size\\nThe only free parameter of the proposed RADAM method is the number of RAEs to be combined, i.e., m. We evaluated\\nmranging from 1 to 32 and the results are shown in Fig. 2 for different backbones in the Outex13 dataset. We observe\\nsigni\\ufb01cant gains for mfrom 1 to 4, while for larger values performance tends to stabilize. These results indicate that\\n4\\u0014m\\u00148is a good approach for a balance between performance and cost since no signi\\ufb01cant gains are achieved 4\\u0014m\\u00148is a good approach for a balance between performance and cost since no signi\\ufb01cant gains are achieved\\nabove that. All the following experiments in this paper are performed using m= 4.\\n6 Table 1: Ablations with positional encoding and different classi\\ufb01ers, using RADAM ( m= 1) with different backbones.\\nResults are measured by classi\\ufb01cation accuracy considering the single train\\/test split of Outex10.\\nOutex10\\nencoding backbone LDA SVM\\nnone ResNet18 77.5 83.4\\npositional ResNet18 79.5 84.7\\nnone ConvNeXt-nano 82.4 89.6\\npositional ConvNeXt-nano 85.9 89.6\\nnone ResNet50 86.0 87.4\\npositional ResNet50 87.4 87.4\\nnone ConvNeXt-T 87.8 91.1\\npositional ConvNeXt-T 89.4 91.1\\n14 8 12 16 20 24 28 32\\nm889092accuracy (%)ConvNeXt-nano\\nConvNeXt-T\\nFigure 2: Impacts of changing the number of RAEs ( m) in the proposed method, considering different backbones on\\nthe Outex13 dataset. All other experiments in this paper consider m= 4.\\nThe effects observed when increasing mare expected considering what is usually seen in model ensembles, or \\u201cmodel\\nsoups\\u201d [45], where the combination of models trained separately may be bene\\ufb01cial. On the other hand, our encoders\\nare random, and each one has different weights. However, even if each encoder creates a different random projection of\\nthe input, the decoders learn to transform the projection back to the same feature space. In other words, the RAEs learn\\ndifferent encoding-decoding functions for the same input that, when combined, provide a better representation in our\\nfeature extraction use case.\\n4.3 Comparison with other pooling techniques\\nTo show the gains of RADAM over the common pooling approach, we performed additional experiments using Global\\nAverage Pooling (GAP) coupled with SVM using the same con\\ufb01gurations considered for RADAM. We use GAP\\napplied over the output of the last layer of the backbone (the usual approach in most CNN architectures), and also GAP\\nagg., which aggregates the GAP from each of the nfeature blocks and returns an image representation with zfeatures\\n(as RADAM). The results are shown in Tab. 2, where it is possible to observe that RADAM overcomes these two\\napproaches by a considerable margin, in all backbones and datasets. GAP agg. usually improves over the regular GAP\\nof only the last convolutional layer, which is to be expected since more features are being added. However, the regular\\nGAP sometimes performs better than this simple aggregation by concatenation. On the other hand, the gains when\\nusing RADAM for aggregation are far more expressive and result in state-of-the-art performance in all benchmarks we\\nconsidered, as we discuss in the following section.\\n4.4 Comparison with literature\\nFinally, we compare RADAM with several state-of-the-art methods on \\ufb01ve challenging texture recognition datasets; all\\nresults are shown in Tab. 3. The table is organized into separate rows according to the different backbones in terms of\\ncomputational budget. We indicate the pre-training dataset used for the backbone; ImageNet-1K [7] was used when not\\nstated. Firstly, we show the results obtained using RADAM and MobileNet V2 (standard or with width multipliers of\\n1.4) as a lightweight alternative (costs are discussed in depth in Sec. 4.5). The \\ufb01rst comparison section contains methods\\nusing ResNet18, and we also included RADAM with ConvNeXt-nano in this section considering that it has a similar\\n7 Table 2: Classi\\ufb01cation accuracy (linear SVM) when using image features obtained from the backbones with different\\ntechniques: the GAP, the aggregation of GAP from each of the nfeature blocks of the backbones (the same way we\\napproach them with RADAM), and the proposed RADAM. We consider two challenging texture benchmarks (DTD and\\nFMD).\\nmethod backbone DTD FMD\\nGAP ResNet18 64.1\\u00061.1 77.1\\u00060.7\\nGAP agg. ResNet18 64.6\\u00061.2 77.2\\u00060.6\\nRADAM ResNet18 68.1\\u00061.0 77.7\\u00060.5\\nGAP ConvNeXt-nano 61.9\\u00061.0 73.8\\u00060.9\\nGAP agg. ConvNeXt-nano 71.7\\u00060.6 84.4\\u00060.4\\nRADAM ConvNeXt-nano 74.9\\u00060.7 87.1\\u00060.4\\nGAP ResNet50 72.6\\u00060.9 84.5\\u00060.6\\nGAP agg. ResNet50 74.6\\u00060.6 83.4\\u00060.4\\nRADAM ResNet50 75.6\\u00061.1 85.3\\u00060.4\\nGAP ConvNeXt-T 66.0\\u00061.0 79.0\\u00060.4\\nGAP agg. ConvNeXt-T 73.7\\u00060.9 83.6\\u00060.5\\nRADAM ConvNeXt-T 77.0\\u00060.7 88.7\\u00060.4\\nGAP ConvNeXt-T in ImageNet-21K 78.4\\u00060.7 91.4\\u00060.3\\nGAP agg. ConvNeXt-T in ImageNet-21K 77.3\\u00060.9 89.9\\u00060.5\\nRADAM ConvNeXt-T in ImageNet-21K 81.4\\u00060.7 93.0\\u00060.3\\nGAP ConvNeXt-B 69.2\\u00061.3 83.6\\u00060.5\\nGAP agg. ConvNeXt-B 73.5\\u00061.0 86.7\\u00060.5\\nRADAM ConvNeXt-B 76.4\\u00060.9 90.2\\u00060.2\\nGAP ConvNeXt-B in ImageNet-21K 79.2\\u00060.6 91.9\\u00060.4\\nGAP agg. ConvNeXt-B in ImageNet-21K 80.4\\u00061.0 92.3\\u00060.3\\nRADAM ConvNeXt-B in ImageNet-21K 82.8\\u00060.9 94.0\\u00060.2\\nGAP ConvNeXt-L 70.9\\u00060.9 84.6\\u00060.4\\nGAP agg. ConvNeXt-L 73.4\\u00060.6 86.4\\u00060.5\\nRADAM ConvNeXt-L 77.4\\u00061.1 89.3\\u00060.3\\nGAP ConvNeXt-L in ImageNet-21K 80.4\\u00060.9 92.2\\u00060.4\\nGAP agg. ConvNeXt-L in ImageNet-21K 81.9\\u00060.6 93.4\\u00060.4\\nRADAM ConvNeXt-L in ImageNet-21K 84.0\\u00061.0 95.2\\u00060.4\\nGAP ConvNeXt-XL in ImageNet-21K 81.3\\u00061.0 92.4\\u00060.3\\nGAP agg. ConvNeXt-XL in ImageNet-21K 82.0\\u00061.1 93.9\\u00060.4\\nRADAM ConvNeXt-XL in ImageNet-21K 83.7\\u00060.9 95.2\\u00060.3\\ncomputational budget. In this scenario, RADAM achieves competitive performance on KTH using ResNet18 but is less\\neffective on other datasets. On the other hand, RADAM with MobileNetV2 1.4 achieves better results than ResNet18 in\\nall cases except GTOS-Mobile and beats all the ResNet18 literature methods on DTD, FMD, and KTH, proving to be\\nan excellent low-cost approach. Using ConvNeXt-nano, RADAM also achieves the best results on all datasets except\\nGTOS and GTOS-Mobile.\\nConsidering the results within the ResNet50 budget, RADAM performs much better compared to ResNet18. Competitive\\nresults are achieved on most datasets using ResNet50, and also the best results on KTH. Using ConvNeXt-T RADAM\\nachieves state-of-the-art on FMD, and also overcomes the compared methods on KTH. Performance is improved even\\nfurther considering ConvNeXt-T pre-trained on ImageNet-21K, achieving the best results on most of the datasets. These\\nresults show the potential gains of better pre-training of the employed backbone.\\nThe results shown in the last block of rows in Tab. 3 concern methods with an increased cost compared to the previous\\nones. Here we compare RADAM using ConvNeXt-B, L, and XL against several works including very recent methods,\\nsuch as ViTs. Our method achieves state-of-the-art results on all datasets in this case, especially with the ConvNeXt-\\nL\\/XL backbones. It is possible to notice again that the better pre-training with ImageNet-21K results in signi\\ufb01cant\\nperformance increases.\\n4.5 Feature extraction cost versus performance\\nAdditional analysis is performed to better understand the balance between the classi\\ufb01cation performance and computa-\\ntional budget of the compared texture recognition methods. We consider the inference costs in terms of GFLOPs and\\n8 Table 3: Classi\\ufb01cation accuracy of different methods on texture benchmarks. The used backbones are separated into row\\nblocks according to their computational budget, the input size is indicated in parentheses (224x224 when not stated),\\nand the two best results in each block are highlighted in bold type. Results in blue show the previous state-of-the-art on\\neach dataset, and red represents our results matching or above that.\\nmethod backbone DTD FMD KTH-2-b GTOS GTOS-Mobile\\nRADAM light (ours) MobileNet V2 69.3\\u00060.8 80.5\\u00060.6 84.8\\u00060.5 81.0\\u00061.3 75.1\\nRADAM light (ours) MobileNet V2 1.4 73.1\\u00060.9 82.6\\u00060.5 86.8\\u00063.1 81.7\\u00061.7 78.2\\nDeepTEN ResNet18 (352) 76.1\\nDEPNet ResNet18 82.2\\nMAPNet ResNet18 69.5\\u00060.8 80.8\\u00061.0 80.9\\u00061.8 80.3\\u00062.6 83.0\\nDSRNet ResNet18 71.2\\u00060.7 81.3\\u00060.8 81.8\\u00061.6 81.0\\u00062.1 83.7\\nCLASSNet ResNet18 71.5\\u00060.4 82.5\\u00060.7 85.4\\u00061.1 84.3\\u00062.2 85.3\\nRADAM (ours) ResNet18 68.1\\u00061.0 77.7\\u00060.5 84.7\\u00063.6 80.6\\u00061.7 79.5\\nRADAM (ours) ConvNeXt-nano 74.9\\u00060.7 87.1\\u00060.4 89.6\\u00063.8 83.7\\u00061.5 81.8\\nDeepTEN ResNet50 (352) 69.6 80.2 \\u00060.9 82.0\\u00063.3 84.5\\u00062.9\\nMAPNet ResNet50 76.1\\u00060.6 85.2\\u00060.7 84.5\\u00061.3 84.7\\u00062.2 86.6\\nDSRNet ResNet50 77.6\\u00060.6 86.0\\u00060.8 85.9\\u00061.3 85.3\\u00062.0 87.0\\nCLASSNet ResNet50 74.0\\u00060.5 86.2\\u00060.9 87.7\\u00061.3 85.6\\u00062.2 85.7\\nDFAEN ResNet50 73.2 86.9 86.3 86.9\\nRADAM (ours) ResNet50 75.6\\u00061.1 85.3\\u00060.4 88.5\\u00063.2 81.8\\u00061.1 81.0\\nRADAM (ours) ConvNeXt-T 77.0\\u00060.7 88.7\\u00060.4 90.7\\u00064.0 84.2\\u00061.7 85.3\\nRADAM (ours) ConvNeXt-T in ImageNet-21K 81.4\\u00060.7 93.0\\u00060.3 91.0\\u00064.9 85.4\\u00061.6 86.5\\nDFAEN Densenet161 76.1 87.6 86.6 86.9\\nMultilayer-FV Ef\\ufb01cientNet-B5 (512) 78.9 88.7 82.9\\nRankGP-3M-CNN++ (3 backbones) 86.2\\u00061.4 91.1\\u00064.5\\n\\ufb01ne-tuning ViT B\\/16 in Bamboo 69m 81.2\\nCLIP zero-shot ViT L\\/14 (336) in WIT 400m 83.0\\n\\u00162Net+ ViT-L\\/16 (384) in ImageNet-21K 82.2\\nRADAM (ours) ConvNeXt-B 76.4\\u00060.9 90.2\\u00060.2 87.7\\u00065.6 84.1\\u00061.6 82.2\\nRADAM (ours) ConvNeXt-B in ImageNet-21K 82.8\\u00060.9 94.0\\u00060.2 91.8\\u00064.1 86.6\\u00061.7 87.1\\nRADAM (ours) ConvNeXt-L 77.4\\u00061.1 89.3\\u00060.3 89.3\\u00063.4 84.0\\u00061.8 85.8\\nRADAM (ours) ConvNeXt-L in ImageNet-21K 84.0\\u00061.0 95.2\\u00060.4 91.3\\u00064.1 85.9\\u00061.6 87.3\\nRADAM (ours) ConvNeXt-XL in ImageNet-21K 83.7\\u00060.9 95.2\\u00060.3 94.4\\u00063.8 87.2\\u00061.9 90.2\\nthe number of parameters according to the backbone used by each method since this is the most resource-demanding\\nstep of every pipeline. One important aspect here is that the input size greatly impacts the FLOP count of the methods\\n(check input sizes in parentheses in Tab. 3). Most works consider 224x224 inputs (the same input size employed by\\nRADAM), and we assume this same size when not stated by the authors. For this analysis, we also are not considering\\nthe preparation of the backbone either in terms of pre-training cost or chosen dataset, nor the \\ufb01ne-tuning of the methods\\nthat do so. The results are shown in Fig. 3 \\ufb01rst for the DTD dataset alone (Fig. 3(a)) since this is the most challenging\\ntask and with more methods to compare, and then as the average for the remaining datasets (Fig. 3(b)). It is possible\\nto notice the superiority of RADAM at different budgets, especially when using MobileNet V2 1.4 and ConvNeXt-T.\\nMoreover, results also scale with backbone complexity.\\nIt is also important to mention the backbone costs of other methods not present in this analysis due to the lack of\\navailable results. For instance, Multilayer-FV achieved the previous state-of-the-art on FMD using Ef\\ufb01cientNet-B5\\nwith an input size of 512 pixels, which yields an approximate inference cost of 12 GFLOPs. This is considerably\\nhigher than the cost of ConvNeXt-T (4.5 GFLOPs), with which RADAM achieves an improvement of 4.3 %in absolute\\nperformance compared to Multilayer-FV . On KTH, RankGP-3M-CNN++ achieves 91.1 %accuracy (previous state-of-\\nthe-art) using three backbones, with a total inference cost of around 30 GFLOPs, while RADAM with ConvNeXt-T\\nachieves comparable results ( \\u00000:1%), and also achieves the state-of-the-art results using ConvNeXt-B (91.8 %with\\ninference cost of 15 GFLOPs), and ConvNeXt-XL (94.4 %with inference cost of 61 GFLOPs). Considering GTOS inference cost of 15 GFLOPs), and ConvNeXt-XL (94.4 %with inference cost of 61 GFLOPs). Considering GTOS\\nand GTOS-Mobile, a competitive cost and performance are also achieved with RADAM using ConvNeXt-T, and\\nstate-of-the-art results with ConvNeXt-B, L, and XL.\\nTo increment the cost analysis, we show in Table 4 the practical running time of RADAM (with m= 4and SVM) in\\ncontrast to the costs of ResNet. Our intuition is to compare the use of RADAM, which requires only a forward pass of\\nthe backbone and an SVM, to the approach in previous works of \\ufb01ne-tuning the backbone. We measured the average\\ntime (in milliseconds) of 100 runs of ResNet18 and ResNet50 over one 224x224 RGB image considering the RADAM\\nmodule, the ResNet forward and backward passes, and the corresponding times when \\ufb01ne-tuning the backbone our\\n9 100101102\\nGFLOPs70758085DTD accuracy (%)MobileNetConvNeXt\\nnanoConvNeXt-TConvNeXt-BConvNeXt-L\\/XL(a) Best methods on the DTD dataset. Text\\ninside the plot indicates backbones used with\\nRADAM.\\nParameters (m.)\\n4.3\\n15.0\\n26.5\\n27.8\\n87.6\\n304.3\\n348.1MAPNet\\nDSRNet\\nCLASSNet\\nDFAEN\\nMultilayer-FV\\nBamboo\\nCLIP\\nNet+\\nRADAM\\n100101102\\nGFLOPs808590average accuracy (%)(b) Average performance of methods with re-\\nsults available on the other four datasets.\\nFigure 3: Computational budget at inference time according to the backbone used by different methods.\\nTable 4: Average running time (in milliseconds) of 100 repetitions using a 224x224 RGB image, performed on a\\nmachine with a GTX 1080ti, Intel Core i7-7820X 3.60GHz processor (using 8 threads), and 64GB of RAM.\\ntime (ms)\\nmethod backbone CPU GPU\\nRADAM module alone ResNet18 2.7\\u00060:2 2.9\\u00060:04\\nbackbone\\u2019s forward pass ResNet18 15.0\\u00061:6 4.3\\u00060:1\\nbackbone\\u2019s backward pass ResNet18 50.8\\u00065:3 9.9\\u00061:5\\nbackbone\\u2019s 1 \\ufb01ne-tuning epoch ResNet18 65.8\\u00066:0 14.2\\u00061:6\\nbackbone\\u2019s forward pass + RADAM + SVM inference ResNet18 19.8\\u00061:6 8.1\\u00060:1\\nRADAM module alone ResNet50 7.0\\u00060:4 5.2\\u00060:4\\nbackbone\\u2019s forward pass ResNet50 34.0\\u00062:6 11.1\\u00060:8\\nbackbone\\u2019s backward pass ResNet50 107.6\\u00065:921.5\\u00062:2\\nbackbone\\u2019s 1 \\ufb01ne-tuning epoch ResNet50 141.7\\u00067:632.6\\u00062:3\\nbackbone\\u2019s forward pass + RADAM + SVM inference ResNet50 50.3\\u00062:6 18.1\\u00060:8\\nusing RADAM for feature extraction with SVM inference. It is important to notice that the cost of RADAM varies\\naccording to the backbone since the number of aggregate features ( z) changes, thus impacting the RAE training cost.\\nConsidering the results in the table, \\ufb01ne-tuning ResNet50 on GTOS-Mobile using only 10 epochs with a batch size of 1\\nwould take around 40 hours on CPU or 9 hours on GPU. On the other hand, extracting features with RADAM followed\\nby SVM inference on the whole GTOS-Mobile dataset takes around 1.3 hours on CPU or half an hour on GPU. The\\ntraining of SVM on the whole dataset (on CPU) takes an additional 15 minutes on average, without hyperparameter\\ntuning. The results demonstrate that the RADAM module is considerably faster than the ResNet backbone, both in terms\\nof inference speed and comparing feature extraction followed by SVM with training\\/\\ufb01ne-tuning. These results extend\\nto ConvNeXt-nano and ConvNeXt-T, considering that the cost is comparable to ResNet18 and ResNet50, respectively,\\ncorroborating our claims that RADAM provides both considerable savings in training time and SOTA results at a similar\\ninference cost. Moreover, considering more costly backbones such as ConvNext-L and XL, the gains in training time\\ncan be even more expressive.\\n5 Conclusion\\nWe presented RADAM, a new feature encoding module for texture analysis. The method consists of randomly encoding\\naggregated deep activation maps from pre-trained DCNN using RAEs. These autoencoders learn to pool activation\\nmaps into a 1-dimensional representation by training on its z-dimentional pixels as sample points. A texture image\\nis then encoded by using the decoder weights learned from its activation maps. The procedure is orderless, but takes\\ninto account the spatial information of the pixels by using a 2D positional encoding. Compared to previous works, our\\nmethod does not require \\ufb01ne-tuning of the backbone, and the encoding module is rather simple. Linear classi\\ufb01cation of\\nthe descriptors is performed with an SVM, and we achieve state-of-the-art performance on several texture benchmarks.\\nRADAM also achieves the best ef\\ufb01ciency considering inference cost and performance using backbones with varying\\ncomputational budgets. These results are impressive also considering that, compared to other methods, no \\ufb01ne-tuning\\nof the backbone is needed for RADAM, causing a lower cost also at training time.\\n10 Our work corroborates a simpler approach to texture recognition where the \\ufb01ne-tuning of costly backbones may not be\\nnecessary to achieve high discriminatory power. For future works, one may explore different backbones or different\\nformulations of our RAE, with multiple layers, more hidden neurons, and other possible improvements. On the other\\nhand, if enough computing resources are available, another approach more similar to previous works would be to\\nexplore our module in an end-to-end manner. Since RADAM is deterministic and a closed-form solution, an alternative\\nwould be adding a linear layer instead of an SVM and optimizing it along with the backbone.\\nAcknowledgements\\nL. Scabini acknowledges funding from the S\\u00e3o Paulo Research Foundation (FAPESP) (Grants #2019\\/07811-0 and\\n#2021\\/09163-6) and the National Council for Scienti\\ufb01c and Technological Development (CNPq) (Grant #142438\\/2018-\\n9). K M. Zielinski acknowledges support from S\\u00e3o Paulo Research Foundation (FAPESP) (Grant #2022\\/03668-1)\\nand Higher Education Personnel Improvement Coordination (CAPES) (Grant #88887.631085\\/2021-00). L. C. Ribas\\nacknowledges support from S\\u00e3o Paulo Research Foundation (FAPESP) (grants #2021\\/07289-2 and #2018\\/22214-6).\\nW. N. Gon\\u00e7alves acknowledges support from CNPq (Grants #405997\\/2021-3 and #305296\\/2022-1). O. M. Bruno\\nacknowledges support from CNPq (Grant #307897\\/2018-4) and FAPESP (grants #2018\\/22214-6 and #2021\\/08325-2).\\nThe authors are also grateful to the NVIDIA GPU Grant Program. This research received funding from the Flemish\\nGovernment under the \\\"Onderzoeksprogramma Arti\\ufb01ci\\u00eble Intelligentie (AI) Vlaanderen\\\" programme.\\nReferences\\n[1]Robert Azencott, Jia-Ping Wang, and Laurent Younes. Texture classi\\ufb01cation using windowed fourier \\ufb01lters. IEEE Transactions\\non Pattern Analysis and Machine Intelligence , 19(2):148\\u2013153, 1997. 1\\n[2]Andr\\u00e9 Ricardo Backes, Dalcimar Casanova, and Odemir Martinez Bruno. Color texture analysis based on fractal descriptors.\\nPattern Recognition , 45(5):1984\\u20131992, 2012. 1\\n[3]B. Caputo, E. Hayman, and P. Mallikarjuna. Class-speci\\ufb01c material categorisation. In Tenth IEEE International Conference on\\nComputer Vision (ICCV\\u201905) Volume 1 , volume 2, pages 1597\\u20131604 V ol. 2, 2005. 6\\n[4]Zhile Chen, Feng Li, Yuhui Quan, Yong Xu, and Hui Ji. Deep texture recognition via exploiting cross-layer statistical\\nself-similarity \\u2021. In2021 IEEE\\/CVF Conference on Computer Vision and Pattern Recognition (CVPR) , pages 5227\\u20135236, 2021.\\n1, 2\\n[5]Mircea Cimpoi, Subhransu Maji, Iasonas Kokkinos, Sammy Mohamed, and Andrea Vedaldi. Describing textures in the wild.\\nIn2014 IEEE Conference on Computer Vision and Pattern Recognition , pages 3606\\u20133613, 2014. 1, 2, 6\\n[6]Mircea Cimpoi, Subhransu Maji, and Andrea Vedaldi. Deep \\ufb01lter banks for texture recognition and segmentation. In 2015\\nIEEE Conference on Computer Vision and Pattern Recognition (CVPR) , pages 3828\\u20133836, 2015. 1, 2\\n[7]Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In\\n2009 IEEE conference on computer vision and pattern recognition , pages 248\\u2013255. Ieee, 2009. 7\\n[8]Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa\\nDehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16\\nwords: Transformers for image recognition at scale. CoRR , abs\\/2010.11929, 2020. 3\\n[9]Andrea Gesmundo. A continual development methodology for large-scale multitask dynamic ml systems. arXiv preprint\\narXiv:2209.07326 , 2022. 3\\n[10] Robert M Haralick. Statistical and structural approaches to texture. Proceedings of the IEEE , 67(5):786\\u2013804, 1979. 1\\n[11] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of\\nthe IEEE conference on computer vision and pattern recognition , pages 770\\u2013778, 2016. 3 the IEEE conference on computer vision and pattern recognition , pages 770\\u2013778, 2016. 3\\n[12] Minh A Hoang, Jan-Mark Geusebroek, and Arnold WM Smeulders. Color texture measurement and segmentation. Signal\\nprocessing , 85(2):265\\u2013275, 2005. 1\\n[13] Guang-Bin Huang, Qin-Yu Zhu, and Chee-Kheong Siew. Extreme learning machine: theory and applications. Neurocomputing ,\\n70(1):489\\u2013501, 2006. 1, 3\\n[14] S. Huovinen, M. Pietik\\u00e4inen, T. Ojala, J. Kyll\\u00f6nen, J. Viertola, and T. M\\u00e4enp\\u00e4\\u00e4. Outex - new framework for empirical evaluation\\nof texture analysis algorithms. In Proceedings of 16th International Conference on Pattern Recognition , volume 1, page 10701,\\nLos Alamitos, CA, USA, aug 2002. IEEE Computer Society. 6\\n[15] Bela Julesz. Visual pattern discrimination. IRE transactions on Information Theory , 8(2):84\\u201392, 1962. 1\\n[16] Jarbas Joaci de Mesquita S\\u00e1 Junior, Lucas Correia Ribas, and Odemir Martinez Bruno. Randomized neural network based\\nsignature for dynamic texture classi\\ufb01cation. Expert Systems with Applications , 135:194\\u2013200, 2019. 3\\n[17] Liyanaarachchi Lekamalage Chamara Kasun, Hongming Zhou, G-B Huang, and Chi Man V ong. Representational learning\\nwith elms for big data. IEEE Intelligent Systems , 28:31\\u201334, 2013. 1, 3, 5\\n[18] Tsung-Yu Lin, Aruni RoyChowdhury, and Subhransu Maji. Bilinear cnn models for \\ufb01ne-grained visual recognition. In 2015\\nIEEE International Conference on Computer Vision (ICCV) , pages 1449\\u20131457, 2015. 2\\n[19] Li Liu, Jie Chen, Paul Fieguth, Guoying Zhao, Rama Chellappa, and Matti Pietik\\u00e4inen. From BoW to CNN: Two decades of\\ntexture representation for texture classi\\ufb01cation. International Journal of Computer Vision , 127(1):74\\u2013109, Nov. 2018. 2\\n11 [20] Zhuang Liu, Hanzi Mao, Chao-Yuan Wu, Christoph Feichtenhofer, Trevor Darrell, and Saining Xie. A convnet for the 2020s.\\nInProceedings of the IEEE\\/CVF Conference on Computer Vision and Pattern Recognition , pages 11976\\u201311986, 2022. 3\\n[21] Lucas O. Lyra, Antonio Elias Fabris, and Joao B. Florindo. Multilayer deep feature extraction for visual texture recognition,\\n2022. 2\\n[22] Rayner H. M. Condori and Odemir M. Bruno. Analysis of activation maps through global pooling measurements for texture\\nclassi\\ufb01cation. Information Sciences , 555:260\\u2013279, 2021. 1, 2\\n[23] E. H. Moore. On the reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society , 26:394\\u2013395,\\n1920. 3\\n[24] Timo Ojala, Matti Pietikainen, and Topi Maenpaa. Multiresolution gray-scale and rotation invariant texture classi\\ufb01cation with\\nlocal binary patterns. IEEE Transactions on pattern analysis and machine intelligence , 24(7):971\\u2013987, 2002. 1\\n[25] Yoh-Han Pao, Gwang-Hoon Park, and Dejan J Sobajic. Learning and generalization characteristics of the random vector\\nfunctional-link net. Neurocomputing , 6(2):163\\u2013180, 1994. 1, 3\\n[26] Y-H Pao and Yoshiyasu Takefuji. Functional-link net computing: theory, system architecture, and functionalities. Computer ,\\n25(5):76\\u201379, 1992. 1, 3\\n[27] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin,\\nNatalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan\\nTejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. PyTorch: An Imperative Style,\\nHigh-Performance Deep Learning Library. In Advances in Neural Information Processing Systems 32 , pages 8024\\u20138035.\\nCurran Associates, Inc., 2019. 6\\n[28] F. Pedregosa, G. Varoquaux, A. Gramfort, V . Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V . Dubourg,\\nJ. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python.\\nJournal of Machine Learning Research , 12:2825\\u20132830, 2011. 6\\n[29] R. Penrose. A generalized inverse for matrices. Mathematical Proceedings of the Cambridge Philosophical Society , 51(3):406\\u2014\\n-413, 1955. 3\\n[30] John Platt et al. Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods. Advances\\nin large margin classi\\ufb01ers , 10(3):61\\u201374, 1999. 6\\n[31] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell,\\nPamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. Learning transferable visual models from natural language\\nsupervision, 2021. 3\\n[32] Lucas C Ribas, Jarbas Joaci de Mesquita S\\u00e1 Junior, Antoine Manzanera, and Odemir M Bruno. Learning graph representation\\nwith randomized neural network for dynamic texture classi\\ufb01cation. Applied Soft Computing , 114:108035, 2022. 3\\n[33] Lucas C. Ribas, Jarbas Joaci Mesquita S\\u00e1 Junior, Leonardo F.S. Scabini, and Odemir M. Bruno. Fusion of complex networks\\nand randomized neural networks for texture analysis. Pattern Recognition , 103:107189, 2020. 3, 5\\n[34] Brian D Ripley. Pattern recognition and neural networks . Cambridge university press, 2007. 6\\n[35] Jarbas Joaci Mesquita S\\u00e1 Junior and Andr\\u00e9 Ricardo Backes. ELM based signature for texture classi\\ufb01cation. Pattern Recognition ,\\n51:395\\u2013401, 2016. 3\\n[36] Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mobilenetv2: Inverted residuals\\nand linear bottlenecks. In Proceedings of the IEEE conference on computer vision and pattern recognition , pages 4510\\u20134520,\\n2018. 4\\n[37] Andrew M Saxe, James L McClelland, and Surya Ganguli. Exact solutions to the nonlinear dynamics of learning in deep linear\\nneural networks. arXiv preprint arXiv:1312.6120 , 2013. 3, 5\\n[38] Leonardo FS Scabini, Rayner HM Condori, Wesley N Gon\\u00e7alves, and Odemir M Bruno. Multilayer complex network neural networks. arXiv preprint arXiv:1312.6120 , 2013. 3, 5\\n[38] Leonardo FS Scabini, Rayner HM Condori, Wesley N Gon\\u00e7alves, and Odemir M Bruno. Multilayer complex network\\ndescriptors for color\\u2013texture characterization. Information Sciences , 491:30\\u201347, 2019. 1\\n[39] Leonardo FS Scabini, Lucas C Ribas, and Odemir M Bruno. Spatio-spectral networks for color-texture analysis. Information\\nSciences , 515:64\\u201379, 2020. 1\\n[40] Wouter F Schmidt, Martin A Kraaijveld, and Robert P W Duin. Feedforward neural networks with random weights. In Pro-\\nceedings., 11th IAPR International Conference on Pattern Recognition. Vol.II. Conference B: Pattern Recognition Methodology\\nand Systems , pages 1\\u20134, 1992. 1, 3\\n[41] Lavanya Sharan, Ruth Rosenholtz, and EH Adelson. Material perception: What can you see in a brief glance? Journal of\\nVision - J VISION , 9:784\\u2013784, 08 2010. 6\\n[42] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, \\u0141ukasz Kaiser, and Illia\\nPolosukhin. Attention is all you need. Advances in neural information processing systems , 30, 2017. 5\\n[43] Zelun Wang and Jyh-Charn Liu. Translating math formula images to latex sequences using deep neural networks with\\nsequence-level training. International Journal on Document Analysis and Recognition (IJDAR) , 24(1):63\\u201375, 2021. 5\\n[44] Ross Wightman. Pytorch image models. https:\\/\\/github.com\\/rwightman\\/pytorch-image-models , 2019. 6\\n[45] Mitchell Wortsman, Gabriel Ilharco, Samir Ya Gadre, Rebecca Roelofs, Raphael Gontijo-Lopes, Ari S Morcos, Hongseok\\nNamkoong, Ali Farhadi, Yair Carmon, Simon Kornblith, et al. Model soups: averaging weights of multiple \\ufb01ne-tuned models\\nimproves accuracy without increasing inference time. In International Conference on Machine Learning , pages 23965\\u201323998.\\nPMLR, 2022. 5, 7\\n[46] Jia Xue, Hang Zhang, and Kristin Dana. Deep texture manifold for ground terrain recognition. In 2018 IEEE\\/CVF Conference\\non Computer Vision and Pattern Recognition . IEEE, June 2018. 1, 2, 6\\n[47] Jia Xue, Hang Zhang, Ko Nishino, and Kristin Dana. Differential viewpoints for ground terrain material recognition. IEEE\\ntransactions on pattern analysis and machine intelligence , PP, 09 2020. 1, 2, 6\\n12 [48] Zhijing Yang, Shujian Lai, Xiaobin Hong, Yukai Shi, Yongqiang Cheng, and Chunmei Qing. Dfaen: Double-order knowledge\\nfusion and attentional encoding network for texture recognition. Expert Systems with Applications , 209:118223, 2022. 1, 2\\n[49] Wei Zhai, Yang Cao, Zheng-Jun Zha, HaiYong Xie, and Feng Wu. Deep structure-revealed network for texture recognition. In\\n2020 IEEE\\/CVF Conference on Computer Vision and Pattern Recognition (CVPR) , pages 11007\\u201311016, 2020. 1, 2\\n[50] Wei Zhai, Yang Cao, Jing Zhang, and Zheng-Jun Zha. Deep multiple-attribute-perceived network for real-world texture\\nrecognition. In 2019 IEEE\\/CVF International Conference on Computer Vision (ICCV) , pages 3612\\u20133621, 2019. 1, 2\\n[51] H. Zhang, J. Xue, and K. Dana. Deep ten: Texture encoding network. In 2017 IEEE Conference on Computer Vision and\\nPattern Recognition (CVPR) , pages 2896\\u20132905, Los Alamitos, CA, USA, jul 2017. IEEE Computer Society. 1, 2\\n[52] Jianguo Zhang and Tieniu Tan. Brief review of invariant texture analysis methods. Pattern recognition , 35(3):735\\u2013747, 2002. 1\\n[53] Yuanhan Zhang, Qinghong Sun, Yichun Zhou, Zexin He, Zhenfei Yin, Kun Wang, Lu Sheng, Yu Qiao, Jing Shao, and Ziwei\\nLiu. Bamboo: Building mega-scale vision dataset continually with human-machine synergy, 2022. 3\\n13 SUPPLEMENTARY MATERIAL FOR RADAM: T EXTURE\\nRECOGNITION THROUGH RANDOMIZED AGGREGATED\\nENCODING OF DEEPACTIVATION MAPS\\n1 RADAM implementation\\nWe give the exact python code for the RADAM module and all its internal mechanisms. In the future, we will make\\nthis implementation and the experiments available in a public repository. First of all, RADAM is applied at the output\\nof a backbone built using the timm library. We do that using the following call:\\n1import timm\\n2net = timm.create_model(model, features_only=True, pretrained=True, output_stride=8)\\nHere, model is a string with the name of the backbone according to the library ( e.g. resnet18, convnext nano, etc).\\nThefeatures only option works for most convolutional architectures and turns the output of the model into a list of\\nactivation maps at different depths. In this sense, the following operations can be used to get some properties from\\nthese activation maps:\\n1import numpy as np\\n2import torch\\n3device = \\\"cuda\\\" if torch.cuda.is_available() else \\\"cpu\\\"\\n4net.to(device)\\n5act_maps = net(torch.zeros(1,3,224,224))\\n6z = sum([d.size()[1] for d in act_maps])\\n7half_depth = int(np.round(len(act_maps)\\/2))\\n8w = act_maps[half_depth].size()[2]\\n9h = act_maps[half_depth].size()[3]\\nThese properties are, respectively, the sum of the number of features of each activation map z, and the spatial dimen-\\nsions at approximately the middle of the architecture (w,h). Notice that these properties can be computed for any input\\nimage size aside from (224,224) if accepted by the backbone. They are then used to create the main class for RADAM:\\n1import torchvision\\n2from einops.layers.torch import Rearrange\\n3class RADAM(torch.nn.Module):\\n4 def __init__(self, device, z, spatial_dims = (28,28), m=4, pos_encoding=True,\\ndim_norm=(2,3)): \\u25c1arrowhookleft\\u2192\\n5 super(RADAM, self).__init__()\\n6 self.q=1\\n7 self.z=z\\n8 self.device = device\\n9 self.rearange = torch.nn.Sequential(lp_norm_layer(p=2.0, dim=dim_norm, eps=1e-10),\\n10 torchvision.transforms.Resize(spatial_dims),\\n11 Rearrange('b c h w -> b c (h w)')\\n12 )\\n13 self.RAEs = []\\n14 for model in range(m):\\n15 self.RAEs.append(RAE(q=self.q, z=z, w=spatial_dims[0], h=spatial_dims[1],\\ndevice=device, seed=model*(self.q*z), pos_encoding=pos_encoding)) \\u25c1arrowhookleft\\u2192 16\\n17 def forward(self, x):\\n18 out = []\\n19 if not isinstance(x, list):\\n20 x = [x]\\n21 device = self.device\\n22 batch, _, _, _ = x[0].size()\\n23 for depth in range(len(x)):\\n24 x[depth] = self.rearange(x[depth])\\n25 for sample in range(batch):\\n26 agg_activation_map = torch.vstack([x[i][sample] for i in range(len(x))])\\n27 pooled_features = torch.zeros(self.q,self.z).to(device)\\n28 for rae in self.RAEs:\\n29 pooled_features += rae.fit_AE(agg_activation_map)\\n30 out.append(pooled_features)\\n31 return torch.stack(out)\\n32\\n33class lp_norm_layer(torch.nn.Module):\\n34 def __init__(self,p=2.0, dim=(1,2), eps=1e-10):\\n35 super().__init__()\\n36 self.p=p\\n37 self.dim=dim\\n38 self.eps=eps\\n39 def forward(self, x):\\n40 return torch.nn.functional.normalize(x, p=self.p, dim=self.dim, eps=self.eps)\\nThe parameters in the code are the ones used by our experiments in the main paper. Additionally, the einops library is\\nused for efficient reorganization of the activation map, and we also use an additional class to perform Lpnormalization\\nin batches as a PyTorch layer.\\nThe implementation of the Randomized Autoencoder (RAE) is shown in the following. It takes as arguments the\\ndimensions of the model, previously passed to the RADAM class. The constructor generates the random encoder and\\nthe positional encoding, and the fitAEmethod computes the decoder weights, i.e, the least-squares training:\\n1class RAE():\\n2 def __init__(self, q, z, w, h, device, pos_encoding, seed):\\n3 self._device = device\\n4 self.pos_encoding=pos_encoding\\n5 self.W = make_orthogonal(LCG(q, z, seed)).to(device)\\n6 if self.pos_encoding:\\n7 self.encoding = positionalencoding2d(z, w, h)\\n8 self.encoding = torch.reshape(self.encoding, (z, w*h)).to(device)\\n9 self._activation = torch.sigmoid\\n10\\n11 def fit_AE(self, x):\\n12 if self.pos_encoding:\\n13 x = torch.add(x, self.encoding)\\n14 g = self._activation(torch.mm(self.W, x))\\n15 return torch.linalg.lstsq(g.t(), x.t()).solution\\nThe remaining functions used for weight initialization and positional encoding are given in the following:\\n1import pickle\\n2def LCG(m, n, seed):\\n3 L = m*n\\n4 # To generate this weight file, use:\\n5 #V = torch.zeros(2**18, dtype=torch.float)\\n6 #V[0], a, b, c = 1, 75, 74, (2**16)+1\\n7 #for x in range(1, (2**18)):\\n8 # V[x] = (a*V[x-1]+b) % c\\n9 #with open('RAE_LCG_weights.pkl', 'wb') as f:\\n10 #pickle.dump(V, f)\\n11 with open('RAE_LCG_weights.pkl', 'rb') as f:\\n2 12 V = pickle.load(f)\\n13 f.close()\\n14 V = V[seed:L+seed]\\n15 V = torch.divide(torch.subtract(V, torch.mean(V)), torch.std(V))\\n16 return V.reshape((m,n))\\n17\\n18def make_orthogonal(tensor):\\n19 rows = tensor.size(0)\\n20 cols = tensor.numel() \\/\\/ rows\\n21 flattened = torch.reshape(tensor, (rows, cols))\\n22 if rows < cols:\\n23 flattened.t_()\\n24 q, r = torch.linalg.qr(flattened)\\n25 d = torch.diag(r, 0)\\n26 ph = d.sign()\\n27 q *= ph\\n28 if rows < cols:\\n29 q.t_()\\n30 return q\\n31\\n32import math\\n33def positionalencoding2d(d_model, height, width):\\n34 pe = torch.zeros(d_model, height, width)\\n35 d_model = int(d_model \\/ 2)\\n36 div_term = torch.exp(torch.arange(0., d_model, 2) *\\n37 -(math.log(10000.0) \\/ d_model))\\n38 pos_w = torch.arange(0., width).unsqueeze(1)\\n39 pos_h = torch.arange(0., height).unsqueeze(1)\\n40 pe[0:d_model:2, :, :] = torch.sin(pos_w * div_term).transpose(0, 1).unsqueeze(1).repeat(1,\\nheight, 1) \\u25c1arrowhookleft\\u2192\\n41 pe[1:d_model:2, :, :] = torch.cos(pos_w * div_term).transpose(0, 1).unsqueeze(1).repeat(1,\\nheight, 1) \\u25c1arrowhookleft\\u2192\\n42 pe[d_model::2, :, :] = torch.sin(pos_h * div_term).transpose(0, 1).unsqueeze(2).repeat(1,\\n1, width) \\u25c1arrowhookleft\\u2192\\n43 pe[d_model + 1::2, :, :] = torch.cos(pos_h * div_term).transpose(0,\\n1).unsqueeze(2).repeat(1, 1, width) \\u25c1arrowhookleft\\u2192\\n44 return pe\\nFinally, the main RADAM class can be simply applied to the output of net, for a given image or batch, to obtain texture\\nrepresentations:\\n1texture_representation = RADAM(device, z, (w,h))(net(input_batch))\\nAs detailed in the main paper, these representations are then used to train a linear SVM for texture classification. For\\nthat, we use the scikit-learn library to build the classifier as follows:\\n1from sklearn import svm\\n2SVM = svm.SVC(kernel='linear')\\n2 Additional Experiments with RADAM\\nHere we present additional experiments not present in the main paper.\\n2.1 Variance caused by random initializer\\nTab. 1 shows the RADAM performance variance caused by changing the LCG configuration ( a,b, and c) for weight\\ninitialization of the RAE. We selected a set of 10 configurations from the table available in1. In general, we observe\\nminimal performance variance concerning these parameters, especially with larger backbones, corroborating that the\\nchoice of the random weights is not critical. We adopted parameters a= 75 ,b= 74 , and c= 216+ 1starting with\\nx= 1(first line in the table), as it uses the smallest integers and is also a classical configuration according to the ZX81\\n1www.wikipedia.org\\/wiki\\/Linear_congruential_generator\\n3 computer from 1981. However, small variations may impact the comparison between RADAM and other methods.\\nNevertheless, it is important to notice that our choice of LCG parameters does not impact the new state-of-the-art\\nresults reported in the main paper for RADAM. In fact, carefully selecting specific configurations in Tab. 1 for each\\ndataset and backbone yields even higher results, but we preferred a single consistent configuration to be used for all\\ncases. Additionally, for applicability purposes, we provide files with the exact values computed by our code for all the\\nLCG configurations (which will be available in our code repository).\\nTable 1: The variance on classification accuracy of RADAM when changing the LCG configuration to generate random\\nweights for the encoder. For the second part of the table (ConvNeXt results), the LCG configurations correspond to\\nthe same rows as for ResNet18.\\nParameters of LCG ResNet18\\nsource a b c DTD FMD KTH Outex13\\nZX81 75 74 216+ 1 68.1 77.7 84.7 89.0\\nNumerical Recipes book 1664525 1013904223 23268.0 77.9 85.0 90.0\\nBorland C\\/C++ 22695477 1 23266.7 79.7 86.0 88.2\\nglibc (used by GCC) 1103515245 12345 23168.0 78.0 84.7 90.0\\nBorland Delphi, Virtual Pascal 134775813 1 23267.9 79.2 85.6 90.1\\nMicrosoft Visual\\/Quick C\\/C++ 214013 2531011 23268.1 78.2 85.1 89.3\\nMicrosoft Visual Basic 1140671485 12820163 22468.1 79.2 85.4 88.7\\nRtlUniform from Native API 2147483629 2147483587 231-1 68.0 78.9 85.1 89.6\\nJava\\u2019s java.util.Random 25214903917 11 24867.9 77.4 85.4 89.7\\ncc65 16843009 826366247 23268.0 78.7 85.6 89.0\\naverage 67.9\\u00b10.478.5\\u00b10.885.3\\u00b10.489.4\\u00b10.6\\nConvNeXt-nano ConvNeXt-XL in ImageNet-21K\\nDTD FMD KTH Outex13 DTD FMD KTH Outex13\\n74.9 87.1 89.6 92.5 83.7 95.2 94.4 92.5\\n75.0 86.9 89.2 92.4 83.9 95.7 94.5 91.8\\n75.1 86.8 89.5 91.2 83.7 95.3 94.6 92.4\\n75.0 86.6 89.8 92.2 83.9 95.4 94.1 92.6\\n75.1 86.4 89.5 92.4 83.8 95.6 94.5 92.4\\n75.1 87.1 89.5 92.5 83.8 95.5 94.6 92.8\\n74.9 87.7 89.4 92.2 83.7 95.3 94.6 92.4\\n75.1 86.9 89.6 91.9 83.8 95.2 94.6 91.9\\n74.9 86.9 89.3 92.6 83.8 95.5 94.8 92.2\\n75.0 87.6 89.0 92.2 83.8 95.5 94.5 92.8\\naverage 75.0\\u00b10.187.0\\u00b10.489.4\\u00b10.292.2\\u00b10.483.8\\u00b10.195.4\\u00b10.294.5\\u00b10.292.4\\u00b10.3\\n4\",\"8\":\" A Computer Vision Enabled damage detection model with\\nimproved YOLOv5 based on Transformer Prediction Head\\nArunabha M. Roy1and Jayabrata Bhaduri2\\n1Aerospace Engineering Department, University of Michigan, Ann Arbor, MI\\n48109, USA\\n2Capacloud AI, Deep Learning &Data Science Division, Kolkata, WB 711103,\\nIndia.\\nAbstract\\nObjective. Computer vision-based up-to-date accurate damage classification and localization\\nare of decisive importance for infrastructure monitoring, safety, and the serviceability of civil\\ninfrastructure. Current state-of-the-art deep learning (DL)-based damage detection models,\\nhowever, often lack superior feature extraction capability in complex and noisy environments,\\nlimiting the development of accurate and reliable object distinction. Method. To this end,\\nwe present DenseSPH-YOLOv5, a real-time DL-based high-performance damage detection\\nmodel where DenseNet blocks have been integrated with the backbone to improve in preserving\\nand reusing critical feature information. Additionally, convolutional block attention modules\\n(CBAM) have been implemented to improve attention performance mechanisms for strong and\\ndiscriminating deep spatial feature extraction that results in superior detection under various\\nchallenging environments. Moreover, an additional feature fusion layers and a Swin-Transformer\\nPrediction Head (SPH) have been added leveraging advanced self-attention mechanism for more\\nefficient detection of multiscale object sizes and simultaneously reducing the computational\\ncomplexity. Results. Evaluating the model performance in large-scale Road Damage Dataset\\n1arXiv:2303.04275v1  [cs.CV]  7 Mar 2023 2\\n(RDD-2018), at a detection rate of 62.4 FPS, DenseSPH-YOLOv5 obtains a mean average\\nprecision (mAP) value of 85 :25%, F1-score of 81 :18%, and precision (P) value of 89 :51%\\noutperforming current state-of-the-art models. Significance. The present research provides\\nan effective and efficient damage localization model addressing the shortcoming of existing\\nDL-based damage detection models by providing highly accurate localized bounding box\\nprediction. Current work constitutes a step towards an accurate and robust automated damage\\ndetection system in real-time in-field applications.\\nKeywords: Automated damage detection; You Only Look Once (YOLOv5) algorithm; Swin\\nTransformer Object Detection (OD); Computer vision; Deep Learning (DL)\\n1. Introduction :\\nIn recent years, automated damage detection plays an important role in various industrial\\napplications including product quality assessment (Agarwal and Singh, 2015; Hanzaei et al.,\\n2017), infrastructure monitoring (Eisenbach et al., 2017; Gopalakrishnan, 2018), safety and\\nthe serviceability of civil infrastructure (Koch et al., 2015; Hartmann and Trappey, 2020).\\nEarly-stage accurate crack detection is critical for pavement damage rating and subsequent\\nsealing or rehabilitation activities (Chen et al., 2021; Chen and Cho, 2022). Therefore, it\\nis important for roadway infrastructure engineers to detect pavement cracks accurately so\\nthat the best cost-effective plans of maintenance and rehabilitation could be employed (Ni\\net al., 2022; Dong et al., 2021). While traditional damage detection techniques mainly include\\nvisual inspection, however, such labor-intensive methods have disadvantages due to their low\\nefficiency, high cost, and individual biases (Xu et al., 2022; Shang et al., 2023). Additionally, it\\nis also limited in reproducibility, reliability, and objectivity due to the requirement of qualified\\npersonnel for domain-specific experience, knowledge, and skill sets (Fang et al., 2020). To\\ncircumvent such issues, more recently, various automatic and semi-automatic crack detection\\nalgorithms have gained significant attraction (Gopalakrishnan, 2018).\\nFor the last three decades, image-based crack detection (Mohan and Poobal, 2018; Koch\\net al., 2015) that include various image processing approaches, such as edge detection (Zhao 3\\net al., 2010; Hanzaei et al., 2017; Li et al., 2022), dynamic thresholding (Oliveira and Correia,\\n2009; Wang et al., 2021b), and different morphological operations (Anitha et al., 2021; Li\\nand Zhao, 2021) have been the central focus for detecting damage in challenging real-world\\nscenarios. However, the aforementioned methods are quite sensitive to noise and varying\\nillumination intensities, and hence, not suitable in real-world complex conditions (Koch et al.,\\n2015). To circumvent such issues, later, conventional machine learning (ML) approaches\\nhave been introduced for damage detection (Mohan and Poobal, 2018; Koch et al., 2015). In\\ngeneral, such methods utilize a trained classifier such as a Support Vector Machine (SVM)\\non local feature descriptors that can be either Local Binary Patterns (LBP) (Varadharajan\\net al., 2014; Quintana et al., 2015) or Histogram of Oriented Gradient (HOG) (Kapela et al.,\\n2015). Although, compared to conventional image processing approaches, ML-based models\\nsignificantly improve the accuracy and efficiency of the damage detection, however, due to large\\nerrors in classification performances remains a serious bottleneck for deploying such models in\\nreal-world applications (Fang et al., 2020).\\nMore recently, deep learning (DL) characterized by multilayer neural networks (NN) (LeCun\\net al., 2015) has shown remarkable breakthroughs in pattern recognition for various fields\\nincluding image classification (Rawat and Wang, 2017; Jamil et al., 2022; Khan et al., 2022b,a),\\ncomputer vision (Voulodimos et al., 2018; Roy and Bhaduri, 2021; Roy et al., 2022c; Roy and\\nBhaduri, 2022; Roy et al., 2022a), object detection (Zhao et al., 2019a; Chandio et al., 2022;\\nRoy et al., 2022b; Singh et al., 2023a), brain-computer interfaces (Roy, 2022b,a,c; Singh et al.,\\n2023b), signal classification (Jamil and Roy, 2023, 2022) and across diverse scientific disciplines\\n(Bose and Roy, 2022; Roy and Bose, 2023b; Roy and Guha, 2022; Roy and Bose, 2023a; Roy\\nand Guha, 2023). Following the success, there is an increasing thrust of research works geared\\ntowards damage classification tasks employing DL techniques, mostly convolutional neural\\nnetworks (CNN), such as ResNet (Bang et al., 2018), AlexNet (Dorafshan et al., 2018; Li\\net al., 2018), VGG-net (Gopalakrishnan et al., 2017; Silva and Lucena, 2018) and various\\nothers (Chow et al., 2020; Nath et al., 2022; Li et al., 2021). Particularly in object localization,\\nDL methods have demonstrated superior accuracy (Han et al., 2018) that can be categorized\\ninto two classes: two-stage and one-stage detector (Lin et al., 2017a). Two-stage detectors\\nincluding Region Convolution Neural Network (R-CNN) (Girshick, 2015), faster R-CNN (Ren 4\\net al., 2016), mask R-CNN (He et al., 2017) etc that have shown a significant improvement\\nin accuracy in object localization. In recent times, You Only Look Once (YOLO) variants\\n(Redmon et al., 2016; Redmon and Farhadi, 2017, 2018; Bochkovskiy et al., 2020) have been\\nproposed that unify target classification and localization. In (Roy et al., 2022c; Roy and\\nBhaduri, 2022, 2021), DenseNet (Huang et al., 2017) blocks attaching Spatial Pyramid Pooling\\n(SPP) (He et al., 2015) with an improved modified Path Aggregation Network (PANet) (Liu\\net al., 2018) has been integrated to enhance the representation of receptive fields and extraction\\nof important contextual features in the original YOLOv4 leading to significant improvement in\\nthe detection speed and accuracy. In order to enhance gradient performance and reduce the\\ncomputational cost, YOLOv4 (Bochkovskiy et al., 2020) designs a cross-stage partial (CSP)\\nnetwork. To further improve detection accuracy, YOLOv4 implements Mish activation (Misra,\\n2020) and CIoU loss (Zheng et al., 2020). Recently, Scaled-YOLOv4 (Wang et al., 2021a)\\ndemonstrated its superior performance in detecting the vast range of linearly scaled objects for\\nvarious applications. As the latest generation of the YOLO series, the YOLOv5 (Jocher et al.,\\n2021) has been rated top among state-of-the-art object detection models which inherit all the\\nabove advantages. Thus, in the present work, YOLOv5 has been considered a benchmark model\\nfor multiclass damage detection. More recently, improved YOLOv5 Based on Transformer\\nPrediction Head (TPH-YOLOv5) (Zhu et al., 2021) has been proposed integrating convolutional\\nblock attention module (CBAM) (Woo et al., 2018) for closely packed object detection and\\nSwin Transformer-enabled YOLOv5 (SPH-YOLOv5) (Gong et al., 2022) has been designed\\nincorporating Normalization-based Attention Modules (NAM) that demonstrate significant\\nimprovement in accuracy while simultaneously reducing the computational complexity of the\\nmodel which are the main motivations for the network architectural development of the current\\nwork.\\n1.1 Related Works :\\nIn this section, some recent and relevant DL works have been highlighted in the field of road\\ndamage detection. In recent years, multiple studies have adopted various ML and DL-based\\napproaches for automated road surface damage classification and detection (Zhang et al., 2017a; 5\\nStricker et al., 2019; Bi\\u0018 cici and Zeybek, 2021) For instance, a smartphone-based supervised deep\\nconvolutional neural network (D-CNN) has been proposed for pavement damage classification\\n(Zhang et al., 2016). Along a similar line, deep neural network (DNN) architecture has been\\nemployed for detecting cracks and potholes (Anand et al., 2018; Silva and Lucena, 2018) as well\\nas pavement condition assessment (Fan et al., 2018). In Nhat-Duc et al. (2018), the superiority\\nof the DCNN-based approach has been demonstrated over edge-detection-based approaches for\\npavement crack detection. In Maeda et al. (2018), a real-time road damage detection model\\nbased on SDD has been proposed that has been trained on a publicly available large-scale\\nroad damage dataset (RDD-2018) for eight different categories of road damages. Due to the\\npopularity of the dataset, various attempts have been made, notably using YOLO (Alfarrarjeh\\net al., 2018), Faster R-CNN (Kluger et al., 2018), Faster R-CNN with ResNet-152 (Wang et al.,\\n2018a), Ensemble models with Faster R-CNN and SSD (Wang et al., 2018b), and RetinaNet\\n(Angulo et al., 2019) to further improve the detection performance. In addition, RetinaNet\\n(Angulo et al., 2019) has been used on a modified RDD-2018 dataset which demonstrates\\nsignificant performance improvement. Following the work of Maeda et al. (2018), progressive\\ngrowing- generative adversarial networks (PG-GANs) (Maeda et al., 2021) with Poisson blending\\nhave been used to generate new training data (RDD-2019) in order to improve the accuracy of\\nroad damage detection. More recently, transfer learning (TL)-based road damage detection\\nmodel (Arya et al., 2021a) has been proposed introducing large-scale open-source dataset\\nRDD2020 (Arya et al., 2020, 2021b) considering multiple countries. In Naddaf-Sh et al. (2020),\\nEfficientDet-D7 has been employed for the detection of asphalt pavement distress. Whereas,\\nYOLO CSPDarknet53 (Mandal et al., 2020) has been used for road damage detection. Similarly,\\nthe YOLO network has been used for detecting pavement distress from high-resolution images\\n(Du et al., 2021). In Majidifard et al. (2020), YOLOv2 model has been utilized for pavement\\ndistress classification from Google street view images. Along a similar line, a CNN-based\\npredictive model trained in Google API images has been employed for detecting potholes\\nin Patra et al. (2021). In a separate work in Guan et al. (2021), a stereo-vision integrated\\nsegmentation-based DL model with modified depth-wise separable convolution U-net has\\nbeen deployed for crack and pothole detection where multi-feature image datasets have been\\nused to train the model. More recently, a semi-supervised DL-based pixel-level segmentation 6\\nmodel (Karaaslan et al., 2021) has been proposed utilizing attention guidance for cracks and\\nspalls localization that reduces computational cost significantly. In separate work, an improved\\nYOLOv5 road damage detection algorithm (Guo and Zhang, 2022) has been proposed leveraging\\nMobileNetV3 as a backbone feature extractor. In Hac\\u0010efendio\\u0015 glu and Ba\\u0018 sa\\u0015 ga (2022), Faster\\nR-CNN has been employed for concrete pavement crack detection under various illumination and\\nweather conditions. Although, there exist several state-of-the-art works for damage detection\\nincluding multi-class damage localization models, however, they often suffer from low accuracy,\\nmissed detection, and relatively large computational overhead (Cao et al., 2020; Azimi et al.,\\n2020; Naddaf-Sh et al., 2020).\\n1.2 Motivations :\\nDespite illustrating outstanding performance in damage detection, current state-of-the-art\\nDL algorithms still require further improvement due to their insufficient fine-grain contextual\\nfeature extraction capability leading to missed detection and false object predictions for various\\ndamages\\/cracks which possess a wide range of textures, shapes, sizes, and colors (Cao et al.,\\n2020; Azimi et al., 2020; Naddaf-Sh et al., 2020). Between various damage classes, accurate\\ndetection and localization tasks can be challenging due to significant variability of lightening\\nconditions, low visibility, the coexistence of multi-object classes with various aspect ratios, and\\nother morphological characteristics (Azimi et al., 2020; Naddaf-Sh et al., 2020). Additionally,\\nvisual similarities, complex backgrounds, and various other critical factors offer additional\\ndifficulties for the state-of-the-art damage detection models (Naddaf-Sh et al., 2020). To\\nthis end, the current works aim to develop an efficient and robust damage classification and\\naccurate damage localization model simultaneously productive in terms of training time and\\ncomputational cost which is currently lacking in the recent state-of-the-art endeavors.\\n1.3 Contributions :\\nTo address the aforementioned shortcomings, in the current study, we present DenseSPH-YOLOv5\\nbased on an improved version of the state-of-art YOLOv5 detection model for accurate real-time 7\\ndamage detection. The major contributions of the present research work can be summarized as\\nfollows:\\n\\u2022In Dense-SPH-YOLOv5, we have attached DenseNet blocks with CSP modules in the\\nCSPDarknet53 to preserve critical feature maps and efficiently reuse the discriminating\\nfeature information.\\n\\u2022Secondly, we have introduced an additional detection head specifically for detecting tiny\\nobjects in the head part of the proposed DenseSPH-YOLOv5 network.\\n\\u2022In addition, the convolutional block attention module (CBAM) has been implemented for\\nthe construction of progressive feature attention with large coverage along both channel\\nand spatial dimensions for strong and discriminating deep spatial feature extraction during\\nobject detection.\\n\\u2022Furthermore, the regular CNN prediction heads (CPH) in YOLOv5 have been upgraded\\nwith Swin transformer Prediction Heads (SPHs) employing Swin transformer (STR)\\nencoder block leveraging advanced self-attention mechanisms for efficient detection of\\nmulti-scale object sizes and simultaneously reducing the computational complexity.\\n\\u2022Spatial Pyramid Pooling (SPP) has been tightly attached to the backbone to enhance\\nthe representation of receptive fields and extraction of important contextual features.\\n\\u2022Finally, an improved modified Path Aggregation Network (PANet) has been utilized\\nto efficiently preserve fine-grain localized information by feature fusion in a multi-scale\\nfeature pyramid map.\\nWith the aforementioned modifications, the detection accuracy of the model has been\\nsignificantly enhanced for multi-scale object detection. An extensive ablation study has been\\nperformed for different combinations of backbone-neck architecture in order to optimize both\\naccuracies of detection and detection speed. The proposed DenseSPH-YOLOv5 has been\\nemployed to detect distinct eight different damage classes that provide superior and accurate\\ndetection under various complex and challenging environments. The present work effectively\\naddresses the shortcoming of existing DL-based crack detection models and illustrates its 8\\nFigure 1: Sample images from RDD-2018 dataset (Maeda et al., 2018): (a) to (g) correspond\\nto each of the eight categories with the legends.\\nsuperior potential in real-time in-field applications. The rest of the paper is organized as follows:\\ndescription of the dataset has been described in Section 2; Section 3 introduces the proposed\\nmethodology for damage detection; the relevant finding and discussion of the proposed model\\nhave been discussed in Sections 4 and 5, respectively. Finally, the conclusions of the present\\nwork have been discussed in section 6.\\n2. Road Damage Dataset :\\nIn the current study, large-scale Road Damage Dataset (RDD-2018) (Maeda et al., 2018)\\nhas been used that consists of 9,053 labeled road damage images of resolution 600 \\u0002600 pixels\\ncontaining a total number of 15,435 annotated bounding boxes for eight different types of\\nroad damage. To avoid biases, the images have been photographed in various weather and\\nillumination conditions from different regions of Japan including Chiba, Muroran, Ichihara,\\nSumida, Nagakute, and Numazu. During annotation, professional road damage expertise\\nhas been employed to verify various damage classes that ensure the reliability of the dataset.\\nVarious damage types and corresponding class identifiers have been listed in Table. 1. Each 9\\nTable 1: Various road damage types and corresponding class identifiers in RDD-2018 dataset\\n(Maeda et al., 2018).\\nClass\\nIdentifierDamage\\ntypeAlignment Details\\nD00 Linear Crack Longitudinal Wheel-marked part\\nD01 Linear Crack Longitudinal Construction joint part\\nD10 Linear Crack Lateral Equal interval\\nD11 Linear Crack Lateral Construction joint part\\nD20 Alligator Crack - Partial pavement, overall pavement\\nD40 Other Crack - Pothole, rutting, separation\\nD43 Other Crack - White line blur\\nD44 Other Crack - Cross walk blur\\ntype of damages have been illustrated in Fig. 1. Primarily, the damage has been classified into\\ncracks or different corruptions. Then, the cracks have been divided into linear and alligator\\ncracks. Whereas, other corruptions include both potholes and rutting as well as other road\\ndamage classes such as blurring of white lines.\\n3. Proposed Methodology for damage detection:\\nIn object detection, target object classification and localization are performed simultaneously\\nwhere the target class has been categorized and separated from the background. The purpose\\nof object localization is to locate objects by drawing bounding boxes (BBs) on input images\\ncontaining the entire object. This is particularly useful for counting endangered species for\\naccurate surveying. To this end, the main goal of the current work is to develop an efficient and\\nrobust damage classification and accurate damage localization model. In this regard, different\\nvariants of YOLO (Redmon et al., 2016; Redmon and Farhadi, 2017, 2018; Bochkovskiy et al.,\\n2020) are some of the best high-precision one-stage object detection models. More recently, 10\\n \\n\\u03b7 \\nd \\nwp \\nhp \\nwgt \\nhgt \\n(a) (b) Damage  detection  \\nInput  N \\u00d7N  grids  \\nBBs+ confidence  score  \\nClass probability   \\n \\n \\n \\nFigure 2: Schematic of (a) YOLO object localization process for damage localization; (b)\\nSchematic of CIoU offset regression for target BBs predictions.\\nYOLOv5 (Jocher et al., 2021) has been introduced that currently achieves the best detection\\nperformance and has four different model variants including YOLOv5s, YOLOv5m, YOLOv5l,\\nand YOLOv5x depend on different model widths and depths. In general, the overall architecture\\nof YOLOv5 consists of the following parts: a backbone for deep feature extraction, followed by\\nthe neck for gathered semantic feature fusion, and finally head network for object classification\\nand localization. The original version of YOLOv5 utilizes CSPDarknet53 (Wang et al., 2020;\\nBochkovskiy et al., 2020) with SPP and PANet (Liu et al., 2018) as backbone and neck,\\nrespectively. Whereas, YOLO detection head (Redmon et al., 2016) has been employed in\\nthe detection head. The YOLO model transforms the object detection task into a regression\\nproblem by generating BBs coordinates and probabilities for each class as shown in Fig. 2.\\nDuring the process, the inputted image size has been uniformly divided into N\\u0002Ngrids where\\nBpredictive BBs have been generated. Subsequently, a confidence score has been assigned if\\nthe target object falls inside that particular grid. It detects the target object for a particular\\nclass when the center of the ground truth lies inside a specified grid. During detection, each\\ngrid predicts NBnumbers of BBs with the confidence value \\b Bas:\\n\\bB=Pr(obj)\\u0002IoUt\\np_P r(obj)20;1 (1) 11\\nwherePr(obj) infers the accuracy of BB prediction, i.e., Pr(obj) = 1 indicates that the target\\nclass falls inside the grid, otherwise, Pr(obj) = 0. The degree of overlap between ground truth\\nand the predicted BB has been described by the scale-invariant evaluation metric intersection\\nover union (IoU) which can be expressed as\\nIoU =Bp\\\\Bgt\\nBp[Bgt(2)\\nwhere BgtandBpare the ground truth and predicted BBs, respectively.\\n3.1 Loss in BBs regression :\\nTo further improve BBs regression and gradient disappearance, generalized IoU (GIoU)\\n(Rezatofighi et al., 2019) and distance-IoU (DIoU) (Zheng et al., 2020) as been introduced\\nconsidering aspect ratios and orientation of the overlapping BBs. More recently, complete IoU\\n(CIoU) (Zheng et al., 2020) has been proposed for improved accuracy and faster convergence\\nspeed in BB prediction which can be expressed as\\nLCIoU= 1 +\\f\\u0018+\\u000b2(bp;bgt)\\n\\u00112\\u0000IoU (3)\\n\\u0018=4\\n\\u00192\\u0012\\ntan\\u00001wgt\\nhgt\\u0000tan\\u00001wp\\nhp\\u00132\\n;\\f=\\u0018\\n(1\\u0000IoU) +\\u00180 (4)\\nwhere bgtandbpdenotes the centroids of BgtandBp, respectively; \\u0018and\\fare the consistency\\nand trade-off parameters, respectively. As shown in Fig. 3 -(b), \\u0011is the smallest diagonal length\\nofBp[Bgt;wgt,wpare widths and hgt,hpare heights of BgtandBp, respectively. With\\nincreasingwp=hp, we get\\u0018!0 from Eq. 4. Therefore, to optimize the influence of \\u0018on the\\nCIoU,wp=hpcan be properly chosen for the detection model. Finally, the best BB prediction\\ncan be obtained from the non-maximum suppression (NMS) (Ren et al., 2016) algorithm from\\nvarious scales. In object detection in the YOLO framework, The total loss function \\u0001 lconsist\\nof BB coordinate prediction error \\u0001 cor, IoU error \\u0001 IoU, classification error term \\u0001 cl. The loss 12\\nFigure 3: Schematic illustration of measuring various performance metrics: (a) Intersection\\nover Union (IoU); (b) precision (P); (c) recall (R) during damage detection process.\\nfunction \\u0001 lfor BBs regression can be formulated as,\\n\\u0001l= \\u0001 cor+ \\u0001 cl+ \\u0001 IoU: (5)\\n3.2 Performance metrics:\\nIn the present work, the performance of the object detection models has been evaluated\\nby common standard measures (Ferri et al., 2009) including average precision (AP), precision\\n(P), recall (R), IoU, F-1 score, mean average precision (mAP), etc. The confusion matrix\\nobtained from the evaluation procedure provides the following interpretations of the test results:\\ntrue positive (TP), false positive (FP), false negative (FN), and true negative (TN). During\\nobject classification, the classified object can be defined as TP for IoU \\u00150:5. Whereas, it can\\nbe classified as FP for IoU <0:5. Based on the aforementioned interpretations, the metric P of 13\\nthe classifier can be defined by its ability to distinguish target classes correctly as :\\nP =TP\\n(TP+FP); (6)\\nThe ratio of the correct prediction of target classes is called R of the classifier which can be\\nevaluated as:\\nR =TP\\n(TP+FN)(7)\\nThe higher values of P and R indicate superior detection capability. Whereas, F-1 score is the\\narithmetic mean of the P and R given as :\\nF1 = 2\\u0002\\u0012P R\\nP + R\\u0013\\n(8)\\nA relatively high F1 score represents a robust detection model. The performance metrics AP\\ncan be defined as the area under a P-R curve (Davis and Goadrich, 2006) as follows\\nAP =Z1\\n0P (R) dR (9)\\nA higher average AP value indicates better accuracy in predicting various object classes. In\\naddition, AP 50:95denotes AP over IoU=0 :50 : 0:05 : 0:95; AP 50and AP 75are APs at IoU\\nthreshold of 50% and 75%, respectively. The AP for detecting small, medium, and large objects\\ncan be measured through AP S, AP M, and AP L, respectively. Finally, mAP can be obtained\\nfrom the average of all APs as:\\nmAP =1\\nNcNX\\ni=1APi: (10)\\nIn YOLOv5, various combinations of activation functions including sigmoid, leaky- ReLU and\\nSiLU (Hendrycks and Gimpel, 2016) can be utilized to improve the performance of the model\\nfor a specific detection task. In addition, bag of freebies and specials (Bochkovskiy et al.,\\n2020) can also be employed to further optimize the detection architecture of YOLOv5. As\\nthe latest generation of the YOLO series, YOLOv5 has been shown to provide state-of-the-art\\ndetection performance. Therefore, it has been chosen as the baseline model for our present study. 14\\n3.3 DenseSPH-YOLOv5 architecture:\\nIn recent years, various attempts have been made on DL-based computer vision models\\nfor damage detection such as Faster R-CNN (Kluger et al., 2018; Wang et al., 2018a), SSD\\n(Maeda et al., 2018; Wang et al., 2018b), RetinaNet (Angulo et al., 2019), YOLO (Alfarrarjeh\\net al., 2018; Mandal et al., 2020), YOLOv2 (Majidifard et al., 2020), YOLOv5 (Guo and\\nZhang, 2022) etc. Although the aforementioned techniques have demonstrated outstanding\\nperformance, however, the damage detection task faces several challenges, in particular, due to\\nthe presence of complex and noisy backgrounds, significant variability of lightening conditions,\\nlow visibility, densely packed classes, and overlap, the coexistence of multi-object classes with\\nvarious aspect ratios, and other morphological characteristics (Azimi et al., 2020; Naddaf-Sh\\net al., 2020). In this regard, YOLOv5 can be a suitable model which demonstrates both superior\\naccuracy and faster detection speed compared to YOLOv4. The state-of-the-art YOLOv5 is\\ncomparatively light, yet effective, which performs multiple downsampling and turns shallow\\nsemantic information into high-level semantic information that effectively reduces the number\\nof parameters. However, this inevitably loses semantic information and makes the network\\nineffective at extracting and fusing discriminating semantic features. Therefore, the original\\nYOLOv5 network requires further improvement due to its insufficient fine-grain contextual\\nfeature extraction capability leading to missed detection and false object predictions for various\\ndamages\\/cracks which possess a wide range of textures, shapes, sizes, and colors (Cao et al., 2020;\\nAzimi et al., 2020; Naddaf-Sh et al., 2020). In order to achieve high accuracy and high efficiency\\nin damage detection, we propose a novel object localization algorithm DenseSPH-YOLOv5\\nbased on a state-of-the-art YOLOv5 network to enhance feature extraction, preserve fine-grain\\nlocalized information and improve feature fusion that provides superior damage detection\\nunder various challenging environments. The overall network of the DenseSPH-YOLOv5 model\\nis shown in Fig. 4. To improve performance in terms of classification accuracy and object\\nlocalization, we perform extensive experiments, and various modifications are proposed which\\nare detailed in the subsequent sections. 15\\nFigure 4: Schematic of the proposed DenseSPH YOLOv5 with fused DenseNet in CSPDarknet53\\nbackbone; CBAM module in the improved neck; four SPHs utilize hierarchical feature maps\\nfrom STR encoder blocks in the modified Neck.\\n3.3.1 DenseNet block : Since YOLOv5 reduces the feature maps from the inputted images\\nthrough convolution and down-sampling procedures that result in significant semantic feature\\nloss during transmission. To this end, we have introduced DenseNet (Huang et al., 2017)\\nin the original CSPDarknet53 of YOLOv5 attached to the CSP module to preserve critical\\nfeature maps and efficiently reuse the discriminative feature information as shown in Fig. 4. In\\nDenseNet, each layer has been connected to other layers in a feed-forward mode where n-th\\nlayer can receive the important feature information \\u0018nfrom all the previous layers \\u00180;\\u00181;:::;\\u0018 n\\u00001\\nas:\\n\\u0018n= \\u0002 n[\\u00180; \\u00181; :::;\\u0018 n\\u00001] (11)\\nwhere \\u0002 nis the feature map function for n-th layer. The schematic of the DenseNet blocks\\nnetwork structure have been shown in Fig. 5-(d). Through our extensive experiments, we found\\nout that DenseNet improves the feature transfer and mitigates over-fitting in the detection\\nnetwork. Therefore, in the proposed Dense-SPH YOLOv5 network, as shown in Fig. 3, we\\nhave introduced four DenseNet blocks: the first block (Dense B-1) has been attached before\\ncross-stage partial block CSP3; the second block (Dense B-2) has been placed before CSP6; 16\\nFigure 5: Schematic of (a) CBAM that has two sequential sub-modules: (b) CAM and (c) SAM\\nfor adaptive refinement of feature map at every intermediate convolution blocks; (d) network\\narchitecture of DenseNet block used in Dense-SPH YOLOv5 detection model. 17\\nwhereas, third (Dense B-3) and fourth (Dense B-4) blocks have been added before CSP6 and\\nCSP3, respectively which results in enhanced feature propagation. Additionally, by reducing\\nredundant feature operations, the Dense-CSPDarknet53 network improves the computational\\nspeed.\\n3.3.2 Convolutional block attention module (CBAM): Convolutional block attention\\nmodule (CBAM) (Woo et al., 2018) is a lightweight, yet effective attention block which can be\\nintegrated into the object detection model in an end-to-end manner that has shown superior\\ndetection accuracy (Zhu et al., 2021). For an inputted feature map, CBAM sequentially\\nemploys the attention map along the channel and spatial dimensions. As shown in Fig. 5-(a),\\nthe network structure of the CBAM module consists of two sequential sub-modules: Channel\\nAttention Module (CAM) and Spatial Attention Module (SAM) for adaptive refinement of the\\nfeature map at every intermediate convolution blocks by multiplying input feature map with the\\nattention map. For inputted feature map FcFcFc2RC\\u0002H\\u0002W, CAM produces 1D channel attention\\nmapMcMcMc2RC\\u00021\\u00021, and sequentially, SAM infers 2D spatial attention map MsMsMs2R1\\u0002H\\u0002Was\\nshown in Figs. 4-(a-c). The overall CBAM attention process can be expressed as:\\nF0\\ncF0\\ncF0\\nc=McMcMc(FcFcFc)\\nFcFcFc;F00\\ncF00\\ncF00\\nc=MsMsMs(F0\\ncF0\\ncF0\\nc)\\nF0\\ncF0\\ncF0\\nc; (12)\\nwhere ;F00\\ncF00\\ncF00\\ncis the CBAM refined output; \\nrepresents element-wise multiplication. More\\nspecifically, CAM produces two different spatial descriptors including average-pooled features\\nFc\\navgFc\\navgFc\\navgand max pooled features Fc\\nmaxFc\\nmaxFc\\nmaxwhich produce channel attention map McMcMc2RC\\u00021\\u00021from\\nmulti-layer perceptron (MLP). The CAM attention can be expressed as:\\nMcMcMc(FcFcFc) =\\u001b(W1W1W1(W0W0W0(Fc\\navgFc\\navgFc\\navg))) +W1W1W1(W0W0W0(Fc\\nmaxFc\\nmaxFc\\nmax)) (13)\\nwhereW0W0W02RC=r\\u0002C, andW1W1W12RC\\u0002C=rare the MLP weights ; \\u001brepresents the sigmoid\\nactivation function. In SAM, inter-spatial attention map MsMsMs(FcFcFc)2RH\\u0002Whas been aggregated\\nto generate 2D maps: Fs\\navgFs\\navgFs\\navg2R1\\u0002H\\u0002WandFs\\nmaxFs\\nmaxFs\\nmax2R1\\u0002H\\u0002W. From these two feature maps, the 18\\nspatial attention can be computed as:\\nMsMsMs(FFF) =\\u001b(fn\\u0002n([Fs\\navgFs\\navgFs\\navg;Fs\\nmaxFs\\nmaxFs\\nmax]) (14)\\nwherefn\\u0002ndenotes a convolution operation with the filter size of n\\u0002nwithn= 7. The\\nimplementation of the CBAM module into the proposed DenseSPH-YOLOv5 significantly\\nimproves the detection accuracy of the damage detection by providing specific attention to the\\ndense objects and confusing noisy areas which proved the effectiveness of this module. Overall,\\nit helps to learn more expressive features that demonstrate significant improvement in detection\\naccuracy for road damage datasets considered herein.\\n3.3.3 Swin transformer encoder blocks: Inspired by the superior performance of the\\nvision transformer (Dosovitskiy et al., 2020) in dense and occluded object detection, in the\\nproposed model, Swin transformer (STR) encoder blocks (Liu et al., 2021) have been fused\\nto all four detection heads of DenseSPH-YOLOv5 architecture as shown in Fig. 3. Such\\nimplementation improves the global semantic feature extraction and contextual information\\nfusion leveraging self-attention mechanism (Vaswani et al., 2017) that demonstrated superior\\nperformance in dense object detection (Gong et al., 2022).\\nFor inputted feature map FsFsFs2RH\\u0002W\\u0002C, it transforms to Qs;Ks;VsQs;Ks;VsQs;Ks;Vs2RN\\u0002C0where\\nN=H\\u0002Wto feed Multi-head Self Attention (MSA) module after linear projection and\\nreshape operations. The output feature map ZsZsZsfrom MSA aggregates global information which\\ncan be expressed as\\nZsZsZs=AsVsAsVsAsVs;AsAsAs= softmaxQsKsTQsKsTQsKsT; (15)\\nwhereAsAsAs2RN\\u0002Nis the self-attention matrix that represents the relationship between feature\\nmap elements with remaining elements. Although, MSA in Transformer is effective in generating\\na self-attention matrix by integrating multiple independent subspaces, however, the global\\ncomputation overhead is significantly high for high-resolution images. Thus, for dense prediction\\nor to tackle high-resolution images, the computational complexity of MSA in Transformer\\nis not suitable which leads to quadratic complexity with respect to the number of tokens 19\\nFigure 6: Schematic of (a) STR encoder architecture that contains (b) regular W-MSA and\\nSW-MSA used in detection heads of DenseSPH-YOLOv5 architecture.\\n(Liu et al., 2021). To this end, STR can significantly improve the computational efficiency\\nof MSA that has linear computational complexity with respect to image size which enhances\\nthe performance of the model in terms of detection speed and accuracy. Therefore, in the\\nproposed DenseSPH-YOLOv5 detection model, the STR encoder module has been fused into\\nfour different prediction heads. As shown in Fig. 6 -(b), each STR encoder contains two\\nsub-layers that include shifted window-based multi-head self-attention (MSA) module, followed\\nby a fully-connected MLP with GeLU nonlinearity. Residual connections are used after each\\nMSA module. Subsequently, Layer Norm (LN) has been added before MSA and MLP. In STR,\\nthe obtained feature map has been dived into non-overlapping separate windows in the W-MSA\\nmodule. Subsequently, self-attention has been computed in each local window. Applying\\nSW-MSA partitioning, the consecutive STR blocks can be computed as:\\n^z^z^zi=W-MSA (LN(zzzi\\u00001)) +zzzi\\u00001;zzzi=MLP (LN(^z^z^zi)) + ^z^z^zi(16)\\n^z^z^zi+1=SW-MSA (LN(zzzi)) +zzzi;zzzi+1=MLP (LN(^z^z^zi+1)) + ^z^z^zi+1(17)\\nwherezzziand^z^z^zirepresent the output features from MLP and SW-MSA modules, respectively.\\nFor a feature map FsFsFs2RH\\u0002W\\u0002Cwithm\\u0002msize of local window, the computational complexity 20\\n\\u0007 can be obtained as\\n\\u0007(MSA ) = 4HWC2+ 2(HW)2C; \\u0007( W-MSA ) = 4HWC2+ 2(HW)M2C (18)\\nwhereHW is the patch number. Evidently, for large HW, the computational overhead of global\\nMSA is exceptionally high, while, W-MSA is scalable. Thus, implementing STR significantly\\nreduces the computational complexity of the model.\\n3.3.4 Receptive field enhancement: One of the requirements of CNN is to have fixed-size\\ninput images. However, due to the different aspect ratios of the images, they have been fixed\\nby cropping and warping during the convolution process which results in losing important\\nfeatures. In this regard, SPP (He et al., 2015) applies an efficient strategy in detecting target\\nobjects at multiple length scales. To this end, we have added an SPP block integrated with\\nDCSP-3 in the Dense-CSPDarknet53 backbone to improve receptive field representation and\\nextraction of important contextual features as shown in Fig. 4.\\n4. Results:\\nIn this section, the performance and detection accuracy of the proposed DenseSPH-YOLOv5\\nframework have been discussed that been evaluated in the RDD-2018 dataset consisting of 8\\ndifferent categories of damage classes. For better clarity in BB representation, each damage type\\nhas been marked with the corresponding class identifiers as shown in Table 1. The performance\\nof the DenseSPH-YOLOv5 network has been optimized through extensive ablation studies.\\nFinally, the performance of the proposed model has been studied in detail and compared with\\nseveral state-of-the-art object detection models.\\n4.1 Training procedure :\\nIn the present work, we have performed an extensive and elaborate study to explore the\\ncomparative performance analysis of the proposed DenseSPH-YOLOv5 models for road damage\\nlocalization tasks. From the RDD-2018 dataset, a total of 80% and 20% images have been 21\\nrandomly chosen for training and validation, respectively. For all the experiments, we have\\nused a Windows 10 Pro (64-bit) based computational system that has Intel Core i5-10210U\\nwith CPU @ 2.8 GHz \\u00026, 32 GB DDR4 memory, NVIDIA GeForce RTX 2080 utilizing\\nGPU parallelization. As part of data augmentation, along with traditional methods such\\nas photometric distortion and geometric distortions, additional data augmentation strategies\\nincluding MixUp (Zhang et al., 2017b), CutMix (Yun et al., 2019) and Mosaic (Bochkovskiy\\net al., 2020) have been combined which help improves the performance of DenseSPH-YOLOv5.\\nIn addition, Mosaic augmentation can significantly enrich the background information. Since\\nsuch a method increases the batch size, therefor, batch normalization has been followed. Unless\\notherwise stated, a batch size set to 16 with a total number of training steps has been kept as\\n80. The initial learning rate has been set to 0.01 with SGD optimizer. The training dataset has\\nbeen trained to utilize the available pre-trained weights-file (Lin et al., 2014).\\n4.2 Optimization of network performance:\\nAt first, we conduct extensive experiments to select proper backbone-neck combination modules\\nto optimize the performance of the proposed DenseSPH-YOLOv5 model in terms of both\\ndetection accuracy and speed. For different combinations of backbone-neck configurations,\\ndetection accuracy in terms of parameters AP, AP 50, AP 75, AP S, AP M, and AP Las well as\\ndetection speed (in FPS) has been reported in Table. 2. From the Table. 2, one can see\\nthe introducing Additional Detection Head (ADH) improves performance by employing an\\nadditional feature fusion mechanism. Notably, there are 2.3% increases in AP 50; and 6.1%\\nincreases in AP Lcompared to the baseline original YOLOv5. Furthermore, implementation\\nof DenseNet with ADH enhances the performance further, in particular, for relatively small\\nobject detection with 3.6 % increase in AP Las shown in Fig. 7-(a). However, this results in a\\nsignification reduction in detection speed (from 69.1 FPS to 55.2 FPS). With the introduction\\nof the CBAM in the neck part of the detection model, the detection performance improves\\nfurther compared to DensNet+ADH configuration for relatively medium and large object sizes\\nwith 3.7% and 3.6% increase in AP Mand AP L, respectively while slightly compromising the\\ndetection speed. Thus, the implementation of CBAM has been proven to be an effective 22\\nTable 2: Performance of various residual and dense block combinations in DenseSPH-YOLOv5\\narchitecture for anchors size of 416 \\u0002416.\\nBackbone\\nadd-inNeck\\nadd-inAP AP 50AP 75APSAPMAPLFPS\\n(YOLOv5) - - 69.2 71.7 83.2 61.2 78.3 81.4 70.2\\n- ADH 70.4 74.1 85.9 67.3 79.7 84.7 69.1\\nDenseNet ADH 74.5 79.7 88.5 70.9 81.2 85.1 55.2\\nDenseNet ADH+CBAM 79.1 83.6 90.2 75.1 84.9 88.7 56.4\\nDenseNet ADH+CBAM+STR 81.4 85.2 91.4 78.2 86.5 89.5 62.4\\nstrategy to retain semantic and delicate spatial information from spatial and channel attention\\nmechanisms that enhance the overall performance of the model. Finally, the best performance\\nhas been achieved when both CBAM and STR respectively have been integrated into the neck\\nand detection head which demonstrate significant improvements in the accuracy parameter, in\\nparticular, AP, AP 50, AP 75, AP Mand AP Lincrease by 12.2%, 13.5%, 8.2%, 8.3%, and 8.1%,\\nrespectively compared to original YOLOv5 configuration. Moreover, we have also observed\\nimprovement in detection speed with the introduction of STR in the detection head by reducing\\nthe computational complexity compared to only ADH+CBAM configuration in the neck as\\nshown in Fig. 7-(a). Therefore, a such configuration in DenseSPH-YOLOv5 provides the\\noptimal performance in terms of detection accuracy and speed for the road-damaged data\\nset considered herein. In summary, together with proper activation function and improved\\nbackbone-neck combination in DenseSPH-YOLOv5 provides an efficient high-performance\\nmodel for damage detection in complex scenarios.\\n4.3 State-of-the-art comparison:\\nIn this section, the detection performance of the proposed DenseSPH-YOLOv5 has been\\ncompared with some of the existing state-of-the-art detection models (Zhao et al., 2019b). For\\nthe performance comparison, we have considered Faster R-CNN (Ren et al., 2016), RetinaNet\\n(Lin et al., 2017b), SSD (Liu et al., 2016), YOLOv4 (Bochkovskiy et al., 2020), Dense-YOLOv4 23\\nFigure 7: Comparison bar chart of precision parameters and detection speed for (a) different\\ncombinations of backbone and neck of the detection model; (b) different state-of-the-art models\\nand DenseSPH-YOLOv5. 24\\nTable 3: Comparison of different performance parameters including P, R, F1, mAP, and\\ndetection speed (in FPS) between DenseSPH-YOLOv5 and other SOAT models where bold\\nhighlights the best performance values.\\nModel P (%) R (%) F1-score (%) mAP (%) Dect. time (ms) FPS\\nFaster R-CNN 52.32 45.39 48.60 52.17 20.50 48.7\\nRetinaNet 59.11 55.67 51.35 54.11 18.87 52.7\\nSSD 62.13 57.19 59.56 60.52 17.92 55.8\\nYOLOv4 73.22 63.35 67.93 67.13 15.31 65.3\\nYOLOv5 79.17 68.47 73.43 71.13 14.24 70.2\\nTPH-YOLOv5 82.37 70.51 74.52 77.62 22.47 44.5\\nDense-YOLOv4 86.75 73.75 78.28 81.87 19.30 51.8\\nDenseSPH-YOLOv5 89.51 76.25 81.18 85.25 16.02 62.4\\n(Roy and Bhaduri, 2022), YOLOv5 (Jocher et al., 2021), and TPH-YOLOv5 (Zhu et al., 2021)\\nthat are trained in RDD-2018 datset. Comparison of different performance parameters including\\nP, R, F1-score, mAP, and detection speed obtained from these models have been presented\\nin Table 3. The comparison reveals that the accuracy values obtained from Faster R-CNN,\\nRetinaNet, and SSD are quite inferior compared to YOLO variants as visually illustrated in\\nthe bar-chart plot in Fig. 7-(b). Between YOLOv4 and YOLOv5, YOLOv5 demonstrated\\nbetter performance with a 5 :95% increase in P and 4 :01% increase in mAP, respectively.\\nWe observe that the performance of Dense-YOLOv4 is superior to the TPH-YOLOv5 with\\n4:38%, 3:24%, 3:76%, and 4 :25% increase in P, R, F1, and mAP, respectively. However, the\\nproposed DenseSPH-YOLOv5 yields the best performance reaching the values of 89 :51%, 76:25%,\\n81:18%, and 85 :25% in P, R, F1, and mAP, respectively as shown in Fig.7- (b). Moreover,\\nDenseSPH-YOLOv5 provides a superior real-time detection speed of 62.4 FPS which is 28 :68%\\nand 16:98% higher than TPH-YOLOv5 and Dense-YOLOv4 models, respectively. In summary,\\nDenseSPH-YOLOv5 outshines some of the best detection models in terms of both detection\\naccuracy and speed illustrating its suitability for automated high-performance damage detection\\nmodels. 25\\n4.4 Comparison with existing state-of-the-art damage detection models:\\nIn addition, the performance of the DenseSPH-YOLOv5 has been compared with several\\nexisting state-of-the-art road damage detection models evaluated in RDD-2018 datasets. As\\nshown in Table 4, we compared the performance from various DL models including SSD\\nInception v2 (Maeda et al., 2018), SSD MobileNet (Maeda et al., 2018), YOLO (Alfarrarjeh\\net al., 2018), Faster R-CNN (Kluger et al., 2018), Faster R-CNN with ResNet-152 (Wang et al.,\\n2018a), Ensemble models with Faster R-CNN and SSD (Wang et al., 2018b), and RetinaNet\\n(Angulo et al., 2019) with our Dense-YOLOv4 and DenseSPH-YOLOv5 models. Note, in\\n(Angulo et al., 2019) additional images were included in the RDD-2018 dataset to improve the\\ndetection performance. From the direct comparison, two current state-of-the-art models Faster\\nR-CNN with ResNet-152 (Wang et al., 2018a) and Ensemble models (Wang et al., 2018b) have\\nreached the F1 value of 62.55% that is 2.63% improvement over SSD MobileNet (Maeda et al.,\\n2018). While RetinaNet (Angulo et al., 2019) illustrated significant performance improvement,\\nDense-YOLOv4 performs better with F1 of 78.28% in the original RDD-2018 dataset. Relative\\nto the aforementioned models, our proposed model has achieved the best F1 value of 81.18 %\\namong current state-of-the-art damage detection models. In terms of other precision metrics,\\nthere are 3.64% and 10.51% improvements in P and R compared to RetinaNet (Angulo et al.,\\n2019), respectively. Comparing detection speed, DenseSPH-YOLOv5 provides competitive\\nperformance compared to SSD Inception v2 (Maeda et al., 2018) elucidates its superiority in\\nreal-time damage detection.\\n4.5 Overall performance of DenseSPH-YOLOv5:\\nFrom section 4.3, we observed that TPH-YOLOv5, Dense-YOLOv4, and DenseSPH-YOLOv5\\nprovide better performance compared to other SOAT models. Therefore, these three models are\\nclosely compared in terms of mAP, F1, IoU, final loss, and average detection time as shown in\\nTable 5. The proposed DenseSPH-YOLOv5 has achieved the highest average IoU value of 0.803\\nindicating superior BB accuracy during target detection compared to the other two models.\\nSimilarly, it has also illustrated better detection performance and accuracy by achieving the 26\\nTable 4: Comparison of different performance parameters between DenseSPH-YOLOv5 and\\nother SOAT road damage detection models evaluated in RDD-2018 dataset where bold highlights\\nthe best performance values.\\nReference Method Performance Det. Speed\\n(FPS)\\nMaeda et al. (2018) SSD Inception v2 F-1: 52.61; P: 81.10; R: 38.97 63.1\\nMaeda et al. (2018) SSD MobileNet F-1: 59.92; P: 81.11; R: 47.52 30.6\\nAlfarrarjeh et al. (2018) YOLO F-1: 62.00; P: -; R: - -\\nKluger et al. (2018) Faster R-CNN F-1: 61.00; P: -; R: - -\\nWang et al. (2018a) Faster R-CNN\\n(ResNet-152)F-1: 62.55; P: -; R: - -\\nWang et al. (2018b) Ensemble\\n(Faster R-CNN+ SSD)F-1: 62.55; P: -; R: - -\\nAngulo et al. (2019)\\u0003RetinaNet F-1: 74.97; P: 85.87; R: 65.75 -\\n(Roy and Bhaduri, 2022) Dense-YOLOv4 F-1: 78.28; P: 86.75; R: 73.75 51.8\\nOurs (current study) DenseSPH-YOLOv5 F-1: 81.18; P: 89.51; R: 76.25 62.4 27\\nTable 5: Overall performance comparison between TPH-YOLOv5, Dense-YOLOv4, and\\nDenseSPH-YOLOv5.\\nDetection model IoU F1 mAP Validation loss Detection\\ntime (ms)Detection\\nspeed (FPS)\\nTPH-YOLOv5 0.740 0.745 0.776 18.07 22.47 44.5\\nDense-YOLOv4 0.781 0.782 0.819 10.13 19.30 51.8\\nDenseSPH-YOLOv5 0.803 0.811 0.852 7.18 16.02 62.4\\nFigure 8: Comparison of (a) P-R curves; (b) loss evolution curves between TPH-YOLOv5,\\nDense-YOLOv4, and DenseSPH-YOLOv5.\\nhighest F1 and mAP values of 0.811 and 0.852 which are 6 :61% and 7:63% improvements over\\nthe TPH-YOLOv5, respectively. Furthermore, the detection speed of 62.4 FPS obtained from\\nDenseSPH-YOLOv5 was found to be higher than both TPH-YOLOv5 and Dense-YOLOv4.\\nThus, it can provide real-time road damage detection with better accuracy compared to the\\nother two state-of-the-art models. In addition, the comparison of P-R curves between the three\\nmodels have been depicted in Fig 8-(a). From the comparison of the P-R curves, one can see\\nthat DenseSPH-YOLOv5 attains a better P value for a particular R. It achieved the highest\\narea under the P-R curve indicating superior detection performance compared to TPH-YOLOv5\\nand Dense-YOLOv4. Next, we compare the loss evolution curves as shown in Fig 8-(b). In the\\ninitial phase, after exhibiting several cycles of fluctuation, the loss in the DenseSPH-YOLOv5 28\\nTable 6: Comparison of detection results for individual classes between TPH-YOLOv5,\\nDense-YOLOv4, and DenseSPH-YOLOv5\\nModel Class !D00 D01 D10 D11 D20 D40 D43 D44 Avg.\\nTPH-YOLOv5P 0.87 0.62 0.84 0.87 0.84 0.87 0.87 0.81 0.82\\nR 0.61 0.89 0.49 0.51 0.71 0.74 0.81 0.88 0.71\\nF1 0.72 0.73 0.62 0.64 0.77 0.79 0.83 0.84 0.75\\nmAP 0.81 0.79 0.78 0.82 0.73 0.78 0.76 0.74 0.77\\nDense-YOLOv4P 0.89 0.69 0.91 0.92 0.89 0.91 0.89 0.84 0.87\\nR 0.65 0.93 0.52 0.51 0.78 0.78 0.86 0.87 0.74\\nF1 0.75 0.79 0.66 0.65 0.83 0.84 0.87 0.85 0.78\\nmAP 0.82 0.81 0.84 0.89 0.77 0.82 0.81 0.79 0.82\\nDenseSPH-YOLOv5P 0.92 0.74 0.93 0.94 0.93 0.92 0.91 0.87 0.89\\nR 0.68 0.95 0.58 0.54 0.81 0.79 0.88 0.87 0.76\\nF1 0.78 0.83 0.71 0.68 0.86 0.85 0.89 0.87 0.81\\nmAP 0.84 0.89 0.81 0.91 0.82 0.85 0.85 0.85 0.85\\nmodel tends to saturate after approximately 50 training steps with a final loss value of 7.18.\\nWhereas, the other two models exhibit higher fluctuation in loss evolution and yield higher final\\nloss value. Evidently, the proposed TPH-YOLOv5 is easier to train with faster convergence\\ncharacteristics demonstrating its efficacy from the computational point of view.\\nTo further gain insight into the performances of these models, a comparison of detection\\nresults containing P, R, mAP, and F-1 values from each individual class between TPH-YOLOv5,\\nDense-YOLOv4, and DenseSPH-YOLOv5 has been presented in Table 6. DenseSPH-YOLOv5\\nhas illustrated significant improvement in P and R values for various classes, in particular,\\nfor detecting longitudianl and lateral linear cracks, alligator cracks, and potholes. Thus,\\nDenseSPH-YOLOv5 efficiently maximizes the TP value while simultaneously reducing FP and\\nFN values for all classes. The proposed model improves 2 :01% in P and 2 :11% in R compared\\nto Dense-YOLOv4. From the overall comparison, we can conclude that DenseSPH-YOLOv5\\ndemonstrated the best performance in detecting various damage classes outperforming the other\\ntwo state-of-the-art models in terms of precision and accuracy.\\n4.6 Detection of various damage classes:\\nIn this section, we have demonstrated the detection results obtained from DenseSPH-YOLOv5\\nfor eight different damage classes and compared them with TPH-YOLOv5 and Dense-YOLOv4. 29\\nFigure 9: Comparison of various damage detection results from (a-e) TPH-YOLOv5; (f-j)\\nDense-YOLOv4; (k-p) DenseSPH-YOLOv5. Detailed detection results with average confidence\\nindexes have been shown in Table 7.\\nTable 7: Detailed detection results from TPH-YOLOv5, Dense-YOLOv4, and\\nDenseSPH-YOLOv5 for different damage classes as shown in Figs. 9- 10.\\nFigs. No Model Detc. False\\/Undetc. Avg. confidence Score\\n9 (a-e) TPH-YOLOv5 13 8 0.71\\n9 (f-j) Dense-YOLOv4 16 5 0.75\\n9 (k-p) DenseSPH-YOLOv5 20 1 0.83\\n10 (a-e) TPH-YOLOv5 13 7 0.67\\n10 (f-j) Dense-YOLOv4 14 6 0.72\\n10 (k-p) DenseSPH-YOLOv5 18 2 0.79 30\\nFigure 10: Comparison of various damage detection results from (a-e) TPH-YOLOv5; (f-j)\\nDense-YOLOv4; (k-p) DenseSPH-YOLOv5. Detailed detection results with average confidence\\nscores have been shown in Table 7.\\nThe visual representations of the detection results have been presented with confined bounding\\nboxes with class identifiers (see Table 1) as shown in Figs. 9- 10. Corresponding detailed\\ndetection results consisting of the number of detected and undetected target classes with\\naverage confidence scores have been reported in Table. 7. From the overall comparison, it can\\nbe concluded that the proposed model shows its efficacy by preciously detecting the target\\nobjects with high average confidence index values. At the same time, it minimizes the false\\nand missed-detection results compared to both TPH-YOLOv5 and Dense-YOLOv4 models.\\nParticularly, when the multiple target objects have a significant degree of overlap between\\nthem, the bounding box prediction from the proposed DenseSPH-YOLOv5 is quite accurate in\\ndetecting each target object as illustrated in Figs. 9-(l, n) and Figs. 10-(k, n). 31\\nFigure 11: Comparison of damage detection results under various challenging scenarios from\\n(a-e) TPH-YOLOv5; (f-j) Dense-YOLOv4; (k-p) DenseSPH-YOLOv5. Detailed detection results\\nwith average confidence indexes have been shown in Table 6.\\n4.7 Detection under challenging scenarios:\\nIn this section, we have extended the detection under some challenging scenarios as shown\\nin Fig. 11 to demonstrate the efficacy of the proposed DenseSPH-YOLOv5. The detailed\\ndetection results that consist of the total number of detected and undetected target classes with\\naverage confidence scores have been reported in Table. 8. Here we consider various complex\\nbackgrounds and challenging environments where the target class is predominately difficult\\nto detect due to the presence of shadows. Additionally, there is some degree of occlusion\\nof overlapping bounding boxes between target classes. It is a challenging task to detect\\ntarget objects individually for an object detection model. Thus, superior feature extraction is\\ncritical to detect various target classes efficiently and correctly. In such cases, the proposed\\nDenseSPH-YOLOv5 has demonstrated its superiority in terms of boundary box precision with 32\\nTable 8: Detailed detection results from TPH-YOLOv5, Dense-YOLOv4, and\\nDenseSPH-YOLOv5 for different damage classes as shown in Fig. 11.\\nFigs. No Model Detc. False\\/Undetc. Avg. confidence Score\\n11 (a)-(e) TPH-YOLOv5 13 10 0.68\\n11 (f)-(j) Dense-YOLOv4 15 8 0.69\\n11 (k)-(p) DenseSPH-YOLOv5 19 4 0.77\\nTable 9: Detection result comparison from the DenseSPH-YOLOv5 for original RGB, greyscale,\\nlow resolution, and various brightness intensity (75 % and 50 %) images as shown in Fig. 12.\\nFigs. No Original Grey scale Low resolution Brightness-75 % Brightness-50 %\\nDet. Undet. Det. Undet. Det. Undet. Det. Undet. Det. Undet.\\nFigs. 12-(a-e) 6 0 5 1 4 2 6 0 5 1\\nFigs. 12-(f-j) 3 1 4 0 2 2 3 1 3 1\\nFigs. 12-(k-p) 5 0 5 0 2 3 5 0 3 2\\na higher average confidence score compared to the other two models. Notably, it significantly\\nreduces the false or missed detection (10 to 4 compared to TPH-YOLOv5) and consequently,\\nthe confidence scores have been improved which is evident from Table 8. Overall, based on the\\ndetection results for the various damage classes, the proposed DenseSPH-YOLOv5 illustrates\\nsuperior detection ability, in particular, in detecting the presence of shadows compared to\\nTPH-YOLOv5 and Dense-YOLOv4 models.\\n4.8 Detection under Greyscale, low resolution, and different illumination intensities:\\nIn this section, the detection accuracy of the DenseSPH-YOLOv5 has been further tested with\\ngrayscale and pixelated low-resolution (100 \\u0002100) images as shown in Fig. 12. Additionally,\\nprediction results under different illumination intensities (i.e., brightness-75% and brightness-\\n50%) have been studied and compared with the detection result for original RGB images. In\\nTable 9, detailed detection results with confidence scores have been reported. For grayscale\\nimages, the proposed model has demonstrated impressive performance with accurate bounding\\nbox prediction with a better bounding box confidence score as shown in Figs. 12-(b, g, 33\\nFigure 12: Detection results for (a, f, k) original RGB images; (b, g, l) corresponding grey\\nscale images; (c, h, m) low resolution images; (d, i, n) 75 % brightness intensity; (e, j, p) 50 %\\nbrightness intensity from the DenseSPH-YOLOv5. Detailed detection results have been shown\\nin Table 6. 34\\nl). For pixelated low-resolution images, the performance of DenseSPH-YOLOv5 has been\\nslightly diminished, however, it can still detect most of the target class with reasonable\\naccuracy. Similarly, the proposed model demonstrates superior detection results under different\\nillumination intensities as demonstrated in Figs. 12-(d, i, n) and Figs. 12-(e, j, p). It can be\\nconcluded from the test results that the proposed model is more adaptive in more challenging\\nenvironments compared to TPH-YOLOv5 and Dense-YOLOv4 models.\\n5. Discussion:\\nFrom the overall detection result, it is evident that DenseSPH-YOLOv5 has higher adaptability\\nand better capability in localizing various damage classes in a complex environment and\\nchallenging conditions compared to current state-of-the-art models. Additionally, it has\\ndemonstrated higher accuracy in bounding box detection and therefore can effectively avoid\\nthe problem of false and missed detection over other detection networks which justifies the\\nusefulness of the proposed model in practical on-field damage detection tasks. Future work\\ncan be geared towards further optimization of detection accuracy as well as detection speed for\\nportable on-field damage detection in a mobile computing platform. The current model can be\\nfurther improved by incorporating some state-of-the-art algorithms such as YOLO-X (Ge et al.,\\n2021), YOLOv7 (Wang et al., 2022), etc. Additionally, it can be assembled with a drone video\\nsurveillance system for real-time accurate damage detection. Furthermore, overlapping and\\noccultation can lead to the wrong prediction which can be further improved by considering\\nshortening the feature stride for better localization and bounding box prediction. In order to\\navoid overlapping boxes, it can be useful to switch from bounding boxes to points (Ribera\\net al., 2019) or masks (Xu et al., 2020). Moreover, one of the potential applications can be\\nassembling object detection framework with semantic segmentation methods such as Mask\\nR-CNN (Bharati and Pramanik, 2020), U-Net (Esser et al., 2018) to extract morphological\\ninformation of various damage classes. Nonetheless, the current work elucidates the possibility\\nof applying cutting-edge DL-based computer vision methodology for multiclass automated\\ndamage detection processes for commercial applications. 35\\n6. Conclusions :\\nSummarizing, in the present work, we have developed an efficient and robust object localization\\nmodel DenseSPH-YOLOv5 based on an improved version of the YOLOv5 network for accurate\\nclassification and localization of damage detection. The proposed DenseSPH-YOLOv5 has\\nbeen employed to detect distinct eight different road damage classes that provide superior and\\naccurate detection under various complex and challenging environments. Evaluated on the\\nRDD-2018 dataset, it has been found that at a detection rate of 62.40 FPS, DenseSPH-YOLOv5\\nhas achieved mean-average precision (mAP), F1, and precision (P) values of 85 :25%, 81:18%,\\nand 89:51%, respectively outperforms existing state-of-the-art damage detection models in\\nterms of both classification accuracy and localized bounding box prediction in detecting all\\ndamage classes. The present work effectively addresses the shortcoming of existing DL-based\\ndamage detection models and illustrates its superior potential in real-time in-field applications.\\nIn short, current work constitutes a step toward a fully automated and efficient DL-based\\ncomputer vision methodology for multi-class damage detection processes.\\nAcknowledgements: The support of the Aeronautical Research and Development Board\\n(Grant No. DARO\\/08\\/1051450\\/M\\/I) is gratefully acknowledged.\\nConflict of interest: The authors declare that they have no known competing financial\\ninterests or personal relationships that could have appeared to influence the work reported in\\nthis paper. 36\\nReferences\\nAgarwal, S. and Singh, D. (2015). An adaptive statistical approach for non-destructive underline\\ncrack detection of ceramic tiles using millimeter wave imaging radar for industrial application.\\nIEEE Sensors Journal , 15(12):7036--7044.\\nAlfarrarjeh, A., Trivedi, D., Kim, S. H., and Shahabi, C. (2018). A deep learning approach for\\nroad damage detection from smartphone images. In 2018 IEEE International Conference on\\nBig Data (Big Data) , pages 5201--5204. IEEE.\\nAnand, S., Gupta, S., Darbari, V., and Kohli, S. (2018). Crack-pot: Autonomous road crack\\nand pothole detection. In 2018 Digital Image Computing: Techniques and Applications\\n(DICTA) , pages 1--6. IEEE.\\nAngulo, A., Vega-Fern\\u0013 andez, J. A., Aguilar-Lobo, L. M., Natraj, S., and Ochoa-Ruiz, G. (2019).\\nRoad damage detection acquisition system based on deep neural networks for physical asset\\nmanagement. In Mexican International Conference on Artificial Intelligence , pages 3--14.\\nSpringer.\\nAnitha, M., Hemalatha, R., and Radha, S. (2021). A survey on crack detection algorithms for\\nconcrete structures. Advances in Smart System Technologies , pages 639--654.\\nArya, D., Maeda, H., Ghosh, S. K., Toshniwal, D., Mraz, A., Kashiyama, T., and Sekimoto, Y.\\n(2021a). Deep learning-based road damage detection and classification for multiple countries.\\nAutomation in Construction , 132:103935.\\nArya, D., Maeda, H., Ghosh, S. K., Toshniwal, D., Omata, H., Kashiyama, T., and Sekimoto, Y.\\n(2020). Global road damage detection: State-of-the-art solutions. In 2020 IEEE International\\nConference on Big Data (Big Data) , pages 5533--5539. IEEE.\\nArya, D., Maeda, H., Ghosh, S. K., Toshniwal, D., and Sekimoto, Y. (2021b). Rdd2020: An\\nannotated image dataset for automatic road damage detection using deep learning. Data in\\nbrief, 36:107133.\\nAzimi, M., Eslamlou, A. D., and Pekcan, G. (2020). Data-driven structural health monitoring\\nand damage detection through deep learning: State-of-the-art review. Sensors , 20(10):2778. 37\\nBang, S., Park, S., Kim, H., Yoon, Y., and Kim, H. (2018). A deep residual network with\\ntransfer learning for pixel-level road crack detection. Network , 93(84.90):89--03.\\nBharati, P. and Pramanik, A. (2020). Deep learning techniques|r-cnn to mask r-cnn: a survey.\\nComputational Intelligence in Pattern Recognition , pages 657--668.\\nBi\\u0018 cici, S. and Zeybek, M. (2021). An approach for the automated extraction of road surface\\ndistress from a uav-derived point cloud. Automation in Construction , 122:103475.\\nBochkovskiy, A., Wang, C.-Y., and Liao, H.-Y. M. (2020). Yolov4: Optimal speed and accuracy\\nof object detection.\\nBose, R. and Roy, A. (2022). Accurate deep learning sub-grid scale models for large eddy\\nsimulations. Bulletin of the American Physical Society .\\nCao, M.-T., Tran, Q.-V., Nguyen, N.-M., and Chang, K.-T. (2020). Survey on performance of\\ndeep learning models for detecting road damages using multiple dashcam image resources.\\nAdvanced Engineering Informatics , 46:101182.\\nChandio, A., Gui, G., Kumar, T., Ullah, I., Ranjbarzadeh, R., Roy, A. M., Hussain, A., and\\nShen, Y. (2022). Precise single-stage detector. arXiv preprint arXiv:2210.04252 .\\nChen, J. and Cho, Y. K. (2022). Crackembed: Point feature embedding for crack segmentation\\nfrom disaster site point clouds with anomaly detection. Advanced Engineering Informatics ,\\n52:101550.\\nChen, Q., Huang, Y., Sun, H., and Huang, W. (2021). Pavement crack detection using hessian\\nstructure propagation. Advanced Engineering Informatics , 49:101303.\\nChow, J. K., Su, Z., Wu, J., Tan, P. S., Mao, X., and Wang, Y.-H. (2020). Anomaly detection\\nof defects on concrete structures with the convolutional autoencoder. Advanced Engineering\\nInformatics , 45:101105.\\nDavis, J. and Goadrich, M. (2006). The relationship between precision-recall and roc curves.\\nInProceedings of the 23rd international conference on Machine learning , pages 233--240. 38\\nDong, J., Meng, W., Liu, Y., and Ti, J. (2021). A framework of pavement management system\\nbased on iot and big data. Advanced Engineering Informatics , 47:101226.\\nDorafshan, S., Thomas, R. J., and Maguire, M. (2018). Sdnet2018: An annotated image dataset\\nfor non-contact concrete crack detection using deep convolutional neural networks. Data in\\nbrief, 21:1664--1668.\\nDosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani,\\nM., Minderer, M., Heigold, G., Gelly, S., et al. (2020). An image is worth 16x16 words:\\nTransformers for image recognition at scale. arXiv preprint arXiv:2010.11929 .\\nDu, Y., Pan, N., Xu, Z., Deng, F., Shen, Y., and Kang, H. (2021). Pavement distress detection\\nand classification based on yolo network. International Journal of Pavement Engineering ,\\n22(13):1659--1672.\\nEisenbach, M., Stricker, R., Seichter, D., Amende, K., Debes, K., Sesselmann, M., Ebersbach,\\nD., Stoeckert, U., and Gross, H.-M. (2017). How to get pavement distress detection ready\\nfor deep learning? a systematic approach. In 2017 international joint conference on neural\\nnetworks (IJCNN) , pages 2039--2047. IEEE.\\nEsser, P., Sutter, E., and Ommer, B. (2018). A variational u-net for conditional appearance and\\nshape generation. In Proceedings of the IEEE conference on computer vision and pattern\\nrecognition , pages 8857--8866.\\nFan, Z., Wu, Y., Lu, J., and Li, W. (2018). Automatic pavement crack detection based on\\nstructured prediction with the convolutional neural network. arXiv preprint arXiv:1802.02208 .\\nFang, F., Li, L., Gu, Y., Zhu, H., and Lim, J.-H. (2020). A novel hybrid approach for crack\\ndetection. Pattern Recognition , 107:107474.\\nFerri, C., Hern\\u0013 andez-Orallo, J., and Modroiu, R. (2009). An experimental comparison of\\nperformance measures for classification. Pattern recognition letters , 30(1):27--38.\\nGe, Z., Liu, S., Wang, F., Li, Z., and Sun, J. (2021). Yolox: Exceeding yolo series in 2021.\\narXiv preprint arXiv:2107.08430 . 39\\nGirshick, R. (2015). Fast r-cnn in proceedings of the ieee international conference on computer\\nvision (pp. 1440--1448). Piscataway, NJ: IEEE.[Google Scholar] .\\nGong, H., Mu, T., Li, Q., Dai, H., Li, C., He, Z., Wang, W., Han, F., Tuniyazi, A., Li, H.,\\net al. (2022). Swin-transformer-enabled yolov5 with attention mechanism for small object\\ndetection on satellite images. Remote Sensing , 14(12):2861.\\nGopalakrishnan, K. (2018). Deep learning in data-driven pavement image analysis and\\nautomated distress detection: A review. Data , 3(3):28.\\nGopalakrishnan, K., Khaitan, S. K., Choudhary, A., and Agrawal, A. (2017). Deep convolutional\\nneural networks with transfer learning for computer vision-based data-driven pavement\\ndistress detection. Construction and building materials , 157:322--330.\\nGuan, J., Yang, X., Ding, L., Cheng, X., Lee, V. C., and Jin, C. (2021). Automated pixel-level\\npavement distress detection based on stereo vision and deep learning. Automation in\\nConstruction , 129:103788.\\nGuo, G. and Zhang, Z. (2022). Road damage detection algorithm for improved yolov5. Scientific\\nreports , 12(1):1--12.\\nHac\\u0010efendio\\u0015 glu, K. and Ba\\u0018 sa\\u0015 ga, H. B. (2022). Concrete road crack detection using deep\\nlearning-based faster r-cnn method. Iranian Journal of Science and Technology, Transactions\\nof Civil Engineering , 46(2):1621--1633.\\nHan, J., Zhang, D., Cheng, G., Liu, N., and Xu, D. (2018). Advanced deep-learning techniques\\nfor salient and category-specific object detection: a survey. IEEE Signal Processing Magazine ,\\n35(1):84--100.\\nHanzaei, S. H., Afshar, A., and Barazandeh, F. (2017). Automatic detection and classification\\nof the ceramic tiles' surface defects. Pattern Recognition , 66:174--189.\\nHartmann, T. and Trappey, A. (2020). Advanced engineering informatics-philosophical\\nand methodological foundations with examples from civil and construction engineering.\\nDevelopments in the built environment , 4:100020. 40\\nHe, K., Gkioxari, G., Doll\\u0013 ar, P., and Girshick, R. (2017). Mask r-cnn. in proceedings of the\\nieee international conference on computer vision.\\nHe, K., Zhang, X., Ren, S., and Sun, J. (2015). Spatial pyramid pooling in deep convolutional\\nnetworks for visual recognition. IEEE transactions on pattern analysis and machine\\nintelligence , 37(9):1904--1916.\\nHendrycks, D. and Gimpel, K. (2016). Gaussian error linear units (gelus). arXiv preprint\\narXiv:1606.08415 .\\nHuang, G., Liu, Z., Van Der Maaten, L., and Weinberger, K. Q. (2017). Densely connected\\nconvolutional networks. In Proceedings of the IEEE conference on computer vision and\\npattern recognition , pages 4700--4708.\\nJamil, S., Abbas, M. S., and Roy, A. M. (2022). Distinguishing malicious drones using vision\\ntransformer. AI, 3(2):260--273.\\nJamil, S. and Roy, A. M. (2022). Robust pcg-based vhd detection model using\\nd-cnns, nature-inspired algorithms, and vision transformer. Available at SSRN:\\nhttp:\\/\\/dx.doi.org\\/10.2139\\/ssrn.4316752 , page 41.\\nJamil, S. and Roy, A. M. (2023). An efficient and robust phonocardiography (pcg)-based\\nvalvular heart diseases (vhd) detection framework using vision transformer (vit). Computers\\nin Biology and Medicine , page 106734.\\nJocher, G., Stoken, A., Borovec, J., Chaurasia, A., Changyu, L., Laughing, A., Hogan, A.,\\nHajek, J., Diaconu, L., Marc, Y., et al. (2021). ultralytics\\/yolov5: v5. 0-yolov5-p6 1280\\nmodels aws supervise. ly and youtube integrations. Zenodo , 11.\\nKapela, R., \\u0013Sniata la, P., Turkot, A., Rybarczyk, A., Po_ zarycki, A., Rydzewski, P., Wycza lek,\\nM., and B loch, A. (2015). Asphalt surfaced pavement cracks detection based on histograms\\nof oriented gradients. In 2015 22nd International Conference Mixed Design of Integrated\\nCircuits & Systems (MIXDES) , pages 579--584. IEEE.\\nKaraaslan, E., Bagci, U., and Catbas, F. N. (2021). Attention-guided analysis of infrastructure\\ndamage with semi-supervised deep learning. Automation in Construction , 125:103634. 41\\nKhan, W., Kumar, T., Cheng, Z., Raj, K., Roy, A. M., and Luo, B. (2022a). Sql and\\nnosql databases software architectures performance analysis and assessments--a systematic\\nliterature review. arXiv preprint arXiv:2209.06977 .\\nKhan, W., Raj, K., Kumar, T., Roy, A. M., and Luo, B. (2022b). Introducing urdu digits\\ndataset with demonstration of an efficient and robust noisy decoder-based pseudo example\\ngenerator. Symmetry , 14(10):1976.\\nKluger, F., Reinders, C., Raetz, K., Schelske, P., Wandt, B., Ackermann, H., and Rosenhahn,\\nB. (2018). Region-based cycle-consistent data augmentation for object detection. In 2018\\nIEEE International Conference on Big Data (Big Data) , pages 5205--5211. IEEE.\\nKoch, C., Georgieva, K., Kasireddy, V., Akinci, B., and Fieguth, P. (2015). A review on\\ncomputer vision based defect detection and condition assessment of concrete and asphalt\\ncivil infrastructure. Advanced Engineering Informatics , 29(2):196--210.\\nLeCun, Y., Bengio, Y., and Hinton, G. (2015). Deep learning. nature , 521(7553):436--444.\\nLi, D., Xie, Q., Gong, X., Yu, Z., Xu, J., Sun, Y., and Wang, J. (2021). Automatic\\ndefect detection of metro tunnel surfaces using a vision-based inspection system. Advanced\\nEngineering Informatics , 47:101206.\\nLi, P., Xia, H., Zhou, B., Yan, F., and Guo, R. (2022). A method to improve the accuracy\\nof pavement crack identification by combining a semantic segmentation and edge detection\\nmodel. Applied Sciences , 12(9):4714.\\nLi, S. and Zhao, X. (2021). Pixel-level detection and measurement of concrete crack using\\nfaster region-based convolutional neural network and morphological feature extraction.\\nMeasurement Science and Technology , 32(6):065010.\\nLi, Y., Li, H., and Wang, H. (2018). Pixel-wise crack detection using deep local pattern\\npredictor for robot application. Sensors , 18(9):3042.\\nLin, T.-Y., Goyal, P., Girshick, R., He, K., and Doll\\u0013 ar, P. (2017a). Focal loss for dense object\\ndetection. In Proceedings of the IEEE international conference on computer vision , pages\\n2980--2988. 42\\nLin, T.-Y., Goyal, P., Girshick, R., He, K., and Doll\\u0013 ar, P. (2017b). Focal loss for dense object\\ndetection. In Proceedings of the IEEE international conference on computer vision , pages\\n2980--2988.\\nLin, T.-Y., Maire, M., Belongie, S., Hays, J., Perona, P., Ramanan, D., Doll\\u0013 ar, P., and Zitnick,\\nC. L. (2014). Microsoft coco: Common objects in context. In European conference on\\ncomputer vision , pages 740--755. Springer.\\nLiu, S., Qi, L., Qin, H., Shi, J., and Jia, J. (2018). Path aggregation network for instance\\nsegmentation. In Proceedings of the IEEE conference on computer vision and pattern\\nrecognition , pages 8759--8768.\\nLiu, W., Anguelov, D., Erhan, D., Szegedy, C., Reed, S., Fu, C., and Berg, A. (2016). Ssd:\\nSingle shot multibox detector, \\/uni2016in european conference on computer vision (eccv).\\nLiu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., Zhang, Z., Lin, S., and Guo, B. (2021). Swin\\ntransformer: Hierarchical vision transformer using shifted windows. In Proceedings of the\\nIEEE\\/CVF International Conference on Computer Vision , pages 10012--10022.\\nMaeda, H., Kashiyama, T., Sekimoto, Y., Seto, T., and Omata, H. (2021). Generative\\nadversarial network for road damage detection. Computer-Aided Civil and Infrastructure\\nEngineering , 36(1):47--60.\\nMaeda, H., Sekimoto, Y., Seto, T., Kashiyama, T., and Omata, H. (2018). Road\\ndamage detection and classification using deep neural networks with smartphone images.\\nComputer-Aided Civil and Infrastructure Engineering , 33(12):1127--1141.\\nMajidifard, H., Jin, P., Adu-Gyamfi, Y., and Buttlar, W. G. (2020). Pavement image datasets:\\nA new benchmark dataset to classify and densify pavement distresses. Transportation\\nResearch Record , 2674(2):328--339.\\nMandal, V., Mussah, A. R., and Adu-Gyamfi, Y. (2020). Deep learning frameworks for pavement\\ndistress classification: A comparative analysis. In 2020 IEEE International Conference on\\nBig Data (Big Data) , pages 5577--5583. IEEE.\\nMisra, D. (2020). Mish: A self regularized non-monotonic activation function. 43\\nMohan, A. and Poobal, S. (2018). Crack detection using image processing: A critical review\\nand analysis. Alexandria Engineering Journal , 57(2):787--798.\\nNaddaf-Sh, S., Naddaf-Sh, M.-M., Kashani, A. R., and Zargarzadeh, H. (2020). An efficient\\nand scalable deep learning approach for road damage detection. In 2020 IEEE International\\nConference on Big Data (Big Data) , pages 5602--5608. IEEE.\\nNath, N. D., Cheng, C.-S., and Behzadan, A. H. (2022). Drone mapping of damage information\\nin gps-denied disaster sites. Advanced Engineering Informatics , 51:101450.\\nNhat-Duc, H., Nguyen, Q.-L., and Tran, V.-D. (2018). Automatic recognition of asphalt\\npavement cracks using metaheuristic optimized edge detection algorithms and convolution\\nneural network. Automation in Construction , 94:203--213.\\nNi, F., He, Z., Jiang, S., Wang, W., and Zhang, J. (2022). A generative adversarial learning\\nstrategy for enhanced lightweight crack delineation networks. Advanced Engineering\\nInformatics , 52:101575.\\nOliveira, H. and Correia, P. L. (2009). Automatic road crack segmentation using entropy and\\nimage dynamic thresholding. In 2009 17th European Signal Processing Conference , pages\\n622--626. IEEE.\\nPatra, S., Middya, A. I., and Roy, S. (2021). Potspot: Participatory sensing based monitoring\\nsystem for pothole detection using deep learning. Multimedia Tools and Applications ,\\n80(16):25171--25195.\\nQuintana, M., Torres, J., and Men\\u0013 endez, J. M. (2015). A simplified computer vision system for\\nroad surface inspection and maintenance. IEEE Transactions on Intelligent Transportation\\nSystems , 17(3):608--619.\\nRawat, W. and Wang, Z. (2017). Deep convolutional neural networks for image classification:\\nA comprehensive review. Neural computation , 29(9):2352--2449.\\nRedmon, J., Divvala, S., Girshick, R., and Farhadi, A. (2016). You only look once: Unified,\\nreal-time object detection. In Proceedings of the IEEE conference on computer vision and\\npattern recognition , pages 779--788. 44\\nRedmon, J. and Farhadi, A. (2017). Yolo9000: better, faster, stronger. In Proceedings of the\\nIEEE conference on computer vision and pattern recognition , pages 7263--7271.\\nRedmon, J. and Farhadi, A. (2018). Yolov3: An incremental improvement.\\nRen, S., He, K., Girshick, R., and Sun, J. (2016). Faster r-cnn: towards real-time object\\ndetection with region proposal networks. IEEE transactions on pattern analysis and machine\\nintelligence , 39(6):1137--1149.\\nRezatofighi, H., Tsoi, N., Gwak, J., Sadeghian, A., Reid, I., and Savarese, S. (2019). Generalized\\nintersection over union: A metric and a loss for bounding box regression. In Proceedings of\\nthe IEEE\\/CVF Conference on Computer Vision and Pattern Recognition , pages 658--666.\\nRibera, J., Guera, D., Chen, Y., and Delp, E. J. (2019). Locating objects without bounding\\nboxes. In Proceedings of the IEEE\\/CVF Conference on Computer Vision and Pattern\\nRecognition , pages 6479--6489.\\nRoy, A. M. (2022a). Adaptive transfer learning-based multiscale feature fused deep convolutional\\nneural network for eeg mi multiclassification in brain--computer interface. Engineering\\nApplications of Artificial Intelligence , 116:105347.\\nRoy, A. M. (2022b). An efficient multi-scale CNN model with intrinsic feature integration for\\nmotor imagery EEG subject classification in brain-machine interfaces. Biomedical Signal\\nProcessing and Control , 74:103496.\\nRoy, A. M. (2022c). A multi-scale fusion cnn model based on adaptive transfer learning for\\nmulti-class mi-classification in bci system. BioRxiv .\\nRoy, A. M. and Bhaduri, J. (2021). A deep learning enabled multi-class plant disease detection\\nmodel based on computer vision. AI, 2(3):413--428.\\nRoy, A. M. and Bhaduri, J. (2022). Real-time growth stage detection model for high degree\\nof occultation using densenet-fused YOLOv4. Computers and Electronics in Agriculture ,\\n193:106694.\\nRoy, A. M., Bhaduri, J., Kumar, T., and Raj, K. (2022a). A computer vision-based object\\nlocalization model for endangered wildlife detection. Ecological Economics, Forthcoming . 45\\nRoy, A. M., Bhaduri, J., Kumar, T., and Raj, K. (2022b). Wildect-yolo: An efficient and\\nrobust computer vision-based accurate object localization model for automated endangered\\nwildlife detection. Ecological Informatics , page 101919.\\nRoy, A. M. and Bose, R. (2023a). Deep learning-accelerated computational framework based\\non physics informed neural network for solution of linear elasticity. Neural Networks , 2:2.\\nRoy, A. M. and Bose, R. (2023b). Physics-aware deep learning framework for linear elasticity.\\narXiv preprint arXiv:2302.09668 .\\nRoy, A. M., Bose, R., and Bhaduri, J. (2022c). A fast accurate fine-grain object detection\\nmodel based on YOLOv4 deep neural network. Neural Computing and Applications , pages\\n1--27.\\nRoy, A. M. and Guha, S. (2022). Elastoplastic physics-informed deep learning approach for j2\\nplasticity. Available at SSRN: https:\\/\\/ssrn.com\\/abstract=4332254 , page 48.\\nRoy, A. M. and Guha, S. (2023). A data-driven physics-constrained deep learning computational\\nframework for solving von mises plasticity. Engineering Applications of Artificial Intelligence ,\\n122:106049.\\nShang, H., Sun, C., Liu, J., Chen, X., and Yan, R. (2023). Defect-aware transformer network\\nfor intelligent visual surface defect detection. Advanced Engineering Informatics , 55:101882.\\nSilva, W. R. L. d. and Lucena, D. S. d. (2018). Concrete cracks detection based on deep\\nlearning image classification. In Proceedings , volume 2, page 489. MDPI AG.\\nSingh, A., Raj, K., Kumar, T., Verma, S., and Roy, A. M. (2023a). Deep learning-based\\ncost-effective and responsive robot for autism treatment. Drones , 7(2):81.\\nSingh, A., Ranjbarzadeh, R., Raj, K., Kumar, T., and Roy, A. M. (2023b). Understanding eeg\\nsignals for subject-wise definition of armoni activities. arXiv preprint arXiv:2301.00948 .\\nStricker, R., Eisenbach, M., Sesselmann, M., Debes, K., and Gross, H.-M. (2019). Improving\\nvisual road condition assessment by extensive experiments on the extended gaps dataset. In\\n2019 International Joint Conference on Neural Networks (IJCNN) , pages 1--8. IEEE. 46\\nVaradharajan, S., Jose, S., Sharma, K., Wander, L., and Mertz, C. (2014). Vision for road\\ninspection. In IEEE winter conference on applications of computer vision , pages 115--122.\\nIEEE.\\nVaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser,  L., and\\nPolosukhin, I. (2017). Attention is all you need. Advances in neural information processing\\nsystems , 30.\\nVoulodimos, A., Doulamis, N., Doulamis, A., and Protopapadakis, E. (2018). Deep learning for\\ncomputer vision: A brief review. Computational intelligence and neuroscience , 2018.\\nWang, C.-Y., Bochkovskiy, A., and Liao, H.-Y. M. (2021a). Scaled-yolov4: Scaling cross stage\\npartial network. In Proceedings of the IEEE\\/cvf conference on computer vision and pattern\\nrecognition , pages 13029--13038.\\nWang, C.-Y., Bochkovskiy, A., and Liao, H.-Y. M. (2022). Yolov7: Trainable bag-of-freebies\\nsets new state-of-the-art for real-time object detectors. arXiv preprint arXiv:2207.02696 .\\nWang, C.-Y., Liao, H.-Y. M., Wu, Y.-H., Chen, P.-Y., Hsieh, J.-W., and Yeh, I.-H. (2020).\\nCspnet: A new backbone that can enhance learning capability of cnn. In Proceedings of\\nthe IEEE\\/CVF conference on computer vision and pattern recognition workshops , pages\\n390--391.\\nWang, W., Li, L., and Han, Y. (2021b). Crack detection in shadowed images on gray level\\ndeviations in a moving window and distance deviations between connected components.\\nConstruction and Building Materials , 271:121885.\\nWang, W., Wu, B., Yang, S., and Wang, Z. (2018a). Road damage detection and classification\\nwith faster r-cnn. In 2018 IEEE international conference on big data (Big data) , pages\\n5220--5223. IEEE.\\nWang, Y. J., Ding, M., Kan, S., Zhang, S., and Lu, C. (2018b). Deep proposal and detection\\nnetworks for road damage detection and classification. In 2018 IEEE International Conference\\non Big Data (Big Data) , pages 5224--5227. IEEE. 47\\nWoo, S., Park, J., Lee, J.-Y., and Kweon, I. S. (2018). Cbam: Convolutional block attention\\nmodule. In Proceedings of the European conference on computer vision (ECCV) , pages\\n3--19.\\nXu, B., Wang, W., Falzon, G., Kwan, P., Guo, L., Chen, G., Tait, A., and Schneider, D. (2020).\\nAutomated cattle counting using mask r-cnn in quadcopter vision system. Computers and\\nElectronics in Agriculture , 171:105300.\\nXu, R., Hao, R., and Huang, B. (2022). Efficient surface defect detection using self-supervised\\nlearning strategy and segmentation network. Advanced Engineering Informatics , 52:101566.\\nYun, S., Han, D., Oh, S. J., Chun, S., Choe, J., and Yoo, Y. (2019). Cutmix: Regularization\\nstrategy to train strong classifiers with localizable features. In Proceedings of the IEEE\\/CVF\\ninternational conference on computer vision , pages 6023--6032.\\nZhang, A., Wang, K. C., Li, B., Yang, E., Dai, X., Peng, Y., Fei, Y., Liu, Y., Li, J. Q.,\\nand Chen, C. (2017a). Automated pixel-level pavement crack detection on 3d asphalt\\nsurfaces using a deep-learning network. Computer-Aided Civil and Infrastructure Engineering ,\\n32(10):805--819.\\nZhang, H., Cisse, M., Dauphin, Y. N., and Lopez-Paz, D. (2017b). mixup: Beyond empirical\\nrisk minimization. arXiv preprint arXiv:1710.09412 .\\nZhang, L., Yang, F., Zhang, Y. D., and Zhu, Y. J. (2016). Road crack detection using deep\\nconvolutional neural network. In 2016 IEEE international conference on image processing\\n(ICIP) , pages 3708--3712. IEEE.\\nZhao, H., Qin, G., and Wang, X. (2010). Improvement of canny algorithm based on pavement\\nedge detection. In 2010 3rd International Congress on Image and Signal Processing , volume 2,\\npages 964--967. IEEE.\\nZhao, Z.-Q., Zheng, P., Xu, S.-t., and Wu, X. (2019a). Object detection with deep learning: A\\nreview. IEEE transactions on neural networks and learning systems , 30(11):3212--3232.\\nZhao, Z.-Q., Zheng, P., Xu, S.-t., and Wu, X. (2019b). Object detection with deep learning: A\\nreview. IEEE transactions on neural networks and learning systems , 30(11):3212--3232. 48\\nZheng, Z., Wang, P., Liu, W., Li, J., Ye, R., and Ren, D. (2020). Distance-iou loss: Faster\\nand better learning for bounding box regression. In Proceedings of the AAAI Conference on\\nArtificial Intelligence , volume 34, pages 12993--13000.\\nZhu, X., Lyu, S., Wang, X., and Zhao, Q. (2021). Tph-yolov5: Improved yolov5 based on\\ntransformer prediction head for object detection on drone-captured scenarios. In Proceedings\\nof the IEEE\\/CVF International Conference on Computer Vision , pages 2778--2788.\",\"9\":\" Patch of Invisibility:\\nNaturalistic Black-Box Adversarial Attacks on Object Detectors\\nRaz Lapid\\nDept. of Computer Science, Ben-Gurion University\\nBeer-Sheva 84105, Israel\\nrazla@post.bgu.ac.il\\nand\\nDeepKeep, Tel-Aviv, Israel\\nMoshe Sipper\\nDept. of Computer Science, Ben-Gurion University\\nBeer-Sheva 84105, Israel\\nsipper@bgu.ac.il\\nAbstract\\nAdversarial attacks on deep-learning models have been\\nreceiving increased attention in recent years. Work in this\\narea has mostly focused on gradient-based techniques, so-\\ncalled \\u201cwhite-box\\u201d attacks, wherein the attacker has access\\nto the targeted model\\u2019s internal parameters; such an as-\\nsumption is usually unrealistic in the real world. Some at-\\ntacks additionally use the entire pixel space to fool a given\\nmodel, which is neither practical nor physical (i.e., real-\\nworld). On the contrary, we propose herein a gradient-free\\nmethod that uses the learned image manifold of a pretrained\\ngenerative adversarial network (GAN) to generate natural-\\nistic physical adversarial patches for object detectors. We\\nshow that our proposed method works both digitally and\\nphysically.\\n1. Introduction\\nDeep Neural Networks (DNNs) are increasingly de-\\nployed in safety-critical applications, many involving some\\nform of identifying humans. The risk of facing deceptive\\nimages\\u2014so called adversarial instances\\u2014grows with the\\nuse of DNN models. In order to trick the DNN into cate-\\ngorizing the input image differently than a human would,\\nan adversarial sample is employed. As \\ufb01rst demonstrated\\nin [28], neural networks are susceptible to such adversity.\\nThe methods used to generate adversarial instances\\nbased on minimal input perturbations were enhanced by\\nsubsequent works [4, 20, 25]. Instead of modifying existingimages, [29] generated unconstrained adversarial instances\\nusing generative adversarial networks (GANs). While many\\nmethods concentrate on digital, white-box attacks, where\\nthe attacker has access to the model, it is crucial to investi-\\ngate and comprehend potential realistic physical attacks in\\nmore depth, i.e., attacks that occur in the physical world.\\nFigure 1. An adversarial patch evolved by our novel gradient-free\\nalgorithm, which conceals people from an object detector.\\nAdversarial attacks are a type of cybersecurity threat\\nthat aims to deceive deep learning (DL) systems by inject-\\ning carefully crafted inputs that are designed to fool them.\\nThese attacks exploit vulnerabilities in DL models and take\\nadvantage of their tendency to make mistakes when pro-\\ncessing data. Adversarial attacks can be used for manipula-\\ntion in a wide range of applications, including image recog-\\nnition, natural language processing, autonomous vehicles,\\nand medical diagnosis [20, 30].\\nThe implications of adversarial attacks can be far-arXiv:2303.04238v2  [cs.CV]  9 Mar 2023 reaching, as they can compromise the security and accu-\\nracy of systems that rely on DL. For instance, an adversar-\\nial attack on a vehicle-mounted, image-recognition system\\ncould cause it to misidentify a stop sign as a speed-limit\\nsign [8], potentially causing the vehicle to crash. As DL be-\\ncomes increasingly ubiquitous, the need to mitigate adver-\\nsarial attacks becomes more pressing. Therefore, research\\ninto adversarial attacks and defenses is a rapidly growing\\narea, with researchers working on developing robust and se-\\ncure models that are less susceptible to such attacks.\\nIn this work we focus on fooling surveillance cameras\\n(both indoor and outdoor), because of their ubiquity and\\nsusceptibility to attack, by creating adversarial patches (Fig-\\nure 1). The next section presents previous work. Our\\nmethod is described in Section 3, followed by experimental\\nresults in Section 4. We discuss our \\ufb01ndings in Section 5,\\nending with concluding remarks in Section 6.\\n2. Previous work\\nThis section offers some background and discusses rele-\\nvant literature on adversarial attacks. We begin with digital\\nadversarial attacks in classi\\ufb01cation tasks, followed by digi-\\ntal attacks on object detectors, ending with physical attacks.\\nIn general, attacks can be divided into three categories:\\nwhite box, black box, and gray box.\\n\\u2022White-box threat models assume that the attacker has\\ncomplete knowledge of the system being attacked.\\nThis includes knowledge of the system\\u2019s architecture,\\nimplementation details, and even access to the source\\ncode. In a white-box threat model, the attacker can in-\\nspect the system\\u2019s internals and use this knowledge to\\nlaunch targeted attacks.\\n\\u2022Black-box threat models assume that the attacker has\\nno prior knowledge or access to the system being\\nattacked\\u2014no knowledge of the system\\u2019s architecture\\nor implementation details. This means that the attacker\\nis only able to observe the system\\u2019s behavior from the\\noutside, without any ability to inspect the internals of\\nthe system.\\n\\u2022Gray-box threat models are a mix of both black-box\\nand white-box models, where the attacker has some\\nknowledge of the system being attacked\\u2014but not com-\\nplete knowledge.\\nHerein we focus on black-box threat models because we\\nonly assume access to the models\\u2019 output.\\nConvolutional Neural Networks (CNNs) have been of\\nparticular focus in the domain of adversarial attacks, due\\nto their popularity and success in imaging tasks [9, 19, 20,\\n27, 34].\\nDigital adversarial attacks on classi\\ufb01cation models\\nare a type of attack on learning models that includes addinga minimal perturbation to the input data, with the goal of\\ncausing the model to predict an incorrect class. There has\\nbeen a plethora of studies over the past several years on\\ncreating and enhancing adversarial attacks on classi\\ufb01cation\\nmodels.\\nThe Fast Gradient Sign Method (FGSM) [11], \\ufb01rst pre-\\nsented in 2015, is one of the earliest and most well-known\\nadversarial attacks. The FGSM is a straightforward and ef-\\nfective white-box technique for generating adversarial at-\\ntacks. It works by computing the gradient of the loss func-\\ntion with respect to the input data and perturbing the input\\ndata according to the gradient\\u2019s sign. FGSM has received\\nmuch attention since its introduction, and has been proven\\nquite successful.\\nProjected Gradient Descent (PGD) [23], a white-box at-\\ntack introduced in 2017, executes several small steps along\\nthe gradient\\u2019s direction and projects the perturbed image\\nback onto a sphere centered around the original image.\\nPGD has been shown to outperform FGSM when applied\\nto models with robust defenses against adversarial attacks.\\nThe Carlini and Wagner (C&W) white-box attack [4],\\nwhich was presented in 2016 as an optimization-based\\nmethod for generating adversarial instances, is another\\nwell-known adversarial attack. In C&W, an optimization\\nproblem is solved to identify the smallest perturbation (in\\nterms ofk\\u0001k ) that leads to a misclassi\\ufb01cation. A wide well-known adversarial attack. In C&W, an optimization\\nproblem is solved to identify the smallest perturbation (in\\nterms ofk\\u0001k ) that leads to a misclassi\\ufb01cation. A wide\\nvariety of learning models\\u2014some with advanced defenses\\nagainst adversarial attacks\\u2014have been demonstrated to be\\nsusceptible to C&W.\\nOverall, a signi\\ufb01cant amount of work has been done on\\ngenerating and enhancing adversarial attacks on classi\\ufb01ca-\\ntion models (mostly in a white-box setting), and over the\\npast several years many ef\\ufb01cient methods have been devel-\\noped. In order to protect the integrity and dependability of\\nmodels deployed in an increasing number of applications it\\nis crucial to continually seek out their weaknesses and cre-\\nate strong defenses against adversarial attacks.\\nDigital adversarial attacks on object-detection mod-\\nels, which are used in a variety of systems, from surveil-\\nlance devices to autonomous cars, have become an increas-\\ning concern. These attacks involve intentionally malicious\\ninputs to fool the model into generating bad predictions,\\nand they can have severe effects, including inaccurate ob-\\nject identi\\ufb01cation or failing to detect.\\nOne of the most commonly used object-detection models\\nis You Only Look Once (YOLO) [26], which is based on a\\nsingle convolutional neural network (CNN) that simultane-\\nously predicts the class and location of objects in an image.\\nSeveral studies have shown that YOLO is vulnerable to ad-\\nversarial attacks [13, 15, 22, 29]. For example, targeted per-\\nturbation attacks can be used to modify an input image in a\\nway that causes YOLO to misidentify or fail to detect cer-\\ntain objects. An untargeted attack, on the other hand, seeks to create adversarial examples that cause general disruption\\nto the model\\u2019s performance. Adversarial attacks can have a\\nsigni\\ufb01cant in\\ufb02uence on object-detection models. Most prior\\nresearch regarding such attacks on object-detection mod-\\nels focus on white-box, gradient-based attacks, which is, by\\nand large, a non-real-world scenario.\\nPhysical adversarial attacks pose an even greater threat\\nthan digital ones. Most digital adversarial attacks need ac-\\ncess to the actual model in order to fool it. Further, many at-\\ntacks use global perturbation over pixel space\\u2014e.g., chang-\\ning the sky\\u2019s pixels\\u2014thus rendering it less realistic. Phys-\\nical adversarial attacks can be engendered in a variety of\\nways, including covering an object with paint or some other\\nmaterial [3, 31], applying stickers [8, 18] or camou\\ufb02age\\n[7, 31], or changing the object\\u2019s shape or texture [14, 33].\\nSuch changes are intended to alter how the object appears\\nto the model while maintaining it recognizable to humans.\\nInterestingly, the attack often looks identical to the naked\\neye\\u2014but notto the model (as we shall also show below).\\nPhysical adversarial attacks can have serious effects,\\ne.g., when an autonomous car\\u2019s failed detection results in a\\ncrash. Physical attacks can also be intentionally used to get\\npast security systems or enter restricted areas without au-\\nthorization. Again, most prior research on physical attacks\\nhas been done using gradients, in a white-box setting.\\nContrarily, we create adversarial patches in a black-\\nbox manner , i.e., without the use of gradients, leveraging\\nthelearned image manifold of GANs [10].\\nGenerative Adversarial Networks (GANs) : The qual-\\nity of generative models used for image generation has im-\\nproved signi\\ufb01cantly with the advent of GANs [10] and Dif-\\nfusion Models [12]. Herein, we focus on GANs, due to their\\nrelatively small latent-space dimension and their fast image\\ngeneration. GANs utilize a learnable loss function, where\\na separate discriminator network is used to distinguish be-\\ntween real and fake images, and the image generator net-\\nwork is trained to synthesize images that are indistinguish-\\nable from real ones. Despite their visually appealing results,\\nGANs often face issues such as instability, vanishing gradi-\\nents, and mode collapse. Researchers have suggested many\\ndifferent GANs to address these issues [1, 16, 24]. Herein\\nwe chose to use BigGAN2.\\n3. Method\\nOur objective is to generate physically plausible adver-\\nsarial patches, which are performant and appear realistic \\u2014\\nwithout the use of gradients . An adversarial patch is a spe-\\nci\\ufb01c type of attack, where an image is modi\\ufb01ed by adding\\na small, local pattern that engenders missclassi\\ufb01cation. The\\ngoal of such an attack is to intentionally mislead a model\\ninto making an incorrect prediction or decision.\\nBy \\u201cphysically plausible\\u201d we mean patches that not\\nonly work digitally, but also in the physical world, e.g.,when printed\\u2014and used. The space of possible adversarial\\npatches is huge, and with the aim of reducing it to afford a\\nsuccessful search process, we chose to use pretrained GAN\\ngenerators.\\nGiven a pretrained generator, we seek an input latent\\nvector , corresponding to a generated image that leads the\\nobject detector to err. We leverage the latent space\\u2019s (rela-\\ntively) small dimension, approximating the gradients using\\nan Evolution Strategy algorithm [32], repeatedly updating\\nthe input latent vector by querying the target object detector\\nuntil an appropriate adversarial patch is discovered.\\nFigure 2 depicts a general view of our approach. We\\nsearch for an input latent vector that, given a pretrained gen-\\nerator, corresponds to a generated image that causes the ob-\\nject detector to mistakenly detect that image as a person.\\nAlgorithm 1 shows the pseudocode of the evolution strat-\\negy. In each iteration t, we sample nGaussian noise values,\\n\\u000f1;\\u000f2;:::;\\u000f n(there is one latent vector per noise value).\\nThen, we scale them using \\u001b, add them to the latent vector egy. In each iteration t, we sample nGaussian noise values,\\n\\u000f1;\\u000f2;:::;\\u000f n(there is one latent vector per noise value).\\nThen, we scale them using \\u001b, add them to the latent vector\\nzt, and feed to the model F. The latent vector zt+1is then\\nupdated using the weighted sum of the loss values Fifor\\neach\\u000fi;i2f1;2;:::ng. This step can be done using any\\narbitrary optimizer\\u2014herein we used Adam [17].\\nAlgorithm 1: Evolution Strategy\\nInput: learning rate \\u000b, noise standard deviation \\u001b, initial\\nlatent vectorz0, number of iterations T,\\npopulation size n\\nfort= 1;:::;T do\\nSample\\u000f1;\\u000f2;:::;\\u000fn\\u0018N (0;I)\\nCompute \\ufb01tness Fi=F(zt+\\u001b\\u000fi)fori= 1;:::;n\\nSetzt+1 \\u000b1\\nn\\u001bPn\\ni=1Fi\\u000fi\\nend\\n3.1. Generating adversarial patches\\nPrevious research optimized adversarial patches in pixel\\nspace . We, on the other hand, focus on a GAN generator\\u2019s\\nlatent space . Our resultant adversarial patch will be closer\\nto the manifold of natural pictures and hence appear more\\nrealistic (because GANs learn a latent space that roughly\\napproximates the manifold of natural images). We employ\\na generatorGthat has been previously trained on ImageNet\\nusing a GAN framework, and we search the space of learned\\nnatural image manifold using an evolution strategy.\\nThe evolution strategy algorithm begins by using an\\nisotropic standard multivariate distribution, N(\\u0016=0;\\u0006=\\nId), which parameterizes the evolved population with \\u0016and\\n\\u0006. We begin with an initial latent vector z0. We then ran-\\ndomly sample nnoises\\u000f1;\\u000f2;:::;\\u000f n2Rdusing the stan-\\ndard multivariate distribution, scale them by \\u001band add them\\nto the initial latent vector z0resulting with Z2Rd\\u0002n\\u2014\\nwhich then are fed to the generatior to create the population Figure 2. Naturalistic Black-Box Adversarial Attack: Overview of framework. The system creates patches for object detectors by using\\nthe learned image manifold of a pretrained GAN ( G) on real-world images (as is often the case, we use the GAN\\u2019s generator, but do not\\nneed the discriminator). We use a pretrained classi\\ufb01er ( C) to force the optimizer to \\ufb01nd a patch that resembles a speci\\ufb01c class, the TV\\ncomponent in order to make the images as smooth as possible, and the detector ( D) for the actual detection loss. Ef\\ufb01cient sampling of the\\nGAN images via an iterative evolution strategy ultimately generates the \\ufb01nal patch.\\nof patchesP=G(Z)2Rn\\u00023\\u0002H\\u0002W\\u2013 wherenstands for\\nthe population size, 3 is the number of channels (RGB), H\\nis the patch\\u2019s height, and Wis the patch\\u2019s width.\\nThen, using gradient approximation through an evolu-\\ntion strategy, we repeatedly search the latent vector zthat\\nbest achieves our objective function, which is:\\nLtotal=Ldet+\\u0015tvLtv+\\u0015clsLcls; (1)\\nwhere:\\n\\u2022Ldet: detection loss of object detector of the speci\\ufb01c\\nclass.\\n\\u2022Ltv: total variation loss, to promote smoothness of\\ngenerated patch.\\n\\u2022Lcls: classi\\ufb01cation loss, to promote more-realistic\\npatch generation.\\n\\u2022\\u0015tvand\\u0015clsare regularization weights.\\nWe expound upon the these terms in below.\\n3.2. Adversarial gradient estimation\\nIn order to generate patches that may deceive the target\\nobject detector, the generator uses adversarial gradient esti-\\nmations as its primary navigation tool. We initially add thepatch onto a given image. In order to compute an adversar-\\nial loss for the detection of bounding boxes (BBs), we feed\\nthe image to the object detector.\\nAdversarial detection loss . Detection can be arbitrarily\\nproduced by object detectors like YOLO [26]. We are in-\\nterested in minimizing two terms for the patch detection i:\\nits objectness probability Di\\nobj, which speci\\ufb01es the model\\u2019s\\ncon\\ufb01dence regarding whether there is an object or not, and\\nthe class probability Di\\ncls, which speci\\ufb01es the model\\u2019s con-\\n\\ufb01dence of a speci\\ufb01c class. In this paper we focused on gen-\\nerating patches that conceal humans. Thus, we want to min-\\nimize both the objectness Di\\nobjand class probabilities Di\\ncls\\nfor our generated patch with respect to the person class, i.e.,\\nminimizing the term:\\nLdet=1\\nnnX\\ni=1detX\\nj=1Dj\\nobj(xi)Dj\\ncls(xi); (2)\\nwherenis the population size and detis the number of\\nhuman detections by the model. By minimizing Ldet, we\\nachieve a patch that minimizes the objectness and class\\nprobabilities of the target class.\\nPhysical transformations . We have no in\\ufb02uence over\\nthe adversarial patch\\u2019s viewpoint, position, or size, with re-\\nspect to the images. In order to enhance the robustness of\\nour adversarial patch, we overlay it onto a human and gen- erate a variety of settings with distinct con\\ufb01gurations. Fur-\\nthermore, we subject our created adversarial patch Pto var-\\nious transformations, including rotation, brightness change,\\nand resizing, to mimic the different visual appearances it\\nmay adopt in real-world situations. The transformations are\\napplied by the Patch Transformer (Figure 2).\\nSmoothness . To promote smoothness in the generated\\nimage we apply the term Ltv, which represents the total\\nvariation loss. The calculation of Ltvfrom a patch Pis\\ndone as follows:\\nLtv(P) =X\\ni;jq\\n(Pi+1;j\\u0000Pi;j)2+ (Pi;j+1\\u0000Pi;j)2;\\n(3)\\nwhere subscripts iandjrefer to the pixel coordinates of\\npatchP. A constant value of \\u0015tv= 0:1was employed in\\nall experiments presented in this paper.\\nClass loss . Empirically, we noticed that the patches gen-\\nerated by the generator are very abstract and apparently far\\nfrom the latent image manifold learned by the generator.\\nThus, we added another loss term in order to make the patch\\nresemble a speci\\ufb01c class, by adding a pretrained classi\\ufb01er,\\ntrained on ImageNet [6]; this adds regularization that makes\\nthe image similar to a speci\\ufb01c class.\\nRealism . To maintain realism we enforce a constraint on\\nthe norm of the latent vectors Z, which should not exceed a\\nthreshold\\u001c. By adjusting this threshold we can balance re-\\nalism versus attack performance. We use k\\u0001k1to constrain\\nz, resulting in:\\nzt=\\u0019(zt\\u00001\\u0000\\u000b~rzLtotal); (4)\\nwhere:\\n\\u0019(z) =fzijzi min(max(zi;\\u0000\\u001c);\\u001c);8zi2zg;(5)\\ntis the timestep (epoch), \\u000bis the step size, and ~rzLtotal is\\nthe gradient approximation of the total loss with respect to\\nthe latent vector z. We used\\u001c= 20 in all experiments.\\n4. Experimental results\\nWe begin by delineating the implementation details of\\nour experiments, followed by the qualitative and quantita-\\ntive experiments themselves, focusing on the effectiveness\\nof the proposed adversarial patch in two settings: 1) dig-\\nital environments, speci\\ufb01cally, the INRIA person dataset\\n[5], where we target images with one person, to facilitate\\noptimization; 2) physical environments, as demonstrated\\nthrough recorded videos in various real-life scenes. We\\nalso perform several ablation studies, aimed at re\\ufb01ning var-\\nious hyperparameters, ultimately leading to the creation of\\na physically generated adversarial patch that appears more\\nnatural, while maintaining comparable attack performance.Additionally, subjective evaluations are conducted to assess\\nthe naturalness of the generated patches.\\n4.1. Ablation studies\\nThe experiments were performed using the Adam [17]\\noptimizer, with a learning rate of 0:02and\\f1= 0:5;\\f2=\\n0:999. If the changes in losses remain below 1e\\u00004for at\\nleast 50 epochs, we reduce the learning rate, following [13].\\nOur generator consists of BigGAN2 [2], which has been\\npretrained on the ImageNet dataset, with a latent vector of\\nsize120, and 128\\u0002128output resolution. As the BigGAN\\ngenerator is class-conditional, the generated patch can be\\nguaranteed to belong to a speci\\ufb01c class. We used two dif-\\nferent object detectors\\u2014Tiny-YOLOv3 and Tiny-YOLOv4\\n[26], trained on the COCO dataset [21]\\u2013with an input reso-\\nlution of 416\\u0002416. We chose these faster, memory-ef\\ufb01cient\\nmodels so as to be able to conduct far more experiments.\\nFor each of the detected bounding boxes, we placed an ad-\\nversarial patch of size 25% of the bounding box. The mean\\nof the false negatives (FNs, failure to identify object, in our\\ncase a person) of all experiments is shown in Table 4.1. In\\nboth models, the FN rate on the original INRIA dataset us-\\ning one-person only, was 0%.\\nTable 1. FN rate for the two objective functions for each of the\\nmodels. A higher FN value means the algorithm better fooled the\\nobject detector.\\nModel Obj only - FN (%) Det + Obj - FN (%)\\nTiny-YOLOv3 21:9 67 :0\\nTiny-YOLOv4 40:4 40 :6\\n4.2. Visual results\\nFigures 3 through 6 show evolved patches. Figures 7\\nand 8 show digital attacks. Figures 9 through 13 show phys-\\nical attacks. Note that the \\ufb01gures show diverse situations\\nand lighting conditions.\\n5. Discussion and 8 show digital attacks. Figures 9 through 13 show phys-\\nical attacks. Note that the \\ufb01gures show diverse situations\\nand lighting conditions.\\n5. Discussion\\n5.1. Impact of loss function\\nWe explored two different loss objectives. The \\ufb01rst one\\nfocuses only on the objectness loss, i.e.,:\\nLdet=1\\nNNX\\ni=1#detX\\nj=1Dj\\nobj; (6)\\nFor Tiny-YOLOv3 (Figure 14), Equation 2 (which takes\\ninto account class probabilities as well) does better than\\nEquation 6. For Tiny-YOLOv4 (Figure 15), the two are\\nsimilar. This result is contrary to the result presented in\\n[29]\\u2014though we haven\\u2019t used any gradients. We think this\\nmight be due to the speci\\ufb01c hyperparameters tested in our Figure 3. Patches evolved by our algorithm, on Tiny-YOLOv3,\\nwith\\u0015cls= 0:1, and (top-left to bottom-right) population sizes of\\n50, 70, 90, and 110, respectively.\\nFigure 4. Patches evolved by our algorithm, on Tiny-YOLOv3,\\nwith\\u0015cls= 0:2, and (top-left to bottom-right) population sizes of\\n50, 70, 90, and 110, respectively.\\nFigure 5. Patches evolved by our algorithm, on Tiny-YOLOv4,\\nwith\\u0015cls= 0:1, and (top-left to bottom-right) population sizes of\\n50, 70, 90, and 110, respectively.\\nexperiments\\u2014further analysis in the future will ascertain\\nthis point.\\nFigure 6. Patches evolved by our algorithm, on Tiny-YOLOv4,\\nwith\\u0015cls= 0:2, and (top-left to bottom-right) population sizes of\\n50, 70, 90, and 110, respectively.\\nFigure 7. Digital examples of our proposed attack on Tiny-\\nYOLOv3. The top-left image shows an example wherein the patch\\nfailed to \\u201cconceal\\u201d the person. In all other images the attack suc-\\nceeded: no person was detected.\\n5.2. Impact of population size\\nWe explored 4 different population sizes: 50;70;90, and\\n110. In order to estimate the gradient well enough, we need\\n2 queries per each coordinate of the latent vector (of size\\n120). Thus, we need 2dqueries to the target model, result-\\ning in 240queries for one gradient estimation. We actually\\nused far less queries, evidencing a strength of our approach.\\nFurther, we surmised that a larger population would yield\\nbetter results, but, empirically, it seems that a population of\\nsize70yielded lower loss. We think this is because when\\nwe estimate a gradient using a relatively small population\\nwe increase the algorithm\\u2019s exploration, resulting in a lower\\nloss value. Figure 8. Digital examples of our proposed attack on Tiny-\\nYOLOv4. The top-left image shows an example wherein the patch\\nfailed to \\u201cconceal\\u201d the person. In all other images the attack suc-\\nceeded: no person was detected.\\nFigure 9. Real-life images\\u2014taken by coauthor\\u2014of Tiny-YOLOv3\\ndetection using no patch. Top: low-lighting condition, Bottom:\\nhigh-lighting condition.\\n6. Concluding remarks\\nWe presented a novel algorithm that generates naturalis-\\ntic, physical adversarial patches for object detectors. The\\npatches can be printed and used in the real world. Our\\napproach involved optimizing a latent vector to minimize\\nprobabilities associated with the appearance of a person in\\nFigure 10. Real-life example of Tiny-YOLOv4 detection using no\\npatch. Top: low-lighting condition, Bottom: high-lighting condi-\\ntion.\\nFigure 11. Real-life example of evolved adversarial patch on Tiny-\\nYOLOv3. Successful patch \\u201cconceals\\u201d the person. Low-lighting\\ncondition. Figure 12. Real-life example of evolved adversarial patch on Tiny-\\nYOLOv3. Successful patch \\u201cconceals\\u201d the person. High-lighting\\ncondition.\\nFigure 13. Real-life example of evolved adversarial patch on Tiny-\\nYOLOv4. Successful patch \\u201cconceals\\u201d the person. Low-lighting\\ncondition.\\nFigure 14. Total loss as function of number of epochs, on Tiny-\\nYOLOv3. Top: Optimization process using Equation 6. Bottom:\\nOptimization process using Equation 2\\nthe detector\\u2019s output, without using any internal information\\nof the model\\u2014a realistic scenario that does not rely on the\\nuse of gradients. We compared different deep models and\\nconcluded that is is possible to generate patches that fool\\nobject detectors. The real-world tests of the printed patches\\ndemonstrated their ef\\ufb01cacy in \\u201cconcealing\\u201d persons, evi-\\ndencing a basic threat to security systems. In future work\\nwe wish to expand our algorithm to other types of attacks,\\nsuch as false-positive attacks (which cause the appearance\\nof a person where none exists), evasion attacks, and more.\\nAcknowledgement\\nThis research was partially supported by the Israeli In-\\nnovation Authority through the Trust.AI consortium. Figure 15. Total loss as function of number of epochs, on Tiny-\\nYOLOv4. Top: Optimization process using Equation 6. Bottom:\\nOptimization process using Equation 2 References\\n[1] Martin Arjovsky, Soumith Chintala, and L \\u00b4eon Bottou.\\nWasserstein generative adversarial networks. In In-\\nternational Conference on Machine Learning , pages\\n214\\u2013223. PMLR, 2017. 3\\n[2] Andrew Brock, Jeff Donahue, and Karen Simonyan.\\nLarge scale gan training for high \\ufb01delity natural image\\nsynthesis. arXiv preprint arXiv:1809.11096 , 2018. 5\\n[3] Tom B Brown, Dandelion Man \\u00b4e, Aurko Roy, Mart \\u00b4\\u0131n\\nAbadi, and Justin Gilmer. Adversarial patch. arXiv\\npreprint arXiv:1712.09665 , 2017. 3\\n[4] Nicholas Carlini and David Wagner. Towards evaluat-\\ning the robustness of neural networks. In 2017 IEEE\\nSymposium on Security and Privacy (SP) , pages 39\\u2013\\n57. Ieee, 2017. 1, 2\\n[5] Navneet Dalal and Bill Triggs. Histograms of oriented\\ngradients for human detection. In 2005 IEEE Com-\\nputer Society Conference on Computer Vision and Pat-\\ntern Recognition (CVPR\\u201905) , volume 1, pages 886\\u2013\\n893. Ieee, 2005. 5\\n[6] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai\\nLi, and Li Fei-Fei. Imagenet: A large-scale hierar-\\nchical image database. In 2009 IEEE Conference on\\nComputer Vision and Pattern Recognition , pages 248\\u2013\\n255. Ieee, 2009. 5\\n[7] Ranjie Duan, Xingjun Ma, Yisen Wang, James Bai-\\nley, A Kai Qin, and Yun Yang. Adversarial cam-\\nou\\ufb02age: Hiding physical-world attacks with natural\\nstyles. In Proceedings of the IEEE\\/CVF Conference\\non Computer Vision and Pattern Recognition , pages\\n1000\\u20131008, 2020. 3\\n[8] Kevin Eykholt, Ivan Evtimov, Earlence Fernandes, Bo\\nLi, Amir Rahmati, Chaowei Xiao, Atul Prakash, Ta-\\ndayoshi Kohno, and Dawn Song. Robust physical-\\nworld attacks on deep learning visual classi\\ufb01cation.\\nInProceedings of the IEEE Conference on Computer\\nVision and Pattern Recognition , pages 1625\\u20131634,\\n2018. 2, 3\\n[9] Weiwei Feng, Baoyuan Wu, Tianzhu Zhang, Yong\\nZhang, and Yongdong Zhang. Meta-attack: Class-\\nagnostic and model-agnostic physical adversarial at-\\ntack. In Proceedings of the IEEE\\/CVF International\\nConference on Computer Vision , pages 7787\\u20137796,\\n2021. 2\\n[10] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza,\\nBing Xu, David Warde-Farley, Sherjil Ozair, Aaron\\nCourville, and Yoshua Bengio. Generative adversarial\\nnetworks. Communications of the ACM , 63(11):139\\u2013\\n144, 2020. 3[11] Ian J Goodfellow, Jonathon Shlens, and Christian\\nSzegedy. Explaining and harnessing adversarial ex-\\namples. arXiv preprint arXiv:1412.6572 , 2014. 2\\n[12] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising\\ndiffusion probabilistic models. Advances in Neural In-\\nformation Processing Systems , 33:6840\\u20136851, 2020.\\n3\\n[13] Yu-Chih-Tuan Hu, Bo-Han Kung, Daniel Stanley Tan,\\nJun-Cheng Chen, Kai-Lung Hua, and Wen-Huang\\nCheng. Naturalistic physical adversarial patch for\\nobject detectors. In Proceedings of the IEEE\\/CVF\\nInternational Conference on Computer Vision , pages\\n7848\\u20137857, 2021. 2, 5\\n[14] Zhanhao Hu, Siyuan Huang, Xiaopei Zhu, Fuchun\\nSun, Bo Zhang, and Xiaolin Hu. Adversarial texture\\nfor fooling person detectors in the physical world. In\\nProceedings of the IEEE\\/CVF Conference on Com-\\nputer Vision and Pattern Recognition , pages 13307\\u2013\\n13316, 2022. 3\\n[15] Jung Im Choi and Qing Tian. Adversarial attack and\\ndefense of yolo detectors in autonomous driving sce-\\nnarios. In 2022 IEEE Intelligent Vehicles Symposium\\n(IV), pages 1011\\u20131017. IEEE, 2022. 2\\n[16] Tero Karras, Samuli Laine, and Timo Aila. A style-\\nbased generator architecture for generative adversar-\\nial networks. In Proceedings of the IEEE\\/CVF Con-\\nference on Computer Vision and Pattern Recognition ,\\npages 4401\\u20134410, 2019. 3\\n[17] Diederik P Kingma and Jimmy Ba. Adam: A\\nmethod for stochastic optimization. arXiv preprint\\narXiv:1412.6980 , 2014. 3, 5\\n[18] Stepan Komkov and Aleksandr Petiushko. Advhat:\\nReal-world adversarial attack on arcface face id sys-\\ntem. In 2020 25th International Conference on Pattern\\nRecognition (ICPR) , pages 819\\u2013826. IEEE, 2021. 3\\n[19] Alexey Kurakin, Ian J Goodfellow, and Samy Bengio.\\nAdversarial examples in the physical world. In Arti-\\n\\ufb01cial Intelligence Safety and Security , pages 99\\u2013112. [19] Alexey Kurakin, Ian J Goodfellow, and Samy Bengio.\\nAdversarial examples in the physical world. In Arti-\\n\\ufb01cial Intelligence Safety and Security , pages 99\\u2013112.\\nChapman and Hall\\/CRC, 2018. 2\\n[20] Raz Lapid, Zvika Haramaty, and Moshe Sipper.\\nAn evolutionary, gradient-free, query-ef\\ufb01cient, black-\\nbox algorithm for generating adversarial instances\\nin deep convolutional neural networks. Algorithms ,\\n15(11):407, 2022. 1, 2\\n[21] Tsung-Yi Lin, Michael Maire, Serge Belongie, James\\nHays, Pietro Perona, Deva Ramanan, Piotr Doll \\u00b4ar,\\nand C Lawrence Zitnick. Microsoft coco: Com-\\nmon objects in context. In Computer Vision\\u2013ECCV\\n2014: 13th European Conference, Zurich, Switzer-\\nland, September 6-12, 2014, Proceedings, Part V 13 ,\\npages 740\\u2013755. Springer, 2014. 5 [22] Xin Liu, Huanrui Yang, Ziwei Liu, Linghao Song,\\nHai Li, and Yiran Chen. Dpatch: An adversar-\\nial patch attack on object detectors. arXiv preprint\\narXiv:1806.02299 , 2018. 2\\n[23] Aleksander Madry, Aleksandar Makelov, Ludwig\\nSchmidt, Dimitris Tsipras, and Adrian Vladu. To-\\nwards deep learning models resistant to adversarial at-\\ntacks. arXiv preprint arXiv:1706.06083 , 2017. 2\\n[24] Xudong Mao, Qing Li, Haoran Xie, Raymond YK\\nLau, Zhen Wang, and Stephen Paul Smolley. Least\\nsquares generative adversarial networks. In Proceed-\\nings of the IEEE International Conference on Com-\\nputer Vision , pages 2794\\u20132802, 2017. 3\\n[25] Seyed-Mohsen Moosavi-Dezfooli, Alhussein Fawzi,\\nand Pascal Frossard. Deepfool: a simple and accurate\\nmethod to fool deep neural networks. In Proceedings\\nof the IEEE Conference on Computer Vision and Pat-\\ntern Recognition , pages 2574\\u20132582, 2016. 1\\n[26] Joseph Redmon, Santosh Divvala, Ross Girshick, and\\nAli Farhadi. You only look once: Uni\\ufb01ed, real-time\\nobject detection. In Proceedings of the IEEE Con-\\nference on Computer Vision and Pattern Recognition ,\\npages 779\\u2013788, 2016. 2, 4, 5\\n[27] Mahmood Sharif, Sruti Bhagavatula, Lujo Bauer, and\\nMichael K Reiter. A general framework for adversar-\\nial examples with objectives. ACM Transactions on\\nPrivacy and Security (TOPS) , 22(3):1\\u201330, 2019. 2\\n[28] Christian Szegedy, Wojciech Zaremba, Ilya Sutskever,\\nJoan Bruna, Dumitru Erhan, Ian Goodfellow, and\\nRob Fergus. Intriguing properties of neural networks.\\narXiv preprint arXiv:1312.6199 , 2013. 1\\n[29] Simen Thys, Wiebe Van Ranst, and Toon Goedem \\u00b4e.\\nFooling automated surveillance cameras: Adversarial\\npatches to attack person detection. In Proceedings of\\nthe IEEE\\/CVF Conference on Computer Vision and\\nPattern Recognition Workshops , pages 0\\u20130, 2019. 1,\\n2, 5\\n[30] Snir Vitrack Tamam, Raz Lapid, and Moshe Sipper.\\nFoiling explanations in deep neural networks. arXiv\\ne-prints , pages arXiv\\u20132211, 2022. 1\\n[31] Jiakai Wang, Aishan Liu, Zixin Yin, Shunchang Liu,\\nShiyu Tang, and Xianglong Liu. Dual attention sup-\\npression attack: Generate adversarial camou\\ufb02age in\\nphysical world. In Proceedings of the IEEE\\/CVF Con-\\nference on Computer Vision and Pattern Recognition ,\\npages 8565\\u20138574, 2021. 3\\n[32] Daan Wierstra, Tom Schaul, Tobias Glasmachers, Yi\\nSun, Jan Peters, and J \\u00a8urgen Schmidhuber. Natural\\nevolution strategies. Journal of Machine Learning Re-\\nsearch , 15(1):949\\u2013980, 2014. 3[33] Chenglin Yang, Adam Kortylewski, Cihang Xie,\\nYinzhi Cao, and Alan Yuille. Patchattack: A black-\\nbox texture-based attack with reinforcement learn-\\ning. In Computer Vision\\u2013ECCV 2020: 16th European\\nConference, Glasgow, UK, August 23\\u201328, 2020, Pro-\\nceedings, Part XXVI , pages 681\\u2013698. Springer, 2020.\\n3\\n[34] Alon Zol\\ufb01, Shai Avidan, Yuval Elovici, and Asaf\\nShabtai. Adversarial mask: Real-world adversarial at-\\ntack against face recognition models. arXiv preprint\\narXiv:2111.10759 , 2021. 2\",\"10\":\" MAP-Elites with Descriptor-Conditioned Gradients and\\nArchive Distillation into a Single Policy\\nMaxence Faldor\\nm.faldor22@imperial.ac.uk\\nImperial College London\\nLondon, United KingdomF\\u00e9lix Chalumeau\\nf.chalumeau@instadeep.com\\nInstaDeep\\nCape Town, South Africa\\nManon Flageat\\nmanon.flageat18@imperial.ac.uk\\nImperial College London\\nLondon, United KingdomAntoine Cully\\na.cully@imperial.ac.uk\\nImperial College London\\nLondon, United Kingdom\\nABSTRACT\\nQuality-Diversity algorithms, such as MAP-Elites, are a branch of\\nEvolutionary Computation generating collections of diverse and\\nhigh-performing solutions, that have been successfully applied\\nto a variety of domains and particularly in evolutionary robotics.\\nHowever, MAP-Elites performs a divergent search based on random\\nmutations originating from Genetic Algorithms, and thus, is limited\\nto evolving populations of low-dimensional solutions. PGA-MAP-\\nElites overcomes this limitation by integrating a gradient-based\\nvariation operator inspired by Deep Reinforcement Learning which\\nenables the evolution of large neural networks. Although high-\\nperforming in many environments, PGA-MAP-Elites fails on several\\ntasks where the convergent search of the gradient-based operator\\ndoes not direct mutations towards archive-improving solutions. In\\nthis work, we present two contributions: (1) we enhance the Policy\\nGradient variation operator with a descriptor-conditioned critic\\nthat improves the archive across the entire descriptor space, (2) we\\nexploit the actor-critic training to learn a descriptor-conditioned\\npolicy at no additional cost, distilling the knowledge of the archive\\ninto one single versatile policy that can execute the entire range\\nof behaviors contained in the archive. Our algorithm, DCG-MAP-\\nElites improves the QD score over PGA-MAP-Elites by 82% on\\naverage, on a set of challenging locomotion tasks.\\n1 INTRODUCTION\\nA fascinating aspect of evolution is its ability to generate a large\\nvariety of different species, each being adapted to their ecological\\nniche. Inspired by this idea, Quality-Diversity (QD) optimization is\\na family of evolutionary algorithms that aims to generate a set of\\nboth diverse and high-performing solutions to a problem [ 3,7,32].\\nContrary to traditional optimization methods that return a single\\nhigh-performing solution, the goal of QD algorithms is to illuminate\\na search space of interest called descriptor space [27]. Producing\\na large collection of diverse and effective solutions enables to get\\nmultiple alternatives to solve a single problem which is useful\\nin robotics to improve robustness, recover from damage [ 6] or\\nreduce the reality gap [ 4]. Furthermore, conventional optimization\\nmethods are prone to get stuck in local optima whereas keeping\\ndiverse solutions to a problem can help to find stepping stones that\\nlead to globally better solutions [ 27,28]. Another benefit of diversity\\nFigure 1: DCG-MAP-Elites performs a standard MAP-Elites\\nloop of selection, variation, evaluation and addition. Two\\ncomplementary variation operators are applied: (1) a stan-\\ndard Genetic Algorithm (GA) variation operator for explo-\\nration, (2) a Descriptor-Conditioned Policy Gradient (PG)\\nvariation operator for fitness improvement. Concurrently to\\nthe critic\\u2019s training, the knowledge of the archive is distilled\\nin the descriptor-conditioned actor as by-product.\\nsearch is efficient exploration in problems where the reward signal\\nis sparse or deceptive [2, 8, 31].\\nMAP-Elites [ 27] is a conceptually simple but effective QD opti-\\nmization algorithm that has shown competitive results in a variety\\nof applications, to generate large collections of diverse skills. How-\\never, MAP-Elites relies on random variations that can cause slow\\nconvergence in large search space [ 5,28,31], making it inadequate\\nto evolve neural networks with a large number of parameters.\\nIn contrast, Deep Reinforcement Learning (DRL) [ 25,26] algo-\\nrithms combine reinforcement learning with the directed search to evolve neural networks with a large number of parameters.\\nIn contrast, Deep Reinforcement Learning (DRL) [ 25,26] algo-\\nrithms combine reinforcement learning with the directed search\\npower of gradient-based methods in order to learn a single solu-\\ntion. DRL can surpass human performance at video games [ 40],\\nbeat world champions in board games [ 35] and control complex\\nrobots in continuous action spaces [ 17], which is a long-standing\\nchallenge in artificial intelligence. Policy Gradient methods have\\nshown state-of-the-art results to learn large neural network policies\\nwith thousands of parameters in high-dimensional state space and\\ncontinuous action space [18, 24, 36].\\nPGA-MAP-Elites [ 28] is an extension of MAP-Elites that inte-\\ngrates the sample efficiency of DRL using the TD3 algorithm [ 15].arXiv:2303.03832v1  [cs.NE]  7 Mar 2023 This algorithm uses a Policy Gradient (PG) variation operator for\\nefficient fitness improvement, coupled with the usual Genetic Algo-\\nrithm (GA) variation operator. The PG variation operator leverages\\ngradients derived from DRL to improve fitness and drive mutations\\ntowards the global optimum and is supported by the divergent\\nsearch of the GA variation operator for both exploration and opti-\\nmization [ 10]. Other recent works have also introduced methods to\\ncombine the strength of QD algorithms with reinforcement learn-\\ning [31, 38] on complex robotics tasks.\\nPGA-MAP-Elites achieves state-of-the-art performances in most\\nof the environments considered so far in the literature [ 28,31,38].\\nHowever, the PG variation operator becomes ineffective in tasks\\nwhere the global optimum is in an area of the search space that is\\nnot likely to produce offspring that are added to the archive. For\\nexample, consider a locomotion task where the fitness is the oppo-\\nsite of the energy consumption and the descriptor is defined as the\\nfinal position of the robot. The global optimum for the fitness is the\\nsolution that does not move in order to minimize energy consump-\\ntion. Thus, the PG variation operator will encourage solutions to\\nstay motionless, collapsing their descriptors to a single point, the\\ndescriptor of the global optimum. Consequently, the PG variation\\noperator generates offspring that are discarded and no interesting\\nstepping stone is found, thereby hindering diversity.\\nIn this work, we introduce Descriptor-Conditioned Gradients\\nMAP-Elites (DCG-MAP-Elites) that builds upon PGA-MAP-Elites\\nalgorithm by enhancing the PG variation operator with a descriptor-\\nconditioned critic that provides gradients depending on a target de-\\nscriptor. The descriptor-conditioned critic takes as input a state and\\na descriptor to evaluate actions. With such a descriptor-conditioned\\ncritic, the PG variation operator can mutate solutions to produce\\noffspring with higher fitness while targeting a desired descriptor,\\nthereby avoiding to collapse the descriptor to a single point.\\nFurthermore, TD3 which is the DRL algorithm used by the PG\\nvariation operator, requires to train an actor and a critic in parallel.\\nWe take advantage of this intertwined actor-critic training to make\\nthe actor descriptor-conditioned as well, allowing it to take actions\\nbased on the current state and on an input descriptor we want to\\nachieve. Thus, instead of taking actions that maximize the fitness\\nglobally, the actor now takes actions that maximize the fitness\\nwhile achieving a desired descriptor. At the end of training, we\\ncan condition the actor on a desired descriptor to execute a policy\\nthat takes actions that achieve the desired descriptor. On half of\\nthe tasks, we observe that the descriptor-conditioned actor can\\nachieve the entire range of descriptors contained in the archive\\nwith a similar QD-score, negating the burden of dealing with a\\ncollection of thousands of solutions.\\nIn summary, we present two contributions: (1) we enhance the\\nPG variation operator with a descriptor-conditioned critic, (2) we\\ndistill the knowledge of the archive into one single versatile policy\\nat no additional cost. We compare our algorithm to four state-of-the-\\nart QD algorithms on four high-dimensional robotics locomotion\\ntasks. The results demonstrate that DCG-MAP-Elites has a QD-\\nscore 82% higher than PGA-MAP-Elites on average.2 BACKGROUND\\n2.1 Problem Statement\\nWe consider an agent sequentially interacting with an environment\\nat discrete time steps \\ud835\\udc61for an episode of length \\ud835\\udc47. At each time step\\n\\ud835\\udc61, the agent observes a state \\ud835\\udc60\\ud835\\udc61, takes an action \\ud835\\udc4e\\ud835\\udc61and receives a\\nscalar reward \\ud835\\udc5f\\ud835\\udc61. We model it as a Markov Decision Process (MDP)\\nwhich comprises a state spaceS, a continuous action spaceA, a\\nstationary transition dynamics distribution \\ud835\\udc5d(\\ud835\\udc60\\ud835\\udc61+1|\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61)and a\\nreward function \\ud835\\udc5f:S\\u00d7A\\u2192 R. In this work, a policy (also called\\nsolution ) is a deterministic neural network parameterized by \\ud835\\udf19\\u2208\\u03a6, stationary transition dynamics distribution \\ud835\\udc5d(\\ud835\\udc60\\ud835\\udc61+1|\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61)and a\\nreward function \\ud835\\udc5f:S\\u00d7A\\u2192 R. In this work, a policy (also called\\nsolution ) is a deterministic neural network parameterized by \\ud835\\udf19\\u2208\\u03a6,\\nand denoted \\ud835\\udf0b\\ud835\\udf19:S\\u2192A . The agent uses its policy to select actions\\nand interact with the environment to give a trajectory of states,\\nactions and rewards. The fitness of a solution is given by \\ud835\\udc39:\\u03a6\\u2192R,\\ndefined as the expected discounted return E\\ud835\\udf0b\\ud835\\udf19\\u0002\\u00cd\\ud835\\udc47\\u22121\\n\\ud835\\udc61=0\\ud835\\udefe\\ud835\\udc61\\ud835\\udc5f\\ud835\\udc61\\u0003\\n.\\nThe objective of QD algorithms in this MDP setting is to find\\nthe highest-fitness solutions in each point of the descriptor space\\nD. The descriptor function \\ud835\\udc37:\\u03a6\\u2192D is generally defined by the\\nuser and characterize solutions in a meaningful way for the type of\\ndiversity desired. With this notation, our objective is to evolve a\\npopulation of solutions that are both high-performing with respect\\nto\\ud835\\udc39and diverse with respect to \\ud835\\udc37.\\n2.2 MAP-Elites\\nMulti-dimensional Archive of Phenotypic Elites (MAP-Elites) [ 27]\\nis a simple yet effective QD algorithm that discretizes the descriptor\\nspaceDinto a multi-dimensional grid of cells called archive Xand\\nsearches for the best solution in each cell, see Alg. 4. The goal of the\\nalgorithm is to return an archive that is filled as much as possible\\nwith high-fitness solutions. MAP-Elites starts by generating ran-\\ndom solutions and adding them to the archive. The algorithm then\\nrepeats the following steps until a budget of \\ud835\\udc3csolutions have been\\nevaluated: (1) a batch of solutions from the archive are uniformly\\nselected and modified through mutations and\\/or crossovers to pro-\\nduce offspring, (2) the fitnesses and descriptors of the offspring are\\nevaluated, and each offspring is placed in its corresponding cell if\\nand only if the cell is empty or if the offspring has a better fitness\\nthan the current solution in that cell, in which case the current\\nsolution is replaced. As most evolutionary methods, MAP-Elites\\nrelies on undirected updates that are agnostic to the fitness objec-\\ntive. With a Genetic Algorithm (GA) variation operator, MAP-Elites\\nperforms a divergent search that may cause slow convergence in\\nhigh-dimensional problems due to a lack of directed search power,\\nand thus, is performing best on low-dimensional search space [ 28].\\n2.3 Deep Reinforcement Learning\\nDeep Reinforcement Learning (DRL) [ 25,26] combines the rein-\\nforcement learning framework with the function approximation\\ncapabilities of deep neural networks to represent policies and value\\nfunctions in high-dimensional state and action spaces. In opposition\\nto black-box optimization methods like evolutionary algorithms,\\nDRL leverages the structure of the MDP in the form of the Bellman\\nequation to achieve better sample efficiency. The objective is to\\nfind an optimal policy \\ud835\\udf0b\\ud835\\udf19, which maximizes the expected return or\\nfitness\\ud835\\udc39(\\ud835\\udf0b\\ud835\\udf19). In reinforcement learning, many approaches try to\\nestimate the action-value function \\ud835\\udc44\\ud835\\udf0b(\\ud835\\udc60,\\ud835\\udc4e)=E\\ud835\\udf0b\\u0002\\u00cd\\ud835\\udc47\\u22121\\n\\ud835\\udc61=0\\ud835\\udefe\\ud835\\udc61\\ud835\\udc5f\\ud835\\udc61|\\ud835\\udc60,\\ud835\\udc4e\\u0003\\n2 defined as the expected discounted return starting from state \\ud835\\udc60,\\ntaking action \\ud835\\udc4eand thereafter following policy \\ud835\\udf0b.\\nThe Twin Delayed Deep Deterministic Policy Gradient (TD3)\\nalgorithm [ 15] is an actor-critic, off-policy reinforcement learn-\\ning method that achieves state-of-the-art results in environments\\nwith large and continuous action space. TD3 indirectly learns a\\npolicy\\ud835\\udf0b\\ud835\\udf19via maximization of the action-value function \\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e).\\nThe approach is closely connected to Q-learning [ 15] and tries\\nto approximate the optimal action-value function \\ud835\\udc44\\u2217(\\ud835\\udc60,\\ud835\\udc4e)in or-\\nder to find the optimal action \\ud835\\udc4e\\u2217(\\ud835\\udc60)=arg max\\ud835\\udc4e\\ud835\\udc44\\u2217(\\ud835\\udc60,\\ud835\\udc4e). How-\\never, computing the maximum over action in max\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e)is in-\\ntractable in continuous action space, so it is approximated using\\nmax\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e)=\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60)). In TD3, the policy \\ud835\\udf0b\\ud835\\udf19takes actions\\nin the environment and the transitions are stored in a replay buffer.\\nThe collected experience is then used to train a pair of critics \\ud835\\udc44\\ud835\\udf031,\\n\\ud835\\udc44\\ud835\\udf032using temporal difference and target networks \\ud835\\udc44\\ud835\\udf031\\u2032,\\ud835\\udc44\\ud835\\udf032\\u2032are\\nupdated to slowly track the main networks. Both critics use a single\\nregression target \\ud835\\udc66, calculated using whichever of the two critics\\ngives a smaller target value and using target policy smoothing by\\nsampling a noise \\ud835\\udf16\\u223cclip(N(0,\\ud835\\udf0e),\\u2212\\ud835\\udc50,\\ud835\\udc50):\\n\\ud835\\udc66=\\ud835\\udc5f(\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61)+\\ud835\\udefemin\\n\\ud835\\udc56=1,2\\ud835\\udc44\\ud835\\udf03\\ud835\\udc56\\u2032\\u0010\\n\\ud835\\udc60\\ud835\\udc61+1,\\ud835\\udf0b\\ud835\\udf19\\u2032(\\ud835\\udc60\\ud835\\udc61+1)+\\ud835\\udf16\\u0011\\n(1)\\nBoth critics are learned by regression to this target and the policy\\nis learned with a delay, only updated every \\u0394iterations simply by\\nmaximizing \\ud835\\udc44\\ud835\\udf031withmax\\ud835\\udf19E\\u0002\\n\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60))\\u0003\\n. The actor is updated\\nusing the deterministic policy gradient:\\n\\u2207\\ud835\\udf19\\ud835\\udc3d(\\ud835\\udf19)=Eh\\n\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60)\\u2207\\ud835\\udf19\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60)i\\n(2)\\n2.4 PGA-MAP-Elites\\nPolicy Gradient Assisted MAP-Elites (PGA-MAP-Elites) [ 28] is an\\nextension of MAP-Elites that is designed to evolve deep neural\\nnetworks by combining the directed search power and sample effi-\\nciency of DRL methods with the exploration capabilities of genetic\\nalgorithms, see Alg. 5. The algorithm follows the usual MAP-Elites\\nloop of selection, variation, evaluation and addition for a budget of\\n\\ud835\\udc3citerations, but uses two parallel variation operators: half of the\\noffspring are generated using a standard Genetic Algorithm (GA)\\nvariation operator and half of the offspring are generated using\\na Policy Gradient (PG) variation operator. During each iteration\\nof the loop, PGA-MAP-Elites stores the transitions from offspring\\nevaluation in a replay buffer Band uses it to train a pair of critics\\nbased on the TD3 algorithm, described in Alg. 6. The trained critic\\nis then used in the PG variation operator to update the selected\\nsolutions from the archive for \\ud835\\udc5agradient steps to select actions\\nthat maximize the approximated action-value function, as described\\nin Alg. 7. At each iteration, the critics are trained for \\ud835\\udc5bsteps of\\ngradients descents towards the target described in Eq. 1 averaged\\nover\\ud835\\udc41transitions of experience sampled uniformly from the replay\\nbufferB. The actor (also named greedy actor [ 28]) learns with a\\ndelay \\u0394via maximization of the critic according to Eq. 2.\\n3 RELATED WORK\\n3.1 Scaling QD to Neuroevolution\\nThe challenge of evolving diverse solutions in a high-dimensional\\nsearch space has been an active research subject over the recentyears. ME-ES [ 5] scales MAP-Elites to high-dimensional solutions\\nparameterized by large neural networks. This algorithm leverages\\nEvolution Strategies to perform a directed search that is more effi-\\ncient than random mutations used in Genetic Algorithms. Fitness\\ngradients are estimated locally from many perturbed versions of\\nthe parent solution to generate a new one. The population tends\\ntowards regions of the parameter space with higher fitness but it\\nrequires to sample and evaluate a large number of solutions, mak-\\ning it particularly data inefficient. In order to use the time step\\nlevel information and hence improve data efficiency, methods that\\ncombine MAP-Elites with Reinforcement Learning [ 28,30,31,38]\\nhave emerged and proved to efficiently evolve populations of high- level information and hence improve data efficiency, methods that\\ncombine MAP-Elites with Reinforcement Learning [ 28,30,31,38]\\nhave emerged and proved to efficiently evolve populations of high-\\nperforming and diverse neural network for complex tasks. PGA-\\nMAP-Elites [ 28] uses policy gradients for part of its mutations, see\\nsection 2.4 for details. CMA-MEGA [ 38] estimates descriptor gradi-\\nents with Evolution Strategies and combines the fitness gradient\\nand the descriptor gradients with a CMA-ES mechanism [ 12,19].\\nQD-PG [ 31] introduces a diversity reward based on the novelty of\\nthe states visited and derives a policy gradient for the maximiza-\\ntion of those diversity rewards which helps exploration in settings\\nwhere the reward is sparse or deceptive. PBT-MAP-Elites [ 30] mixes\\nMAP-Elites with a population based training process [ 21] to opti-\\nmize hyper-parameters of diverse RL agents. Interestingly, recent\\nwork [ 37] scales the algorithm CMA-MAE [ 13] to high-dimensional\\npolicies on robotics tasks with pure Evolution Strategies while\\nshowing comparable data efficiency to QD-RL approaches. It shows\\ncompetitiveness but is still outperformed by PGA-MAP-Elites.\\n3.2 Conditioning the critic\\nNone of the above methods takes a descriptor into account when\\nderiving policy gradients used to mutate solutions. In other words,\\nthey do not use descriptor-conditioned policies nor descriptor-\\nconditioned critics as our method DCG-MAP-Elites does. The con-\\ncept of descriptor-conditioned critic is related to the concept of\\nUniversal Value Function Approximators [ 33] and the most related\\nfield to Quality-Diversity that uses it is Skill Discovery Reinforce-\\nment Learning [ 1]. In VIC, DIAYN, DADS, SMERL [ 9,16,23,34],\\nconditioned actor-critic are used but the condition is a sampled prior\\nand does not correspond to a real posterior like in DCG-MAP-Elites.\\nFurthermore, those methods use diversity at the step level and not\\nexplicitly at the trajectory level like ours. Finally, they do not use an\\narchive to store their population, resulting in much smaller sets of\\nfinal policies. Ultimately, it has been shown that behaviors evolved\\nby QD methods are competitive with skills learned by this family of\\nmethods [ 1], in regards to their use for adaptation and hierarchical\\nlearning.\\n3.3 Archive distillation\\nDistilling the knowledge of an archive into a single neural model\\nis an alluring process that reduces the number of parameters out-\\nputted by the algorithm and enables generalization and interpo-\\nlation\\/extrapolation. Although distillation is usually referring to\\npolicy distillation \\u2014 learning the observation\\/action mapping from\\na teacher policy \\u2014 we present archive distillation as a general term\\nreferring to any kind of knowledge transfer from an archive to\\n3 another model, should it be the policies, transitions experienced in\\nthe environment, full trajectories or discovered descriptors.\\nTo the best of our knowledge, two QD-related works use the\\nconcept of archive distillation. Go-Explore [ 8] stores an archive of\\nreached states and trains a goal-conditioned policy to reproduce the\\ntrajectory of the policy that reached that state. Another interesting\\napproach to archive distillation is to learn a generative policy net-\\nwork [ 22] over the policy contained in the archive. Our approach\\nDCG-MAP-Elites distills the experience of the archive into a single\\nversatile policy.\\n4 DCG-MAP-ELITES\\nOur new method Descriptor-Conditioned Gradients MAP-Elites\\n(DCG-MAP-Elites) overcomes limitations of PGA-MAP-Elites by\\nleveraging a descriptor-conditioned critic to improve the PG varia-\\ntion operator and concurrently distills the knowledge of the archive\\nin a single versatile policy as a by-product of the actor-critic train-\\ning. The pseudocode is provided in Alg. 1. The algorithm follows\\nthe usual MAP-Elites loop of selection, variation, evaluation and\\naddition for a budget of \\ud835\\udc3citerations. Two complementary and inde-\\npendent variation operators are used in parallel: 1) a standard GA\\noperator 2) a descriptor-conditioned PG operator. At each iteration,\\nthe transitions from the evaluation step are stored in a replay buffer\\nand used to train an actor-critic pair based on TD3.\\nContrary to PGA-MAP-Elites, the actor-critic pair is descriptor-\\nconditioned. In addition to the state \\ud835\\udc60and action\\ud835\\udc4e, the critic\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\n\\ud835\\udc51)also depends on the descriptor \\ud835\\udc51and estimates the expected dis-\\ncounted return starting from state \\ud835\\udc60, taking action \\ud835\\udc4eand thereafter\\nfollowing policy \\ud835\\udf0bandachieving descriptor \\ud835\\udc51. Achieving descrip-\\ntor\\ud835\\udc51means that the descriptor of the trajectory generated by the\\npolicy\\ud835\\udf0bshould have descriptor \\ud835\\udc51. In addition to the state \\ud835\\udc60, the\\nactor\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51)also depends on the descriptor \\ud835\\udc51and maximizes the\\nexpected discounted return conditioned on achieving descriptor \\ud835\\udc51.\\nThus, the goal of the descriptor-conditioned actor is to achieve the\\ninput descriptor \\ud835\\udc51while maximizing fitness.\\n4.1 Descriptor-Conditioned Critic\\nInstead of estimating the action-value function with \\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e), we\\nwant to estimate the descriptor-conditioned action-value function\\nwith\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51). When a policy \\ud835\\udf0binteracts with the environment\\nfor an episode of length T, it generates a trajectory \\ud835\\udf0f, which is a\\nsequence of transitions:\\n(\\ud835\\udc600,\\ud835\\udc4e0,\\ud835\\udc5f0,\\ud835\\udc601),...,(\\ud835\\udc60\\ud835\\udc47\\u22121,\\ud835\\udc4e\\ud835\\udc47\\u22121,\\ud835\\udc5f\\ud835\\udc47\\u22121,\\ud835\\udc60\\ud835\\udc47)\\nwith descriptor \\ud835\\udc37(\\ud835\\udf0b)=\\ud835\\udc51. We extend the definition of a transition\\n(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f,\\ud835\\udc60\\u2032)to include the descriptor \\ud835\\udc51of the policy(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f,\\ud835\\udc60\\u2032,\\ud835\\udc51).\\nThus, a trajectory \\ud835\\udf0fwith descriptor \\ud835\\udc51gives a sequence of transitions:\\n(\\ud835\\udc600,\\ud835\\udc4e0,\\ud835\\udc5f0,\\ud835\\udc601,\\ud835\\udc51),...,(\\ud835\\udc60\\ud835\\udc47\\u22121,\\ud835\\udc4e\\ud835\\udc47\\u22121,\\ud835\\udc5f\\ud835\\udc47\\u22121,\\ud835\\udc60\\ud835\\udc47,\\ud835\\udc51)\\nHowever, the descriptor is only available at the end of the episode,\\ntherefore the transitions can only be augmented with the descriptor\\nafter the episode is done. In all the tasks we consider, the reward\\nfunction is positive \\ud835\\udc5f:S\\u00d7A\\u2192 R+and hence, the fitness function\\n\\ud835\\udc39and action-value function are positive as well. Thus, for any\\nsampled descriptor \\ud835\\udc51\\u2032\\u2208D, we define the descriptor-conditioned\\ncritic as equal to the normal action-value function when the policy\\nachieves the sampled descriptor \\ud835\\udc51\\u2032and as equal to zero when theAlgorithm 1 DCG-MAP-Elites\\nInput: batch size\\ud835\\udc4f, number of GA variations \\ud835\\udc54\\u2264\\ud835\\udc4f\\nInitialize archiveXwith\\ud835\\udc4frandom solutions and replay buffer B\\nInitialize critic networks \\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032and actor network \\ud835\\udf0b\\ud835\\udf19\\n\\ud835\\udc56\\u21900\\nwhile\\ud835\\udc56<\\ud835\\udc3cdo\\ntrain_actor_critic (\\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032,\\ud835\\udf0b\\ud835\\udf19,B)\\n\\ud835\\udf0b\\ud835\\udf131,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc4f\\u2190selection(X)\\n\\ud835\\udf0bb\\ud835\\udf131,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc54\\u2190variation_ga(\\ud835\\udf0b\\ud835\\udf131,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc54)\\n\\ud835\\udf0bb\\ud835\\udf13\\ud835\\udc54+1,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc4f\\u2190variation_pg(\\ud835\\udf0b\\ud835\\udf13\\ud835\\udc54+1,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc4f,\\ud835\\udc44\\ud835\\udf031,B)\\naddition(\\ud835\\udf0bb\\ud835\\udf131,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc4f,X,B)\\n\\ud835\\udc56\\u2190\\ud835\\udc56+\\ud835\\udc4f\\nfunction addition (X,B,\\ud835\\udf0b\\ud835\\udf19,\\ud835\\udf0bb\\ud835\\udf13...)\\nfor\\ud835\\udc51\\u2032\\u2208D sampled from \\ud835\\udc4fsolutions inXdo\\n(\\ud835\\udc53,transitions)\\u2190\\ud835\\udc39(\\ud835\\udf0b\\ud835\\udf19(.|\\ud835\\udc51\\u2032))\\ninsert(B,transitions)\\nfor\\ud835\\udf0bb\\ud835\\udf13...do\\n(\\ud835\\udc53,transitions)\\u2190\\ud835\\udc39(\\ud835\\udf0bb\\ud835\\udf13),\\ud835\\udc51\\u2190\\ud835\\udc37(\\ud835\\udf0bb\\ud835\\udf13)\\ninsert(B,transitions)\\nifX(\\ud835\\udc51)=\\u2205or\\ud835\\udc39(X(\\ud835\\udc51))<\\ud835\\udc53then\\nX(\\ud835\\udc51)\\u2190\\ud835\\udf0bb\\ud835\\udf13\\npolicy does not achieve the sampled descriptor \\ud835\\udc51\\u2032. Given a transition\\n(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f,\\ud835\\udc60\\u2032,\\ud835\\udc51), and\\ud835\\udc51\\u2032\\u2208D,\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\u2032)=(\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e),if\\ud835\\udc51=\\ud835\\udc51\\u2032\\n0, if\\ud835\\udc51\\u2260\\ud835\\udc51\\u2032(3) insert(B,transitions)\\nifX(\\ud835\\udc51)=\\u2205or\\ud835\\udc39(X(\\ud835\\udc51))<\\ud835\\udc53then\\nX(\\ud835\\udc51)\\u2190\\ud835\\udf0bb\\ud835\\udf13\\npolicy does not achieve the sampled descriptor \\ud835\\udc51\\u2032. Given a transition\\n(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f,\\ud835\\udc60\\u2032,\\ud835\\udc51), and\\ud835\\udc51\\u2032\\u2208D,\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\u2032)=(\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e),if\\ud835\\udc51=\\ud835\\udc51\\u2032\\n0, if\\ud835\\udc51\\u2260\\ud835\\udc51\\u2032(3)\\nHowever, with this piecewise definition, the descriptor-conditioned\\naction-value function is not continuous and violates the universal\\napproximation theorem continuity hypothesis [ 20]. To address this\\nissue, we introduce a similarity function \\ud835\\udc46:D2\\u2192]0,1]defined\\nas\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)=\\ud835\\udc52\\u2212||\\ud835\\udc51\\u2212\\ud835\\udc51\\u2032||D\\n\\ud835\\udc59 to smooth the descriptor-conditioned critic\\nand relax Eq. 3 into:\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\u2032)=\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e)=\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)E\\ud835\\udf0b\\\"\\ud835\\udc47\\u22121\\u2211\\ufe01\\n\\ud835\\udc61=0\\ud835\\udefe\\ud835\\udc61\\ud835\\udc5f\\ud835\\udc61\\f\\f\\f\\f\\f\\ud835\\udc60,\\ud835\\udc4e#\\n=E\\ud835\\udf0b\\\"\\ud835\\udc47\\u22121\\u2211\\ufe01\\n\\ud835\\udc61=0\\ud835\\udefe\\ud835\\udc61\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\ud835\\udc5f\\ud835\\udc61\\f\\f\\f\\f\\f\\ud835\\udc60,\\ud835\\udc4e#\\n(4)\\nWith Eq. 4, we demonstrate that learning the descriptor-conditioned\\ncritic is equivalent to scaling the reward by the similarity \\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\nbetween the descriptor of the trajectory \\ud835\\udc51and the sampled descrip-\\ntor\\ud835\\udc51\\u2032. Therefore, the critic target in Eq. 1 is modified to include the\\nsimilarity scaling and the descriptor-conditioned actor:\\n\\ud835\\udc66=\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\ud835\\udc5f(\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61)+\\ud835\\udefemin\\n\\ud835\\udc56=1,2\\ud835\\udc44\\ud835\\udf03\\ud835\\udc56\\u2032\\u0010\\n\\ud835\\udc60\\ud835\\udc61+1,\\ud835\\udf0b\\ud835\\udf19\\u2032(\\ud835\\udc60\\ud835\\udc61+1|\\ud835\\udc51\\u2032)+\\ud835\\udf16|\\ud835\\udc51\\u2032\\u0011\\n(5)\\nIf the sampled descriptor \\ud835\\udc51\\u2032is approximately equal to the observed\\ndescriptor\\ud835\\udc51of the trajectory \\ud835\\udc51\\u2248\\ud835\\udc51\\u2032then we have \\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\u22481so the\\nreward is unchanged. However, if the descriptor \\ud835\\udc51\\u2032is very different\\nfrom the observed descriptor \\ud835\\udc51then, the reward is scaled down to\\n\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\ud835\\udc5f(\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61)\\u22480. The scaling ensures that the magnitude of the\\nreward depends not only on the quality of the action \\ud835\\udc4ewith regards\\nto the fitness function \\ud835\\udc39, but also on achieving the descriptor \\ud835\\udc51\\u2032.\\nGiven one transition (\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f,\\ud835\\udc60\\u2032,\\ud835\\udc51), we can generate infinitely many\\ncritic updates by sampling \\ud835\\udc51\\u2032\\u2208D. This is leveraged in the new\\n4 actor-critic training introduced with DCG-MAP-Elites, which is\\ndetailed in Alg. 2 and section 4.3.\\nAlgorithm 2 Descriptor-conditioned Actor-Critic Training\\nfunction train_actor_critic (\\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032,\\ud835\\udf0b\\ud835\\udf19,B)\\nfor\\ud835\\udc61=1\\u2192\\ud835\\udc5bdo\\nSample\\ud835\\udc41transitions(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e),\\ud835\\udc60\\u2032,\\ud835\\udc51,\\ud835\\udc51\\u2032)fromB\\nSample smoothing noise \\ud835\\udf16\\n\\ud835\\udc66\\u2190\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e)+\\ud835\\udefemin\\n\\ud835\\udc56=1,2\\ud835\\udc44\\ud835\\udf03\\u2032\\n\\ud835\\udc56\\u0000\\ud835\\udc60\\u2032,\\ud835\\udf0b\\ud835\\udf19\\u2032(\\ud835\\udc60\\u2032|\\ud835\\udc51\\u2032)+\\ud835\\udf16|\\ud835\\udc51\\u2032\\u0001\\nUpdate both critics by regression to \\ud835\\udc66\\nif\\ud835\\udc61mod\\u0394then\\nUpdate actor using the deterministic policy gradient:\\n1\\n\\ud835\\udc41\\u00cd\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\u2032)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51\\u2032)\\u2207\\ud835\\udf19\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51\\u2032)\\nSoft-update target networks \\ud835\\udc44\\ud835\\udf03\\ud835\\udc56\\u2032and\\ud835\\udf0b\\ud835\\udf19\\u2032\\n4.2 Descriptor-Conditioned Actor and\\nArchive Distillation\\nThe training of the critic requires to train an actor \\ud835\\udf0b\\ud835\\udf19to approx-\\nimate the optimal action \\ud835\\udc4e\\u2217as explained in section 2.3. However,\\nin this work, the action-value function estimated by the critic\\nis conditioned on a descriptor \\ud835\\udc51. Hence, we don\\u2019t want to esti-\\nmate the best action globally, but the best action given that the\\npolicy should achieve descriptor \\ud835\\udc51by the end of the trajectory.\\nTherefore, the actor is extended to a descriptor-conditioned policy\\n\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51), that maximizes the descriptor-conditioned critic\\u2019s value\\nwithmax\\ud835\\udf19E\\u0002\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51)|\\ud835\\udc51)\\u0003\\n. The actor is updated using the\\ndeterministic policy gradient, see Alg. 2:\\n\\u2207\\ud835\\udf19\\ud835\\udc3d(\\ud835\\udf19)=1\\n\\ud835\\udc41\\u2211\\ufe01\\n\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\u2032)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51\\u2032)\\u2207\\ud835\\udf19\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51\\u2032)(6)\\nThe policy \\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51)learns to suggest actions \\ud835\\udc4ethat optimize\\nreward\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e)while generating a trajectory achieving descriptor \\ud835\\udc51.\\nConsequently, the descriptor-conditioned actor can exhibit a wide\\nrange of descriptors, effectively distilling some of the capabilities\\nof the archive into a single versatile policy.\\n4.3 Actor-Critic Training\\nIn section 4.1, we show that the new descriptor-conditioned critic\\ntarget in Eq. 5 requires a sampled descriptor \\ud835\\udc51\\u2032. At each iteration,\\nthe transitions(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e),\\ud835\\udc60\\u2032,\\ud835\\udc51)generated from the evaluation of\\noffspring are stored in a replay buffer B. To learn the relation\\nbetween being in state \\ud835\\udc60, taking action \\ud835\\udc4eand achieving descriptor\\n\\ud835\\udc51\\u2032, the critic needs to be trained on \\u201cpositive\\u201d samples i.e. samples\\nwith\\ud835\\udc51\\u2032=\\ud835\\udc51and \\u201cnegative\\u201d samples i.e. samples with \\ud835\\udc51\\u2032\\u2260\\ud835\\udc51. Positive\\nsamples are easy to generate because we can sample \\ud835\\udc51\\u2032as equal\\nto the observed descriptor \\ud835\\udc51of the trajectory. However, negative\\nsamples require to define an arbitrary sampling strategy that can\\nimpact the performance of the algorithm.\\nMoreover, learning to act from data without active interactions in\\nthe environment is difficult [ 29]. If the actor only learns passively on\\nrollouts performed by offspring from policies stored in the archive,\\nstrong deviation from the state-action distribution can happen,\\npreventing convergence. To solve this passive learning problem\\nand generate negative samples at the same time, we evaluate the\\nactor\\ud835\\udf0b\\ud835\\udf19in the environment. At each iteration, we sample a batch of\\ud835\\udc4fdescriptors\\ud835\\udc51\\u2032inDand evaluate \\ud835\\udf0b\\ud835\\udf19(.|\\ud835\\udc51\\u2032). The actor evaluation\\nstep gives\\ud835\\udc4ftrajectories with transitions (\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61,\\ud835\\udc5f\\ud835\\udc61,\\ud835\\udc60\\ud835\\udc61+1,\\ud835\\udc51,\\ud835\\udc51\\u2032)with\\n\\ud835\\udc51the observed descriptor achieved by the trajectory and \\ud835\\udc51\\u2032the\\ninput descriptor that is sampled for evaluation. In general, the\\nobserved descriptor and the sampled descriptor are different, \\ud835\\udc51\\u2032\\u2260\\n\\ud835\\udc51, providing negative samples to the critic training. To get the\\ndescriptors\\ud835\\udc51\\u2032, we uniformly sample with replacement \\ud835\\udc4fsolutions\\n\\ud835\\udf0b\\ud835\\udf13from the archive and take their descriptors with an additional\\nGaussian noise. The solutions have descriptors \\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)=\\ud835\\udc51\\ud835\\udf13and we\\ncompute a batch of descriptors \\ud835\\udc51\\u2032=\\ud835\\udc51\\ud835\\udf13+N(0D,\\ud835\\udf0e\\ud835\\udc51\\ud835\\udc3c). The negative\\nsamples from the actor evaluation are stored in the replay buffer\\nalongside the positive samples from offspring evaluation for which\\nwe take\\ud835\\udc51\\u2032=\\ud835\\udc51, as detailed in the addition function of Alg. 1.\\n4.4 Descriptor-Conditioned PG variation\\nOnce the critic \\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51)is learned, it can be used to improve the\\nfitness of solutions in the archive, as described in Alg. 3. A parent\\nsolution\\ud835\\udf0b\\ud835\\udf13with descriptor \\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)=\\ud835\\udc51\\ud835\\udf13is selected from the archive.\\nThen, we apply the PG variation operator, using the descriptor \\ud835\\udc51\\ud835\\udf13\\nto condition the critic, and thus, to apply \\ud835\\udc5adescriptor-conditioned solution\\ud835\\udf0b\\ud835\\udf13with descriptor \\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)=\\ud835\\udc51\\ud835\\udf13is selected from the archive.\\nThen, we apply the PG variation operator, using the descriptor \\ud835\\udc51\\ud835\\udf13\\nto condition the critic, and thus, to apply \\ud835\\udc5adescriptor-conditioned\\ngradient steps using the deterministic policy gradient:\\n\\u2207\\ud835\\udf13\\ud835\\udc3d(\\ud835\\udf13)=1\\n\\ud835\\udc41\\u2211\\ufe01\\n\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\ud835\\udf13)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60)\\u2207\\ud835\\udf13\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60) (7)\\nThe goal is to improve the fitness of the solution while remaining\\nin the same cell as the parent.\\nAlgorithm 3 Descriptor-conditioned PG Variation\\nfunction variation_pg (\\ud835\\udf0b\\ud835\\udf13...,\\ud835\\udc44 \\ud835\\udf031,B)\\nfor\\ud835\\udf0b\\ud835\\udf13...do\\n\\ud835\\udc51\\ud835\\udf13\\u2190\\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)\\nfor\\ud835\\udc56=1\\u2192\\ud835\\udc5ado\\nSample\\ud835\\udc41transitions(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e),\\ud835\\udc60\\u2032,\\ud835\\udc51,\\ud835\\udc51\\u2032)fromB\\nUpdate actor using the deterministic policy gradient:\\n1\\n\\ud835\\udc41\\u00cd\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\ud835\\udf13)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60)\\u2207\\ud835\\udf13\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60)\\nreturn\\ud835\\udf0bb\\ud835\\udf19...\\n5 EXPERIMENTS\\n5.1 Evaluation Tasks\\nWe evaluate DCG-MAP-Elites on four QDGym tasks [ 28] imple-\\nmented in Brax [ 14] and derived from standard DRL benchmarks\\nfor robot locomotion. We call these tasks \\u201cAnt-Omni\\u201d, \\u201cAntTrap-\\nOmni\\u201d, \\u201cHexapod-Omni\\u201d, and \\u201cWalker-Uni\\u201d. Three omnidirectional\\ntasks and one unidirectional task are considered for evaluation.\\nAnt-Omni, AntTrap-Omni and Hexapod-Omni are omnidirectional\\ntasks in which the objective is for the robot to reach all possible\\npoints in the plane while minimizing energy consumption. Walker-\\nUni is a unidirectional task in which the objective is for the robot\\nto discover all possible ways to walk forward while maximizing a\\ntrade-off between speed and energy consumption. Walker-Uni is\\nexactly the same tasks used in PGA-MAP-Elites paper [ 28], Ant-\\nOmni and Hexapod-Omni have been used by Flageat and Lim et\\nal. [11] and AntTrap-Omni is adapted from QD-PG paper [ 31]. The\\ndifference is the elimination of the forward reward term in the\\nreward function. The goal of AntTrap-Omni is to reach every point\\n5 in the plane while minimizing energy consumption in a deceptive\\nenvironment. This new task is designed to show that our algorithm\\nperforms well in environment where there is a discontinuity on the\\nfitness landscape in descriptor space caused by the trap. The trap\\nis causing a discontinuity that the descriptor-conditioned critic has\\nto learn. Indeed, points on both sides of the trap will be close in the\\ndescriptor space, but distant in terms of trajectory to achieve these\\ndescriptors. We detail these tasks in Tab. 1.\\nPGA-MAP-Elites has previously shown state-of-the-art results\\non some of these tasks, in particular Walker-Uni but tends to strug-\\ngle in omnidirectional tasks. In those tasks, the global optimal of\\nthe fitness function is a solution that prevents the robot to move,\\nwhich is directly opposed to the objective of discovering how to\\nreach different locations. Hence, most of the offspring generated by\\nthe PG variation will tend to move less and travel a shorter distance.\\nInstead, DCG-MAP-Elites aims to improve the energy consumption\\nwhile maintaining the ability to reach distant locations.\\nTable 1: Evaluation Tasks\\nAnt AntTrap Hexapod Walker\\nState Position and velocity of body and joints\\nAction Torques for each joint\\nType Omni Omni Omni Uni\\nDescriptor Final(\\ud835\\udc65,\\ud835\\udc66)position Feet contact\\nFitness Surviving reward + Energy consumption penalty\\nState dim 29 29 48 19\\nAction dim 8 8 18 6\\nEpisode len 250 250 250 1000\\nParameters 21,384 21,384 25,106 19,846\\n5.2 Baselines\\nWe compare DCG-MAP-Elites with four algorithms: CVT-MAP-\\nElites [ 39], MAP-Elites-ES [ 5], PGA-MAP-Elites [ 28] and QD-PG [ 31].\\nWe could not compare to PBT-MAP-Elites [ 30] as its implementa-\\ntion is not yet available. We use QDax implementations and each\\nexperiment is replicated 20 times with random seeds, over one\\nmillion evaluations. The source code of DCG-MAP-Elites is avail-\\nable at github.com\\/adaptive-intelligent-robotics\\/DCG-MAP-Elites\\nincluding a container in which all the experiments can be replicated.\\n5.3 Evaluation Metrics\\nWe consider three main metrics to evaluate the archives and two\\nadditional metrics to evaluate the descriptor-conditioned actor. We\\ncompute p-values based on the Wilcoxon rank-sum test with a\\nBonferroni correction for all metrics.\\n\\u2022QD-score: The sum of fitness of all solutions stored in the archive.\\nIn the task considered, fitnesses are always positive, which avoid\\npenalizing algorithms for finding additional solutions. This score\\nencompasses both the quality and diversity of the population.\\n\\u2022Coverage: The proportion of filled cells in the archive, indicating\\nthe success of descriptor space illumination.\\u2022Maximum fitness: The fitness of the best solution in the archive.\\nWe also consider two additional metrics to evaluate the descriptor-\\nconditioned policy. The purpose of these metrics is to quantify how\\nsimilar the archive and the descriptor-conditioned policy are in\\nterms of QD-score and coverage of the descriptor space.\\n\\u2022Descriptor-conditioned QD-score: The sum of fitness achieved\\nby the descriptor-conditioned policy evaluated on each descriptor\\nof the solutions in the archive.\\u2211\\ufe01\\n\\ud835\\udf0b\\ud835\\udf13\\u2208X\\ud835\\udc39(\\ud835\\udf0b\\ud835\\udf19(.|\\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)))\\n\\u2022Descriptor error mean: (i) for the archive, the average differ-\\nence over all solutions, between the stored descriptor and the\\nreevaluated descriptor, (ii) for the descriptor-conditioned policy,\\nthe average difference over all solutions, between the stored de-\\nscriptor and the reevaluated descriptor achieved by the policy\\nconditioned on the stored descriptor. ( |X|number of solutions in\\nthe archive).\\n1\\n|X|\\u2211\\ufe01\\n\\ud835\\udf0b\\ud835\\udf13\\u2208X||\\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf19(.|\\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)))\\u2212\\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)||D\\n5.4 Results\\n5.4.1 Archive. The experimental results on Fig. 2 demonstrate that\\nDCG-MAP-Elites achieves the best archive in the two Ant tasks in\\nterms of QD-score ( \\ud835\\udc5d<10\\u22126) and coverage ( \\ud835\\udc5d<10\\u22124) compared\\nto all other baselines. On the two other tasks, the QD-score of\\nour algorithm is similar to PGA-MAP-Elites, the previous state-\\nof-the-art, while achieving a significantly better coverage than to all other baselines. On the two other tasks, the QD-score of\\nour algorithm is similar to PGA-MAP-Elites, the previous state-\\nof-the-art, while achieving a significantly better coverage than\\nall baselines ( \\ud835\\udc5d<10\\u22123). The coverage metric shows that DCG-\\nMAP-Elites surpasses the exploration capabilities of QD-PG on\\nall tasks (\\ud835\\udc5d<10\\u22124). We also show that our method still benefits\\nfrom the exploration power of the GA operator even in deceptive\\nenvironment like AntTrap-Omni.\\n30\\n 20\\n 10\\n 0 10 20 30\\nx position30\\n20\\n10\\n0102030y positionPGA-MAP-Elites\\n30\\n 20\\n 10\\n 0 10 20 30\\nx positionDCG-MAP-Elites\\n6006507007508008509009501000\\nFigure 3: Final archives for PGA-MAP-Elites and DCG-MAP-\\nElites on Ant-Omni after one million evaluations.\\nThe experimental results confirm that DCG-MAP-Elites is able\\nto overcome the limits of PGA-MAP-Elites on omnidirectional tasks\\nwhile still performing on par when evaluated on the unidirectional\\ntask (Walker-Uni) where no significant improvement was expected.\\nThus confirming the interest of having a descriptor-conditioned\\ngradient to make the PG variation operator fruitful in a wider range\\n6 0.00.20.40.60.8QD-Score (1e6)Ant-Omni\\n0.00.20.40.60.8AntTrap-Omni\\n0.00.20.40.6Hexapod-Omni\\n0.00.51.01.5Walker-Uni\\n020406080Coverage (%)\\n020406080\\n020406080\\n020406080\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e67008009001000Maximum fitness\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e67008009001000\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e6950100010501100\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e60100020003000\\nDCG-MAP-Elites PGA-MAP-Elites QD-PG MAP-Elites MAP-Elites ESFigure 2: QD-score, coverage and maximum fitness for DCG-MAP-Elites and the baselines on all tasks. Each experiment is\\nreplicated 20 times with different random seeds. The solid line is the median and the shaded area is the first and third quartiles.\\nof tasks. Overall, DCG-MAP-Elites shows best (or competitive) per-\\nformance on all metrics and tasks, hence proving to be the first\\nsuccessful effort in the QD-RL literature to achieve well on both\\nthe unidirectional and omnidirectional tasks. Previous efforts were\\nusually adapted to either one or the other [ 28,31,38]. Nevertheless,\\nour results show that there is still room for improvement. In partic-\\nular, we can see that DCG-MAP-Elites was not able to outperform\\nthe baselines with the same margin on Hexapod-Omni that it has\\nbeen able to do on the other omnidirectional tasks. We hypothesize\\nthat the environment dynamics and the higher-dimensional search\\nspace of Hexapod-Omni makes it much more challenging.\\nTable 2: Median of the QD-score and Descriptor Error Mean\\n(DEM) after one million evaluations for the archive and for\\nthe descriptor-conditioned policy over 20 replications.\\nMetric Type Ant AntTrap Hexapod Walker\\nQD-score\\n(105)Archive 8.7 7.2 6.5 9.8\\nPolicy 8.3 7.0 0.96 5.2\\nDEMArchive 4.04 3.44 0.078 0.12\\nPolicy 6.25 5.8 1.39 0.27\\n5.4.2 Descriptor-Conditioned Policy. In Tab. 2 we provide the final\\nQD-score and Descriptor Error Mean (DEM) of the archive and\\nthe descriptor-conditioned policy, see section 5.3 for definitions.\\nFirst, on Ant-Omni and AntTrap-Omni, the descriptor-conditioned\\npolicy achieves the same QD-score as the archive, measured by\\nthe descriptor-conditioned QD-score (decreases by less than 3%).\\nThis shows that the single-conditioned policy is able to restore\\nthe quality of the archive although having compressed the infor-\\nmation in a single network. Second, it is interesting to cross this\\nobservation with the fact that, at the same time, the Descriptor\\nError Mean obtained by the conditioned policy is reasonable. We\\nobserve a DEM of 6.25, which is less than 8% of the typical size ofthe environment (i.e. the diagonal of the square considered). The\\nreader can refer to Fig. 3 to get an idea of what that represents. In\\naddition, the DEM of the archive is of 4.04 (4.7% of size of the envi-\\nronment), showing an inevitable stochasticity in the evaluation that\\nis related to the simulator used [ 14]. Those two combined observa-\\ntions (QD-score and DEM) tend to show that our archive and our\\ndescriptor-conditioned policy have similar properties on Ant-Omni\\nand AntTrap-Omni and hence that the versatile policy could be a\\npractical summary of our archive. Third, we can observe that on the\\nHexapod-Omni and Walker-Uni tasks, the descriptor-conditioned\\npolicy struggles to recover the QD-score of the archive (reduced\\nby more than 40% in both cases). On the Hexapod-Omni, the DEM\\nof the descriptor-conditioned policy is significantly larger than the\\ntypical error obtained by the archive (1.39 vs 0.078). This task is\\nmore challenging than Ant-Omni because of a higher number of\\ndimensions (see Tab. 1) and we hypothesize that the optimization\\nalgorithm used (TD3) could be the limitation. For Walker-Uni, the\\nDEM of the archive (0.12) shows that there is also an inherent vari-\\nability of the descriptor, we are hence less surprised by the DEM\\nobtained by the single-conditioned policy (0.27) although it could\\nmake it less usable in practice. ability of the descriptor, we are hence less surprised by the DEM\\nobtained by the single-conditioned policy (0.27) although it could\\nmake it less usable in practice.\\nOverall, those results shows that our single descriptor-conditioned\\npolicy can already be seen as a promising summary of our archive,\\nshowing very similar properties on half our tasks. The other tasks\\nprove that there is still room for improvement of our method.\\n5.4.3 Ablations. We perform additional ablation experiments to\\nshow the importance of three components of DCG-MAP-Elites,\\nnamely: using negative samples for the critic\\u2019s training, actively\\ncollecting experiences with the actor, and conditioning the actor\\non the descriptor. In Ablation (1) , we remove the actor evaluation\\nin the environment. This allows to study the impact of both pas-\\nsive learning and of lack of negative samples, see section 4.3 for a\\n7 0.00.20.40.60.8QD-Score (1e6)Ant-Omni\\n0.00.20.40.60.8AntTrap-Omni\\n0.00.20.40.6Hexapod-Omni\\n0.00.51.01.5Walker-Uni\\n020406080Coverage (%)\\n020406080\\n020406080\\n020406080\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e699099510001005Maximum fitness\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e699099510001005\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e6111011201130\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e6100020003000\\nDCG-MAP-Elites Ablation 1 Ablation 2 Ablation 3Figure 4: QD-score, coverage and maximum fitness for DCG-MAP-Elites and the ablations on all tasks. Each experiment is\\nreplicated 20 times with different random seeds. The solid line is the median and the shaded area is the first and third quar-\\ntiles. Ablation (1) is DCG-MAP-Elites without actor evaluation and without negative samples, Ablation (2) is DCG-MAP-Elites\\nwithout actor evaluation but with negative samples, Ablation (3) is DCG-MAP-Elites without a descriptor-conditioned actor.\\ndetailed explanation. The lack of active learning deprives the learn-\\ning of negative samples that are necessary to learn the descriptor-\\nconditioned action-value function. We observe that ablation (1) is\\nperforming worse than DCG-MAP-Elites on all omnidirectional\\ntasks (\\ud835\\udc5d<0.01). However it performs similarly on Walker-Uni,\\nwhich can be explained by the fact that non-conditioned PG emitter\\nalready prove efficient in solving this task. This first ablation result\\nhighlight the importance of active learning and negative examples\\nfor the critic\\u2019s training. To complement this result, in Ablation (2) ,\\nwe remove the actor evaluation but we generate negative samples\\nartificially by sampling in the descriptor space. We show that ab-\\nlation (2) is performing significantly worse than DCG-MAP-Elites\\non Ant tasks ( \\ud835\\udc5d<10\\u22123) but performs similarly on Hexapod-Omni\\nand Walker-Uni. However, ablation (2) is performing significantly\\nbetter than ablation (1) on Ant tasks ( \\ud835\\udc5d<10\\u22125), demonstrating the\\nimportance of both active learning andnegative samples. Finally, in\\nAblation (3) we replace the descriptor-conditioned actor \\ud835\\udf0b\\ud835\\udf19(.|\\ud835\\udc51)\\nwith a normal actor \\ud835\\udf0b\\ud835\\udf19(.). Ablation (3) is performing significantly\\nworse on all omnidirectional tasks ( \\ud835\\udc5d<0.05), demonstrating the\\nimportance of the descriptor-conditioned actor for the critic\\u2019s train-\\ning. Ablation (3) performs equivalently to DCG-MAP-Elites on\\nWalker-Uni.\\n6 CONCLUSION\\nIn this work, we introduce DCG-MAP-Elites and demonstrate the\\nbenefits of having descriptor-conditioned gradients to evolve popu-\\nlations of large neural networks. We concurrently train a descriptor-\\nconditioned actor, as a by-product of the critic\\u2019s training, that can\\nachieve a wide range of descriptors. Our method performs better\\nthan PGA-MAP-Elites by a significant margin in omnidirectional\\ntasks while maintaining similar performance to PGA-MAP-Elitesin unidirectional tasks. We also show that the synergy between the\\nfitness improvement capabilities of the PG variations and the explo-\\nration capabilities of the GA variations is preserved, even in decep-\\ntive environments. On some tasks, the descriptor-conditioned pol-\\nicy demonstrates properties that are similar to the discrete archive,\\nsummarizing its capabilities into one single neural network and\\nacting as a continuous archive. We think that distilling the archive\\ninto a single policy is a promising method as it enables to have\\nless redundancy compared to a discrete archive in which most of\\nthe solutions can be similar, especially between close cells. The\\ndescriptor-conditioned policy could also negate the burden of deal-\\ning with archive of thousands of solutions in practical applications.\\nThe benefits of combining DRL methods with MAP-Elites come\\nwith the limitations of MDP settings. Specifically, we are limited\\nto evolving differentiable solutions and the foundations of DRL\\nalgorithms rely on the Markov property and full observability. In\\nthis work in particular, we face challenges with the Markov property to evolving differentiable solutions and the foundations of DRL\\nalgorithms rely on the Markov property and full observability. In\\nthis work in particular, we face challenges with the Markov property\\nbecause the descriptors depend on full trajectories. Thus, the scaled\\nreward introduced in our method depends on the full trajectory\\nand not only on the current state and action. The performance of\\nthe descriptor-conditioned policy also shows that there is room for\\nimprovement to better distill the knowledge of the archive.\\nFor future work, we would like to investigate the generaliza-\\ntion capabilities of the descriptor-conditioned policy trained with\\nDCG-MAP-Elites and try to produce solutions with descriptors\\nthat are not in the archive, effectively performing descriptor space\\ninterpolation. In our method, the critic attempts to mutate solu-\\ntions to produce offspring with higher fitness while keeping their\\ndescriptors constant. We think that we could use the descriptor-\\nconditioned critic to mutate solutions to produce offspring towards\\ndifferent descriptors, thereby explicitly promoting diversity.\\n8 REFERENCES\\n[1]Felix Chalumeau, Raphael Boige, Bryan Lim, Valentin Mac\\u00e9, Maxime Allard,\\nArthur Flajolet, Antoine Cully, and Thomas Pierrot. 2022. Neuroevolution is a\\nCompetitive Alternative to Reinforcement Learning for Skill Discovery. https:\\n\\/\\/doi.org\\/10.48550\\/arXiv.2210.03516 arXiv:2210.03516 [cs].\\n[2]Felix Chalumeau, Thomas Pierrot, Valentin Mac\\u00e9, Arthur Flajolet, Karim Beguir,\\nAntoine Cully, and Nicolas Perrin-Gilbert. 2022. Assessing Quality-Diversity\\nNeuro-Evolution Algorithms Performance in Hard Exploration Problems. http:\\n\\/\\/arxiv.org\\/abs\\/2211.13742 arXiv:2211.13742 [cs].\\n[3]Konstantinos Chatzilygeroudis, Antoine Cully, Vassilis Vassiliades, and Jean-\\nBaptiste Mouret. 2020. Quality-Diversity Optimization: a novel branch of sto-\\nchastic optimization. http:\\/\\/arxiv.org\\/abs\\/2012.04322 arXiv:2012.04322 [cs, math,\\nstat].\\n[4]Konstantinos Chatzilygeroudis, Vassilis Vassiliades, and Jean-Baptiste Mouret.\\n2018. Reset-free Trial-and-Error Learning for Robot Damage Recovery. Robotics\\nand Autonomous Systems 100 (Feb. 2018), 236\\u2013250. https:\\/\\/doi.org\\/10.1016\\/j.\\nrobot.2017.11.010 arXiv:1610.04213 [cs].\\n[5]C\\u00e9dric Colas, Joost Huizinga, Vashisht Madhavan, and Jeff Clune. 2020. Scal-\\ning MAP-Elites to Deep Neuroevolution. CoRR abs\\/2003.01825 (2020), 67\\u201375.\\narXiv:2003.01825 https:\\/\\/arxiv.org\\/abs\\/2003.01825\\n[6]Antoine Cully, Jeff Clune, Danesh Tarapore, and Jean-Baptiste Mouret. 2015.\\nRobots that can adapt like animals. Nature 521, 7553 (May 2015), 503\\u2013507. https:\\n\\/\\/doi.org\\/10.1038\\/nature14422 arXiv:1407.3501 [cs, q-bio].\\n[7]Antoine Cully and Yiannis Demiris. 2017. Quality and Diversity Optimization: A\\nUnifying Modular Framework. http:\\/\\/arxiv.org\\/abs\\/1708.09251 arXiv:1708.09251\\n[cs].\\n[8]Adrien Ecoffet, Joost Huizinga, Joel Lehman, Kenneth O. Stanley, and Jeff Clune.\\n2021. Go-Explore: a New Approach for Hard-Exploration Problems. http:\\n\\/\\/arxiv.org\\/abs\\/1901.10995 arXiv:1901.10995 [cs, stat].\\n[9]Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine. 2018.\\nDiversity is All You Need: Learning Skills without a Reward Function. http:\\n\\/\\/arxiv.org\\/abs\\/1802.06070 arXiv:1802.06070 [cs].\\n[10] Manon Flageat, Felix Chalumeau, and Antoine Cully. 2022. Empirical analysis of\\nPGA-MAP-Elites for Neuroevolution in Uncertain Domains. http:\\/\\/arxiv.org\\/\\nabs\\/2210.13156 arXiv:2210.13156 [cs].\\n[11] Manon Flageat, Bryan Lim, Luca Grillotti, Maxime Allard, Sim\\u00f3n C. Smith,\\nand Antoine Cully. 2022. Benchmarking Quality-Diversity Algorithms on\\nNeuroevolution for Reinforcement Learning. http:\\/\\/arxiv.org\\/abs\\/2211.02193\\narXiv:2211.02193 [cs].\\n[12] Matthew C. Fontaine and Stefanos Nikolaidis. 2021. Differentiable Quality Diver-\\nsity. https:\\/\\/doi.org\\/10.48550\\/arXiv.2106.03894 arXiv:2106.03894 [cs].\\n[13] Matthew C. Fontaine and Stefanos Nikolaidis. 2023. Covariance Matrix\\nAdaptation MAP-Annealing. https:\\/\\/doi.org\\/10.48550\\/arXiv.2205.10752\\narXiv:2205.10752 [cs].\\n[14] C. Daniel Freeman, Erik Frey, Anton Raichuk, Sertan Girgin, Igor Mordatch, and\\nOlivier Bachem. 2021. Brax \\u2013 A Differentiable Physics Engine for Large Scale\\nRigid Body Simulation. http:\\/\\/arxiv.org\\/abs\\/2106.13281 arXiv:2106.13281 [cs].\\n[15] Scott Fujimoto, Herke van Hoof, and David Meger. 2018. Addressing Function\\nApproximation Error in Actor-Critic Methods. http:\\/\\/arxiv.org\\/abs\\/1802.09477\\narXiv:1802.09477 [cs, stat].\\n[16] Karol Gregor, Danilo Jimenez Rezende, and Daan Wierstra. 2016. Variational\\nIntrinsic Control. http:\\/\\/arxiv.org\\/abs\\/1611.07507 arXiv:1611.07507 [cs].\\n[17] Shixiang Gu, Ethan Holly, Timothy Lillicrap, and Sergey Levine. 2016. Deep\\nReinforcement Learning for Robotic Manipulation with Asynchronous Off-Policy\\nUpdates. http:\\/\\/arxiv.org\\/abs\\/1610.00633 arXiv:1610.00633 [cs].\\n[18] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. 2018. Soft\\nActor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a\\nStochastic Actor. http:\\/\\/arxiv.org\\/abs\\/1801.01290 arXiv:1801.01290 [cs, stat].\\n[19] Nikolaus Hansen. 2016. The CMA Evolution Strategy: A Tutorial. http:\\/\\/arxiv. Stochastic Actor. http:\\/\\/arxiv.org\\/abs\\/1801.01290 arXiv:1801.01290 [cs, stat].\\n[19] Nikolaus Hansen. 2016. The CMA Evolution Strategy: A Tutorial. http:\\/\\/arxiv.\\norg\\/abs\\/1604.00772 arXiv:1604.00772 [cs, stat].\\n[20] Kurt Hornik, Maxwell Stinchcombe, and Halbert White. 1989. Multilayer feedfor-\\nward networks are universal approximators. Neural Networks 2, 5 (1989), 359\\u2013366.\\nhttps:\\/\\/doi.org\\/10.1016\\/0893-6080(89)90020-8\\n[21] Max Jaderberg, Valentin Dalibard, Simon Osindero, Wojciech M. Czarnecki, Jeff\\nDonahue, Ali Razavi, Oriol Vinyals, Tim Green, Iain Dunning, Karen Simonyan,\\nChrisantha Fernando, and Koray Kavukcuoglu. 2017. Population Based Training\\nof Neural Networks. https:\\/\\/doi.org\\/10.48550\\/arXiv.1711.09846 arXiv:1711.09846\\n[cs].\\n[22] Marija Jegorova, St\\u00e9phane Doncieux, and Timothy Hospedales. 2019. Behavioural\\nRepertoire via Generative Adversarial Policy Networks. In 2019 Joint IEEE 9th\\nInternational Conference on Development and Learning and Epigenetic Robotics\\n(ICDL-EpiRob) . https:\\/\\/doi.org\\/10.1109\\/ICDL-EpiRob44920.2019 arXiv:1811.02945\\n[cs, stat].\\n[23] Saurabh Kumar, Aviral Kumar, Sergey Levine, and Chelsea Finn. 2020. One\\nSolution is Not All You Need: Few-Shot Extrapolation via Structured MaxEnt RL.\\nhttp:\\/\\/arxiv.org\\/abs\\/2010.14484 arXiv:2010.14484 [cs].[24] Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom\\nErez, Yuval Tassa, David Silver, and Daan Wierstra. 2019. Continuous control with\\ndeep reinforcement learning. http:\\/\\/arxiv.org\\/abs\\/1509.02971 arXiv:1509.02971\\n[cs, stat].\\n[25] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis\\nAntonoglou, Daan Wierstra, and Martin Riedmiller. 2013. Playing Atari with\\nDeep Reinforcement Learning. http:\\/\\/arxiv.org\\/abs\\/1312.5602 arXiv:1312.5602\\n[cs].\\n[26] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness,\\nMarc G. Bellemare, Alex Graves, Martin Riedmiller, Andreas K. Fidjeland, Georg\\nOstrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen\\nKing, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. 2015.\\nHuman-level control through deep reinforcement learning. Nature 518, 7540 (Feb.\\n2015), 529\\u2013533. https:\\/\\/doi.org\\/10.1038\\/nature14236 Number: 7540 Publisher:\\nNature Publishing Group.\\n[27] Jean-Baptiste Mouret and Jeff Clune. 2015. Illuminating search spaces by mapping\\nelites. http:\\/\\/arxiv.org\\/abs\\/1504.04909 arXiv:1504.04909 [cs, q-bio].\\n[28] Olle Nilsson and Antoine Cully. 2021. Policy gradient assisted MAP-Elites. In\\nProceedings of the Genetic and Evolutionary Computation Conference . ACM, Lille\\nFrance, 866\\u2013875. https:\\/\\/doi.org\\/10.1145\\/3449639.3459304\\n[29] Georg Ostrovski, Pablo Samuel Castro, and Will Dabney. 2021. The Difficulty of\\nPassive Learning in Deep Reinforcement Learning. http:\\/\\/arxiv.org\\/abs\\/2110.\\n14020 arXiv:2110.14020 [cs].\\n[30] Thomas Pierrot and Arthur Flajolet. 2023. Evolving Populations of Diverse RL\\nAgents with MAP-Elites. https:\\/\\/openreview.net\\/forum?id=CBfYffLqWqb\\n[31] Thomas Pierrot, Valentin Mac\\u00e9, F\\u00e9lix Chalumeau, Arthur Flajolet, Geoffrey\\nCideron, Karim Beguir, Antoine Cully, Olivier Sigaud, and Nicolas Perrin-Gilbert.\\n2022. Diversity Policy Gradient for Sample Efficient Quality-Diversity Opti-\\nmization. In Proceedings of the Genetic and Evolutionary Computation Conference .\\n1075\\u20131083. https:\\/\\/doi.org\\/10.1145\\/3512290.3528845 arXiv:2006.08505 [cs].\\n[32] Justin K. Pugh, Lisa B. Soros, and Kenneth O. Stanley. 2016. Quality Diversity:\\nA New Frontier for Evolutionary Computation. Frontiers in Robotics and AI 3\\n(2016). https:\\/\\/www.frontiersin.org\\/articles\\/10.3389\\/frobt.2016.00040\\n[33] Tom Schaul, Daniel Horgan, Karol Gregor, and David Silver. 2015. Universal Value\\nFunction Approximators. In Proceedings of the 32nd International Conference on\\nMachine Learning (Proceedings of Machine Learning Research, Vol. 37) , Francis\\nBach and David Blei (Eds.). PMLR, Lille, France, 1312\\u20131320. https:\\/\\/proceedings.\\nmlr.press\\/v37\\/schaul15.html Machine Learning (Proceedings of Machine Learning Research, Vol. 37) , Francis\\nBach and David Blei (Eds.). PMLR, Lille, France, 1312\\u20131320. https:\\/\\/proceedings.\\nmlr.press\\/v37\\/schaul15.html\\n[34] Archit Sharma, Shixiang Gu, Sergey Levine, Vikash Kumar, and Karol Hausman.\\n2020. Dynamics-Aware Unsupervised Discovery of Skills. http:\\/\\/arxiv.org\\/abs\\/\\n1907.01657 arXiv:1907.01657 [cs, stat].\\n[35] David Silver, Aja Huang, Chris J. Maddison, Arthur Guez, Laurent Sifre, George\\nvan den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershel-\\nvam, Marc Lanctot, Sander Dieleman, Dominik Grewe, John Nham, Nal Kalch-\\nbrenner, Ilya Sutskever, Timothy Lillicrap, Madeleine Leach, Koray Kavukcuoglu,\\nThore Graepel, and Demis Hassabis. 2016. Mastering the game of Go with\\ndeep neural networks and tree search. Nature 529, 7587 (Jan. 2016), 484\\u2013489.\\nhttps:\\/\\/doi.org\\/10.1038\\/nature16961 Number: 7587 Publisher: Nature Publishing\\nGroup.\\n[36] David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and\\nMartin Riedmiller. 2014. Deterministic Policy Gradient Algorithms. In Proceedings\\nof the 31st International Conference on Machine Learning (Proceedings of Machine\\nLearning Research, Vol. 32) , Eric P. Xing and Tony Jebara (Eds.). PMLR, Bejing,\\nChina, 387\\u2013395. https:\\/\\/proceedings.mlr.press\\/v32\\/silver14.html\\n[37] Bryon Tjanaka, Matthew C. Fontaine, Aniruddha Kalkar, and Stefanos Nikolaidis.\\n2022. Training Diverse High-Dimensional Controllers by Scaling Covariance\\nMatrix Adaptation MAP-Annealing. https:\\/\\/doi.org\\/10.48550\\/arXiv.2210.02622\\narXiv:2210.02622 [cs].\\n[38] Bryon Tjanaka, Matthew C. Fontaine, Julian Togelius, and Stefanos Nikolaidis.\\n2022. Approximating Gradients for Differentiable Quality Diversity in Reinforce-\\nment Learning. https:\\/\\/doi.org\\/10.48550\\/arXiv.2202.03666 arXiv:2202.03666\\n[cs].\\n[39] Vassilis Vassiliades, Konstantinos Chatzilygeroudis, and Jean-Baptiste Mouret.\\n2017. Using Centroidal Voronoi Tessellations to Scale Up the Multi-dimensional\\nArchive of Phenotypic Elites Algorithm. http:\\/\\/arxiv.org\\/abs\\/1610.05729\\narXiv:1610.05729 [cs].\\n[40] Oriol Vinyals, Igor Babuschkin, Wojciech M. Czarnecki, Micha\\u00ebl Mathieu, An-\\ndrew Dudzik, Junyoung Chung, David H. Choi, Richard Powell, Timo Ewalds,\\nPetko Georgiev, Junhyuk Oh, Dan Horgan, Manuel Kroiss, Ivo Danihelka, Aja\\nHuang, Laurent Sifre, Trevor Cai, John P. Agapiou, Max Jaderberg, Alexander S.\\nVezhnevets, R\\u00e9mi Leblond, Tobias Pohlen, Valentin Dalibard, David Budden, Yury\\nSulsky, James Molloy, Tom L. Paine, Caglar Gulcehre, Ziyu Wang, Tobias Pfaff,\\nYuhuai Wu, Roman Ring, Dani Yogatama, Dario W\\u00fcnsch, Katrina McKinney,\\nOliver Smith, Tom Schaul, Timothy Lillicrap, Koray Kavukcuoglu, Demis Has-\\nsabis, Chris Apps, and David Silver. 2019. Grandmaster level in StarCraft II\\nusing multi-agent reinforcement learning. Nature 575, 7782 (Nov. 2019), 350\\u2013\\n354. https:\\/\\/doi.org\\/10.1038\\/s41586-019-1724-z Number: 7782 Publisher: Nature\\nPublishing Group.\\n9 SUPPLEMENTARY MATERIAL \\u2014 MAP-ELITES\\nWITH DESCRIPTOR-CONDITIONED\\nGRADIENTS AND ARCHIVE DISTILLATION\\nINTO A SINGLE POLICY\\nA ALGORITHMS\\nA.1 MAP-Elites\\nAlgorithm 4 MAP-Elites\\nInput: batch size\\ud835\\udc4f\\nInitialize archiveXwith\\ud835\\udc4frandom solutions\\n\\ud835\\udc56\\u21900\\nwhile\\ud835\\udc56<\\ud835\\udc3cdo\\n\\ud835\\udc651,...,\\ud835\\udc65 \\ud835\\udc4f\\u2190selection(X)\\nb\\ud835\\udc651,...,b\\ud835\\udc65\\ud835\\udc4f\\u2190variation(\\ud835\\udc651,...,\\ud835\\udc65 \\ud835\\udc4f)\\naddition(X,b\\ud835\\udc651,...,b\\ud835\\udc65\\ud835\\udc4f)\\n\\ud835\\udc56\\u2190\\ud835\\udc56+\\ud835\\udc4f\\nfunction addition (X,b\\ud835\\udc65... ) :\\nforb\\ud835\\udc65... do\\n\\ud835\\udc53\\u2190\\ud835\\udc39(b\\ud835\\udc65),\\ud835\\udc51\\u2190\\ud835\\udc37(b\\ud835\\udc65)\\nifX(\\ud835\\udc51)=\\u2205or\\ud835\\udc39(X(\\ud835\\udc51))<\\ud835\\udc53then\\nX(\\ud835\\udc51)\\u2190b\\ud835\\udc65\\nA.2 PGA-MAP-Elites\\nAlgorithm 5 PGA-MAP-Elites\\nInput: batch size\\ud835\\udc4f, number of GA variations \\ud835\\udc54\\u2264\\ud835\\udc4f\\nInitialize archiveXwith\\ud835\\udc4frandom solutions and replay buffer B\\nInitialize critic networks \\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032and actor network \\ud835\\udf0b\\ud835\\udf19\\n\\ud835\\udc56\\u21900\\nwhile\\ud835\\udc56<\\ud835\\udc3cdo\\ntrain_actor_critic (\\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032,\\ud835\\udf0b\\ud835\\udf19,B)\\n\\ud835\\udf0b\\ud835\\udf131,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc4f\\u22121\\u2190selection(X)\\n\\ud835\\udf0bb\\ud835\\udf131,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc54\\u2190variation_ga(\\ud835\\udf0b\\ud835\\udf131,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc54)\\n\\ud835\\udf0bb\\ud835\\udf13\\ud835\\udc54+1,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc4f\\u22121\\u2190variation_pg(\\ud835\\udf0b\\ud835\\udf13\\ud835\\udc54+1,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc4f\\u22121,\\ud835\\udc44\\ud835\\udf031,B)\\naddition(\\ud835\\udf0bb\\ud835\\udf131,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc4f\\u22121,\\ud835\\udf0b\\ud835\\udf19,X,B)\\n\\ud835\\udc56\\u2190\\ud835\\udc56+\\ud835\\udc4f\\nfunction addition (X,B,\\ud835\\udf0bb\\ud835\\udf13...)\\nfor\\ud835\\udf0bb\\ud835\\udf13...do\\n(\\ud835\\udc53,transitions)\\u2190\\ud835\\udc39(\\ud835\\udf0bb\\ud835\\udf13),\\ud835\\udc51\\u2190\\ud835\\udc37(\\ud835\\udf0bb\\ud835\\udf13)\\ninsert(B,transitions)\\nifX(\\ud835\\udc51)=\\u2205or\\ud835\\udc39(X(\\ud835\\udc51))<\\ud835\\udc53then\\nX(\\ud835\\udc51)\\u2190\\ud835\\udf0bb\\ud835\\udf13\\nAlgorithm 6 Actor-Critic Training\\nfunction train_actor_critic (\\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032,\\ud835\\udf0b\\ud835\\udf19,B)\\nfor\\ud835\\udc61=1\\u2192\\ud835\\udc5bdo\\nSample\\ud835\\udc41transitions(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e),\\ud835\\udc60\\u2032)fromB\\nSample smoothing noise \\ud835\\udf16\\n\\ud835\\udc66\\u2190\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e)+\\ud835\\udefemin\\n\\ud835\\udc56=1,2\\ud835\\udc44\\ud835\\udf03\\u2032\\n\\ud835\\udc56\\u0000\\ud835\\udc60\\u2032,\\ud835\\udf0b\\ud835\\udf19\\u2032(\\ud835\\udc60\\u2032)+\\ud835\\udf16\\u0001\\nUpdate both critics by regression to \\ud835\\udc66\\nif\\ud835\\udc61mod\\u0394then\\nUpdate actor using the deterministic policy gradient:\\n1\\n\\ud835\\udc41\\u00cd\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60)\\u2207\\ud835\\udf19\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60)\\nSoft-update target networks \\ud835\\udc44\\ud835\\udf03\\ud835\\udc56\\u2032and\\ud835\\udf0b\\ud835\\udf19\\u2032Algorithm 7 PG Variation\\nfunction variation_pg (\\ud835\\udf0b\\ud835\\udf13...,\\ud835\\udc44 \\ud835\\udf031,B)\\nfor\\ud835\\udf0b\\ud835\\udf13...do\\nfor\\ud835\\udc56=1\\u2192\\ud835\\udc5ado\\nSample\\ud835\\udc41transitions(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e),\\ud835\\udc60\\u2032)fromB\\nUpdate actor using the deterministic policy gradient:\\n1\\n\\ud835\\udc41\\u00cd\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60)\\u2207\\ud835\\udf13\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60)\\nreturn\\ud835\\udf0bb\\ud835\\udf13...\\nB HYPERPARAMETERS\\nIn Tab. 3 we provide the hyperparameters used for DCG-MAP-Elites\\nand the baselines on all tasks.\\n10 Table 3: Hyperparameters for DCG-MAP-Elites and all baselines\\nParameter MAP-Elites MAP-Elites ES PGA-MAP-Elites QD-PG DCG-MAP-Elites\\nNumber of centroids 1024 1024 1024 1024 1024\\nEvaluation batch size \\ud835\\udc4f 256 1050 256 256 256\\nPolicy networks [128, 128,|A|] [128, 128,|A|] [128, 128,|A|] [128, 128,|A|] [128, 128,|A|]\\nNumber of GA variations \\ud835\\udc54 256 0 128 86 128\\nGA variation param. 1 \\ud835\\udf0e1 0.005 0 .005 0 .005 0 .005\\nGA variation param. 2 \\ud835\\udf0e2 0.05 0 .05 0 .05 0 .05\\nActor network [256, 256,|A|] [256, 256,|A|] [256, 256,|A|]\\nCritic network [256, 256, 1] [256, 256, 1] [256, 256, 1]\\nTD3 batch size \\ud835\\udc41 100 100 100\\nQuality critic training steps \\ud835\\udc5b 3000 3000 3000\\nDiversity critic training steps \\ud835\\udc5b 300\\nPG training steps \\ud835\\udc5a 150 150 150\\nPolicy learning rate 5\\u00d710\\u221235\\u00d710\\u221235\\u00d710\\u22123\\nActor learning rate 3\\u00d710\\u221243\\u00d710\\u221243\\u00d710\\u22124\\nCritic learning rate 3\\u00d710\\u221243\\u00d710\\u221243\\u00d710\\u22124\\nReplay buffer size 106106106\\nDiscount factor \\ud835\\udefe 0.99 0 .99 0 .99\\nActor delay \\u0394 2 2 2\\nTarget update rate 0.005 0 .005 0 .005\\nSmoothing noise var. \\ud835\\udf0e 0.2 0 .2 0 .2\\nSmoothing noise clip 0.5 0 .5 0 .5\\nSample number 1000\\nSample sigma 0.02\\nlengthscale \\ud835\\udc59 0.008\\nDescriptor sigma \\ud835\\udf0e\\ud835\\udc51 0.0004\\n11\",\"11\":\" Multiplexed gradient descent: Fast online training of\\nmodern datasets on hardware neural networks without\\nbackpropagation\\nA. N. McCaughan1, B. G. Oripov2, N. Ganesh1, S. W. Nam1,\\nA. Dienstfrey1, S. M. Buckley1\\n1National Institute of Standards and Technology, Boulder, CO 80305\\n2University Colorado Boulder, Boulder, CO 80309\\nAbstract\\nWe present multiplexed gradient descent (MGD), a gradient descent framework\\ndesigned to easily train analog or digital neural networks in hardware. MGD utilizes\\nzero-order optimization techniques for online training of hardware neural networks. We\\ndemonstrate its ability to train neural networks on modern machine learning datasets,\\nincluding CIFAR-10 and Fashion-MNIST, and compare its performance to backprop-\\nagation. Assuming realistic timescales and hardware parameters, our results indicate\\nthat these optimization techniques can train a network on emerging hardware platforms\\norders of magnitude faster than the wall-clock time of training via backpropagation on\\na standard GPU, even in the presence of imperfect weight updates or device-to-device\\nvariations in the hardware. We additionally describe how it can be applied to existing\\nhardware as part of chip-in-the-loop training, or integrated directly at the hardware\\nlevel. Crucially, the MGD framework is highly \\rexible, and its gradient descent pro-\\ncess can be optimized to compensate for speci\\fc hardware limitations such as slow\\nparameter-update speeds or limited input bandwidth.\\n1arXiv:2303.03986v1  [cs.LG]  5 Mar 2023 1 Introduction\\nMachine learning has proven an invaluable tool for a variety of applications [1]. However,\\nmachine learning on traditional digital hardware is ine\\u000ecient, leading to a signi\\fcant e\\u000bort\\ntowards building custom hardware that can perform machine learning tasks at high speeds\\nwith lower energy costs [2]. A number of hardware platforms have emerged using analog [3],\\ndigital [4, 5], or mixed-signal processing [6] that will potentially o\\u000ber increased operational\\nspeeds and\\/or reduced energy costs [7]. However, many of the most promising hardware\\ninstantiations only perform the inference part of the machine learning algorithm. Meanwhile\\nthe larger portion of the energy cost is spent training on datasets [8], usually via gradient\\ndescent. Backpropagation is by far the most commonly used method of computing the gra-\\ndient for gradient descent, but has proved to be challenging to implement in novel hardware\\nplatforms [9].\\nThough often con\\rated, training via gradient descent does not require backpropagation\\n{ backpropagation is only used to calculate the gradient. Other methods for computing the\\ngradient in neural networks exist, but are much less e\\u000ecient in software than backpropagation\\nand so are rarely used in today's machine learning applications. This is not generally true in\\nhardware, where backpropagation may not only be challenging to implement, but also may\\nnot be the most e\\u000ecient way to compute the gradient.\\nOf particular interest in hardware are model-free methods, in which we require no knowl-\\nedge of the internal structure of the network (e.g topology, activation function, derivatives,\\netc), only the ability to perturb the network's parameters and measure the network's re-\\nsponse. The simplest example of such a method is \\fnite-di\\u000berence [10], which has been\\nemployed for chip-in-the-loop training [11]. However, \\fnite-di\\u000berence has several other dis-\\nadvantages that prevent its widespread implementation in hardware, including the require-\\nments for extra memory at every synapse and global synchronization. Fortunately, there\\nare a variety of other model-free methods that overcome some of the issues associated with\\n\\fnite-di\\u000berence [12, 13].\\n2 In this paper, we show that model-free perturbative methods can be used to e\\u000eciently\\ntrain modern neural network architectures in a way that can be implemented natively within\\nemerging hardware. These methods were investigated for training VLSI neural networks\\nbeginning in the 1990s [14, 15, 16, 17, 18, 19, 20, 21, 22, 23], and more recently on memristive\\ncrossbars [24] and photonic hardware [25], but all these demonstrations have been very\\nlimited in scale, comprising small datasets with only a few neurons. Below we describe a\\nframework for applying these techniques to existing hardware at much larger scales, with\\nan emphasis on creating simple, highly-localized circuits that could be implemented on-chip\\nif desired. The framework is also extensible to training existing hardware systems via a\\nchip-in-the-loop technique. We note that these methods have also been adapted in forward\\ngradient approaches using auto-di\\u000berentiation, which have attracted recent interest in the\\nmachine learning literature [26, 27, 28].\\nWe show that under realistic assumptions of the operating timescales of analog and digital\\nhardware neural networks, one can train hardware to solve modern datasets such as CIFAR-\\n10 faster than training a software network on a GPU, even in the presence of signal noise\\nand device-to-device variations in the hardware. A major advantage of this framework is\\nthat it can be used to perform online training of hardware platforms originally designed only\\nfor inference while making minimal hardware modi\\fcations.\\n2 Multiplexed gradient descent\\n2.1 Computing the gradient with perturbations\\nWe begin with the basic assumption that we have some hardware with programmable pa-\\nrameters (e.g. weights and biases) that can perform inference. Our goal is to augment the\\nhardware minimally such that it can also be trained via gradient descent. We will show\\nhow to con\\fgure the hardware such that the network as a whole automatically performs\\ngradient descent, without backpropagation. As an example, assume we have a hardware\\n3 instantiation of a feedforward multi-layer neural network as shown in Fig. 1. The hardware\\ntakes time-varying inputs x(t), training target ^ y(t), has variable parameters \\u0012, outputs the\\ninferencey(t), and computes a cost C(y(t);^y(t)). To allow us to compute the gradient of\\nsuch a system, we \\frst add a small time-varying perturbation ~\\u0012i(t) to each parameter base\\nvalue\\u0012i(Fig. 1a, inset). This perturbation will slightly modulate the cost C, and that mod-\\nulation will be fed back to the parameters. This process will ultimately allow us to extract\\nthe gradient of the system.\\nC\\nneuron\\nsynapse\\n...\\n\\u0283\\n update \\n(a)\\n(c)(b)\\n...... x1\\nx2\\nx3sinusoidalt\\nsequential\\ncode\\nFigure 1: (a) Schematic diagram showing the operation of the MGD framework in a feed-\\nforward neural network using example sinusoidal perturbations. (a, inset) Each parameter\\niis modulated slightly from its base value \\u0012iby the perturbation ~\\u0012i. The result of these\\nperturbations causes a modulation in the cost ~C, which is globally broadcast back to all\\nthe parameters. (b) A homodyne detection process is used to compute the partial gradient\\napproximations Gifrom the integrated product of \\u0012iand ~C. This partial gradient is then\\nused to update \\u0012iin the approximate direction of the gradient. (c) Example perturbation\\ntypes that can be used with this process.\\nAlthough the perturbations can take a variety of di\\u000berent forms, we will \\frst describe\\nthis process by using sinusoidal perturbations as they are conceptually straightforward to\\nunderstand. In this scenario, each parameter \\u0012iis slightly modulated at a unique frequency\\n!iand amplitude \\u0001 \\u0012, giving the perturbation ~\\u0012i(t) = \\u0001\\u0012sin(!it). As each parameter is\\n4 modulated, it slightly changes y(t) which in turn changes the cost. Thus, if the parameters\\nare modulated by frequencies !1,!2,!3, etc, those same frequencies will necessarily appear\\nas small modulations in the cost ~C(t) on top of the baseline (unperturbed) cost value C0,\\nsuch that\\nC(t) =C0+~C(t) =C0+X\\ni\\u0001Cisin(!it) (1)\\nIf we remove C0, we are left with a time varying signal ~C(t) =P\\ni\\u0001Cisin(!it) corre-\\nsponding only to the e\\u000bects of our parameter perturbations. The amplitude \\u0001 Ciis simply\\nthe amplitude of change in the cost due to ~\\u0012i(t), the perturbation of parameter i.\\nSince the gradient with respect to the cost dC=d\\u0012 is composed solely from the partial\\ngradientsdC=d\\u0012 = (@C=@\\u0012 1; @C=@\\u0012 2; :::), if we can extract \\u0001 Cifor each parameter we\\ncan produce an estimate of the complete gradient G= (\\u0001C1=\\u0001\\u00121;\\u0001C2=\\u0001\\u00122;:::). Now the\\ntask becomes to extract individual \\u0001 Ciout of the summed signal ~C(t). Fortunately, to\\nextract a given \\u0001 Ci, all we need to do is integrate the product of the input perturbation\\n~\\u0012i(t) with ~C(t). The integration takes the form of a homodyne detection, where unwanted\\nperturbations (frequencies) from other parameters are eliminated via integration:\\nGi=1\\n\\u0001\\u00122\\ni1\\nTZT\\nt=0X\\nk\\u0001Cksin(!kt)\\u0001\\u0012isin(!it)dt\\n=\\u0001Ci\\n\\u0001\\u0012iasT!1(2)\\nwhere 1=\\u0001\\u00122\\niis a normalization constant.\\nThe valueGiis the approximation for the partial gradient for parameter i.Gapproaches\\nthe exact gradient when both T!1 and the amplitude of the perturbation \\u0001 \\u0012iapproaches\\nzero, and is only an approximation otherwise. Fortunately, even at realistic timescales and\\namplitudes, Gcontains meaningful information and can be used to perform gradient de-\\nscent [12].\\n5 For illustrative purposes we have described the algorithm using sinusoidal parameter\\nperturbations. However, any collection of orthogonal, mean zero perturbations can be used\\n[13], including a variety of analog and discrete perturbations as shown in Fig. 1c. In general,\\nwe will be integrating the product ei(t) = ~C(t)~\\u0012i(t)=\\u0001\\u00122\\ni, which we refer to as the error\\nsignal, and Giwill be given by1\\nGi=ZT\\nt=0~C(t)~\\u0012i(t)\\n\\u0001\\u00122\\nidt (3)\\nWe discuss the e\\u000bects of changing the perturbation type in Section 3.4. We also note that\\nalthough many of the quantities described here are time-varying, in the following sections\\nwe will drop the explicit time dependence notation for the sake of brevity.\\n2.2 Gradient descent in the MGD framework\\nHere we describe the practical implementation of a model-free gradient descent framework\\nin hardware, which we term multiplexed gradient descent (MGD). To better understand\\nthe algorithm from a hardware perspective, we will run through the same computation\\npreviously described, but from the viewpoint of a single parameter (e.g. a synapse weight in\\na hardware neural network). The process begins with the application of a local perturbation\\n~\\u0012ithat slightly modi\\fes the base value of the parameter \\u0012i(Fig. 1a, inset). As previously\\ndescribed, this perturbation { and any other perturbations from other parameters { induce\\na change in the cost ~Con top of the baseline cost C0such that the cost at the output is\\nC=C0+~C.~Cmay be extracted from Ceither by direct subtraction of C0or, in some\\nanalog scenarios, by a simple highpass \\flter. The resulting ~Csignal is broadcast globally\\nto all parameters, so our parameter \\u0012ihas access to it. (Note that although Fig. 1 shows a\\nwireless broadcast tower for purposes of clarity, in most hardware platforms this will be a\\n1Note that here and in the simulation results, Giis being accumulated with time and is not normalized by\\n1=T, unlike Eq. 2. As described later, this allows us to vary the integration time without greatly impacting\\nthe rate of training{equivalently, one can integrate for a long time resulting in a large step down the gradient,\\nor one can take a series of shorter steps instead and travel approximately the same distance along the gradient.\\n6 wired connection). However, we must assume that parameters other than the ith are also\\ncausing modulations in the cost as well. To our parameter \\u0012i, these other modulations are\\nunwanted and must be \\fltered out. As shown in Fig. 1b, for the parameter ito extract only\\nits own e\\u000bect on the cost, it can just integrate the product of its local perturbation ~\\u0012iand\\nthe global cost signal ~Cit receives. This has the e\\u000bect of isolating the contribution from \\u0012i\\ndue to the pairwise orthogonality of the perturbation signals. From Eq. 3, this integration\\nproduces the partial gradient approximation Gi\\/\\u0001Ci=\\u0001\\u0012i. The parameter can then use\\ntheGivalue to directly to reduce the cost by updating itself according to a gradient descent\\nstep\\n\\u0012i!\\u0012i\\u0000\\u0011Gi (4)\\nwhere\\u0011is the learning rate. When all the parameters of a system perform this operation\\nin parallel, the resulting weight update corresponds to gradient descent training of the entire\\nnetwork. Because all the parameters are being perturbed and updated simultaneously, we\\ncall the framework multiplexed.\\nLooking at this process from the hardware perspective, one must also examine several\\npractical considerations such as when to perform the weight update, how long to integrate\\nthe gradient, and so forth. We introduce the following time constants to provide a framework\\nfor managing these considerations in the MGD context.\\n\\u2022\\u001cpis the timescale over which perturbations occur. In a digital system, the pertur-\\nbations of each parameter would be updated to new values every \\u001cp. In an analog\\n(continuous) system, \\u001cpcorresponds to the characteristic timescale of the perturba-\\ntions. For instance, if using sinusoidal perturbations, it corresponds approximately to\\nthe total bandwidth the frequencies occupy.\\n\\u2022\\u001c\\u0012is the gradient-integration time (i.e. Tin Eq. 3). It sets how often parameter-\\nupdates occur and also determines how accurate the gradient approximation will be.\\n7 For each time period \\u001c\\u0012, the gradient approximation Gis integrated, and at the end of\\nthat period the parameters are updated according to Eq. 4.\\n\\u2022\\u001cxcontrols how often new training samples x, ^yare shown to the hardware. After\\neach\\u001cxperiod, the old sample is discarded and a new one is applied, generating new\\noutputsyand costC.\\n0 4 8 0 4 8 0 4 8 0 4 8Finite difference Coordinate descent SPSA Analog(a) (b) (c) (d)\\n0\\n0\\n0\\n0\\n0\\n0 0\\n0 0\\n0\\n0\\n0\\n0\\n0\\n0\\n0\\n0\\n0\\n0\\n00\\n0\\n0\\nTime Time Time Time320 2020 20\\n33 3\\nFigure 2: Di\\u000berent optimization algorithms can be implemented in the MGD framework\\nby changing the values the time constants, \\u001cp,\\u001c\\u0012and\\u001cx, and varying the perturbation\\ntype. Shown here are the parameter values \\u0012that are updated every \\u001c\\u0012, perturbation ~\\u0012\\nwith characteristic time constant \\u001cp, computed cost C, and cost-modulation ~C. Varying\\u001c\\u0012\\nand\\u001cxand perturbation type allows MGD to implement a variety of optimization algorithms\\nincluding (a) \\fnite-di\\u000berence (b) coordinate descent (c) simultaneous perturbation stochastic\\napproximation (SPSA) and (d) an analog implementation. Each line color in the upper two\\nplots corresponds to one of the three parameters in this simulation.\\nThe values of the time constants \\u001c\\u0012and\\u001cphave a large impact on the particulars of the\\ntraining, and particular choices of \\u001c\\u0012and\\u001cpallow the implementation of certain conventional\\nalgorithms in numerical analysis. For instance, consider the implementation of the forward\\n\\fnite-di\\u000berence algorithm within this framework. To do this we start with discrete pertur-\\nbations, such that every \\u001cponly a single parameter is perturbed by \\u0001 \\u0012, and the parameters\\nare perturbed sequentially as shown in Fig. 1c. When parameter iis perturbed by \\u0001 \\u0012i, the\\ncost changes by \\u0001 Cand the resulting partial gradient \\u0001 C=\\u0001\\u0012i\\u0019@C=@\\u0012iis stored in Gi.\\n8 Now if we set \\u001c\\u0012=P\\u001cp, wherePis the number of parameters in the network, this is ex-\\nactly equivalent to computing a \\fnite-di\\u000berence approximation of the gradient: each \\u001cp, one\\nelement of the gradient is approximated, and after P\\u001cp, every partial gradient Gihas been\\nmeasured and stored. Since \\u001c\\u0012=P\\u001cp, the weight update only comes after all the partials\\nofGhas been collected, just as in \\fnite-di\\u000berence. This process is shown schematically in\\nFig. 2a.\\nSimilarly, other optimization algorithms can be implemented simply by modifying the\\nvalues of the time constants. For example, using same procedure as above but with the\\nintegration time reduced to a single timestep, i.e. \\u001c\\u0012=\\u001cp. This corresponds exactly to\\ncoordinate descent. In this case, rather than storing each Giuntil all the partials of the\\ngradient are fully assembled, the weight update is applied immediately after each Giis\\ncomputed. This may be more appealing from a hardware perspective than a \\fnite-di\\u000berence\\napproach, as Gican be used for the weight update and immediately discarded { unlike \\fnite-\\ndi\\u000berence, it does not require a per-parameter memory to store Giuntil the weight update\\nis executed. This coordinate-descent process is shown schematically in Fig. 2b.\\nAs a third example, it is possible to implement simultaneous-perturbation stochastic\\napproximation (SPSA) [12] by only changing the values of the time constants and the form\\nof the perturbation. In this case, \\u001c\\u0012=\\u001cpand a random, discrete f+\\u0001\\u0012;\\u0000\\u0001\\u0012gperturbation\\nis applied to every parameter every \\u001cp, as shown in Fig. 2c. Similar to coordinate descent,\\nthis con\\fguration avoids the need for additional per-parameter memories, as Givalues do not\\nneed to be stored. This method avoids the need for global synchronization of the parameters\\n{ the perturbations do not need to be sequential, and instead can be generated locally and\\nrandomly at the parameter.\\nIn addition to being able to choose these speci\\fc optimization algorithms, by varying the\\ntime constants \\u001cpand\\u001c\\u0012one can also implement entirely new optimization algorithms. For\\ninstance, in Fig. 2d we show an MGD implementation on an analog system using sinusoidal\\nperturbations that does not correspond to any of the aforementioned methods. In this\\n9 case,\\u001cpcorresponds to the timescale 1 =\\u0001f, where \\u0001fis the perturbation bandwidth, the\\ndi\\u000berence between the maximum and minimum perturbation frequency. Additionally, in the\\nanalog case there is no discrete update of the parameters and instead \\u001c\\u0012is an integration\\ntime constant. Unlike the discrete case which accumulates Gfor\\u001c\\u0012time then resets it to\\nzero,\\u0012is continuously updated with the output of an lowpass \\flter with time constant \\u001c\\u0012\\n(see Algorithm 2).\\nBecause modern machine learning datasets are composed of many training examples\\n(often tens of thousands), \\u001cx, the time constant that controls how often training samples\\nare changed, is critical. In fact, by setting \\u001cxappropriately, mini-batching can even be\\nimplemented in hardware that only allows 1 sample input at a time. The batch size is\\ndetermined by \\u001c\\u0012=\\u001cx, the ratio of the gradient integration time \\u001c\\u0012and the sample update\\ntime\\u001cx. When\\u001cxis shorter than \\u001c\\u0012, multiple samples are shown to the network during a\\nsingle gradient integration period. As the sample changes, the gradient approximation G\\nwill then include gradient information from each of those samples. This integration-in-time\\nprocess is arithmetically identical to summing multiple examples in parallel (as is often done\\non a GPU). As a concrete example, for hardware which accepts one input example at a time,\\nif\\u001c\\u0012=\\u001cxthe batch size is 1, but if \\u001c\\u0012= 4\\u001cx, the batch size is 4. This is shown in Fig. 3 for\\nthe same network as in Fig. 2 and using simultaneous discrete perturbations. In Section 3\\nwe demonstrate that batch sizes as large as 1000 function correctly in MGD.\\nIn summary, setting just three time constants and the perturbation type allows the MGD\\nframework to implement a wide range of variations of gradient descent and, depending on\\nthe desired approach, one can tailor their training to the needs of the problem and the\\nhardware. For example, MGD can match backpropagation by making \\u001c\\u0012arbitrarily long2,\\nas we show in Section 3.2. At the same time, we show that short \\u001c\\u0012values can work equally\\nwell. In practice, hardware considerations such as write-speed limitations may require an\\n2The MGD approximation to a partial derivative converges to the true partial derivative as \\u001c\\u0012becomes\\narbitrarily long. Thus the various parameter update steps presented above can be made arbitrarily close to\\ntheir analytical de\\fnitions as implemented using digital arithmetic and backpropagation.\\n10 Time 0\\nx\\n40 80 120Figure 3: Illustration of batching in a small network with three parameters and two x\\ninputs, training on a dataset with four samples. The parameters \\u0012are updated every \\u001c\\u0012, and\\nduring that time all four training samples are shown to the network and integrated into the\\ngradient approximation (batch size \\u001c\\u0012=\\u001cx= 4). The gradient approximation Gaccumulates\\neach timestep, and is reset during the weight-update process after each \\u001c\\u0012period. Note that\\nthe updates to \\u0012occurs in the opposite direction of G, per Eq. 4.\\nintermediate \\u001c\\u0012value, as we discuss in Section 4. These features allow the MGD framework\\nto leverage the same training techniques that have been so successful for training machine\\nlearning models, while also being compatible with new and emergent hardware.\\n3 Simulation of MGD\\n3.1 Intro\\nTo characterize the utility of the MGD framework, we \\frst simulated its performance on\\nmodern machine learning datasets. The goal of the simulator was not to perform gradient\\ndescent as fast as possible on a CPU or GPU, but rather to emulate hardware implement-\\ning MGD and evaluate its potential performance in a hardware context. In particular, we\\nused the simulator to estimate the speed, accuracy, and resilience to noise and fabrication\\nimperfections. The simulator was written in the Julia language and can be run on a CPU\\n11 or GPU and is available online3. The algorithms used in the simulation are shown in the\\nAlgorithm 1 and Algorithm 2 boxes, with the parameters and their types in software shown\\nin Table 1.\\nAlgorithm 1 Discrete algorithm\\n1:Initialize parameters \\u0012\\n2:forninnum iterations do\\n3: if(nmod\\u001cx= 0)then\\n4: Input new training sample x, ^y\\n5: if(nmod\\u001cx= 0) or (nmod\\u001c\\u0012= 0)then\\n6: Set perturbations to zero ~\\u0012 0\\n7: Update baseline cost C0 C(f(x;\\u0012);^y)\\n8: if(nmod\\u001cp= 0)then\\n9: Update perturbations ~\\u0012\\n10: Compute output y f(x;\\u0012+~\\u0012)\\n11: Compute cost C C(y;^y)\\n12: Compute change in cost ~C C\\u0000C0\\n13: Compute instantaneous error signal e ~C~\\u0012=\\u0001\\u00122\\n14: Accumulate gradient approximation G G+e\\n15: if(nmod\\u001c\\u0012= 0)then\\n16: Update parameters \\u0012 \\u0012\\u0000\\u0011G\\n17: Reset gradient approximation G 0\\nAlgorithm 2 Analog algorithm\\n1:Initialize parameters \\u0012\\n2:fort= 0toTstepdtdo\\n3: if(tmod\\u001cx= 0)then\\n4: Input new training sample x, ^y\\n5: Update perturbations ~\\u0012\\n6: Compute output y f(x;\\u0012+~\\u0012)\\n7: Compute cost C(t) C(y;^y)\\n8: Compute discretized highpass ~C(t) \\u001chp\\n\\u001chp+dt\\u0010\\n~C(t\\u0000dt) +C(t)\\u0000C(t\\u0000dt)\\u0011\\n9: Compute instantaneous error signal e(t) ~C~\\u0012dt=\\u0001\\u00122\\n10: Update gradient approximation G(t) dt\\n\\u001c\\u0012+dt\\u0000\\ne(t) +\\u001c\\u0012\\ndtG(t\\u0000dt)\\u0001\\n11: Update parameters \\u0012 \\u0012\\u0000\\u0011G\\n3Code available at https:\\/\\/github.com\\/amccaugh\\/multiplexed-gradient-descent-paper\\n12 Description Symbol analog or digital\\nChange in the cost due to\\nperturbation~C both\\nPerturbation to parameters ~\\u0012 both\\nParameters \\u0012 both\\nInput sample x both\\nTarget output ^y both\\nNetwork output y both\\nCost C both\\nUnperturbed baseline cost C0 digital\\nGradient approximation G both\\nInstantaneous error signal e both\\nLearning rate \\u0011 both\\nPerturbation amplitude \\u0001\\u0012 both\\nInput-sample change time\\nconstant\\u001cx both\\nParameter update time\\nconstant\\u001c\\u0012 both\\nPerturbation time constant \\u001cp digital\\nHighpass \\flter time\\nconstant\\u001chp analog\\nTable 1: Algorithm parameters and variables used in the simulations\\n3.2 Equivalence to backpropagation\\nWe \\frst veri\\fed that the simulation was able to minimize the cost for a sample problem, and\\nthat it was equivalent to gradient descent via backpropagation with appropriate parameter\\nchoices. For this initial comparison, we chose to solve the 2-bit parity problem by training a\\n2-2-1 feedforward network with 9 parameters (6 weights, 3 biases). We simulated the problem\\nwith a very large value for \\u001c\\u0012and\\u001c\\u0012=\\u001cxsuch that we achieved a very good approximation\\nof the gradient in Gfor each training sample. We then ran the same problem using a \\u001c\\u0012\\nvalue of 1 so that the gradient approximation Gfor each sample was relatively poor. We\\nmeasured both the number of epochs and the amount of time (number of iterations of the\\nsimulation) for the two experiments, and the results are shown in Fig. 4. Here, an epoch was\\nde\\fned in the typical way { equivalent to the network being shown all training examples of\\na dataset.\\nComparing the two scenarios in terms of epochs in Fig. 4a, one can observe that a value\\n13 0.3\\n0\\n10\\n0\\n0.3\\n0\\n0\\n2\\n1\\n(a)\\n10\\n1\\n10\\n2\\n10\\n3\\n10\\n4\\n10\\n6\\n10\\n5\\nx10\\n6\\nbackprop\\n= 1\\n= 1000\\n \\nx\\n=\\n \\nx\\n=\\n= 1\\n= 1000\\n \\nx\\n=\\n \\nx\\n=\\n(b)\\nMean cost\\nMean cost\\nEpochs\\nTime (iterations)Figure 4: Solving the 2-bit parity (XOR) problem using a 2-2-1 network with 9 parameters,\\n\\u001cp= 1, and batchsize \\u001c\\u0012\\/\\u001cx= 1. (a) Mean cost over dataset versus epochs the on a 2-2-1\\nfeedforward network averaged over 1000 random initializations and trained with MGD with\\ndi\\u000berent gradient integration times \\u001c\\u0012(solid lines). The dashed line shows the same network\\nand dataset trained using backpropagation. (b) The same MGD training data as in part (a)\\nplotted versus time (number of perturbations \\u001cp) instead of epochs.\\nof\\u001c\\u0012=\\u001cx= 1000 resulted in the system following a nearly identical training trajectory as\\nbackpropagation. This is as expected { for each sample shown to the network, the gradient\\napproximation Ghas 1000 timesteps to integrate a very accurate estimate that should be\\nvery close to the true gradient computed by backpropagation. When \\u001c\\u0012=\\u001cx= 1, however,\\neach sample only has a single timestep to estimate the gradient before moving on to the\\nnext sample. As a result, the samples have to be shown to the network many more times to\\nminimize the cost, resulting in a much larger number of epochs.\\nHowever, while the \\u001c\\u0012=\\u001cx= 1 case uses the sample data less e\\u000eciently (requiring more\\nepochs), there is actually a tradeo\\u000b for data e\\u000eciency and run time. If we plot the cost\\nversus iterations rather than versus epoch, we get a more accurate estimate of how long it\\nwill take hardware to train in terms of real time. As shown in Fig. 4b, one can see that the\\nshorter\\u001c\\u0012and\\u001cxvalues take approximately half the time to minimize the cost as the longer\\nvalues. These examples serve to highlight that while longer integration times produce a more\\naccurate gradient approximation, integration times as short as \\u001cpmay also be used to train\\na network, consistent with the \\fndings of Ref. [12]. In fact, shorter integration times may\\n14 even be faster to train in terms of operational time.\\n100101102103104100\\n02040608090\\u00ba\\n2-bit parity\\n4-bit parity\\nAngle (deg)\\nTimeNIST7x7 dataset\\nFigure 5: Angle between the gradient approximation Gand the true gradient versus time.\\nThen-bit parity networks used n-n-1 networks, while the NIST7x7 networks used a 49-4-4\\nnetwork . The networks have 9, 25 and 220 parameters respectively, and the datasets have\\n4, 16 and 44136 training examples respectively. The solid line shows the median angle value\\nfor 100 random initializations for the n-bit parity and and 15 random initializations for the\\nNIST7x7 dataset. The shaded regions show the 1st to 3rd quartile values.\\nTo quantify the e\\u000bect of longer integration times on the accuracy of the gradient ap-\\nproximation, we measured how the gradient approximation Gconverged to the true gradient\\n@C=@\\u0012 (as computed by backpropagation) as a function of time. For this experiment, we ran\\nthe simulation with \\u001c\\u0012=1and\\u001cx=\\u001cp= 1, so that it continuously integrated Gwithout\\never resetting or updating the parameters. As the simulation ran, we repeatedly computed\\nthe angle between the true gradient @C=@\\u0012 and the approximation G. The results are shown\\nin Fig. 5, showing the solution to the 2-bit parity, 4-bit parity, and NIST7x7 problems. The\\nNIST7x7 dataset is a small image recognition problem based on identifying the letters N,\\nI, S, and T on a 7 \\u00027 pixel plane. The dataset has the property that it cannot be solved\\nto greater than 93% with a linear solve for a 49-4-4 feedforward network with sigmoidal\\nactivation functions. We also chose this network and dataset because it was small enough\\nto perform many di\\u000berent simulations to acquire statistics, and the problem space is large\\nenough that random solutions are unlikely.\\n15 As expected, the angle decreased with time as Galigned with the true gradient. The\\ntime axis is in units of \\u001cp, which is the minimum discrete timestep in this system. For a\\nreal hardware platform, this timestep is approximately the inference time of the system. In\\ngeneral, the more parameters the network has, the longer it takes to converge to the true\\ngradient.\\nBatch size              =  1\\nBatch size              =  4\\n0 200 400 600 800150550x103Training time\\n0 200 400 600Max \\u03b7100101\\nMinimum training time104105\\n(a) (b)\\nBatch size              =  1\\nBatch size              =  4\\nFigure 6: E\\u000bect of \\u001c\\u0012on the training time of the 2-bit parity (XOR) problem. (a) Training\\ntime as a function of \\u001c\\u0012and batch size. (b) E\\u000bect of \\u001c\\u0012on the maximum achievable learning\\nrate\\u0011and corresponding minimum training time.\\n3.3 Details of mini-batching\\nWe next investigated in more detail how \\u001c\\u0012and\\u001cxa\\u000bect the training time. Longer \\u001c\\u0012values\\nresult in a more accurate gradient approximation, but reduce the frequency of parameter\\nupdates. Using a \\fxed, low \\u0011value, we trained a 2-2-1 network to solve 2-bit parity (XOR)\\nfor 100 di\\u000berent random parameter initializations, varying \\u001c\\u0012but keeping the batch size \\u001c\\u0012=\\u001cx\\nconstant at either 4 or 1. Since the 2-bit parity dataset is composed of four ( x, ^y) pairs,\\n\\u001cx= 4\\u001c\\u0012is analogous to gradient descent { all four samples are integrated into the gradient\\napproximation Gbefore performing a weight update. When \\u001cx=\\u001c\\u0012, the network performs\\nstochastic gradient descent (SGD) with a batch size of 1. Fig. 6a shows the training time as\\na function of \\u001c\\u0012for these two cases. Here, training time corresponds to the time at which\\n16 the total cost Cdropped below 0.04, indicating the problem was solved successfully. In the\\ncase where the batch size was 1, we observed that increasing \\u001c\\u0012increased the training time.\\nHowever, when the batch size was 4, increasing \\u001c\\u0012had little e\\u000bect on the training time.\\nAs with any training process, the training can become unstable at higher \\u0011values and\\nfail to solve the task. Since the results in Fig. 6a were only for a \\fxed learning rate, we\\nalso wanted to examine the e\\u000bect of \\u001c\\u0012on the maximum achievable \\u0011. Here, we de\\fned\\nthe \\\\max\\u0011\\\" as the maximum learning rate where the network successfully solved the 2-bit\\nparity problem for at least 50 out of 100 random initializations. As seen in Fig. 6b, as \\u001c\\u0012\\nis increased the max \\u0011decreases, resulting in longer minimum training times. This was\\ntrue whether the batch size was large or small, although the larger batch sizes had higher\\nachievable\\u0011values overall.\\nFrom these results, we infer that a poor gradient approximation taken with respect to all\\ntraining examples is more useful than collecting an accurate gradient with respect to a single\\nexample. Stated another way, waiting a long time for an extremely accurate gradient then\\ntaking a large step is less productive than taking a series of short (but less accurate) steps.\\nThis is an important conclusion for hardware, as it shows that implementing an e\\u000bective\\ngradient descent process in MGD does not necessarily require additional memory to store\\naccurate, high-bit-depth gradient values. Note that in our implementation Gaccumulates\\nwith time and so the size of the parameter update \\u0011Gfrom Eq. 4 grows proportionally to\\nthe integration time. This meant that when \\u001c\\u0012was larger the e\\u000bective step in the direction\\nof the gradient was also larger, and so for \\fxed \\u0011the rate of training therefore remains\\napproximately constant. If this was not the case, whenever \\u001c\\u0012was doubled we would also\\nneed to reduce \\u0011by half to maintain the same approximate rate of training.\\n3.4 Analog and digital perturbations\\nThe parameter perturbations can take many di\\u000berent forms, as long as they are low-amplitude\\nand their time averages are pairwise orthogonal or, in a statistical setting, are uncorre-\\n17 lated [13]. We have implemented four types of perturbations: sinusoidal perturbations,\\nsequential discrete perturbations, discrete code perturbations, and random code perturba-\\ntions. Sinusoidal perturbations are like those shown in Fig. 1a, where each parameter is\\nassigned a unique frequency. Sequential discrete perturbations refer to the case where where\\nparameters are sequentially perturbed, one at a time, by +\\u0001 \\u0012, as described in the \\fnite-\\ndi\\u000berence implementation of Section 2.2. \\\\Code\\\" perturbations are simultaneous discrete\\nperturbations off\\u0000\\u0001\\u0012;+\\u0001\\u0012gfor every parameter every \\u001cptimesteps. We call them code\\nperturbations due to their similarity to code-multiplexing (spread-spectrum) techniques used\\nin wireless communication technologies. There are two \\ravors of code-perturbations: the \\frst\\nconsists of a prede\\fned set of pairwise-orthogonal square wave functions that take the values\\noff\\u0000\\u0001\\u0012;+\\u0001\\u0012g. Each of these perturbation patterns is a deterministic sequence, and no\\ntwo parameters have the same sequence. One example of these are the Walsh codes used in\\nmodern cell-phone communications. The second consists of randomly-generated sequences\\noff\\u0000\\u0001\\u0012;+\\u0001\\u0012gthat are pairwise uncorrelated. We call these \\\\statistically orthogonal.\\\"\\nThe statistically orthogonal case is slightly less e\\u000ecient than the deterministic orthogonal\\ncodes since perturbations from multiple parameters interfere with each other more in ~C{\\nany \\fnite sample of the perturbations will have a non-zero correlation that decreases to zero\\nwith sample size. However, the use of the statistically orthogonal version allows the per-\\nturbations to be generated locally and randomly. These perturbations may be very useful\\nin hardware implementations, as they are spread-spectrum and single-frequency noise from\\nexternal sources is unlikely to corrupt the feedback signals.\\nTo compare the training performance between di\\u000berent perturbation types, we applied\\nfour di\\u000berent perturbation types to the same 2-2-1 network described in the last section in\\norder to solve the 2-bit parity problem. In particular, we aimed to show that training can\\nhappen in both a purely analog or purely digital way. In practice, many systems have hybrid\\nanalog-digital features, and a combination approach may be used.\\nFig. 7 shows the training time distributions for the 2-bit parity problem using the dif-\\n18 x 105\\nTraining time\\nsinusoidal code sequential8\\n6\\n4\\n2Figure 7: Demonstrating equivalence between the various perturbation types by measuring\\ntheir time to train a 2-2-1 network on the 2-bit parity (XOR) problem. Each box plot\\nrepresents the results of 100 random initializations. The hyperparameters were \\u001cx= 250,\\n\\u001c\\u0012= 1,\\u0011= 0:05,\\u001cp= 1 (discrete), \\u0001 f= 0:3 (analog)\\n.\\nferent perturbation types. The bandwidth for sinusoidal perturbations was set to be 1 =2\\u001cp.\\nAs expected, we found that the di\\u000berent perturbation types are approximately equivalent\\nin terms of speed of training. This equivalence makes sense when one considers that the\\nfeedback from ~Chas a \\fnite bandwidth that must be shared between all the parameters {\\nno matter the encoding (perturbation) scheme, the information carried in that feedback to\\nthe parameters will be limited by that \\fnite bandwidth.\\n3.5 Operation on noisy or imperfect hardware\\nThe fabrication defects and signal noise present in emerging hardware platforms pose signi\\f-\\ncant challenges for current training techniques in hardware. For example, weight parameters\\nmay di\\u000ber from design or update non-deterministically, causing a discrepancy between the\\nmodel and actual hardware that can negatively a\\u000bect training [29, 30, 9]. For example,\\nin Ref. [30], a simulated network achieves 97.6% accuracy for the MNIST dataset, but the\\nhardware di\\u000bractive optical network implementation only obtains 63.9% accuracy after di-\\n19 rect transfer of the pre-trained model to their hardware without some in-situ training, while\\nRef. [31] shows that a 3% component variation can lead to a 7% drop in classi\\fcation accu-\\nracy without error correction. Ref. [9] shows how small discrepancies in modeled parameters\\ncan blow-up in general hierarchical problems. In their toy problem, a 0.5% error in a model\\nparameter leads to a 30% error in output after 20 layers. The solutions to these non-idealities\\ncan be cumbersome for o\\u000fine training. For example, the actual value of the weights may\\nhave to be regularly measured during training [29], imperfections may be carefully measured\\nand accounted for in the models [32, 31], or weights may be quantized with low bit-depth,\\nwhere the bit depth is chosen such that the system can be modeled and controlled accu-\\nrately despite device-to-device variations. This is common in hardware platforms such as\\nmemristive crossbar arrays [33] or phase change materials [34], where bit depths of 6-8 can\\nbe achieved.\\nHere we investigate the e\\u000bects of three di\\u000berent types of imperfections and noise that\\ncould a\\u000bect hardware systems: (1) stochastic noise on the output cost Cnoise, (2) stochastic\\nnoise on the parameter update \\u0012noise, (3) per-neuron defects in the activation function, where\\neach neuron has a randomly scaled and o\\u000bset sigmoidal activation function that is static in\\ntime. These tests were performed on the NIST7x7 dataset using the previously described\\n49-4-4 network with 220 parameters and with \\u001cx=\\u001c\\u0012= 1 unless otherwise stated.\\nIn the \\frst test, we added Gaussian noise with mean zero and standard-deviation \\u001bCto\\nthe cost, applied every timestep such that noise C(t) =Cideal(t) +Cnoise(t;\\u001bC). For example,\\nin optical hardware, this could be noise due to laser \\ructuations. Noise in the cost is also\\nequivalent to any other gaussian mean-zero noise a\\u000becting the accumulated gradient G.\\nFig. 8a shows the e\\u000bect on the training time for di\\u000berent learning rates as \\u001bCwas increased.\\nFor a given learning rate, there is a threshold amount of noise below which the training\\ntime is minimally changed. However, as cost noise increases, the training time eventually\\nincreases and ultimately stops converging.\\nTo see how this noise would a\\u000bect the minimum training time for optimized learning\\n20 0 0.08\\n\\u03b7 = 0.1\\n\\u03b7 = 1\\n107\\n107\\n106\\n105100\\n10-1\\n10-2101\\n0.20\\n106\\nTraining time\\nMax \\u03b7 \\nMinimum training time\\n(a) (b)Figure 8: E\\u000bect of noise applied to the cost signal. (a) Training time versus \\u001bC, the amplitude\\nof the Gaussian noise added to the cost signal C(normalized to the perturbation magnitude\\nj~\\u0012j). The training time is the median time to >80% accuracy for 10 random parameter\\ninitializations. (b) E\\u000bect of noise on maximum \\u0011and training time. The maximum \\u0011is\\nthe largest learning rate for which >80% of the 10 random initializations converged; the\\ncorresponding training time at that maximum learning rate is shown on the right axis.\\nrates, we also measured the maximum achievable \\u0011value for a range of \\u001bC. Fig. 8b shows\\nthis maximum \\u0011value versus cost noise, and corresponding minimum training time. The\\ntrend indicates that the lower the cost noise \\u001bC, the higher we can make \\u0011and the faster we\\ncan train. Stated in reverse, one can compensate for large amounts of cost noise in a system\\nsimply by reducing the learning rate.\\nIn the next test, we analyzed the e\\u000bect of noisy parameter updates on the training\\nprocess. For this experiment, whenever any parameter was updated, the update included a\\nrandomly-applied deviation. Thus, the update rule became\\n\\u0012 \\u0012\\u0000\\u0011G+\\u0012noise (5)\\nwhere\\u0012noisewas Gaussian with mean zero and standard deviation \\u001b\\u0012, normalized by \\u0001 \\u0012,\\nsuch that\\u0012noise\\u0018N(0;\\u001b\\u0012=\\u0001\\u0012). This kind of noise is often seen in analog memories without\\nclosed-loop feedback [35, 36].\\n21 1.0\\n00 14 2 4 6 8 10 12\\n38\\n4567x105\\n38\\n4567x105\\n0 2 4 6 0 2 4 6\\n= 1\\n= 1\\n= 100Converged fraction(a)\\n(d)Training time\\nTraining time\\nConverged fraction1.0\\n00 14 2 4 6 8 10 12\\n= 0\\n= 0.1\\n= 0.3\\n= 100(b)\\n(c)Figure 9: E\\u000bect of noisy (stochastic) parameter updates on solving XOR in a 2-2-1 feedfor-\\nward network, measured for various noise amplitudes \\u001b\\u0012. (a,b) E\\u000bect of parameter update\\nnoise on the probability of the training converging, as a function of learning rate \\u0011. Noise\\nwas much more likely to prevent convergence for (a) \\u001c\\u0012= 1 than (b) \\u001c\\u0012= 100. Convergence\\nwas de\\fned as reaching 93% accuracy within 5 \\u0002107iterations. Statistics are from 25 ran-\\ndom parameter initializations. (c,d) E\\u000bect of parameter update noise on training time as a\\nfunction of the learning rate \\u0011for integration times (c) \\u001c\\u0012= 1 and (d) \\u001c\\u0012= 100. Only \\u0011\\nvalues sets with >50% convergence are shown { in (c) \\u001b\\u0012= 0.3 did not have \\u0011values that\\nmet this criteria.\\nWe found that larger values of \\u001b\\u0012can prevent convergence entirely (Fig. 9a). Curiously, in\\nthe presence of this noise, increasing \\u0011can actually improve the convergence of the problem,\\nas highlighted by the \\u001b\\u0012= 0:1 and\\u001b\\u0012= 0:3 lines in Fig. 9a. We believe this is likely because\\nat very small \\u0011values\\u0012noisewill overwhelm the the very small \\u0011Gin Eq. 5. By making the\\n\\u0011term larger, one can prevent \\u0011Gfrom being drowned out by \\u0012noise. Obviously, at very\\nlarge\\u0011values, the usual gradient-descent instability starts to dominate and the convergence\\napproaches zero. For a given \\u0011, we found that small values of \\u001b\\u0012marginally increase the\\ntraining time, but the e\\u000bect is less signi\\fcant than simply changing the learning rate \\u0011\\n(Fig. 9c).\\n22 Another way to reduce the impact of \\u0012noiseis to increase the integration time of the\\ngradient. When \\u001c\\u0012is increased, the value of Gis accumulated for a longer time and becomes\\nproportionally larger. For instance, when \\u001c\\u0012is 100 times larger, the value of \\u0011Gbecomes\\n100 times larger, meaning the e\\u000bect of \\u0012noisein Eq. 5 is relatively 100 times smaller. The\\nresult can be seen in Fig. 9b,d, where even the largest \\u001b\\u0012value has little e\\u000bect on the result.\\n0 0.25\\u03b7 = 0.1\\n\\u03b7 = 1 106Training time\\n(a) (b)\\n0.2\\n1 Converged fraction\\n0 0.5\\n\\u03b7 = 0.1\\n\\u03b7 = 1\\nFigure 10: E\\u000bect of adding random o\\u000bsets and scaling to each neuron's sigmoid activation\\nfunction. (a) Training time versus \\u001ba, the standard deviation of activation-function o\\u000bsets\\nand scaling. (b) Converged fraction versus the standard deviation of the logistic function\\nparameters. The results are statistics of 25 di\\u000berent random network initializations and\\nactivation-function randomizations. Convergence is de\\fned as reaching 80% accuracy within\\nthe time of the simulation.\\nIn the \\fnal test, we analyzed the e\\u000bect of including \\\\defects\\\" in the neuronal activation\\nfunctions. Here, neuronal activation functions were no longer identical sigmoid functions,\\nbut had \\fxed random o\\u000bsets and scaling that were static in time. These variations were\\nmeant to emulate device-to-device variations that may be found in hardware, for instance in\\nanalog VLSI neurons [37]. The sigmoid activation function for each neuron kwas modi\\fed\\nto a general logistic function fk(a) =\\u000bk(1\\u0000e\\u0000\\fk(a\\u0000ak))\\u00001+bk. The variations were all\\nGaussian, and the scaling factors \\u000bk,\\fkhad a standard deviation \\u001baand a mean of 1, while\\nthe o\\u000bset factors ak,bkalso had a standard deviation of \\u001babut were mean-zero.\\nAs seen in Fig. 10a, adding defects to the network's activation functions had a relatively\\nsmall e\\u000bect on the training time. Even with relatively large variations in the activation\\n23 Setup Parameters Accuracy\\nTask Networkj\\u0012j\\u001c\\u0012\\u001cp\\u0011 batch size 104steps 105steps 106steps 107steps backprop\\n2-bit parity 2-2-1 9 1 15 1 100% 100% 100% 100% 100%\\nN-I-S-T 49-4-4 220 1 13 1 38% 81% 94% 97.7 % 99.8%\\nN-I-S-T 49-4-4 220 1 10.5 1 22% 45% 93% 98.7 % 99.8%\\nFashion-MNIST 2-layer CNN 14378 1 19 1000 34.2% 66.3% 79.3 % 83.5% 88.6%\\nFashion-MNIST 2-layer CNN 14378 10 19 1000 34.3% 66.3% 79.2 % 83.4% 88.6%\\nFashion-MNIST 2-layer CNN 14378 100 19 1000 35.3% 66.3% 77.7 % 84.7% 88.6%\\nFashion-MNIST 2-layer CNN 14378 1000 19 1000 35.3% 59.6% 79.1 % 86.1% 88.6%\\nCIFAR-10 3-layer CNN 26154 1 19 1000 12% 23% 43.8% 60.7% 68%\\nTable 2: Performance of MGD training on four di\\u000berent datasets. The setup, MGD param-\\neters and achieved test accuracies are shown. The best accuracy achieved with training via\\nbackpropagation for the same network is shown in the \\fnal column for comparison.\\nfunctions (\\u001ba= 0:25), the network only took about twice as much time to fully train the\\nNIST7x7 dataset. However, deviations that were too large ( \\u001ba>0:25) caused issues with\\nthe training being able to converge. This is likely due to the output neurons being so scaled\\nand o\\u000bset that it was no longer possible for them to produce the correct output.\\nOverall, we found that MGD is robust against di\\u000berent types of noise and fabrication\\nimperfections of a reasonable scale as those that would be seen in a real hardware system. In\\ngeneral, these non-idealities a\\u000bected the training speed of the network, but did not prevent\\nthe desired problem from being solved.\\n3.6 Modern dataset results\\nTo assess the scalability and relevance of MGD for larger machine learning problems, we com-\\npared MGD and backpropagation on a variety of tasks for di\\u000berent network architectures\\nand hyperparameters. Table 2 compares the accuracies obtained with MGD and backpropa-\\ngation for di\\u000berent datasets and various hyperparameter choices ( \\u001c\\u0012,\\u001cp,\\u0011, batch size), with\\n\\u001cx\\fxed at 1. To make an e\\u000bective comparison, for the backpropagation results we used a\\nbasic stochastic gradient descent (SGD) optimizer without momentum. Both strategies used\\na mean squared error (MSE) cost function. These choices allowed us to avoid confounding\\ne\\u000bects from more complex strategies, although we note that MGD is capable of implement-\\ning several of these more complicated feature (e.g., momentum, dropout, etc.). In all cases\\nwe used the same network architectures for both sets of results. The networks architectures\\n24 used were not selected to reach state-of-the-art accuracy values, but instead smaller networks\\nwere chosen for purposes of rapid testing and statistical analysis. To improve the accuracy\\nfurther, more advanced network architectures (e.g. more layers) or optimizers (e.g. adding\\nmomentum) would be needed. For the purposes of meaningful comparison, we measured the\\naccuracy obtained in the MGD process after \\fxed numbers of timesteps. This allowed us to\\nestimate actual training time for various hardware con\\fgurations in Section 4.1.\\nThe CIFAR-10 network was composed of 3 convolution and max-pool layers followed\\nby a fully-connected layer. The convolutional \\flters were 3 \\u00023 with 16, 32 and 64 output\\nchannels respectively, stride 1, and relu activation function outputs. Each convolution was\\nfollowed by a 2\\u00022 max-pool layer. The \\fnal fully-connected layer converted the 256 outputs\\nof the convolutional layers to the 10 classes, and no softmax was used. The Fashion-MNIST\\nnetwork was similarly composed, comprising two convolution and max-pool layers followed\\nby a (32\\u000210) fully-connected layer and no softmax. The networks used to solve NIST7x7\\nand XOR were fully connected feedforward networks with sigmoidal activation functions. To\\nmake the comparisons in Table 2 as fair as possible, the backpropagation accuracies of each\\nrow were maximized by training the networks with the listed batch size and a variety of \\u0011\\nvalues. The backpropagation accuracy reported is the highest median accuracy obtained for\\nall the\\u0011values tested, with the median taken over \\fve random initializations that each ran\\nfor 2500 epochs (long after the accuracy had converged).\\nBy training networks on the 2-bit parity, NIST7x7, Fashion-MNIST, and CIFAR-10\\ndatasets, we found that MGD approached the solution-accuracy of backpropagation. How-\\never, even after 107steps we observed the MGD-trained networks still had slightly lower\\naccuracy than could be achieved by backpropagation. This is likely related to convergence\\ncriteria of the gradient approximation not being met, due to \\fxed \\u0011values { for instance,\\nthe theory underlying the SPSA process only guarantees convergence with a learning rate\\nthat asymptotically approaches zero [12]. In general, we observed that while larger learning\\nrates achieved higher accuracies earlier on, lower learning rates achieved higher accuracies at\\n25 longer times (see e.g. NIST7x7 result). Therefore custom learning rates are likely to achieve\\nmore optimal training time and accuracy.\\nThe simulations also indicate that changing \\u001c\\u0012had a marginal e\\u000bect on the maximum\\naccuracy of MGD for a \\fxed \\u0011value for the larger datasets shown here, with increased values\\nof\\u001c\\u0012leading to slightly higher \\fnal accuracies. This may be because longer \\u001c\\u0012values produce\\nmore-accurate gradient estimates and allow the network to better optimize the weights as the\\nsystem approaches a local minima in the cost, although more experiments and statistics are\\nneeded to verify if this e\\u000bect is occurs in general. Fortunately, our results show this e\\u000bect is\\nrelatively small, allowing the hardware-designer \\rexibility in how often weight updates need\\nto be performed.\\nThese results show explicitly that MGD is a viable technique for training emerging hard-\\nware platforms on real machine learning datasets. Previous theoretical studies have focused\\non convergence proofs and computation of gradient approximations to varying degrees of\\naccuracy [12, 13] or on simpli\\fed problems, such as linear activation functions [38]. While\\nthese theoretical results are clearly important and necessary, they are not su\\u000ecient to give\\ninsight into the applicability of these optimization techniques to real machine learning prob-\\nlems on experimental hardware platforms. Meanwhile, previous experimental and simulation\\nstudies have for the most part focused on small scale problems, for which 100% accuracy is\\ntrivial, such as XOR [14, 15], few-bit parity [16, 20, 21], the iris dataset [24] or other similar\\ntoy problems [17, 18, 25]. These are not representative of larger machine learning datasets,\\nwhich typically have on the order of tens of thousands of training examples and are trained\\non networks with even more parameters.\\nThese results go further than any of these previous studies, by elucidating the util-\\nity of these techniques for larger, modern, machine learning datasets, and implementing a\\nframework that can connect di\\u000berent optimization techniques (SPSA, \\fnite-di\\u000berence, ap-\\nproximate gradient descent) via speci\\fc hardware parameters.\\nIt is also worth noting that the random weight change (RWC) algorithm has been imple-\\n26 mented in memristor networks [23, 39] previously, and while super\\fcially similar to MGD is\\nfunctionally very di\\u000berent. RWC is not an approximate gradient descent technique, since the\\nweight update is not scaled by the magnitude of the change in the cost, but rather random\\nperturbations are either kept or discarded based on whether or not they improve the cost.\\nBecause of this, it scales more poorly with number of parameters than the optimization\\ntechniques we have implemented in the MGD framework.\\n4 Hardware considerations\\n4.1 Time constants\\nHW1 HW2 HW3 backprop\\n\\u001cx(input-sample update time) 100 ns 1 ns 10 ps n\\/a\\n\\u001cp(perturbation time) 1 ms 10 ns 200 ps n\\/a\\n\\u001c\\u0012(parameter-update time) 1 ms 1\\u00b5s 200 ps n\\/a\\n2-bit parity training time ( 104steps) 20 s 200\\u00b5s 4\\u00b5s 70 msy\\nFashion-MNIST training time ( 106steps) 33 min 20 ms 400\\u00b5s 54 s\\u0003\\nCIFAR-10 training time ( 107steps) 5.6 hours 200 ms 4 ms 480 s\\u0003\\nExamples of hardwarechip-in-the-loop,\\nintegrated photonics\\nwith thermo-optic tuning[40, 11]Mem-compute devices [41],\\nanalog VLSI [42]superconducting devices [43],\\nathermal resonant\\nsilicon photonic modulator [44]yCPU\\/\\u0003GPU\\nTable 3: Approximate training time using MGD for estimated hardware time constants. Data\\nshown is estimated using the same dataset and network architectures as used in Table 2. The\\n\\fnal column shows the time it takes to get to that same accuracy using backpropagation on\\na standard GPU or CPU.\\nWhile the MGD framework is general, individual hardware platforms may have di\\u000berent\\npractical considerations that dictate the optimal choices for the di\\u000berent time constants.\\nHere we discuss potential issues for implementation on hardware, and point out tradeo\\u000bs\\nthat may exist.\\nPerturbation speed ( \\u001cp):In many hardware platforms, perturbing parameters directly\\nmay be di\\u000ecult or undesired either due to memory-endurance or update-speed concerns.\\nHowever, it is not necessary to to perturb parameters directly{instead, one may perturb\\na separate element that is in series with the parameter. For instance, in a photonic MZI\\narray, the weights may be implemented with slow ( \\u0018millisecond) thermal or mechanical phase\\nshifters, but the small perturbation could be implemented with a fast ( \\u0018nanosecond) electro-\\n27 optic modulator such as a depletion [44] or opto-mechanical [45] modulator. Alternatively,\\nin a memristor system, the perturbation could be implemented with a single sub-threshold\\ntransistor in series with the memristor.\\nFor perturbative-style techniques like MGD to function properly, \\u001cpshould be slower than\\nthe system's inference time [13] which includes the cost calculation and broadcast processes.\\nIf\\u001cpis too short, it can introduce misalignment between the input perturbation and resulting\\ncost feedback.\\nParameter update ( \\u001c\\u0012);When performing gradient descent on hardware, parameters\\nmust be updated multiple times. For a training period of length T, the weights will be\\nupdatedT=\\u001c \\u0012times. For some hardware platforms it may be advantageous to reduce the\\nnumber of parameter updates, for instance if parameters have a limited lifetime, take a\\nlong time to update, or the update process requires a large amount of energy. Fortunately,\\nTable 2 shows that for networks with large numbers of parameters, \\u001c\\u0012can be increased\\nwithout greatly impacting the overall training speed or \\fnal accuracy.\\nChanging training examples ( \\u001cx):The value of \\u001cxis limited by the speed with which\\ntraining examples can be input to the network. For most technologies, \\u001c\\u0012or\\u001cxwill not be\\nlimiting factors, as xand ^ysamples can be provided by a conventional computer or FPGA\\nat nanosecond timescales. However, it should be noted that MGD will not function correctly\\nif\\u001cxis shorter than the inference time of the hardware, and this will likely be the practical\\nlimit for\\u001cx.\\u001cxis also used as a way of controlling the batch size (given by \\u001c\\u0012=\\u001cx). The\\nparticulars of the task and dataset may determine the desired batch size, and this may have\\nsome e\\u000bect on \\u001cx, for example if the minimum allowable value of \\u001c\\u0012is very long.\\nCalculation of cost: The calculation of the cost is performed only at the output of the\\nsystem, and therefore it is acceptable that the hardware used in its computation be more\\nexpensive and complex. In integrated electronic systems, the cost can be straightforward\\nto implement, although it may occupy more relatively more chip real-estate than other\\nelements. However, for very exotic hardware platforms, this cost computation may still be\\n28 di\\u000ecult to perform on-chip. This can be addressed by using a non-standard cost function\\nor by using a computer to calculate the cost o\\u000b-chip. However, the input and output to an\\no\\u000b-chip computation may limit the speed of the cost computation and also limit the speed of\\nthe global broadcast of ~C, and may be another speed bottleneck in some hardware systems.\\nBased on the literature, some estimates of plausible time constants that could be imple-\\nmented in hardware are shown in Table 3 with the corresponding times to solve benchmark\\ntasks, based on the results from Table 2.Examples from the literature of hardware platforms\\nthat could implement these sets of parameters are shown in the \\fnal row. By comparison\\nof these results with the \\fnal column in Table 3, we see that using realistic estimates for\\nemerging hardware, MGD could be signi\\fcantly faster than current implementations using\\nbackpropagation.\\n4.2 Analog and digital implementations\\nThere are a few notable di\\u000berences for an analog versus a digital hardware implementation of\\nthe MGD framework. The discrete case requires one memory element at the network output\\nto storeC0(sample-and-hold), and a simple subtraction operation to compute ~C=C\\u0000C0.\\nThe network also requires some timesteps to be devoted to the measurement of C0. This\\nmeans that for the simple case of \\u001c\\u0012=\\u001cp, only a single additional memory element is required\\nfor training the entire hardware system, located at the network cost output.\\nHowever, in the case where \\u001c\\u0012>\\u001cp, an additional memory element is required for every\\nparameter (for instance, an analog capacitor or discrete memory) to perform the gradient\\nintegration. The analog case requires a lowpass \\flter at every parameter element, and a single\\nhighpass \\flter on the network output to convert Cto~C. These could be created with simple\\nanalog circuits like RC or LR \\flters. However, we note the use of a continuous highpass \\flter\\ncan cause implementation issues is some cases. For instance, if \\u0011is too large, rapid changes\\nin\\u0012can generate unwanted frequency components that mix with the perturbation input and\\nmay negatively a\\u000bect the gradient approximation. Moreover, if the dataset is discrete, jumps\\n29 inxcan propagate high frequency noise through Cand ~C, again negatively a\\u000becting the\\ngradient approximation. The simulations demonstrate that in principle MGD can be used\\non di\\u000berent hardware platforms of either an analog or digital nature for network training.\\n5 Discussion\\nAn interesting analogy can be drawn between the MGD training framework and wireless\\ncommunication systems. In wireless communications, cell phone users must be able to extract\\nthe signal that is relevant to their own communication from a received broadcast that contains\\nmultiple users' signals. MGD operates similarly{each parameter is analogous to an individual\\ncell phone user and the global cost broadcast ~Cis analogous to the cell tower transmitter.\\nThe broadcasted signal ~Cis available to all parameters, and each parameter extracts its\\nrelevant (gradient) information { analogous to voice data { from that broadcasted signal.\\nThe encoding and multiplexing techniques used in wireless communications are broadly\\ntermed \\\"multiple access\\\" techniques in communications, and there are several varieties of\\nthem including frequency, time, and code multiplexing. Similarly, in MGD, these multi-\\nplexing techniques are directly analogous to the di\\u000berent perturbation types discussed in\\nSection 3.4. For instance, sinusoidal perturbations can be considered a type of frequency-\\nmultiplexing: each \\\\user\\\" (parameter) has a unique frequency containing information rel-\\nevant to it, and must perform a time-integration to extract that information from the ~C\\nsignal which contains many other, irrelevant signals. Likewise, as mentioned in Section 3.4,\\nthe discretef\\u0000\\u0001\\u0012;+\\u0001\\u0012gperturbation type is analogous to the code-multiplexing techniques\\nused in modern cell phones.\\nIt is important to note that for a \\fxed bandwidth broadcast, all multiplexing techniques\\nhave the same nominal spectral e\\u000eciency. However, the particulars of real systems can\\ngreatly a\\u000bect which multiplexing techniques are most e\\u000bective. For example, because code-\\nmultiplexing encodes a user's signal out over a wide time window and a spread spectrum, it\\n30 has been shown to have an improved robustness to multipath distortion and timing errors\\ncompared to time-multiplexing.\\nSimilarly, the various perturbation types described earlier may operate better or worse\\ndepending on the particulars of the hardware network and environment. Newly developed\\ncommunications techniques may also be directly applicable to MGD training in real hardware\\nsystems. Key to its usage in hardware, each parameter can operate like a cell phone |\\nas an independent device with a receiver, processor and memory using only information\\navailable locally and from the global broadcast. This contrasts with most other training\\nalgorithms, where error signals must propagate through the entire network [46]. While trivial\\nin software, this type of upstream information-\\row means that hardware neurons must (1)\\nhave additional memory to process the error signal, (2) perform di\\u000berent operations for\\nforward-inference and reverse error propagation, and (3) operate in a synchronized fashion\\nto propagate the error correctly.\\nIn systolic arrays \\\\broadcast\\\" is often discouraged because the time and energy of com-\\nmunication is proportional to the number of inputs by CV2(with C growing with the number\\nof inputs). Furthermore, as the capacitance is larger, then a larger ampli\\fer is needed for\\nthe communication which also requires energy and area. However in other types of analog\\nor emergent hardware, broadcast may be a more energy and space e\\u000ecient process. For\\nexample, in optical hardware, a global optical signal could be accessible in a simple slab\\nwaveguide mode. Alternatively, in analog electrical hardware, a single electrical plane could\\nbe used to carry the ~Csignal back to the individual parameters.\\nThe MGD framework may also be much more biologically-plausible than other algo-\\nrithms. There are a host of algorithms being developed to address the mystery of how the\\nbrain learns, and to use biology as inspiration to \\fnd more hardware-friendly approaches\\n[46, 47, 48]. One of the major issues with backpropagation from both a practical and bio-\\nplausibility standpoint is the need for a separate type of operation in the forward (inference)\\nand backward (learning) phases. This is not the case for MGD: in MGD, the training is\\n31 done in the same con\\fguration used for inference. In a biological analogy, the a global cost\\nsignal ~Ccould be modeled as some type of quasi-global chemical signal in the brain.\\nIn future, the MGD algorithm could be extended such that the cost is not global to every\\nsynapse, but rather to speci\\fc clusters of neurons\\/synapses. There are chemical signals in\\nthe brain with these types of properties. The time scales needed for perturbation may line\\nup with biologically observed e\\u000bects such as short-term plasticity (minutes) [49, 50] and\\nsynaptic noise [51]. MGD is also consistent with \\\\three-factor\\\" learning rules, for which\\nthere is mounting experimental evidence [50]. Additionally, although not explored in this\\nwork, MGD should work in a spiking model. Similar algorithms that use perturbations in\\nneuron conductance [52], probabilistic synapse transmission [53] or the stochastic nature\\nof spike-timing arrival [54] for reward-based learning have been explored in small spiking\\nnetworks.\\n6 Conclusions\\nIn this paper we show that with realistic timescales for emerging hardware, training via\\nMGD could be orders of magnitude faster than backpropagation in terms of wall-clock time-\\nto-solution on a standard GPU\\/CPU. The MGD framework allows the implementation of\\nmultiple optimization algorithms using a single, global, cost signal and local parameter\\nupdates. The style of algorithm used (e.g. \\fnite-di\\u000berence, coordinate-descent, SPSA, etc.)\\ncan be adjusted via the tuning of the MGD time-constants, and can even be adjusted during\\ntraining if desired. Because it is a model-free perturbative technique (sometimes called\\nzeroth order optimization), it is applicable to a wide range of systems { it can be applied to\\nboth analog and digital hardware platforms, and it can be used in the presence of noise and\\ndevice imperfections. This overcomes a major barrier to using hardware platforms based on\\nemerging technologies, which are often di\\u000ecult to train.\\nGoing forward, there are many opportunities to explore the application of MGD on both\\n32 large and small hardware systems. The most direct next step will be to test the training per-\\nformance in existing hardware, for example on photonic [40] or memristive crossbar hardware\\nusing a chip-in-the-loop process. This can occur without even redesigning the hardware, as\\nMGD only requires the ability to input samples, capture inference output, and modify pa-\\nrameters when in a chip-in-the-loop con\\fguration. When used in this fashion, the speed will\\nmost likely be limited by system I\\/O. For example, perturbations can be injected directly\\nto the hardware from an external computer, and that same computer could capture the\\nchanges in cost and perform the gradient approximation and calculate the weight updates.\\nThis would allow testing of the algorithm without any changes to the hardware. Ultimately,\\nto overcome I\\/O limitations local circuits can be designed be implemented for autonomous\\nonline training. Although not examined in detail here, in this case many of the hardware\\nplatforms examined would likely have several orders of magnitude improvement in terms of\\nenergy usage as well.\\nThere is also lots of room for improvement on tuning the technique and its hyper-\\nparameters. In this paper we performed very little hyperparameter optimization or reg-\\nularization, and so there are likely opportunities for further optimization by examining tech-\\nniques such as dropout, momentum, and regularization. Also of interest is examining the\\nnode-perturbation version of the algorithm [55] on large datasets, which could signi\\fcantly\\nreduce the number of perturbative elements and speed up the training process.\\nWhile in this paper we only examined feed forward networks, it has already been demon-\\nstrated that is possible to use perturbative techniques to train recurrent neural networks [56],\\nspiking networks [53] and other non-standard networks at small scale; future work remains\\nto demonstrate their utility on problems of modern interest. This will have applicability to\\na wide range of neuromorphic hardware and other physical neural networks.\\nUltimately, MGD is well-suited to implementation directly on-chip with local, autonomous\\ncircuits. Although signi\\fcant work remains before autonomous learning can be truly imple-\\nmented without the participation of an external computer, such a device would be truly\\n33 enabling for remote and edge-computing applications, allowing training in-the-\\feld on real-\\ntime data.\\n7 Acknowledgements\\nThe authors acknowledge helpful discussions with the NIST NCSG community. This re-\\nsearch was funded by NIST (https:\\/\\/ror.org\\/05xpvk416) and University Colorado Boulder\\n(https:\\/\\/ror.org\\/02ttsq026).\\nReferences\\n[1] G Hinton Y LeCun, Y Bengio. Deep learning. Nature , 521(7553):436{444, may 2015.\\n[2] Catherine D. Schuman, Thomas E. Potok, Robert M. Patton, J. Douglas Birdwell,\\nMark E. Dean, Garrett S. Rose, and James S. Plank. A Survey of Neuromorphic\\nComputing and Neural Networks in Hardware. may 2017.\\n[3] Wilfried Haensch, Tayfun Gokmen, and Ruchir Puri. The Next Generation of Deep\\nLearning Hardware: Analog Computing. Proceedings of the IEEE , 107(1):108{122, jan\\n2019.\\n[4] Paul A Merolla, John V Arthur, Rodrigo Alvarez-icaza, Andrew S Cassidy, Jun Sawada,\\nFilipp Akopyan, Bryan L Jackson, Nabil Imam, Chen Guo, Yutaka Nakamura, Bernard\\nBrezzo, Ivan Vo, Steven K Esser, Rathinakumar Appuswamy, Brian Taba, Arnon Amir,\\nMyron D Flickner, William P Risk, Rajit Manohar, and Dharmendra S Modha. A\\nmillion spiking-neuron integrated circuit with a scalable communication network and\\ninterface. Science , 345:668, 2014.\\n[5] Mike Davies, Narayan Srinivasa, Tsung-Han Lin, Gautham Chinya, Yongqiang Cao,\\nHarsha Choday, Georgios Dimou, Prasad Joshi, Nabil Imam, Shweta Jain, Yuyun Liao,\\n34 Chit-Kwan Lin, Andrew Lines, Ruokun Liu, Deepak Mathaikutty, Steve Mccoy, Arnab\\nPaul, Jonathan Tse, Guruguhanathan Venkataramanan, Yi-Hsin Weng, Andreas Wild,\\nYoonseok Yang, and Hong Wang. Loihi: A neuromorphic manycore processor with on-\\nchip learning; loihi: A neuromorphic manycore processor with on-chip learning. IEEE\\nMicro , 38, 2018.\\n[6] Karlheinz Meier. A mixed-signal universal neuromorphic computing system. In 2015\\nIEEE International Electron Devices Meeting (IEDM) , 2015.\\n[7] A. Mehonic and A. J. Kenyon. Brain-inspired computing needs a master plan. Nature ,\\n604:255{260, 4 2022.\\n[8] Neil C Thompson, Kristjan Greenewald, Keeheon Lee, and Gabriel F Manso. The\\ncomputational limits of deep learning. arXiv:2007.05558, 2020.\\n[9] Logan G. Wright, Tatsuhiro Onodera, Martin M. Stein, Tianyu Wang, Darren T.\\nSchachter, Zoey Hu, and Peter L. McMahon. Deep physical neural networks trained\\nwith backpropagation. Nature , 601:549{555, 2022.\\n[10] J. Kiefer and J. Wolfowitz. Stochastic Estimation of the Maximum of a Regression\\nFunction. The Annals of Mathematical Statistics , 23(3):462{466, 1952.\\n[11] Yichen Shen, Nicholas C. Harris, Scott Skirlo, Mihika Prabhu, Tom Baehr-Jones,\\nMichael Hochberg, Xin Sun, Shijie Zhao, Hugo Larochelle, Dirk Englund, and Marin\\nSoljacic. Deep learning with coherent nanophotonic circuits. Nature Photonics , 11:441{\\n446, 2017.\\n[12] James C. Spall. Multivariate stochastic approximation using a simultaneous perturba-\\ntion gradient approximation. IEEE Transactions on Automatic Control , 37:332{341,\\n1992.\\n35 [13] Amir Dembo and Thomas Kailath. Model-Free Distributed Learning. IEEE Transac-\\ntions on Neural Networks , 1(1):58{70, 1990.\\n[14] T. Matsumoto and M. Koga. Novel learning method for analogue neural networks. ElL,\\n26(15):1136, sep 1990.\\n[15] Marwan Jabri and Barry Flower. Weight Perturbation: An Optimal Architecture and\\nLearning Technique for Analog VLSI Feedforward and Recurrent Multilayer Networks.\\nIEEE Transactions on Neural Networks , 3(1):154{157, 1992.\\n[16] J. Alspector, R. Meir, B. Yuhas, A. Jayakumar, and D. Lippe. A parallel gradient\\ndescent method for learning in analog VLSI neural networks | Proceedings of the 5th\\nInternational Conference on Neural Information Processing Systems. In NIPS'92: Pro-\\nceedings of the 5th International Conference on Neural Information Processing Systems ,\\npages 836{844, 1992.\\n[17] David B. Kirk and Douglas Kerns. Analog VLSI Implementation of Multi-dimensional\\nGradient Descent. In Advances in Neural Information Processing Systems 5 (NIPS\\n1992) , pages 789{796, 1992.\\n[18] Yutaka Maeda, Hiroaki Hirano, and Yakichi Kanata. A learning rule of neural net-\\nworks via simultaneous perturbation and its hardware implementation. Neural Net-\\nworks , 8(2):251{259, jan 1995.\\n[19] Gert Cauwenberghs. An analog VLSI recurrent neural network learning a continuous-\\ntime trajectory. IEEE Transactions on Neural Networks , 7(2):346{361, 1996.\\n[20] Hiroshi Miyao, Kazuhiro Noguchi, Masafumi Koga, and Takao Matsumoto. Multifre-\\nquency oscillation learning method for analog neural network: Its implementation in a\\nlearning LSI. Electronics and Communications in Japan (Part III: Fundamental Elec-\\ntronic Science) , 80(5), 1997.\\n36 [21] Antonio J. Montalvo, Ronald S. Gyurcsik, and John J. Paulos. Toward a general-\\npurpose analog VLSI neural network with on-chip learning. IEEE Transactions on\\nNeural Networks , 8(2):413{423, 1997.\\n[22] Yutaka Maeda and Toshiki Tada. FPGA implementation of a pulse density neural\\nnetwork with learning ability using simultaneous perturbation. IEEE Transactions on\\nNeural Networks , 14(3):688{695, may 2003.\\n[23] Shyam Prasad Adhikari, Hyongsuk Kim, Ram Kaji Budhathoki, Changju Yang, and\\nLeon O. Chua. A circuit-based learning architecture for multilayer neural networks\\nwith memristor bridge synapses. IEEE Transactions on Circuits and Systems I: Regular\\nPapers , 62(1):215{223, jan 2015.\\n[24] Chunhua Wang, Lin Xiong, Jingru Sun, and Wei Yao. Memristor-based neural networks\\nwith weight simultaneous perturbation training. Nonlinear Dynamics , 95(4):2893{2906,\\n2019.\\n[25] Saumil Bandyopadhyay, Alexander Sludds, Stefan Krastanov, Ryan Hamerly, Nicholas\\nHarris, Darius Bunandar, Matthew Streshinsky, Michael Hochberg, and Dirk Englund.\\nSingle chip photonic deep neural network with accelerated training. pages 1{21, aug\\n2022.\\n[26] At\\u0010l\\u0010m G\u007f unes Baydin, Barak A. Pearlmutter, Don Syme, Frank Wood, and Philip Torr.\\nGradients without backpropagation, 2022.\\n[27] David Silver, Anirudh Goyal, Ivo Danihelka, Matteo Hessel, and Hado van Hasselt.\\nLearning by directional gradient descent. In Proceedings of the International Conference\\non Learning Representations (ICLR) , 2022.\\n[28] Mengye Ren, Simon Kornblith, Renjie Liao, and Geo\\u000brey Hinton. Scaling forward\\ngradient with local losses, 2022.\\n37 [29] Brian Hoskins, Mitchell Fream, Matthew Daniels, Jonathan Goodwill, Advait Mad-\\nhavan, Jabez Mcclelland, Osama Yousuf, Gina Adam, Wen Ma, Tung Hoang, Mark\\nBranstad, Muqing Liu, Rasmus Madsen, Martin Lueker-Boden, and Martin Lueker. A\\nsystem for validating resistive neural network prototypes. International Conference on\\nNeuromorphic Systems 2021 , 5.\\n[30] Tiankuang Zhou, Xing Lin, Jiamin Wu, Yitong Chen, Hao Xie, Yipeng Li, Jingtao Fan,\\nHuaqiang Wu, Lu Fang, and Qionghai Dai. Large-scale neuromorphic optoelectronic\\ncomputing with a recon\\fgurable di\\u000bractive processing unit. Nature Photonics , 15:367{\\n373, 2021.\\n[31] Saumil Bandyopadhyay, Ryan Hamerly, Ryan Hamerly, and Dirk Englund. Hardware\\nerror correction for programmable photonics. Optica, Vol. 8, Issue 10, pp. 1247-1255 ,\\n8:1247{1255, 2021.\\n[32] Amir Khosrowshahi, Casimir Wierzynski, Michael R. DeWeese, Michael Y.-S. Fang, and\\nSasikanth Manipatruni. Design of optical neural networks with component imprecisions.\\nOptics Express, Vol. 27, Issue 10, pp. 14009-14029 , 27:14009{14029, 2019.\\n[33] Miao Hu, Catherine E. Graves, Can Li, Yunning Li, Ning Ge, Eric Montgomery, Noraica\\nDavila, Hao Jiang, R. Stanley Williams, J. Joshua Yang, Qiangfei Xia, and John Paul\\nStrachan. Memristor-based analog computation and neural network classi\\fcation with\\na dot product engine. Advanced Materials , 30:1705914, 2018.\\n[34] Stefano Ambrogio, Pritish Narayanan, Hsinyu Tsai, Robert M. Shelby, Irem Boybat,\\nCarmelo Di Nolfo, Severin Sidler, Massimo Giordano, Martina Bodini, Nathan C.P.\\nFarinha, Benjamin Killeen, Christina Cheng, Yassine Jaoudi, and Geo\\u000brey W. Burr.\\nEquivalent-accuracy accelerated neural-network training using analogue memory. Na-\\nture, 558:60{67, 2018.\\n38 [35] Stefano Ambrogio, Simone Balatti, Antonio Cubeta, Alessandro Calderoni, Nirmal Ra-\\nmaswamy, and Daniele Ielmini. Statistical \\ructuations in hfox resistive-switching mem-\\nory: Part i-set\\/reset variability. IEEE Transactions on Electron Devices , 61:2912{2919,\\n2014.\\n[36] Jubin Hazra, Maximilian Liehr, Karsten Beckmann, Sarah Ra\\fq, and Nathaniel Cady.\\nImproving the memory window\\/resistance variability trade-o\\u000b for 65nm cmos integrated\\nhfo2 based nanoscale rram devices. volume 2019-October. Institute of Electrical and\\nElectronics Engineers Inc., 10 2019.\\n[37] Cory Merkel and Dhireesha Kudithipudi. Comparison of o\\u000b-chip training methods\\nfor neuromemristive systems. volume 2015-February, pages 99{104. IEEE Computer\\nSociety, 2 2015.\\n[38] Justin Werfel, Xiaohui Xie, and H. Sebastian Seung. Learning Curves for Stochastic\\nGradient Descent in Linear Feedforward Networks. Neural Computation , 17(12):2699{\\n2718, dec 2005.\\n[39] Changju Yang, Hyongsuk Kim, Shyam Prasad Adhikari, and Leon O. Chua. A Circuit-\\nBased Neural Network with Hybrid Learning of Backpropagation and Random Weight\\nChange Algorithms. Sensors 2017, Vol. 17, Page 16 , 17(1):16, dec 2016.\\n[40] Alexander N. Tait, Thomas Ferreira De Lima, Ellen Zhou, Allie X. Wu, Mitchell A.\\nNahmias, Bhavin J. Shastri, and Paul R. Prucnal. Neuromorphic photonic networks\\nusing silicon photonic weight banks. Scienti\\fc Reports , 7, 2017.\\n[41] Geo\\u000brey W Burr, Robert M Shelby, Abu Sebastian, Sangbum Kim, Seyoung Kim, Sev-\\nerin Sidler, Kumar Virwani, Masatoshi Ishii, Pritish Narayanan, Alessandro Fumarola,\\nLucas L Sanches, Irem Boybat, Manuel Le Gallo, Kibong Moon, Jiyoo Woo, Hyunsang\\nHwang, and Yusuf Leblebici. Neuromorphic computing using non-volatile memory.\\n2:89{124, 2017.\\n39 [42] Y. Kohda, Y. Li, K. Hosokawa, S. Kim, R. Khaddam-Aljameh, Z. Ren, P. Solomon,\\nT. Gokmen, S. Rajalingam, C. Baks, W. Haensch, and E. Leobandung. Unassisted true\\nanalog neural network training chip. Technical Digest - International Electron Devices\\nMeeting, IEDM , 2020-December:36.2.1{36.2.4, 12 2020.\\n[43] Michael Schneider, Emily Toomey, Graham Rowlands, Je\\u000b Shainline, Paul Tschirhart,\\nand Ken Segall. Supermind: a survey of the potential of superconducting electronics\\nfor neuromorphic computing. Superconductor Science and Technology , 35:053001, 2022.\\n[44] Erman Timurdogan, Cheryl M. Sorace-Agaskar, Jie Sun, Ehsan Shah Hosseini, Alek-\\nsandr Biberman, and Michael R. Watts. An ultralow power athermal silicon modulator.\\nNature Communications , 5:1{11, 2014.\\n[45] Mark Dong, Genevieve Clark, Andrew J Leenheer, Matthew Zimmermann, Daniel\\nDominguez, Adrian J Menssen, David Heim, Gerald Gilbert, Dirk Englund, and Matt\\nEichen\\feld. High-speed programmable photonic circuits in a cryogenically compatible,\\nvisible-near-infrared 200 mm cmos architecture. Nature Photonics , 16:59{65, 2022.\\n[46] Timothy P. Lillicrap, Adam Santoro, Luke Marris, Colin J. Akerman, and Geo\\u000brey\\nHinton. Backpropagation and the brain. Nature Reviews Neuroscience 2020 21:6 ,\\n21(6):335{346, apr 2020.\\n[47] Friedemann Zenke and Emre O. Neftci. Brain-inspired learning on neuromorphic sub-\\nstrates. Proceedings of the IEEE , 109:935{950, 5 2021.\\n[48] Benjamin Scellier and Yoshua Bengio. Equilibrium propagation: Bridging the gap\\nbetween energy-based models and backpropagation. Frontiers in Computational Neu-\\nroscience , 11:24, 5 2017.\\n[49] Andrea Soltoggio. Short-term plasticity as cause{e\\u000bect hypothesis testing in distal\\nreward learning. Biological Cybernetics , 109:75{94, 2 2015.\\n40 [50] Eligibility Traces and Plasticity on Behavioral Time Scales: Experimental Support of\\nNeoHebbian Three-Factor Learning Rules. Frontiers in Neural Circuits , 0:53, jul 2018.\\n[51] Benjamin A. Rowland, Anthony S. Maida, and Istvan S. N. Berkeley. Synaptic noise as\\na means of implementing weight-perturbation learning. 18(1):69{79, mar 2007.\\n[52] Ila R. Fiete and H. Sebastian Seung. Gradient learning in spiking neural networks by\\ndynamic perturbation of conductances. Phys. Rev. Lett. , 97:048104, Jul 2006.\\n[53] H. Sebastian Seung. Learning in Spiking Neural Networks by Reinforcement of Stochas-\\ntic Synaptic Transmission. Neuron , 40(6):1063{1073, dec 2003.\\n[54] Xiaohui Xie and H. Sebastian Seung. Learning in neural networks by reinforcement of\\nirregular spiking. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related\\nInterdisciplinary Topics , 69(4):10, 2004.\\n[55] Barry Flower and Marwan Jabri. Summed weight neuron perturbation. In NIPS'92:\\nProceedings of the 5th International Conference on Neural Information Processing Sys-\\ntems, pages 212{219, 1992.\\n[56] Gert Cauwenberghs. A Fast Stochastic Error-Descent Algorithm for Supervised Learn-\\ning and Optimization. In Advances in Neural Information Processing Systems 5 (NIPS\\n1992) , pages 244{251, 1992.\\n41\",\"12\":\" 1 Wang et al. Evolutionary Deep Nets for Non-Intrusive Load Monitoring Abstract\\u2014Non-Intrusive Load Monitoring (NILM) is an energy efficiency technique to track electricity consumption of an individual appliance in a household by one aggregated single, such as building level meter readings. The goal of NILM is to disaggregate the appliance from the aggregated singles by computational method. In this work, deep learning approaches are implemented to operate the desegregations. Deep neural networks, convolutional neural networks, and recurrent neural networks are employed for this operation. Additionally, sparse evolutionary training is applied to accelerate training efficiency of each deep learning model. UK-Dale dataset is used for this work.   Index Terms\\u2014deep learning, load disaggregation, non-intrusive load monitoring, sparse evolutionary training.    I. INTRODUCTION Non-Intrusive Load Monitoring (NILM, alternative to energy disaggregation) is a computational technique to track the electricity consumption at appliance level based on a single sensor at building level, instead of imbedding any sensor on an individual appliance [1]. This work concentrates NILM in aspect of residential electricity consumption. Significances of NILM for residential energy efficiency are summarized: as compared with intrusive monitoring, NILM performs without direct sensor network installation to individual appliance; consequently, NILM is an effective implementation to physical and computing cost, and a securable monitoring technology for privacy of the residential consumers [2]. As the benefits of NILM, first, the residential consumers receive energy saving services that enhance electricity billing with specific appliance level advice, mobile-application-based energy audits, and real time consumption information; second, utility provides service of energy trading and planning and regulatory incentives; third, home service sector delivers maintenance for Heating, Ventilation and Air Conditioning (HVAC) systems, retrofits, advertising and electricity security services [3].  A framework of NILM approach is given by Zoha et al. [4]. Accordingly, NILM undertakes three modules: first, data acquisition, feature engineering, and inference and learning. The data acquisition module collects global (or aggregated) data, such as voltage, current, and power at different frequencies in accordance of requirements. The feature engineering module extracts characteristics of the data which indicate the activations of the appliances. Furthermore, the features are classified in two types: steady-state (for example, Voltage-Current trajectory) and transient-sate (for example, start-up current waveforms). The residential appliances are categorized in four types: two-state (for example, lamp, which has on and off state), multi-state (for example, washing machine), continuously variable devices (CVD) (for example, power drill and dimmer light), and permanent devices (for example, smoker detector). The two types of feature are extracted to describe each type of appliance. The inference and learning module disaggregate induvial appliance data from the aggregated data and identifies the appliance in specific based on the features. Furthermore, ground truth data of each appliance are collected for the learning system training.  Common NILM learning methods are HMM-based methods. Hidden Markov Model (HMM) is applied when the concept of NILM is originally proposed by Hart [5]. Factorial Hidden Markov Model (FHMM) [6], which is employed for disaggregation in NILM commonly [7]. Additionally, other well-known learning methods, such as support vector machine (SVM) [8] [9], neural networks [10] [11], and Bayesian models [12] [13], are evaluated as outperformed approaches to NILM. Accordingly, first, based on steady and transient features, SVM appears efficiency to disaggregate four types of appliance, while neural networks appears efficiency to appliance type of two- and multi-state and CVD; second, HMM-based and Bayesian methods perform appears efficiency to disaggregate four types of appliance, while neural networks appears efficiency to appliance type of two- and multi-state and CVD; second, HMM-based and Bayesian methods perform disaggregation of two- and multi-state appliances by steady-state features. In summary, above-mentioned framework and learning methods are feature-based approaches. Deep learning methods are applied and evaluated as efficiency approach to NILM instead of feature engineering [1]. This work focuses on the deep learning approach.  The objective of this work is formed to improve disaggregation accuracy and computing efficiency of NILM by deep learning approach. Accordingly, by observing above-mentioned methods, even high accuracies (97%-99%) [4] are achieved, a general method to disaggregate all types of appliance is not defined. Two- and multi-state appliances are identified at a high accurate because steady-state features are sufficient to characterize such types of appliances with lower frequency samples; however, CVD and permanent devices challenge the disaggregation performance because high-frequency samples that describe detail significance of transient state to the consumption event is required to train the models. When high-frequency sample is employed, cost of data acquisition increases. Computing efficiency gets adverse impact on the high-frequency condition. Moreover, feature engineering requires human-labor involvement and additional extraction and selection model, which cause more computing Evolutionary Deep Nets for Non-Intrusive Load Monitoring Jinsong Wang, and Kenneth A. Loparo, Life Fellow Preprint DOI: 10.13140\\/RG.2.2.25983.28324 cost for large volume of data at high-frequency. Due to above issues, this work employs three deep learning models: deep neural networks (DNN) [14], convolutional neural networks (CNN) [15], and recurrent neural networks (RNN) [16], to NILM, instead of feature-based methods. Sparse evolutionary training (SET) [17] are applied to optimize the computing efficiency of each deep learning model.  II. DATA The UK-DALE [18], which is stand for United Kingdom domestic appliance-level electricity demand, dataset is used in this work. The dataset records active power (kilowatts or kW) which is collected at very 6 second for 5 years in file name by data acquisition channel number. Each channel dataset is the raw data of the electricity consumption of each appliance.  Channel datasets of dishwasher, washing machine, microwave and fridge selected for the NILM experiment in this work. They correspond to four types of appliance, respectively and conduct the ground truth of the disaggregation.  Training and testing dataset are a synthesis aggregated power dataset constructed by the selected appliance due to the issues that the power of ground truth aggregated data is much larger than the sum of the selected appliance power at one time step, which is large noise to the classification target appliance. The training and testing dataset are constructed in following steps: Step 1: within each individual appliance channel dataset, length of a time window is set up as 1 hour, in which 600 data points are included, and length of a forward moving step is setup as 5 minutes, in which 50 data points are included. Each time window conducts one data entry. Step 2: validity of data in the time windows is defined by following condition: if the appliance appears that active power consumption events add up to more than 10 minutes with an hour (one time window), the time window is a valid data entry. A single power measurement greater than zero indicates one active power consumption event.  Step 3: the validated data entries are combined into a matrix. Four appliance channels correspond to the matrixes, data_M1~4, respectively, and the number of the validated data entries of each matrix is calculated as num_1~4.  Step 4: a binary (0 or 1) variable \\ud835\\udc63\\ud835\\udc52\\ud835\\udc50_01 and a numeric integer variable \\ud835\\udc63\\ud835\\udc52\\ud835\\udc50_\\ud835\\udc56\\ud835\\udc5b are initiated.  \\ud835\\udc63\\ud835\\udc52\\ud835\\udc50_01 is a 4-dimensional vector, in which four binary values are random generated. \\ud835\\udc63\\ud835\\udc52\\ud835\\udc50_\\ud835\\udc56\\ud835\\udc5b is a 4-dimensional vector, in which four elements are randomly selected from the range of 0 to num_1~4, respectively. Each integer element represents a valid data entry in the corresponding matrix. The subjective of this step is to construct activation combination of the appliances randomly. Step 5: synthesis aggregated power data is generated. The synthesis aggregated power is sum of the power of four selected appliances. In one iteration, a single value in the data entries that has the corresponding representatives of valid data entries in \\ud835\\udc63\\ud835\\udc52\\ud835\\udc50_\\ud835\\udc56\\ud835\\udc5b associated with \\u201c1\\u201d of the activation status in \\ud835\\udc63\\ud835\\udc52\\ud835\\udc50_01 are summed. Having 10000 repetitions, randomly combined synthesis aggregated power data emeries are generated as the featured input dataset.  III. METHODS 1) Deep Neural Networks  Artificial neural networks (ANN) [19] takes inspiration of biological neural networks is a general name of the neural network family. It powerfully conducts all types of machine learning tasks: supervised learning [20], unsupervised learning [20], and reinforcement learning [21]. ANN is a mathematical model constructed by neurons connected by weights. The neurons only connect to the ones in immediate next consecutive layer. The single neuron composed by incoming and outcoming weights and an activation function. Neurons assemble in layers of ANN. If an ANN appears a fully-connected structure, the neurons in one layer are connected to every single neuron in the consecutive layer.  ANN models are built in various architectures according to their purposes. appears a fully-connected structure, the neurons in one layer are connected to every single neuron in the consecutive layer.  ANN models are built in various architectures according to their purposes. Multi-layer perceptron (MLP) [22] is a common ANN architecture that is a feed-forward training model mapping an input set to the targeted output sets, nonlinearly. \\u201cmulti-layer\\u201d describes that the model is formed by an input and output layer and one or more hidden layer(s). The information flow (datasets or features) starts at the input layer, passes through the hidden layers towards the output layer. Thus in usage of supervised learning, MLP is evaluated as an universal estimator for regression and classification cases.  Convolutional neural networks (CNN) are constructed by a MLP in which a convolutional layer is added before the first hidden layer in. The convolutional layer filters the input information flow into a small group of receptive fields which are the down-sampled feature maps. A fully-connected layer are applied after the convolutional layer to train the input feature maps [14].  Recurrent neural network (RNN) trains the neural networks in cycle procedures that the output from previous neurons is assigned to the input to neuron at current time step. Hidden states are labeled in order that the history of outputs vector is computed, and the shared weights are across the procedures [14]. 3 Wang et al. Evolutionary Deep Nets for Non-Intrusive Load Monitoring In recent years, rapidly developed processing power enables even hundred-layer-depth of hidden layers in neural network, which is entitled as \\u201cdeep learning\\u201d. Such MLP is named deep neural networks (DNN) in general [14]. A larger number of hidden layers achieves higher accuracy while results more computing cost and complexity for a fully-connected DNN; therefore, SET is applied to improve the computing efficiency.   2) Sparse Evolutionary Training (SET) Sparse Evolutionary Training (SET) give expressions to three straightforward significances: the nets have relatively few connections, few hubs and short paths. Applying SET to the deep learning models is motivated by its efficiency to simplify fully-connected networks to adaptive sparse connected networks and optimize training time with limit computing resource [17]. The general training procedure of SET is that:  First, having initialized a fully-connected ANN model, each fully-connected layer (hidden layer) is replaced by a sparse connected layer. The neurons in the fully-connected layer randomly connect to next layer by Erdod-Renyi topology [23]. Each Then the connection weights are represented in a sparse matrix. The matrix is assigned a coefficient that indicates sparse level. Additionally, each weight is assigned a fraction that indicates the validity of the weight. The fractions are computed and updated at each epoch.  Second, for each epoch during the training phase, first, a feed-forward training with weights update is processed; then, in the sparse weight matrix, the weights with smallest positive and the largest negative fraction are removed and replace by new weights. Then the training is process again with the updated weigh matrix.    IV. EXPERIMENT 1) Set Up  Three deep learning methods: deep-MLP (or DNN), CNN, and RNN are implemented for NILM models. DNN is composed by one input layer, two hidden layers, and one, output layer. CNN contains a one-dimension convolution layer in addition to DNN. Batch normalization function [24] is used to normalize data after the convolution function in order to fit the data to next fully-connected layer. A maxpooling1d layer is added to reduce the complexity of the output and prevent overfitting of the data. 0.2 dropout layer is added, that randomly assigns 0 weights to the neurons in the network by a rate. With this operation, the network becomes less sensitive to react to smaller variations in the data. Furthermore, it increase accuracy on unseen datasets [25]. RNN is constructed by a simple recurrent type: ono-to-one. The input data is the synthesis aggregated power data, which is divided to 8:2 for training and testing, respectively. The output are the classification of the appliance. SET is applied to accelerate each deep learning models. Disaggregation performances of the standard DNN, CNN, and RNN are compared with SET boosted models: SET- DNN, SET-CNN, and SET-RNN, by their losses and accuracies plots. Evaluation matrixes of mean absolute error [14], precision [14], and recall [14] to compare the results.  2) Results and Discussions Having investigated the accuracy and loss plots, SET-CNN performs the best disaggregation operation. CNN achieves highly similar performance as SET-CNN. DNN appears unstable obvious shifts of the accuracy and loss curves in its disaggregation performance, graphically, for example, the accuracy and loss plots in cases of fridge and dishwasher.  As observed from the cases of microwave and washing machine, according to the type of deep learning model, convolution-based methods achieve similar accuracy among same model, and higher accuracy than the standard-MLP-based models (DNN and SET-DNN). Moreover, DNN and SET-DNN operate disaggregation in similar performance. The convolution layer filters the large input of active power data to small samples for the hidden layers in training procedures, that is the reason to the outperformances of the convolution-based models. Additionally, RNN and SET-RNN large input of active power data to small samples for the hidden layers in training procedures, that is the reason to the outperformances of the convolution-based models. Additionally, RNN and SET-RNN perform disaggregation operation below other model. RNN models have advantage to sequential data; however, this work apply time window which does not require time series.   Effect of SET is obviously presented in cases of fridge, in Fig. 1.  plots of loss and accuracy. Preprint DOI: 10.13140\\/RG.2.2.25983.28324 \\nwhich SET-accelerated learning models achieve better performance. Additionally, the cases of microwave and washing machine indicate a slight effect of SET on appliance consumption disaggregation. In case of dishwasher, SET performs the worst accuracy of disaggregation, which is even lower than the standard DNN. The reason of the unstable effect to the learning models is that the models are constructed at relative simple architectures. Two hidden layers does not indicate high density and complexity required optimization. Moreover, simple models is optimized with unsuitable sparse level; consequently, the network has less connected neuron and decrease training efficiency.  Furthermore, the plot of evaluation matrixes demonstrated above discussions and present graphical comparison of each models. Moreover, accordingly, type of appliance has impact on the disaggregation accuracy. Microwave and washing machine have significant and longer power consumption events, which make the model operate disaggregation efficiently.  V. CONCLUSION NILM models are implemented by DNN, CNN, and RNN. SET is used to accelerate the training efficiency. The highest disaggregation accuracy is 97.6%. CNN and SET-CNN outperforms other deep learning models. From the result investigations, first, model has significant effect to the disaggregation performance; second, SET has less effect to neural networks with relative simple architectures; appliance type and consumption behavior that indicate long and significant consumption event are efficient to disaggregate. Future work is required to improve the disaggregation accuracy, apply SET to complex neural network to demonstrate the efficiency acceleration, and applied SET-based deep \\nFig. 2.  plots of evaluation matrix. 5 Wang et al. Evolutionary Deep Nets for Non-Intrusive Load Monitoring learning to all types of appliances with real power data.     REFERENCES  [1]  J. Kelly and W. Knottenbelt, \\\"Neural NILM: Deep Neural Networks Applied to Energy Disaggregation,\\\" in ACM BuildSys, Seoul, 2015.  [2]  K. C. ARMEL, A. GUPTA, G. SHRIMALI and A. ALBERT, \\\"IS DISAGGREGATION THE HOLY GRAIL OF ENERGY EFFICIENCY? THE CASE OF ELECTRICITY,\\\" Precourt Energy Efficiency Center, Stanford, 2012. [3]  D. Christensen, L. Earle and B. Sparn, \\\"NILM Applications for the Energy-Efficient Home,\\\" National Renewable Energy Laboratory, Golden, 2012. [4]  A. Zoha, A. Gluhak, M. A. Imran and S. Rajasegarar, \\\"Non-Intrusive Load Monitoring Approaches for Disaggregated Energy Sensing: A Survey,\\\" Sensors, pp. 16838-16866, 2012.  [5]  G. Hart, \\\"Nonintrusive appliance load monitoring,\\\" IEEE Proc., vol. 80, p. 1870\\u20131891, 1992.  [6]  Z. Ghahramani and M. I. Jordan, \\\"Factorial hidden markov models,\\\" Machine Learning , vol. 29, no. (2\\u20133), p. 245\\u2013273, 1997.  [7]  H. Kim, M. Marwah, M. Arlitt, G. Lyon and J. Han, \\\"Unsupervised disaggregation of low frequency power measurements,\\\" in the SIAM Conference on Data Mining, 2011.  [8]  S. Patel, T. Robertson, J. Kientz, M. Reynolds and G. Abowd, \\\"At the Flick of a Switch: Detecting and Classifying Unique Electrical Events on the Residential Power Line,\\\" in the 9th International Conference on Ubiquitous Computing, Innsbruck, 2007.  [9]  M. Marceau and R. Zmeureanu, \\\"Nonintrusive load disaggregation computer program to estimate the energy consumption of major end uses in residential buildings,\\\" Energ. Convers. Manag, vol. 41, p. 1389\\u20131403, 2000 .  [10]  D. Srinivasan, W. Ng and A. Liew, \\\"Neural-network-based signature recognition for harmonic source identification,\\\" IEEE Trans. Power Del., vol. 21, p. 398\\u2013405, 2006 .  [11]  A. Ruzzelli, C. Nicolas and A. Schoofs, \\\"O\\u2019Hare, G.M.P. Real-Time Recognition and Profiling of Appliances through a Single Electricity Sensor,\\\" in the 7th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks, Boston, 2010.  [12]  A. Marchiori, D. Hakkarinen, Q. Han and L. Earle, \\\"Circuit-level load monitoring for household energy management,\\\" IEEE Pervas. Comput, vol. 10, p. 40\\u201348, 2011.  [13]  M. Zeifman, \\\"Disaggregation of home energy display data using probabilistic approach,\\\" IEEE Trans. Consum. Electron , vol. 58, pp. 23-31, 2012.  [14]  I. Goodfellow, Y. Bengio and A. Courville, Deep Learning, Amherst: MIT Press, 2016.  [15]  Y. B. L. B. Y. &. H. P. LeCun, \\\"Gradient-based learning applied to document recognition,\\\" Proc. IEEE , vol. 86, p. 2278\\u20132324 , 1998.  [16]  A. e. a. Graves, \\\"A novel connectionist system for unconstrained handwriting recognition,\\\" IEEE Trans. Pattern Anal. Mach. Intell., vol. 31 , p. 855\\u2013868, 2009.  [17]  D. C. Mocanu, E. Mocanu, P. Stone, P. H. Nguyen, M. Gibescu and A. Liotta, \\\"Scalable training of artificial neural networks with adaptive sparse connectivity inspired by network science,\\\" NATURE COMMUNICATIONS, vol. 9, p. 2383, 2018.  [18]  J. Kelly and W. Knottenbelt1, \\\"The UK-DALE dataset, domestic appliance-level electricity demand and whole-house demand from five UK homes,\\\" SCIENTIFIC DATA, vol. 7, 2015.  [19]  C. M. Bishop, Pattern Recognition and Machine Learning, Secaucus: Springer-Verlag New York, 2006.  [20]  T. T. R. &. F. J. Hastie, The Elements of Statistical Learning, New York: Springer New York , 2001.  [21]  R. S. &. B. A. G. Sutton, Introduction to Reinforcement Learning, Cambridge: MIT Press, 1998.  [22]  F. Rosenblatt, Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms, Washington: Spartan , 1962.  [23]  P. &. R. A. Erd\\u00f6s, \\\"On random graphs i,\\\" Publ. Math.-Debr, vol. 6 , p. 290\\u2013297, 1959.  [24]  S. Ioffe and C. Szegedy, \\\"Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift,\\\" 2 2015. [Online]. Available: https:\\/\\/arxiv.org\\/pdf\\/1502.03167.pdf. [25]  N. Srivastava, G. C. Szegedy, \\\"Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift,\\\" 2 2015. [Online]. Available: https:\\/\\/arxiv.org\\/pdf\\/1502.03167.pdf. [25]  N. Srivastava, G. Hinton, A. Krizhevsk, I. Sutskever and R. Salakhutdinov, \\\"Dropout: A Simple Way to Prevent Neural Networks from Overfitting,\\\" Journal of Machine Learning Research, no. 15, pp. 1929-1958, 2014.\",\"13\":\" SYMBOLIC SYNTHESIS OF NEURAL NETWORKS\\nEli Whitehouse\\nNew York, NY 10024\\neliw55@gmail.com\\nABSTRACT\\nNeural networks adapt very well to distributed and continuous representations, but struggle to learn\\nand generalize from small amounts of data. Symbolic systems commonly achieve data ef\\ufb01cient\\ngeneralization by exploiting modularity to bene\\ufb01t from local and discrete features of a representation.\\nThese features allow symbolic programs to be improved one module at a time and to experience\\ncombinatorial growth in the values they can successfully process. However, it is dif\\ufb01cult to design\\ncomponents that can be used to form symbolic abstractions and which are highly-overparametrized\\nlike neural networks, as the adjustment of parameters makes the semantics of modules unstable.\\nI present Graph-based Symbolically Synthesized Neural Networks (G-SSNNs), a form of neural\\nnetwork whose topology and parameters are informed by the output of a symbolic program. I\\ndemonstrate that by developing symbolic abstractions at a population level, and applying gradient-\\nbased optimization to such neural models at an individual level, I can elicit reliable patterns of\\nimproved generalization with small quantities of data known to contain local and discrete features.\\nThe paradigm embodied by G-SSNNs offers a route towards the communal development of compact\\nand composable abstractions which can be \\ufb02exibly repurposed for a variety of tasks and high-\\ndimensional media. In future work, I hope to pursue these bene\\ufb01ts by exploring more ambitious\\nG-SSNN designs based on more complex classes of symbolic programs. The code and data associated\\nwith the reported results are available here.\\nKeywords neural networks\\u0001symbolic programs \\u0001graph neural networks \\u0001library learning\\u0001distributional program\\nsearch\\n1 Introduction\\nMost conventional modes of human communication naturally occur in a high-dimensional medium such as text, audio,\\nor images. When processing these media, slight errors can easily accrue across many dimensions of the input where\\nfeatures are distributed. With adequate data, neural networks adapt effectively to these patterns by adjusting many\\ninterdependent real-valued parameters. In their basic form, however, neural models are data inef\\ufb01cient. In settings\\nwhere more data cannot be sourced, pretraining has arisen as the most general and data-driven answer to this challenge.\\nPretraining allows practitioners to repurpose data that is not speci\\ufb01cally suited to a task to enrich a representation or\\ninitialize parameters of a neural model. While pretraining grants a model exposure to out-of-distribution data which\\nmay be irrelevant or inappropriate to the task at hand, the bene\\ufb01ts of this exposure can outweigh the costs.\\nIn symbolic systems, modularity is often used to stimulate greater data ef\\ufb01ciency and generalization. By examining the\\nparticular dimensions and values at which systems fail, developers can trace issues back to speci\\ufb01c modules. When a\\nfailure in one module is isolated and corrected, the system bene\\ufb01ts from this across all combinations of values and\\ndimensions, witnessing potentially exponential growth in the number of inputs on which it succeeds. This approach\\nrelies on the locality and discreteness of the representations associated with the task so that modular functionalities\\ncan be developed and debugged. Unfortunately, it is dif\\ufb01cult to design a fundamental unit of a system that gains the\\nbene\\ufb01ts of modularity and of neural networks, as it is dif\\ufb01cult to compose stable abstractions from components that\\ncontain many adjustable parameters. Nonetheless, this is a more direct solution to the problem of data inef\\ufb01ciency\\nand inadequate generalization in circumstances where high-dimensional data are known to exhibit both locality and\\ndiscreteness.arXiv:2303.03340v1  [cs.NE]  6 Mar 2023 Symbolic Synthesis of Neural Networks\\nIn this work, I propose a technique for structuring modular neural networks that utilizes symbolic programs to\\ninform the network\\u2019s topology and to determine the values of some of its parameters. This allows the symbolic\\nprograms themselves to be developed so as to exhibit increasing degrees of abstraction at a population level, while\\nallowing resulting individual networks to be trained and evaluated with standard techniques. I present Graph-based\\nSymbolically-Synthesized Neural Networks (G-SSNNs) and apply populations of G-SSNNs to a binary prediction task\\nover high-dimensional data containing many local and discrete features. I show that these populations exhibit reliable\\npatterns of improved generalization from small quantities of data compared to a population of baseline models which\\nare not structured according to the output of symbolic programs.\\n2 Related Work\\n2.1 Neural Architecture Search\\nNeural Architecture Search (NAS) is the process of using automated methods to explore the discrete space of neural\\nnetwork topologies. Though varying in the design of their search space [Jin et al., 2022], search strategies [Real\\net al., 2020], and model evaluation strategies [Abdelfattah et al., 2021], all approaches I reviewed assume that model\\nparameters learnable through gradient-based optimization will be learned in this fashion once search is complete. That\\nis, they serve to explore a (potentially hierarchical) space of hyperparameters in which topology often plays a central\\nrole. This differs from G-SSNNs, in which symbolic programs play a role in determining both the topology and the\\nparameters of a model.\\n2.2 Embedding Spaces and Preprocessing\\nThe act of adding \\ufb01xed transformations into a neural network that could otherwise be performed with the help of\\nlearnable parameters seems more analogous to an act of preprocessing, as might be performed with word embeddings\\n[Mikolov et al., 2013] or positional encoding [Vaswani et al., 2017]. However, I found no works in which similar acts\\nof preprocessing were designed with the help of automated program synthesis.\\n2.3 Program Synthesis\\nThe program synthesis apparatus used in this work is inspired by the DreamCoder system [Ellis et al., 2021] and uses\\nthe same cycle of library learning and distributional program search between iterations of G-SSNN training, though I\\nutilize the more recent STITCH tool [Bowers et al., 2023] and heap search algorithm [Matricon et al., 2022] for these\\nrespective functionalities. G-SSNNs have a deeper analogy with the DreamCoder system in that the process of training\\na G-SSNN on a dataset generalizes the evaluation mechanisms for tasks and symbolic programs used in DreamCoder\\u2019s\\ndomains. The greater generality of G-SSNNs introduces concerns about performance on unseen data that are not\\nrelevant when measuring the performance of purely symbolic programs, and must be addressed by separate mechanisms.\\nHowever, in principle, nothing prevents the multitask Bayesian program learning framework of DreamCoder from being\\napplied to the development of multiple, parallel populations of G-SSNNs.\\nIn the broader neurosymbolic literature, I did not \\ufb01nd other examples of systems in which adjacent parametric and\\nnonparametric modules are separately optimized through gradient-based learning and the development of increasingly\\nhierarchical symbolic abstractions.\\n3 Methods\\n3.1 Evolutionary Framework\\nWhile the parameters of a G-SSNN may be optimized with ordinary training methods, we must utilize a separate\\nmechanism to optimize their symbolic structure. Since a G-SSNN is characterized by a single symbolic program, we\\nmust retain a population of G-SSNNs in order to conduct program search. This search is guided in two respects: by the\\nstructure of the DSL, and by the parameters of a distribution over programs. The structure of the DSL is determined by structure of the DSL, and by the parameters of a distribution over programs. The structure of the DSL is determined by\\nthe results of library learning performed with STITCH [Bowers et al., 2023]. STITCH\\u2019s library learning functionality\\noperates by extracting novel primitives from a corpus according to which provides the greatest degree of compression\\n(i.e. reduction in aggregate program size). A distribution over programs may also be inferred from such a corpus by\\ncounting how many times n-grams of primitives from the DSL are used.1Thus, if we are able to select an approximately\\noptimal corpus of symbolic programs based on the training performance of a population of G-SSNNs, we may both\\n1In experiments, I utilize a unigram distribution.\\n2 Symbolic Synthesis of Neural Networks\\nbias compression towards the discovery of modules common to successful programs and bias program search towards\\nthe discovery of increasingly complex programs which utilize those modules.\\nBecause there is always the possibility that a G-SSNN performs differently on unseen data, there is no heuristic that is\\nunquestionably best for encouraging the greatest generalization. Perhaps the simplest and most generally agreeable,\\nhowever, are those based on discarding G-SSNNs with the Lowest Training Performance (LTP). I refer to the heuristic I\\nuse in the experiments presented here as rank-LTP-50, which involves discarding the bottom half of G-SSNNs in terms\\nof the rank of their training performance among the population.\\n3.2 G-SSNNs\\nThere are many ways that the output of a symbolic program could be incorporated into the topology and parameter\\nvalues of a neural network. However, the use of learned abstractions forces a consideration: how do we make symbolic\\nprograms increasingly sophisticated without simply making their corresponding neural networks larger? In order that\\nour models don\\u2019t become increasingly costly to train as the symbolic programs that structure them utilize increasingly\\ncomplex modules, the number of parameters optimized with gradient-based learning should remain \\ufb01xed regardless of\\nhow the symbolic program \\ufb01xes their topology or values. This can be achieved by repurposing generic graph neural\\nnetworks (GNNs).\\nLets2Skbe relational structures whose elements are accompanied by symbolic feature vectors v2Nk. let\\ne:Nk\\u0002Rm!Rnbe an embedding function that maps each symbolic feature vector vand a vector x2Rmto\\nanother real-valued feature vector. Lastly, let graph_map :Sk\\u0002Rm!Gnbe the function that maps from relational\\nstructures and input vectors to graphs with the corresponding real-valued features given by e(v; x)for each v2s. We\\nmay then de\\ufb01ne a G-SSNN as a model m:Rm!Rqconsisting of a GNN mg:Gn!Rq, a relational structure s,\\nand an embedding function e, for which\\nm(x) =mg(graph_map (e; x; s )):\\nProvided that eis differentiable with respect to x, a G-SSNN may function as a generic neural module.\\nThere are many ways one could design the class of relational structures Skand the embedding function edepending on\\nthe degree and forms of control we want to grant symbolic programs over the parameter space and topology of the\\nnetwork. For the experiments conducted in this paper, I adopt a simple design: I utilize a subset of the class S2of\\nsingle-component undirected graphs with node and edge features. The natural numbers associated with each graph\\nelement correspond to a unique ID and a neighborhood rank. IDs are 0-indexed and re\\ufb02ect the order in which the\\nelements were created. I denote the number of unique IDs in a graph with d\\u0003= 1+max v2sv2. The neighborhood ranks\\nare natural numbers in a limited range which distinguish the edges of a particular neighborhood. The neighborhood rank\\nof a node is 0, while the neighborhood rank of an edge ranges from 1 to a \\ufb01xed maximum degree. Since neighborhood\\nranks function to distinguish edges, even if their endpoints are the same, this class of relational structures could be\\nunderstood as permitting multiple edges. I denote the number of distinct neighborhood ranks with r\\u0003= 1 + max v2sv2.\\nTo process these symbolic features, I utilize the embedding function\\ne(v; x) =tile_expand (xidx(v1):idx(v1+1)) +2d_pos (v1; v2)\\nwhere idx(w) = ( mmodl) +w\\u0001lforl=bm=d\\u0003c. The function tile_expand :Rt!Rnimplements the\\ndifferentiable operation of concatenating dn=tecopies of its input and truncating the result to a dimensionality of n,\\nand the function 2d_pos :N2!Rnproduces computes a vector b=2d_pos (d; r)such that2\\nbi=8\\n>><\\n>>:sin(reorder_1 (d; r)=10000imod ( n=4)) 1 < i\\u0014n=4\\ncos(reorder_1 (d; r)=10000imod ( n=4))n=4< i\\u00142n=4\\nsin(reorder_2 (d; r)=10000imod ( n=4)) 2n=4< i\\u00143n=4\\ncos(reorder_2 (d; r)=10000imod ( n=4)) 3n=4< i\\u0014n: cos(reorder_1 (d; r)=10000imod ( n=4))n=4< i\\u00142n=4\\nsin(reorder_2 (d; r)=10000imod ( n=4)) 2n=4< i\\u00143n=4\\ncos(reorder_2 (d; r)=10000imod ( n=4)) 3n=4< i\\u0014n:\\nThis embedding function constructs node and edge features by selecting values from a subset of the dimensions of x\\nand adding them to a bias vector, where both the subset of dimensions and the value of the vector are chosen based\\non the unique values of each graph element\\u2019s symbolic feature vector. This incentivizes greater modularity in mby\\nensuring that the parameters of mgfunction differently when applied to the same values depending on the dimensions\\n2The functions reorder_1 andreorder_2 are an implementation detail due to mistaken indexing which did not affect the validity\\nof the experiment. These functions are de\\ufb01ned as reorder_1 (d; r) = (d\\u0001r\\u0003+r) mod d\\u0003andreorder_2 (d; r) =b(d\\u0001r\\u0003+r)=d\\u0003c\\nwhere d\\u0003andr\\u0003are the maximal values of dandr. These functions may be replaced by f1(d; r) =dandf2(d; r) =rrespectively\\nto simplify the computation of 2d_pos .\\n3 Symbolic Synthesis of Neural Networks\\nofxin which they appear. Similarly, this behavior incentivizes antecedent neural modules to exhibit greater modularity\\nby ensuring that their parameters function differently based on the dimensions of xthat they in\\ufb02uence. These incentives\\nrely crucially on the bias term of e, which allows the symbolic program generating the relational structure sto in\\ufb02uence\\nthe parameter space of m. Without this term, the symbolic program would still in\\ufb02uence the topology of mthrough the\\nstructure of the graph convolutions applied by mgand the distribution of values from speci\\ufb01c dimensions of xinto\\nspeci\\ufb01c node and edge features under e.3However, this alone would create no obligation for different parameters of\\nmgor antecedent neural modules to behave differently with regard to the various dimensions of x. In general, while\\ntopology may allow or disallow particular parameters from in\\ufb02uencing each other, it cannot guarantee that they behave\\ndifferently with respect to the output for the same input. This can only be achieved by systematically varying the values\\ndifferent parameters may take.\\n3.3 Symbolic Programs\\nThe symbolic programs used to generate the relational structures described above are of type S2!S2and are all\\nevaluated by application to the same initial relational structure. The primitive operations of the initial DSL are such that\\nsymbolic programs may only make decisions based on the information contained in the relational structure. Since the\\ninitial relational structure is uninformative, symbolic programs therefore only have access to information about their\\nown semantics as it manifests in the relational structure they construct. More detailed information on the operations of\\nthe initial DSL is available in the documentation of the antireduce-graphs library.\\nDuring program search, symbolic programs are considered novel if they generate a relational structure that is unique\\nunder isomorphism when symbolic features are not considered. Isomorphism testing is performed with the VF2++\\nalgorithm [J\\u00fcttner and Madarasi, 2018].\\n4 Experiments\\n4.1 Setup & Hyperparameters\\nTo facilitate comparison, I utilize a base model architecture consisting of a transformer with a convolutional stem [Xiao\\net al., 2021]. I use subpixel convolution to perform downsampling from 128\\u0002128to8\\u00028[Shi et al., 2016], followed\\nby a pointwise convolution and three residual convolutional units. The patches are then \\ufb02attened and fed to a simpli\\ufb01ed\\nVision Transformer (ViT) [Beyer et al., 2022] with 2 transformer blocks. Each block has an MLP dimension of 512 and\\n4 128-dimensional attention heads. Global average pooling is then applied to produce an output of the target dimension\\nfor baseline models, or with the dimensionality of a graph element for experimental models in which the output is fed\\nto a G-SSNN unit.\\nTo construct G-SSNNs, I use the Graph Isomorphism Network with Edge features (GINE) of Hu et al. [2020] as\\nimplemented in the DGL library [Wang et al., 2019]. I use 512-dimensional node and edge representations, which\\nare passed through three GINE units parametrized by an MLP layer with identical input and output dimensionalities.\\nBetween GINE units, I apply 30% droput [Srivastava et al., 2014]. To produce the \\ufb01nal output, I apply average pooling\\nfollowed by another MLP layer which projects its input to the target output.\\nFor both baseline and experimental models I use a batch size of 8, train each model for 16 epochs with the Adam\\noptimizer [Kingma and Ba, 2015], and reduce the learning rate by a factor of 1\\/2 when a new low in the loss has not\\nbeen experienced in the \\ufb01rst 50 batches, or in the most recent 65 batches since the learning rate was last reduced.\\nAcross iterations of evolutionary selection, I retain a population of at most 50 and apply the rank-LTP-50 heuristic if\\nthe size of the population is greater than or equal to 25. I run distributional program search for a 15 second period. In the size of the population is greater than or equal to 25. I run distributional program search for a 15 second period. In\\nthe course of this run, I retain only the most likely 50\\u0000population_size novel programs; however, if programs are\\ndiscovered that are shorter than, but semantically equivalent to, programs currently in the population, these shorter\\nreformulations are kept and their equivalent\\u2019s are set aside. I do not consider programs that consist of more than 150\\nprimitives from the most updated DSL. Prior to program search, I reweight the parameters of the program distribution\\nsuch that the log likelihoods of the most and least likely primitives differ by no more than 0.5. I implemented this\\ntransformation such that it preserves the relative distance of each primitive\\u2019s log likelihood from the mean and therefore\\n3The latter of these two behaviors is included both because it makes greater use of the relational structure s, and also because it\\naverts a potential perverse incentive that could be created by distributing the entirety of xwithin each node and edge representation.\\nNamely, this has the consequence of making more copies of the information in xavailable if a graph has more elements, making\\nit less important to depend on the contribution of e\\u2019s bias term. This is especially undesirable since library learning acts as an\\nindiscriminate force for increased complexity in relational structure; thus, to an extent, this effect would be dif\\ufb01cult to avoid whether\\nor not this perverse incentive was directly appealing.\\n4 Symbolic Synthesis of Neural Networks\\n0 10 20 30 40 50\\nRank (Descending)0.650.700.750.800.850.900.951.00AccuracyBaseline\\nDataset\\ntraining\\nvalidation\\nvalidation\\n0.0 0.2 0.4 0.6 0.8 1.0\\nRank (Descending)0.600.650.700.750.800.850.900.95AccuracyExperimental (Iteration 1)\\nDataset\\ntraining\\nvalidation\\n0 1 2 3 4 5 6\\nRank (Descending)0.60.70.80.9AccuracyExperimental (Iteration 2)\\nDataset\\ntraining\\nvalidation\\n0 10 20 30 40\\nRank (Descending)0.40.50.60.70.80.91.0AccuracyExperimental (Iteration 3)\\nDataset\\ntraining\\nvalidation\\nvalidation\\n0 10 20 30 40 50\\nRank (Descending)0.50.60.70.80.9AccuracyExperimental (Iteration 4)\\nDataset\\ntraining\\nvalidation\\nvalidation\\n0 10 20 30 40 50\\nRank (Descending)0.40.50.60.70.80.9AccuracyExperimental (Iteration 5)\\nDataset\\ntraining\\nvalidation\\nvalidation\\nFigure 1: Training and validation set performances of populations of neural networks. Each model of a population is\\nrepresented in both trend lines of a plot at parallel points with respect to the x-axis. The baseline plot depicts the spread\\nof training set and validation set accuracies for 50 instantiations of a baseline model. The experimental plots depict\\nthe spread of training set and validation set accuracies for populations of G-SSNNs at various stages of evolution. At\\nstage n, the population has undergone nrounds of heuristic selection\\/culling and program synthesis. At iterations 1-5,\\nthe populations were of sizes 2, 7, 44, 50 and 50, respectively. Dotted lines represent unsmoothed data and solid lines\\nrepresent smoothed data with a window size of 6. I present validation set accuracies in both smooth and unsmoothed\\nform except where the population of models seem too small to meaningfully bene\\ufb01t from smoothing ( n <15).\\ndoes not alter the DSL\\u2019s total probability mass.4During library learning, I perform up to three rounds of compression,\\neach producing a primitive of arity 2 or less. All other parameters of the STITCH compression routine were allowed to\\ntake their default values.5\\n4.2 Dataset\\nThe RA VEN dataset [Zhang et al., 2019] is an arti\\ufb01cial dataset consisting of visual puzzles known as Raven\\u2019s Progressive\\nMatrices (RPMs). Originated as a psychometric test, RPMs have since been extensively studied as tests of Abstract\\nVisual Reasoning (A VR) in both symbolic and neural AI systems. An RPM is a 9\\u00029matrix of panels depicting simple\\narrangements of polygons. Within each row, panels exhibit one of a handful of consistent patterns with respect to each\\nattribute\\u2013size, shape, color, angle, position, and multiplicity\\u2013of their polygons. Due to the independence between these\\ncharacteristics, their limited sets of values, and the segregated placement of the polygons, RPMs are naturally local and\\ndiscrete high-dimensional representations.\\nClassically, the \\ufb01nal panel of the last sequence (bottom right) is omitted and some number of candidate panels are\\nprovided, only one of which contains polygons which obey the same patterns exhibited in the \\ufb01rst two rows for all\\nattributes. The goal of the solver is then to distinguish the correct panel from the pool of candidates. Odouard and\\nMitchell [2022] have demonstrated that several recent high-performing neural models from the RA VEN literature\\nperform poorly when more exhaustively evaluated on their understanding of the various instantiations of an individual\\npattern. This inspired the Prediction of Constant Color Pattern (PCCP) task, a binary prediction task in which a model\\nexamines a complete and correct RPM and predicts whether it exhibits constancy of polygon color across rows. The\\ntwo classes in this task\\u2013constant and non-constant\\u2013were relatively evenly-distributed among generated data without\\n4This and other domain-general aspects of program search are implemented in the antireduce library.\\n5Explanation of these settings may be found in the documentation of the stitch_core Python package.\\n5 Symbolic Synthesis of Neural Networks\\nadditional constraints on pattern instantiation or attribute values.6As with any binary RPM prediction task based on the\\npresence of a single pattern-attribute pairing, indiscernible spurious correlation with the presence or absence of other\\npattern-attribute pairings is unlikely, even with small data.\\nThe RA VEN-like data used in my experiments were generated with a reimplementation of the original RA VEN data\\ngeneration code extended to support creation of RPMs at lower resolutions than were originally accommodated. I use a\\ntraining dataset size of 128 RPMs and a validation set of 1200 RPMs. These sizes were well suited to demonstrating the\\ngeneralization performance of the baseline and experimental models and were also helpful for practical performance\\nwhen serially training populations of G-SSNNs. In general, the G-SSNNs of a population are not interdependent and\\nmay be trained in parallel. Serial training was instead a logistical limitation of the current study.\\n4.3 Results\\nFigure 1 presents comparative results for populations of baseline and experimental models. Across these plots, we see\\nthat validation set accuracy tends to be especially high for models whose training set accuracy is in the range of 80-90%.\\nAmongst baseline models less than 20% fall in this range, whereas after iteration 1, 30% or more of the population\\nof experimental models fall in this range. The populations of experimental models are centered on this \\u201csweet spot\\\"\\nof training performances, and this portion of the distribution gets \\ufb02atter and broader across iterations 3-5. These data\\ndemonstrate that the use of G-SSNNs reduces the performance gap between seen and unseen data for at least one form\\nof local and discrete high-dimensional representation. Moreover, the models for which this results in peak validation set\\naccuracies are signi\\ufb01cantly more numerous and fall in a reliable part of the distribution of validation set accuracies.\\nThis is a markedly different pattern of improved generalization which is easier to anticipate and exploit.\\nThe distributions of validation set accuracies illustrated in these data suggest that it is actually models with a training\\nset performance of a more intermediate rank that are most likely to exhibit peak generalization. In future work and\\napplications of G-SSNNs, it may therefore be bene\\ufb01cial to explore the space of Intermediate Training Performance\\n(ITP) heuristics for selecting from the population. This pattern also suggests that in successful G-SSNNs, symbolic\\nprograms offer a form of regularization. It is possible for a relational structure to permit a G-SSNN to over\\ufb01t, and\\nalso for it to prevent a G-SSNN from \\ufb01tting very well at all; the optimal relational structures would then be those that\\ndo not inhabit either extreme, making performance on seen and unseen data more similar without lowering training\\nperformance more than necessary.\\n5 Discussion\\nAcross high-dimensional media, there are tasks that involve extracting information that manifests in local and discrete\\nfeatures. In adapting to such tasks, modular models may exhibit an advantage as they are more likely to specialize in\\ndetecting a small number of patterns. In the case of G-SSNNs, we encourage modularity by requiring that models\\ncontextualize data-independent symbols in a task-speci\\ufb01c manner. This has the auxiliary bene\\ufb01t of contributing\\ntransferable knowledge about how symbols should be composed. In a world where symbolically-informed, modular\\nneural models are widely used, the community of model developers would bene\\ufb01t from the sharing of novel abstractions\\nin a way that mirrors present-day communal processes of software development. However, unlike current practice,\\nthese abstractions could be \\ufb02exibly adapted to the circumstances in which they are applied in ways that cannot always\\nbe concisely expressed with symbols. In advancing this possibility, I believe that G-SSNNs capture much of what be concisely expressed with symbols. In advancing this possibility, I believe that G-SSNNs capture much of what\\nis intuitively desirable about the promise of common sense: the ability to quickly adapt and elaborate on collective\\nknowledge in a wide range of circumstances.\\nIn future work, I intend to investigate other G-SSNN designs incorporating more complex embedding functions and\\nclasses of relational structures, as well as more complex classes of symbolic programs. Alternative embedding functions\\ncould draw parameter values from more interesting sources, such as pretrained models, and could embed these values\\ninto more interesting topological structures such as convolutions or attention mechanisms. These choices could be\\nsupported by more complex logic on the part of symbolic programs, which could be enhanced with access to richer\\nexternal databases of symbolic information.\\n6Some constraints on pattern instantiation and attribute values were necessary to ensure legibility at the desired resolution of\\n128\\u0002128. Speci\\ufb01cally, polygon size was kept large and not allowed to vary and oblique angles of polygon rotation were disallowed.\\nNeither of these constraints contribute to determining the color constancy of an RPM.\\n6 Symbolic Synthesis of Neural Networks\\n6 Conclusion\\nI have presented G-SSNNs, a class of neural modules whose topology and parameter space are in\\ufb02uenced by the output\\nof a modular symbolic program. I presented the results of applying neural models containing G-SSNN modules to a\\nbinary prediction task over Raven\\u2019s Progressive Matrices (RPMs), psychometric tests of Abstract Visual Reasoning\\n(A VR) which contain local and discrete features. These results demonstrate that G-SSNNs produce reliable patterns of\\nimproved generalization that are not exhibited by baseline models. In addition to their potential to improve performance\\non tasks involving local and discrete features across high-dimensional media, G-SSNNs offer a route towards communal\\ndevelopment of compact and novel abstractions that can be \\ufb02exibly applied to numerous tasks. In future work, I intend\\nto investigate more complex G-SSNN designs and more complex classes of associated symbolic programs.\\nReferences\\nCharles Jin, Phitchaya Mangpo Phothilimthana, and Sudip Roy. Neural architecture search using property guided\\nsynthesis. Proc. ACM Program. Lang. , 6(OOPSLA2):1150\\u20131179, 2022. doi:10.1145\\/3563329. URL https:\\n\\/\\/doi.org\\/10.1145\\/3563329 .\\nEsteban Real, Chen Liang, David R. So, and Quoc V . Le. Automl-zero: Evolving machine learning algorithms from\\nscratch. CoRR , abs\\/2003.03384, 2020. URL https:\\/\\/arxiv.org\\/abs\\/2003.03384 .\\nMohamed S. Abdelfattah, Abhinav Mehrotra, Lukasz Dudziak, and Nicholas D. Lane. Zero-cost proxies for lightweight\\nNAS. CoRR , abs\\/2101.08134, 2021. URL https:\\/\\/arxiv.org\\/abs\\/2101.08134 .\\nTom\\u00e1s Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Ef\\ufb01cient estimation of word representations in vector\\nspace. In Yoshua Bengio and Yann LeCun, editors, 1st International Conference on Learning Representations, ICLR\\n2013, Scottsdale, Arizona, USA, May 2-4, 2013, Workshop Track Proceedings , 2013. URL http:\\/\\/arxiv.org\\/\\nabs\\/1301.3781 .\\nAshish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia\\nPolosukhin. Attention is all you need. CoRR , abs\\/1706.03762, 2017. URL http:\\/\\/arxiv.org\\/abs\\/1706.03762 .\\nKevin Ellis, Catherine Wong, Maxwell I. Nye, Mathias Sabl\\u00e9-Meyer, Lucas Morales, Luke B. Hewitt, Luc Cary,\\nArmando Solar-Lezama, and Joshua B. Tenenbaum. Dreamcoder: bootstrapping inductive program synthesis with\\nwake-sleep library learning. In Stephen N. Freund and Eran Yahav, editors, PLDI \\u201921: 42nd ACM SIGPLAN\\nInternational Conference on Programming Language Design and Implementation, Virtual Event, Canada, June\\n20-25, 2021 , pages 835\\u2013850. ACM, 2021. doi:10.1145\\/3453483.3454080. URL https:\\/\\/doi.org\\/10.1145\\/\\n3453483.3454080 .\\nMatthew Bowers, Theo X. Olausson, Lionel Wong, Gabriel Grand, Joshua B. Tenenbaum, Kevin Ellis, and Armando\\nSolar-Lezama. Top-down synthesis for library learning. Proc. ACM Program. Lang. , 7(POPL):1182\\u20131213, 2023.\\ndoi:10.1145\\/3571234. URL https:\\/\\/doi.org\\/10.1145\\/3571234 .\\nTh\\u00e9o Matricon, Nathana\\u00ebl Fijalkow, Guillaume Lagarde, and Kevin Ellis. Deepsynth: Scaling neural program\\nsynthesis with distribution-based search. J. Open Source Softw. , 7(78):4151, 2022. doi:10.21105\\/joss.04151. URL\\nhttps:\\/\\/doi.org\\/10.21105\\/joss.04151 .\\nAlp\\u00e1r J\\u00fcttner and P\\u00e9ter Madarasi. VF2++ - an improved subgraph isomorphism algorithm. Discret. Appl. Math. , 242:\\n69\\u201381, 2018. doi:10.1016\\/j.dam.2018.02.018. URL https:\\/\\/doi.org\\/10.1016\\/j.dam.2018.02.018 .\\nTete Xiao, Mannat Singh, Eric Mintun, Trevor Darrell, Piotr Doll\\u00e1r, and Ross B. Girshick. Early con-\\nvolutions help transformers see better. In Marc\\u2019Aurelio Ranzato, Alina Beygelzimer, Yann N. Dauphin,\\nPercy Liang, and Jennifer Wortman Vaughan, editors, Advances in Neural Information Processing Systems\\n34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14,\\n2021, virtual , pages 30392\\u201330400, 2021. URL https:\\/\\/proceedings.neurips.cc\\/paper\\/2021\\/hash\\/\\nff1418e8cc993fe8abcfe3ce2003e5c5-Abstract.html . 2021, virtual , pages 30392\\u201330400, 2021. URL https:\\/\\/proceedings.neurips.cc\\/paper\\/2021\\/hash\\/\\nff1418e8cc993fe8abcfe3ce2003e5c5-Abstract.html .\\nWenzhe Shi, Jose Caballero, Ferenc Huszar, Johannes Totz, Andrew P. Aitken, Rob Bishop, Daniel Rueckert, and\\nZehan Wang. Real-time single image and video super-resolution using an ef\\ufb01cient sub-pixel convolutional neural\\nnetwork. In 2016 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2016, Las Vegas, NV ,\\nUSA, June 27-30, 2016 , pages 1874\\u20131883. IEEE Computer Society, 2016. doi:10.1109\\/CVPR.2016.207. URL\\nhttps:\\/\\/doi.org\\/10.1109\\/CVPR.2016.207 .\\nLucas Beyer, Xiaohua Zhai, and Alexander Kolesnikov. Better plain vit baselines for imagenet-1k. CoRR ,\\nabs\\/2205.01580, 2022. doi:10.48550\\/arXiv.2205.01580. URL https:\\/\\/doi.org\\/10.48550\\/arXiv.2205.\\n01580 .\\n7 Symbolic Synthesis of Neural Networks\\nWeihua Hu, Bowen Liu, Joseph Gomes, Marinka Zitnik, Percy Liang, Vijay S. Pande, and Jure Leskovec. Strategies\\nfor pre-training graph neural networks. In 8th International Conference on Learning Representations, ICLR 2020,\\nAddis Ababa, Ethiopia, April 26-30, 2020 . OpenReview.net, 2020. URL https:\\/\\/openreview.net\\/forum?id=\\nHJlWWJSFDH .\\nMinjie Wang, Lingfan Yu, Da Zheng, Quan Gan, Yu Gai, Zihao Ye, Mufei Li, Jinjing Zhou, Qi Huang, Chao Ma,\\nZiyue Huang, Qipeng Guo, Hao Zhang, Haibin Lin, Junbo Zhao, Jinyang Li, Alexander J. Smola, and Zheng Zhang.\\nDeep graph library: Towards ef\\ufb01cient and scalable deep learning on graphs. CoRR , abs\\/1909.01315, 2019. URL\\nhttp:\\/\\/arxiv.org\\/abs\\/1909.01315 .\\nNitish Srivastava, Geoffrey E. Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout:\\na simple way to prevent neural networks from over\\ufb01tting. J. Mach. Learn. Res. , 15(1):1929\\u20131958, 2014.\\ndoi:10.5555\\/2627435.2670313. URL https:\\/\\/dl.acm.org\\/doi\\/10.5555\\/2627435.2670313 .\\nDiederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun,\\neditors, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015,\\nConference Track Proceedings , 2015. URL http:\\/\\/arxiv.org\\/abs\\/1412.6980 .\\nChi Zhang, Feng Gao, Baoxiong Jia, Yixin Zhu, and Song-Chun Zhu. RA VEN: A dataset for relational and\\nanalogical visual reasoning. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2019,\\nLong Beach, CA, USA, June 16-20, 2019 , pages 5317\\u20135327. Computer Vision Foundation \\/ IEEE, 2019.\\ndoi:10.1109\\/CVPR.2019.00546. URL http:\\/\\/openaccess.thecvf.com\\/content_CVPR_2019\\/html\\/Zhang_\\nRAVEN_A_Dataset_for_Relational_and_Analogical_Visual_REasoNing_CVPR_2019_paper.html .\\nVictor Vikram Odouard and Melanie Mitchell. Evaluating understanding on conceptual abstraction benchmarks. In\\nJos\\u00e9 Hern\\u00e1ndez-Orallo, Lucy Cheke, Joshua Tenebaum, Tomer D. Ullman, Fernando Mart\\u00ednez-Plumed, Danaja\\nRutar, John Burden, Ryan Burnell, and Wout Schellaert, editors, Proceedings of the Workshop on AI Evaluation\\nBeyond Metrics co-located with the 31st International Joint Conference on Arti\\ufb01cial Intelligence (IJCAI-ECAI\\n2022), Vienna, Austria, July 25th, 2022 , volume 3169 of CEUR Workshop Proceedings . CEUR-WS.org, 2022. URL\\nhttp:\\/\\/ceur-ws.org\\/Vol-3169\\/paper8.pdf .\\n8\",\"14\":\" Noname manuscript No.\\n(will be inserted by the editor)\\nOptimizing L1 Cache for Embedded Systems through\\nGrammatical Evolution\\nJosefa D\\u0013 \\u0010az \\u0013Alvarez \\u0001J. Manuel Colmenar \\u0001Jos\\u0013 e L. Risco-Mart\\u0013 \\u0010n \\u0001Juan\\nLanchares \\u0001Oscar Garnica\\nReceived: date \\/ Accepted: date\\nAbstract Nowadays, embedded systems are provided\\nwith cache memories that are large enough to in\\ruence\\nin both performance and energy consumption as never\\noccurred before in this kind of systems. In addition, the\\ncache memory system has been identi\\fed as a compo-\\nnent that improves those metrics by adapting its con-\\n\\fguration according to the memory access patterns of\\nthe applications being run. However, given that cache\\nmemories have many parameters which may be set to\\na high number of di\\u000berent values, designers face to a\\nwide and time-consuming exploration space.\\nIn this paper we propose an optimization framework\\nbased on Grammatical Evolution (GE) which is able\\nto e\\u000eciently \\fnd the best cache con\\fgurations for a\\ngiven set of benchmark applications. This metaheuris-\\ntic allows an important reduction of the optimization\\nruntime obtaining good results in a low number of gen-\\nerations. Besides, this reduction is also increased due\\nto the e\\u000ecient storage of evaluated caches. Moreover,\\nwe selected GE because the plasticity of the grammar\\neases the creation of phenotypes that form the call to\\nJosefa D\\u0013 \\u0010az \\u0013Alvarez\\nCentro Universitario de M\\u0013 erida, Universidad de Ex-\\ntremadura, 06800 M\\u0013 erida, Spain\\nTel.: +34-924-387268\\nE-mail: mjdiaz@unex.es\\nJ. Manuel Colmenar\\nDept. of Computer Science and Statistics, Universidad Rey\\nJuan Carlos, 28933 M\\u0013 ostoles, Spain\\nE-mail: josemanuel.colmenar@urjc.es\\nJos\\u0013 e L. Risco-Mart\\u0013 \\u0010n \\u0001Juan Lanchares\\u0001\\u0013Oscar Garnica\\nDept. of Computer Architecture and Automation, Universi-\\ndad Complutense de Madrid, 28040 Madrid, Spain\\nE-mail:fjlrisco,julandan,ogarnica g@dacya.ucm.esthe cache simulator required for the evaluation of the\\ndi\\u000berent con\\fgurations.\\nExperimental results for the Mediabench suite show\\nthat our proposal is able to \\fnd cache con\\fgurations\\nthat obtain an average improvement of 62% versus a\\nreal world baseline con\\fguration.\\n1 Introduction\\nEmbedded computing systems have experienced a great\\ndevelopment in last decades, becoming one of the cru-\\ncial driving forces in technology. Portable devices such\\nas smartphones, digital cameras, GPS, etc. are con-\\nstantly incorporating new functionalities. Most of these\\nnew features are devoted to support new services and\\nmultimedia applications, despite that embedded sys-\\ntems have limited resources. Portable systems, for in-\\nstance, run on batteries, which are limited in capacity\\nand size, because of design constraints. Moreover, they\\nhave a limited cooling ability and, hence, a low energy\\nconsumption is a main requirement. Besides, portable\\nsystems usually execute multimedia applications, which\\nrequire high performance and, therefore, are energy in-\\ntensive tasks. As a consequence, one of the main work-\\nforces of embedded systems designers is the search for\\nthe right balance between increasing performance and\\nreducing energy consumption at a low cost.\\nPrevious studies have identi\\fed on-chip cache mem-\\nory as one of the most energy consuming components,\\nranging between 20% and 30% of the total power of the\\nchip in embedded processors [31]. In addition, the de-\\nsign of the cache memory has a high in\\ruence on energy\\nconsumption and performance, because of the di\\u000berent\\nhardware complexity that it implies. For instance, a di-\\nrect mapped cache design presents the fastest hit times,arXiv:2303.03338v1  [cs.NE]  6 Mar 2023 2 Josefa D\\u0013 \\u0010az \\u0013Alvarez et al.\\nbut requires higher cache size to obtain comparable per-\\nformance than an N-way set associative cache. On the\\nother hand, set associative caches present higher energy\\nconsumption given that they access to a set of Ntags\\non each operation [15].\\nThe cache memory behavior is not only conditioned\\nby structural parameters (capacity, block size, associa-\\ntivity, etc.), but also by parameters such as algorithms\\nfor search, prefetch and replacement, write policies, etc.\\nAll these features form the so-called cache con\\fgura-\\ntion. Finding optimal values for these parameters will\\nbring us the best performance and energy consumption\\nand consequently, the best cache con\\fguration.\\nBesides, applications have di\\u000berent behavior and hence\\nhave distinct cache requirements to meet the speci\\fc\\nobjectives of energy consumption and performance. Thus,\\nwe need to take into account the behavior of the target\\napplication in order to \\fnd the best cache con\\fguration\\nthat improves performance or energy consumption.\\nIn summary, an optimization scheme to determine\\nthe best cache con\\fguration should: (1) travel along the\\ndesign space of cache con\\fgurations, which is de\\fned as\\nthe set of all combinations of values for all cache param-\\neters; (2) take into account the energy and performance\\nof target applications for each candidate cache con\\fgu-\\nration, because of di\\u000berent memory access patterns.\\nThe evaluation of a cache con\\fguration requires a\\ncache simulator to process a trace of the cache accesses\\nfrom a previous execution of the target program. This\\nprocess will allow to calculate the execution time and\\nenergy consumption for the application under study.\\nThis is a slow process because program traces usually\\nrecord billions of memory operations, which spends tens\\nof seconds of simulator runtime. Hence an exhaustive\\nsearch on this design space will take an una\\u000bordable\\namount of time given the high number of cache con\\fg-\\nurations that can be de\\fned.\\nOn the other hand, heuristic techniques reduce the\\nnumber of evaluations while performing the search to-\\nwards the optimal and, therefore, \\ft well in this kind\\nof problems. Therefore, we present in this paper an\\noptimization scheme based on Grammatical Evolution\\n(GE) [11] that obtains, using program traces, the op-\\ntimized cache con\\fguration in terms of execution time\\nand energy consumption for a given target application.\\nCompared with the exhaustive approach, the execution\\ntime is highly reduced due to two reasons: (1) the meta-\\nheuristic algorithm converges even with a short number\\nof generations and population size; (2) we have added\\na hashed map to store the objective values of each eval-\\nuated cache. In addition, the plasticity of the gram-\\nmar in GE help us to generate the parameters that the\\ncache simulator call requires for each evaluation, mak-ing more easy to communicate it with the optimization\\nalgorithm.\\nIn order to test our proposal, we have run experi-\\nments that \\fnd the best cache con\\fguration for a set\\nof multimedia applications taken from the Mediabench\\nbenchmarks [18], since they are representative for im-\\nage, audio and video processing, which are typical ap-\\nplications of embedded systems. We have assumed in\\nthese experiments a hardware architecture based on the\\nARM9 processors family [2], broadly used in multime-\\ndia embedded devices. As will be shown later, the av-\\nerage reduction of execution time and energy consump-\\ntion of the optimized cache con\\fgurations is 75% and\\n96% respectively in relation to a baseline cache. In ad-\\ndition, the optimization runtime is a\\u000bordable and we\\nhave estimated an average reduction of 94% in relation\\nto a GE implementation with no storage of evaluations.\\nThe rest of the paper is organized as follows. Section\\n2 describes the main issues about cache design and de-\\n\\fnes the search space of the problem. Section 3 reviews\\nthe related work on the cache optimization techniques.\\nSection 4 describe both performance and energy models nes the search space of the problem. Section 3 reviews\\nthe related work on the cache optimization techniques.\\nSection 4 describe both performance and energy models\\nadopted in this work to compute the goodness of the\\ncandidate solutions. Section 5 describes the whole op-\\ntimization framework, detailing the o\\u000b-line processes as\\nwell as the Grammatical Evolution we manage. Then,\\nSection 6 analyzes our experimental results. Finally,\\nSection 7 draws conclusions and future work.\\n2 Cache design and search space\\nIn order to clarify the scope of the problem under study,\\nwe devote this section to introduce some cache design\\nconcepts and related issues to be addressed. These con-\\ncepts will also help us determine the size of the search\\nspace.\\nFirstly, it must be noticed that there exist difer-\\nent design features, also named design parameters, that\\nneed to be tuned in order to decide a cache con\\fgura-\\ntion. The values given to those parameters in\\ruence on\\nthe cache behavior in both execution time and energy\\nconsumption. However, those parameters may be dif-\\nferent depending on the processor where the cache will\\nbe included.\\nIn this regard, we \\frst have to decide which type of\\ncache to select for this work. Taking into account that\\nwe focus our approach on embedded systems, we have\\nselected the cache of the ARM9 processors family [2].\\nARM processors, because of their low power con-\\nsumption and cost, have a prevalent position in the\\nmarket of portable devices and they are widespread on\\nmultimedia embedded devices. In fact, the Apple A7, Optimizing L1 Cache for Embedded Systems through Grammatical Evolution 3\\nA8 and A8X processors included in the popular iPhone\\n5S, iPhone 6 and iPad Air 2 do implement the ARMv8-\\nA instruction set.\\nIn addition, the ARM9 processors family is present\\nin devices such as game consoles (GP32, Nintendo DS),\\ncalculators (HP 49\\/50), mobile phones (HTC TyTN, K\\nand W series of Sony Ericsson) and car GPS (Samsung\\nS3C2410), for example.\\nProcessors belonging to this family allow the instal-\\nlation of a cache memory with sizes between 4 KB and\\n128 KB. For instance, the ARM920T, ARM922T and\\nARM940T processors (ARM9TDMI family) implement\\na separated 16KB, 8KB and 4KB instruction and data\\ncache, respectively. Although revisions on ARM9E fam-\\nily allow sizes up to 1MB, such as the ARM946E-S pro-\\ncessor.\\nGiven that the separation into data and instruction\\ncaches is common in the more recent ARM9 processors,\\nwe do consider this two-caches approach in our work.\\nHence, the cache parameters that can be tuned in\\nthe instruction and data caches of this processor are\\nthe following: cache size, line size, replacement algo-\\nrithm, associativity and prefetch algorithm for both the\\ninstruction and data caches, and also write policy for\\ndata cache. Here, cache size is the memory cache capac-\\nity in bytes, block size represents the amount of data\\nread or written in each cache access, replacement al-\\ngorithm is in charge of taking the decision to evict a\\nblock from cache memory, and prefetch algorithm de-\\ncides how blocks are carried to the cache memory. Fi-\\nnally, the write policy decides when data stored in the\\ncache shall be written to main memory, and it only af-\\nfects the data cache.\\nIn brief, the problem we address consists on select-\\ning the values for the cache parameters that optimize\\nthe execution time and energy consumption of a given\\napplication. Therefore, the search space is formed by all\\nthe combinations of di\\u000berent values for each one of the\\ncache parameters. Thus, in our target system, a cache\\ncon\\fguration is formed by 11 parameters: 5 for the in-\\nstructions cache and 6 for the data cache. According to\\nthe typical cache con\\fgurations of embedded systems,\\nwe selected a range of permitted values for each param-\\neter, which are shown in Fig. 1.\\nFor instructions cache (I-Cache), memory size may\\ntake 8 di\\u000berent values ranging between 512 bytes and 64\\nKB, block size can take 4 di\\u000berent values, 3 replacement\\nalgorithms and 3 prefetch algorithms may be selected,\\nand 8 associativities are available. This results in 2304\\npossible con\\fgurations for instructions cache.\\nData cache (D-Cache) have the same parameters\\nas instructions cache, but adding the 2 possible writepolicies. Hence, 4608 possible data cache con\\fgurations\\ncan be selected.\\nI-Cache D-Cache \\nBlock Size\\n8   16   32   64\\nPrefetch\\nAlgorithm\\nMISS  ON-DEMAND  ALWAYSAssociativity\\n  1   2   4  8  16 32  64 128\\nReplacement\\nAlgorithm\\n  LRU  FIFO RANDOMSize\\n512 1K 2K 4K 8K 16K 32K 64KCache Memory\\nCOPY-BACK   WRITE-TRGH.Write Policy\\nFig. 1 Cache con\\fguration parameters. Both instruction and\\ndata caches must be customized with available values.\\nThus, considering all the possible combinations of\\nboth caches, the size of the search space is 2304 \\u0002\\n4608 = 10616832 con\\fgurations. However, some of the\\nconstraints are related to infeasible parameters combi-\\nnations. For example, a cache con\\fguration of 512B,\\n64-ways set associative with 32 bytes block size is not\\nfeasible. Then, if this combination is selected for in-\\nstruction or data caches, it shall be invalidated by any\\noptimization algorithm.\\n3 Related work\\nIt is well proven that the cache memory is a component\\nthat plays a crucial role to improve performance and en-\\nergy consumption. In recent decades, many researchers\\nhave focused their work on improving di\\u000berent charac-\\nteristics of the cache memory. Several optimization ap-\\nproaches have been addressed to increase performance\\nand\\/or reduce energy consumption, develop simulation teristics of the cache memory. Several optimization ap-\\nproaches have been addressed to increase performance\\nand\\/or reduce energy consumption, develop simulation\\nand evaluation tools, etc. Next, we analyze some of the\\nworks related to our proposal that we have separted\\ninto two di\\u000berent cathegories: (1) works that deal with\\ncache recon\\fguration, and (2) works based on evolu-\\ntionary techniques.\\n3.1 Cache recon\\fguration\\nOne of the main goals for years has been to con\\fg-\\nure the cache memory according to the running work- 4 Josefa D\\u0013 \\u0010az \\u0013Alvarez et al.\\nload, in order to mainly improve performance and\\/or\\nenergy consumption. This requires both hardware sup-\\nport for cache recon\\fguration, and algorithms to decide\\nthe proper values of the parameters that obtain the best\\nperformance taking into account the running applica-\\ntion. In this way, many research works have analyzed\\ncache memory parameters and proposed new recon\\fg-\\nuration techniques from di\\u000berent points of view.\\nNaz et al. [20] proposed a design with split data\\ncaches for embedded systems. The data cache was split\\nin scalar and array caches aiming to take advantage of\\ntemporal and spatial localities respectively. Their op-\\ntimization was based on dealing with prede\\fned cache\\ncon\\fgurations, where the values of the parameters were\\n\\fxed before the optimization. Hence, the search space\\nin this case is small compared with a space where all\\nthe parameter values are taken into account.\\nChen and Zou [8] presented an e\\u000ecient recon\\fgura-\\ntion management algorithm for embedded systems. The\\ncache con\\fguration automatically changed after identi-\\nfying a phase change. The search space was composed\\nby these parameters: cache size, block size and associa-\\ntivity. From our point of view, this work do not consider\\nenough number of cache parameters.\\nGordon-Ross et al. [14] applied a method with an\\no\\u000b-line phase classi\\fcation and an on-line predictor phase.\\nPhase classi\\fcation breaks applications execution into\\n\\fxed size intervals grouped according to its similar be-\\nhavior pattern. The phase predictor decides the cache\\ncon\\fguration for the next interval. Exploration space\\nwas de\\fned by cache size, block size and associativity,\\nwhich means that only three parameters were consid-\\nered to de\\fne the search space. In this case, given the\\non-line prediction phase, a low number of parameters\\nwas required.\\nWang et al. [33] proposed dynamic cache recon\\fgu-\\nration for real time embedded systems. They minimize\\nenergy consumption performing a static analysis at run-\\ntime. This proposal is based in previous works [32,34]\\nwhere they addressed an uni\\fed two-level cache hierar-\\nchy and multi-level cache hierarchy respectively. A com-\\nbination of static analysis and dynamic tuning of cache\\nparameters, without time overhead were performed in\\nboth cases. However, a few number of parameters were\\noptimized: cache size (1, 2 or 4KB), line-size (16, 32 or\\n64 bytes) and associativity (1-way, 2-way or 4-way).\\nNew hardware technologies and core-based proces-\\nsor technologies, such as those proposed, for example,\\nby ARM [3], allow changing the cache con\\fguration for\\neach application. Changes a\\u000bect the main parameters:\\ncapacity, block size and associativity. However, an e\\u000e-\\ncient algorithm is needed to determine optimal values\\nfor each application.3.2 Evolutionary Techniques\\nEvolutionary techniques have been widely applied in\\ndesign optimization, both for hardware and software\\noptimization with di\\u000berent objectives.\\nThe creators of the Grammatical Evolution tech-\\nnique published a work where a caching algorithm was\\nautomatically generated [22]. However, they only con-\\nsidered the generation of the caching algorithm and\\nkept \\fxed some other parameters like cache size, for\\ninstance. Again, many parameters were missed.\\nPalesi and Givargis [25] presented a multi-objective\\noptimization approach to explore the design space of a\\nsystem-on-a-chip (SoC) architecture to \\fnd the pareto\\noptimal con\\fgurations of the system. They performed\\nthe optimization through a genetic algorithm where the\\nevaluation is done directly in the SoC. However, the\\nnumber of parameters they study is lower than ours,\\nand their approach is not escalable to current multime-\\ndia applications given the low performance of the SoC.\\nFilho et al. [30] presented an approach based on the\\nNSGA-II algorithm to evaluate cache con\\fgurations on\\na second level cache in order to improve energy con-\\nsumption and performance, optimizing cache size, line NSGA-II algorithm to evaluate cache con\\fgurations on\\na second level cache in order to improve energy con-\\nsumption and performance, optimizing cache size, line\\nsize and associativity. However, again, the number of\\nparameters is lower than in our proposal and, there-\\nfore, the search space is smaller.\\nDani et al. [9] applied a genetic algorithm to \\fnd an\\noptimal cache con\\fguration for a chip multiprocessor\\n(CMP) system. They deal with a complex system like\\na CMP. However, they do not manage a higher num-\\nber of cache parameters and, given that they encode\\nthe con\\fguration of both the chip and the cache into\\na chromosome, the decodi\\fcation process is complex.\\nThis issue makes di\\u000ecult the application of the same\\nmethod to a di\\u000berent microarchitecture.\\nRisco et al. [28] presented a framework based on\\nGrammatical Evolution to design custom dynamic mem-\\nory managers for multimedia applications in embedded\\nsystems. They optimized memory accesses, memory us-\\nage and energy consumption. However, they do not deal\\nwith cache parameters but optimizing the operating\\nsystem memory manager instead of dealing with cache\\ndesign parameters.\\nFinally, performing a deep search in the literature,\\nand specially in one of the main hubs of Grammatical\\nEvolution publications [1], we have not found any recent\\nwork which applied this technique to optimize the cache\\ndesign of a microarchitecture. Optimizing L1 Cache for Embedded Systems through Grammatical Evolution 5\\n3.3 Summary\\nAs seen, the approaches that make use of recon\\fgura-\\ntion require extra hardware complexity in the design\\nof the memory subsystem. Also, in the majority of the\\ncases, this complexity adds an overhead in execution\\ntime. In our work we propose a di\\u000berent approach fo-\\ncused on \\fnding the best cache con\\fguration after ex-\\nploring the search space of cache designs. This strategy\\navoids the cache recon\\fguration which does not add\\nhardware complexity to the cache memory design.\\nIn addition, previous works de\\fne their design space\\nby cache size, block size and associativity of either in-\\nstruction cache or data cache, but we have not found\\nany proposal that considers both caches at the same\\ntime on embedded systems. Our motivation is di\\u000berent\\nsince we consider the optimization of instruction and\\ndata caches together, and we selected all the parameters\\nthat can be tunned: cache size, block size, associativity,\\nreplacement algorithm and prefetch algorithm for both\\ncaches; and write policy for data cache. Therefore, we\\nexplore a wider search space.\\nMoreover, to the best of our knowledge, none of the\\nprevious works considers the optimization of as many\\nparameters as we tackle in our proposal. Regarding the\\nworks using evolutionary techniques, most of the cited\\npapers have focused their space exploration on typi-\\ncal structural cache parameters such as cache size, line\\nsize and associativity and, in addition, they usually deal\\nwith short ranges for parameter values. One of the most\\nused technique in this kind of works, the genetic algo-\\nrithm, presents the drawback of the codi\\fcation process\\nwhich has to be customized for every di\\u000berent cache\\nparadigm and simulator software.\\nOn the other hand, the methodology we propose\\ncan be applied to any kind of cache and processor ar-\\nchitectures. To this aim, the main and unique change\\nin our framework will be the cache simulator and the\\ngrammar, as it will be shown later.\\n4 Performance and energy models\\nIn this research work we consider the execution time\\nand energy consumption of a given cache con\\fguration\\nwhen running a benchmark application. As explained\\nbefore, we deal with program traces that are processed\\nby a cache simulator that accounts for the number of\\nhits and misses of each execution, which represent the\\ncache behavior.\\nAs stated before, we selected the ARM9 processors\\nfamily [2] as our target embedded system where cache\\ncon\\fguration will be optimized. Here, the L1 cache isTable 1 Main memory (DRAM) features.\\nSize 64 Mb\\nAccess time 3:9889\\u000210\\u00009sec.\\nBandwidth 6:7108864\\u0002109bytes\\/sec.\\nAccess power 1:051 W\\ndivided into instructions cache and data cache, and an\\nembedded DRAM acts as main memory.\\nIn addition to the target architecture, we consider\\nthat instructions cache is read-only, and a main memory\\nis available. The main memory features are displayed in\\nTable 1 and will be taken into account in the evluation\\nof a cache con\\fguration.\\nOnce the hardware features are set, we need a model\\nable to translate the number of hits and misses of the\\nbenchmarks runs into execution times and energy con-\\nsumption. To this aim, we chose the performance and\\nenergy models described in [16] for our optimization\\nframework.\\nThe authors of the paper proved, with a 100% ac-\\ncuracy, that the models were equivalent to the results\\nobtained from a cache simulator. Therefore, given this\\nperformance, and taking into account that the expres-\\nsions of the models could be evaluated very fast, they\\n\\ft very well into an optimization algorithm like the one\\nwe present.\\nHence, considering these models, we do account the\\nperformance and energy model of the cache memory\\nsubsystem. The model of the rest of the components\\nof the chip (CPU, main memory, etc.) is not taken\\ninto account given that our optimization scheme tack-\\nles only the cache subsystem. Next we brie\\ry describe\\nthese cache models.\\n4.1 Performance model of the chip (CPU, main memory, etc.) is not taken\\ninto account given that our optimization scheme tack-\\nles only the cache subsystem. Next we brie\\ry describe\\nthese cache models.\\n4.1 Performance model\\nThe cache performance model allows to obtain the ex-\\necution time. This model is based on the number of\\nhits and misses on the cache memory system and the\\ntime needed to solve them. We show in (1) the equa-\\ntion applied to compute execution time. Each one of its\\ncomponents is described bellow, but a wider explana-\\ntion can be found in [16].\\nT=Icache access\\u0002Icache access time +\\nIcache miss\\u0002DRAM access time +\\nIcache miss\\u0002Icache line size\\u00021\\nDRAM bwidth+\\nDcache access\\u0002Dcache access time +\\nDcache miss\\u0002DRAM access time +\\nDcache miss\\u0002Dcache line size\\u00021\\nDRAM bwidth(1) 6 Josefa D\\u0013 \\u0010az \\u0013Alvarez et al.\\nEach term in the equation represents:\\n{Icache access andDcache access are the number of\\ncache memory accesses to the instruction and data\\ncaches, respectively.\\n{Icache missandDcache missare the number of cache\\nmisses. For each one, data must be copied from the\\nmain memory.\\n{Icache access timeandDcache access timerepresent the\\ntime needed for each access to the instruction and\\ndata caches, respectively.\\n{DRAM access timeis the main memory latency time.\\n{Icache linesizeandDcache linesizecorrespond to the\\nline size (block size) for instruction and data caches,\\nrespectively.\\n{DRAM bwidth is the transfer capacity of the DRAM.\\nTherefore, the \\frst and fourth parts in this equa-\\ntion represented by Icache access\\u0002Icache access timeand\\nDcache access\\u0002Dcache access timecompute the total time\\nneeded to solve all instruction and data cache accesses,\\nrespectively. Icache miss\\u0002DRAM access time andDca-\\nchemiss\\u0002DRAM access timeare the total time spent by\\nmain memory accesses in response to instruction and\\ndata cache misses. The third and sixth parts, speci-\\n\\fed by Icache miss\\u0002Icache linesize\\u00021\\nDRAM bwidthand\\nDcache miss\\u0002Dcache linesize\\u00021\\nDRAM bwidthcalculate\\nthe total time needed to \\fll in an instruction and data\\ncaches line when a miss happens, respectively.\\n4.2 Energy model\\nThe energy model is de\\fned in (2), and also detailed in\\n[16]. This equation is used to determine the dynamic\\nenergy consumption for a cache con\\fguration.\\nE=Total time\\u0002CPU power +\\nIcache access\\u0002Icache access energy +\\nDcache access\\u0002Dcache access energy +\\nIcache miss\\u0002Icache access energy \\u0002Icache line size+\\nDcache miss\\u0002Dcache access energy \\u0002Dcache line size+\\nIcache miss\\u0002DRAM access power +\\n(DRAM access time +Icache line size\\u00021\\nDRAM bwidth) +\\nDcache miss\\u0002DRAM access power +\\n(DRAM access time +Dcache line size\\u00021\\nDRAM bwidth) (2)\\nMost of the components of the equation have been\\nbrie\\ry described above. The rest of the components are\\nthe following:\\n{DRAM access power represents the power consump-\\ntion for each DRAM access.{Icache access energy andDcache access energy corres-\\npond to energy consumption for each instruction\\nand data caches access, respectively.\\nTheIcache access\\u0002Icache access energy andDcache -\\naccess\\u0002Dcache access energy terms calculate the energy\\nconsumption due to instruction and data caches, re-\\nspectively. Icache miss\\u0002Icache access energy\\u0002Icache -\\nlinesizeandDcache miss\\u0002Dcache access energy\\u0002Dcache -\\nlinesizeis the energy cost of writing data to the instruc-\\ntion and data caches when misses occur. The two last\\nterms calculate the energy cost of the DRAM when re-\\nsponding to cache misses.\\nIn our approach, we have removed the \\frst term of\\nthe energy equation Total time\\u0002CPU power because of\\nthree reasons: (1) the term CPU power is constant and\\nthe term Total timecorresponds to the total time of the\\nsystem, not only the cache subsystem, (2) it represents\\nthe amount of energy consumed by the CPU and we are\\noptimizing just the performance and energy consumed\\nby the memory subsystem, and (3) this term could be\\nlarge enough to hide the di\\u000berences because of memory\\ncache operations. Thus, the energy equation is reduced\\nto:\\nE=Icache access\\u0002Icache access energy +\\nDcache access\\u0002Dcache access energy +\\nIcache miss\\u0002Icache access energy\\u0002Icache line size+\\nDcache miss\\u0002Dcache access energy\\u0002Dcache line size+\\nIcache miss\\u0002DRAM access power +\\n(DRAM access time +Icache line size\\u00021\\nDRAM bwidth) +\\nDcache miss\\u0002DRAM access power +\\n(DRAM access time +Dcache line size\\u00021\\nDRAM bwidth) (3)\\nAll the equations use seconds, watts, joules, bytes\\nand bytes\\/sec as units to measure time, power, energy,\\ncache line size and bandwidth, respectively.\\nNext, we describe the whole process of optimization\\nshowing where these models are applied.\\n5 Optimization methodology\\nAs explained before, in this work we propose an ap-\\nproach to identify the best cache con\\fguration for a\\ngiven set of applications. We de\\fne the best cache con-\\n\\fguration as the one which minimizes the execution proach to identify the best cache con\\fguration for a\\ngiven set of applications. We de\\fne the best cache con-\\n\\fguration as the one which minimizes the execution\\ntime and energy consumption for a given benchmark.\\nWith this aim in mind, in this section we describe\\nthe optimization framework that we propose to opti-\\nmize the cache memory, in this case, of multimedia Optimizing L1 Cache for Embedded Systems through Grammatical Evolution 7\\nApplication \\nTracesCacti\\nAccess Time \\n&\\n Energy \\/ accessOff-line Profiling Off-line  Characterization\\nGrammatical\\nEvolutionI- and D-Cache Configuration\\n# Hits\\n# MissesMediaBenchApplicationsCacheParameters\\nDinero IVTrimaran\\nARM\\nOPTIMAL Cache\\nConfiguration\\n(I-Cache & D-Cache)Start\\nFig. 2 Processes involved in the cache con\\fguration opti-\\nmization.\\nembedded systems. Fig. 2 presents an overview of the\\nwhole process.\\nWe divided our optimization process into three dif-\\nferent stages. Firstly, two processes are run just once\\nbefore the optimization. These are called o\\u000b-line pro-\\ncesses and perform: (1) cache characterization and (2)\\napplication pro\\fling. The third stage is the optimiza-\\ntion, which is formed by the evolutionary process run-\\nning GE and calling the cache simulator.\\nThe optimization is performed by the Grammatical\\nEvolution (GE) module, as seen in Fig. 2. Each time a\\ncon\\fguration has to be evaluated, the GE calls Dinero\\nIV, which is a trace-driven cache simulator [12]. It re-\\nceives a cache con\\fguration from GE and returns the\\nnumber of cache hits and misses for the target applica-\\ntion traces. These date are included in the models to\\nevaluate the con\\fguration. Next, we give more detail of\\nthis optimization \\row.\\n5.1 O\\u000b-line process\\nTwo tasks are run o\\u000b-line: the characterization of the\\npossible cache con\\fgurations, and the pro\\fling of the\\ntarget benchmarks. Both tasks are slow, but can be\\nrun just once, because their results are stored in data\\n\\fles that are read by the cache simulator and the GE\\nmodule.\\nThe characterization process, on the upper left part\\nof Fig. 2, has the speci\\fc task of combining all the pa-\\nrameter values for cache con\\fgurations. That is, gen-\\nerates all the possible cache con\\fgurations and obtains\\ntheir access time and energy consumption per access.\\nHence, this process provides the time and energy val-\\nues used for the terms in equations (1) and (3). For\\nthis purpose we used Cacti [19] considering 32nm tech-\\nnology to compute the DRAM and cache access timesand dynamic energy consumed for each cache access.\\nCacti is an analytical model widely applied to estimate\\nenergy and power consumption, performance and area\\nof cache memories. This characterization was run only\\nonce, and we selected for this task the web tool pro-\\nvided by the Cacti organization. The most important\\nparameters required by Cacti are those mainly related\\nto hardware: cache size, line size and associativity, and\\nthere is no distinction between instructions and data\\ncaches. Hence, this is a slow but a\\u000bordable task.\\nThe pro\\fling process, on the upper right part of\\nFig. 2, is in charge of the generation of the traces from\\nthe target applications. Again, this is a process that\\nhas to be run just once. The pro\\fling takes each bench-\\nmark application and obtains the sequence of cache hits\\nand misses from a run, generating a memory trace \\fle.\\nThis process is carried out by the open-source compiler\\nand performance monitoring infrastructure called Tri-\\nmaran [7], which provides the resources required to ob-\\ntain accurate application traces. Given that the ARM\\nprocessors are not directly addressed by Trimaran, we\\nhave modi\\fed it by incorporating code from the ar-\\nchitectural simulator SimpleScalar [5], which is able to\\nmodel a wide set of di\\u000berent architectures, being the\\nARM among them.\\nBesides, it has to be mentioned that input and out-\\nput data of all the benchmarks under study are repre-\\nsented in the trace \\fle. Therefore, in order to capture\\nthe mean behavior of each application, we will select the\\nrecommended inputs indicated in the documentation of\\neach benchmark.\\n5.2 Grammatical Evolution\\nThe characterization of the cache ends up with two kind\\nof results. On the one hand, the time and energy re-\\nquired per access is obtained for each cache con\\fgu-\\nration and, on the other hand, the traces with all the\\nmemory operations of each benchmark application have quired per access is obtained for each cache con\\fgu-\\nration and, on the other hand, the traces with all the\\nmemory operations of each benchmark application have\\nbeen obtained. Therefore, our framework is ready to\\nevaluate the performance of any candidate cache con-\\n\\fguration, which will be generated in the optimization\\nmodule, based on Grammatical Evolution.\\nGrammatical Evolution [23,24,21,11] is an evolu-\\ntionary technique that is able to represent individuals\\nby means of grammars. GE is a form of Genetic Pro-\\ngramming (GP) [17] which evolves programs that are\\nevaluated according their \\ftness values. In this regard,\\nGE represents each program, that is, each individual,\\nthrough an expression generated by a grammar de\\fned\\nby the researcher.\\nGE performs evolutionary process on variable-length\\nbinary strings, which are mapped to generate a program 8 Josefa D\\u0013 \\u0010az \\u0013Alvarez et al.\\nby selecting production rules in a Backus-Naur form\\n(BNF) grammar de\\fnition. Given that strings are sim-\\nilar to chromosomes, GE allows using genetic operators\\nlike selection, crossover and mutation. Problems such\\nas \\fnancial modeling [4], game strategies [13], dynamic\\ngame environment [26], 2D platforms games design [29],\\n3D design [6] among others, have been successfully ad-\\ndressed by using GE.\\nSo, the GE traverses the search space of cache con-\\n\\fgurations according to the grammar de\\fnition. For\\neach individual, it calls the cache simulator to obtain\\nthe number of hits and misses for the target bench-\\nmarks. Then, using the access time and energy per\\naccess previously computed by Cacti, and the perfor-\\nmance and energy models described in Section 4, the\\n\\ftness is computed. This process is repeated till the se-\\nlected number of generations for GE is reached. As a\\nresult, an optimized cache con\\fguration is obtained for\\neach one of the benchmark programs.\\nAs seen in Section 2, the search space of this problem\\nis de\\fned by the set of cache con\\fgurations. However,\\nGE works with individuals that represent those con\\fg-\\nurations under a proper codi\\fcation. In fact, GE uses\\na linear genome rather than a tree-based structure, as\\nin GP. Linear genome facilitates the application of ge-\\nnetic operators. Here, the chromosome of each individ-\\nual consists of a string of integer values corresponding\\nto codons. Each codon stores a value of 8 bits, hence\\nforming the genotype. The genotype is decoded in a\\nprocess called mapping, where a grammar is used to\\ntranslate the codons into the phenotype of the individ-\\nual.\\nOne individual represents a particular cache con\\fg-\\nuration that must be evaluated in order to measure its\\nexecution time and energy consumption. In this regard,\\nwe have decided to consider the same weight for both\\nfeatures of the cache. Therefore, for a given cache con-\\n\\fguration ci, the evaluation process is conducted by the\\n\\ftness function fde\\fned in (4).\\nf(ci) = 0:5\\u0002T(ci)\\nT(Baseline )+ 0:5\\u0002E(ci)\\nE(Baseline )(4)\\nEach component of the \\ftness, that is, execution\\ntime and energy consumption, are normalized with re-\\nspect to a baseline con\\fguration. By normalizing to a\\nbaseline we are able to correctly scale and operate the\\nunits of di\\u000berent measures like time and energy. As seen\\nin (4), both terms are weighted to 50% seeking the best\\nbalance between the improvement of the performance\\nand energy consumption. The best cache con\\fguration\\nin GE will be the one that minimizes the \\ftness value.N = { <DineroParams >, <CacheSizeB >, <LineSizeB >,\\n<ReplAlg >, <Assoc >, <PrefAlg >, <WritePol > }\\nT = { 1, 2, 4, 8, 16, 32, 64, 128 512 , 1024 , 2048 ,\\n4096 , 8192 , 16384 , 32768 , 65536 , l, f, r, m,\\nd, a, n }\\nS = <DineroParams >\\nP = { I <DineroParams > ::= -l1 - isize <CacheSizeB >\\n-l1 - ibsize <LineSizeB >\\n-l1 - irepl <ReplAlg >\\n-l1 - iassoc <Assoc >\\n-l1 - ifetch <PrefAlg >\\n-l1 - dsize <CacheSizeB >\\n-l1 - dbsize <LineSizeB >\\n-l1 - drepl <ReplAlg >\\n-l1 - dassoc <Assoc >\\n-l1 - dfetch <PrefAlg >\\n-l1 - dwback <WritePol >\\nII <CacheSizeB > ::= 512 | 1024 | 2048 | 4096\\n| 8192 | 16384 | 32768\\n| 65536\\nIII <LineSizeB > ::= 8 | 16 | 32 | 64\\nIV <ReplAlg > ::= l | f | r\\nV <Assoc > ::= 1 | 2 | 4 | 8 | 16 | 32 | 64\\n| 128\\nVI <PrefAlg > ::= m | d | a\\nVII <WritePol > ::= a | n }\\nFig. 3 Grammar for cache con\\fguration description.\\nGiven that the evaluation of a cache con\\fguration\\ninvolves a call to the cache simulator, we have designed\\na grammar that produces phenotypes which form the\\nparameters of the call to the simulator. In fact, one\\nof the bene\\fts of GE is the ability to create complex\\nphenotypes from a genotype formed by integer values.\\nFig. 3 shows, in BNF format, the grammar we have\\ndesigned for the cache optimization under the previ-\\nously de\\fned search space. As stated before, the gram-\\nmar produces the string of parameters that will be sent\\nto the cache simulator. In fact, we have adapted the pa- ously de\\fned search space. As stated before, the gram-\\nmar produces the string of parameters that will be sent\\nto the cache simulator. In fact, we have adapted the pa-\\nrameters to the requirements of the simulator. As seen\\nin the \\fgure, the replacement algorithm will take values\\nlikel,forr, which correspond in the simulator, respec-\\ntively, to LRU,FIFO and RANDOM . The same applies for\\nprefetching and write policy.\\nAccording to the terminology, a grammar is described\\nas a tuplefN, T, S, Pg, where Nrepresents the non-\\nterminals symbols, Tis the set of terminals, Srepre-\\nsents the start symbol belonging to N, and Pde\\fnes\\nthe set of production rules that map the values of Nto\\nT. The \\\\j\\\" symbol separates the di\\u000berent choices within\\na single production rule. In our grammar, Pis formed\\nby production rules from ItoVII.\\nNext we show an example of the mapping process for\\nthe problem at hand using the grammar in Fig. 3. Let's\\nconsider the genotype shown in the upper part of Fig. 4\\nas a candidate solution. This individual is a genotype\\nthat represents a cache con\\fguration. However, it has to\\nbe decoded in order to obtain the values of each one of Optimizing L1 Cache for Embedded Systems through Grammatical Evolution 9\\nthe 11 parameters of the cache con\\fguration previously\\ndescribed.\\nThe mapping or decodi\\fcation process is performed\\nby applying the modulus operation between the codon\\nvalue and the number of options of each production\\nrule corresponding to the non-terminal symbol being\\nprocessed.\\n<CacheSizeB> <LineSizeB> <ReplAlg> <Assoc> <PrefAlg>\\n819264r20357196\\n<CacheSizeB> <LineSizeB> <ReplAlg> <Assoc> <PrefAlg> <WritePol>1 m123\\n2048 16 f210 137 7 5\\n4 a20\\nn35 ( Wrap )\\nDecoded phenotype:Genotype:\\n  20    35    71    96    123     210    137    7    5\\nMapping:Phenotype after initial symbol is decoded:\\n-l1-isize <CacheSizeB> -l1-ibsize <LineSizeB> -l1-irepl <ReplAlg>-l1-iassoc <Assoc> -l1-ifetch <PrefAlg> -l1-dsize <CacheSizeB>-l1-dbsize <LineSizeB> -l1-drepl <ReplAlg> -l1-dassoc <Assoc>-l1-dfetch <PrefAlg> -l1-dwback <WritePol>  \\n-l1-isize 8192  -l1-ibsize 64 -l1-irepl r\\n-l1-iassoc 1 -l1-ifetch m\\n-l1-dsize 2048  -l1-dbsize 16 -l1-drepl f\\n-l1-dassoc 4 -l1-dfetch a -l1-dwback n \\nFig. 4 Decoding process in GE that, starting from the geno-\\ntype above, obtains the phenotype by mapping through the\\nproposed grammar.\\nThen, considering the example individual, the de-\\ncoding process is performed as shown in Fig. 4. It begins\\ndecoding the starting symbol, <DineroParams> , which\\nis substituted by its production, that corresponds to the\\nparameterized call to the cache simulator. Here, several\\nnon-terminal symbols are present, and the \\frst one is\\n<CacheSizeB> . Hence, the \\frst gene of the genotype is\\nselected, 20. Given that the production for <CacheSizeB>\\n(rule II) has 8 di\\u000berent options, the selected value is\\noption 4 (20 MOD 8 = 4), which corresponds to 8192 .\\nNotice that the \\frst option of the rule is linked to 0,\\nthe second to 1 and so on. Next, the second gene is read\\nand its value, 35, is used to decode the following non-\\nterminal symbol, <LineSizeB> . The production rule for\\nthis symbol is III, and has 4 options. Then, after the\\nmodulus operation (35 MOD 4 = 3) <LineSizeB> is\\nreplaced by 64. The following non-terminal symbol to\\nbe decoded is <ReplAlg> , whose production rule, IVhas\\n3 di\\u000berent options. Then, the codon value, 71, is pro-cessed. Given that 71 MOD 3 = 2, the selected value is\\nr. In an analogous way, the next codon, 96 for <Assoc>\\nis turned into 1 given that 96 MOD 8 = 0. The last non-\\nterminal corresponding to the instructions cache pa-\\nrameters, <PrefAlg> , is determined by the codon value\\n123 which is replaced by m, once that 123 MOD 3 = 0.\\nNext, the mapping process continues with the codon\\n210 and the second <CacheSizeB> symbol, which maps\\n2048 and corresponds to the data cache size. Then, 137\\nmaps <LineSizeB> to 16, gene 7 maps <ReplAlg> tof\\nand 5 maps <Assoc> to 4.\\nAt this point, all the genes have been used but the\\nexpression is still partially decoded. However, GE is\\nable to work with variable length chromosomes and, if\\nthe decodi\\fcation process reaches the end of the chro-\\nmosome and there are still non-terminal symbols, GE\\ngoes to the \\frst codon and allows the decodi\\fcation to\\n\\fnish. This process is called wrapping . Hence, in our\\nexample a wrap takes place at this point. Therefore,\\n<PrefAlg> is decoded with the \\frst gene, 20, obtaining\\na. Finally, decodi\\fcation ends mapping <WritePol> to\\nnwith gene value 35.\\nAs a result, our genotype example returns the phe-\\nnotype shown in the lower part of Fig.4, which rep-\\nresents a cache con\\fguration where instructions cache\\nsize is 8192, instructions cache block size is 64, and so\\non.\\nOnce decoded, the phenotype is used to call the\\ncache simulator in order to evaluate the corresponding\\ncon\\fguration on the target application, obtaining both\\nthe execution time and energy consumption according\\nto the \\ftness de\\fned in (4). At the end of the selected\\nnumber of generations, the GE returns the cache con-\\n\\fguration that obtains the best \\ftness value.\\nAs seen, the grammar determines both the concrete number of generations, the GE returns the cache con-\\n\\fguration that obtains the best \\ftness value.\\nAs seen, the grammar determines both the concrete\\nparameters to be optimized, as well as the communica-\\ntion with the cache simulator, which provides the \\ft-\\nness value. Therefore, changing the grammar will allow\\ndi\\u000berent optimization approaches: \\fx the value of some\\nparameters, add new parameters, change the cache sim-\\nulator, use variable length arguments in the cache sim-\\nulator (for custom algorithms, for example), an so on.\\nIn this way, given both an invariable number (and\\ntype) of parameters and a cache simulator, the gram-\\nmar forces the evolution of a \\fxed structure, and GE\\nacts optimising the parameters in this structure. Hence,\\nthis problem could be tackled with a \\fxed-length algo-\\nrithm like a GA, PSO and so on. However, in this kind\\nof approaches, any change in the number or type of pa-\\nrameters, or even in the simulator, will require the mod-\\ni\\fcation of an important part of the code, mainly in the\\ndecodi\\fcation and \\ftness evaluation modules. Hence,\\nour approach based on GE provides a great \\rexibility 10 Josefa D\\u0013 \\u0010az \\u0013Alvarez et al.\\nas well as extensibility features because those changes\\ncould be incorporated with a simple change of the gram-\\nmar.\\nThe GE algorithm was developed in Java by our\\nresearch team, and is publicly available as the JECO\\nlibrary [27].\\n6 Experimental results\\nIn order to test our experimental framework, we se-\\nlected twelve benchmarks from the Mediabench suite [18],\\nwhich are representative of multimedia applications.\\nNext, we describe the features of these benchmarks.\\nThe full description can be found on the Mediabench\\nwebsite.\\n{JPEG is a C software to compress ( cjpeg ) and un-\\ncompress ( djpeg ) full-color and gray-scale JPEG im-\\nages.\\n{MPEG is an optimized implementation for trans-\\nmitting high quality digital video. The mpegenc bench-\\nmark is in charge of encode process, and the mpegdec\\nbenchmark to the decode process.\\n{GSM implements an algorithm to reliable voice recog-\\nnition. The algorithm is based on the implementa-\\ntion of the European GSM 06.10 provisional stan-\\ndard for full-rate speech transcoding. The gsmenc\\nbenchmark performs the compression frames pro-\\ncess, and the gsmdec performs the decompression\\nprocess.\\n{Pegwit implements the process for public key en-\\ncryption and authentication. The process uses an\\nelliptic curve over GF(2255), the codi\\fcation algo-\\nrithm SHA1 for hashing, and the symmetric block\\ncipher square. Pegwitenc andpegwitdec perform en-\\ncoding and decoding respectively.\\n{EPIC (E\\u000ecient Pyramid Image Coder) performs\\nimage data compression ( epic) and decompression\\n(unepic ). The algorithms are based on a biorthogo-\\nnal critically-sampled dyadic wavelet decomposition\\nand a combined run-length\\/Hu\\u000bman entropy coder.\\n{ADPCM (Adaptive Di\\u000berential Pulse Code Modu-\\nlation) implements speech compression and decom-\\npression algorithms. Rawcaudio performs the com-\\npression while rawdaudio performs the decompres-\\nsion. The algorithm they use achieves a compression\\nrate of 4:1 on 16 bits linear PCM samples.\\nOur experiments were carried out in a machine pro-\\nvided with an Intel i5 660 processor at 3.3 GHz. Given\\nthat this is a 4-core processor, we run a maximum of 4\\nexperiments at the same time. The available RAM in\\nthe machine was 8 Gb, and the operating system was\\nUbuntu Desktop 14.04.6.1 Experimental setup\\nAs stated before, our experimental framework is pa-\\nrameterized according to the ARM architecture. Then,\\nto evaluate our approach we consider a \\frst level cache\\nmemory with a range of sizes between 512B and 64KB,\\nincreased in powers of two. With this set of possible\\ncache sizes we cover many of the sizes for some devices\\nof the ARM9 processor family. According to these fea-\\ntures, the search space for GE was described in Section\\n2 combining both instruction and data caches. However,\\nthe cache characterization is performed for individual\\nmemories taking into account only the hardware pa-\\nrameters in the o\\u000fine characterization process. In our\\nsearch space, these parameters correspond to 8 possible\\ncache sizes, 8 di\\u000berent associativity degrees and 4 possi-\\nbles values for block size, as shown in Fig. 1. Hence, the\\nnumber of cache characterizations to be run are 256.\\nUsing our methodology for o\\u000b-line pro\\fling, we have\\ngenerated the Mediabench traces using Trimaran tools [7]\\nin connection with SimpleScalar [5] to evaluate the ARM\\narchitecture. In the light of the above, the appropriate\\nchanges have been made for both SimpleScalar and Tri-\\nmaran tools to generate application traces able to be\\nprocessed by the Dinero IV cache simulator.\\nThis kind of experiments require to manage full ap-\\nplication traces or execute a number of appropriate in-\\nstructions which re\\rects the behavior of the full pro-\\ngram. This way prevents to capture a partial or phase\\nbehavior. For instance, phase behavior happens at the\\nbeginning of the application run, when all data blocks\\nmust be copied from the main memory, so all accesses\\nto cache memory are misses. Thus, a small number of beginning of the application run, when all data blocks\\nmust be copied from the main memory, so all accesses\\nto cache memory are misses. Thus, a small number of\\ninstructions can provide results that cannot be consid-\\nered as signi\\fcant or conclusive. In this regard, we have\\nsimulated every application a maximum of 7 :5\\u0002107in-\\nstructions, or even the full application, if the total num-\\nber of instructions for the complete execution is lower\\nthan this value. The aim is to reach a balance between\\nthe simulation time, the size of the program traces and\\na proper number of instructions.\\nIn addition, GE has been run 10 times for each tar-\\nget application with the aim of reducing the probability\\nto get a local optimum.\\nTable 2 shows the parameters for GE. We have also\\napplied simple dominance, single point crossover and\\ninteger \\rip mutation operator with probabilities accord-\\ning to recommendations in [10].\\nAs stated in (4) (see Section 5.2), our algorithm re-\\nquires a baseline cache con\\fguration in order to cal-\\nculate the goodness of an individual. For these exper-\\niments we selected a baseline cache con\\fguration that\\nis similar to the cache of the \\frst core in the GP2X Optimizing L1 Cache for Embedded Systems through Grammatical Evolution 11\\nTable 2 Parameters for GE.\\nNumber of generations 100\\nPopulation size 50\\nProbability of crossover 0:9\\nProbability of mutation 0:01\\nportable game console, which incorporates a dual 200\\nMHz ARM940T processor.\\nTable 3 shows the size of the cache, the size of the\\nblock, the number of ways (associativity degree), re-\\nplacement policy, search algorithm and, for the data\\ncache, the allocation policy of the baseline con\\fgura-\\ntion.\\nExecution time and energy consumption for the base-\\nline cache was computed following the models described\\nin Section 4.\\n6.2 Objective function\\nFirstly, we analyze the results regarding the value of\\nthe objective function. We would like to recall that ob-\\njective value 1 corresponds to the execution time and\\nenergy consumption of the baseline cache con\\fguration\\nrunning each one of the bechmarks. Therefore, we min-\\nimize the objective function in order to reduce both\\nfeatures.\\nTable 4 shows the following statistics, averaged for\\neach one of the benchmarks: objective value, deviation\\npercentage with respect to the best objective found,\\nand number of best results found.\\nAs seen in the table, the average objective values\\nrange from 0 :47 to 0 :30, which mean improvements from\\n53% to 70%, with an average improvement of 62% in\\nthe objective value. Hence, the optimization obtains re-\\nally good objective improvements. In addition, the low\\nvalues of the standar deviation show that GE obtains\\nvery close results across the optimization runs. In fact,\\nthe number of times the best result is found is higher or\\nequal to 70% in 7 out of the 12 benchmarks. In partic-\\nular, GE obtains great results in pegwitdec ,pegwitenc ,\\nrawcaudio and rawdaudio benchmarks. On the other\\nhand, despite that GE obtains the best value only once\\nindjpeg andunepic , the deviation is below 3 :1% and\\n0:7% respectively, which again is a very good result.\\nThese results prove that a metaheuristic like GE,\\neven with a low number of generations and population\\nsize, is able to \\fnd solutions where the objetive function\\nis reduced above the 62% for the cache optimization\\nproblem.6.3 Optimization runtime\\nWe have made another analysus regarding the opti-\\nmization runtime (RT). Table 5 shows the average opti-\\nmization runtime in hours, the average number of eval-\\nuations in relation to the maximum expected for GE,\\nthe average runtime for the evaluation of one cache con-\\n\\fguration in seconds, the maximum optimization run-\\ntime of GE if all evaluations were performed, and the\\nruntime savings of the experiments in relation to that\\ntime.\\nAt the same time the optimization was run, we mea-\\nsured the time spent for the evaluation of each cache\\ncon\\fguration. As seen in the table, the average evalua-\\ntion time is high is some of the benchmarks like djpeg ,\\nmpegdec and mpegenc . Taking into account this time\\nvalues, we have calculated the optimization time that\\na GE execution will spent if the 5000 evaluations were\\nrun, that is, if the algorithm will not implement mem-\\nory for evaluated individuals.\\nAs seen in the savings columns, the execution time\\nis reduced in more than 93 :6%. Therefore, it is clear\\nthat our GE implementation is able to run in reason-\\nable amounts of time due to the memory of evaluated\\nindividuals.\\nThe reduced optimization runtimes are caused by\\ntwo factors. On the one hand, the number of evalua-\\ntions is low because the exploration of the GE algorithm\\nreaches an area of minimum values making the popu-\\nlation to converge to that minimum. In other words,\\nthe individuals of the population are becoming simi-\\nlar over the generations. On the other hand, given that\\nour algorithm recalls the evaluation of each individual,\\nthe previously evaluated con\\fgurations are not sent to\\nthe external simulator, but read from the GE memory.\\nThis is a very fast process because we implemented it\\nthrough hashed maps.\\n6.4 Execution time and energy consumption the external simulator, but read from the GE memory.\\nThis is a very fast process because we implemented it\\nthrough hashed maps.\\n6.4 Execution time and energy consumption\\nAs stated in 5.2, the objective function incorporates\\nboth the execution time and energy consumption val-\\nues of a given cache con\\fguration. Then, once we have\\nobtained the optimized cache con\\fgurations, we show\\nhow these results are interpreted in terms of execution\\ntime and energy consumption.\\nFig.5 plots the average enery consumption of the\\nruns for each benchmark considering that 100% is the\\nenergy consumption of the baseline cache. On average,\\nthe solutions we have obtained reach the 3 :04% of the\\nbaseline.\\nFig.6 plots the average execution time of the runs\\nfor each benchmark considering that 100% is the en- 12 Josefa D\\u0013 \\u0010az \\u0013Alvarez et al.\\nTable 3 Baseline cache con\\fguration. It is formed by parameters of instructions and data caches.\\nInstructions Cache\\nSize Block Sz. Assoc. Replac. Search Alg.\\n16 KB 32 B 4 LRU On-Demand\\nData Cache\\nSize Block Sz. Assoc. Replac. Search Alg. Alloc.\\n16 KB 32 B 4 LRU On-demand Copy-Back\\nTable 4 Objective function values.\\nBenchmark Avg. Obj. Dev. (%) # Best\\ncjpeg 0.3986650026 0.74% 8\\ndjpeg 0.4746031647 3.06% 1\\nepic 0.3365548961 1.24% 7\\nunepic 0.3977187056 0.65% 1\\ngsmdec 0.3857290316 1.15% 7\\ngsmenc 0.4442631856 1.82% 4\\nmpegdec 0.3682433302 1.09% 6\\nmpegenc 0.3631493163 0.89% 3\\npegwitdec 0.3663486721 0.01% 9\\npegwitenc 0.3568943181 0.00% 10\\nrawcaudio 0.3052829963 0.00% 8\\nrawdaudio 0.31533024 0.56% 9\\nTable 5 Runtime values. For each benchmark, it shows the average runtime (hours), average number of evaluations vs max.\\nGE, average runtime for one cache evaluation (seconds), maximum runtime for GE (no evaluation recall) and runtime savings\\nin relation to maximum GE.\\nBenchmark RT (h.) # Eval. (%) Eval. (sec.) Max GE (h.) RT Sav.\\ncjpeg 2.185 5.96% 26.200 36.389 93.99%\\ndjpeg 2.647 5.74% 33.064 45.923 94.24%\\nepic 1.405 5.87% 17.148 23.816 94.10%\\nunepic 0.350 5.79% 4.340 6.028 94.19%\\ngsmdec 0.004 6.33% 0.043 0.060 93.67%\\ngsmenc 0.003 5.58% 0.041 0.057 94.41%\\nmpegdec 2.275 5.69% 28.805 40.006 94.31%\\nmpegenc 2.031 5.80% 25.209 35.013 94.20%\\npegwitdec 0.945 6.38% 10.667 14.815 93.62%\\npegwitenc 0.869 5.98% 10.466 14.537 94.02%\\nrawcaudio 0.372 5.95% 4.483 6.227 94.03%\\nrawdaudio 0.274 5.89% 3.343 4.642 94.10%\\nergy consumption of the baseline cache. On average,\\nthe solutions we have obtained reach the 24 :74% of the\\nbaseline.\\nIn brief, we have improved the cache behavior by\\n97% in energy consumption and, at the same time, by\\n75% in execution time considering a commercial base-\\nline cache. These results were obtained in very short\\nruntimes compared with the expected time that a GE\\nwith no evaluation recall will reach.For the sake of clarity, we would like to remark that\\nthe baseline con\\fguration acts as a reference point from\\nwhich the GE process optimizes. In fact, given that our\\nframework uses the characterization from a cache em-\\nulator (Cacti in our case), any baseline con\\fguration\\ncould be optimized by providing the framework with\\nthe corresponding baseline characterization values.\\nIn addition, given that the benchmark results for\\nthe baseline cache are not optimised, the improvements Optimizing L1 Cache for Embedded Systems through Grammatical Evolution 13\\nFig. 5 Average energy consumption for the solutions obtained on each benchmark in relation to baseline (100%).\\nFig. 6 Average execution time for the solutions obtained on each benchmark in relation to baseline (100%).\\nobtained by GE (which are optimised) should be taken\\nas indicative of potential performance gains.\\n7 Conclusions and future work\\nThe signi\\fcant increase of multimedia applications in\\nembedded devices such as tablets, smartphones, GPS\\nnavigation systems, etc. has caused researchers focus\\ntheir works on improving either the execution time and\\nenergy consumption, or both. In order to improve the\\nexecution time, current embedded systems include a\\ncache hierarchy similar to desktop computers. However,the wide range of possible cache con\\fgurations compli-\\ncates the search of the best cache con\\fguration. Not\\nto mention the fact that the execution time and energy\\nconsumption are directly a\\u000bected by the memory access\\npatterns of applications. Therefore, new and e\\u000ecient\\ntechniques have been proposed for years to explore and\\ndecide the best cache con\\fguration for di\\u000berent appli-\\ncations. However, we have not found in literature any\\nwork that optimize, at the same time, all the parame-\\nters that con\\fgure both the instructions and the cache\\nmemory for a given set of applications.\\nIn this paper we have presented an optimization\\nframework based on static pro\\fling and metaheuristics 14 Josefa D\\u0013 \\u0010az \\u0013Alvarez et al.\\nto automatically search for the best cache con\\fgura-\\ntion given a set of applications to be run on a target\\nembedded system. Our optimization scheme considers\\n11 di\\u000berent parameters that con\\fgure both the instruc-\\ntion and the data caches. The framework is divided into\\ntwo phases. The \\fst one is an o\\u000b-line phase in charge to\\nprovide the program traces and the features of the set\\nof possible hardware con\\fgurations for cache memories.\\nThe second phase is an optimization process that ap-\\nplies GE obtaining the cache con\\fguration that would\\nobtain the best behavior for each target application. In\\norder to evaluate a cache con\\fguration, GE obtains the\\nperformance of each con\\fguration with the help of the\\nDinero IV cache simulator. In this way, the grammar\\nof GE eases the communication with Dinero IV given\\nthat the phenotype is formed by the parameters and\\nformats required by the simulator call.\\nIn addition, our approach allows to modify the num-\\nber of parameters to be optimized, as well as the change\\nof the cache simulator by simply modifying the gram-\\nmar at hand. Besides, if a cache simulator will allow\\nvariable length parameters, an appropriate grammar\\nwill be able to generate correct simulator calls without\\nany modi\\fcation in our optimization framework.\\nWe have tested our proposal with twelve applica-\\ntions from the Mediabench benchmarks considering an\\nactual cache con\\fguration as baseline. Our results av-\\neraged an improvement of 97% in energy consumption\\nand 75% in execution time versus the baseline cache.\\nOur future plans include e\\u000borts to simplify processes\\nand design an e\\u000bective framework that allow customize\\nthe optimization process for a set of target applications\\nat a time. In addition, we would take into account the\\ndi\\u000berent load that memory operations represent on the\\ntarget applications in order to precisely balance their\\nweight in the optimization. In fact, a natural step for-\\nward after this research will be to analyze the in\\ruence\\nbetween execution time and energy results, and con-\\nsider a multi-objective optimization.\\nReferences\\n1. http:\\/\\/www.grammatical-evolution.org\\/pubs.html\\n(2015)\\n2. ARM Ltd.: ARM9 Processor Family.\\nhttp:\\/\\/www.arm.com\\/products\\/processors\\/classic\\/arm9\\/\\n(2013)\\n3. ARM946E-S TM: Technical reference manual.\\nhttp:\\/\\/www.arm.com\\/products\\/processors\\/classic\\/arm9\\/\\narm946.php (2014)\\n4. Brabazon, A., O'Neill, M.: Biologically Inspired Algo-\\nrithms for Financial Modelling. Springer (2006)\\n5. Burger, D., T.M.Austin: The simplescalar tool set, ver-\\nsion 2.0. In: Technical Report, pp. CS{TR{97{1342. Uni-\\nversity of Wisconsin-Madison (1997)6. Byrne, J., McDermott, J., Galva\\u0013 on-Lo\\u0013 opez, E., O'Neill,\\nM.: Implementing an intuitive mutation operator for in-\\nteractive evolutionary 3d design. In: Evolutionary Com-\\nputation (CEC), 2010 IEEE Congress on, pp. 1{7 (2010).\\nDOI 10.1109\\/CEC.2010.5586485\\n7. Chakrapani, L.N., Gyllenhaal, J., mei W. Hwu, W.,\\nMahlke4, S.A., Palem1, K.V., Rabbah5, R.M.: Trimaran:\\nAn infrastructure for research in instruction-level par-\\nallelism. In: In Instruction-Level Parallelism. Lecture\\nNotes In Computer Science (2005). URL http:\\/\\/www.\\ntrimaran.org\\/\\n8. Chen, L., Zou, X., Lei, J., Liu, Z.: Dynamically recon-\\n\\fgurable cache for low-power embedded system. In:\\nNatural Computation, 2007. ICNC 2007. Third Interna-\\ntional Conference on, vol. 5, pp. 180 {184 (2007). DOI\\n10.1109\\/ICNC.2007.346\\n9. Dani, A., Srikant, Y., Amrutur, B.: E\\u000ecient cache explo-\\nration method for a tiled chip multiprocessor. In: High\\nPerformance Computing (HiPC), 2012 19th International\\nConference on, pp. 1{6 (2012). DOI 10.1109\\/HiPC.2012.\\n6507524\\n10. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast\\nand elitist multiobjective genetic algorithm: Nsga-ii. Evo-\\nlutionary Computation, IEEE Transactions on 6(2), 182{\\n197 (2002). DOI 10.1109\\/4235.996017\\n11. Dempsey, I., O'Neill, M., Brabazon, A.: Foundations\\nin Grammatical Evolution for Dynamic Environments, lutionary Computation, IEEE Transactions on 6(2), 182{\\n197 (2002). DOI 10.1109\\/4235.996017\\n11. Dempsey, I., O'Neill, M., Brabazon, A.: Foundations\\nin Grammatical Evolution for Dynamic Environments,\\nStudies in Computational Intelligence , vol. 194. Springer\\n(2009). URL http:\\/\\/www.springer.com\\/engineering\\/\\nbook\\/978-3-642-00313-4\\n12. Edler, J., Hill, M.D.: Dinero iv trace-driven uniproces-\\nsor cache simulator (1998). URL http:\\/\\/pages.cs.wisc.\\nedu\\/ ~markhill\\/DineroIV\\/\\n13. Galv\\u0013 an-L\\u0013 opez, E., Swa\\u000bord, J., O'Neill, M., Brabazon,\\nA.: Evolving a ms. pacman controller using grammat-\\nical evolution. In: C. Chio, S. Cagnoni, C. Cotta,\\nM. Ebner, A. Ekrt, A. Esparcia-Alcazar, C.K. Goh,\\nJ. Merelo, F. Neri, M. Preu, J. Togelius, G. Yannakakis\\n(eds.) Applications of Evolutionary Computation, Lec-\\nture Notes in Computer Science , vol. 6024, pp. 161{\\n170. Springer Berlin Heidelberg (2010). DOI 10.1007\\/\\n978-3-642-12239-2 n17. URL http:\\/\\/dx.doi.org\\/10.\\n1007\\/978-3-642-12239-2_17\\n14. Gordon-Ross, A., Lau, J., Calder, B.: Phase-based cache\\nrecon\\fguration for a highly-con\\fgurable two-level cache\\nhierarchy. In: Proceedings of the 18th ACM Great Lakes\\nsymposium on VLSI, GLSVLSI '08, pp. 379{382. ACM,\\nNew York, NY, USA (2008). DOI 10.1145\\/1366110.\\n1366200. URL http:\\/\\/doi.acm.org\\/10.1145\\/1366110.\\n1366200\\n15. Hennessy, J.L., Patterson, D.A.: Computer Architecture:\\nA Quantitative Approach. Morgan Kaufmann (2011)\\n16. Janapsatya, A., Ignjatovic, A., Parameswaran, S.: Find-\\ning optimal l1 cache con\\fguration for embedded systems.\\nIn: Design Automation, 2006. Asia and South Paci\\fc\\nConference on, p. 6 pp. (2006). DOI 10.1109\\/ASPDAC.\\n2006.1594783\\n17. Koza, J.R., Poli, R.: Introductory tutorials in optimiza-\\ntion, search and decision support (2003)\\n18. Lee, C., Potkonjak, M., Mangione-Smith, W.H.: Medi-\\nabench: a tool for evaluating and synthesizing multi-\\nmedia and communicatons systems. In: Proceedings of\\nthe 30th annual ACM\\/IEEE international symposium\\non Microarchitecture, MICRO 30, pp. 330{335. IEEE\\nComputer Society, Washington, DC, USA (1997). URL\\nhttp:\\/\\/dl.acm.org\\/citation.cfm?id=266800.266832 Optimizing L1 Cache for Embedded Systems through Grammatical Evolution 15\\n19. Mamidipaka, M., Dutt, N.: eCACTI: An enhanced power\\nestimation model for on-chip caches. Tech. Rep. TR-04-\\n28, CECS, UC Irvine (2004)\\n20. Naz, A., Kavi, K., Rezaei, M., Li, W.: Making a case for\\nsplit data caches for embedded applications. SIGARCH\\nComput. Archit. News 34(1), 19{26 (2005). DOI 10.\\n1145\\/1147349.1147355. URL http:\\/\\/doi.acm.org\\/10.\\n1145\\/1147349.1147355\\n21. O'Neill, M., Hemberg, E., Gilligan, C., Bartley, E., Mc-\\nDermott, J., Brabazon, A.: GEVA - grammatical evolu-\\ntion in Java. SIGEVOlution 3(2), 17{22 (2008). DOI\\nhttp:\\/\\/doi.acm.org\\/10.1145\\/1527063.1527066\\n22. O'Neill, M., Ryan, C.: Automatic generation of caching\\nalgorithms. In: Evolutionary Algorithms in Engineering\\nand Computer Science, pp. 127{134. John Wiley & Sons\\n(1999)\\n23. O'Neill, M., Ryan, C.: Grammatical evolution. IEEE\\nTrans. Evolutionary Computation 5(4), 349{358 (2001)\\n24. O'Neill, M., Ryan, C.: Grammatical Evolution: Evolu-\\ntionary Automatic Programming in an Arbitrary Lan-\\nguage. Kluwer Academic Publishers (2003)\\n25. Palesi, M., Givargis, T.: Multi-objective design space ex-\\nploration using genetic algorithms. In: Hardware\\/Soft-\\nware Codesign, 2002. CODES 2002. Proceedings of the\\nTenth International Symposium on, pp. 67{72 (2002).\\nDOI 10.1109\\/CODES.2002.1003603\\n26. Perez, D., Nicolau, M., O'Neill, M., Brabazon, A.: Evolv-\\ning behaviour trees for the mario ai competition using\\ngrammatical evolution. In: C. Chio, S. Cagnoni, C. Cotta,\\nM. Ebner, A. Ekrt, A. Esparcia-Alc\\u0013 azar, J. Merelo,\\nF. Neri, M. Preuss, H. Richter, J. Togelius, G. Yan-\\nnakakis (eds.) Applications of Evolutionary Computa-\\ntion, Lecture Notes in Computer Science , vol. 6624,\\npp. 123{132. Springer Berlin Heidelberg (2011). DOI\\n10.1007\\/978-3-642-20525-5 n13. URL http:\\/\\/dx.doi.\\norg\\/10.1007\\/978-3-642-20525-5_13\\n27. Risco-Mart\\u0013 \\u0010n, J.L., Colmenar, J.M.: Java Evolution-\\nary COmputation library (JECO). Available at:\\nhttps:\\/\\/sourceforge.net\\/projects\\/jeco (2013)\\n28. Risco-Mart\\u0013 \\u0010n, J.L., Colmenar, J.M., Hidalgo, J.I., Lan-\\nchares, J., D\\u0013 \\u0010az, J.: A methodology to automatically op-\\ntimize dynamic memory managers applying grammatical\\nevolution. Journal of Systems and Software 91(0), 109\\n{ 123 (2014). DOI http:\\/\\/dx.doi.org\\/10.1016\\/j.jss.2013.\\n12.044. URL http:\\/\\/www.sciencedirect.com\\/science\\/\\narticle\\/pii\\/S016412121400017X\\n29. Shaker, N., Nicolau, M., Yannakakis, G., Togelius, J.,\\nO'Neill, M.: Evolving levels for super mario bros using\\ngrammatical evolution. In: Computational Intelligence\\nand Games (CIG), 2012 IEEE Conference on, pp. 304{\\n311 (2012). DOI 10.1109\\/CIG.2012.6374170\\n30. Silva-Filho, A., Bastos-Filho, C., Falcao, D., Cordeiro,\\nF., Castro, R.: An optimization mechanism intended for\\ntwo-level cache hierarchy to improve energy and perfor-\\nmance using the nsgaii algorithm. In: Computer Archi-\\ntecture and High Performance Computing, 2008. SBAC-\\nPAD '08. 20th International Symposium on, pp. 19 {26\\n(2008). DOI 10.1109\\/SBAC-PAD.2008.9\\n31. Varma, A., Debes, E., Kozintsev, I., Jacob, B.:\\nInstruction-level power dissipation in the intel xscale\\nembedded microprocessor. In: In SPIEs 17th Annual\\nSymposium on Electronic Imaging Science & Technology\\n(2005)\\n32. Wang, W., Mishra, P.: Dynamic recon\\fguration of two-\\nlevel caches in soft real-time embedded systems. In:\\nVLSI, 2009. ISVLSI '09. IEEE Computer Society An-nual Symposium on, pp. 145{150 (2009). DOI 10.1109\\/\\nISVLSI.2009.22\\n33. Wang, W., Mishra, P., Gordon-Ross, A.: Dynamic cache\\nrecon\\fguration for soft real-time systems. ACM Trans.\\nEmbed. Comput. Syst. 11(2), 28:1{28:31 (2012). DOI\\n10.1145\\/2220336.2220340. URL http:\\/\\/doi.acm.org\\/\\n10.1145\\/2220336.2220340\\n34. Wang, W., Mishra, P., Ranka, S.: Dynamic cache re-\\ncon\\fguration and partitioning for energy optimization in\\nreal-time multi-core systems. In: Proceedings of the 48th\\nDesign Automation Conference, pp. 948{953. ACM, New\\nYork, NY, USA (2011). DOI 10.1145\\/2024724.2024935.\\nURL http:\\/\\/doi.acm.org\\/10.1145\\/2024724.2024935\",\"15\":\" Using a Variational Autoencoder  to Learn Valid Search Spaces of \\nSafely Monitored Autonomous Robots for Last-M ile Delivery \\nPeter J. Bentley \\nDepartment of Computer \\nScience, UCL, Autodesk \\nResearch \\nLondon, United Kingdom  \\n p.bentley@cs.ucl.ac.u k Soo Ling Lim \\n Department of Computer \\nScience, UCL  \\n London, United \\nKingdom  \\n s.lim@cs.ucl.ac.uk  Paolo Arcaini \\n National Institute of \\nInformatics  \\n Tokyo, Japan  \\n arcaini@nii.ac.jp  Fuyuki Ishikawa \\nNational Institute of \\nInformatics  \\n Tokyo, Japan  \\n f-ishikawa@nii.ac.jp  \\nABSTRACT  \\nThe use of autonomous robots for delivery of goods to customers \\nis an exciting new way to provide a reliable and sustainable \\nservice. However, in the real world, autonomous robots still require human supervision for safety reasons. We tackle the real -\\nworld problem of optimizing autonomous robot timings to maximize deliveries, while ensuring that there are never too m any \\nrobots running simultaneously  so that they can be monitored \\nsafely . We assess the use of a recent  hybrid machine- learning -\\noptimization approach COIL (constrained optimization in learned \\nlatent space) and compare it with a baseline  genetic algorithm for \\nthe purposes of exploring variations of this problem. We also investigate new methods for improving the speed and efficiency \\nof COIL . We show that only COIL can find valid solutions where \\nappropriate  numbers of robots run simultaneou sly for all problem \\nvariations tested. We also show that when COIL has learned its latent representation, it can optimize 10% faster than the GA, \\nmaking it a good choice for daily re -optimization of robots where \\ndelivery requests for each day are allocated to robots while \\nmaintaining safe numbers of robots running at once.  \\nCCS CONCEPTS \\n\\u2022 Computing methodologies~Search methodologies  \\u2022 Computing \\nmethodologies~Learning latent representations  \\u2022 C omputing \\nmethodologies~Robotic planning  \\nKEYWORDS \\nVariational autoencoder , autonomous robots, scheduling, learning \\nlatent representations , genetic algorithm  \\n \\n  \\n 1 INTRODUCTION  \\nHome delivery of groceries and other goods is rapidly becoming a \\nmajor new industry worldwide, accelerated by the COVID \\npandemic. The use of automobiles for last -mile delivery (from \\nstore to customer) caus es environmental concerns [1] and in \\ncountries such as Japan  where our problem originates, there may \\nbe a lack of labor. O ur industry partner , a major electronics \\ncorporation, is currently trialing their solution: an automatic \\ndelivery service performed by autonomous robots. Customers \\norder goods which are then collected from a supermarket or drug \\nstore by a robot and delivered to the customer. The robots are \\nautonomous but have human monitors that intervene should a \\nproblem occur, such as a potential collision with a pedestrian , \\nFigure 1. The use of robots for this purpose is in the \\nexperimentation phase, with details such as number of robots and human monitors  still under consideration.  \\n \\n  \\nFigure 1: Home delivery service using low speed robots with \\nhuman operators performing safety monitoring.  Map shows \\nsection  of actual location. P. Bentley  et al.  \\n \\n \\n In this real -world problem there is a clear conflict between \\nfulfilling orders and maintaining  safety. More robots operating \\nsimultaneously will mean more orders could be fulfilled, yet too \\nmany robots operating at once would be dangerous as the human \\noperators can only monitor a limited number of robots. The \\nproblem of how best to schedule  different numbers of robots such \\nthat maximum orders are delivered safely is therefore difficult.  It \\nis also a problem that must be solved repeatedly for every new set of customer orders.  This problem is likely to become increasingly \\nrelevant as autonomous robot deliveries become more prevalent, \\nyet safety is rarely considered in the literature.  \\nWe address this problem by using a hybrid machine learning  \\nand evolutionary computation approach  [2, 3]. This method  uses a \\nvariational autoencoder to learn the valid  regions of the search \\nspace, i.e., the areas of the search space where safe numbers of robots are scheduled to work simultaneously. A genetic algorithm \\n(GA) is then used to search in this learned latent space to find \\noptimal timings for robots suc h that the maximum number of \\norders can be delivered safely. We compare this approach with a \\nbaseline  genetic algorithm optimizing robot timings  while solving \\nthe constraint, without the use of machine learning. We investigate how well both approaches  enable us to examine \\ndifferent design variations and we assess suitability for long term \\nuse once the design variables are determined . \\nThe contributions of this work can be summarized as follows:  \\n\\u2022 The first study to explore real-world autonomous robot \\nrouting  while satisfying a safety monitoring requirement. \\n\\u2022 Adapting the hybrid ML -EA approach (COIL)  [2, 3] to a \\nreal-world problem for the first time.  \\n\\u2022 Evaluation of COIL by comparing to  a baseline GA to \\nexplore problem variations,  showing better  solution \\nquality and optimization speed for daily robot scheduling.  \\n\\u2022 COIL performance is enhanced on th e problem through  \\nreduction of the training set and number of latent \\nvariables.  \\nThe rest of the paper is organized  as follows: Section 2 \\nprovides a literature review of relevant research, Section 3 describes the method, Section  4 describes the experiments , results  \\nand discusses findings for the problem . We conclude in Section 5. \\n2 BACKGROUND  \\n2.1 Autonomous Robots for Home Delivery \\nAutonomous delivery robots have the potential to significantly reduce energy consumption and CO\\n2 emissions in urban areas  [1]. \\nThere are several examples of solutions. Mercedes -Benz Vans \\npartnered with Starship Technologies to propose the concept of \\nautonomous del ivery robots launched from trucks [4]. The truck \\ncan replenish robots at decentralized depots to launch more  until \\nall its customers are supplied. Boysen et al. [4] developed  \\nscheduling procedures which determine the truck route, such that \\nthe weighted number of late custom er deliveries is minimized.  \\nYu et al. [5] considered a variant of this problem and  proposed \\nan LV -SAV (large vehicle - small full y automated ground vehicles) model where multiple LVs cooperate with their \\nassociated SAVs. Safety was one of their concerns, and having \\nslow moving SAVs is safer in urban delivery, and was considered \\nto be safer than using drones in the HVDRP (hybrid vehi cle-drone \\nrouting problem) proposed by Karak and Abdelghany [6]. \\nChen et al. [7] introduced another variant called  a new vehicle \\nrouting problem with time windows and delivery robots (VRPTWDR). They showed that c onsiderable operational time \\nsavings can be achieved by dispatching delivery robots to serve \\nnearby customers while a driver is also serving a customer . \\nSince delivery robots share sidewalks with pedestrians, Bakach \\net al. investigated the option to ch oose paths for them that avoid \\nzones with high pedestrian  density  [8]. They  considered  path Since delivery robots share sidewalks with pedestrians, Bakach \\net al. investigated the option to ch oose paths for them that avoid \\nzones with high pedestrian  density  [8]. They  considered  path \\nflexibility given the presence of zones with varying pedestrian \\nlevel of service .  \\nThe study of vehicle routing problems (VRP) continues to \\ndevelop, with  many  variants includ ing real-life constraints and \\nassumptions, which makes the models more realistic and the \\napproaches more applicable in practice [9]. Autonomous delivery \\nvehicles introduce  safety concerns  whilst driving autonomously \\non the public road networks and performance risk s while \\ndelivering  parcels (e.g., risk of malfunctioning of the technology) \\n[10]. A study of  public perceptions and acceptance of autonomous \\nvehicles found that people expressed high concerns around safety, \\ncyber -security, legal liability, and regulat ion issues  [11]. \\nIn this work we focus on safety  and a key component of the \\nsolution created by our i ndustry collaborator. New to this field, f or \\nour problem, human operators monitor the robots so that they can \\nmanually operate the m when unexpected obstacles are detected , \\nor they can talk with the users.  Similar monitoring will be \\nrequired for many other real -world examples, so methods to solve \\nthis problem are likely to be widely useful. \\n2.2 Evolving Latent Variables  \\nAutoencoder s [12] are neural network s first used for \\ndimensionality reduction . Since their inception they have become \\npopular for  learning generative models of the data. An \\nautoencoder co mprises  two parts - an encoder \\ud835\\udc5d, which maps the \\nobservations \\ud835\\udc65 to a (lower dimensional) embedding space \\ud835\\udc67 and a \\ndecoder \\ud835\\udc5e, which  maps the embeddings back to the original \\nobservation space.  When trained, the autoencoder minimizes the \\nerror in reconstruction of output compared to input.  The \\nvariational autoencoder (VAE) is a probabilistic autoencoder [13, \\n14]. Instead of encoding an observation as a single point, VAEs \\nencode it as a distribution over the latent space.  \\nRecent work in the field of evolutionary computation makes \\nuse of VAE s to learn representations and then evolve using those \\nrepresentations, searching in the latent space  (LVE \\u2013 latent \\nvariable evolution). Game levels  [15], fingerprints to foil security \\nsystems [16], and program synthesis  [17] include some of the \\napplications. Researchers also make use of quality -diversity \\napproaches  [18] with LVE  to improve optimi zation in high \\ndimensional problems, for example DDE -Elites  [19] which uses a \\ncombination of the direct encoding with a learned latent encoding to accelerate search. Learn ing Valid Search Spaces of Autonomous Robots   \\n \\n Most recently, COIL (constrained optimization in learned \\nlatent space) [2] and SOLVE (search space optimization with \\nlatent variable evolution) [3] present an LVE approach to learn a \\nlatent rep resentation that is biased towards solutions that satisfy a \\nconstraint or additional objective. This is very relevant to the real -\\nworld problem we investigate in our work. However, while COIL  \\n[2] and SOLVE  [3] have been demonstrated on simple constraints \\nand benchmark functions, the technique has not been tested on a \\nreal-world problem. In this work we app ly the concepts in COIL \\nfor the first time to the problem of scheduling a safe number of \\nautonomous robots for home delivery of a maximum number of \\norders by customers .  \\n3 METHOD  \\nWe base our approach on the system under development  by our \\nindustry partner. Their automated delivery service will enable \\ncustomers to order goods from a local store which are delivered \\nby autonomous robots. Customers submit requests through  an app. \\nEach request is scheduled centrally with a robot allocate d to serve \\nit. Robots are monitored by human operator s who can intervene if \\nthe robot encounters problems, helping to initiate the appropriate response in the robot or talk to users , Figure 1. \\nWe focus on  two stages to this problem: \\n1. Investigate possible design variations, with different numbers of robots, human operators, request durations . \\n2. On completion of the first stage, provide an efficient method for daily optimizatio n of robot timings.  \\nFor the first stage,  we investigate optimization methods \\ncapable of exploring design variations . The algorithms must \\nsuccessfully  schedul e the robots to maximize delivery of requests \\nwhile ensuring there are never too many robots run ning \\nsimultaneously, such that the human operators can safely monitor \\nthem. For the second stage , with the design variables now \\ndetermined,  we investigate the fastest method of scheduling \\nrobots to meet the human operator constraint, daily , because  this \\ndecision- making process reoccurs every day with new customer \\nrequests.  \\nWe formulate the problem as a constrained optimization \\nproblem.  Given a randomly generated set Reqs  comprising Rq \\ncustomer requests, each with duration Rqdur: \\n Reqs  = {Rqdur\\n1, .., Rqdur Rq} \\n \\nwhere Rqdur j = random[ 60..dr] and given a set of robots Rbts \\ncomprising Rb pairs of robot starting time Rst and running time  \\nRrt: \\n \\nRbts = {Rst1, Rrt1, .., R stRb, RrtRb} \\n \\nallocate each request  Rqdur j in order from j = 1 to Rq (i.e., on a \\nfirst-come, first -served basis) to the most appropriate robot using \\na simple scheduler created for this work , Algorithm 1. (Should \\nthis solution be commissioned for use on the real robots, this \\nalgorithm will be replaced with the industrial partner\\u2019s internal \\nscheduler.)  Algorithm 1: First -come first -served scheduler used to \\nallocate customer requests to robots. totalRm  count s the total \\nnumber of requests met out of a maximum of Rq. \\ntotalRm  = 0 \\nfor i = 1 to Rb \\n    remaining_dur i = Rrti\\u00d710 \\nendfor  \\nfor j = 1 to Rq \\n    if there exists a remaining_dur k such that  \\n        0 <= ( remaining_dur k  - Rqdur j ) < 10 and \\n        there are no other  closer matches then \\n            remaining_dur k  \\u2212= Rqdur j \\n            totalRm  ++ \\n            break \\n    else find largest remaining_dur k such that  \\n        ( remaining_dur k  - Rqdur j ) > 0 then \\n        remaining_dur k  \\u2212= Rqdur  \\n        totalRm  ++ \\n            break  \\nendfor  \\n The task is then to use an optimizer to find a suitable set of \\nRbts such that the number of requests fulfilled by the robots is \\nmaximized by efficiently fitting robot start times and running \\ntimes to the set Reqs while ensuring that the number of robots \\nrunning simultaneously at any point in time remains below the maximum robot threshold RT. \\nWe assume that robots operate 12 hours in a day (e.g., 7am to running simultaneously at any point in time remains below the maximum robot threshold RT. \\nWe assume that robots operate 12 hours in a day (e.g., 7am to \\n7pm)  and they run at a fixed speed . The minimum running time \\nfor a robot is one hour. Robots only operate for a single period per day. We divide the 12 -hour period into 10 -minute time slots such \\nthat every start time and duration for each robot may be defined by an integer from 0 to 66, wh ere Rst\\ni = 0 corresponds to a robot \\nstart time of 7am and Rrti = 0 corresponds to the minimum \\nduration of 1 hour. The set of requests Reqs is built from Rq \\nrandom integers from 60 to 180 (i.e., customer requests take \\nrobots anything from 60 to 180 minutes  to fulfil). Customer \\nrequests must be fulfilled within the 7am to 7pm working period \\nof robots and goods may be delivered at any point during that \\nperiod for all requests . By default Rq = 120 to present a \\nsignificant challenge for the robots; for experimen ts with shorter \\nrequest durations we double Rq to 240.  \\nIn this work we measure the success of solutions for the \\nproblem using the objective function  to be maximised : \\n\\ud835\\udc53(\\ud835\\udc65\\u0305)=\\ud835\\udc61\\ud835\\udc5c\\ud835\\udc61\\ud835\\udc4e\\ud835\\udc59\\ud835\\udc45\\ud835\\udc5a  \\nwhere totalRm  is the total number of requests from Reqs  allocated \\nby the scheduler to the current set of robots Rbts. \\nWe use a single constraint:  \\n\\ud835\\udc61\\ud835\\udc5c\\ud835\\udc61\\ud835\\udc4e\\ud835\\udc59\\ud835\\udc45\\ud835\\udc4f\\ud835\\udc61\\ud835\\udc60 \\u2264\\ud835\\udc45\\ud835\\udc47 \\nwhere totalRbts  is the total number of robots working \\nsimultaneously in any one timeslot.  As a huma n operator must \\nmonitor each robot in this problem, RT directly correlates  with the \\nnumber of human operators. Figure 2 illustrates the calculation of  \\ntotalRbts. P. Bentley  et al.  \\n \\n \\n Rbts  0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 100-110 110-120 \\n1 0 0 0 1 1 1 1 0 0 0 0 0 \\n2 0 0 0 0 1 1 1 1 1 1 1 1 \\n3 1 1 1 1 1 0 0 0 0 0 0 0 \\n4 1 1 1 1 1 1 1 1 1 1 1 1 \\n+ 2 2 2 3 4 3 3 2 2 2 2 2 \\nFigure 2: Example calculation showing  totalRbts  = 4, for four \\nrobots in a two -hour period (ignoring the minimum duration \\nof 60 for this example) . If the threshold RT = 2 then the \\nconstraint would not be met on 4 out of the 12 possible \\ntimeslots , marked in bold. When totalRbts <= RT  for every \\ntimeslot for all robots, the constraint is met fully. In this case \\nthe constraint score being minimized is 4 - 2 = 2.  With the \\nconstraint not satisfied, this is not a valid solution.  \\n3.1 Optimization with Genetic Algorithm \\nOur baseline approach to address this problem is to use a genetic algorithm to evolve start times and durations for a set of robots \\nsuch that the constraint is met and the objective is  maximized for \\na given set of customer requests Reqs . \\nOur search variables \\n\\ud835\\udc65\\u0305 comprise:  \\n\\ud835\\udc65\\u0305=[\\ud835\\udc65!\\\"#,\\ud835\\udc65$\\\"#,..,\\ud835\\udc65!\\\"%&,\\ud835\\udc65$\\\"%&] \\nwhere we map \\ud835\\udc65!\\\"',\\ud835\\udc65$\\\"' to Rsti, Rrti respectively  for every i. Should \\nany i exist such that \\ud835\\udc65!\\\"'+\\ud835\\udc65$\\\"'>65 then the corresponding values \\nof Rrti is corrected such that the robot ceases its shift at the end of \\nthe working day.  Values are corrected during this mapping stage \\nrather than modifying  the evolving parameters as this has been \\nshown to be more conducive to search [20]. During evolution the \\nfitness and constraint is calculated as described above. This is the \\nworst -case scenario for the constraint as it assumes that every \\nrobot will run for its maximum permitted time. In reality, after requests have been scheduled some robots may run for shorter \\ndurations depending on which requests were allocated. Thus , for \\naccuracy, o ur reported results for all experiments first updates  \\nrobot durations  according to the scheduler  in Algorithm 1 : \\n\\u2200\\ud835\\udc56\\u2208{1..\\ud835\\udc45\\ud835\\udc4f}\\t\\ud835\\udc45\\ud835\\udc5f\\ud835\\udc61'\\u2212=E\\ud835\\udc5f\\ud835\\udc52\\ud835\\udc5a\\ud835\\udc4e\\ud835\\udc56\\ud835\\udc5b\\ud835\\udc56\\ud835\\udc5b \\ud835\\udc54($'\\/10K \\nand then measure s the actual number of scheduled robots that ran \\nsimultaneously, as shown in Figure 2. \\nWe use a constrained optimization algorithm [21] using DEAP  \\n[22] with tournament fitness (see [2] for the algorithm) to enable \\nthe equal contribution of constraint and objective to evolution.  \\nThis is a widely used approach in constraint -handling used by \\nnumerous other researchers [23]. In this method, fitness of each \\nindividual is the number of times its separate criteria win a series \\nof tournaments held between randomly selected subgroups  of 3 \\nindividuals. Within each tournament, the fitness  of an individual \\nis increased if its solution is better (lower) for the number of times \\nthe constraint is not satisfied, and also when its solutio n is better \\n(higher) for the objective  \\ud835\\udc53(\\ud835\\udc65\\u0305) compared to the others in the \\ntournament.  We use a Gaussian  Integer creep mutation. \\n   \\nFigure 3: Constrained optimization in learned latent space \\n(COIL) applied to real -world autonomous robot scheduling \\nproblem.  \\n3.2 Optimization with COIL  \\nWe are tackling a challenging constrained optimization problem, so our proposed approach to be compared with the baseline is \\nCOIL  [2] \\u2013 a new hybrid machine learning and optimization \\nmethod  designed to learn valid regions of the search space and \\nimprove the ability of optimizers to find valid solutions. COIL has \\nthe advantage that , unlike other constrained optimizers, it does not \\nrely on complex operators, weight -tuning, or specialized \\nexpertise. So far COIL has been tested only on simple test \\nfunctions for constrained optimization ; this is the first research to \\napply this approach to a real -world problem.  \\nCOIL operates using three steps  (Figure 3): \\n1. Dataset generation from constraint  \\n2. Learning new latent represen tation from dataset \\n3. Optimization using latent representation  using \\nconstraint and objective  \\nFor the first step, we use a simple genetic algorithm (using 2. Learning new latent represen tation from dataset \\n3. Optimization using latent representation  using \\nconstraint and objective  \\nFor the first step, we use a simple genetic algorithm (using \\nDEAP) with a population size of 200 for running for at most 200 \\ngenerations. Our search variables \\n\\ud835\\udc65\\u0305 are exactly as described for \\nthe baseline GA.  The objective function for this data- generation \\nGA is minimization of the number of times the constraint  is not \\nsatisfied , with any solution that satisfies the constraint added to \\nthe dataset, and the GA restarting. The threshold RT, which \\ndetermines how many robots may work simultaneously , is varied \\nduring experiments. Solutions that specify robot durations that \\nfinish beyo nd the 12- hour working period  are corrected as \\ndescribed in 3.1 before being normalized and added to the dataset.  \\nTo encourage the dataset -generating GA not to \\u201ccheat\\u201d and only \\nprovide solutions with minimal robot running times (the simplest \\nway to satisfy  the constraint), we also add a selective pressure \\ntowards longer robot durations by adding 100 \\u2211 \\ud835\\udc65$\\\"' )%\\n#\\u2044  to the \\nfitness term. We run this GA repeatedly until DS vectors  are \\ngenerated. Each vector represents a valid set of robot start times \\nand duration s such that the constraint is satisfied. The overall \\nobjective to find robot times and durations that enable the most \\ncustomer requests is not used at this point.  \\nIn the second step, we provide  the dataset to a simple \\nvariational autoencoder [2] with 4 linear layers , a prior  of N(0,I), \\nKLD = 1, for 200 epochs using the Adam optimizer with a \\nlearning rate of 0.001. We run the VAE 10 times and choose the \\nlearned model with the lowest error. This is our learned latent \\nrepresentation , with latent variables:  \\n\\ud835\\udc67\\u0305=[\\ud835\\udc67#,\\ud835\\udc67*,..,\\ud835\\udc67*\\u00d7,-.\\/0] Learn ing Valid Search Spaces of Autonomous Robots   \\n \\n In the third step, we use the same GA as described in section \\n3.1, to ensure a completely fair comparison. For this GA, the \\nsearch variables \\ud835\\udc67\\u0305 are encoded as real -valued variables with \\nranges between -2.0 and 2.0, and a Gaussian creep mutation. We \\nconvert \\ud835\\udc67\\u0305 into \\ud835\\udc65\\u0305 by using the learned VAE model  to express the \\nvalues , and then  perform a scalar inverse transform, \\nunnormalization, and conversion to intege r to convert the VAE \\noutput into the desired 0..66 range.  We then map \\ud835\\udc65!\\\"',\\ud835\\udc65$\\\"' to Rsti, \\nRrti respectively  for every i as described in section 3.1  with the \\nsame robot duration updates performed according to the scheduler  \\nand use the identical tournament fitness approach to measure \\nfitness for the objective and constraint. \\n4 EXPERIMENTS \\n4.1 Experiments  \\nWe perform experiments to investigate the following research \\nquestions, which reflect the two stages to this problem : \\n \\nRQ1: Does COIL help us explore valid design variations more \\neffectively than the baseline GA?  \\nWe explore this question in several experiments . \\nE1.1: Can the baseline GA and  COIL help us explore valid \\nsolutions ? Before we can explore different design variations we \\nneed to check that we can solve this problem at all. Experiment 1 \\ntackles this by directly comparing the output of the baseline GA with COIL.  Following the training of the VAE within COIL, the \\nsame learn ed model is used in all repeated runs. We use a small \\npopulation size of 20 for just 50 generations following  [2, 3]. Both \\nGAs are run 100 times and mean results are shown.  A new set of \\nrequests Reqs  is randomly generated for every run.  \\nE1.2: Can the GA and COIL explore  valid solutions varying \\nRT? In this experiment we explore variants  of the problem by \\nvarying RT to the values: 10, 15, 20. This explores solutions \\nwhere we permit more robots to run simultaneously (at the cost of \\nrequiring more human monitoring.)  \\nE1.3: Can the GA and COIL explore  valid solutions varying \\nRb? In the second problem variant, we fix RT to 10 and vary the \\nnumber of robots Rb to the values: 2 0, 25, 30.  This explores \\nsolutions where we have fewer robots overall, with at most 10 running simultaneously.  \\nE1.4: Can the GA and COIL explore  valid solutions varying \\ndr? In our final set of problem variants, w e vary the duration of \\nrequests; instead of random from 60 to 180, we try 60 to dr, where \\ndr \\n\\u2208 {60, 80, ..,  360}.  \\nFor all RQs, parameters other than those being varied remain \\nunchanged. COIL is run 100 times for each setting, with a new set Reqs  of random requests each run.  \\n \\nRQ2: Is COIL an efficient algorithm for daily optimization?  \\nOnce the company decides on the number of robots and operators to use, we arrive at the second stage. Here we investigate the \\nsuitability of COIL for use as a daily optimizer, investigating \\nwhether the method can be tuned with this goal in mind.  E2.1: Can we use smaller  dataset size s for COIL without \\naffect ing its ability to generate usef ul latent representations?  We \\nneed a fast way of optimizing robot schedules every day for the new set of requests, while always meeting the constraint. O ne \\ndrawback of COIL is the need to generate a dataset first  and train \\nthe VAE. Although this is a one -off, offline computation, it is still \\nsignificant . This experiment  addresses this by varying the dataset \\nsize DS to four different values: 2500, 5000, 7500, 10000. All \\nother parameters remain unchanged. COIL is run 100 times (with \\na new set Reqs  of random requests each run) and mean results are \\nshown. We also measure the difference in data generation and \\ntraining times and the error rate of the VAE.  \\nE2.2: Can we imp rove the performance of COIL by reducing \\nthe number of latent variables?  Here we investigate an idea for \\nimproving the performance of COIL.  While COIL was described E2.2: Can we imp rove the performance of COIL by reducing \\nthe number of latent variables?  Here we investigate an idea for \\nimproving the performance of COIL.  While COIL was described \\nas a method to improve the search space while keeping the number of latent variables the sa me as the number of input \\nvariables, other work has shown that VAEs may be able to provide the additional benefit of reducing the search  space size \\n[24]. We investigate this question with the number of robots Rb = \\n30 (giving us 60 problem variables) by varying maxlv  to the \\nvalues: 5, 10, 15, 20, 25, 30 (i.e., 10 to 60 latent variables) . All \\nother parameters remain unchanged.  COIL is run 100 times for \\neach setting, with a new set Reqs  of random requests each run.  \\nTable 1 summarizes the parameters investigated in each \\nexperiment. All processing was performed on a MacBook Air \\n2020 M1 with 16Gb memory. All code was implemented in \\nPython and is available\\n1. \\n \\nTable 1. Parameters for the experiments  \\nExp DS maxlv  RT Rb dr \\nE1.1 10000 30 10 30 180 \\nE1.2 10000 30 10,15,20  30 180 \\nE1.3  10000 30 10 20,25,30  180 \\nE1.4  10000 30 10 30 60..360  \\nE2.1 2500..10000  30 10 30 180 \\nE2.2  10000 5..30  10 30 180 \\n \\n4.2 Results  and Analysis \\nIn this section we describe the results for  research question RQ1 \\n(experiments E1.1, E1.2, E1.3 and E1.4) and for RQ2 (experiments E2.1 and E2.2).  \\n Table 2. E1. 1: COIL vs GA.  Better results in bold.  \\n COIL  GA \\navg objective  (stdv) 66.43 (6.80) 93.6 (5.55)  \\nmin objective 51 76 \\nmax objective  77 107 \\navg constraint  (stdv) 23.93 ( 8.43) 44.7 (3.89) \\nmin constraint  0 35 \\nmax constraint  30 52 \\n \\n1 GitHub link temporarily removed for purposes of paper anonymization. P. Bentley  et al.  \\n \\n \\n  \\nFigure 4: Example s ingle best run baseline GA. \\n \\n \\nFigure 5: Example best run COIL . \\n \\nE1.1: Can the GA and COIL help us explore valid solutions ? \\nIn this experiment we compare the output of the baseline GA with \\nCOIL to check that both can find valid solutions for our default \\nparameter values. Table 2 summarizes the results. The results \\nshow that only  COIL is able to provide valid solutions for this \\nvariant of the problem. COIL provides superior results in terms of constraint satisfaction, with the best results being a perfect zero, \\nand average of 23.9, compared t o the baseline GA best of 35 \\n(worse than the worst score of COIL) and average of 44.7. The scores for the objective shows that GA allocates more tasks on \\naverage than COIL, managing 107 at best compared to COIL\\u2019s much lower 77. However, the scores are mean ingless when the \\nconstraint is broken \\u2013 the baseline GA cheats by not meeting the \\nconstraint in order to allocate more requests, and such solutions \\nare not viable or useful.  \\nExamining a single best run for the baseline GA and COIL, \\nFigure 4 and Figure 5 show the difference in evolution. The GA \\nattempts to improve the constraint while keeping request \\nallocation constant but never achieves a single valid solution \\u2013 \\nshowing a failure of this algorithm to tackle the problem effectively. In contrast COIL start s better and optimizes the \\nconstraint to zero very rapidly while keeping the number of requests allocated constant. To assess whether the GA could be \\ncoerced into focusing more on the constraint, we increased the \\nweighting by 10 times in the tournament sel ection algorithm for the constraint; we also tried much larger population sizes and number of generations. The results were barely changed and the \\nGA still failed to find any solutions that satisfied the constraint. Figure 4 and Figure 5 (grey line) also show how effectively \\n\\ud835\\udc65\\u0305 is \\nbeing optimized, with both algorithms ensuring that  91% of the \\nrobots\\u2019 time is being used.  While COIL does not find perfect \\nsolutions every time for this difficult problem (unlike findings of [2, 3]) it is clear that its learned latent representation is biased \\ntowards solutions  that satisfy the constraint more, with Figure 6 \\ntop left showing that the baseline GA results (orange cir cles) are \\nall worse than the COIL results (blue circles).  \\n \\nE1.2 : Number of simultaneously running robots  RT \\nWhen we increase the value of RT, we make the constraint easier \\nas more robots are permitted to run simultaneously. The results show that in all problem variants, COIL successfully finds valid \\nsolutions to the problem. However, as the optimization problem becomes easier, the baseline GA begins to function more usefully \\n(Figure 6 top left). When RT = 10, COIL performs better than the \\nbaseline GA. When RT = 15, COIL is able to find more solutions \\nthat satisfy the constraints, while the baseline GA still finds none. But when RT = 20 (the problem is no longer diffic ult) there is no \\nlonger any significant difference between the GA and COIL, with COIL finding slightly more valid solutions, but the GA finding a \\nfew solutions that satisfy the constraint and fulfil the objective \\nslightly better.  \\n \\nE1.3 : Total n umber of robots  Rb \\nSimilar to the previous experiment, when we reduce the value of Rb, we make the constraint easier as fewer robots in total are \\nrunning, so fewer are likely to run simultaneously. However, with RT fixed we do not see improvements in objective scores (keeping \\nthe number of simultaneously running robots constant, limits the number of requests that can be served). When reducing Rb from \\n30 to 25, the baseline GA comes closer to finding valid solutions. When Rb = 20, the GA finds several valid solutions. H owever, for \\nall values tested, COIL always finds valid solutions to the \\nproblem, becoming more consistent as the problem is easier \\n(Figure 6 top right).  \\n E1.4 : Changi ng request durations dr all values tested, COIL always finds valid solutions to the \\nproblem, becoming more consistent as the problem is easier \\n(Figure 6 top right).  \\n E1.4 : Changi ng request durations dr \\nIf we alter the request durations (increasing the number of requests from 120 to 240 to make up for smaller durations), we change the difficulty of the problem in a new way: shorter tasks \\nare easier to allocate while longer tasks are more difficult. Figure \\n7 shows how the average number of constraints broken slightly \\nimproves (lowers) as the request duration increases for COIL, \\nsuggesting that COIL scales better to gre ater problem difficulty. \\nAs expected, the number of requests that can be allocated falls as the task durations increase, with the baseline GA always allocating more (because it cheats by not satisfying the \\nconstraint). Figure 6: Best result at final generation for 100 runs  for each algorithm , plotted with objective score against constraint . Lower \\nmeans constraint is better satisfied, more to the right means more tasks are allocated. Only solutions on y = 0 satisfy the c onstraint \\nfully and thus are valid. Top left: E1.1 Comparing GA and COIL and E1.2 Varying RT. Top right: E. 1.3 Varying Rb. Bottom left: \\nE2.1 Varying  COIL  DS. Bottom right: E2.2 Varying  COIL  maxlv . \\n \\n \\nFigure 7: E1.4: Altering maximum random request durations.  \\n \\nE2.1: Reducing dataset size DS \\nThe previous  experiments showed that COIL clearly outperforms \\nthe baseline GA for this real -world problem  \\u2013 only COIL has \\npermitted us to explore valid solutions for all design variations \\ntested . However, there is a difference in computation time \\nrequired. When evolving the solutions for E1.1, the baseline GA \\ntook 1 .89 minutes to complete 100 runs. COIL took a slightly \\nshorter 1.81  minutes \\u2013 despite the additional need to express the \\nevolving latent variables into the problem variables using the VAE. However, COI L also required a one -off pre -computation to generate valid data and use the VAE to learn the latent \\nrepresentation. This computation was substantial, requiring 25.80 \\nhours  to generate 10,000 datapoints, and 8.95 minutes for the \\nVAE to learn (running 10 ti mes and choosing the best learned \\nmodel, which had loss of 0.9114).  These times reduced when the \\nproblem was easier, for example in E1.2, 10,000 valid points were generated in just 19.28 minutes when RT = 15, and just 8.08 \\nminutes when RT = 20.  \\nIn the next experiment, we investigate the effects of the dataset \\nsize to see if COIL can still achieve acceptable results when the VAE is trained with smaller datasets.  We achieved this by simply \\nrunning the data generator each time with DS = 2500, 5000, 7500, \\nand 10000. The datasets were checked to see if they contained any redundancy through duplication; analysis revealed that every \\npoint in each dataset is unique and not repeated.  \\nFigure 8 shows the differences in computation time vs the \\nVAE loss , indicating that the dataset of size 5000 has the smallest \\nloss; detailed results from COIL for the constraint and objective \\nfor each dataset size  are provided in Supplementary Materials . \\nWhile the best average constraint satisfaction is achieved using \\nthe largest dataset, performance is still relatively unaffected for \\neven the smallest size. Figure 6 (bottom left) shows that valid \\nsolutions on y = 0 are generated by all dataset sizes, and all also \\nhave a similar trend in the solution space. P. Bentley  et al.  \\n \\n \\n  \\nE2.2 : Modifying num ber of  latent variables  maxlv  \\nFinally we investigate whether we can improve the performance \\nof COIL in terms of the quality its solutions. Applying COIL with \\ndifferent numbers of latent variables shows an important effect on \\nthe distribution of solutions, Figure 6 (bottom right) and \\nSupplementary Materials . Reducing the number of latent variables \\ngenerally improves the ability of COIL to find solutions that satisfy the constraint.  Fewer latent vari ables mean faster training \\ntime for the VAE, and faster optimization for the GA using the smaller latent representations. But as the number of latent \\nvariables decreases to 10, the VAE loss becomes worse ( Figure \\n9). The smaller search space also appears to impact the objective, \\nwith fewer tasks being allocated.  \\nIn contrast, increasing to 50 or more latent variables appears to \\nprovide an unwelcome bias away from valid so lutions on y = 0. \\nFor this problem, there appears to be a \\u201csweet spot\\u201d between \\nmaxlv=15 and maxlv=25 (30 and 50 latent variables  respectively), \\nwhere solutions with high number of request allocations and perfect constraint scores are generated ( Figure 6 bottom right). \\nFor our problem it appears that having fewer latent variables \\ncompared to problem variables is advantageous.  \\n \\n \\nFigure 8: Computation time and VAE loss  for E2.1 . Bar chart \\nusing left y-axis: Data generation (hours), VAE training \\n(minutes), blue line using right y-axis: VAE loss. \\n \\n \\nFigure 9: Computation time and VAE loss fo r E2.2 . Bar chart \\nusing left y-axis: VAE training (minutes), blue line using right \\ny-axis: VAE loss. \\n 4.3 Discussion of Findings Relating to Problem  \\nCOIL has enabled a useful exploration of the design space for this \\nreal-world problem. Our findings indicated that the amount of \\navailable time spent by robots delivering requests rarely changes (usually at around 90%). This appears to be the best use of the \\nrobots\\u2019 time as determined by the scheduler , and both optimizers \\nare always able to find appropriate start times and durations for \\nthe autonomous robots to reach this efficiency. However, results \\nfrom the baseline GA were often unusable as the safety constraint \\nwas not met. COIL provided usable results for all setups. The experiments also showed that when customer requests were likely \\nto take less time, more would be delivered successfully by the robots. The safety constraint was slightly more likely to be \\nsatisfied by COIL for longer duration requests; for the baseline \\nGA there was no difference.  \\nOver all, COIL\\u2019s exploration of the problem space  indicate s \\nthat when we keep the total number of robots Rb constant but \\nallow more to run simultaneously (i.e., more human monitors are employed), the number of customer requests tha t can be satisfied \\nwill increa se. In contrast, if we keep the number of simultaneously \\nrunning robots RT constant and reduce the overall numbers of \\nrobots that are running, there is little change to the number of \\ncustomer requests met. Thus, with the fixed relationship between \\nthe numb er of simultaneously running robots and the number of \\nhuman monitors, there appears to be a direct correlation between the number of human monitors and customer satisfaction.   \\n5 CONCLUSIONS \\nThe problem of scheduling autonomous robots safely such that sufficient human monitors are available is challenging , and \\nrelatively  unexplored in the literature . Our work shows that a  \\nstandard genetic algorithm using a well-established method  for \\nconstrained optimization (the baseline GA ) was unable to find \\nsolutions tha t satisfied the constraint except for variations where \\nthe problem was relatively easy. In contrast a new ML -EA hybrid \\napproach (constrained optimization in learned latent space: COIL ) \\nwas able to find valid solutions for all problem variants, enabling \\nus to perform a useful investigation of the effects of each approach (constrained optimization in learned latent space: COIL ) \\nwas able to find valid solutions for all problem variants, enabling \\nus to perform a useful investigation of the effects of each \\nparameter on solutions. This work has also shown for the first \\ntime that the data -generation and training stage of COIL can be \\nimproved by reducing the dataset size and number of latent \\nvariables. This means that once suitable parameters are selected, \\nand the initial one -off training is performed  to learn the latent \\nrepresentation , COIL is able to optimize new schedules rapidly \\nand reliably. Indeed, with these improvements, COIL is 10% \\nfaster to run than  the baseline GA. COIL is thus an ideal choice \\nfor an everyday optimizer for this problem.  \\nFuture work will examine further problem variables of interest \\nto our industrial partner such as robot speed and will integrate \\nwith other schedulers. Other  improvements to COIL will be \\nexamined, such as the use of more advanced VAEs and optimizers such as MAP -Elites for the data generation and optimization \\nstages, to further improve speed and efficiency. Learn ing Valid Search Spaces of Autonomous Robots   \\n \\n REFERENCES \\n[1] M. Figliozzi and D. Jennings, \\\"Autonomous delivery robots and their potential \\nimpacts on urban freight energy consumption and emissions,\\\" Transportation \\nresearch procedia, vol. 46, pp. 21 -28, 2020. \\n[2] P. J. Bentley, S. L. Lim, A. Gaier, and L. Tran, \\\"COIL: Constrained \\nOptimi zation in Learned Latent Space --Learning Representations for Valid \\nSolutions,\\\" in ACM Genetic and Evolutionary Computation Conference \\n(GECCO'22) Companion , 2022, pp. 1870 \\u20131877.  \\n[3] P. J. Bentley, S. L. Lim, A. Gaier, and L. Tran, \\\"Evolving through the look ing \\nglass: Learning improved search spaces with variational autoencoders,\\\" in \\nInternational Conference on Parallel Problem Solving from Nature (PPSN) , \\n2022, pp. 371 -384. \\n[4] N. Boysen, S. Schwerdfeger, and F. Weidinger, \\\"Scheduling last -mile deliveries \\nwith truck -based autonomous robots,\\\" European Journal of Operational \\nResearch, vol. 271, no. 3, pp. 1085 -1099, 2018.  \\n[5] S. Yu, J. Puchinger, and S. Sun, \\\"Two -echelon urban deli veries using \\nautonomous vehicles,\\\" Transportation Research Part E: Logistics and \\nTransportation Review, vol. 141, p. 102018, 2020.  \\n[6] A. Karak and K. Abdelghany, \\\"The hybrid vehicle -drone routing problem for \\npick-up and delivery services,\\\" Transportation Research Part C: Emerging \\nTechnologies, vol. 102, pp. 427 -449, 2019.  \\n[7] C. Chen, E. Demir, Y. Huang, and R. Qiu, \\\"The adoption of self -driving \\ndelivery robots in last mile logistics,\\\" Transportation research part E: logistics \\nand transportation review, vol. 146, p. 102214, 2021.  \\n[8] I. Bakach, A. M. Campbell, and J. F. Ehmke, \\\"Robot -Based Last -Mile \\nDeliveries With Pedestrian Zones,\\\" Frontiers in Future Transportation, p. 32, \\n2022.  \\n[9] K. Braekers, K. Ramaekers, and I. Van Nieuwenhuyse, \\\"The vehicle routing  \\nproblem: State of the art classification and review,\\\" Computers & industrial \\nengineering, vol. 99, pp. 300 -313, 2016.  \\n[10] S. Kapser and M. Abdelrahman, \\\"Acceptance of autonomous delivery vehicles \\nfor last- mile delivery in Germany \\u2013Extending UTAUT2 with ri sk perceptions,\\\" \\nTransportation Research Part C: Emerging Technologies, vol. 111, pp. 210 -\\n225, 2020.  \\n[11] H. Liu, R. Yang, L. Wang, and P. Liu, \\\"Evaluating initial public acceptance of highly and fully autonomous vehicles,\\\" International Journal of Human \\u2013\\nComputer Interaction, vol. 35, no. 11, pp. 919 -931, 2019.  \\n[12] G. E. Hinton and R. R. Salakhutdinov, \\\"Reducing the dimensionality of data \\nwith neural networks,\\\" Science, vol. 313, no. 5786, pp. 504 -507, 2006.  [13] D. P. Kingma and M. Welling, \\\"Auto -encoding  variational bayes,\\\" in \\nProceedings of the International Conference on Learning Representation (ICLR), 2014.  \\n[14] \\u00d6. Yeniay, \\\"Penalty function methods for constrained optimization with genetic \\nalgorithms,\\\" Mathematical and Computational Applications, vol. 10, no. 1, pp. \\n45-56, 2005. \\n[15] V. Volz, J. Schrum, J. Liu, S. M. Lucas, A. Smith, and S. Risi, \\\"Evolving Mario \\nlevels in the latent space of a deep convolutional generative adversarial \\nnetwork,\\\" in Proceedings of the Genetic and Evolutionary Computation \\nConference (GECCO) , 2018, pp. 221 -228. \\n[16] P. Bontrager, A. Roy, J. Togelius, N. Memon, and A. Ross, \\\"Deepmasterp rints: \\nGenerating masterprints for dictionary attacks via latent variable evolution,\\\" in \\nProceedings of the 2018 IEEE 9th International Conference on Biometrics \\nTheory, Applications and Systems (BTS) , 2018, pp. 1 -9. \\n[17] P. Liskowski, K. Krawiec, N. E. Toklu, and J. Swan, \\\"Program synthesis as latent continuous optimization: Evolutionary search in neural embeddings,\\\" in \\nProceedings of the 2020 Genetic and Evolutionary Computation Conference , \\n2020, pp. 359 -367. \\n[18] J. K. Pugh, L. B. Soros, and K. O. Stanley , \\\"Quality diversity: a new frontier for \\nevolutionary computation,\\\" Frontiers in Robotics and AI, vol. 3, p. 40, 2016. 2020, pp. 359 -367. \\n[18] J. K. Pugh, L. B. Soros, and K. O. Stanley , \\\"Quality diversity: a new frontier for \\nevolutionary computation,\\\" Frontiers in Robotics and AI, vol. 3, p. 40, 2016.  \\n[19] A. Gaier, A. Asteroth, and J. -B. Mouret, \\\"Discovering representations for \\nblack -box optimization,\\\" in Proceedings of the Genetic and  Evolutionary \\nComputation Conference (GECCO) , 2020, pp. 103 -111. \\n[20] T. Yu and P. Bentley, \\\"Methods to evolve legal phenotypes,\\\" in Proceedings of \\nthe International Conference on Parallel Problem Solving from Nature (PPSN) , \\n1998, pp. 280 -291. \\n[21] K. Deb,  \\\"An efficient constraint handling method for genetic algorithms,\\\" \\nComputer Methods in Applied Mechanics and Engineering, vol. 186, no. 2 -4, \\npp. 311 -338, 2000.  \\n[22] F.-M. De Rainville, F. -A. Fortin, M. -A. Gardner, M. Parizeau, and C. Gagn\\u00e9, \\n\\\"Deap: A python framework for evolutionary algorithms,\\\" in Proceedings of the \\n14th annual conference companion on Genetic and evolutionary computation , \\n2012, pp. 85 -92. \\n[23] C. A. C. Coello, \\\"Theoretical and numerical constraint -handling techniques \\nused with evolutionary  algorithms: a survey of the state of the art,\\\" Computer \\nmethods in applied mechanics and engineering, vol. 191, no. 11 -12, pp. 1245 -\\n1287, 2002.  \\n[24] X. Ding, Z. Zou, and C. L. Brooks III, \\\"Deciphering protein evolution and fitness landscapes with latent s pace models,\\\" Nature communications, vol. 10, \\nno. 1, pp. 1 -13, 2019. SUPPLEMENTA RY MATERIA LS  \\n \\n \\nTable S1: E1.2 : Modifying RT:  number of simultaneously running robots , COIL . \\nCOIL 10 robots at once  15 robots at once  20 robots at once  \\navg objective (stdv)  66.43 (6.80)  73.94 (3.99)  89.86 (3.14)  \\nmin objective  51 63 83 \\nmax objective  77 84 98 \\navg constraint (stdv)  23.93 (8.43)  9.91 (10.78)  0.81 (2.86)  \\nmin constraint  0 0 0 \\nmax constraint  30 29 16 \\n \\nTable S2: E1.2 : Modifying RT:  number of simultaneously running robots , GA. \\nGA 10 robots at once  15 robots at once  20 robots at once  \\navg objective (stdv)  93.6 (5.55)  91.09 (5.43)  92.24 (4.32)  \\nmin objective  76 80 80 \\nmax objective  107 108 103 \\navg constraint (stdv)  44.7 (3.89)  26.69 (5.56)  0.8 (2.97)  \\nmin constraint  35 10 0 \\nmax constraint  32 41 17 \\n Table S3: E1.3: Modifying Rb: total number of robots , COIL . \\nCOIL 20 total robots  25 total robots  30 total robots  \\navg objective (stdv)  50.67 ( 3.63) 57.96 ( 7.59) 66.43 (6.80)  \\nmin objective  44 47 51 \\nmax objective  62 72 77 \\navg constraint (stdv)  4.4 (8.4) 15.39 ( 13.13)  23.93 (8.43)  \\nmin constraint  0 0 0 \\nmax constraint  27 32 30 \\n Table S4: E1.3: Modifying Rb: total number of robots , GA. \\nGA 20 total robots  25 total robots  30 total robots  \\navg objective (stdv)  63.47 ( 5.33) 81.01 ( 6.49)  93.6 (5.55)  \\nmin objective  51 64 76 \\nmax objective  75 102 107 \\navg constraint (stdv)  23.71 ( 7.97) 38.35 ( 4.71) 44.7 (3.89)  \\nmin constraint  0 26 35 \\nmax constraint  41 48 32 \\n \\nTable  S5: E2.1 : Modifying dataset size  DS \\n 2500 data points  5000 data points  7500 data points  10000 data points  \\navg objective (stdv)  66.75 (5.87)  68.2 (8.15)  65.45 (8.09)  66.43 (6.80)  \\nmin objective  51 49 48 51 \\nmax objective  81 82 80 77 \\navg constraint (stdv)  26.52 (4.22)  25.66 (8.02)  23.28 (8.18)  23.93 (8.43)  \\nmin constraint  0 0 0 0 \\nmax constraint  31 33 31 30 \\n \\nTable  S6: E2.2 : Modifying number of latent variables  maxlv  \\n 10 latent variables 20 latent variables 30 latent variables 40 latent variables 50 latent variables 60 latent variables \\navg objective (stdv)  50.1 (2.26)  52.08 (2.40)  53.19 (2.86)  53.26 (3.38)  53.47 (3.56)  66.43 (6.80)  \\nmin objective  45 47 47 44 47 51 \\nmax objective  56 59 61 63 70 77 \\navg constraint (stdv)  9.25 (5.64)  1.96 (3.03)  4.69 (6.11)  4.41 (6.78)  6.69 (7.83)  23.93 (8.43)  \\nmin constraint  0 0 0 0 0 0 \\nmax constraint  19 13 24 25 32 30\",\"16\":\" Vectorial Genetic Programming { Optimizing\\nSegments for Feature Extraction\\nPhilipp Fleck1;2?[0000\\u00000001\\u00008290\\u00006356], Stephan Winkler1;2[0000 \\u00000002\\u00005196\\u00004294],\\nMichael Kommenda1;3[0000 \\u0000003\\u00002049\\u0000723X], and Michael\\nA\\u000benzeller1;2[0000 \\u00000001\\u00005692\\u00005940]\\n1Heuristic and Evolutionary Algorithms Laboratory (HEAL),\\nUniversity of Applied Sciences Upper Austria, Softwarepark 11, 4232 Hagenberg, Austria\\n2Institute for Symbolic Arti\\fcial Intelligence,\\nJohannes Kepler University, Altenberger Stra\\u0019e 69, 4040 Linz, Austria\\n3Josef Ressel Center for Symbolic Regression,\\nUniversity of Applied Sciences Upper Austria, Softwarepark 11, 4232 Hagenberg, Austria\\nAbstract. Vectorial Genetic Programming (Vec-GP) extends GP by\\nallowing vectors as input features along regular, scalar features, using\\nthem by applying arithmetic operations component-wise or aggregating\\nvectors into scalars by some aggregation function. Vec-GP also allows\\naggregating vectors only over a limited segment of the vector instead of\\nthe whole vector, which o\\u000bers great potential but also introduces new\\nparameters that GP has to optimize. This paper formalizes an optimiza-\\ntion problem to analyze di\\u000berent strategies for optimizing a window for\\naggregation functions. Di\\u000berent strategies are presented, included ran-\\ndom and guided sampling, where the latter leverages information from\\nan approximated gradient. Those strategies can be applied as a simple\\noptimization algorithm, which itself ca be applied inside a specialized\\nmutation operator within GP. The presented results indicate, that the\\ndi\\u000berent random sampling strategies do not impact the overall algorithm\\nperformance signi\\fcantly, and that the guided strategies su\\u000ber from be-\\ncoming stuck in local optima. However, results also indicate, that there\\nis still potential in discovering more e\\u000ecient algorithms that could out-\\nperform the presented strategies.\\nKeywords: Genetic Programming \\u00b7Vectorial \\u00b7Optimization \\u00b7Gradient\\n1 Introduction\\nVectorial Genetic Programming (Vec-GP) for Symbolic Regression (SR) is a ex-\\ntension of GP, where the space of input features is extended to vectors alongside\\nregular scalars[1]. This allows vectorial GP to directly handle higher dimensional\\ndata, such as time series, without the need of prior feature engineering to ex-\\ntract scalar features. Instead, vectorial GP models themselves extracts scalar\\n?corresponding author\\nphilipp.fleck@fh-hagenberg.atarXiv:2303.03200v1  [cs.NE]  3 Mar 2023 2 P. Fleck et al.\\nvalues from the vectors by applying aggregation functions, such as the arith-\\nmetic mean. Compared to traditional feature engineering, where scalar features\\nare usually only extracted from the raw vector features, vectorial GP can also\\nextract features from interacting vectors by performing arithmetic operations on\\nvectors before aggregation takes place. Fig. 1 shows an example, where the vector\\nvariable for temperature and pressure are divided prior to calculating the arith-\\nmetic mean to obtain a scalar. The literature already indicates, that vectorial\\nGP outperforms regular GP with feature engineering in various benchmarks[2].\\nFig. 1: An example dataset containing both scalar and vector input features to\\npredict a scalar target value (the quality). It also shows a GP model that uses a\\nwindowed-aggregation function to aggregate a vector into a scalar.\\nIn case of a vector representing a time series, it might be bene\\fcial to aggre-\\ngate only over a certain time frame. In this sense, vectorial GP can be further\\nextended by adding the ability to aggregate not only over a whole vector, but\\nover a sub-segment of the vector, where GP is also in charge of optimizing the\\nsegment bounds. For instance, as also shown in Fig. 1, aggregation might only\\nbe done on a speci\\fc time frame, for example, between 3 \\u0014t\\u001410.\\n2 Problem Description\\nWhen using an aggregation function with an aggregation window, GP is now\\nrequired to optimize two additional parameters for each windowed aggregation\\nfunction within a GP model. While GP could handle this by allowing the indices\\nto change via mutation, random mutation is not particularly well suited for this\\ntask, especially for longer vectors. Instead of relying on mutation as GP's only\\nway of optimizing aggregation windows, this paper is a \\frst step towards iden-\\ntifying other strategies that are more tailored towards optimizing aggregation\\nwindows, similar to applying e\\u000ecient gradient-based algorithms for optimizing\\ncontinuous, numerical parameters of a \\fxed GP model[4].\\nTo focus on the problem of optimizing the aggregation window, we use a\\nsimpli\\fed optimization problem that omits any GP related aspect, only keeping\\nthe core problem of optimizing an aggregation window over some \\fxed data. Vectorial GP { Optimizing Segments for Feature Extraction 3\\nThis Segment Optimization Problem (SOP) is an integer problem, optimizing\\nthe parameters ^ a;^b2fi2Nji < Ngrepresenting the start and end index with,\\nminimizing\\narg min\\n^a;^bMX\\ni=0\\u0010\\nfagg\\u0000\\nvi[^a:^b]\\u0001\\n|{z}\\noptimized bounds\\u0000fagg\\u0000\\nvi[a:b]\\u0001\\n|{z}\\nknown bounds\\u00112\\n; (1)\\nwhere f aggis a \\fxed aggregation function, vi2RM\\u0002Nare the Msamples of\\nvectors with length N,aandbrepresenting the known aggregation indices and\\nvi[a:b] being the sub-vector of a selected vector vi. In essence, the goal of the\\nSOP is to \\fnd the minimum deviation when aggregating using the free indices\\n^aand^bcompared to aggregating using the known indices aandb.\\n3 Method\\nInstead of relying on GPs mutation to identify good aggregation windows, our\\ngoal is to de\\fne a specialized mutation operator for GP that is tailored to-\\nwards optimizing the start and end indices of an aggregation window e\\u000eciently.\\nStart and end indices can either be optimized simultaneously, i. e. as a multi-\\ndimensional optimization problem, or only a single dimension at a time, i. e.\\nas a single-dimensional optimization problem. We will later analyze which form\\nyields better results.\\nLooking at the \\ftness landscape of some of the benchmark instances in Fig. 2,\\nwe can clearly observe, that there is gradient information that could be lever-\\naged for e\\u000ecient optimization. Therefore, we \\frst approximate the gradient by\\nusing a \\fve-point stencil[5] on the neighboring indices ( h= 1) and then use\\nit to guide the search. The proposed optimization method in this paper is an\\niterative algorithm with two steps: First, the gradient is calculated and used to\\nguide search towards promising regions. Second, potential solutions are sampled\\nfrom that promising region, evaluated and the best one is selected for the next\\nFig. 2: The \\ftness landscape of two benchmark instances (x3 and x6) with lower\\n(brighter) values indicating better regions in the search space. 4 P. Fleck et al.\\niteration. This procedure is continued until a prede\\fned number of total solution\\nevaluations is exceeded.\\nThe \\frsts guiding step in the optimization procedure is responsible for se-\\nlecting a promising region, from which samples can be drawn in the second step\\nafterwards. We de\\fne three di\\u000berent guiding strategies, shown in Fig. 3: Full\\nrepresents no strategy at all, where the whole solution space is utilized. This\\nstrategy serves will serve as a baseline. Direction uses the approximate gradient\\nonly to guide whether an index should be increased or decreased and does not\\ntake the magnitude of the gradient into account. Range uses the approximate\\ngradient information to de\\fne a promising region by a de\\fned range, where the\\nmagnitude of the gradient also in\\ruences the distance from the current index.\\nFig. 3: Three di\\u000berent guiding strategies for de\\fning the search space, given a\\nfree start index and a \\fxed end index.\\nWhile the random direction simply looks at the direction of gradient, the\\nguided range mimics a traditional gradient descent algorithm with two modi\\f-\\ncations, shown in Fig. 4. First, we round the gradient step towards the nearest\\ninteger, to ensure that the next index will also be valid index. Additionally, we\\nintroduce a new parameter, the search-range, that de\\fnes how many samples\\naround the next index are considered promising and are thus eligible for being\\nsampled afterwards.\\nFig. 4: Adoption of a gradient descent step to integer indices, with an additional\\nsearch-range parameter.\\nAfter de\\fning the search space, the second step is responsible of drawing\\nthe samples from this search space. For this paper, we have considered three Vectorial GP { Optimizing Segments for Feature Extraction 5\\nsampling mechanisms, shown in Fig. 5: Exhaustive draws all samples from the\\nsolution space. This is typically not feasible for a multi-dimensional search space,\\ndue to a potentially high number of possible solutions. However, it could be a\\nviable option for a single dimensional space. Random draws samples completely\\nat random (without repetition), where the number of samples are de\\fned by an\\nadditional sample-size parameter. Orthogonal selects the samples equally spaced\\nalong each dimension, by \\frst selecting a prede\\fned number of equally spaced\\npoints on a continuous line of the search space and then rounding towards the\\nnearest indices and removing any duplicates.\\nFig. 5: Di\\u000berent sampling strategies for a given search space.\\n4 Experiment Setup\\nTo identify good parameters of the search strategies de\\fned in the previous sec-\\ntion, we performed a grid search of the parameters over multiple benchmark\\ninstances. However, certain combinations were skipped due to infeasible run-\\ntimes, e. g. exhaustive search on multi-dimensional search spaces. Each parame-\\nter combination was repeated 20 times, and each execution of the optimization\\nalgorithm was allowed a total of 100,000 sample evaluations, meaning the total\\nnumber of iterations varied, depending on the sample-size, for instance.\\nThe generated benchmark instances have di\\u000berent characteristics to represent\\ndi\\u000berent challenges and di\\u000eculties. For instance, shown in Fig. 6, the second\\ninstance shows an overall increasing behavior where the noise stays constant.\\nThe most right benchmark, however, does not only have more varying slopes,\\nbut also the noise is increasing over time. The benchmark instances were created\\nin multiple steps by creating randomized control values \\frst, that are then used\\nas arguments for other distributions that are \\fnally used to create the vectors.\\nFor instance, the most right benchmark is based on the control values m=\\nU(2;5) +U(0:2;4)=80\\u0001tands=U(0;0:001) +U(0:002;0:01)\\u0001twhereUdenotes\\na uniform random variable and tdenotes the time or index. The \\fnal vector\\nis then created using a normal distribution N(m; s2) with the prior de\\fned\\ncontrol values. Thus, for this benchmark instance, the mean value and also the\\nnoise increases with higher t. Since the parameters of the distributions change\\nwith t, the distributions from which a sample is drawn are created for each\\ntindependently. The full list and de\\fnition of the benchmark instances can\\nfound in the additional materials online at https:\\/\\/dev.heuristiclab.com\\/wiki\\/\\nAdditionalMaterial. For the presented results, we used vectors of length 1,000. 6 P. Fleck et al.\\nFig. 6: Some exemplary benchmark instances, where each line represent one row\\nin the dataset.\\n5 Result\\nTo compare di\\u000berent optimization strategies, we analyze the run-length distribu-\\ntions, as de\\fned in the black-box optimization benchmarking (BBOB)[3]. This\\nallows an evaluation of not only an algorithms ability of \\fnding a good solution,\\nbut also shows how fast an algorithm converges. This is crucial, since a good\\noptimization strategy for the segment optimization problem would be used in-\\nside a mutation operator within GP, thus, long runtimes would multiply in the\\ncontext of GP.\\nIn addition to the presented results, we also analyzed additional parameters,\\nsuch as the e\\u000bect of random versus orthogonal sampling, however they showed\\nno signi\\fcant di\\u000berences, thus are not discussed further in this paper. We will\\n\\frst look at whether approaching the optimization problem as single- or multi-\\ndimensional performs better, shown in Fig. 7. Overall, the results suggest, that\\noptimizing all dimensions simultaneously performs slightly better, however, for\\nsome problem instances, such as x1, using a single dimension yielded better\\nresults. This behavior was suspected, since it shows, that some of the benchmark\\nare easy enough to be separable and optimized dimension by dimension.\\nFig. 7: In\\ruence of single- versus multi-dimension on algorithm performance.\\nNext, we analyze directional search space reduction, where we compare lim-\\niting the search space towards a direction randomly versus using the gradient\\nto guide the direction, shown in Fig. 8. The result suggests, that the bene\\fts\\ndepend on the problem instance. While for problem instance x5 using a guided\\ndirection seems to slow down convergence, overall, the bene\\fts are not conclu-\\nsive. We suspect, that by using the gradient always to move towards the best\\noption, it forces the algorithm towards local optima without any way of escaping. Vectorial GP { Optimizing Segments for Feature Extraction 7\\nFig. 8: Performance of random versus guided direction.\\nNext, we analyze limiting the search space within a speci\\fc range. Because\\nthe range strategy has two additional parameter, the step-size and search-width,\\nwe will compare the random version to di\\u000berent settings of the parameters,\\nboth shown in Fig. 9. Since the step-size controls the jumping distance from the\\ncurrent index, having low values are expected to yield bad results, since jumping\\ntoo little makes the search ine\\u000ecient or halt completely due to the rounding in\\nthe gradient step. The e\\u000bect of the step-size on instance x3 suggest that this\\nassumption holds, since lower step-sizes do not perform well, as shown in Fig. 9a\\nOn the other hand, higher step-sizes tend to perform better for instance x3 and\\nmight even outperform the random range strategy in \\fnding the best solutions\\nquicker, however, this is not the case for all instances. Along the step-size, the\\nsearch-width de\\fnes how far the search space extends around the next index.\\nThe results in Fig. 9b suggests, that having a lower search-width generally causes\\nthe algorithm to converge slower. This is especially true for combinations with a\\nlow step-size, since the search-range is the only mechanism of changing an index\\nwhen the gradient step is rounded to zero. Similar to the guided direction, where\\nwe suspect that random direction outperforms the guided direction because the\\nalgorithm becomes stuck at a local optimum, the same could also be true for\\nguided range to a certain degree. Having a large search-width could help breaking\\nout of the local optimum, but still, the result suggest that a random range would\\nbe the better option in most cases.\\n(a) In\\ruence of the step-size.\\n (b) In\\ruence of the search-width.\\nFig. 9: Performance of random versus guided search range reduction. 8 P. Fleck et al.\\n6 Conclusion\\nThe results presented in the previous section suggests, that simple random sam-\\npling is generally better than using a guided strategy such as guided direction\\nor guided range. We believe that this is mainly caused by the guided strategies\\ntend to work similar to greedy algorithms, and can quickly become stuck in local\\noptima. However, when applying the presented methods in the context of GP,\\npreliminary results suggests, that the guided strategies tend to outperform their\\nrandom counterparts. We suspect that GP can easily break free of local optima\\nby regular crossover and mutation operations, thus becoming stuck in a local\\noptima is not as critical.\\nDespite the current result suggesting that the gradient-based strategies do\\nnot work well, we still believe that the information of the approximate gradient\\nis extremely useful, because most \\ftness landscapes of the benchmark instances\\nclearly indicate that there is a gradient that could be exploited for fast optimiza-\\ntion. Therefore, we will continue working on leveraging the gradient for a fast\\noptimization algorithm for the segment optimization problem.\\nAcknowledgments This work was carried out within the Dissertationspro-\\ngramm der Fachhochschule O \u007fO #875441 Vektor-basierte Genetische Program-\\nmierung f\u007f ur Symbolische Regression und Klassi\\fkation mit Zeitreihen (Sym-\\nRegZeit) , funded by the Austrian Research Promotion Agency FFG. The au-\\nthors also gratefully acknowledge support by the Christian Doppler Research\\nAssociation and the Federal Ministry of Digital and Economic A\\u000bairs within the\\nJosef Ressel Centre for Symbolic Regression .\\nReferences\\n1. Azzali, I., Vanneschi, L., Silva, S., Bakurov, I., Giacobini, M.: A Vectorial Approach\\nto Genetic Programming. EuroGP pp. 213{227 (2019)\\n2. Fleck, P., Winkler, S., Kommenda, M., A\\u000benzeller, M.: Grammar-based Vecto-\\nrial Genetic Programming for Symbolic Regression. In: Banzhaf, W., Trujillo, L.,\\nWinkler, S., Worzel, B. (eds.) Genetic Programming Theory and Practice XVIII.\\nSpringer Nature Singapore Pte Ltd. (2021)\\n3. Hansen, N., Auger, A., Ros, R., Finck, S., Po\\u0014 s\\u0013 \\u0010k, P.: Comparing results of 31 algo-\\nrithms from the black-box optimization benchmarking bbob-2009. In: Proceedings\\nof the 12th annual conference companion on Genetic and evolutionary computation.\\npp. 1689{1696 (2010)\\n4. Kommenda, M., Burlacu, B., Kronberger, G., A\\u000benzeller, M.: Parameter identi\\f-\\ncation for symbolic regression using nonlinear least squares. Genetic Programming\\nand Evolvable Machines 21(3), 471{501 (2020)\\n5. Sauer, T.: Numerical solution of stochastic di\\u000berential equations in \\fnance. In:\\nHandbook of computational \\fnance, pp. 529{550. Springer (2012)\",\"17\":\" This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n Low -discrepancy S ampling  in the \\nExpanded Dimensional Space: A n \\nAcceleration Technique for Particle Swarm \\nOptimization  \\n \\nFeng Wu *, Yuelin Zhao , Jianhua Pang *, Jun Yan,  and Wanxie Zhong  \\n \\nState Key Laboratory of Structural Analysis, Optimization and CAE Software for \\nIndustrial Equipment , Department  of Engineering Mechanics , Faculty of Vehicle \\nEngineering and Mechanics, Dalian University of Technology, Dalian 116023, \\nP.R.China ; Guangdong Ocean University, Zhanjiang 524088, China ; Shenz hen \\nInstitute of Guangdong Ocean University, Shenzhen 518120, China  \\nEmail:,  wufeng_chn@163.com , zhaoyl9811@163.com , pangjianhua@gdou.edu.cn , \\nyanjun@dlut.edu.cn, wxzhong @dlut.edu.cn  \\n \\n \\nProject supported by  the National Natural Science Foundation of China grants (Nos. 11472076, \\n51609034 and U1906233), the National Key R&D program of  China (No. 2021YFA1003501), the \\nScience Foundation of Liaoning Province of China (No. 2021 -MS-119), the Dalian Youth Science \\nand Technology Star project (No. 2018RQ06), and the Fundamental Research Funds for the \\nCentral Universities grant (Nos. DUT20RC(5)009 and DUT20GJ216).  \\n \\n \\n*Corresponding author: wufeng_chn@163.com ; yanjun@dlut.edu.cn  \\nMarch 2023 This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n Abstract : Compared with random sampling,  low-discrepancy sampling  is more effective in \\ncovering the search space. However, the existing research cannot definitely state  whether the \\nimpact of a  low-discrepancy samp le on particle swarm optimization (PSO) is positive or negative.  \\nUsing Niderreiter \\u2019s theorem, this study  completes an error analysis of PSO, which  reveals  that the \\nerror bound of PSO at each iteration depends on the dispersion  of the sample  set in an expanded \\ndimensional space. Based on this error analysis , an acceleration technique  for PSO-type \\nalgorithms  is proposed with low-discrepancy sampling  in the expanded dimensional space.  The \\nacceleration technique can generate  a low-discrepancy sample set with  a smaller dispersion, \\ncompared with  a random sampling, in the expanded dimensional space;  it also  reduce s the error at \\neach iteration, and hence improve s the convergence speed. The acceleration technique is \\ncombined with the standard PSO and  the c omprehensive learning particle swarm optimization, \\nand the performance of the improved algorithm is compared with the o riginal algorithm. The \\nexperimental results show that the two improved algorithms have significantly faster convergence \\nspeed under the same accuracy requirement.  \\n \\nKeywords : Evolutionary algorithm; Low -discrepancy sample; Particle swarm optimization; \\nComprehensive learning particle swarm optimization; Swarm intelligence; Convergence speed  \\n1. Introduction  \\narticle swarm optimization (PSO) is a population -based stochastic optimization technology. \\nSince it was proposed by Kennedy and Eberhart in  1995 [1, 2] , it has attracted great attention.  \\nPSO is an excellent evolutionary algorithm that is easy to implement, can quickly converge to the \\noptimal solution, and has been proven to perform well i n many fields, such as neural networks [3, \\n4], photovoltaic system [5], feature selection [6, 7], operational research [8], etc. \\nHowever, it is not easy for PSO to consider both the convergence speed and calculation \\naccuracy when solving complex optimization problems. Many excellent PSO variants are \\ncommitted to solving this problem, and can be divided based on three aspects. The fi rst aspect is \\nto enhance the PSO by adjusting the parameter changes in the algorithm; Shi and Eberhart [9] \\nintroduce d the inertia weight, which can balance the local development and global search \\ncapabilities of PSO.  Clerc and Kennedy  [10] proposed  a set of coefficients to control the \\nconvergence trend of the system.  Zhan et al.  [11] developed an  adaptive PSO  and improved the \\nsearch efficiency. There are also a sea of enhancement approaches for the PSO iteration formula . \\nLiang et al.  [12] proposed a new comprehensive learning particle swarm optimizer  (CLPSO), This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n which introduces a learning strategy to update the particle\\u2019s velocity that  significantly improves \\nthe performance of PSO. Lynn and Suganthan [13] proposed heterogeneous comprehensive \\nlearning particle swarm optimization  (HCLPSO) , which divides the population into two \\nsubpopulations, and each subpopulation only focuses on exploration or expl oitation to maintain \\nthe diversity of the population.  Zhan et al.  [14] proposed orthogonal learning particle swarm \\noptimization, which  employs an orthogonal learning strategy  that can  guide particles t o fly in an \\noptimal direction.  Liu et al. [15] proposed an improved quantum particle swarm optimization with \\nL\\u00e9 vy flight and straight flight, which is more suitable for high -dimensional situations.  Mendes et \\nal. [16] and Zeng  et al.  [17] showed that replacing the best position among all the particles with \\nthe best position in some smaller number  of adjacent particles can also improve the search ability \\nof PSO. The third aspect focuses on population initialization [18, 19] . In general, the population is \\ngenerated by random samples [20, 21] . However, quite a few researchers believe that the \\nuniformity of the initial population obviously impacts the search ability of PSO. Employing \\nnumerical experiments,  Richards and Ventura [22] concluded that  low-discrepancy sampling \\n(LDS)  can be used to improve the calculation accuracy and the convergence  speed of PSO in high -\\ndimensional space.  Morrison  [23] verified,  based on numerical simulations , that the LDS \\ninitialization method can reduce the variation  of the search results without losing accuracy and \\nperformance.  Nevertheless, some studies stil l disagree with this view. Li  et al.  [24] showed \\nthrough the numerical results that random initialization is most suitable for PSO.  Tharwat and \\nSchenck [25] compared the effects of five famous population initialization methods and concluded \\nthat PSO is not sensitive to initialization methods under sufficient numbers of iterations.  \\nIt can be said that the existing studies did not give a clear conclu sion on whether the \\nuniformity of the initial sample set  has a positive or negative impact on the PSO algorithm and \\nwhy this impact occurs. The author has also done experiments, and the result show that PSO is not \\nsensitive to initialization methods. The a uthor has further verified the effect of using LDS directly \\nin each iteration, and the result is worse . However, it is undeniable that many experiments observe \\nthat in the early iterations process , the population generated by  LDS  has a better convergence \\nspeed. Then, why can\\u2019t the advantages of using an LDS initialization at the early iterations be \\npreserved during the entire iteration of PSO? Furthermore, what is the relationship between the This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n error of the entire iteration process of PSO and the sample unifo rmity? These unclear conclusions \\nand contradictions motivated us to investigate more deeply. First, we try to find the answers to \\nthese two questions from the existing theoretical analysis.  \\nCurrently, the theoretical study of the PSO algorithm mainly focu ses on two aspects. One is \\nthe movement of the single particle or the particle swarm with time; Kennedy [26] simulated \\ndifferent particle trajectories by simplifying the particle evolution model. Ozcan  and Mohan [27] \\npointed out that particles move along a path defined by a sinusoidal wave in a simplified one -\\ndimensional PSO system . van den Bergh [28] developed a model that could describe the behavior \\nof a single particle to investigate the local convergence properties of PSO. Liang et al.  [12] \\nanalyzed the particle swarm system\\u2019s search beha vior and concluded that the CLPSO algorithm \\ncan search a greater number of potential regions. Another aspect of theoretical analysis is studying \\nPSO\\u2019s stability and convergence.  Clerc and Kennedy [10] first analyzed the stability of PSO . \\nTrelea [29], Emara and Fattah  [30] determined the parameter range that can ensure the stability or \\nconvergen ce of the algorithm through a similar analysis.  Garc\\u00ed a -Gonzalo and Fern\\u00e1 ndez -Mart\\u00ed nez \\n[31] provided an analytical expression for the upper bound of the second -order stable regions of \\nthe particle traje ctories. Kadirkamanathan et al.  [32] analyzed the stability of the particle \\ndynamics under stochastic conditions using a Lyapunov stability analysis and the concept of \\npassive systems.  \\nFrom the current  work, it is easy to see that although many studies have focused on a \\ntheoretical analysis of the PSO algorithm, an error analysis of PSO urgently needs research \\nattention. Moreover, the existing theoretical analysis cannot explain the two questions raised  in the \\nprevious article. Therefore, this study attempts to answer these two questions. Using  Niderreiter\\u2019s \\ntheorem, this study  analyzes the error of PSO and propose s a new acceleration technique. Some \\nrepresentative LDSs are selected for numerical validat ion. At present, there are many types  of \\nLDSs. For example, there are sampling methods based on orthogonal experimental design , \\ninclud ing the Latin hypercube [33], orthogonal array [34], uniform experimental design [35] and \\nso on. There are also sampling methods based on number theory, which often have relatively \\nsimple generation formats, such as the Hua -Wang sampl ing (HWS) [36-38], Halton sampling  (HS) \\n[39], Sobol sampl ing (SS) [40] and so on. In addition, there are some other sampling methods, This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n such as Chaos sampling [41], Poisson disk sampling [42, 43] , optimized Halton sampling (OHS) \\n[44], and dynamics evolution sampling (DES) [45]. The OHS uses an evolutiona ry algorithm to \\noptimize HS, and the DES uses a dynamic evolutionary algorithm to optimize the existing \\nsamples.  Both methods generate a large number of samples, which can be downloaded directly on \\nthe internet.  We choose HWS, DES, SS and Halton  scramble  sampling  (HSS) [46], a modified \\nversion of the famous H S for numerical verification. The research contributions of this study are \\nas follows:  \\n1) Based on Niderreiter \\u2019s theorem, an error analysis of PSO is given, and the influence of \\nLDS initialization on the PSO calculation results is clarified. The relationship among sample \\nuniformity, calculation error and the convergence rate is found, which provides a new theoretical \\nbasis for improving PSO.  \\n2) Based on th e theoretical derivation in 1), a new acceleration technique, LDS in the \\nexpanded dimensional space (LDSEDS), is given . According to different sample construction \\nmethods, two versions of LDSEDS are proposed . These two versions are combined with PSO to \\ngenerate two more effective algorithms. In the simulation studies, several benchmark functions are \\nperformed, and the performances of the proposed algorithms are compared with the original PSO \\nto demonstrat e the superiority of these two algorithms.  At the sam e time, the computational \\nperformance of these two algorithms is compared, and a more effective algorithm is selected for \\nhigher dimension promotion.  \\n3) The LDSEDS acceleration technique proposed in 2) continues to be extended to other \\nPSO series algorithms. Considering that the theoretical derivation in 1) needs preconditions, \\nCLPSO is ultimately selected for an improvement to form CLPSO -LDSEDS, and the superiority \\nof CLPSO -LDSEDS is also verified.  \\nThe rest of this paper is organized as follows. Section  2 briefly introduces the preliminaries. \\nSection  3 gives the error analysis of PSO. Section 4 proposes a new acceleration technique \\nLDSEDS to form two acceleration algorithms. In Section 5, the acceleration algorithms are \\nverified numerically. In Section  6, this  acceleration technique is extended to CLPSO. Finally, \\nSection 7 draws a conclusion and gives future work directions. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n 2. Preliminaries  \\n2.1 Particle Swarm Optimization  \\nWithout loss of generality, we assume a \\n-D dimensional optimization problem to be \\nexpressed as  \\n \\n() ( ) 12 min , , , D\\ni i if f x x x\\na x b= \\uf0ec\\n\\uf0ed\\uf0a3\\uf0a3\\uf0eex\\n . (1) \\nwhere \\n()fx  is the cost  function, and \\nia  and \\nib  represent the boundary of the search space. To \\nsolve Eq. (1), we use PSO, which  simulates the swarming habits of animals such as birds [1]. The \\npositions and velocities of all the particles at the \\n-thg  iteration are expressed as  \\n \\n\\uf05b \\uf05d \\uf05b \\uf05d ,1 ,2 , ,1 ,2 ,, , , , , , , g g g g N g g g g N==X x x x V v v v\\n , (2) \\nwhere \\nN  is the number of particles. The position and velocity of the \\n-thi particle  are updated \\nusing  \\n \\n() ()( )() ()( )12\\n1, , 1 , 2 , , , ,\\n1, , 1,l gg i g i g i g i g i g i g i\\ng i g i g icc \\uf077 +\\n++= + \\u2212 + \\u2212\\n=+vv \\u03b5 px \\u03b5 px\\nx x v\\n , (3) \\nwhere \\n\\uf077 , \\n1c and \\n2c  are coefficients that can be constants or change with iteration, \\n  denotes the \\nHadamard product, \\n()1\\n,gi\\u03b5  and \\n()2\\n,gi\\u03b5  are two \\n1D\\uf0b4  random vectors of numbers distributed between 0 \\nand 1, and  \\n \\n()()()() ,, ,1 ,1 1arg min , arg minl gl i l i gii N l g l gff\\n\\uf0a3 \\uf0a3 \\uf0a3 \\uf0a3 \\uf0a3 \\uf0a3==p x p x . (4) \\n2.2 Comprehensive Learning Particle Swarm Optimizer  \\nCLPSO is a variant of PSO  that uses a new learning strategy to update the particle\\u2019s velocity \\n[12]. The position and velocity in  CLPSO  are updated using  \\n \\n( ) 1, , , , , 1, , 1, , g i g i g i g ij g i g i g i g i c \\uf077 + + += + \\u2212 = + vv \\u03b5 p x x x v\\n , (5) \\nwhere \\n\\uf077  and \\nc  are coefficients that can change with the iterations. \\n,gi\\u03b5  is a \\n1D\\uf0b4  random \\nvector of numbers distributed between 0 and 1. The \\n,g ijp  is determined as follows: for each \\nparticle \\ni , randomly select two other particles from the whole population, compare the fitness of \\nthe two particles and choose the better one \\n,gjji \\uf0b9p\\uff0c  as the candidate particle . CLPSO [12] This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n determines whether to learn the information of other particles by comparing the random number \\nand the given probability \\n,ciP . \\nA random number \\n\\uf065  is ge nerated for each dimension and compared with its \\n,ciP . If \\n\\uf065  \\nsmaller than \\n,ciP  value, the \\n-thi  particle is guided by other particle\\u2019s position \\n,gjp . If \\n\\uf065 is larger \\nthan \\n,ciP , the particle will follow its own \\n,gip  for that dimension. Therefore, the \\n,g ijp  is a new \\nposition where each dimension learns from several particles\\u2019 positions.  \\nWe can see from the probability \\n,ciP  in [12] that the random numbers are greater than \\n,ciP  in \\nmost cases. That is, in the updated information of each particle, only a tiny part of the dimensional \\ninformation comes from other particles\\u2019 historical best. I nteresting ly, although the particle in \\nCLPSO updates its position with only a small amount  of information from other particles, the \\nsearch capabilit y is significantly improved.  \\n2.3 Niderreiter  Theorem  \\nLet \\n()h\\u03b8  be a function with the argument \\nE\\uf0ce\\u03b8 ; the modulus of continuity of \\n()h\\u03b8  is \\ndefined by  \\n \\n \\n()\\n( )()()\\n12\\n1212\\n,\\n,sup h\\nE\\ndhh\\n\\uf065\\uf077\\uf065\\n\\uf0ce\\n\\uf0a3=\\u2212\\n\\u03b8\\u03b8\\n\\u03b8\\u03b8\\u03b8\\u03b8 \\uff0c (6) \\nwhere sup is the abbreviation of supremum, which represents the minimum upper bound in \\nmathematics . \\nFor a sample set  \\n\\uf05b \\uf05d 12, , , N =P\\u03b8\\u03b8\\u03b8\\n  with \\niE\\uf0ce\\u03b8  and \\n1iN\\uf0a3\\uf0a3 , the dispersion of \\nP  is \\ndefined by  \\n \\n() ( )\\n1sup min , NnnNEdd\\n\\uf0a3\\uf0a3\\uf0ce=\\n\\u03b8P \\u03b8\\u03b8  (7) \\nwhere \\n(),d\\uf0d7\\uf0d7  denotes the euclidean  distance. The dispersion is a metric used to measure the \\nuniformity of the sample set . The smaller the dispersion is, the better the uniformity. According to \\nNiderreiter\\u2019s theorem in [47], we have  \\n \\n() ( ) ()( ) () ( )\\n10 min min i h Ni N Eh h d \\uf077\\n\\uf0a3 \\uf0a3 \\uf0ce\\uf0a3 \\u2212 \\uf0a3\\n\\u03b8\\u03b8 \\u03b8 P  (8) \\nwhere \\n() ( )\\n1min iiNh\\n\\uf0a3\\uf0a3\\u03b8  is the approximate value of best solution in space  \\nE, which corresponds \\nto the fitness of the best solution in the field of evolutionary algorithm.  \\n()( ) min\\nEh\\n\\uf0ce\\u03b8\\u03b8  corresponds to \\nthe true optimal solution. Therefore,\\n() ( ) ()( )\\n1min min ii N Ehh\\n\\uf0a3 \\uf0a3 \\uf0ce\\u2212\\n\\u03b8\\u03b8 \\u03b8  represents the relative error between \\nthe fitness of the best solution and the true optimal solution. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n 3. Effect of population initializers  on HCLPSO  \\nIn order to explore the relationship between the error of the entire iteration process of PSO \\nand the sample uniformity, this section presents the error analysis on PSO. The error of the result \\nobtained by using PSO is defined by  \\n \\n(),1min g g iiNfM\\n\\uf0a3\\uf0a3\\uf044 = \\u2212 x \\uff0c (9) \\nwhere \\nM  denotes the exact global optimal fitness value, and \\n(),1min giiNf\\n\\uf0a3\\uf0a3x  denotes the optima \\nof all the particle fitness values at the \\n-thg  iteration. In the error analysis, we assume that PSO \\ncan converge to \\nM  with iteration.  \\n3.1 Error at the  initialize iteration  \\nSuppose the position and velocity of the \\n-thi  particle at the first iteration are  \\n \\n( ) ( ) 0, 1, 0, min 2, max min , i i i i= + \\u2212 = + \\u2212xa \\u03b5 b a v v \\u03b5 vv\\n , (10) \\nwhere  \\n \\n( ) ( )\\n( )\\n( )TT\\n1 2 1 2\\nT\\nmax max,1 max,2 max,\\nT\\nmin min,1 min,2 min,, , , , , , ,\\n, , ,\\n, , ,DD\\nD\\nDa a a b b b\\nv v v\\nv v v==\\n=\\n=ab\\nv\\nv\\n \\uff0c (11) \\nmax, iv\\n and \\nmin,iv  are the maximum and minimum velocities, respectively, and \\n1,i\\u03b5  and \\n2,i\\u03b5  are \\ntwo random vectors in \\n\\uf05b\\uf05d : 0,1DE= .  \\nConsidering  (10), \\n() ( ) ( ) 0, 1,ii ff = + \\u2212xa \\u03b5 ba\\n , which means  the fitness value \\n()0,i fx  is \\nactually a function of \\n1,i\\u03b5 , denoted by \\n() ( ) ( ) 0 1, 1, :ii hf = + \\u2212\\u03b5 a\\u03b5 ba\\n . The \\n1,i\\u03b5  of all the particles \\nform a sample set  \\n\\uf05b \\uf05d 0 1,1 1,2 1, , , , N =P\\u03b5\\u03b5\\u03b5\\n . According to Niderreiter\\u2019s theorem (8), the error at the \\nfirst ite ration satisfies  \\n \\n() () () ( )0\\n10 1, 0 01min min i h Ni N Eh h d \\uf077\\n\\uf0a3 \\uf0a3 \\uf0ce\\u2212\\uf0a3\\n\\u03b5\\u03b5 \\u03b5 P \\uff0c (12) \\n \\n() ( ) 0 1,1sup min , NnnNEdd\\n\\uf065\\uf0a3\\uf0a3\\uf0ce=P \\u03b5\\u03b5 \\uff0c (13) \\nwhere \\n()0 NdP  is the dispersion of \\n0P . As \\n()\\n10 min\\nEhM\\n\\uf0ce=\\n\\u03b5\\u03b5 , we have This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n  \\n() () () ( )0\\n10 0 1, 0 01min min i h Ni N Eh h d \\uf077\\n\\uf0a3 \\uf0a3 \\uf0ce\\uf044 = \\u2212 \\uf0a3\\n\\u03b5\\u03b5 \\u03b5 P  (14) \\nObviously, the better the uniformity of \\n0P , the smaller the dispersion and the smaller the \\nerror. According to  (14), if a more uniformly distributed sample set  is used to generate the initial \\npopulation, the error at the initialize iteration will be smaller.  \\n3.2 Error at the  first iteration  \\nAccording to  (3), we have \\n \\n( ) \\uf05b \\uf05d\\n() ()( ) ( ) \\uf05b \\uf05d1, min 2, max min\\n2 02 1,0,\\n1, 0, 1,ii\\ni i\\ni i ic\\uf077= + \\u2212\\uf0ec\\n\\uf0ef+ \\u2212 + \\u2212\\uf0ed\\n\\uf0ef=+\\uf0eevv \\u03b5vv\\n\\u03b5 pa \\u03b5ba\\nx x v\\n . (15) \\nAs shown in  (15), for all the particles, the parameters \\n()0min max 1 2 , , , , , , , cc\\uf077 a b v v p  are the \\nsame, and the random vectors \\n1,i\\u03b5 , \\n2,i\\u03b5  and \\n()1\\n1,i\\u03b5  are different. For convenience of discussion, let \\n()01 min max 1 2: { ; ; ; ; ; ; ; } cc\\uf077 =\\u03bb a b v v p\\n, and \\n() ()( )12\\n1, 2, 0, : ; ; ii i i =\\u03b8\\u03b5\\u03b5\\u03b5 . Therefore, the fitness value \\n()1,i fx  \\ncan be treated as a function of the random argument \\n()1\\ni\\u03b8 , denoted b y \\n()()1\\n1 ih\\u03b8 . For each particle \\ni , \\nthere is a random argument \\n()1\\ni\\u03b8 , and all these random arguments form a sample set  \\n() () () 1 1 1\\n1 12: [ , , , ]N =P\\u03b8\\u03b8\\u03b8\\n. Similarly, using Niderreiter\\u2019s theorem yields the following error bound  \\n \\n() ( ) ()\\n\\uf05b\\uf05d()( ) 1\\n31\\n1 1 110, 1, sup min ,\\nDh N N iiNd d d\\uf077\\n\\uf0a3\\uf0a3\\uf0ce\\uf044 \\uf0a3 =\\n\\u03b8PP \\u03b8\\u03b8  (16) \\nwhere \\n()1 NdP  denotes the dispersion of \\n1P . (16) gives the error bound at the second iteration. \\nThe better the uniformity of \\n1P  is, the smaller \\n()1 NdP  and the smaller the error. \\n1P  contains three \\ngroups of sequences, in which \\n\\uf07b\\uf07d1,i\\u03b5  and \\n\\uf07b\\uf07d2,i\\u03b5  are the samples for the positions and velocities of \\nthe initial population, and  \\n()\\uf07b\\uf07d2\\n0,i\\u03b5  is required in the velocity updating at the second iteration. In \\nprevious studies, the initial population was  often  generated by using LDS, and hence , \\n\\uf07b\\uf07d1,i\\u03b5  and \\n\\uf07b\\uf07d2,i\\u03b5\\n are of low -discrepancy, while \\n()\\uf07b\\uf07d2\\n0,i\\u03b5  was still generated randomly, which reduces the \\nuniformity of \\n1P . If \\n\\uf07b\\uf07d1,i\\u03b5 , \\n\\uf07b\\uf07d2,i\\u03b5  and \\n()\\uf07b\\uf07d2\\n0,i\\u03b5  are all  generated b y using LDS, the dispersion \\n()1 NdP  \\nand the error can be reduced. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n 3.3 Error at the  g-th iteration  \\nAccording to  (3), we have  \\n \\n() ()( )() ()( )12 12, 1, 1 1, 2 1, 1, 1, 1,\\n2, 1, 2,l\\ni i i i i i i\\ni i icc \\uf077 \\uf0ec = + \\u2212 + \\u2212 \\uf0ef\\uf0ed=+ \\uf0ef\\uf0eevv \\u03b5 px \\u03b5 px\\nx x v\\n , (17) \\nwhich shows that the particle position \\n2,ix  depends on \\n\\uf077 , \\n1c, \\n2c, \\n()\\n1,l\\nip , \\n()1p , \\n()1\\n1,i\\u03b5 , \\n()2\\n1,i\\u03b5 , \\n1,ix  and \\n1,iv\\n. According to the updating formula  (15), \\n()\\n1,l\\nip , \\n1,ix  and \\n1,iv  depend on \\na , \\nb, \\nminv , \\nmaxv , \\n\\uf077, \\n1c\\n, \\n2c, \\n()0p , \\n1,i\\u03b5, \\n2,i\\u03b5  and \\n()\\uf07b\\uf07d2\\n0,i\\u03b5 . For all particles, \\na , \\nb, \\nminv , \\nmaxv , \\n\\uf077, \\n1c, \\n2c, \\n()0p , and \\n()1p  are \\nthe same, and \\n1,i\\u03b5 , \\n2,i\\u03b5 , \\n()\\uf07b\\uf07d2\\n0,i\\u03b5 , \\n()1\\n1,i\\u03b5  and \\n()2\\n1,i\\u03b5  are different. Let \\n() ()\\uf07b \\uf07d012 min max 1 2: ; ; ; ; ; ; ; ; cc\\uf077 =\\u03bb a b v v p p\\n, and \\n() ()()()( )2 2 1 2\\n1, 2, 0, 1, 1, : ; ; ; ; ii i i i i =\\u03b8\\u03b5\\u03b5\\u03b5\\u03b5\\u03b5 . Therefore the fitness \\nvalue \\n()2,i fx  is a function of the argument \\n()2\\ni\\u03b8 , denoted by \\n()()2\\n2 ih\\u03b8 . Similarly, using \\nNiderreiter \\u2019s theorem yields the error bound at the third iteration  \\n \\n() ( )2 22 hNd\\uf077 \\uf044\\uf0a3 P , (18) \\n() () () 2 2 2\\n2 12, , ,N \\uf0e9\\uf0f9=\\uf0eb\\uf0fbP\\u03b8\\u03b8\\u03b8\\n and \\n()\\n\\uf05b\\uf05d()( )\\n52\\n210, 1sup min ,\\nDN iiNdd\\n\\uf0a3\\uf0a3\\uf0ce=\\n\\u03b8P \\u03b8\\u03b8  is the dispersion of \\n2P . The smaller \\n()2 NdP\\n is, the smaller \\n2\\uf044  is. Using the same analysis process, the error bound at the \\n-thg  \\niteration can be expressed as  \\n \\n() ( )\\n()\\n\\uf05b\\uf05d( )()( )\\n21 10, 1sup min ,g\\ngDg h N g\\ng\\nNg iiNd\\ndd\\uf077\\n+ \\uf0a3\\uf0a3\\uf0ce\\uf044\\uf0a3\\n=\\n\\u03b8P\\nP \\u03b8\\u03b8 , (19) \\nwhere \\n \\n() () ()\\n() ()()()()() () ()( )12\\n2 1 2 1 2 1 2\\n1, 2, 0, 1, 1, 2, 2, 1, 1,, , ,\\n; ; ; ; ; ; ; ; ;g g g\\ng N\\ng\\nii i i i i i i g i g i \\u2212\\u2212\\uf0e9\\uf0f9=\\uf0eb\\uf0fb\\n=P\\u03b8\\u03b8\\u03b8\\n\\u03b8\\u03b5\\u03b5\\u03b5\\u03b5\\u03b5\\u03b5\\u03b5\\u03b5\\u03b5\\n . (20) \\n(19) shows that the error bound \\ng\\uf044  depends on the dispersion of \\ngP , which is a sample set in \\nthe \\n( ) 21gD+ -dimensional space. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n 4. PSO With Low -discrepancy S ampling  in the Expanded \\nDimensional Space  \\nIn this s ection, a new acceleration technique, LDS in the expanded dimensional space \\n(LDSEDS) is given. According to the different sample construction methods, two versions of \\nLDSEDS are proposed. These two techniques are combined with PSO to generate two more \\neffective algorithms.  \\n4.1 Inspiration from Error Analysis  \\nIn the introduction, we raised two questions . The first being, why can\\u2019t the advantages of \\nusing LDS initialization at the early iterations be preserved during the entire iteration s of the PSO? \\nFurthermore, what is the relationship between the error of the entire iteration process of PSO and \\nthe samp le uniformity?  The error analysis in Section  3 explains the second question.  As shown in  \\n(19), the error bound of PSO at each iteration is dependent on the dispersion of the sample set \\ngP . \\nThe answer to the first question is also revealed ; in the iterative process of PSO, it is necessary to \\nsample dynamically in the space with increasing dimensions . Generating the initial populati on \\nwith LDS only ensures that the sample sets \\n\\uf07b\\uf07d1,i\\u03b5  and \\n\\uf07b\\uf07d2,i\\u03b5  in the first two dimension s of \\ngP  are \\nof low-discrepancy but ignore  the other dimensions. With  the increase in iterations, \\ngP  contains an \\nincreasing number of  random numbers, becom ing a random sample set, and the advantages \\nintroduced by the low -discrepancy of \\n\\uf07b\\uf07d1,i\\u03b5  and \\n\\uf07b\\uf07d2,i\\u03b5  are completely destroye d.  \\nAt the same time,  the error analysis in Section 3 also explain s why the effect of using LDS  in \\neach iteration step is  worse . This is because the existing LDSs  are mostly deterministic point sets, \\nso the uniformity of the whole iteration process is destroyed , so the calculation error of the \\nalgorithm increases,  resulting in  the worse calculation results .  \\nPSO requires  the sample set \\ngP  in \\n( ) 21gD+ -dimensional space . Suppose we use LDS \\ninstead of random samples to generate \\ngP . In that case,  the dispersion of \\ngP  will be reduced, the \\nerror of PSO at each iteration will be reduced and the convergence speed will be improved. This is \\nthe critical technique of the accelerated PSO proposed in this paper, which we name low -\\ndiscrepancy sampling  in the expanded dimensional space (LDSEDS). The PSO acceleration This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n algorithm combined with this technique is called PSO with LDS in the expan ded dimensional \\nspace (P SO-LDSEDS ). \\n4.2 Low-discrepancy Sampling in Expanded Dimensional Space  \\nBased on Number Theory  \\nThe LDSEDS based on number theory directly generates the low -discrepancy sample set  \\nGP \\nin the ultrahigh dimension . The dimension of \\nGP  is so high that some LDSs  based on optimization \\nalgorithm s, e.g., DE S and OH S, are unsuitable because of the enormous  computational cost \\nrequired to generate the samples. Some LDSs based on number theory stand out in this case , such \\nas HWS , SS and H SS. For such high -dimensional space, we used HS S, a modified version of the \\nfamous H S. By adding some perturbation to the or iginal H S, HSS alleviat es the strip distribution  \\nof the sample set  form which H S, as well as HW S and S S, suffers.  \\nThen, this technique  is combined  with PSO. If the maximum number of iterations is \\nG , the \\niteration requires  \\nN samples \\n() ()\\n1 : , ,GG\\nG N \\uf0e9\\uf0f9=\\uf0eb\\uf0fb\\nP\\u03b8\\u03b8  in \\n( ) 21GD+ dimensional space. The first \\n2D  \\ndimensional numbers in \\nGP  are used for the generation of the initial positions  and velocities. The \\nnumbers in dimensions \\n2g  and \\n21g+  are used to generate \\n()1\\ng\\u03b5  and \\n()2\\ng\\u03b5 , respectively. This \\nimproved PSO algorithm is called  PSO-LDSEDS  based on number theory ( PSO-LDSEDS 1). \\n4.3 Low-discrepancy Sampling  in Expanded Dimensional Space  \\nBased on  a Combination of Low-dimensional Samples  \\nGenerally, the difference between the low -discrepancy sample set  generated by number \\ntheory and that of a random sample set increase s with the number of samples  but decreases with \\nthe dimension.  Considering that the dimension of the sample set \\nGP  involved in PSO is so high \\nthat directly generating it with number theory method may not decrease its discrepancy when the \\nnumber of particles is not too large. Therefore, we propose another LDSEDS based on a \\ncombination of low -dimensional samples.  \\nSuppo sing we have obtained a low -discrepancy sample set \\n\\u0398  with \\nN  samples in \\n-D This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n dimensional  space,  \\n \\n11 12 1 1,:\\n21 22 2 2,:\\n1 2 ,:N\\nN\\nD D DN D\\uf071 \\uf071 \\uf071\\n\\uf071 \\uf071 \\uf071\\n\\uf071 \\uf071 \\uf071\\uf0e9 \\uf0f9 \\uf0e9 \\uf0f9\\n\\uf0ea \\uf0fa \\uf0ea \\uf0fa\\n\\uf0ea \\uf0fa \\uf0ea \\uf0fa==\\uf0ea \\uf0fa \\uf0ea \\uf0fa\\n\\uf0ea \\uf0fa \\uf0ea \\uf0fa\\uf0eb \\uf0fb \\uf0eb \\uf0fb\\u03b8\\n\\u03b8\\u0398\\n\\u03b8\\n , (21) \\nand now we want to generate \\nN  samples in \\n2-D dimensional  space. For the number array \\n\\uf05b \\uf05d 1, 2, , D\\n, we make a random permutation , denoted by \\n\\uf05b \\uf05d 12, , , D \\uf070 \\uf070 \\uf070=\\u03c0\\n . Using \\n\\u03c0  can yield \\na new low-discrepancy sample set  \\n\\uf070\\u0398 \\n \\n1 1 1 1\\n2 2 2 21 2 ,:\\n1 2 ,:\\n1 2 ,:D D D DN\\nN\\nN\\uf070 \\uf070 \\uf070 \\uf070\\n\\uf070 \\uf070 \\uf070 \\uf070\\n\\uf070\\n\\uf070 \\uf070 \\uf070 \\uf070\\uf071 \\uf071 \\uf071\\n\\uf071 \\uf071 \\uf071\\n\\uf071 \\uf071 \\uf071\\uf0e9 \\uf0f9 \\uf0e9 \\uf0f9\\n\\uf0ea \\uf0fa \\uf0ea \\uf0fa\\n\\uf0ea \\uf0fa \\uf0ea \\uf0fa==\\uf0ea \\uf0fa \\uf0ea \\uf0fa\\n\\uf0ea \\uf0fa \\uf0ea \\uf0fa\\uf0eb \\uf0fb \\uf0eb \\uf0fb\\u03b8\\n\\u03b8\\u0398\\n\\u03b8\\n . (22) \\nClearly, \\n\\uf05b \\uf05d;\\u03c0\\u0398\\u0398  forms a new low-discrepancy sample set  with \\nN  samples in \\n2D -\\ndimensional  space.  Then, we apply this technique to PSO.  Using this combining method can \\neasily yield \\nGP  in \\n( ) 22GD+\\uf0b4 -dimensional  space,  \\n \\n\\uf05b \\uf05d1 2 2; ; ;G G + = \\u03c0 \\u03c0 P\\u0398\\u0398\\u0398\\n , (23) \\nwhere \\ni\\u03c0  represents the random permutation. By combining the low -dimensional samples \\ngenerated using different LDSs  (such as HW S, DES, HSS and S S) in a random permutation al way, \\nit is quite convenient  to generate the samples in the dynamically expanded dimensional  space. \\nThis more efficient PSO algorithm is called  PSO-LDSEDS , and is based on a c ombination of low -\\ndimensional samples  (PSO-LDSEDS 2). \\n5. Numerical Test of PSO -LDSEDS  \\nThis section uses numerical experiments to verify the correctness of the  error  analysis  in \\nSection 3 and the effectiveness of the two improved algorithms proposed in Section 4. \\n5.1 Measurement of Performance  \\nIn this section, the PSO s with two  different  samplings are considered as two different \\nalgorithms. To compare the performanc e of the different algorithms, we used the following \\nmetrics:  \\nError tolerance (\\ntol\\uf065 ): When the relative error between the fitness of the best solution and the This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n true optimal solution is smaller than \\ntol\\uf065 , the algorithm is considered convergent.  \\nConvergence speed  (CS) : This metric indicates  the number of  iterat ions after which  the \\nalgorithm converges. For  the two algorithms, the smaller CS  is, the more powerful  the search \\nability of the algorithm. Due to the randomness existing in PSO, 60 runs are performed for each \\ntest function and for each algorithm. The average convergence curve of 60 runs is recorded, and \\nthe CS for a given  error tolerance  is evaluated based on the average convergence curve. If the CS \\nis greater than the maximum number of iteration s \\nG, it is c laimed that the algorithm fails to find \\nthe global optimum under the given error tolerance , and it is represented by \\\" -\\\" in the table.  \\n5.2 Test Functions  \\nIn this section, 15 benchmark functions selected from the 30 wi dely used benchmark \\nfunctions provided by the CEC 2017 special session [48] are used as the test functions. The name, \\ntopology, and tr ue optimal solutions  (\\n*Z) of these 15 test functions are listed  in Table I. For the \\nother 15 functions the original PSO cannot converge within \\n7500G=  iterations,  and they are \\ndiscarded . \\nTABLE I  \\nBENCHMARK FUNCTIONS  \\nTopology  No. Function name  \\n*Z  \\nUnimodal Fns.  \\n1F  Shifted and rotated Zakharov function  300 \\nSimple \\nmultimodal  \\nFns. \\n2F Shifted and rotated Rosenbrock\\u2019s function  400 \\n3F\\n Shifted and rotated Rastrigin\\u2019s function  500 \\n4F\\n Shifted and rotated Expaned Schaffer F6 function  600 \\n5F\\n Shifted and rotated Lunacek Bi -Ratrigin\\u2019s function  700 \\n6F\\n Shifted and rotated Non-continuous Rastrigin\\u2019s function  800 \\n7F\\n Shifted and rotated Lvey function  900 \\nHybrid Fns.  \\n8F  Zakharov, Rosenbrock, Rastrigin  1100  \\n9F\\n High -conditioned elliptic; Ackley; Schaffer; Rastrigin  1400  \\n10F\\n Bent Cigar; HGBat; Rastrigin; Rosenbrock  1500  \\n11F\\n Expanded schaffer; HGBat; Rosenbrock; Modified Schwefel; Rastrigin  1600  \\n12F\\n Katsuura; Ackely; Expanded Griewank plus Rosenbrock; Schwefel; Rastrigin  1700  \\n13F\\n High -conditioned elliptic; Ackley; Rastrigin; HGBat; Dicus  1900  \\n \\n14F  Bent Cigar; Griewank plus Rosenbrock; Rastrigin; Expanded Schaffer  2000  \\nComposite F ns. \\n15F  Griewank; Rastrigin; Modified schwefel  2200  \\nFns. Functions This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n 5.3 Statistical Analysis  \\nThe modified Friedman test  (MFT) [49] and the Nemenyi test  (NT) [50] were used here to \\ncompare the performance of  the different algorithms. The MFT justifies whether  there are \\nsignificant differences between the considered algorithms in terms of the Friedman statictics \\nF\\uf074  \\n \\n( )\\n( ) ( )( )22\\n22\\n211 12,1 1 4F\\nF i F\\ni Fm k k mRm k k k\\uf063\\uf074\\uf063\\uf063\\uf0e9\\uf0f9 \\u2212+= = \\u2212 \\uf0ea\\uf0fa\\u2212 \\u2212 +\\uf0eb\\uf0fb\\uf0e5 , (24) \\nwhere \\nm  is the number of test functions, \\nk  is the number of algorithms considered, and \\niR  is the \\naverage rank of the \\n-thi  algorithm. T he smaller \\niR  is, the better the performance of the \\n-thi\\nalgorithm.  If \\nF\\uf074  is greater than the critical value \\nc\\uf074  for a given s ignificance  level \\n\\uf061 , it is claimed \\nthat there are significant differences between the considered algorithms. If signific ant differences \\nare observed, the NT is used to justify which algorithm is signifi cantly superior to  the other \\nalgorithms. This test states that the performance of  the two algorithms is significantly different if \\nthe corresponding average ranks differ by at least a critical difference , defined as follows : \\n \\n( )1\\n6kkCD qm\\uf061+= , (25) \\nwhere \\nq\\uf061  can be obta ined by querying the studentized range statistic  table [50] with the \\nsignificance  level \\n\\uf061 . In all the experiments, the confidence level is set to \\n0.05\\uf061= . \\n5.4 Performance of PSO -LDSEDS1  \\nIn this section, a comparison between PSO and PSO -LDSEDS1 is presented using the test \\nfunctions with \\n10D= . The parameters involved are the same for both algorithms except for the \\nsample sets.  The maximum number of iterations is \\n7500G= . \\nw decreases linearly with iteration s \\nin the range of 0.9 -0.4. \\n1c  and \\n2c  vary linearly with iteration s in the range of 2.5 -0.5 and 0.5 -2.5, \\nrespectively. To compare the CSs of differ ent algorithms at different  error tolerances, \\ntol1% \\uf065=  \\nand \\n5%  are used.  \\nPSO-LDSEDS1 directly us es the HS S method to generate \\nGP  with dimensions  of \\n( ) 22GD+ . \\nObviously,  \\nGP is an ultrahigh dimensional LDS, so it may need a large number of samples to \\nmatch the dimensions to achieve the ideal uniformity effect . Here, three groups of c omparative \\nexperiment s with \\n40N= , 100 , and 160, are given to compare the CSs of the original PSO This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n (denoted by Rand) and PSO -LDSEDS1 (denoted by HS S) at \\ntol1% \\uf065=  and \\n5% . The CSs  and \\nranks of Rand and HS S with different population sizes  at different tolerance errors and population \\nsizes are listed in Table II. \\nAs shown in Table II, the A vgRks  of PSO-LDSEDS1 are smaller than th ose of PSO in almost \\nall simulation conditions, ex cept \\n40N=  and \\ntol5% \\uf065= . For both \\ntol1% \\uf065=  and \\n5% , the Friedman  \\nstatistics \\nF\\uf074  increase as \\nN  increase s. This is because the LDS  covers the search  space better than \\nthe random sample set,  and the advantage becomes more evident as the population  size increases.  \\nWhen  \\n160N= , the Friedman  statistics  at \\ntol1% \\uf065=  and \\n5%  are both far greater than the critical \\nvalue \\nc4.600 \\uf074= . Therefore, it can be stated that when \\n160N= , PSO -LDSEDS1 converge s \\nsignificantly faster than PSO at the significance  level \\n0.05\\uf061= . \\nThe experimental result shows that , although directly generating the low -discrepancy sample \\nGP\\n can improve the numerical performance of PSO, the po pulation size required need s to be large \\nenough, which increases the computational cost. Next, we test the performance of PSO -LDSEDS2.  \\nTABLE II \\nCSS AND  RANKS OF DIFFERENT ALGORITHMS WITH DIFFERENT POPULATION SIZES  AT \\nDIFFERENT TOLERANCE ERRORS  \\n \\ntol5% \\uf065=  \\ntol1% \\uf065=  \\n40N=\\n \\n100N=  \\n160N=  \\n40N=  \\n100N=  \\n160N=  \\nRand HSS Rand HSS Rand HSS Rand HSS Rand HSS Rand HSS \\n1F\\n 1515 (2) 1281 (1) 1385(2)  1223(1)  1336(2)  1201(1)  1626 (2) 1388 (1) 1492(2)  1314(1)  1436(2)  1289(1)  \\n2F\\n 189(1) 353(2) 82(1)  117(2)  52(2)  42(1)  2006 (2) 1611 (1) 1799(2)  1563(1)  1688(2)  1406(1)  \\n3F\\n 948(2) 747(1) 835(2)  811(1)  745(1)  770(2)  -(1.5)  -(1.5)  2765(1)  4674(2)  2252(1)  2306(2)  \\n4F\\n 24(2) 17(1) 16(2)  15(1)  15(2)  13(1)  474(1) 479(2) 314(1)  346(2)  254(2)  233(1)  \\n5F\\n 1420 (2) 1098 (1) 1299(2)  1072(1)  1244(2)  1048(1)  -(1.5)  -(1.5)  -(1.5)  -(1.5)  -(1.5)  -(1.5)  \\n6F\\n 582(2) 494(1) 327(1)  454(2)  325(2)  287(1)  2472 (1) -(2) 1928(1)  2008(2)  1716(2)  1683(1)  \\n7F\\n 562(2) 511(1) 418(2)  415(1)  281(2)  272(1)  908(2) 729(1) 803(2)  704(1)  675(2)  633(1)  \\n8F\\n 350(1) 398(2) 228(2)  208(1)  172(2)  160(1)  1061 (2) 905(1) 914(2)  821(1)  860(2)  809(1)  \\n9F\\n 1054 (1) 1121 (2) 753(2)  739(1)  705(2)  632(1)  -(1.5)  -(1.5)  5713(1)  -(2) 2715(2)  2651(1)  \\n10F\\n 1995 (1) 2070 (2) 1394(2)  1283(1)  1305(2)  1205(1)  3767 (1) -(2) 2588(2)  2559(1)  2310(2)  2128(1)  \\n11F\\n 452(1) 460(2) 233(1)  261(2)  191(2)  169(1)  906(2) 843(1) 601(1)  683(2)  601(2)  563(1)  \\n12F\\n 392(2) 328(1) 162(1)  208(2)  137(2)  134(1)  1730 (2) 1720 (1) 1360(2)  1286(1)  1261(2)  1243(1)  \\n13F\\n 1656 (1) -(2) 1019(2)  888(1)  793(2)  774(1)  3230 (1) -(2) 1726(2)  1703(1)  1552(2)  1526(1)  \\n14F\\n 165(2) 147(1) 83(1)  90(2)  65(2)  64(1)  1200 (2) 1140 (1) 1116(2)  972(1)  956(2)  893(1)  \\n15F\\n 814(2) 723(1) 396(1)  458(2)  200(2)  22(1)  -(1.5)  -(1.5)  -(1.5)  -(1.5)  -(2) 736(1)  \\nAvgR.  1.60(2) 1.40(1) 1.60(2) 1.40(1) 1.93(2) 1.07(1) 1.60(2) 1.40(1) 1.60(2) 1.40(1) 1.90(2) 1.10(1) \\nF\\uf074\\n 0.583  0.583  42.250  0.583 0.583  24.889  \\nAvgR.  Average ranks This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n 5.5 Performance  of PSO -LDSEDS 2 \\nThe comparison between PSO and PSO -LDSEDS2 using the \\n10D=  dimensional test \\nfunctions is  presented  first. The adopted parameters are the same as those used previously , except \\nfor the sample sets  \\n40N= . PSO -LDSEDS2 uses the combining method introduced in Section 4 \\nto generate \\nGP . As shown b y (23), the seed sample set \\n\\u0398  was generated using four LDSs , i.e., \\nHWS, DES, HSS, and S S. For convenience, PSO -LDSEDS2 , which uses certain type s of LDSs  to \\ngenerate the seed sample set , is denoted by the name of the corresponding LDS . As the original \\nPSO generates \\nGP  by random sampling, it is denoted by Rand. The CSs , the computation times  \\n(CTs) and their ranks of Rand, HW S, DES, HSS, and S S at different tolerance errors are listed in \\nTable III and Table IV.  \\nThe comparison in Table III and Table IV shows that under the same accuracy requirements, \\nthe number of iterations  and the CTs required by PSO -LDSEDS2 is significantly less than that \\nrequired by PSO. According to Table III and Table IV, for both \\ntol5% \\uf065=  and 1%, \\nF\\uf074  is greater \\nthan the critical value \\nc2.537 \\uf074=  in the significance level \\n0.05\\uf061= . According to NT and  (25), the \\ncritical difference  \\n1.575 CD= . Therefore, the A vgRks  of PSO -LDSEDS2 with different LDS s are \\nalways significantl y smaller than those  of PSO at both \\ntol5% \\uf065=  and 1%.  \\nNext, the performance of PSO -LDSEDS2 is evaluated using the \\n30D=  dimensional test \\nfunctions. When  \\n30D= , there are many test functions in Table I for which both PSO and PSO -\\nLDSEDS2 fail to find solutions with relative errors smaller than \\ntol5% \\uf065= . Therefore, \\ntol20% \\uf065=  \\nand \\ntol10% \\uf065=  were used. In addition, there are still five test functions for which the adopted \\nalgorithms fail at \\ntol20% \\uf065= , and we adopted only the re maining  ten test functions. The \\npopulation size is \\n100N= . The o ther parameters are the same as those used previously . The CSs , \\nthe CTs  and the ir ranks of  the considered algorithms at different tolerance errors are listed in Table \\nV and Table  \\u2165. \\nThe comparison in Table V and Table  \\u2165 shows that for 30 -dimensional problem s, PSO-\\nLDSEDS2 still performs significantly better than PSO under the same error tolerance.  For both \\ntol20% \\uf065=\\n and 10%, \\nF\\uf074  are greater than the critical value \\nc2.634 \\uf074=  in the significance level  \\n0.05\\uf061=\\n. Therefore, there are significant differences between th e CSs and the CTs  of the five \\nalgorithms  considered.  The A vgRks  of HW S, DES, HS S and SS are all smaller than that of Rand. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n According to NT and  (25), \\n1.929 CD= . Therefore, HW S, DES and HS S perform significantly \\nbetter than Rand a t\\ntol20% \\uf065= , and  HWS and DES perform significantly better than Rand at  \\ntol10% \\uf065=\\n.Although the difference between the A vgRks  of Rand and S S is not greater than  \\nCD , \\nSS actually performs better than Rand for almost all  the test functions . \\nTABLE III \\nCSS AND  RANKS OF THE FIVE ALGORITHMS AT DIFFERENT TOLERANCE ERRORS  \\n \\ntol5% \\uf065=  \\ntol1% \\uf065=  \\nRand HWS  DES HSS SS Rand HWS  DES  HSS SS \\n1F\\n 1515 (5) 943(2) 977(4) 963(3) 916(1) 1626 (5) 1032 (2) 1064 (4) 1056 (3) 997(1) \\n2F\\n 189(5) 62(1) 142(4) 94(3) 73(2) 2006 (5) 1532 (2) 1800 (3) 1813 (4) 1451 (1) \\n3F\\n 948(5) 428(1) 496(3) 520(4) 488(2) -(5) 4202 (4) 3474 (3) 2347 (1) 3271 (2) \\n4F\\n 24(5) 20(2) 21(3) 19(1) 22(4) 474(5) 128(1) 136(3) 149(4) 130(2) \\n5F\\n 1420 (5) 856(2) 804(1) 870(3) 901(4) -(3) -(3) -(3) -(3) -(3) \\n6F\\n 582(5) 192(2) 195(3) 179(1) 197(4) 2472 (5) 1717 (2) 1786 (4) 1702 (1) 1724 (3) \\n7F\\n 562(5) 163(2) 178(4) 171(3) 152(1) 908(5) 381(3) 384(4) 376(2) 335(1) \\n8F\\n 350(5) 169(3) 154(1.5)  154(1.5)  196(4) 1061 (5) 625(3) 610(1) 652(4) 612(2) \\n9F\\n 1054 (5) 833(1) 874(2) 941(4) 899(3) -(3) -(3) -(3) -(3) -(3) \\n10F\\n 1995 (4) 2068 (5) 1841 (3) 1766 (2) 1747 (1) 3767 (5) 3666 (2) 3633 (1) 3765 (4) 3668 (3) \\n11F\\n 452(5) 308(3) 280(2) 345(4) 263(1) 906(5) 831(4) 672(2) 749(3) 624(1) \\n12F\\n 392(5) 229(3) 251(4) 207(2) 168(1) 1730 (3) 2328 (4) 2947 (5) 1676 (1) 1702 (2) \\n13F\\n 1656 (4) 1499 (1) 1646 (3) 1501 (2) 2115 (5) 3230 (4) 3185 (3) 3127 (2) 3030 (1) 4001 (5) \\n14F\\n 165(5) 158(4) 121(2) 118(1) 124(3) 1200 (5) 976(3) 858(1) 914(2) 979(4) \\n15F\\n 814(5) 193(3) 256(4) 131(1) 140(2) -(3) -(3) -(3) -(3) -(3) \\nAvgR.  4.87(5) 2.33(1) 2.90(4) 2.37(2) 2.53(3) 4.40(5) 2.800 (3.5)  2.800 (3.5)  2.600 (2) 2.400 (1) \\nF\\uf074\\n 11.728  4.817 \\n  \\nTABLE IV  \\nCTS AND  RANKS  OF THE FIVE ALGORITHMS AT DIFFERENT TOLERANCE ERRORS  \\n \\ntol5% \\uf065=  \\ntol1% \\uf065=  \\nRand HWS  DES HSS SS Rand HWS  DES  HSS SS \\n1F\\n 0.101(5) 0.064(1) 0.070(3) 0.071 (4) 0.065(2) 0.109(5) 0.069(2) 0.071(3) 0.074 (4) 0.067(1) \\n2F\\n 0.010 (4) 0.004(1) 0.014(5) 0.006(3) 0.004 (2) 0.097(5) 0.069(1) 0.086(3) 0.091 (4) 0.074(2) \\n3F\\n 0.064(5) 0.033(1) 0.034 (3) 0.038 (4) 0.033 (2) 0.321 (5) 0.229(1) 0.256(3) 0.247 (2) 0.284(4) \\n4F\\n 0.004 (5) 0.003 (1) 0.004 (4) 0.003 (2) 0.004(3) 0.060 (5) 0.017 (2) 0.019(4) 0.019(3) 0.017 (1) \\n5F\\n 0.101 (5) 0.054 (1) 0.061(3) 0.067(4) 0.056(2) 0.592 (5) 0.570(2) 0.543(1) 0.591(4) 0.586(3) \\n6F\\n 0.041(5) 0.016(4) 0.016(3) 0.015 (2) 0.012 (1) 0.234 (5) 0.157(2) 0.152(1) 0.162(3) 0.173 (4) \\n7F\\n 0.041(5) 0.010 (1) 0.011(2) 0.015(4) 0.011 (3) 0.065 (5) 0.025 (2) 0.026(3) 0.029(4) 0.024 (1) \\n8F\\n 0.022(5) 0.010(2) 0.009 (1) 0.011(3) 0.012(4) 0.068(5) 0.034(1) 0.040(2) 0.054 (4) 0.044 (3) \\n9F\\n 0.076 (5) 0.064(2) 0.059(1) 0.074 (4) 0.066 (3) 0.443 (5) 0.403 (1) 0.426(3) 0.428 (4) 0.404(2) \\n10F\\n 0.148(4) 0.137(2) 0.152(5) 0.136(1) 0.145(3) 0.255(5) 0.214 (1) 0.229(2) 0.239 (4) 0.230 (3) \\n11F\\n 0.030(4) 0.023 (3) 0.017 (1) 0.030 (5) 0.020 (2) 0.062 (5) 0.040(1) 0.042(2) 0.048 (4) 0.043(3) \\n12F\\n 0.042 (5) 0.031 (3) 0.036(4) 0.024 (1) 0.030(2) 0.642 (4) 0.654 (5) 0.521 (3) 0.442(1) 0.446 (2) \\n13F\\n 0.514(3) 0.368(1) 0.535 (4) 0.489(2) 0.630 (5) 1.208 (3) 1.114(1) 1.215(4) 1.127(2) 1.281(5) This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n \\n14F 0.024 (5) 0.022 (4) 0.015 (2) 0.017 (3) 0.015 (1) 0.319(5) 0.259 (2) 0.206 (1) 0.261 (3) 0.317(4) \\n15F\\n 0.125(5) 0.044 (3) 0.051(4) 0.042(2) 0.039 (1) 1.293 (4) 1.311 (5) 1.276(3) 1.212 (1) 1.232(2) \\nAvgR.  4.667(5) 2.000 (1) 3.000 (4) 2.933 (3) 2.400 (2) 4.733 (5) 1.933 (1) 2.533 (2) 3.133 (4) 2.667(3) \\nF\\uf074\\n 9.900  11.430  \\nTABLE V \\nCSS AND RANKS OF THE FIVE ALGORITHMS AT DIFFERENT TOLERANCE ERRORS  \\n \\ntol20% \\uf065= \\ntol10% \\uf065=  \\nRand HWS  DES  HSS SS Rand HWS  DES  HSS SS \\n1F\\n 3702(5)  3491(3)  3456(1)  3487(2)  3545(4)  3787(5)  3595(3)  3571(1)  3572(2)  3644(4)  \\n3F\\n 1843(5)  1498(3)  1455(1)  1497(2)  1519(4)  2803(5)  2011(2)  2228(4)  2048(3)  1949(1)  \\n4F\\n 2(3) 2(3) 2(3) 2(3) 2(3) 21(5)  20(3.5)  19(1.5)  19(1.5)  20(3.5)  \\n5F\\n 1985(5)  1581(1)  1598(3)  1596(2)  1704(4)  -(3) -(3) -(3) -(3) -(3) \\n6F\\n 1665(5)  1122(2)  1195(4)  1120(1)  1139(3)  2016(5)  1580(2)  1606(3)  1538(1)  1647(4)  \\n7F\\n 1691(5)  1371(4)  1357(3)  1289(1)  1356(2)  2020(5)  1744(2)  1852(3)  1725(1)  1963(4)  \\n8F\\n 1409(5)  1105(3)  1066(1)  1122(4)  1079(2)  2001(2)  1963(1)  2595(3)  2602(4)  2703(5)  \\n12F\\n 1168(5)  970(2)  956(1)  988(3)  1033(4)  2962(5)  2275(3)  2000(2)  1856(1)  2633(4)  \\n14F\\n 1038(5)  732(1)  770(2)  1032(4)  857(3)  1761(4)  1443(2)  1394(1)  3224(5)  1636(3)  \\n15F\\n 823(5)  324(2)  188(1)  387(4)  375(3)  1402(5)  531(2)  310(1)  940(4)  553(3)  \\nAvgR.  4.800(5) 2.400(2) 2.000(1) 2.600(3) 3.200(4) 4.400(5) 2.350(2) 2.250(1) 2.550(4) 3.450(3) \\nF\\uf074\\n 8.308  4.534 \\n \\nTABLE \\u2165 \\nCTS AND  RANKS  OF THE FIVE ALGORITHMS AT DIFFERENT TOLERANCE ERRORS  \\n \\ntol20% \\uf065= \\ntol10% \\uf065=  \\nRand HWS  DES  HSS SS Rand HWS  DES  HSS SS \\n1F\\n 1.866(5) 1.492 (1) 1.559 (2) 1.566 (3) 1.570 (4) 1.744 (5) 1.423(1) 1.484 (3) 1.462(2) 1.487 (4) \\n3F\\n 1.253 (5) 0.916 (2) 0.792(1) 0.939(3) 0.944(4) 3.095(5) 2.304(2) 2.081(1) 2.465(4) 2.375(3) \\n4F\\n 0.007 (2) 0.007 (3) 0.007(1) 0.007 (4) 0.008(5) 0.028 (5) 0.023 (2) 0.022 (1) 0.025 (3) 0.026 (4) \\n5F\\n 1.648 (5) 1.140 (1) 1.215(2) 1.222 (3) 1.340(4) 5.422(5) 5.055(2) 5.015 (1) 5.197(4) 5.078(3) \\n6F\\n 1.171 (5) 0.806(3) 0.624(1) 0.831(4) 0.771 (2) 1.636(5) 1.321(3) 1.107 (1) 1.151(2) 1.355 (4) \\n7F\\n 1.182(5) 0.873(2) 0.963(3) 0.968(4) 0.829 (1) 1.462(3) 1.254(1) 1.742 (5) 1.314 (2) 1.488(4) \\n8F\\n 0.913 (5) 0.655(3) 0.576 (1) 0.761(4) 0.627(2) 1.785(2) 1.827 (3) 1.706(1) 2.367 (5) 2.127 (4) \\n12F\\n 1.912(5) 1.338(2) 1.582 (3) 1.756 (4) 1.273 (1) 4.454 (5) 3.842 (1) 4.097(3) 3.907(2) 4.369 (4) \\n14F\\n 1.397 (5) 0.998(1) 1.031(2) 1.352 (4) 1.169(3) 3.323 (1) 3.431(2) 3.448(3) 6.032(5) 3.774(4) \\n15F\\n 1.952(5) 0.761(3) 0.217 (1) 0.834(4) 0.456(2) 1.928 (5) 0.773(2) 0.451 (1) 0.996(4) 0.838(3) \\nAvgR.  4.700(5) 2.100(2) 1.700(1) 3.700(4) 2.800 (3) 4.100 (5) 1.900 (1) 2.000(2) 3.300 (3) 3.700 (4) \\nF\\uf074\\n 13.059  6.000 \\n \\n5. Disscussio ns \\nFrom previous experiments, we can conclud e that generating \\nGP  with LDS s improve s the \\nconvergence  speed of PSO. For PSO -LDSEDS1, when  \\n40N= , its performance is similar to that \\nof PSO. This is because the population size is so small that the uniformity of the high -dimensional This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n sample set generated by directly using the LDS  is not better than that of the random sample sets. \\nHowever, with an increase in the population size, PSO -LDSEDS1 performs increasingly  better. \\nWhen  \\n160N= , the convergence speed of PSO -LDSEDS1 is significantly superior to that of PSO. \\nThe PSO-LDSEDS2  can substantially improve the convergence speed without reducing the search \\nability of the original PSO for the 10 - and 30 -dimensional problems. We adopted four sampling \\nmethods to generate the seed low -discrepancy sample set, and HW S and DES are more rob ust \\nthan the other  two samplings. Tables  III-\\u2165 show  that the combin ed sampling technique  is more \\nefficient than direct sampling based on  number theory, and both technique s are more efficient than  \\nrandom sampling.  \\nIt is natural  to question whether  the LDSEDS technique  can be extended to other modified \\nversions of PSO. In the next section, the LDSE DS was combined with a widely used modified \\nPSO, i.e., CLPSO, and experiments and discussions were presented for testing the performance of \\nCLPSO -LDSEDS.  \\n6. Extended to  CLPSO  \\nIt seems that LDSEDS  can be extended to other PSO variants. According to the error analysis \\nin Section 3, there is a  precondition that the  position updating of each particle depends on the \\nrandom variables that are independent of the other particles and a global best that is the same for \\nall the particles . Nevertheless, among the many variants of PSO, few algorithms can strictly meet \\nthese preconditions except for PSO , which  directly changes the control parameter.  For some  PSO \\nvariants that change the population topologies , each particle learn s a lot from  the adjacent particles , \\nwhich undoubtedly does not meet the preconditions of the analysis in Section 3. However, there \\nare some excellent PSO variants, such as CLPSO and HCLPSO, in which each particle updates its \\nposition with only a small amount of information from  the other particles  and mainly depend s on \\nthe information of the particle itself. These types of algorithms c ould be expected to accelerate the \\nconvergence speed by using the LDSE DS. In this section, we choose the classic CLPSO among \\nthese variants as the carrier for the next promotion.  \\nAs shown in Section 5, the combining sampling based on random permutations of a low -\\ndimensional  low-discrepancy sample set is more efficient,  and it will be adopted for CLPSO. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n Unlike PSO, the velocity updating formula of CLPSO does not involve global optima. Therefore, \\nin each iteration of CLPSO, it is only necessary  to sample the random vector \\n,gi\\u03b5  once.  Within \\nG  \\niterations, the dimension of the total sample set \\nGP  is \\n( )1GD+\\uf0b4 . For a seed low -discrepancy \\nsample set \\n\\u0398 , combining its random permutations yields  \\n \\n\\uf05b \\uf05d12; ; ;G G + = \\u03c0 \\u03c0 P\\u0398\\u0398\\u0398\\n  (26) \\nwhere \\ni\\u03c0  represents the random permutation, as shown in Section 4. The modified CLPSO is \\ndenoted as CLPSO -LDSEDS.   \\n6.1 Test Functions  \\nThe test functions were still provided by  the CEC 2017 special session  [48]. As the search \\ncapability of CLPSO is better than that of PSO, we can select 19 test functions, listed in Table V, \\nfor which CLPSO can find optimal solutions with relative errors smaller than \\ntol5% \\uf065= . The o ther \\n11 test functions are dis carded. \\nTABLE VII \\nBENCHMARK FUNCTIONS  \\nTopology  No. Function name  \\n*Z  \\nUnimodal Fns.  \\n \\n1F Shifted and rotated Zakharov function  300 \\nSimple \\nmultimodal \\nFns. \\n2F Shifted and rotated Rosenbrock\\u2019s function  400 \\n3F\\n Shifted and rotated Rastrigin\\u2019s function  500 \\n4F\\n Shifted and rotated Expaned Schaffer F6 function  600 \\n5F\\n Shifted and rotated Lunacek Bi -Ratrigin\\u2019s function  700 \\n6F\\n Shifted and rotated Non -continuous Rastrigin\\u2019s function  800 \\n7F\\n Shifted and rotated Lvey function  900 \\nHybrid Fns.  \\n8F  Zakharov, Rosenbrock, Rastrigin  1100  \\n9F\\n Bent Cigar, Rosenbrock; Lunacek bi -Rastrigin  1300 \\n10F\\n High -conditioned elliptic; Ackley; Schaffer; Rastrigin  1400  \\n11F\\n Bent Cigar; HGBat; Rastrigin; Rosenbrock  1500  \\n12F\\n Expanded schaffer; HGBat; Rosenbrock; Modified Schwefel; Rastrigin  1600  \\n13F\\n Katsuura; Ackely; Expanded Griewank plus Rosenbrock; Schwefel; Rastrigin  1700  \\n14F\\n High -conditioned elliptic; Ackley; Rastrigin; HGBat; Dicus  1900  \\n15F\\n Bent Cigar; Griewank plus Rosenbrock; Rastrigin; Expanded Schaffer  2000  \\nComposite \\nFns. \\n16F Katsuura; Ackley; Rastrigin; Schaffer; Modified Schwefel  2100  \\n17F\\n Griewank; Rastrigin; Modified schwefel  2200  \\n18F\\n Ackley; Griewank; Rastrigin; High -conditioned Elliptic  2400  \\n19F\\n Modified Schwefel; Rastrigin; Rosenbrock; Griewank; Expanded schaffer  2600 This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n 6.2 Parameters Setting  \\nThe CLPSO code is downloaded from https:\\/\\/github.com\\/ zmdsn\\/ALFRAME . The inertia \\nweight \\nw  decreases linearly with iteration s in the range of 0.9 -0.2. The acceleration coefficient is  \\n1.49445c=\\n. The maximum number of iterations is \\n7500G= . When \\n10D= , \\n40N= , and  \\ntol1% \\uf065=\\n and \\n5%  were used . When \\n30D= , \\n100N= , and  \\ntol10% \\uf065=  and 20% were used.  \\n6.3 Numerical Comparisons and Discussions  \\nA comparison between CLPSO and CLPSO -LDSEDS with 10 dimensional test functions is \\nfirst presented. HW S, DES, HS S and S S were adopted  to generate \\n\\u0398 . The CS s, the CTs and the ir \\nranks  of the results obtained by using the original CLPSO (denoted by Rand) and the CLPSO with \\ndifferent LDSs . (denoted by the name of the sampling method) are listed in Table \\u2167 and Table  \\n\\u2168. \\nAs shown in Table \\u2167 and Table  \\u2168, \\nF\\uf074 is greater than the critical value \\nc2.537 \\uf074=  for both \\ntol5% \\uf065=\\n and \\ntol1% \\uf065= , which means  that there are significant differences between the \\nperformances of all five algorithms considered. According to the NT,  \\n1.399 CD= , the AvgRks  of \\nthe CLPSO -LDSEDSs with HWS, DES , and S S are significantly smaller than th ose of CLPSO, \\nwhich means that these three algorithms converge significantly faster than CLSPO at the same \\nerror tolerance. Th e AvgRk of CLPSO -LDSEDS with HSS is also smaller than that of the original \\nCLPSO.   \\nTABLE  \\u2167 \\nCSS AND  RANKS OF THE FIVE ALGORITHMS AT DIFFERENT TOLERANCE ERRORS  \\n \\ntol5% \\uf065=  \\ntol1% \\uf065=  \\nRand HWS  DES  HSS SS Rand HWS  DES  HSS SS \\n1F\\n 2285 (5) 2243 (3) 2203 (2) 2272 (4) 2116 (1) 2603(5)  2602(4)  2527(2)  2589(3)  2405(1)  \\n2F\\n 437(5) 418(3) 409(2) 433(4) 385(1) 2850(5)  2755(2)  2774(4)  2770(3)  2275(1)  \\n3F\\n 972(5) 857(2) 876(3) 956(4) 785(1) 3732(3.5)  3761(5)  3566(1)  3732(3.5)  3655(2)  \\n4F\\n 127(5) 111(1.5) 120(3) 122(4) 111(1.5) 510(5)  442(3)  438(2)  509(4)  427(1)  \\n5F\\n 1568 (5) 1461 (3) 1409 (2) 1566 (4) 1252 (1) -(3) -(3) -(3) -(3) -(3) \\n6F\\n 387(3) 385(2) 400(4) 462(5) 349(1) 2898(5)  2728(3)  2812(3)  2869(4)  2597(1)  \\n7F\\n 549(4) 527(3) 519(2) 561(5) 487(1) 846(4)  785(2)  764(2)  896(5)  729(1)  \\n8F\\n 468(5) 446(2) 455(4) 452(3) 411(1) 1180(5)  1149(3)  1110(2)  1167(4)  1086(1)  \\n9F\\n 6898 (5) 4419 (3) 5961 (4) 4376 (2) 4137 (1) -(3) -(3) -(3) -(3) -(3) \\n10F\\n 2423 (4) 1925 (1) 2065 (2) 2215 (3) 2755 (5) -(4.5)  7499(2.5)  7499(2.5)  7196(1)  -(4.5) This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n \\n11F 2254 (5) 2042 (3) 1829 (1) 2162 (4) 2030 (2) 6976(5)  5874(3)  5076(1)  5597(2)  6703(4)  \\n12F\\n 601(5) 506(1) 591(3) 597(4) 524(2) 1128(5)  1024(1)  1118(4)  1097(3)  1078(2)  \\n13F\\n 315(4) 306(3) 268(1) 344(5) 274(2) 2416(5)  2373(3)  2324(2)  2401(4)  2307(1)  \\n14F\\n 1246 (5) 864(1) 950(2) 1221 (4) 1023 (3) 3294(5)  2494(1)  2806(3)  2928(4)  2775(2)  \\n15F\\n 252(5) 238(3) 235(2) 245(4) 198(1) 1330(5)  1110(1.5)  1177(3)  1299(4)  1110(1.5)  \\n16F\\n -(4) -(4) 6884 (2) -(4) 3407 (1) -(3) -(3) -(3) -(3) -(3) \\n17F\\n 683(5) 677(4) 584(1) 627(2) 641(3) -(3) -(3) -(3) -(3) -(3) \\n18F\\n 4931 (4) 4634 (2) -(5) 4746 (3) 3072 (1) -(3) -(3) -(3) -(3) -(3) \\n19F\\n -(4) -(4) 5858 (2) 5811 (1) -(4) -(3) -(3) -(3) -(3) -(3) \\nAvgR.  4.58(5) 2.55(3) 2.47(2) 3.63(4) 1.76(1) 4.21(5) 2.74(3) 2.61(2) 3.29(4) 2.16(1) \\nF\\uf074\\n 17.287  5.947  \\n \\nTABLE \\u2168 \\nCTS AND  RANKS  OF THE FIVE ALGORITHMS AT DIFFERENT TOLERANCE ERRORS  \\n \\ntol5% \\uf065=  \\ntol1% \\uf065=  \\nRand HWS  DES  HSS SS Rand HWS  DES  HSS SS \\n1F\\n 1.029 (5) 0.807(1) 0.914 (3) 0.922 (4) 0.832 (2) 1.152(5) 0.967 (1) 1.090 (3) 1.097 (4) 0.998 (2) \\n2F\\n 0.286 (5) 0.222(1) 0.284(4) 0.280(3) 0.248 (2) 1.246(4) 1.146 (1) 1.239(3) 1.284(5) 1.226(2) \\n3F\\n 0.445 (5) 0.314 (1) 0.404 (3) 0.422 (4) 0.362(2) 1.706(5) 1.355(1) 1.641(4) 1.617(3) 1.512 (2) \\n4F\\n 0.193 (4) 0.165 (2) 0.181(3) 0.205(5) 0.157 (1) 0.227 (5) 0.165 (1) 0.196(3) 0.221(4) 0.191(2) \\n5F\\n 0.670(5) 0.575 (2) 0.652(3) 0.679(4) 0.552 (1) 3.7680(5) 3.184(1) 3.702(3) 3.736(4) 3.677(2) \\n6F\\n 0.215 (5) 0.144 (1) 0.204(4) 0.203 (3) 0.188(2) 1.320 (5) 1.108(1) 1.275(3) 1.282(4) 1.269 (2) \\n7F\\n 0.234(5) 0.178 (1) 0.215 (3) 0.228 (4) 0.194 (2) 0.413(4) 0.316(1) 0.386 (3) 0.419(5) 0.349(2) \\n8F\\n 0.177 (5) 0.144 (1) 0.172 (3) 0.173 (4) 0.161 (2) 0.516(4) 0.415(1) 0.462(3) 0.534(5) 0.442 (2) \\n9F\\n 1.795 (5) 1.294 (1) 1.713(3) 1.756(4) 1.575(2) 3.257(5) 2.698(1) 3.235 (4) 3.109 (2) 3.139(3) \\n10F\\n 0.868 (5) 0.633(1) 0.858(4) 0.822(2) 0.822 (3) 2.784 (5) 2.123 (1) 2.574 (2) 2.757 (4) 2.700(3) \\n11F\\n 0.932(5) 0.631 (1) 0.695(2) 0.865 (4) 0.786 (3) 2.153(5) 1.796(1) 2.013 (2) 2.035(3) 2.044(4) \\n12F\\n 0.234 (5) 0.157(1) 0.226 (4) 0.214 (3) 0.196 (2) 0.426 (4) 0.333 (1) 0.403(2) 0.430(5) 0.419 (3) \\n13F\\n 0.144 (5) 0.108(1) 0.140(3) 0.143(4) 0.130 (2) 1.153(5) 0.962(1) 1.057 (3) 1.123 (4) 1.030(2) \\n14F\\n 0.687(5) 0.439(1) 0.623(3) 0.652(4) 0.614 (2) 2.330(5) 1.701(1) 1.903(2) 2.065(3) 2.182 (4) \\n15F\\n 0.098(5) 0.073(1) 0.081(2) 0.087 (4) 0.087(3) 0.585(5) 0.439(1) 0.522(2) 0.579 (4) 0.543 (3) \\n16F\\n 2.540(3) 2.817 (5) 2.258(2) 2.621 (4) 1.804(1) 3.930 (4) 3.382 (1) 3.881 (3) 3.864 (2) 3.964(5) \\n17F\\n 0.423(5) 0.328 (1) 0.362 (2) 0.398(4) 0.387 (3) 4.311(3) 3.781(1) 4.240(2) 4.344(5) 4.333(4) \\n18F\\n 2.477(4) 2.314(2) 3.129(5) 2.393(3) 1.424 (1) 4.353 (5) 3.747 (1) 4.352(4) 4.261(2) 4.348 (3) \\n19F\\n 3.095(5) 2.878 (2) 3.054(3) 2.659(1) 3.078 (4) 4.660(5) 4.004 (1) 4.510 (3) 4.514(4) 4.421 (2) \\nAvgR.  4.790(5) 1.421 (1) 3.105 (3) 3.579(4) 2.105 (2) 4.632(5) 1.000 (1) 2.842 (3) 3.790(4) 2.737(2) \\nF\\uf074\\n 39.000  50.689  \\n \\nNext, the 30-dimensional test functions were used to test the performance of CLPSO -\\nLDSEDS. In addition, some functions in Table V were discar ded, as CLPSO cannot find their \\noptimal solutions with relative errors smaller than 20%. The CSs , the CTs  and the ir ranks of  \\ndifferent algorithms are compared in Table \\u2169 and \\u216a . It can be observed again that CLPSO -\\nLDSEDSs perform better than CLSPO. The Friedman statistics \\nF\\uf074  are larger than the critical \\nvalue \\nc2.537 \\uf074=  for both the CSs and the CTs , which shows the significant differences between \\nthe considered five algorithms. According to the NT,  when \\ntol20% \\uf065= , the AvgRks  of CSs of all This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n four CLPSO -LDSEDSs are significantly smaller than that of CLPSO; when \\ntol10% \\uf065= , the \\nAvgRks  of CSs of CLPSO -LDSEDSs with HWS and DES are significantly smaller than that of \\nCLPSO. The AvgRks  of CTs of  with HWS and DES are significantly smaller than that of CLPSO; \\nwhen \\ntol10% \\uf065= , The AvgRks  of CTs of CLPSO -LDSEDSs with HWS, DES , and SS  are \\nsignificantly smaller than that of CLPSO . \\nAccording to Tables \\u2165-\\u2166, it can be concluded that the proposed LDSEDS techniqu e can \\nsubstantially improve the convergence speed without reducing the search ability of the  original \\nCLPSO . Among  the four adopted LDSs , the HWS and DES are more robust than the other two \\nsamplings and can significantly improve the convergence speed of CLPSO.  \\nTABLE \\u2169 \\nCSS AND RANKS OF THE FIVE ALGORITHMS AT DIFFERENT TOLERANCE ERRORS  \\n \\ntol20% \\uf065=  \\ntol10% \\uf065=  \\nRand HWS  DES  HSS SS Rand HWS  DES  HSS SS \\n2F\\n 4086(5)  3568(1)  3700(2)  3719(3)  3827(4)  -(3) -(3) -(3) -(3) -(3) \\n3F\\n 2906(5)  2567(1)  2611(2)  2801(3)  2854(4)  6139(5)  5800(2)  5619(1)  5951(3)  6084(4)  \\n4F\\n 28(5)  19(2)  2(1) 25(3)  27(4)  335(5)  267(2)  261(1)  294(3)  312(4)  \\n5F\\n 3338(5)  2693(1)  2868(2)  3061(3)  3254(4)  -(3) -(3) -(3) -(3) -(3) \\n6F\\n 1633(5)  1364(1)  1403(2)  1426(3)  1595(4)  4044(5)  3571(1)  3578(2)  3671(3)  3825(4)  \\n7F\\n 3715(5)  3079(1)  3237(2)  3560(3)  3687(4)  4454(5)  3727(1)  3855(2)  4176(3)  4373(4)  \\n8F\\n 2195 (5) 1943(2)  1933(1)  2020(3)  2091(4)  3749(5)  3434(2)  3361(1)  3604(3)  3670(4)  \\n11F\\n 4371(5)  4072(1)  4200(2)  4251(3)  4291(4)  5845(5)  5479(2)  5776(4)  5448(1)  5634(3)  \\n13F\\n 1179(5)  1074(2)  1081(3)  1056(1)  1155(4)  2406(5)  2137(3)  2131(2)  2116(1)  2402(4)  \\n14F\\n 3434(5)  2990(1)  3087(2)  3145(3)  3350(4)  3922(5)  3491(1)  3623(3)  3584(2)  3827(4)  \\n15F\\n 1279(5)  959(1)  1091(3)  1162(4)  1070(2)  3578(4)  2605(1)  3542(4)  7270(5)  2810(2)  \\n16F\\n 735(5)  655(1)  656(2)  673(4)  657(3)  -(4.5)  5321(2)  -(4.5)  6380(3)  4162(1)  \\n17F\\n 1938(5)  1910(4)  1630(2)  1561(1)  1753(3)  3333(3)  6182(5)  2718(2)  2609(1)  3421(4)  \\n18F\\n 4986(3)  7310(5)  4553(1)  6482(4)  4643(2)  -(3) -(3) -(3) -(3) -(3) \\n19F\\n -(3.5)  6262(1)  -(3.5)  -(3.5)  -(3.5)  -(3) -(3) -(3) -(3) -(3) \\nAvgR.  4.767 (5) 1.667 (1) 2.033 (2) 2.967 (4) 2.533 (3) 4.233 (5) 2.267 (1) 2.500 (2) 2.667 (3) 3.333 (4) \\nF\\uf074\\n 22.416  4.817 \\n \\nTABLE \\u216a \\nCTS AND  RANKS  OF THE FIVE ALGORITHMS AT DIFFERENT TOLERANCE ERRORS  \\n \\ntol20% \\uf065=  \\ntol10% \\uf065=  \\nRand HWS  DES  HSS SS Rand HWS  DES  HSS SS \\n2F\\n 6.872(5) 4.621 (1) 5.753 (3) 5.627(2) 6.491(4) 10.030(5) 8.381 (1) 9.844 (2) 9.640 (4) 9.724 (3) \\n3F\\n 4.454(5) 3.263(1) 3.933(2) 4.114 (3) 4.262 (4) 10.067(5) 8.237 (1) 9.543(2) 9.025(3) 9.661(4) \\n4F\\n 0.027(3) 0.019(1) 0.019 (2) 0.0290(5) 0.028(4) 0.522(5) 0.409(1) 0.461 (3) 0.486(2) 0.498(4) \\n5F\\n 6.023 (5) 4.219(1) 4.959 (2) 5.378 (3) 5.747 (4) 13.71 3(5) 11.91 0(1) 13.37 4(2) 13.300(4) 13.319(3) \\n6F\\n 2.351 (5) 1.638 (1) 1.970(2) 2.085(4) 2.043 (3) 6.224(5) 4.825(1) 5.628 (3) 5.791(2) 5.841(4) \\n7F\\n 5.163(5) 3.651(1) 4.484 (2) 4.769(3) 4.973(4) 6.589(5) 4.963 (1) 5.886(3) 6.276 (2) 6.514(4) This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n \\n8F 2.600 (5) 1.962(1) 2.435 (2) 2.467(3) 2.527 (4) 5.127(5) 4.180 (1) 4.629(3) 4.885(2) 5.102(4) \\n11F\\n 6.391(5) 4.914 (1) 5.998(2) 6.364(4) 6.162 (3) 8.559(5) 6.553 (1) 7.573 (4) 8.353 (2) 8.118(3) \\n13F\\n 1.803 (5) 1.544(1) 1.702 (2) 1.783 (4) 1.780(3) 4.631 (5) 3.436(2) 3.303(3) 3.926(1) 4.162(4) \\n14F\\n 11.42 9(5) 9.370(1) 9.894(2) 10.597(3) 10.640(4) 13.41 2(5) 11.339(1) 11.800(3) 12.429(2) 13.189(4) \\n15F\\n 2.431(5) 1.640 (1) 2.297 (3) 2.341(4) 1.9610(2) 7.141(3) 4.166(1) 8.088(5) 10.86 1(4) 5.849(2) \\n16F\\n 1.186 (5) 0.969 (1) 1.168 (4) 1.126(3) 1.055 (2) 13.40 1(4) 12.222(3) 15.839(2) 11.73 3(5) 10.59 1(1) \\n17F\\n 3.572(5) 3.190 (4) 2.506(1) 2.831(2) 3.103(3) 5.295 (4) 5.971(5) 4.082(2) 4.801 (1) 5.284 (3) \\n18F\\n 15.678(2) 15.812(3) 15.388(1) 17.48 2(4) 17.598(5) 17.609(2) 16.21 3(1) 17.62 3(5) 17.89 2(3) 17.82 3(4) \\n19F\\n 21.539(4) 18.564(2) 18.008(1) 21.41 1(3) 21.916(5) 22.149(5) 20.67 4(1) 22.100(3) 22.096(4) 22.057(2) \\nAvgR. 4.600(5)  1.400 (1) 2.067(2) 3.333 (3) 3.600 (4) 4.533(5) 1.467(1) 2.733 (2) 3.000 (3) 3.268(4) \\nF\\uf074\\n 25.573  13.155  \\n \\n7. Conclusions  \\nMany researchers have tried to use LDS to initialize the population, hoping to accelerate the \\nconvergence  of PSO by taking advantage of the excellent uniformity of the low-discrepancy \\nsample set . However, the effect of this improvement direction seems inconclusive, and it also \\nlacks corresponding theoretical support.  \\nBased on Niderreiter\\u2019s theorem, this paper reveals  the relationship between the uniformity of \\nthe sample set and the error bound  of PSO, which  provides theoretical support for using LDS  in \\nthe expanded dimensional space  to accelerate PSO . On this basis, a new acceleration technique, \\nLDSEDS , is provided. Two different sample construction methods  are proposed to generate the \\nlow-discrepancy sample sets in the expanded dimensional space, and correspondingly,  two more \\neffective PSO algorithms are proposed, namely PSO-LDSEDS 1 and PSO-LDSEDS 2. \\nThe computa tional performance of PSO -LDSEDS1 and PSO -LDSEDS2 is studied, and the \\nfollowing conclusions are obtained based on numerical verification . Using PSO -LDSEDS1 to \\ncalculate the test function s can indeed converge faster than PSO . However, this advantage is only  \\nobvious when the  population size  \\nN is large  due to  the construction method of the low-\\ndiscrepancy sample set  in PSO -LDSEDS1. Therefore, we prefer to use PSO -LDSEDS2, which has \\nthe same advantages and requires fewer samples. Numerical experiments show that the proposed \\nPSO-LDSEDS2 not only improve s the convergence speed of PSO in solving low dimensional \\nproblems but also improve s its convergence speed in solving high -dimensional problems . \\nIn addition, t he improve d direction asserted in  this paper can be extended to other PSO -type \\nalgorithms. The CLPSO -LDS EDS generated by combining with CLPSO also has a fast This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n conver gence speed in low and high dimensions.   \\nThe effects of the different LDSs are different but on the whole,  HWS  and DES  are more \\nrobust. There is always a significant difference between the improved algorithm using HWS (or \\nDES ) and the original algorithm. Therefore, it is recommended to give priority to HWS and DES \\nin practical application.  \\nFinally, it is worth further emphasizing that the acceleration method in this paper may be \\nextended to other PSO -type optimization algorithms  (including multiobjective optimization \\nproblems) and to other types of evolutionary algorithms, which is also a direction need ing further \\nstudy.  \\n \\n \\nREFERENCE  \\n [1] J. Kennedy and R. Eberhart, \\\"Particle swarm optimization,\\\" in Proceedings of ICNN'95 - International Conference on Neural \\nNetworks , Perth, WA, Australia, 1995, pp. 1942 -1948.  \\n [2] R. Eberhart and J. Kennedy, \\\"A new optimizer using particle swarm theory,\\\" i n MHS'95. Proceedings of the Sixth \\nInternational Symposium on Micro Machine and Human Science , Nagoya, Japan, 1995, pp. 39 -43. \\n [3] W. Jiang  et al. , \\\"Query -Efficient generation of adversarial examples for defensive DNNs via Multi -Objective optimization,\\\" \\nIEEE Trans. Evol. Comput. , pp. 1. 2023, doi: 10.1109\\/TEVC.2022.3231460.  \\n [4] J. H. Huang, B. Xue, Y . N. Sun, M. J. Zhang, and G. G. Yen, \\\"Particle swarm optimization for compact neural architecture \\nsearch for image classification,\\\" IEEE Trans. Evol. Comput. , early access , doi: 10.1109\\/TEVC.2022.321  \\n7290.  \\n [5] K. Ishaque, Z. Salam, M. Amjad, and S. Mekhilef, \\\"An improved particle swarm optimization (PSO) -based MPPT for PV \\nwith reduced Steady -State oscillation,\\\" IEEE Trans. Power Electr. , vol. 27, no. 8, pp. 3627 -3638, Aug. 2012, doi: \\n10.1109\\/TPEL.2012.2185713.  \\n [6] B. Ahadzadeh  et al. , \\\"SFE: A simple, fast and efficient feature selection algorithm for High -Dimensional data,\\\" IEEE Trans. \\nEvol. Comput. , early access , doi: 10.1109\\/TEVC.2023.3238420.  \\n [7] Y . Hu, Y . Zhang, and D. W. Gong, \\\"Multiobjective particle swarm optimization for feature selection with fuzzy cost,\\\" IEEE \\nTrans. Cybern ., vol. 51, no. 2, pp. 874 -888, Feb. 2021, doi: 10.1109\\/TCYB.2020.3015756.  \\n [8] B. B. Li, L. Wang, and B. Liu, \\\"An effective PSO -Based hybrid algorithm for multiobjective permutation flow shop \\nscheduling,\\\" IEEE Trans . on Systems, Man, and Cybern .- Part A: Systems and Humans , vol. 38, no. 4, pp. 818 -831, Jul. 2008, \\ndoi: 10.1109\\/TSMCA.2008.923086.  \\n [9] Y . H. Shi and R. C. Eberhart, \\\"A modified particle swarm optimizer,\\\" in IEEE International Conference on Evolutionary \\nComputation , Anchorage, AK, 1998, pp. 69 -73. \\n[10] M. Clerc and J. Kennedy, \\\"The particle swarm - explosion , stability, and convergence in a multidimensional complex space,\\\" \\nIEEE Trans. Evol. Comput. , vol. 6, no. 1, pp. 58 -73, Feb. 2002, doi: 10.1109\\/4235.985692.  \\n[11] Z. H. Zhan, J. Zhang, and L. Y un, \\\"Adaptive particle swarm optimization,\\\" in IEEE Trans . on Sy stems, Man, and Cybern ., \\nPart B: Cybernetics , USA, 2009, pp. 1362 -1381.  \\n[12] J. J. Liang, A. K. Qin, P. N. Suganthan, and S. Baskar, \\\"Comprehensive learning particle swarm optimizer for global \\noptimization of multimodal functions,\\\" IEEE Trans. Evol. Comput ., vol. 10, no. 3, pp. 281 -295, Jul. 2006, doi: \\n10.1109\\/TEVC.2005.857610. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n [13] N. Lynn and P. N. Suganthan, \\\"Heterogeneous comprehensive learning particle swarm optimization with enhanced \\nexploration and exploitation,\\\" Swarm Evol . Comput ., vol. 24, pp. 11 -24, Oct. 2015, doi: 10.1016\\/j.swevo.2015.05.002.  \\n[14] Z. H. Zhan, J. Zhang, Y . Li, and Y . H. Shi, \\\"Orthogonal learning particle swarm optimization,\\\" IEEE Trans. Evol. Comput. , \\nvol. 15, no. 6, pp. 832 -847, Dec. 2011, doi: 10.1109\\/TEVC.2010.2052054.  \\n[15] X. Y . Liu, G. G. Wang, and L. Wang, \\\"LSFQPSO: Quantum particle swarm optimization with optimal guided L\\u00e9 vy flight \\nand straight flight for solving optimization problems,\\\" Eng. Comput. -Germany , vol. 38, no. S5, pp. 4651 -4682, Dec. 2022, doi: \\n10.1007\\/s00366 -021-01497 -2. \\n[16] R. Mendes, J. Kennedy, and J. Neves, \\\"The fully informed particle swarm: Simpler, maybe better,\\\" IEEE Trans. Evol. \\nComput. , vol. 8, no. 3, pp. 204 -210. 2004, doi: 10.1109\\/TEVC.2004.826074.  \\n[17] N. Y . Zeng  et al., \\\"A dynamic Neighborhood -Based switching particle swarm optimization algorithm,\\\" IEEE Trans. Cybern. , \\nvol. 52, no. 9, pp. 9290 -9301, Sep. 2022, doi: 10.1109\\/TCYB.2020.3029748.  \\n[18] B. Kazimipour, X. D. Li, and A. K. Qin, \\\"A review of population initia lization techniques for evolutionary algorithms,\\\" in \\nIEEE Congr . Evol. Comput . (CEC) , Beijing, China, 2014, pp. 2585 -2592.  \\n[19] D. S. Wang, D. P. Tan, and L. Liu, \\\"Particle swarm optimization algorithm: An overview,\\\" Soft Comput. , vol. 22, no. 2, pp. \\n387-408, Jan. 2018, doi: 10.1007\\/s00500 -016-2474 -6. \\n[20] S. Rana, S. Jasola, and R. Kumar, \\\"A review on particle swarm optimization algorithms and their applications to data \\nclustering,\\\" Artif. Intell. Rev. , vol. 35, no. 3, pp. 211 -222, Nov. 2011, doi: 10.1007\\/ s10462 -010-9191 -9. \\n[21] B. Niu, Y . L. Zhu, X. X. He, and H. Wu, \\\"MCPSO: A multi -swarm cooperative particle swarm optimizer,\\\" Appl. Math. \\nComput. , vol. 185, no. 2, pp. 1050 -1062, Feb. 2007, doi: 10.1016\\/j.amc.2006.07.026.  \\n[22] M. Richards and D. V entura, \\\"Choosing a starting configuration for particle swarm optimization,\\\" in IEEE Inter . Joint \\nConfer . Neural Networks , Budapest, Hungary, 2004, pp. 2309 -2312.  \\n[23] R. W. Morrison, Dispersion -Based population initialization , Berlin, Heidelberg: Springer Berlin Heidelberg, 2003, pp. 1210 -\\n1221.  \\n[24] Q. Li, S. Y . Liu, and X. S. Yang, \\\"Influence of initialization on the performance of metaheuristic optimizers,\\\" Appl. Soft \\nComput. , vol. 91, pp. 106193, Jun. 2020, doi: 10.1016\\/j.asoc.2020.1 06193.  \\n[25] A. Tharwat and W. Schenck, \\\"Population initialization techniques for evolutionary algorithms for single -objective \\nconstrained optimization problems: Deterministic vs. Stochastic techniques,\\\" Swarm Evol . Comput ., vol. 67, pp. 100952, Dec. \\n2021, doi: 10.1016\\/j.swevo.2021.100952.  \\n[26] J. Kennedy, \\\"The behavior of particle,\\\" in International Conference on Evolutionary Programming VII. Springer -Verlag , \\nSan Diego, CA, USA, 1998, pp. 581 -589. \\n[27] E. Ozcan and C. K. Mohan, \\\"Analysis of a simple particl e swarm optimization system,\\\" in Intell . Engineer . Systems Artif . \\nNeural Networks , 1998, pp. 253 -258. \\n[28] F. van den Bergh, \\\"An analysis of particle swarm optimizers,\\\" Ph.  D. dissertation thesis, University of Pretoria, South Africa, \\n2002.  \\n[29] I. C. Trelea, \\\"The particle swarm optimization algorithm: Convergence analysis and parameter selection,\\\" Inform. Process. \\nLett., vol. 85, no. 6, pp. 317 -325, Mar. 2003, doi: 10.1016\\/S0020 -0190(02)00447 -7. \\n[30] H. M. Emara and H. A. A. Fattah, \\\"Continuous s warm optimization technique with stability analysis,\\\" in American Control \\nConfer ., Piscataway NJ, Evanston IL, 2004, pp. 2811 -2817.  \\n[31] E. Garc\\u00ed a -Gonzalo and J. L. Fern\\u00e1 ndez -Mart\\u00ed nez, \\\"Convergence and stochastic stability analysis of particle swarm Confer ., Piscataway NJ, Evanston IL, 2004, pp. 2811 -2817.  \\n[31] E. Garc\\u00ed a -Gonzalo and J. L. Fern\\u00e1 ndez -Mart\\u00ed nez, \\\"Convergence and stochastic stability analysis of particle swarm \\noptimi zation variants with generic parameter distributions,\\\" Appl. Math. Comput. , vol. 249, pp. 286 -302, Dec. 2014, doi: \\n10.1016\\/j.amc.2014.10.066.  \\n[32] V . Kadirkamanathan, K. Selvarajah, and P. J. Fleming, \\\"Stability analysis of the particle dynamics in particl e swarm \\noptimizer,\\\" IEEE Trans. Evol. Comput. , vol. 10, no. 3, pp. 245 -255, Jun. 2006, doi: 10.1109\\/TEVC.2005.8570  \\n77. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n [33] M. D. Mckay, R. J. Beckman, and W. J. Conover, \\\"A comparison of three methods for selecting values of input variables in \\nthe analysis of output from a computer code,\\\" Technometrics , vol. 2, no. 21, pp. 239 -245. 1979, doi: 10.2307\\/1268522.  \\n[34] Y . W. Leung and Y . P. Wang, \\\"An orthogonal genetic algorithm with quantization for global numerical optimization,\\\" IEEE \\nTrans. Evol. Comput. , vol. 5, no. 1, pp. 41 -53, Feb. 2001, doi: 10.1109\\/4235.910464.  \\n[35] L. Peng, Y . Z. Wang, G. M. Dai, and Z. S. Cao, \\\"A novel di fferential evolution with uniform design for continuous global \\noptimization,\\\" Journal Comput ., vol. 7, no. 1. 2012, doi: 10.4304\\/jcp.7.1.3 -10. \\n[36] L. G. Hua and Y . Wang, \\\"On uniform distribution and numerical analysis ( \\u2170) (Number -Theoretic method),\\\" Science China , \\nvol. 16, no. 4, pp. 483 -505, Nov. 1973  \\n[37] L. G. Hua and Y . Wang, \\\"On uniform distribution and numerical analysis ( \\u2171) (Number -Theoretic method),\\\" Science China , \\nvol. 17, no. 3, pp. 331 -348, Jun. 1974  \\n[38] L. G.  Hua and Y . Wang, \\\"On uniform distribution and numerical analysis ( \\u2172) (Number -Theoretic method),\\\" Science China , \\nvol. 18, no. 2, pp. 184 -198, Mar. 1975  \\n[39] J. H. Halton, \\\"On the efficiency of certain quasi -random sequences of points in evaluating multi -dimensional integrals,\\\" \\nNumer. Math. , vol. 2, no. 1, pp. 84 -90, Dec. 1960, doi: 10.1007\\/BF01386213.  \\n[40] I. M. Sobol, \\\"On quasi -Monte Carlo integrations,\\\" Math. Comput. Simulat. , vol. 2 -5, no. 47, pp. 103 -112, Aug. 1998, doi: \\n10.1016\\/S0378 -4754(98)00096 -2. \\n[41] A. Tharwat and A. E. Hassanien, \\\"Chaotic antlion algorithm for parameter optimization of support vector machine,\\\" Appl. \\nIntell. , vol. 48, no. 3, pp. 670 -686, Mar. 2018, doi: 10.1007\\/s10489 -017-0994 -0. \\n[42] R. L. Cook, \\\"Stochastic sampling in computer g raphics,\\\" ACM Transaction on Graphics , vol. 1, no. 5, pp. 51 -72, Jan. 1986, \\ndoi: 10.1145\\/7529.8927.  \\n[43] R. Bridson, \\\"Fast poisson disk sampling in arbitrary dimensions,\\\" presented at the ACM SIGGRAPH 2007 Sketches \\nProgram, 2007.  \\n[44] F. De Rainville, C. G agn\\u00e9 , O. Teytaud, and D. Laurendeau, \\\"Evolutionary optimization of low -discrepancy sequences,\\\" \\nACM Trans. Model . Comput . Simul., vol. 22, no. 2, pp. 1 -25, Mar. 2012, doi: 10.1145\\/2133390.2133393.  \\n[45] F. Wu, Y . L. Zhao, K. Zhao, and W. X. Zhong, \\\"A multi -body dynamical evolution model for generating the point set with \\nbest uniformity,\\\" Swarm Evol . Comput ., no. 73, pp. 101121, Aug. 2022, doi: 10.1016\\/j.swevo.2022.101121.  \\n[46] L. Kocis and W. J. Whiten, \\\" Computational investigations of low -discrepancy sequences,\\\" ACM Trans. Math. Software , vol. \\n23, no. 2, pp. 266 -294, Jun. 1997, doi: 10.1145\\/264029.264064.  \\n[47] H. Niederreiter, A quasi -Monte Carlo method for the approximate computation of the extreme value s of a function : \\nBirkh\\u00e4 user Basel, 1983.  \\n[48] G. H. Wu, R. Mallipeddi, and P. N. Suganthan, \\\"Problem Definitions and Evaluation Criteria for the CEC 2017 Competition \\nand Special Session on Constrained Single Objective Real -Parameter Optimization,\\\". 2017  \\n[49] R. L. Iman and J. M. Davenport, \\\"Approximations of the critical region of the fbietkan statistic,\\\" Commun . statistics. Theory \\nmethods , vol. 9, no. 6, pp. 571 -595, Jul. 1980, doi: 10.1080\\/03610928008827904.  \\n[50] P. Nemenyi, Distribution -free multiple comparisons , WASHINGTON: International Biometric Soc 1441 I ST, 1962, p. 263.  \\n \\nFeng Wu  (Member, IEEE ) received his B.Sc degrees in Civil Engineering from Guangxi \\nUniversity in 2007. And from the same university, he received the M.Eng in Structural Engineering in \\n2010. He obtained the Ph.D degree in Engineering Mechanics from Dalian University of Technology in \\n2015. From 2015 to 2017, he worked as a post -doctor in the School of Naval Architecture & Ocean 2010. He obtained the Ph.D degree in Engineering Mechanics from Dalian University of Technology in \\n2015. From 2015 to 2017, he worked as a post -doctor in the School of Naval Architecture & Ocean \\nEngineering at Dalian Univ ersity of Technology. Since 2017, he has been an associate professor in \\nDalian University of Technology.  His research direction is the  dynamics and control . This work has been submitted to the IEEE for possible publication. Copyright may be transferred without \\nnotice, after which this version may no longer be accessible.  \\n \\n Yuelin Zhao (Student Member , IEEE)  is working as a postgraduate student in Computational \\nmechanics with the faculty of vehicle engineering and mechanics, Dalian University of Technology. Her \\nresearch interests include the construction and application of the low -discrepancy sample s. \\n  \\n \\n \\n \\nJianhua Pang  received the Ph.D degree in Ship and Ocean Engineering from Dalian University of \\nTechnology in 2016. Currently, he is the leader of the marine intelligent equipment and system team of \\nGuangdong Ocean University Shenzhen Research Institute, the post doctoral supervisor of ma rine \\nengineering technology of Tsinghua University International Graduate School, the young scientist of \\nGuangdong Province, the outstanding young scholar of Guangdong Ocean University's top talents, the \\nhigh-level leading reserve talents of Shenzhen. His research direction is the theory and method of \\nintelligent ocean technology.  \\n \\nJun Yan  received the Ph.D degree in Engineering Mechanics from Dalian University of Technology \\nin 2007 . In 2015, he was selected into the growth plan of outstanding young schola rs in colleges and \\nuniversities of Liaoning Province. He is the deputy director of the Department of Engineering Mechanics \\nof Dalian University of Technology, the member of the Education Work Committee of the Chinese \\nMechanics Society, the member of the Fo reign Exchange and Cooperation Work Committee, and the \\ndirector of key funds of the National Natural Science Foundation of China. His research direction is \\nstructure and multidisciplinary optimization.  \\n \\nWanxie Zhong  graduated from the Department of Bridge  and Tunnel, Tongji University in 1956.  In \\n1984, he was elected as the chairman of China Association of Computational Mechanics; In 1993, he \\nwas awarded honorary professor by the University of Wales and the University of Hong Kong; In the \\nsame year, he was  elected as an academician of Chinese Academy of Sciences.  He has put forward \\nimportant theories and methods in group theory, limit analysis, parametric variational principle, etc.  , and \\norganizes and develops various large -scale structural simulation  systems.\",\"18\":\" 1   S  \\n \\nEmerging AI Technologies Inspiring the \\nNext Generation of E -textiles \\nFRANCES CLEARY1,2, WITAWAS SRISA- AN3, David C. Henshall.1 and Sasitharan \\nBalasubramaniam 3 \\n1Physiology and Medical Physics,  RCSI Uni- versity of Medicine Health Sciences,  Dublin,  Ireland  \\n2Walton Institute,  South East Technological University,  Waterford,  Ireland  \\n3School of Computing,  University of Nebraska,  Lincoln,  United States  \\n \\nABSTRACT  The smart textile and wearables sector is looking towards advancing technologies to meet \\nboth industry,  consumer  and new emerging  innovative  textile  application  demands,  within  a fast paced  textile \\nindustry. In parallel inspiration based on the biological neural workings of the human brain is driving the \\nnext generation of artificial intelligence. Artificial intelligence inspired hardware (neuromorphic computing) \\nand software modules mimicking the pr ocessing capabilities and properties of neural networks and the \\nhuman nervous system are taking shape. The textile sector needs to actively look at such emerging and \\nnew technologies  taking  inspiration  from  their workings  and processing methods  in order  to stimulate  new \\nand innovative  embedded intelligence  advancements  in the etextile  world.  This emerging  next generation  of \\nArtificial intelligence(AI)  is rapidly  gaining interest  across  varying industries  (textile,  medical,  automotive, \\naerospace, military ). It brings the promise of new innovative applications enabled by low size, weight  \\nand processing power technologies. Such properties meet the need for enhanced performing integrated circuits (IC\\u2019s) and complex machine learning algorithms. How such propertie s can inspire and drive \\nadvancements within the etextiles sector needs to be considered. This paper will provide an insight into \\ncurrent nanotechnology and artificial intelligence advancements in the etextiles domain before focusing \\nspecifically  on the future  vision  and direction  around the potential  application  of neuromorphic  computing \\nand spiking neural network inspired AI technologies within the textile sector. We investigate the core architectural  elements  of artificial neural  networks,  neuromorphic  computing (2D and 3D structures ) \\nand how such neuroscience inspired technologies could impact and inspire change and new research developments within the e -textile sector.  \\n \\nKeywords  Artificial Intelligence,  Etextiles,  Neural  Networks,  Neuromorphic  Computing.  \\n \\n \\nI. INTRODUCTION  \\nMART  clothing  traditionally  refers  to a garment  with the \\ncapability  to enable\\/disable  a function  such as monitor - \\ning a person\\u2019s physical condition [5], whereas an e -textile \\nprovides  an added layer  of intelligence  such as connection to \\na peripheral  or embedded electronic  device into the garment \\nor fabric, providing added value to the person that wears  \\nthe item.  Smart  clothing  leveraging  embedded intelligent \\ne-textiles with computational and memory capabilities are \\nforeseen  as the next big market  mover  in the Internet  of things \\nspace. Demand for usable and wearable technology is con - \\nstantly growing and the expected market traction was fore - \\ncasted  etextiles  growth from 2981  Million  Dollars  (2022)  to 8508.1 Million  Dollars  (2028)  stated  by Absolute  reports . \\nThree generations of smart textiles have evolved over the \\nyears (1) 1st generation of smart textiles: little integration \\nbetween  the electronics  and the textile  (2) 2nd generation of \\nsmart  textiles: evolved with the adaptation  of traditional tex- \\ntile fabrication methods to include additional functionality. \\neg sewn in conductive thread into textiles. (3) 3rd genera-  \\ntion integration of electronic sensing properties into textile \\nmaterials. What will the next generation of etextiles and \\nsmart  cloth ing look like. Advancements  of ICT technologies \\n(artificial intelligence  (AI))  intertwined  with nanotechnology materials. What will the next generation of etextiles and \\nsmart  cloth ing look like. Advancements  of ICT technologies \\n(artificial intelligence  (AI))  intertwined  with nanotechnology \\n(nano- textiles  and wearable  sensing  nanomaterials ) are key \\nenabling drivers  of the next generation  of smarter  and more 2    \\n  \\n \\n \\nadvanced etextiles  driving AI inspired computing fabrics. \\nRapidly Advancing ICT technologies such as artificial in - \\ntelligent spiking neural networks (SNN) and neuromorphic \\ncomputing core technologies, computational capabilities as \\nwell as their architectural structure brin g the potential to \\ninspire new and fresh innovations in this domain.  \\nThe textile  sector  has been  experiencing  a digital transfor-  \\nmation predominately within its textile manufacturing pro - \\ncesses  and production  industry,  where  AI-enabled  technolo - \\ngies are being adopted for production line fabric inspection \\nand defect detection, enhancing output quality [3] [4]. [14] \\nprovides a survey of state of art technological interventions \\nthat meet automatic fabric defect detection aligning to the \\nindustry 4.0 initiat ive, detailing traditional(statistical meth - \\nods, structural methods, model based methods) as well as learning  based  methods  (machine learning  or deep  learning). \\nIntelligent clustering and classification techniques adopted \\nand utilised in the textiles indu stry are summarised in [13], \\nhighlighting both supervised and unsupervised learning types supporting production planning, fabric fault detection, per - \\nformance  and predictive  models.  In this paper  we will exam - \\nine in more  detail  artificial intelligence  ICT advancements  fo- \\ncusing specifically  on neuromorphic  computing  and spiking \\nneural network artificial intelligence, assessing their archi - \\ntectural  structure,  vision,  capabilities  and how these  elements \\ncould be of relevance to inspire  future  research  advancements \\nin the e -textiles sector. Healthcare is emerging as one of the \\nkey sectors where e -textiles and new advances in artificial \\nintelligence (AI) driving embedded textile intelligence and on-body computation, can be leveraged and utilised in the \\nnear future both within a clinical environment such as a hospital and also support enhanced remote monitoring of \\npatients  from  the comfort  of their own home.  [37] details \\na smart garment MyWear that monitors and collects phys - \\niological data ( muscle  activity, stress levels and heart rate \\nvariations ) processing the data in the cloud and providing \\npredictions  to the user based  on abnormalities  detected.  Such \\ne-textile applications and services utilising these AI tech - \\nnologies bring the added value of a more effective real time monitoring and analysis for varying health conditions such \\nas cancer care, cardiovascular and neurological disorders, \\nleading  when  required  to early  interventions  as critical health \\nconcerns  are detected.  The \\u2018Internet  of Smart  Clothing \\u2019 [12] \\npushes the boundaries around smart garment inter commu-  \\nnication, their interaction with environmental objects and how they actively communicate with remote servers for the \\nprovision of advanced  services.  The next generation  of smart \\nclothing and e -textiles brings more intelligent embedded \\ntechnological  layers  than before  and hence has requirements \\nfor more flexible, modular, integrated, seamless and usable \\nfunctionality to meet end user needs.  \\nThe rest of the paper  is organized as follows:  Section  2 provides an overview of advancements in nano technology and nano materials.  Section  3 details  AI intelligence  currently \\nimpacting  the textile  sector  and highlights  four core technical \\nproperties  required  to be considered  linked  to data flow com- \\nmunication  and process  methodology.  Section  4 discusses the \\ncore architectural  properties  of a spiking neural  network  and \\nsection 5 discusses the core architectural properties of neu - \\nromorphic computing with the objective to provoke thought around how inspiration  can be taken  from  these  elements  and \\nactively fed into next generation of etextiles.  \\n \\nII. NANOTECH TEXTILE  RESEARCH ADVANCEMENTS  \\nNanotech fabric that involves the integration of electron - actively fed into next generation of etextiles.  \\n \\nII. NANOTECH TEXTILE  RESEARCH ADVANCEMENTS  \\nNanotech fabric that involves the integration of electron - \\nics with textiles are pushing the boundaries in an exciting \\nnew phase  of interactive  smart  garments  [39][38].  Nanotech - \\nnology involves the manipulation of individual atoms and molecules in order to create nano -scale machines that open \\nthe potential for a variety of applications  across  many  sectors \\n(Sensors, Repel liquids, sensor display technology, odour \\ncontrol,  fabric  strength,  wrinkle  resistant ). [9] highlights  the \\nadvantages of Wearable memories and Computing devices \\n(WMCs) and recent advances in nanotech nology and mate - \\nrials science. The link between WMC and the human brain \\ncould enable  fast operation  along with interface  complexity, \\ndirectly  mapping the continuous  states  available  to biological \\nsystems.  \\nAdvances in micro -sensor technology and nano materials \\nare feeding new and disruptive innovations around how we \\nwear and actively engage with our clothing. Nanomaterials \\nare enabling the next generation  union of textiles,  electronics \\nand information processing [Fig 1]. Through this combina - \\ntion or h ybrid approach this opens the door to integrate new \\ncapabilities  such as antimicrobial properties  or the integration \\nof biological functions. Nanomaterials can be added during \\nthe fibre  production phase  or also during the finishing  phase \\nonto the fibre  surface.  The aim is to utilise  nanomaterials \\nin order to achieve enhanced flexibility, usability while also being more fashionable. Nanomaterials consist of various \\nmaterials  such as graphene,  carbon nanotubes,  ploymers,  di- \\nelectric  elastomers  and composites.  Based  on various  stimuli \\nto such nanomaterials  this creates  different  characteristic be- \\nhaviours that can in turn be utilised for varying applications \\nspecifically in the healthcare sector related to patient care \\napplications. Biosensors that look at utilising nanomaterials \\nintegrates varying disciplines ranging from molecular engi - \\nneering, material science, biotechnology\\/chemistry offering high sensitivity towards the recognition of various disease biomarkers at an early stage[40].  \\nIn 2004 researchers  succeeded  in isolating  graphene  sheets \\n(exfoliated graphene). This two dimensional material con - \\nsists of carbon  atoms  arranged  in a hexagonal  lattice.  It is the \\nthinnest  material  known in the world  (one atom  thick ) and 3   \\n  \\nopens the door to multiple opportunities in the design and \\ndevelopment  of intelligent and smart  garments  and etextiles. \\nGraphene  is light,  has greater  elasticity  and conductivity and \\nbrings the potential to replace synthetic fibres( polyester, \\nnylon).  Chemical  sensing  properties  through graphene  fibers \\ncan be introduced into textile based materials. Cutecircuit (https:\\/\\/cutecircuit.com ) creat ed a black  dress that contained \\ngraphene. The dress had the capability to change colour in \\nsynch with the wearers  breathings.  This was completed  using \\nmini LED lights with graphene used to power the lights and \\nas a sensor to record the wearers breathing. G raphene based \\ncomposites have also been used in 3D printing technologies \\nto investigate the development of highly stretchable and \\nsensitive  strain  sensors.  Such  an application  can be leveraged \\nfor breathing pattern monitoring and measurement [41].  \\nNanopart icles are emerging  as diagnostic  sensors.  Textile \\nadvances  at nanometer  level encourage a new era of improved \\ntextile  functions  and capabilities  for example enhanced com- \\nmunications , antibacterial, sensing capabilities of textiles or \\nmoisture absorbing. Garment communication through eas - \\nily integrated textile antennas and interconnected integrated wearable textile systems are completed through the use of \\nelectroconductive fibres\\/yar ns and embroidery techniques. \\nResearch  in this area has been  actively  investigating  material \\nproperties  (flexibility,  conductiveness,  weight)  leading  to the \\nemergence of nanoengineered functional  textiles  and the ad- \\ndition  of nanoparticles  to fabrics  to increase textile  properties \\nand capabilities.  (Ghadremani  et al 2016)  details  the addition \\nof silver  nano particle  in order  to improve  the static  properties \\nof a textile. (Liu et al 2010) highlights the addition of Ag (Silver ) nanoparticles  on sock garments  that in turn limit the \\ngrowth of fungi and bacteria, reducing odour and itching of the feet. Nanomaterial skin -like wearable sensors, flexible \\nelectronic  substrates  are the focus  of development  at industry \\nlevel.  There  has been  ongoing attempts  to mimic  human skin \\nflexibility and stretch -ability in the etextile world. Sample \\nSmart skin developments include  \\n\\u2022 (Tianzhao  Bu et al) developed  a stretchable  triboelectric - \\nphotonic  skin with multidimensional tactile  and gesture \\nsensing.  \\n\\u2022 (Liu et al) completed a review on Lab -on-Skin, in par - \\nticular flexible and stretchable electronics for wearable health monitoring.  \\n\\u2022 (Yang Et al) presents a novel structured fibre sensor \\nembedded in silicone for the precise measurement of \\nlow-pressure in sm art skin.  \\nSuch  nanotechnology,  nanomaterial  and smart  skin advance-  \\nments  are attracting  alot of attention  both within  the research \\nworld and also with businesses as they see the economical \\nimpact  and the potential end user commercial  applications  of \\npromi se that are emerging.  In parallel  the next generation  of \\nartificial  intelligence  is seeing  the emergence  of Spiking  Neu- \\nral Networks(SNN) and neuromorphic computing, that will \\nno doubt in the future trigger research directions focused on \\na fusion of nanotechnology AI-driven embedded  E-textiles  innovations. The remainder of this paper will delve further \\ninto the AI ( SNN) and neuromorphic engineering technolo - \\ngies mapping and highlighting key architectural aspects to be considered further within the textile and smart garment \\nresearch and innovation domain.  \\n \\nIII. ARTIFICIAL INTELLIGENCE  IMPACTING  TEXTILE \\nSECTOR  \\nEtextile  research  doma in experts,  suppliers  and manufac-  \\nturers are starting to investigate at a deeper level the im - \\npact AI and machine learning technologies can have across varying sectors [6]. Digital transformation through the use \\nof artificial intelligence is currently im pacting the textile pact AI and machine learning technologies can have across varying sectors [6]. Digital transformation through the use \\nof artificial intelligence is currently im pacting the textile \\nindustry through the creation  of a more  sustainable  digi- \\ntal supply chain and smart intelligent textile manufacturing optimization( production planning and operational process \\nmanagement ) right through to fabric defect identification \\n[2], pattern inspection analytics and much more.(Chaudhari \\net al) provide an insight into the application of Artificial \\nneural network (ANN) in fabric engineering, focusing on  \\nthe application of ANN in the textile domain to support \\nfabric cl assification, fabric wrinkle, automatic recognition  \\nof fabric patterns and fabric comfort. How such enhanced and smart technologies can aide the textile industry towards \\na sustainable and circular economy is of high priority and \\ngaining a substantial amount  of attention.  Textile  fabric  based \\ndesign software is also seeing the adoption and usage of AI \\nbased software tools in pattern design, making and cutting, \\nproviding  a superior  level  of tools  with inbuilt 3D visualisa - \\ntions features [1].  \\nThe adoption and use of AI within fabric based textiles \\nrequires  a structured  and methodological  process  taking  into \\naccount  technical  properties  of importance  along with the end \\nuser functional and form factors (Giuseppe et al)[13]. Such \\nkey etextile  embedded functionality  technical  properties  that \\nare required to be considered when contemplating taking \\ninspiration  from  advanced  AI technologies  into a fabric  based \\nenvironment include  \\n \\n\\u2022 Fabric  based textile  computational  consideration s \\n(Data Acquisition and Data Processing)  \\n\\u2022 Monitoring  and measuring  capabilities  and techniques \\nsuitable for a fabric based environment . \\n\\u2022 Suitable  communications  methods  and techniques . \\n\\u2022 Energy  considerations  of relevance for textile -driven \\nintelligence . \\nAll these  elements  directly  relate to the data flow commu - \\nnication and processing methodology. In this paper we will \\nadopt  these  elements  in order  to map across  new AI technol - \\nogy properties to the textile domain within sect ion 4 and 5. \\nCurrently  within  the healthcare medical  sector  such technical \\nfunctional properties are applied and demonstrated through \\nintelligent e -textiles for patient centric garment- based wear - \\nables[19].  Such  considerations  in the design and development \\nenable the possibility  to gather  human  monitored  health 4   \\n  \\n \\n \\n \\nFIGURE  2. Fabric  based  Neural  Network  Workflow  . \\n \\n \\nrelated  datasets  such as Electromyography (EMG) or Electro - \\ncardiography (EKG) data -sets through textile based sensors \\nin wearable garments.  Such  data collection  type garments  and \\ntextiles  need  to be adaptable  to the users needs  for ease of use \\nat varying levels.  Through  the active  collection  of these  data- \\nsets, t his then allows for the transmission, processing and \\nextraction of key analysis and results for effective decision \\nmaking, this is aided through the development and imple - \\nmentation of intelligent algorithms. The use of Artificial intelligent based algorithms and techniques enables an era  \\nof intelligent textiles utilising real time and accurate data knowledge in dynamically changing healthcare monitoring type environments.When considering such embedded  AI in a \\ntextile environment, there is a need to also look at the state of art of current AI formal methods and how they must be \\nadvanced to adapt. Key challenges and questions arise that \\nwill require  further  investigation  and research  as technology \\nadvances and emerges in this space.  \\nPushing the boundar ies around the use of conductive \\nthreads  and embedded  smart  wearable intelligence  and func- \\ntionality [48] were successful at their attempts to store data \\nin fabric,  this was completed  through the magnetic  properties \\nof conductive thread and this allowed f or applications such \\nas utilising a garment to store a passcode to open a door to \\ngain access to a building through the swipe of the garment arm that holds the passcode.  \\nWe must ask ourselves what future fabric -based AI al - \\ngorithms will look like, how will they be developed and integrated in a functioning manner in the very core of a \\nfabric and fibre environment. Not only will new Fabric AI design fabrications emerge but also new fully fabric driven \\ndata acquisition and data processing driven approaches po - \\ntentially  adopting  or inspired  by new AI best practice,  stan-  \\ndards and methodologies. Key features of fabric -based AI \\nalgorithms need to consider during implementation speed \\n(response  time),  processing  capabilities,  complexity\\/size  and \\nalso learning. [Fig 2] provides an overview of the key com - \\nponents for consideration for a textile driven AI workflow facilitating a fabric based data acquisition, data processing \\nfabric -based engine, execution in a fabric environment and \\nalso deployment.  \\nThere is an em erging research interest around the appli - \\ncation  of artificial intelligent technologies  in smart  wearable \\ngarments and the internet of smart clothing. Research ques - \\ntions  are being investigated  around how such AI intelligence \\ncan be embedded  into the very core of a fabric  environment, \\nwith AI intelligence applications seamlessly embedded invis - \\nible to the human eye. [6] provide a comprehensive review \\nof advancements in Smart textile integrated microelectronic systems,highlighting the core properties of importance 1) \\nflexibility allowing for effective drape on 3D curvilinear \\nsurface such as a human body and 2) structural transforma - \\ntion of textiles resulting in low fibre strain and fabric life cycle longevity. [11] convey a vision of moving from fibre devices to fabric computers , where the fabric fibers have \\ninbuilt capability to perform sensory,storage,processing and \\npowering capabilities providing a fabric based computing \\nenvironment. Such a powerful fabric fibre based processing capability  enables  the execution  of fabric  based  programs  that \\ncan activate  fiber  sensors,  processing  and storing  data within \\nthe fabric computer. Work has been ongoing around the de - \\nvelopment  of such new fibres  with specific  focus  on scalable \\nprocesses using thermal drawing, melt spinning, coating to \\nprovide  fibre  structures  that can house  and deliver  computing \\nfunctionality[7][8][10]. Investigative methods into the digital processes using thermal drawing, melt spinning, coating to \\nprovide  fibre  structures  that can house  and deliver  computing \\nfunctionality[7][8][10]. Investigative methods into the digital \\nfabrication  of fibres  is being researched  where  inbuilt func- 5   \\n  \\ntionality provides in -fibre storage programs, data storage, \\nsensors and digital communication [11], such a fabrication \\nstructured process proposes uniform placement of discrete \\nin-fibre  electronic  devices  that will carry  out such function al- \\nity. Researchers are actively thinking outside the box about new and potentially disruptive innovative ways to fuse AI \\nwith etextiles  moving away  from  the traditional textile  world \\nas we know it. This is sparking a renewed interest in this \\ndomain and s hows promise of real impact across multiple \\nsectors.  \\n \\nIV. SPIKING  NEURAL  NETWORK  PROPERTIES  \\nArtificial neural networks are seeing the emergence of new \\nspiking neural network (SNN) technologies that simulate functionality using electronics components replicating and \\nmimicking human brain biological workings of neurons, \\nsynapses and neural networks. Core architectural properties \\nof an SNN include  \\n\\u2022 Neurons  that emit a spike  once  a set threshold has been \\nmet. \\n\\u2022 Learning  in the neural  network is completed  by altering \\nthe synaptic weight. Random weight change algorithm is one of the most adopted and simple algorithms used \\nduring the learning  phase.  For this algorithm  the correct \\noutput  is known and the error  increased  or decreased  as \\nrequired.  \\n\\u2022 Results obtained depends  on the neuron spiking  activity \\nand also the neural node inputs.  \\n[15] detail and explain a single neuron level, giving 1D \\nneuron model examples such as Leaky integrate and fire (LIF) [16],the  Spike  Response  Model  (SRM)[17]  as well \\nas more c omplex and biologically feasible artificial neurons \\nsuch as the the Hodgkin and Huxley model[18].  We will now \\nassess the core computational, monitoring \\/measurements, communications  and energy  of such SNN,  highlighting struc - \\ntural and processing paradigms inspired by the human brain \\nand the potential they could bring to the etextiles domain. [Fig 2] details the adoption of this data flow communica - \\ntion and processing technology methodology to the etextile domain highlighting a fabric based workflow of relevance towards the implementation, validation and deployment of \\nfabric driven AI ( neural network ) intelligent etextile wear - \\nables.  \\n \\nA. COMPUTATIONAL ELEMENTS OF SNN ENABLING \\nFABRIC-DRIVEN DATA ACQUISITION  AND PROCESSING  \\nData acquis ition refers to the methodology and process of \\nacquiring data and performing analysis in order to interpret \\nit. This involves  the use of varying  techniques  and tools  used \\nto sample  data,  convert  the data into a format  that can in turn \\nbe used for further a nalysis and processing. From a neural \\nnetwork point of view, we will now investigate further the \\nmain computational elements with a focus on highlighting \\narchitectural aspects of importance for consideration in a \\ntextile  environment  1) Artificial fabric  neurons  2) Artificial  fabric  synapses  3) Artificial fabric  neural  networks required \\nto perform data acquisition and processing and the potential for adoption into a fabric environment.  \\n \\n \\n1) Etextile  Artificial  Fabric  Neuron  \\nNeurons are the core building blocks of neural networks. The workings of a neuron include synapses represented by \\nweights, a threshold and an output spike that in turn resets the neuron. Each neuron has a membrane potential. This \\nmembrane potential is the equivalent of a volta ge and when \\nthat voltage passes a defined threshold a spike ( or action \\npotential ) is emitted and hence this generated spike is the \\nmethod by which one neuron communicates to another neu-  \\nron in a SNN. Taking these aspects, how can we begin to \\nconsider a fa bric based neuron, its workings to replicate not \\nonly an individual neurons functionality but having the ca - \\npability  to be extended  to implement  multiple  interconnected \\nneural  nodes  in a fabric  environment.  To work  towards  such only an individual neurons functionality but having the ca - \\npability  to be extended  to implement  multiple  interconnected \\nneural  nodes  in a fabric  environment.  To work  towards  such \\na goal, we have to delve into the artificial electronic neuron representations  currently  defined  and that could inspire  future \\nfabric based neural implementations. Here we will consider \\nthe Leaky integrate and Fire Model and the Hodgkin and \\nHuxley Model. Further enhancements to these models and \\nother  models  exist,  but this is outside  the scope  of this paper. \\nLeaky Integrate and Fire Model (LIF) : The most simplistic \\nmodel of a neuron is the Leaky Integrate and Fire neuron \\nartificial electronic circuit b ased on the logic that if the  \\nspike (driving current) goes beyond the defined threshold then the neuron emits its own spike and resets. The model \\noperates based on a resistor and capacitor (RC circuit) [Fig 3]. Limitations  of the LIF model  is that no memor y of spiking  \\n \\n \\nFIGURE  3. Simple  Integrate  and Fire artificial  (RC) neural  node \\n \\nactivity is retained as the membrane potential is reset after each spike.  We need  to consider  how such a LIF model  could \\nbe replicated to produce an event driven spiking neuron in  \\na fabric environment leveraging the leaky integrate and fire \\nneuron model. Soft fabric based resistors can be fabricated \\nusing conductive  thread  (Zigzag  machine stitch  to create)  and \\nongoing research  is active  around the development  of textile \\nbased  capacitors[20].[51]  investigate  methods  for the capac- 6   \\n  \\nity increasing of textile capacitors for planar and sandwich \\ntype textile capacitors using hybrid conductive threads and \\nconductive  textiles.  Experimental  research  has also been  on- \\ngoing into the development  of wearable fabric  Brain  enabling \\non-garment edge- based sensor data processing inspired by \\nSNN architectural techniques and LIF model [56]. Such advancements  will open up options  and new methods  to work \\ntowards  functional  fabric  based  neural  nodes  based  on an RC \\ncircuit and LIF model, capable of processing \\u2018event\\u2019 spike driven  activity  replicating  a neural  node,  that can be extended \\ninto a basic working event driven spiking neural network.  \\nAdditional  embedded transistor  gated  circuitry  is required \\nin order to implement the threshold level Vth for the mem - \\nbrane potential, when this threshold has been reached this \\nproduces an action potential spike. [22] detail an orga nic \\nfield-effect transistor that is realized on a flexible film that \\ncan be applied,  after the assembly,  on textiles.  [23] showcase \\na fully inkjet -printed  2D-material  active  heterostructures  with \\ngraphene and hexagonal -boron nitride (h -BN) inks, which \\nare used to fabricate an inkjet -printed flexible and wash - \\nable field -effect transistors on textile. Such advancements  \\nin textile based FET\\u2019s bring the capability to incorporate textile  threshold level  gates  enabling  the possibility  of vary- \\ning threshold levels for the various neural nodes in a fabric \\nspiking neural network.  \\n \\nHodgkin and Huxley neuron model  is a more  complex model \\nto replicate the generation of an action potential of a neural node. The model can describe the time behavior of the \\nmembrane  potential and currents  through potassium  (K) and \\nsodium (Na) channels using differential equations[Fig 4. \\nThey  were  able to observe  the generation of action  potential \\nas well as the refractory period [21]. In this circuit the capacitor is representative of the cell membrane, the circuit \\nhas variable resistors that represent the voltage -dependent \\nK+ and Na+ conductance\\u2019s and there is also a fixed resistor \\nrepresenting the voltage -independent leakage conductance. \\nThis model has three power batteries for the reverse po - \\ntentials for the corresponding conductance\\u2019s. Generation of \\nthis model in a fabric environment would require the tex-  \\ntile resistor and capacitor as well as the requirement for a \\nvariable resistor interconnected using co nductive thread. To \\ndate variable textile resistors have been created in the form \\nof fabric based potentiometers.Such fabric potentiometers \\ncontain a conductive wiper function as well as a resistive \\ntrack where its ends has measurement points included. The  \\nconductive wiper acts as a means to set and measure a variable  resistance through  adjustment of the sliding  wiper.  \\n[49] detail and demonstrate a zipper based potentiometer. \\nOther variable resistance elements that could be considered \\nto produce such a vari able resistor include include Eeonyx \\nStretchy Variable Resistance Sensor Fabric ( Adafruit) that \\ncan be utilised to make soft sensors that are required to be \\nmovable and adaptable. Such stretch fabric sensors using \\nstretchable conductive fabric enables chan ges in resistance \\nwhen  stress  is applied.Further  research  is required  into how textile and fabric based variable resistance can be leveraged and potentially be utilised in the design of a fabric neural \\nnode.  \\n \\n \\nFIGURE  4. Hodgkin  and Huxley  electronic  Artificial  neural  node \\n \\nKey textile components are in existence to enable the \\ncreation of a fabric base Hodgkin Neural Node, how such \\nelements can come together from a design perspective to \\nproduce  a working  neural  node  is the key challenge  here \\nin order to produce a working Hodgkin and Huxley Neural \\nnode.  \\n \\n2) Etextile  Artificial  Fabric  synapse produce  a working  neural  node  is the key challenge  here \\nin order to produce a working Hodgkin and Huxley Neural \\nnode.  \\n \\n2) Etextile  Artificial  Fabric  synapse \\nCurrent learning in SNN are dependent on the capability to alter and interchange  the synapse  weights  for each of the neu- \\nral network nodes.  When  a neuron threshold is reached  it fires \\nand produces  an action  potential.  This is a result  of the sum of \\nexcitatory and inhibitory potentials and these are connected to the neuron through the synapse. We refer to the synapse as a synaptic  weight.  This is the strength  of a connection be- \\ntween  two nodes  in a neural  network.  [25] focus  on non-static \\nneural  networks  at a signficant distance  from  each other  and \\nhow through their implementation of an all -optical synapse \\nstemming  from  wavelength  division multiplexed  visible  light \\ncommunications  can overcome  real-time weight  adjustments.  \\n[52] investigated yarn coated with reduced graphene oxide (RGO) to produce two -terminal memristor -based artificial \\nsynapses  suitable  for use in wearable neuromorphic  comput - \\ning systems. [53] researched the design and development of \\na one-dimensional  organic  artificial multi- synapses  enabling \\nelectronic  textile  neural  network  for wearable neuromorphic \\napplications, where the multi- synapses comprising of ferro-  \\nelectric organic transistors fabricated on a silver ( Ag ) wire.  \\nTo replicate a SNN in a fabric environment we need to \\nconsider  the functionality  required  for the synapse  and how to \\nembed this in a workable manner into a fabric environment. \\nAs the weight influences the fir ing of a neuron, in a SNN,  \\nat a basic level this can be replicated by embedding the option to be able to connect  and interchange  from  one 7   \\n  \\nTABLE  1. Types  of Neural  Network  more  relevant  to SNN\\u2019s  \\n \\nSNN  Type  Data  Flow  Memory  Prediction  Weights  \\nFeedF  orward  Input  to \\nOutput  No Memory  Poor Weight  matrix  \\ninputs,  \\nproduces  \\noutput  \\nRecurrent  Looping  \\nInformation  \\ncycles  Has \\nMemory - \\nlearns  Good  Weights  \\napplied  current  \\nand previous  \\ninputs  \\n \\n \\nconductive  thread -based  resistor  in a fabric  environment \\nto another conductive thread resistor. Such textile resistors \\ncan in turn be utilized as synaptic fabric -based weights. \\nFurther  advancements  with the introduction  of memristors  as \\nsynaptic weights are emerging and need to considered from a fabric  synapse  implementation  point  of view,  these  will be \\ncovered in section 5 neuromorphic computing properties.  \\n \\n \\n3) Etextile  Artificial  Fabric  Spiking Neural  Network  \\nThere is multiple types of neural networks percept ron, \\nmultilayered perceptron, feed -forward, recurrent, fully con - \\nnected, Convolution, Radial Basis Functional,Long Short - \\nTerm Memory (LSTM), Sequence to Sequence Models and Modular  Neural  Network.  Table  1 provides  an overview  of a \\nFeed -forward  and re-currant neural  network  properties  of rel- \\nevance when  considering a SNN  type to adopt  and conform  to \\n[42][43][44]. When designing an embedded neural network in a fabric environment key fabric and end user properties need to be accounted for including aesthetics, durability, \\ncomfort and maintenance. Based on the overall size of the \\nSNN  and the number  of hidden layers  it incorporates,  this de- \\nciphers the number of textile artificial neural nodes,synapse interconnections and interconnected fabric multi -layers re-  \\nquired.  \\nWe can then begin to investigate  the best possible  design, \\nlayout and functionality integration methods around how to \\naccommodate and embed into a fabric environment.  \\n \\nB. SPIKING  NEURAL  NETWORK MONITORING\\/ \\nMEASURING  \\nA SNN  can learn  by supervised learning,  where  you have  an \\ninput  and an output  variable  and the algorithmic  computation \\nwithin  the neural  network  learns  from  a training  dataset,  once \\nan acceptable level of performance is achieved the learning stops. An unsupervised method in comparison has input \\nvariables but no output variables that are used to support \\ntraining  and learning  of the neural  network.  [24] present \\nan overview of the varying training methods for SNN\\u2019s  \\nsuch as conventional deep networks, constrained training , \\nspiking variants of back- propagation and variants of Spike \\ntime dependant  plasticity  (STDP)  in order  to categorise SNN \\ntraining methods and also highlight their advantages and \\ndisadvantages.  Creating  such a training  or learning  process  in \\na fabric  etextile  is a challenge that has not yet been  achieved.  The use of nanotech is the obvious initial best approach to \\nattempt to embed such a learning element into an intelligent \\nfabric garment.  \\nDepending on the structure of a SNN, this identifies its \\nclassificatio n. For this paper we will focus on a fully con-  \\nnected  multi- layer  neural  network.  Such  a multilayered  Spik- \\ning neural network consists of multiple layers of artificial neural nodes (usually has three of more layers and utilizes a \\nnonlinear activation function). From a design perspective in a fabric  environment  once  we have  identified  the core textile \\ncomponents required to implement a working neural node  \\nas well as a functional method for interchangeable synaptic \\nweights,  the next step is to progress  towards  the identification \\nof a most  practical  and feasible fabric -driven  design in order \\nto incorporate  multi- layers,  their interconnections  and how to \\nbe capable  of validating  and modifying during the execution \\nphase.  \\nSeveral layers of fabric woven and stacked produces a \\nmultilayer  fabric,  secured  together  with connecting yarns \\nin a third (Z direction) dimension. Such woven fabrication phase.  \\nSeveral layers of fabric woven and stacked produces a \\nmultilayer  fabric,  secured  together  with connecting yarns \\nin a third (Z direction) dimension. Such woven fabrication \\ntechniques  along  with layered  and interwoven  fabric  manip - \\nulations  are design options  that need to be assessed  to identify \\nsuitable  and best practice  design  for the development  of SNN \\ntechnical  textiles  [Fig 5]. Weaving  multiple  layers  in a fabric \\nprovide the opportunity to embed neural network nodes and interconnected neural networks in an embedded fabric en-  \\nvironment . Research into best approach ,best methodology to adopt and also core components and their re -usability  \\nstill remain under investigation, but as textile c omponents \\nand intelligence along with nanotechnology advances, new opportunities are emerging pushing towards this vision of a \\nFabric AI Driven intelligence. In section 5 we will delve a \\nlittle further into 2D\\/3D stacked layered techniques taking \\ninspiration from neuromorphic computing and advancements \\nhere.  \\n \\n \\n \\nFIGURE  5. Multilayered conceptual  design approaches  inspired by stacked \\nneural  networks 8   \\n  \\nTABLE  2. Neuronal  Information  Encoding  Types  \\n \\nCoding  Type  Details  \\nRateCoding  Rate of spikes  in a set time interval.Can  \\nbe used at single  neuron  level  or interpre - \\ntation spike trains.  \\nBinaryCoding  A neuron  is active  or inactive  in a set \\ntime interval.It  fires 1 or more  spikes  \\nwithin that timeframe.  \\nT imetofirstspikecoding  Method  is used to encode  information  \\nfor fast responses  ( in milliseconds).  it is \\nbased on first -spike patterns).  \\nFullyT  emporalcoding  Precise  timing  of all spikes  generated  \\nare part of the encoding.  Timings  are \\nimportant.  \\nPhaseCoding  Convert  inputs  into binary  representa - \\ntions (eg a \\u20191\\u2019 equals a spike gener - \\nated).Information is added by assigning  \\ndifferent  weights  to each bit represented  \\nin the phase  \\nBurstCoding  A burst  of Spikes  (Short  and High  fre- \\nquency  of Train  of spikes).Assessment  of \\nthe number  of pattern  of spikes  in a burst  \\ncoding  \\nLatencycoding  Number  of spikes  is not the prior- \\nity,instead  the time between  an event  and \\nwhen  the first spike  triggers  is important.  \\nStimulus is encoded based on a rank - \\norder  where  neurons  in a group  generate  \\ntheir first spikes  \\n \\n \\nC. SPIKING  NEURAL  NETWORK  COMMUNICATIONS  \\nSNN drives the adoption of brain- inspired computing, pro-  \\nviding not only fast but also a large substantial amount of \\nevent -driven data processing. An SNN neural computation \\nand communication is defined through the generation of spikes enabling neurons to communication from one to an - \\nother via such triggered spikes. Research is ongoing around the types of neuronal information encoding\\u2019s.[29]  summarise \\nthe signal encoding schemes for a spiking neural network. \\nNeuronal  encoding and decoding is the information  and com- \\nmunication process where for example an external variable or stimulus triggers neural activity within the brain. Such \\nstimuli (e .g touch stimuli) produce varying neural activity \\npatterns in the brain.[27] provides an overview of neural \\ncoding with bursts and new methods for their analysis. [28] \\nintroduce spiking autoencoders with temporal coding and \\npulses, trained using backpropagation to store and recon - \\nstruct images with high fidelity from compact representa - \\ntions.  \\nClassification of the spike train pattern and what this \\nmeans, enables active decoding of the pattern. One such example is the classification of spike train through active \\nmatching of the spike train patterns to templates. Such a \\ntemplate  would be a set word,  meaning  or result.  Within \\nan etextile  world  neural  information  encoding,  decoding and \\nthe creation and validation of potential Fabric SNN clas - \\nsification templates mapping to external sensing embedded \\ntextile sensors linked to fabric based neural networks, could \\nenable the communication and processing of fabric based \\nSNN encoding. Such Fabric SNN classification templates \\ncould correspond to an alert notification  raising  awareness  around a critical health monitoring scenario where such a \\ntemplate could be used in conjunction with a SNN fabric \\nSmart garment to assess the health status and provide feed - \\nback to the wearer based on the use of such classification templates to raise alerts to the wearer as required. Core to \\nthe fundamental working of a SNN is the manner in which \\nthe network nodes interconnect and how information flow s \\nand communication between the nodes is enabled. Multiple \\nartificial neural  network types  exist.  [26] provides  a compar - \\native overview of neural coding in a spiking neural network with in-depth detail  on rate coding,  time-to-first spike  (TTFS) \\ncoding, phase coding, and burst coding. Table 2 provides a substantial list of the types  of neuronal  information  encoding \\ntechniques utilised  to-date for consideration. It highlights \\ntheir key elements for consideration when investigating the potential for Fabric -driven SNN neuronal encoding and de - techniques utilised  to-date for consideration. It highlights \\ntheir key elements for consideration when investigating the potential for Fabric -driven SNN neuronal encoding and de - \\ncoding.  \\n \\nD. SPIKING  NEURAL  NETWORKS  ENERGY \\nCONSIDERATIONS  \\nSpiking neural networks bring the promise  of enhanced en - \\nergy efficiency. As an SNN is a dynamic system this suits more  dynamic  driven processes  and applications.  Research  is \\nongoing to investigate  how to effectively  lower  synaptic  oper- \\nations  and hence the computational  performance of the neural \\nnetwork.  [50] focus  on the optimization  of energy  consump-  \\ntion of SNN  for neuromorphic  applications  through a hybrid \\ntraining  strategy  that also accounts  for energy  cost stemming \\nfrom  the networks  computations.  From  a structure  and archi - \\ntectural  point  of view  SNN  have  typically  fewer  neural  nodes \\nthan more  traditional artificial neural  networks  along with the \\nfact that SNN  can implement \\u2019 node  connection pruning\\u2019[Fig \\n6] in order to reduce processing power and improve overall \\nthe working functionality  and energy  efficiency  of the SNN.  \\n[30] develop  a pruning method  for SNNs  by exploiting  the \\n \\n \\n \\nFIGURE  6. Difference conveying no pruning in SNN versus  pruning  in SNN 9   \\n  \\noutput  firing  characteristics  of neurons,  which  can be applied \\nduring network training. [31] detail the process of pruning \\nSTDP- based connections as well as quantizing the weights \\nof critical synapses at regular intervals during the training process. They validate a classification accuracy of 90.1 per- \\ncent and show an improvement in energy efficiency. When implementing  a SNN  in a textile  environment,  the capability \\nto be able to disconnect and reconnect fabric neural nodes \\nneeds  to be considered in order  to be able to prune  the Fabric \\nSNN  in an efficient  manner  enhancing  the fabric  SNN  energy \\noperational  functionality.  From  a computational  point  of view \\nthe SNN  has the capability  to operate more  quickly  due to the \\nneurons sending spike impulses. As SNN\\u2019s adopt temporal \\ninformation retrieval this increases the overall processing \\ntime and productivity and hence has a very positive end impact on energy consumption in the SNN.  \\n \\nV. NEUROMORPHIC COMPUTING  PROPERTIES  \\nNeuromorphic computing concept originated in the 1980\\u2019s. Taking inspiration from computer science, mathematics to \\nbio-inspired  models  of neural  network.  This emerging  inter- \\ndisciplinary research field has the potential to disrupt tradi - \\ntional computing methods  and architectural  implementations \\nleading to a mor e centralized and combined memory and \\ncomputational driven approach, moving away from the von Neumann architectural approach with separate memory and \\ncomputing capabilities  and high compute  power  needs.  Such \\ninspiration coming from the working of the human brain \\npaves the way for new and more fault tolerant layered and \\nparallel architectural designs and layouts.  \\n \\n \\nFIGURE  7. Timeline  of technological  advancements  in Neuromorphic  \\nComputing  In order to understand what aspects of neuromorphic \\ncomputing can inspire innovative advancements in the next generation of smart etextiles and on -garment edge based \\nintelligence,  we will first highlight  the core key architectural \\nelements  of importance  within neuromorphic  computing  and \\nassess  their potential within  an etextiles  domain.  Neuromor - \\nphic computing core architecture  is based  on the concept \\nof communicating through event driven spikes generated \\nthrough simple  processing structures  represented  by synapses \\nand neurons. Ongoing research is pushing the production \\npossibilities using complementary metal oxide semiconduc - \\ntor (CMOS) technology to develop neuromorphic spiking \\nneural  network hardware  implementations  [45][46][47].  Key \\nproperties  such as size, weight,  low power  consumption,  and \\nmodular design (scalability) are dominating the research \\nareas of focus linked to such technologies. Over the years \\nadvancements in CMOS technology has driven smaller and \\nmore power efficient systems with the capabi lity to mass \\nproduce.  Such  technology combined  with advanced  machine \\nlearning techniques has directly lead to the simulation and \\nimplementation of silicon based neurons, otherwise defined \\nas neuromorphic computing.  \\n[Fig 7] highlights advancements, with Bra inchip \\n(https:\\/\\/brainchip.com\\/) announcement in 2022 claiming to \\nbe the worlds first commercial producer of a Neuromorphic \\nAI processor \\u2018Akida \\u2019 that has the capability to mimic the \\nworking of the human brain and process data with high \\nprecision  and energy  efficiency.  Akida  being  an event -based \\nAI neural processor featuring 1.2 million neurons and 10 billion synapse . \\n \\nA. NEUROMORPHIC COMPUTING  COMPUTATION  \\n1) Crossbar  Array  Architectural  Properties  \\nAs neuromorphic computing moves away from the tradi - \\ntional Von Neumenn  architecture towards  a more  focused  in- \\nmemory computational architecture, the processing occurs \\ninside the memory functionality elements, hence reducing \\ndata transfer  time and energy.  Hardware  architectural  design \\nconsiderations  need  to take into account  1) synapse  intercon - inside the memory functionality elements, hence reducing \\ndata transfer  time and energy.  Hardware  architectural  design \\nconsiderations  need  to take into account  1) synapse  intercon - \\nnections between neural nodes and 2) how this can be im - \\nplemented  in order  to complete a fully connected  neural  net- \\nwork . From  a hardware design perspective the cross bar array \\narchitecture has been  adopted in neuromorphic  computing in \\norder to implement a full complement of interconnections \\nrequired for to meet the neural network structure require - \\nments. The cross bar array architecture includes a number  \\nof rows (word lines) and columns (bit lines) with memory \\ndevices interconnected between both the row and column. \\nAdvancements have been made through the development of \\nresistive  memory  devices  known as memristors  (one transis - \\ntor and 1 resistor combination).  The operational  functionality \\nof the crossbar  array  is based  in input  current  (voltage  pulse ) \\nto selected rows which in turn activates selected columns  \\nvia a voltage pulse, depending on the activation of varying \\ncells in the crossbar array. For active cells in a particular \\nrow\\/column vertical  line in the crossbar  array,  the sum of 10   \\n  \\nTABLE  3. Taking  Inspiraton  from SNN  and Neuromorphic  advancements,  highlighting  key considerations  to feed into future  etextile neural  network  research  and \\nprototypes  \\n \\nHuman Brain - NN properties\\/Neuromorphic Computing  Properties  Inspired Etextiles  Considerations  \\nSynapse \\/Memristor: component  that regulates  electric  current  flow remem - \\nbering  the previous current flowthrough.  \\n \\n  \\n \\n \\n \\n \\n \\nSoma \\/Neuristor:  Device  to capture  the properties  of a neuron,  Spike  or \\nimpulse generation when threshold reached.  How to represent weights in a fabric environment to simulate synapse  \\ninterconnections between fabric -based neurons. Conductive thread -based  \\nresistors, surface mount devices or nanodevices in a fabric environment to  \\nsimulate the workings of the synapse weights. Interchangeability of these  \\nweights in a fabric environment need  to be considered and generation of a  \\nnew method and fabric -based process around how to design , develop and  \\nvalidate. How to embed in a workable manner a wearable memory aspect  \\nin a fabric environment, in a seamless functioning manner. What would a \\nfabric  based memristor look like, how could this be completed and validated  \\nin a fabric environment.  \\n \\nReplication of a Neuron in a fabric environment.Take inspiration from  \\ncurrent electronic artificial neurons (LIF, Hodgki  Huxley).Considerations  \\naround how to create textile and fabric -based components to replace hard  \\ncomponent elements (conductive thread -based resistors, textile- based ca-  \\npacitors).  The aim being  to investigate  the most  suitable  way to create  \\na working fabric neuron but with limited hard components instead using  \\ntextile  versions.  \\n \\n Axon \\/Circuit  interconnections  and signal  conditioning.  In order  to maint ain the  focus  on a textile -based  implementation  the use  of \\nconductive  thread  (embroidered  conductive  thread  neuron  interconnections  \\ndesign  pattern)  as a means  to easily  reproduce  and create  such fabric  neuron  \\ninterconnections.  \\nDendrite \\/3D architectural design implementation, pattern detection and sub  \\nthreshold  filtering.  \\n \\n \\nFan In,Fan  Out\\/Implemented  using  crossbar  array,  but this has limitations  \\nwith regard scaling. Higher radix interconnections are being considered.  \\n(Loihi2 pr ovides faster and higher radix interfaces).  Taking  inspiration  from  the 3D architectural  design  used in Neuromorphic  \\ncomputing , can this motivate and inspire new 3D fabric manipulation and  \\n3D fabric layering type designs in a garment structure , to embed and  \\ninterconnect such fabric driven neural network functionality.  \\nConductive  thread,  with insulation  bridges  to eliminate  short  circuits  in \\na fabric environment , can be utilized to embed such a crossbar array  \\ndesign in a textile environment. Dependi ng on the type of synapse utilized \\n( conductive thread resistor, SMD resistor, memristor or other) a physical  \\nconnection to this synapse type will need to be considered to assess best  \\napproach in order to interconnect to the fabric -based crossbar array.  \\n \\n \\n \\n \\n \\nFIGURE  8. Conceptual  Jacket  design  with embedded neural  network  using  \\ncrossbar array architectural design.  \\n \\n \\ncurrents  equals  the output  current,  calculated  using  Ohms  law \\nand Kirchhoff\\u2019s  law. Research  into memristor  crossbar  arrays \\nfor brain inspired computing neural networking has been \\ninvestigated and summarised by [35]. [54] report a 64x64 \\npassive crossbar circuit that demonstrat es approximately 99 \\npercent  nonvolatile  metal -oxide  memristors,  enabling  the ac- \\ntive storing  of large  neural  network models  on neuromorphic  chips.  \\nFrom a functionality and design perspective, how can \\ninspiration be taken and mapped to an etextiles fabric envi - \\nronment in order to progress towards an embedded neural From a functionality and design perspective, how can \\ninspiration be taken and mapped to an etextiles fabric envi - \\nronment in order to progress towards an embedded neural \\nnetwork. The crossbar row\\/column structure design is a key element to consider, how ca n this architectural design be \\naccommodated  in a fabric  material  in order  to recreate such a \\nneural  network [Fig 8]. Can embroidery based  techniques  and \\npatterns using conductive thread , fabric tape or embedded woven conductive elements into a fabric envi ronment be \\nexperimented with in order to recreate such a crossbar array \\ntype architectural  structure  in a fabric  material.  This is a key \\ndesign element for further exploration and research.  \\n \\nB. NEUROMORPHIC COMPUTING MONITORING\\/ \\nMEASURING  \\nMemristors also known as resistive  switching  random  access \\nmemory devices that have the capability to change their re - \\nsistance  state and act as non volatile  memories  for embedded \\nmemory based devices are showing promise in the neuro - \\nmorphic  world  as key components  to implement  high-density \\nmemory.  Properties  of memristors  include  small  device\\/high \\ndensity integration, low power, high speed and highly scal - \\nable( Zhang et al., 2020). New research is focusing on the \\npotential to enable  controls  for resistive  filament switc hing in \\nsynapse  applications, as well as further investigation  around 11   \\n  \\nvarying  memristor  materials  for artificial synapses  with spe- \\ncific focus on the synaptic behaviors of organic materials, \\n2D materials, emerging materials ( halide perovskites ) and \\nlow-dimensional materials [33][34].The memristor is very \\nsuitable for analog based circuits as well as hardware multi - \\nstate neuromorphic appli cations due to its high and low \\nresistance state. Interconnections between the neural nodes \\nin the human brain have a joint strength represented by the \\nsynapse. Memristive synapses are ideal candidates to create \\nan artificial synaptic  device helping it mimic interconnection \\nstrengths  between  artificial neural  nodes.  A core requirement \\nis the need to enable and alter resistance states. When we \\nconsider  fabric  smart  material,  how wearable memory  can be \\nincorporated into a fabric environment is a key element t hat \\nrequires extensive investigation and research. Taking inspi - \\nration from memristor -based analog memory circuits, what \\nproperties  and elements  need  to be considered  when  consid - \\nering the link between fabric materials and the application \\nfunctionality.A nalog memristors  exhibit  a gradual  change  in \\nresistance and hence  are more  suitable  for analog  circuits  and \\nneuromorphic  system  applications.  Bi-stable  memristors  act \\nas binary memory\\/switches and digital logic circuits. Multi-  \\nstate memistors are used as multi -bit memories, reconfig - \\nurable  analog  circuits,  and neuromorphic  circuits  [32].Ames \\nResearch Center in California have implemented a method \\nof weaving flexible computer memory into garments. This \\nflexible memory is woven together using strands of copper and copper -oxide  wires.  [57] demonstrate  advanced  research \\ninto the development  of a textile  memristor  using  a robust  fi- \\nbre through an electric  field assembly  method that weaves  the \\nfibres  into a scalable  textile  memristor.  This exciting  research \\nera will see advances through the fusion of nanotechnology level memristor devices.  \\n \\nC. NEUROMORPHIC COMPUTING  COMMUNICATIONS \\nAND ENERGY EFFICIENCY  \\nBuilding  on the crossbar  array  design,  neuromorphic  chip ad- \\nvancements  look to implement energy  efficient  lower  power \\nconsumption  architectures  supporting  the required  precision \\ncommunication. In order to accomplish this research is on - \\ngoing around the design and development of smaller and \\nmultiple arrays. Such multiple arrays are emerging as either \\nhaving a lateral 2D layout or a 3D vertical stacking layout. \\nCircuit designs  are required  to be efficient  in order  to enable \\ndata flow between each layer in such 3D pas sive arrays. 3D \\nmemristive  neural  networks  are taking  inspiration  from  string \\nstacking for 3D NAND flash (Xia et al).  \\nFrom  a design perspective when  considering how to embed \\na multilayered neural network in a fabric environment, it is \\nvital to consider key properties of the fabric as well as key \\ndesign and usability  functionality  requirements  for end users. \\nWe already  touched  on possible  layered  and woven concep - \\ntual design layout  approaches  in [Fig 5], but if adopting a 3D \\nstacked  layered  fabric  approach,  how we can interconnect  the \\nlayered neural node connections also need to be considered. \\nHow do we interconnect  from  one fabric  layer  to another  fabric layer in an energy efficient, low power and reduced \\nsize capacity to ensure a high operational  standard for the \\nfabric  based  neural  network.  A modular  fabric  design- based \\napproach with inter -changeable neural nodes and hidden \\nneural  layers  may prove  to be a more  suitable  option availing \\nof the capability to interconnect, remove and replace neural \\nnodes  using for example  snap connectors  or other  connector \\nmethod options as described in [36].  \\n \\nVI. AI TEXTILE  INTELLIGENCE  USE CASE  EXAMPLES  \\nEmbedded AI intelligence in a fabric based environment method options as described in [36].  \\n \\nVI. AI TEXTILE  INTELLIGENCE  USE CASE  EXAMPLES  \\nEmbedded AI intelligence in a fabric based environment  \\nhas the potential to be applied across many sector based \\napplications. Here we briefly provide 2 such examples in \\norder to convey the possibilities of such an advanced fab - \\nric computing and fabric AI intelligence driven era, 1) the \\nhealthcare\\/rehabilitation sector application space and 2) the \\nunman ned aircraft\\/drone sector where textile -  driven drone \\ncontrol intelligence applications could be exploited.  \\n \\nA. HEALTHCARE  AI SMART  TEXTILE  USE CASE  \\nSmart  garment  applications  can greatly  contribute  towards  re- \\nmote monitoring, where individual and personalised health-  \\ncare provides enhanced real time assessment and early in - \\ntervention. Embedded seamless AI in a textile environment adds another layer of real time intelligent wearable point of \\ncare going beyond c urrent state of the art. The application  \\nof AI intelligence in a fabric wearable environment bring  \\nthe potential for enhanced quality of life for end users. Here \\nwe provide two such conceptual end user -centric scenario \\nexamples.  \\n\\u2022 Aisling  is concerned  about  getting  a variant  of COVID - \\n19 and is looking for a new means  to be able to monitor \\nand track her general health without it impacting on  \\nher daily  activities.  Aisling  purchased  an AI monitoring \\npackage ( textile sensors and Fabric AI patch intelli - \\ngence)  that can be fitted  in a modular  manner  into \\nher latest modular clothing garment. Aisling now has \\nembedded  artificial  intelligence  in her everyday  clothing \\nand can monitor  her breathing and temperature,  analyse \\nthe data in real time and be alerted about abnormali - \\nties that occur, allowing her to respond in an efficient manner,  detect  symptoms  early,  take a COVID test and \\nrestrict her movements if needs be.  \\n\\u2022 Jim suffers from Epileptic seizures. He was diagno sed \\nas having tonic -clonic seizures which cause symptoms \\nsuch as his muscles to stiffen, reduced breathing capa - \\nbility, loss of consciousness (prone to falls) and rapid \\njerking  of the arms  and legs. Due to not knowing  when \\nsuch a seizure can occur, Jim has become very anxious \\nabout leaving the house on this own to run errands. \\nThis has a huge impact on Jim\\u2019s quality of life. Jim\\u2019s \\ndoctor introduced him to the wearable modular smart \\nAI garment with interchangeable Fabric AI embedded, \\nthat can help monitor Jim\\u2019s temperature, breathing and \\nselected arm\\/leg muscle Electromyography (EMG) re - \\nsponse through embedded textile based sensors in the 12   \\n  \\n \\n \\nFIGURE  9. Core  building modules  and emerging technologies  that are contributing towards  the next smart  textiles  generation incorporating Fabric  AI inspired  \\ne-textiles.  \\n \\nwearable modular  garment  interconnected  to a Fabric  AI \\nneural  network computational  interchangeable  patch.  If \\nJim wears this garment when leaving the house to run \\nhis errands,  the sensors  will monitor  his measurements, \\ndepending on the assessment completed via the Fabric \\nAI neural network patch, this can decipher in real time \\nif Jim is in a pre -seizure state or not. I t was explained \\nto Jim that this analysis  is completed  in the intelligence \\nembedded in the core fabric and yarn of the garment \\nand is not visible to the human eye. If a pre -seizure \\nstate is detected a simple visual alert via an embedded \\nlight alert system  in the sleeve of jims garment  will alert \\nhim to the pre -seizure and give jim time to try to find  \\na safe zone in an attempt to reduce potential falls and \\ninjuries. In parallel the embedded fabric AI would also \\nraise the alert externally  to Jims carer,  informing them  of \\nthe pre-seizure  state.  This form  of monitoring  provided \\nthe wearer with a sense of e mpowerment and control.  \\n \\nB. TEXTILE -DRIVEN DRONE  CONTROL  USE CASE  \\nThe application of etextiles across multiple domains and \\nsectors is gaining traction. New innovative ideas extending beyond the norm of healthcare are starting to be considered and emerging.  Advances  in fabric  based  AI intelligence  open \\nthe opportunity to extend  applications  of etextiles  fusing non \\ntraditional techniques  and new technologies.  An example  of \\none such area is the application and use of a based fabric textile  intelligence  with unmanned  Ariel vehicles  (UAV).  the \\nfollowing provides example use cases for consideration  \\u2022 Taking a modular designed dynamic field pro - \\ngrammable or Fabric AI -driven smart garment with in - \\ntelligent embedded  control  logic  functionality  [55][58], \\nopens up opportunities towards the use of such a smart \\ngarment as a control device of the UAV\\u2019s based on \\nhuman  control  activated  movements  linked  to the smart \\ngarment triggering smart textile sensors as actuators directing the movement and control of the UAV. Such \\nfabric  AI driven haptic  wearable devices  can have  mul- \\ntiple applications for varying devices providing a more seamless embedded control options for end users. This \\nhas numerous innovative applications in construction, \\ndefence and more.  \\n \\nVII. CONCLUSION  \\nIts evident  that AI driven technology advancements  are mov- \\ning at a rapid pace. Vast research stemming from architec-  \\nturally inspired specification and design properties of SNN and neuromorphic computing provide valuable inspiration \\ntowards new techniques, methodologies and designs that  \\ncan be applied across to d rive emerging innovations in the \\netextiles  domain.  This paper  has delved  into key architectural \\nproperties of SNN ( artificial neurons and synapses )as well \\nas neuromorphic  computing (cross  bar analysis,  memristors, \\nstacked and layered design based approache s) to stem such \\nexperimental research avenues.Table 3 and [Fig 9] provide  \\na summarised visual of core aspects to inspire and drive  \\nnew and novel innovations in the next generation fabric AI inspired etextiles. 13   \\n  \\nContinued research is required in this area. Key research \\nquestions  and challenges  still remain  unanswered  hence  val- \\nidating  the need for further research in this space. Such \\nchallenges and future research investigations include the \\nfollowing  \\n\\u2022 Advancements  in the specification,  design and verifica - \\ntion of Fabric AI.  \\n\\u2022 Consideration around the identification and develop - \\nment of a  Fabric AI based development language.  \\n\\u2022 Investigation  into how AI algorithms  can be embedded \\nin an operational manner in a Fabric AI environment.  \\n\\u2022 In a textile  environment  what  methods  or processes  can \\nbe applied to enable ML based data abstraction and proce ssing.  \\n\\u2022 Specification  and formalisation  of textile  driven  proper - \\nties to support fabric AI systems.  \\n\\u2022 The need for further investigate research around the \\nverification of Fabric AI approaches delving into trust - \\nworthy and explainable Fabric AI.  \\nAI technologies  have  developed  at a much  quicker  pace over \\nthe past few years,  its now time for the etextiles  domain \\nto embrace such advancements and build on core defined elements and properties in order to stem new and exciting \\nresearch driven innovations in the etextiles domain.  \\n \\nREFERENCES  \\n[1] Hong,  Y., Zeng,  X., Brunixaux,  P., Chen,  Y. (2018).  Evaluation  of Fashi on \\nDesign Using Artificial Intelligence Tools. In Artificial Intelligence for  \\nFashion Industry in the Big Data Era (pp. 245 -256). Springer, Singapore.  \\n[2] Banumathi, D.P., Sree, T.S.,  Priya, S. (2015). Artificial Intelligence  \\nTechniques in Textile Fabric Inspe ction.  \\n[3] Hanbay, K., Talu, M. F.,  \\u00d6zg\\u00fcven, \\u00d6. F. (2016). Fabric defect detection  \\nsystems and methods \\u2014A systematic literature review. Optik, 127(24),  \\n11960 -11973.  \\n[4] Yuldoshev, Nuritdin Tursunov, Bobir  Qozoqov, Saidmuhtor. (2018).  \\nUse of artificial intelligence methods in operational planning of textile  \\nproduction.  Journal  of Process  Management.  New Technologies.  6. 41-51. \\n10.5937\\/jouproman6- 17221.  [5] Jiang, Siqi Stange, Oliver  B\\u00e4tcke, Fynn  Sultanova, Sabina  Sabantina,  \\nLilia.  (2021).  Applications of Smart  Clothing \\u2013 a Brief  Overview.  Commu - \\nnications  in Development  and Assembling  of Textile  Products.  2. 123-140. \\n10.25367\\/cdatp.2021.2.p123- 140. \\n[6] Shi J, Liu S, Zhang L, Yang B, Shu L, Yang Y, Ren M, Wang Y, Chen J,  \\nChen  W, Chai  Y, Tao X. Smart  Textile -Integrated  Microelectronic  Systems  \\nfor Wearable Applications. Adv Mater. 2020 Feb;32(5):e1901958. doi: \\n10.1002\\/adma.201901958. Epub 2019 Jul 5. PMID: 31273850.  \\n[7] Loke, G., Yan, W., Khudiyev, T., Noel, G.,  Fink, Y. (2020). Recent  \\nprogress and perspectives  of thermally drawn  multimaterial  fiber electron - \\nics. Advanced Materials, 32(1), 1904911.  \\n[8] Park, S., Loke, G., Fink, Y.,  Anikeeva, P. (2019). Flexible fiber -based  \\noptoelectronics for neural interfaces. Chemical Society Reviews, 48(6),  \\n1826- 1852.  \\n[9] Rajan,  Krishna,  Erik Garofalo,  and Alessandro  Chiolerio.  2018.  \\\"Wearable  \\nIntrinsically Soft, Stretchable, Flexible Devices for Memories and Com - \\nputing\\\" Sensors 18, no. 2: 367. https:\\/\\/doi.org\\/10.3390\\/s18020367 \\n[10] Zeng,  W., Shu, L., Li, Q., Chen,  S., Wang,  F. and Tao, X.-M. \\n(2014), Fiber -Based Wearable Electronics: A Review of Materials,  \\nFabrication, Devices, and Applications. Adv. Mater., 26: 5310-5336.  \\nhttps:\\/\\/doi.org\\/10.1002\\/adma.201400633 \\n[11] Loke, Gabriel  Khudiyev, Tural  Wang, Brian  Fu, Stephanie  Payra, \\nSyamantak Shaoul, Yorai Fung, Johnny Chatziveroglou, Ioannis Chou,  \\nPin-Wen Chinn, Itamar  Yan, Wei  Gitelson-Kahn, Anna  Joannopou - \\nlos, John  Fink,  Yoel.  (2021).  Digital  electronics  in fibres  enable  \\nfabric -based machine -learning inference. Nature Communica tions. 12.  \\n10.1038\\/s41467 -021-23628 -5. \\n[12] Fern\\u00e1ndez -Caram\\u00e9s, Tiago M., and Paula Fraga -Lamas. 2018. \\\"Towards  \\nThe Internet  of Smart  Clothing: A Review  on IoT Wearables  and Garments 10.1038\\/s41467 -021-23628 -5. \\n[12] Fern\\u00e1ndez -Caram\\u00e9s, Tiago M., and Paula Fraga -Lamas. 2018. \\\"Towards  \\nThe Internet  of Smart  Clothing: A Review  on IoT Wearables  and Garments \\nfor Creating Intelligent Connected E -Textiles\\\" Electronics 7, no. 12: 405.  \\nhttps:\\/\\/doi.org\\/10.3390\\/electronics7120405 \\n[13] Pelin Yildirim Taser, Vahid Khalilpour Akram, Machine Learning Tech - \\nniques  for IoT-Based  Indoor  Tracking and Localization,  Emerging Trends  \\nin IoT and Integration  with Data Science,  Cloud  Computing,  and Big Data \\nAnalytics, 10.4018\\/978 -1-7998 -4186 -9.ch007, (123-145), (2022).  \\n[14] Li, Jun Li, Yafei Li, Lingmin He, Xiaokang Fu, Jingjing Chen, and  \\nXiaokang Zhou. 2021. Fabric Defect Detection in Textile Manufacturing: \\nA Survey of the State of the Art. Sec. and Commun. Netw. 2021 (2021).  \\nhttps:\\/\\/doi.org\\/10.1155\\/2021\\/9948808 \\n[15] Taherkhani  A, Belatreche  A, Li Y, Cosma  G, Maguire  LP, McGinnity TM. \\nA review of learning in biologically plausible spiking neural networks.  \\nNeural Netw. 2020 Feb;122:253 -272. doi: 10.1016\\/j.neunet.2019.09.036.  \\nEpub 2019 Oct 11. PMID: 31726331.  \\n[16] Koch, C., Segev, I. (Eds.). (1998). Methods in neuronal modeling: from  \\nions to networks. MIT press.  \\n[17] Gerstner, W., Sprekeler, H.,  Deco, G. (2012). Theory and simulation in  \\nneuroscience. Science , 338(6103), 60\\u2013 65. \\n[18] Izhikevich, E. M. (2003). Simple model of spiking neurons. IEEE Trans - \\nactions on neural networks, 14(6), 1569- 1572.  \\n[19] Andreoni, G., Standoli, C. E.,  Perego, P. (2016). Defining Requirements  \\nand Related  Methods for Designing  Sensorized  Garments.  Sensors  (Basel,  \\nSwitzerland),  16(6),  769. https:\\/\\/doi.org\\/10.3390\\/s16060769  \\n[20] Qiang, Siyu and Carey, Tian and Arbab, Adrees and Song, Weihua and  \\nWang, Chaoxia and Torrisi, Felice(2019), Wearable solid -state capacitors  \\nbased on two- dimensional material all -textile heterostructures,Nanoscale  \\njournal,Vol 11, Issue 20 .  \\n[21] Beeman D. (2014) Hodgkin -Huxley Model. In: Jaeger D., Jung R. (eds)  \\nEncyclopedia of Computational Neuroscience. Springer, New York, NY.  \\n[22] A. Bonfiglio et al., \\\"Organic  field effect  transistors for textile  applications,\\\"  \\nin IEEE Transactions on Information Technology in Biomedicine, vol. 9,  \\nno. 3, pp. 319 -324, Sept. 2005.  \\n[23] Carey, T., Cacovich, S., Divitini, G. et al. Fully inkjet -printed two - \\ndimensional material field-effect heterojunctions for wearable and textile  \\nelectronics. Nat Commun 8, 1202 (2017).  \\n[24] Pfeiffer M, Pfeil T. Deep Learning With Spiking Neurons: Oppor - \\ntunities and Challenges. Front Neurosci. 2018 Oct 25;12:774. doi:  \\n10.3389\\/fnins.2018.00774. PMID: 30410432; PMCID: PMC6209684.  \\n[25] M. Hulea, Z. Ghassemlooy and S. Rajbhandari, \\\"A Spiking Neural Net - \\nwork with Visible Light Communications,\\\" 2018 11th International Sym - \\nposium on Communication Systems, Networks  Digital Signal Processing  \\n(CSNDSP), 2018, pp. 1 -5, doi: 10.1109\\/CSNDSP.2018.8471811.  \\n[26] Guo W, Fouda  ME, Eltawil  AM, Salama  KN. Neural  Coding in \\nSpiking Neural  Networks:  A Comparative  Study  for Robust Neu- 14   \\n  \\nromorphic Systems. Front Neur osci. 2021 Mar 4;15:638474. doi: \\n10.3389\\/fnins.2021.638474. PMID: 33746705; PMCID: PMC7970006.  \\n[27] Zeldenrust, F., Wadman, W. J.,  Englitz, B. (2018). Neural Coding With  \\nBursts- Current State and Future Perspectives. Frontiers in computational  \\nneuroscience, 12, 48. https:\\/\\/doi.org\\/10.3389\\/fncom.2018.00048 \\n[28] Coms\\u00b8a, I. M., Versari, L., Fischbacher, T., Alakuijala, J. (2021). Spik - \\ning Autoencoders With Temporal Coding. Frontiers in neuroscience, 15,  \\n712667.  https:\\/\\/doi.org\\/10.3389\\/fnins.2021.712667 \\n[29] Auge, D., Hille, J., Mueller, E.,  Knoll, A. (2021). A survey of encoding  \\ntechniques for signal processing in spiking neural networks. Neural Pro - \\ncessing Letters, 53(6), 4693- 4710.  \\n[30] Shi Y, Nguyen L, Oh S, Liu X, Kuzum D. A Soft-Pruning Method  \\nApplied During Training of Spiking Neural Networks for In-memory  \\nComputing Applications. Front Neurosci. 2019 Apr 26;13:405. doi: \\n10.3389\\/fnins.2019.00405. PMID: 31080402; PMCID: PMC6497807.  \\n[31] Rathi,  N., Panda,  P., Roy,  K. (2019).  STDP -Based  Pruning of Connections \\nand Weight  Quantization  in Spiking  Neural  Networks  for Energy -Efficient \\nRecognition.  IEEE Transactions  on Computer -Aided Design of Integrated  \\nCircuits and Systems, 38, 668-677.  \\n[32] Xu, W., Wang, J.,  Yan, X. (2021). Advances in Memristor -Based Neural  \\nNetworks. Frontiers in Nanotechnology.  \\n[33] Wang,  Z. Y., Wang,  L. Y., Nagai,  M., Xie, L. H., Yi, M. D., Huang,  W., \\nAdv.  Electron. Mater.  2017, 3,  1600510.  \\n[34] Kim, S. G., Han, J. S., Kim, H., Kim, S. Y., Jang, H. W., Adv. Mater. \\nTechnol. 2018 , 3, 1800457.  \\n[35] Xia, Q.,  Yang, J. J. (2019). Memristive crossbar arrays for brain -inspired \\ncomputing. Nature Materials, 18(4), 309 \\u2013323. \\n[36] Castano, L. M.,  Flatau, A. B. (2014). Smart fabric sensors and e -textile  \\ntechnologies: a review. Smart Materials and struc tures, 23(5), 053001.  \\n[37] Sethuraman,  S. C., Kompally,  P., Mohanty,  S. P., Choppali,  U. (2021).  My- \\nWear: A novel smart garment for automatic continuous vital monitoring.  \\nIEEE Transactions on Consumer Electronics, 67(3), 214 -222. \\n[38] Ali, N. A.,  Abu-Elkheir, M. (2015, October). Internet of nano -things  \\nhealthcare applications: Requirements, opportunities, and challenges. In  \\n2015 IEEE 11th International Conference on Wireless and Mobile Com - \\nputing, Networking and Communications (WiMob) (pp. 9 -14). IEEE.  \\n[39] Dong, K., Peng, X., Wang, Z. L. (2020). Fiber\\/fabric -based piezoelectric  \\nand triboelectric  nanogenerators  for flexible\\/stretchable  and wearable  elec- \\ntronics and artificial intelligence. Advanced Materials, 32(5), 1902549.  \\n[40] Xu, K., Huang,  J., Ye, Z., Ying,  Y., Li, Y. (2009).  Recent  development of \\nnano -materials used in DNA biosensors. Sensors, 9(7), 5534 -5557.  \\n[41] Wood,  J. (2018).  Revolutions in wearable  technology for apparel.  In High - \\nperformance apparel (pp. 325 -339).  \\n[42] She, X. (2022). Design and Optimization of Heterogeneous Feedforward  \\nSpiking Neural Network For Spatiotemporal Data Processing (Doctoral  \\ndissertation, Georgia Institute of Technology).  \\n[43] Shen, J., Liu, J. K.,  Wang, Y. (2021). Dynamic Spatiotemporal Pattern  \\nRecognition With Recurrent Spiking Neural Ne twork. Neural Computa - \\ntion, 33(11), 2971 -2995.  \\n[44] Stoliar, P., Schneegans, O.,  Rozenberg, M. J. (2021). Implementation of  \\na Minimal Recurrent Spiking Neural Network in a Solid-State Device.  \\nPhysical Review Applied, 16(3), 034030.  \\n[45] Davies et al., (2018)\\\"Loihi: A Neuromorphic Manycore Processor with  \\nOn-Chip Learning,\\\" in IEEE Micro, vol. 38, no. 1, pp. 82 -99. \\n[46] A. L. et. al (2018), Loihi Asynchronous Neuromorphic Research \\nChip.24th IEEE International Symposium on Asynchronous Circuits and \\nSystems  (ASYNC).  \\n[47] Cheng,  H., Wen,  W., Wu, C., Li, S., Li, H.H.,  & Chen,  Y. (2017).  \\nUnderstanding the design of IBM neurosynaptic system and its tradeoffs: Systems  (ASYNC).  \\n[47] Cheng,  H., Wen,  W., Wu, C., Li, S., Li, H.H.,  & Chen,  Y. (2017).  \\nUnderstanding the design of IBM neurosynaptic system and its tradeoffs:  \\nA user perspective. Design, Automation & Test in Europe Conference &  \\nExhibition (DATE), 2017, 139 -144. \\n[48] Chan,  J., Gollakota,  S. (2017,  October).  Data  storage  and interaction  using \\nmagnetized fabric. In Proceedings of the 30th Annual ACM Symposium  \\non User Interface Software and Technology (pp. 655 -663).  \\n[49] Lindrupsen,  N. (2021).  Exploring  Textile  Controllers  for Computer  Music  \\n(Master\\u2019s  thesis).  \\n[50] Sorbaro M, Liu Q, Bortone M and Sheik S (2020) Optimizing the Energy Consumption of Spiking Neural Networks for Neuromorphic Applica - \\ntions. Front. Neurosci. 14:662. doi: 10.3389\\/fnins.2020.00662 \\n[51] Blecha, T.,  Moravcova, D. (2022, Ma y). Methods for the Capacity In - \\ncreasing of Textile Capacitors. In 2022 45th International Spring Seminar  \\non Electronics Technology (ISSE) (pp. 1-5). IEEE.  \\n[52] Park,  Y., Park,  M.-J., Lee, J.-S., Adv.  Funct.  Mater.  2018,  28, 1804123.  \\nhttps:\\/\\/doi.org\\/10.1002\\/adfm.201804123.  [53] Ham S, Kang M, Jang S, Jang J, Choi S, Kim TW, Wang G. One - \\ndimensional organic artificial multi -synapses enabling electronic textile  \\nneural network for wearable neuromorphic applications. Sci Adv. 2020 \\nJul 10;6(28) :eaba1178. doi: 10.1126\\/sciadv.aba1178. PMID: 32937532.  \\n[54] Kim, H., Mahmoodi, M.R., Nili, H. et al. 4K -memristor analog -grade \\npassive crossbar circuit. Nat Commun 12, 5198 (2021).  \\n[55] Rajappan  A, Jumet  B, Shveda  RA, Decker  CJ, Liu Z, Yap TF, Sanchez  V, \\nPreston DJ . Logic -enabled textiles. Proc Natl Acad Sci U S A. 2022 Aug  \\n30;119(35):e2202118119. doi: 10.1073\\/pnas.2202118119. Epub 2022  \\nAug 22. PMID: 35994641; PMCID: PMC9436326.  \\n[56] F. Cleary  et al., \\\"Wearable  Fabric  Brain  Enabling  On-Garment  Edge -Based  \\nSensor Data Proc essing,\\\" in IEEE Sensors Journal, vol. 22, no. 21, pp.  \\n20839 -20854, 1 Nov.1, 2022, doi: 10.1109\\/JSEN.2022.3207912.  \\n[57] Liu, Y., Zhou,  X., Yan,  H., Zhu, Z., Shi, X., Peng,  Y., Chen,  L., Chen,  \\nP., Peng,  H., Robust Memristive  Fiber  for Woven  Textile  Memristor.  Adv.  \\nFunct.  Mater.  2022,  32, 2201510.  https:\\/\\/doi.org\\/10.1002\\/adfm.202201510 \\n[58] Cleary, F., Henshall, D. C.,  Balasubramaniam, S. (2021). On -Body Edge  \\nComputing Through E -Textile Programmable Logic Array. Frontiers in  \\nCommunications and Networks, 18.\",\"19\":\" Networks\\u2019 modulation: How di\\ufb00erent structural\\nnetwork properties a\\ufb00ect the global\\nsynchronization of coupled Kuramoto oscillators.\\nJuliette Courson1;2;3, Thanos Manos2, and Mathias Quoy2;4\\n1Laboratoire de Physique Th\\u00e9orique et Mod\\u00e9lisation (LPTM), CNRS, UMR 8089,\\nCY Cergy Paris Universit\\u00e9, Cergy-Pontoise Cedex, France\\njuliette.courson@cyu.fr\\n2Equipes Traitement de l\\u2019Information et Syst\\u00e8mes (ETIS), CNRS, UMR 8051,\\nENSEA, CY Cergy Paris Universit\\u00e9, Cergy-Pontoise Cedex, France\\nthanos.manos@cyu.fr\\n3Department of Computer Science, University of Warwick, Coventry, UK\\n4IPAL CNRS Singapore\\nmathias.quoy@cyu.fr\\nAbstract. Inalargevarietyofsystems(biological,physical,socialetc.),\\nsynchronizationoccurswhendi\\ufb00erentoscillatingobjectstunetheirrhythm\\nwhen they interact with each other. The di\\ufb00erent underlying network\\nde\\ufb01ningtheconnectivitypropertiesamongtheseobjectsdrivestheglobal\\ndynamics in a complex fashion and a\\ufb00ects the global degree of synchrony\\nof the system. Here we study the impact of such types of di\\ufb00erent net-\\nwork architectures, such as Fully-Connected, Random, Regular ring lat-\\ntice graph, Small-World and Scale-Free in the global dynamical activity\\nof a system of coupled Kuramoto phase oscillators. We \\ufb01x the external\\nstimulation parameters and we measure the global degree of synchrony\\nwhendi\\ufb00erentfractionsofnodesreceivestimulus.Thesenodesarechosen\\neither randomly or based on their respective strong\\/weak connectivity\\nproperties (centrality, shortest path length and clustering coe\\ufb03cient).\\nOur main \\ufb01nding is, that in Scale-Free and Random networks a sophis-\\nticated choice of nodes based on their eigenvector centrality and average\\nshortest path length exhibits a systematic trend in achieving higher de-\\ngree of synchrony. However, this trend does not occur when using the\\nclustering coe\\ufb03cient as a criterion. For the other types of graphs consid-\\nered, the choice of the stimulated nodes (randomly vs selectively using\\nthe aforementioned criteria) does not seem to have a noticeable e\\ufb00ect.\\n1 Introduction\\nComplex networks\\u2019 theory is a powerful tool in various \\ufb01elds that allow us to\\ninvestigate and understand the real world [1,2]. For example, di\\ufb00erent ensembles\\nof neurons connected by synapses coordinate their activity to perform certain\\ntasks (in biology), infrastructures like the Internet are formed by routers and\\ncomputer cables and optical \\ufb01bers (in hardware communication) and the human\\npersonal or professional relationships (in social sciences) to name a few [3].arXiv:2303.03099v1  [nlin.AO]  24 Feb 2023 1. INTRODUCTIONCourson et al.\\nNonlinearity is a very important feature in complex systems giving a rich\\nrepertoire of di\\ufb00erent activity patterns, such as stable, unstable, periodic etc. A\\nmodi\\ufb01cation of some parameter might also produce a change in their stability,\\nand therefore in the dynamics of the system. Furthermore, such systems may\\nhave a high sensitivity to initial conditions, or to any external input, that could\\ncompletely change their dynamics [4].\\nSuchdynamicsoftenyieldtoaself-organisedcoherentactivity,i.e.tosynchro-\\nnization. The latter can be loosely de\\ufb01ned as the capacity of di\\ufb00erent oscillating\\nobjects to adjust their rhythm due to their interaction and plays a key role in a\\nlarge variety of systems, whether biological, physical, or even social (see e.g. [5]).\\nIn a more formal way, synchronization emerges from the interaction of several\\nautonomous oscillators, also called self-sustained oscillators. That is, nonlinear\\ndynamicalsystemsthatproduceoscillationswithoutanyneedofexternalsource.\\nTheir dynamics is given by a nonlinear di\\ufb00erential equation or, in the case of\\nmultiple coupled oscillators, by several coupled di\\ufb00erential equations.\\nThe relative way that autonomous oscillators are connected within a given\\nnetwork can a\\ufb00ect their global activity and synchronization properties. Neural\\nnetworks can be represented as a graph of connections between the di\\ufb00erent neu-\\nrons. Since the introduction of small-world networks and scale-free networks (see\\ne.g. [6,8]), the \\ufb01eld of network graph analysis has attracted the attention of many\\nstudies aimed to better understand complex systems (see e.g. [9,10,11,12,13]).\\nFurthermore, modern network connectivity techniques allow us to capture var-\\nious aspects of their topological organization, as well as to quantify the local\\ncontributions of individual nodes and edges to network\\u2019s functionality (see e.g.\\n[14]).\\nIn neuroscience, synchronization plays a very important role. The human\\nbrain is a very large and complex system whose activity comprises the rapid\\nand precise integration of a gigantic amount of signals and stimulus to perform\\nmultiple tasks (see e.g. [14,15,16]). One example occurs in epileptic seizures,\\nwhere periods of abnormal synchronization in the neural activity can spread\\nwithin di\\ufb00erent regions of the brain, and cause an attack in the a\\ufb00ected person\\n(see e.g. [17]). More examples are found in other brain diseases such as Parkinson\\ndisease, where an excessively synchronized activity in a brain region correlates\\nwith motor de\\ufb01cit (see e.g. [18,19] and references therein) or tinnitus (see e.g.\\n[20,21,22] and references therein).\\nIn this study, we focus at a rather theoretical framework. We set out to in-\\nvestigate the impact of di\\ufb00erent network architectures, such as Fully-Connected,\\nRandom, Regular ring lattice graph, Small-World and Scale-Free in the global\\ndynamical activity of a system of coupled Kuramoto phase oscillators [23]. The\\nKuramoto model has been broadly used to study various types of oscillatory\\ncomplex activity, see e.g. [24,25,26] (to name only a few) and references therein.\\nOur goal is to investigate the impact of the network (graph) structure in the sys-\\ntem\\u2019sglobaldegreeofsynchronizationwhenapplyingidenticaland\\ufb01xedexternal\\nstimulus to di\\ufb00erent subsets of nodes which are chosen according to various net-\\nwork connectivity criteria. We \\ufb01nd that, in scale-free and random networks, a\\n2 Courson et al.\\n2. METHODS AND MATERIALS\\nsophisticated choice of nodes based on graph connectivity properties exhibits a\\nsystematic trend in achieving higher degree of synchrony. For the other types of\\ngraphs considered, the choice of the stimulated nodes (randomly vs selectively\\nusing the aforementioned criteria) seems to not have a noticeable e\\ufb00ect.\\n2 Methods and Materials\\n2.1 Connectivity measurements\\nWe here study the dynamics of phase oscillators coupled via binary, undirected\\ngraphsG= (V;E), containing a set of NverticesV=fv2J1 :NKgand a set\\nE=f(v;w)2J1 :NK2gof edges. Let Abe the corresponding adjacency matrix,\\nwithAvw= 1if there is a connection between node vand nodew,0otherwise.\\nSelf-connections are excluded, so Av;v= 0for any vertex v. For our analysis\\nlater on, we will use the following graph connectivity measurements [27]:\\n\\u2013 Shortest path length. The shortest path length Lv;wbetween any two\\nnodesvandwis the number of connections on the shortest path going\\nfrom one to another, computed following Dijkstra\\u2019s algorithm. We de\\ufb01ne\\nthe shortest path length of a node vas the average shortest path between v\\nand any other node of the network:\\n<Lv>=X\\nw2VLv;w\\nN: (1)\\nNote thatLv;wmight not be de\\ufb01ned if there is no way connecting node vto\\nnodew. The lower the shortest path length, the fastest the information goes\\nfrom one node to another. For example, when building a subway network\\n(that is, a graph where di\\ufb00erent stations are interconnected), one might\\nwant to minimize the stations\\u2019 average shortest path length so the users can\\neasily navigate across the city.\\n\\u2013 Centrality. The eigenvector centrality is used to quantify the importance\\nof a node in the network. Let \\u0015be the highest eigenvalue for A, so that all\\nthe corresponding eigenvector\\u2019s components are non null. The eigenvector\\ncentralityxvof vertexvis de\\ufb01ned as the vthcomponent the eigenvector,\\nnamely:\\nxv=1\\n\\u0015X\\nw2VAw;vxw: (2)\\nKeeping in mind the subway network example, a station with a high cen-\\ntrality would be densely connected to other stations, in particular to other\\ncentral ones.\\n\\u2013 Clustering. Letkv=P\\nwAvwbe the degree of node v. In the case of a\\nundirected graph,kv(kv\\u00001)\\n2edges can exist in the direct neighborhood of v.\\nWithnvthe number edges that actually exist in this neighborhood, the local\\nclustering coe\\ufb03cient is de\\ufb01ned as [16]:\\nCv=2nv\\nkv(kv\\u00001): (3)\\n3 2. METHODS AND MATERIALSCourson et al.\\nThat is, in a subway network where stations BandCare the next stop after\\nstationAon their line, clustering would give the probability that there exists\\na line directly connecting BandC.\\n2.2 Neural networks as graphs\\nWe investigate synchronization properties in various network con\\ufb01gurations that\\nexhibit in general di\\ufb00erent characteristics. In more detail we here employ the\\nneural networks described in the following list (see e.g. [6]):\\n\\u2013 Fully-Connected networks. They contains (N\\u00001)2edges connecting\\nevery node in one layer to every node in the other layer.\\n\\u2013 Regular networks. They consist of a lattice of Nnodes, each being con-\\nnected to their knearest neighbors.\\n\\u2013 Small-World networks. A Small-World network is constructed from a\\nRegular one after multiple random rewiring phases: going clockwise over the\\nlattice, a vertex and the edge to its nearest neighbor are selected. The edge\\nis removed, and the vertex reconnected to a random node with probability\\np, without duplicating any existing edge. This random rewiring is repeated,\\nconsidering the following nearest neighbour, until having rewired with prob-\\nabilitypall edges of the network. Choosing pin an adequate range of values,\\nthe built network exhibits both high mean clustering and low mean charac-\\nteristic path length, that is small-worldness.\\n\\u2013 Random networks. Using the same procedure as for Small-World net-\\nworks, setting p= 1produces a Random graph, with all edges being system-\\natically randomly rewired.\\n\\u2013 Scale-Free networks. They are networks whose degree distribution follows\\na power-law:\\nP(k)\\/k\\u0000\\r: (4)\\nwithkthe node degrees, \\ra real constant. The Barab\\u00e1si-Albert model gives\\na procedure for the construction of scale-free networks [7], by starting with\\na small Fully-Connected network of m0nodes then adding one by one the\\nN\\u0000m0remaining nodes, connecting them to the malready present nodes\\nwith probability\\npv=kvP\\nwkw; (5)\\nw2J0 :m\\u00001K. Scale-free networks exhibit nodes with degrees that are\\nseveral standard deviations away from the average degree of the network.\\nThesehighlyconnectednodesarecalled hubs.Notethat,however,forsmaller\\nnetworks with size N < 100, the scale-free property might not be properly\\nobservable.\\nFig. 1 provides a visual representation of the above mentioned graphs. Note\\nthat for visualization purposes we here show only a small fraction of the actual\\nnetworks that we use later in our simulations where the number of nodes is set\\nbeN= 500.\\n4 Courson et al.\\n2. METHODS AND MATERIALS\\n(a) Fully-Connected\\n (b) Scale-Free\\n(c) Regular\\n (d) Swall-World\\n (e) Random\\nFig.1. Network graphs. Small graphs of size N= 20showing the di\\ufb00erent network\\nstructures: (a)Fully-Connected graph, (b)Scale-Free graph with initial size m0= 5,\\n(c)Regular graph with node degree k= 4,(d)Small-World graph with initial node\\ndegreek= 4and rewiring probability p= 0:2and(c)Random graph with initial node\\ndegreek= 4. See text for more details.\\n2.3 The Kuramoto model\\nWe use of the Kuramoto model to study the neural activity of the coupled\\nsystem. To this end we consider a population of Nphase oscillators [23]:\\n_\\u0012i=!i+F\\u000eisin(\\nt+\\u0012i) +K\\nkiX\\njAijsin(\\u0012j\\u0000\\u0012i); (6)\\nwhere\\u0012idenotes thephaseof the i\\u0000th oscillator, !iits respectivefrequency (Hz)\\ndrawn from a Lorentz probability distribution g(!)of scale parameter \\r= 0:5,\\ncentered in x0= 1.Ais the binary adjacency matrix coupling the oscillators, ki\\nthedegreeofoscillator iandKistheglobalcouplingconstant.Weapplyexternal\\nstimulus in a subset of the oscillators with \\ufb01xed amplitude Fand frequency\\n\\n. The term \\u000eiis a binary function indicating this subset of nodes where the\\nstimulation is applied in di\\ufb00erent realization in our simulations,\\n\\u000ei=\\u001a1if nodeiis in the stimulated subset\\n0 else.(7)\\nWe set the time-step at 0:01s and we integrated the system with an Euler scheme\\n(no noise is considered).\\n5 3. RESULTSCourson et al.\\nThe system\\u2019s degree of synchrony is measured using its order parameter r\\n[23]:\\nrei =1\\nNNX\\nj=1ei\\u0012j; (8)\\nwhere\\tdenotes the population\\u2019s mean phase. The order parameter rtends to 1\\nforaperfectlysynchronizedpopulationandto 0intheabsenceofsynchronization\\nrespectively. Due to the presence of strong \\ufb02uctuations, all rtime-series shown\\nin this paper are determined using a moving average on r, on time windows of\\nlength 2s sliding each 0:1s. The \\ufb01nal states of a population, rf, are computed by\\naveraging these moving-averaged rtime-series over a 15s time-window where the\\nsystem has reached its stable state. The system\\u2019s degree of synchrony depends\\non the coupling strength K\\u2019s relative position to a critical coupling strength Kc,\\nwhose value depends on the network con\\ufb01guration (see e.g. [28,29]). Here, we\\nset this value at K= 0:2so that all considered networks are desynchronized in\\nthe absence of any external stimulation.\\n3 Results\\nNetwork modulation. In order to adequately tune the stimulus\\u2019 amplitude\\nand frequency values such that they can lead the system into a synchronous\\nstate, we \\ufb01rst perform a systematic analysis in the parameter space (F;\\n).\\nHence, we begin by applying external stimulus to all the nodes for each pair of\\nparameters and measure the \\ufb01nal order parameter rf. Such a parameter map\\nreveals the presence of the well-known Arnold tongues [5], namely regions in the\\nplane (F;\\n)for which the system gets synchronized.\\nIn Fig. 2, we present the di\\ufb00erent synchronization regions (presence of sev-\\neral Arnold tongues) for a Regular network of a relatively small size N= 20and\\nmean neighborhood k= 4. For every other studied network, the maps depicts\\nsimilar features with large synchronization regions at relatively small amplitudes\\nof the external current, F < 200. These tongues get thinner with higher values of\\nF. Inside the main Arnold tongues, the oscillators are phase-locked at the forc-\\ning frequency \\nandrf\\u00191. Inside zones of weaker degree of synchrony rf<1,\\nsome oscillators are phase-locked, while the oscillators of higher natural frequen-\\ncies keep rotating independently. The white star symbol in the bottom-left part\\nof the \\ufb01gure indicates the chosen parameters (F;\\n) = (5;1)for our forthcoming\\nsimulations, resulting in a partial phase-locking of the network. Note that we\\nhave performed similar analysis with larger sizes but smaller parameter grid size\\nand the overall picture turns out to be consistent. We have also prepared similar\\nplots for all considered network con\\ufb01gurations (\\ufb01gures not shown here).\\nSimulation protocol. We set the values F= 5,\\n= 1Hz for the stimulus in-\\ntensity and frequency in Eq. (6), so that the network is weakly entrained without\\nbeing completely phase-locked. We then measure the degree of synchronization\\n6 Courson et al.\\n3. RESULTS\\nFig.2. Synchronization regions in the stimulation frequency-amplitude pa-\\nrameterspaceforaRegularnetwork. FinalorderparameterreachedforaRegular\\nnetwork of size N= 20, and degree k= 4where all nodes are stimulated, for di\\ufb00erent\\npairs of stimulus intensity ( F) and frequency ( \\n) values in Eq. (6). Each data point\\ncorresponds to a single simulation over 30s, the \\ufb01nal order parameter being averaged\\nover the last 15s. The color map shows large main synchronization regions, as well\\nas small higher-order synchronization areas. The white star symbol at the bottom-left\\npart of the \\ufb01gure indicates the chosen parameters (F;\\n ) = (5;1)for the forthcoming\\nsimulations.\\n(with the order parameter) in di\\ufb00erent networks described in Sect. 2.2. The sys-\\ntem starts evolving for 4s without any external input before we start applying\\nthe stimulation to a subset of nodes until 30s. More precisely, we stimulate dif-\\nferent fractions of nodes, i.e., 25%,50%and75%in each given network. These\\nnodes can be either chosen randomly, or depending on particular connectivity\\nproperties (as described in Sect. 2.1). For the latter case, we \\ufb01rst sort the nodes\\naccording to their connectivity relative measurements (from higher to lower),\\ni.e. the eigenvector centrality, average shortest path length and clustering coef-\\n\\ufb01cient. The resulting time series are smoothed with a moving average, and their\\nrfis averaged over the last 15s. In order to obtain a statistically relevant value\\nof the \\ufb01nal value of the order parameter rf, we performed 20simulations for\\neach di\\ufb00erent network-setup (randomizing the initialization\\/generation of the\\nnetworks, the natural frequencies and the initial conditions for each simulation).\\nIn Fig.3, we show representative time-series for various stimulation setups for\\nScale-Free networks of size N= 500and initial size m0= 5. Each panel corre-\\nsponds to a di\\ufb00erent initialization, from which the order parameter evolution is\\ncomputed depending on the amount of stimulated nodes and the way they are\\nselected. Thin solid lines correspond to randomly selected nodes. In that case,\\na \\ufb01rst subset containing 25 %of the nodes is created. Then for the stimulation\\nof larger in size subsets, another 25 %of nodes is successively added, so that\\nlarger subsets of random nodes always contains the smaller one. The bold solid\\n7 3. RESULTSCourson et al.\\n(a) Eigenvector centrality\\n (b) Shortest path length\\n (c) Clustering coe\\ufb03cient\\nFig.3.Representative rtime-series for di\\ufb00erent stimulation setups of Scale-\\nFree networks. The order parameter as a function of time, for N= 500oscillators,\\nwhenstimulating0%(blue),25%(orange),50%(red)and75%(black)ofthenodes.The\\nstimulated nodes are chosen randomly, then based on their (a)eigenvector centrality\\n(b)average shortest path length and (c)clustering coe\\ufb03cient values. Bold solid lines\\n(resp. dashed lines) correspond to the stimulation of nodes with the highest (resp.\\nlowest) values, while thin solid lines correspond to the stimulation of randomly chosen\\nnodes.\\nlines (resp. dashed lines) show the time-series when the stimulated nodes are\\nthe ones with the highest (resp. lowest) eigenvector centrality (panel (a)), aver-\\nage shortest path length (panel (b)) and clustering coe\\ufb03cient (panel (c)). Note\\nthat higher values of the connectivity measurement do not necessarily lead to\\nstronger synchrony. In particular, lower values for the nodes\\u2019 average shortest\\npath length depicts shorter connections to the rest of the network, and therefore\\na more e\\ufb03cient synchronization.\\nOptimization of global synchronization In Fig. 4, we present the main\\n\\ufb01nding of our work, namely a systematic comparison of the synchronization ef-\\n\\ufb01ciency when applying identical stimulus in di\\ufb00erent types of graph networks.\\nEachpanelissplitinto3columnsshowingthestatisticalsummaryfortheensem-\\nbles of di\\ufb00erent realizations and choosing to stimulate di\\ufb00erent subsets of nodes,\\ni.e. randomly (middle-orange boxplots in the legends) or with highest (upper-red\\nboxplots in legends) or lower (lower-blue boxplots in the legends) connectivity\\nmeasurement. Panel (a) refers to a Small-World network of size N= 500, initial\\ndegreek= 4and rewiring probability p= 0:2, (b) to a Random network of\\nsizeN= 500, initial degree k= 4and rewiring probability p= 1and (c) to a\\nScale-Free network of size N= 500and initial size m0= 5respectively. Note\\nthis analysis is not performed on Regular and Fully-Connected networks, since\\nall nodes of such networks have identical connectivity properties and does not\\nallow any connectivity-based selection.\\n8 Courson et al.\\n3. RESULTS\\n(a) Small-World network\\n (b) Random network\\n(c) Scale-Free network\\nFig.4. Synchronization e\\ufb03ciency comparison for di\\ufb00erent types of graph\\nnetworks. Final order parameter obtained, for (a)Small-World networks (b)Ran-\\ndom networks (c)Scale-Free networks of size N= 500. The \\ufb01nal value of the order\\nparameterrfis computed for di\\ufb00erent stimulation subset sizes, composed of randomly\\nchosen nodes (middle-orange boxplots in the legends), nodes with the highest connec-\\ntivity measurement (left-red boxplots)and nodes with thelowest connectivity measure-\\nment(right-blueboxplots). rfshownareanaverageover 20simulations.Theanalysisis\\nperformed with three di\\ufb00erent connectivity measurements: eigenvector centrality (left\\ncolumn in each panel), average shortest path length (central column in each panel) and\\nclustering (right column in each panel).\\nInScale-Freenetworks,theglobalorderparameterreacheshighervalueswhen\\nthe stimulus is applied to the nodes with higher eigenvector centrality or lower\\naverage shortest path length. In such networks, a small fraction of the nodes\\n9 4. SUMMARY AND DISCUSSIONCourson et al.\\nhave signi\\ufb01cantly higher connectivity, and stimulating preferentially these nodes\\nenables strong synchronization compared to stimulating random nodes.\\nIn Random networks, selecting stimulated nodes according to their lowest av-\\nerage shortest path length instead of randomly enhances synchronization. How-\\never, there is no bene\\ufb01t in choosing more central nodes. Indeed, the high degree\\nof randomness (induced by a rewiring probability p= 1, see Sect. 2.2) in these\\nnetworks\\u2019connectionscausesdisparityinthenodes\\u2019averageshortestpathlength,\\nwithout creating any node of way higher centrality. In Small-World networks,\\nthe connections are distributed in a more homogeneous way, and hence the node\\nselection as no substantial impact on the system\\u2019s \\ufb01nal synchrony.\\nFor all three aforementioned networks and all stimulation subset sizes, the\\nselection of the nodes according to their clustering coe\\ufb03cient does not show\\nany advantage over a simple random choice. Note that, we also checked for a\\npotential overlap of clustering coe\\ufb03cient values between the randomly chosen\\nnodes and those chosen with higher respective values. It turns out that they\\ndi\\ufb00er signi\\ufb01cantly. Our working hypothesis is that, by picking nodes with higher\\nclustering coe\\ufb03cient, we induce higher synchrony locally (member nodes of the\\nsub-network) while we disfavour other nodes outside of it. However, by choosing\\nnodes randomly, we end up stimulating simultaneously nodes within and outside\\nclusters eventually providing a better global synchronization. Finally, for all net-\\nworks and stimulation subset selection methods, although they overall achieve\\nhigher degree of synchrony, larger stimulated subsets containing 75% of the pop-\\nulation do not allow to observe any clear advantage in particular selection of the\\nnodes, since all three possible subsets largely overlap.\\n4 Summary and Discussion\\nIn this study, we investigated the impact of structure and connectivity properties\\nin a modulated network. We sought out to identify e\\ufb03cient ways to synchronize\\na population of Kuramoto phase oscillators using nodes\\u2019 stimulation with \\ufb01xed\\nsmall amplitude and frequency. To this end, we \\ufb01rst performed a parameter\\nsweep exploration for stimulus amplitude and frequency parameters to identify\\nsettings that allow the system to synchronize. Then, we computed the evolution\\nof characteristic networks of Kuramoto oscillators, where external stimulation\\nis applied to di\\ufb00erent subpopulations with identical \\ufb01xed low amplitude and\\nfrequency. In order to measure the system\\u2019s synchrony of each network-type and\\nstimulationcon\\ufb01guration,wecalculatedtheglobalorderparameterforensembles\\nof di\\ufb00erent random system initializations.\\nWe showed that by choosing this subpopulations based on their respective\\nnetwork connectivity properties (i.e. high eigenvector centrality and lower short-\\npath length), we were able to enhance the networks\\u2019 global degree of synchro-\\nnization in comparison to the one achieved by randomly choosing them. This is\\nnot the case when using the clustering coe\\ufb03cient as a selection criterion. How-\\never, this direction deserves further investigation, e.g. detect di\\ufb00erent network\\n10 Courson et al.\\n4. SUMMARY AND DISCUSSION\\ncommunities, measure their local synchronization and study its impact on the\\nglobal synchrony when such nodes are chosen to be stimulated.\\nFrom a neuroscience point of view Scale-Free networks play an important\\nrole in the structure and function of mammal brains, see for example [30] (a\\nstudy on the scale-free dynamics and the emergence of collective organisation\\noccurs in rodents) or [31] (investigating the fractal structure of the human brain\\nand its dynamics). Furthermore, in Alzheimer patients\\u2019 brain, the functional\\nconnectivity structure is found to exhibit properties similar to Random network\\ngraphs (see e.g. [32,33] and references therein). Thus, understanding how to\\noptimally synchronize systems with similar network structures can improve the\\noverall expected performance of a given external simulation protocol.\\nAcknowledgements\\nJC was supported by the LABEX MME DII (ANR-16-IDEX-0008) PhD grant\\nand CY cognition grant of the CY Cergy-Paris University. TM was supported\\nby the Labex MME DII (ANR-11-LBX-410 0023-01) French national funding\\nprogram. MQ was supported by a CNRS \\ufb01nancing for a research semester in\\nthe IPAL Lab in Singapore. We also acknowledge the use of the Osaka compute-\\ncluster resources of CY Cergy-Paris University.\\nCode and data availability\\nAll data generated or analysed during this study are included in this manuscript.\\nOur code is written in Pythonand we used the NetworkX python package. Our\\ncode is available upon request.\\nReferences\\n1. Barrat, A., Barth\\u00e9lemy, M., Vespignani, A.: Dynamical processes on complex net-\\nworks. Cambridge: Cambridge University Press (2008)\\n2. Nicolis, G., Nicolis, C.: Foundations of Complex Systems: Emergence, Information\\nand Prediction, World Scienti\\ufb01c, 2nd ed., (2012)\\n3. Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks. From biological nets to\\nthe internet and WWW. Oxford: Oxford University Press (2003)\\n4. Strogatz, S.H.: Nonlinear dynamics and chaos: with applications to physics, biology,\\nchemistry,andengineering.Secondedition.Boulder,CO:WestviewPress,amember\\nof the Perseus Books Group, (2015)\\n5. Pikovsky, A., Rosenblum, M.G., Kurths, J.: Synchronization, A Universal Concept\\nin Nonlinear Sciences. Cambridge University Press, Cambridge (2001)\\n6. Watts, D.J., Strogatz, S.H.: Collective dynamics of \\u2018small-world\\u2019 networks. Nature\\n393(6684), 440\\u2013442 (1998)\\n7. Barab\\u00e1si, A.L.: The barab\\u00e1si-albert model. In: Network science. Cambridge Univer-\\nsity Press, Cambridge (2016)\\n8. Barab\\u00e1si,A.,R\\u00e9ka,A.:Emergenceofscalinginrandomnetworks.Science286(5439),\\n509\\u2013512 (1999)\\n11 4. SUMMARY AND DISCUSSIONCourson et al.\\n9. Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barab\\u00e1si, A.L.: The large-scale\\norganization of metabolic networks. Nature 407(6804), 651\\u2013654 (2000)\\n10. Strogatz, S.H.: Exploring complex networks. Nature 410(6825), 268\\u2013276 (2001)\\n11. Li, W., Cai, X.: Statistical analysis of airport network of china. Phys. Rev. E 69,\\n046106 (2004)\\n12. Bassett, D.S., Bullmore, E.: Small-world brain networks. The Neuroscientist 12(6),\\n512\\u2013523 (2006)\\n13. Berry, H., Quoy, M.: Structure and dynamics of random recurrent neural networks.\\nAdaptive behavior 14, 129\\u2013137 (2006)\\n14. Sporns, O.: Networks of the Brain. The MIT Press (2010)\\n15. Chialvo, D.: Emergent complex neural dynamics. Nature Phys 6, 744\\u2013750 (2010)\\n16. Fornito, A., Zalesky, A., Bullmore, E.: Fundamentals of brain network analysis.\\nElsevier\\/Academic Press (2016)\\n17. Wong, R.K., Traub, R.D., Miles, R.: Cellular basis of neuronal synchrony in\\nepilepsy. Advances in Neurology 44, 583\\u2013592 (1986)\\n18. Brown, P.: Oscillatory nature of human basal ganglia activity: Relationship to the\\npathophysiology of parkinson\\u2019s disease. Movement Disorders 18(4), 357\\u2013363 (2003)\\n19. Manos, T., Diaz-Pier, S., Tass, P.A.: Long-term desynchronization by coordinated\\nreset stimulation in a neural network model with synaptic and structural plasticity.\\nFrontiers in Physiology 12 (2021)\\n20. Eggermont, J.J., Tass, P.A.: Maladaptive neural synchrony in tinnitus: Origin and\\nrestoration. Frontiers in Neurology 6 (2015)\\n21. Manos, T., Zeitler, M., Tass, P.A.: Short-term dosage regimen for stimulation-\\ninduced long-lasting desynchronization. Frontiers in Physiology 9 (2018)\\n22. Manos, T., Zeitler, M., Tass, P.A.: How stimulation frequency and intensity impact\\non the long-lasting e\\ufb00ects of coordinated reset stimulation. PLOS Computational\\nBiology 14(5), 1\\u201331 (2018)\\n23. Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Dover Books on\\nChemistry Series, Dover Publications (2003)\\n24. Acebr\\u00f3n, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The kuramoto\\nmodel: A simple paradigm for synchronization phenomena. Reviews of modern\\nphysics 77(1), 137 (2005)\\n25. Rodrigues, F.A., Peron, T.K.D., Ji, P., Kurths, J.: The kuramoto model in complex\\nnetworks. Physics Reports 610, 1\\u201398 (2016)\\n26. Popovych, O.V., Jung, K., Manos, T., Diaz-Pier, S., Ho\\ufb00staedter, F., Schreiber, J.,\\nYeo, B.T., Eickho\\ufb00, S.B.: Inter-subject and inter-parcellation variability of resting-\\nstate whole-brain dynamical modeling. NeuroImage 236, 118201 (2021)\\n27. Newman, M.E.J.: Networks: an introduction. Oxford University Press, Oxford;\\nNew York (2010)\\n28. Mirollo, R., Strogatz, S. The Spectrum of the Partially Locked State for the Ku-\\nramoto Model. J Nonlinear Sci 17, 309\\u2014347 (2007)\\n29. Chiba, H., Medvedev, G. S., Mizuhara, M. S. Bifurcations in the Kuramoto model\\non graphs. Chaos (Woodbury, N.Y.), 28(7), 073109 (2018)\\n30. Ribeiro, T.L., Chialvo, D.R., Plenz, D.: Scale-free dynamics in animal groups and\\nbrain networks. Frontiers in Systems Neuroscience 14 (2021)\\n31. Grosu, G.F., Hopp, A.V., Moca, V.V., B\\u00e2rzan, H., Ciuparu, A., Ercsey-Ravasz,\\nM., Winkel, M., Linde, H., Mures ,an, R.C.: The fractal brain: scale-invariance in\\nstructure and dynamics. Cerebral Cortex (2022)\\n32. Stam, C.J., De Haan, W., Da\\ufb00ertshofer, A., Jones, B.F., Manshanden, I., Van\\nCappellen van Walsum, A.N., Montez, T., Verbunt, J.P.A., De Munck, J.C., Vn\\n12 Courson et al.\\n4. SUMMARY AND DISCUSSION\\nDijk,B.W.,Berendse,H.W.,Scheltens,P.:Graphtheoreticalanalysisofmagnetoen-\\ncephalographic functional connectivity in Alzheimer\\u2019s disease. Brain 132, 213\\u2014224\\n(2009)\\n33. Kang, M., Petrasek, Z.: Random Graphs: Theory and Applications from Nature\\nto Society to the Brain. Internationale Mathematische Nachrichten 227 1\\u201324 (2014)\\n13\",\"20\":\" On Modifying a Neural Network\\u2019s Perception\\nManuel de Sousa Ribeiro ,Jo\\u02dcao Leite\\nNOV A LINCS, NOV A University Lisbon, Portugal\\nmad.ribeiro@campus.fct.unl.pt, jleite@fct.unl.pt\\nAbstract\\nArti\\ufb01cial neural networks have proven to be ex-\\ntremely useful models that have allowed for mul-\\ntiple recent breakthroughs in the \\ufb01eld of Arti\\ufb01cial\\nIntelligence and many others. However, they are\\ntypically regarded as black boxes , given how dif-\\n\\ufb01cult it is for humans to interpret how these mod-\\nels reach their results. In this work, we propose a\\nmethod which allows one to modify what an arti-\\n\\ufb01cial neural network is perceiving regarding spe-\\nci\\ufb01c human-de\\ufb01ned concepts, enabling the gener-\\nation of hypothetical scenarios that could help un-\\nderstand and even debug the neural network model.\\nThrough empirical evaluation, in a synthetic dataset\\nand in the ImageNet dataset, we test the proposed\\nmethod on different models, assessing whether the\\nperformed manipulations are well interpreted by\\nthe models, and analyzing how they react to them.\\n1 Introduction\\nIn this paper, we investigate how to modify a neural network\\u2019s\\nperception regarding speci\\ufb01c human-de\\ufb01ned concepts not di-\\nrectly encoded in its input, with the goal of being able to\\nbetter understand how such concepts affect a models\\u2019 pre-\\ndictions. Our results suggest that this is possible to do with\\nfew labeled data, and without need to change or retrain the\\nexisting neural network model.\\nIn the last few decades, arti\\ufb01cial neural networks have en-\\nabled major advances in multiple \\ufb01elds, from text, image,\\nvideo and speech processing [Khan et al. , 2020 ], to medicine\\n[Shahid et al. , 2019 ], economics and \\ufb01nance [Ghoddusi et\\nal., 2019 ], and many others. Numerous successful neural\\nnetwork-based applications have recently been implemented\\n[Abiodun et al. , 2018 ], rendering these models ubiquitous.\\nDespite their popularity, neural networks lack interpretabil-\\nity, as they supply no direct human-interpretable indication\\nof why a given result was provided [Hitzler et al. , 2020 ].\\nThis led to the development of multiple methods focused\\non solving this shortcoming (cf. Section 6). Most popular\\nmethods typically focus on pointing out which inputs con-\\ntributed most for the output, or on substituting the model for\\none that is inherently interpretable [Kim et al. , 2018 ]. How-\\never, such methods leave to the users the burden of under-standing why the provided explanation justi\\ufb01es the output,\\ne.g, users must interpret why a particular set of features, e.g.,\\npixels in an image, and their values, leads to the output. Var-\\nious user studies [Adebayo et al. , 2020; Chu et al. , 2020;\\nShen and Huang, 2020 ]show that the explanations given by\\nthese methods are often disregarded or unhelpful to end users.\\nOne reason is that humans do not typically reason with fea-\\ntures at a very low level, such as individual pixels in an image\\n\\u2013 they typically reason with higher-level human de\\ufb01ned con-\\ncepts. For example, a useful explanation from the standpoint\\nof a human as to why a network classi\\ufb01ed a particular picture\\nof a train as being one of a passenger train will probably refer\\nto the fact that the wagons had windows rather than pointing\\nout speci\\ufb01c pixels in the image.\\nAdditionally, when humans attempt to determine the\\ncauses for some non-trivial phenomena, they often resort\\nto counterfactual reasoning [Miller, 2019 ], trying to under-\\nstand how different scenarios would lead to different out-\\ncomes. This approach seems also helpful for interpreting ar-\\nti\\ufb01cial neural networks, as it emphasizes the causes \\u2013 what is\\nchanged \\u2013 and the effects \\u2013 what mutates as a result of the\\nchanges. For example, to better interpret how a neural net-\\nworks is classifying a particular picture of a train, the user\\ncould ask what would have been the output had the picture\\ncontained a passenger wagon. We would like to be able to\\ngenerate such counterfactual scenarios based on how different could ask what would have been the output had the picture\\ncontained a passenger wagon. We would like to be able to\\ngenerate such counterfactual scenarios based on how different\\nhuman-de\\ufb01ned concepts impact a model\\u2019s predictions. The\\nuse of human-de\\ufb01ned concepts \\u2013 the concept of passenger\\nwagon in the previous example \\u2013 is important as it provides\\nsemantics with which to describe the counterfactuals, while\\nensuring that these concepts are meaningful and understand-\\nable for the users of a particular model.\\nThere has been work on developing methods to gener-\\nate counterfactuals for arti\\ufb01cial neural networks [Guidotti,\\n2022 ], with some even allowing for generating counterfac-\\ntual samples with respect to particular attributes [Yang et\\nal., 2021 ]. However, to our knowledge, they focus on how\\nparticular changes to the input samples might affect a mod-\\nels\\u2019 output, neglecting what the model is actually perceiving\\nfrom those samples. Furthermore, current methods to pro-\\nduce counterfactuals are typically complex to implement, of-\\nten requiring the training of an additional model to produce\\nthe counterfactuals [Stepin et al. , 2021 ], and the use of spe-\\nci\\ufb01c neural network architectures, e.g., invertible neural net-arXiv:2303.02655v1  [cs.AI]  5 Mar 2023 works [Hvilsh\\u00f8j et al. , 2021 ].\\nIn this work, we address the issue of counterfactual gen-\\neration at a different level of abstraction, focusing on what\\na model is perceiving from a given input regarding speci\\ufb01c\\nhuman-de\\ufb01ned concepts, and how that affects the model\\u2019s\\noutput. The idea would be to \\u201c convince \\u201d the model that it\\nis perceiving a particular concept \\u2013 for example a passenger\\nwagon \\u2013 without producing a speci\\ufb01c image containing one,\\nand checking its effect on the model\\u2019s output. By abstracting\\naway from generating particular counterfactual samples and\\ninstead focusing on generating counterfactuals with regards\\nto what a model is perceiving about human-de\\ufb01ned concepts,\\nwe allow for a better understanding of how what is encoded in\\na neural network impacts a models\\u2019 predictions in a human-\\nunderstandable manner. By manipulating what a model per-\\nceives regarding speci\\ufb01c concepts of interest, we allow users\\nto explore how the output of a neural network depends on\\nwhat it is perceiving regarding such concepts.\\nIn order for it to be possible to generate counterfactuals\\nbased on what a model is perceiving instead of what the\\nmodel is fed, we need to be able to understand what a neu-\\nral network model is perceiving and be able to manipulate\\nwhat the model is perceiving with respect to a speci\\ufb01c con-\\ncept. We \\ufb01nd inspiration in the research conducted in the\\n\\ufb01eld of neuroscience, where highly selective neurons that\\nseemed to represent the meaning of a speci\\ufb01c stimulus can be\\nidenti\\ufb01ed [Reddy and Thorpe, 2014 ]. These neurons, known\\nasconcept cells , seemed to \\u201cprovide a sparse, explicit and\\ninvariant representation of concepts\\u201d [Quiroga et al. , 2005;\\nQuiroga, 2012 ]. The discovery of these cells contributed for\\na better understanding of how the brain \\u2013 a complex and sub-\\nsymbolic system \\u2013 works and how it relates different concepts\\n[Gastaldi et al. , 2022 ]. Additionally, through the technique of\\noptogenetics, it is possible to manipulate the activity of spe-\\nci\\ufb01c neurons and learn their purpose [Okuyama, 2018 ].\\nAnalogously, being able to assign meaning to speci\\ufb01c neu-\\nrons in a neural network could provide us with a better un-\\nderstanding of what information is encoded in a given model,\\nand how it might be associating different concepts. Moreover,\\ngiven that typically one has access to a neural networks\\u2019 inter-\\nnals, we could manipulate the outputs of the neurons to which\\nmeaning has been assigned, and examine how such changes\\naffect the model under different circumstances, thus generat-\\ning counterfactual scenarios.\\nBased on the existing evidence that neurons with similar\\nproperties to concept cells seem to emerge in arti\\ufb01cial neural\\nnetworks [Goh et al. , 2021 ], we hypothesize that by identi-\\nfying which neurons in a neural network act as concept cells\\nto speci\\ufb01c human-de\\ufb01ned concepts and by manipulating the\\nactivations of such neurons, we should be able to modify a\\nneural network\\u2019s perception regarding those concepts.\\nIn this paper, we propose and test a method to gener-\\nate counterfactuals scenarios for arti\\ufb01cial neural networks\\nby modifying what they are perceiving regarding speci\\ufb01c\\nhuman-de\\ufb01ned concepts. In Section 2, we present the pro-\\nposed method, with Section 2.1 discussing how to pinpoint\\nwhich neurons identify a particular concept in a neural net-\\nwork, and Sections 2.2 to 2.4 addressing different aspects of\\nthe proposed method and providing experimental evidence tosupport our claims. Sections 3 and 4 illustrate possible ap-\\nplications of the proposed method. In Section 5, we apply\\nand test the method in the setting of the ImageNet dataset. In\\nSection 6, we discuss related work, concluding in Section 7.\\n2 A Method to Manipulate a Neural\\nNetwork\\u2019s Perception\\nIn order to generate counterfactuals regarding what a neural\\nnetwork model is perceiving about human-de\\ufb01ned concepts,\\nwe propose a method composed by three main steps. For each\\nconcept of interest:\\na) Estimate how sensitive each neuron is to that concept, network model is perceiving about human-de\\ufb01ned concepts,\\nwe propose a method composed by three main steps. For each\\nconcept of interest:\\na) Estimate how sensitive each neuron is to that concept,\\ni.e., how well its activations separate samples where that\\nconcept is present from those where it is absent;\\nb) Based on the neurons\\u2019 sensitivity values, select which\\nneurons are considered as \\u201cconcept cell-like\\u201d, which we\\nwill refer to as concept neurons ;\\nc) For each concept neuron, compute two activation values,\\nrepresenting, respectively, the output of that neuron for\\nsamples where that concept is present and absent.\\nConsider a neural network model M:I!O, which is the\\nmodel being analyzed, where Iis the input data space and O\\ndenotes the model output space. For each considered human-\\nde\\ufb01ned concept C, we assume a set of positive PC\\u0012Iand\\nnegative NC\\u0012Isamples where the concept is, respectively,\\npresent\\/absent. Let aM\\ni:I!Rrepresent the output of the\\nithneuron of neural network model Maccording to some\\narbitrary ordering of the neurons.\\nThe \\ufb01rst step of our method consists in estimating how sen-\\nsitive each neuron is to each considered concept, i.e., how\\nwell the activations of each neuron separate samples where a\\nconcept is present from those where it is not. We denote by\\nrM\\ni: 2I\\u00022I![0;1]the function which, for a given neuron\\niof a modelM, takes a set PCandNCand provides a value\\nrepresenting how sensitive iis to concept C. In Section 2.1,\\nwe consider different implementations of this function.\\nThe second step is to select, for each concept of interest C,\\nwhich neurons ofMare to be considered as concept neurons\\nforC. LetsC:R!f0;1gbe a threshold function indicating,\\nfor a given concept Cand sensitivity value, whether that value\\nis high enough to be considered as a concept cell. Then, for\\neach concept of interest C, we determine the set of selected\\nneurons as SC=fi:sC(rM\\ni(PC; NC)) = 1 ; i2[1; k]g,\\nwhere kis the number of neurons in M. In all experiments,\\nwe set the threshold value by gradually decreasing it, while\\ntesting the model performance in a 100sample validation set\\nfor each concept. Once more than 3neurons have been added,\\nby decreasing the threshold, and no performance improve-\\nment is seen in the validation set, we set the threshold to the\\nbest found value.\\nLastly, for each neuron selected as a concept neuron for\\nsome concept C, we compute two activation values repre-\\nsenting, respectively, when that concept is present and ab-\\nsent. We consider a function cM\\ni: 2I!R, which computes\\nan activation value for the ithneuron ofM, and use it com-\\npute an activation value representing the presence of concept\\nC,cM\\ni(PC), and one representing its absence, cM\\ni(NC). In TypeA\\u0011WarTraintEmptyTrain 9has :FreightWagonu9has :PassengerCaru9has :EmptyWagonvMixedTrain\\nTypeB\\u0011PassengerTraintLongFreightTrain 9has :(PassengerCaruLongWagon )t(\\u00152 has :PassengerCar )vPassengerTrain\\nTypeC\\u0011RuralTraintMixedTrain 9has :ReinforcedCaru9has :PassengerCarvWarTrain\\nLongFreightTrain\\u0011LongTrainuFreightTrain (\\u00152 has :LongWagon )t(\\u00153 has :Wagon )vLongTrain\\nEmptyTrain\\u00118has :(EmptyWagontLocomotive )u9has :EmptyWagon (\\u00152 has :FreightWagon )vFreightTrain\\nFigure 1: Subset of the XTRAINS dataset ontology\\u2019s axioms, describing how the trains\\u2019 representations are classi\\ufb01ed.\\nFigure 2: Sample images of the XTRAINS dataset.\\nSection 2.2, we discuss different implementations of cM\\niand\\ncompare how they impact the method\\u2019s results.\\nSubsequently, when some input x2Iis fed to neural net-\\nworkM, to generate a counterfactual scenario where concept\\nCis present or absent, we only need to replace the activa-\\ntion value aM\\ni(x)of each neuron iinSCwithcM\\ni(PC)or\\ncM\\ni(NC), respectively. We refer to this step, as injecting a\\nconcept into a neural network model.\\nTo test our method, we consider the Explainable Abstract\\nTrains Dataset (XTRAINS) [de Sousa Ribeiro et al. , 2020 ], a\\nsynthetic dataset composed of representations of trains, such\\nas those shown in Figure 2. This dataset contains labels re-\\ngarding various visual concepts and a logic-based ontology\\ndescribing how each of these concepts is related. Figure 1\\nshows a subset of the dataset\\u2019s accompanying ontology il-\\nlustrating how different concepts are related to each other.\\nThis ontology speci\\ufb01es, for example, that TypeA is either\\nWarTrain orEmptyTrain , and that WarTrain encompasses\\nthose having a ReinforcedCar and a PassengerCar . These\\nconcepts have a visual representation, e.g, a ReinforcedCar\\nis shown as a car having two lines on each wall, such as the\\n\\ufb01rst two cars of the leftmost image in Figure 2.\\nIn the experiments described below, we adopt a neural net-\\nwork developed in [Ferreira et al. , 2022 ], trained to identify\\ntrains of TypeA \\u2013 referred to asMA. This neural network was\\ntrained to achieve an accuracy of about 99% in a balanced test\\nset of 10 000 images. We also consider the de\\ufb01nition of rele-\\nvancy given in [de Sousa Ribeiro and Leite, 2021 ]to establish\\nwhich concepts are related to the task of a given neural net-\\nwork (w.r.t. the dataset\\u2019s accompanying ontology).\\n2.1 Identifying Concept Cell-like Neurons\\nIn order to be able to modify a neural network\\u2019s perception re-\\ngarding speci\\ufb01c human-de\\ufb01ned concepts, it is \\ufb01rst necessary\\nto determine which neurons in a model are identifying these\\nconcepts. Based on the evidence that neural networks seem\\nto have information encoded in their internals regarding con-\\ncepts which are related with their tasks [de Sousa Ribeiro and\\nLeite, 2021 ], and that neurons with \\u201cconcept cell-like\\u201d prop-\\nerties seems to emerge in arti\\ufb01cial neural networks [Goh et\\nal., 2021 ], we hypothesize that neural networks have neurons\\nwhich act as concept cells for concepts related with the tasks\\nthey perform. In this section, we investigate how to determine\\n\\u22122 0 2 4\\nActivation0.00.51.0DensityConcept\\u00acConcept\\n\\u22122 0 2 4Figure 3: Probability density function of two neurons for samples\\nwhere a given concept is present\\/absent.\\nsuch neurons in a neural network model.\\nWe consider a neuron to be \\u201cconcept-cell like\\u201d for some\\nconcept Cif it is possible to separate samples where Cis\\npresent from those where it is absent based on its activations.\\nFigure 3, illustrates the estimated probability density func-\\ntion of two neurons for some PCandNCsets. The neuron\\non the left side seems to act as a concept cell for C, showing\\nwell separated distributions for both sample sets. On the other\\nhand, the neuron on the right side does not act as a concept\\ncell for this concept, given that its activations do not distin-\\nguish between both sets of samples.\\nWhile there are methods, such as TCA V [Kim et al. , 2018 ]\\nand Mapping Networks [de Sousa Ribeiro and Leite, 2021 ], guish between both sets of samples.\\nWhile there are methods, such as TCA V [Kim et al. , 2018 ]\\nand Mapping Networks [de Sousa Ribeiro and Leite, 2021 ],\\nwhich allow for an understanding of whether a given model\\nis sensitive to a certain concept, here we are interested in un-\\nderstanding whether individual neurons are sensitive to that\\nconcept. Our goal is not to check whether the model is sen-\\nsitive to a concept, for which only some neurons might be\\nsuf\\ufb01cient, but rather to \\ufb01nd all neurons that are sensitive to\\nthat concept, so that we can manipulate them.\\nWe consider three different implementations of rM\\nito eval-\\nuate the adequacy of a neuron iofMas concept cell for C:\\n\\u2022 Spearman rank-order correlation between a neuron\\u2019s ac-\\ntivations and the dataset\\u2019s labels, computed as j1\\u0000\\n6Pd2\\nj\\nn(n2\\u00001)j, where dj=aM\\ni(PCj)\\u0000aM\\ni(NCj)andn=\\njPCj, assumesjPCj=jNCj;\\n\\u2022 Accuracy of a linear binary classi\\ufb01er ( b) predicting the\\ndataset\\u2019s labels from a neuron\\u2019s activations, computed as\\njfx:x2NC\\\\b(aM\\ni(x))=0)gj+jfx:x2PC\\\\b(aM\\ni(x))=1)gj\\njNCj+jPCj;\\n\\u2022 Probability density function intersection of a neu-\\nron\\u2019s activations for PCandNC, computed as 1\\u0000R\\nmin(fPC\\ni(x); fNC\\ni(x))dxwhere fPC\\niis the estimated\\nprobability density function of the activation value of\\nneuron ifor the positive samples in PCandfNC\\nifor the\\nnegative samples, assumes the samples to be indepen-\\ndent and identically distributed.\\nTo test our hypothesis that neural networks should typically Spearman Accuracy Intersection\\nEmptyTrain 13 20 16\\n9has:PassengerCar 15 19 18\\n9has:ReinforcedCar 18 22 21\\nWarTrain 13 15 9\\nFigure 4: Amount of concept neurons in MAfor each concept.\\nhave concept neurons for concepts which are relevant for\\ntheir tasks, we compute the three described metrics for four\\nrandom relevant concepts: EmptyTrain ,9has:PassengerCar ,\\n9has:ReinforcedCar , and WarTrain ; and for four non-\\nrelevant concepts: 9has:LongWagon ,9has:OpenRoofCar ,\\nLongFreightTrain , and MixedTrain . According to our hy-\\npothesis, we would expect that for relevant concepts we\\nshould be able to \\ufb01nd neurons with high values in the de-\\nscribed metrics, while for non-relevant concepts we should\\nbe unable to do so. For each considered metric, we de\\ufb01ne a\\nthreshold value above which a neuron is considered as a con-\\ncept neuron for this particular experiment: 0:85for Spearman\\nrank-order correlation, 0:95for the linear classi\\ufb01er\\u2019s accu-\\nracy, and 0:9for the probability density function intersection.\\nFigure 4 shows the amount of identi\\ufb01ed concept neurons\\naccording to each metric for all relevant concepts. For non-\\nrelevant concepts, none of the metrics found any concept neu-\\nrons. For example, for the concept of EmptyTrain , we iden-\\nti\\ufb01ed 13concept neurons based on the Spearman rank-order\\ncorrelation and 20based on the linear classi\\ufb01er\\u2019s accuracy. In\\nSection 2.2, we explore the importance of the selected con-\\ncept neurons for the overall performance of our method.\\nRegarding the amount of selected concept neurons, while\\nthese are dependent on the speci\\ufb01c threshold value, it should\\nbe noted thatMAcontains about 2\\u0002106neurons. Thus,\\nas one might expect the amount of selected concept neurons\\nis minuscule (about 0:001% ) in comparison with the total\\namount of neurons in a model. Interestingly, we found all\\nselected neurons to be in the dense part of the model, with\\nthe more abstract and complex concepts being focused on the\\nlater dense layers of the model.\\nThese results seem to con\\ufb01rm our hypothesis, indicating\\nthat it is possible to identify neurons in a neural network\\nmodel whose activations are highly correlated with human-\\nde\\ufb01ned concepts, as long as those concepts are relevant for\\nthe task being performed by the model.\\n2.2 Manipulating a Neural Networks\\u2019 Perception\\nWe now proceed to test our main hypothesis: that it is pos-\\nsible to manipulate a neural network\\u2019s perception regarding\\nspeci\\ufb01c human-de\\ufb01ned concepts by manipulating the activa-\\ntions of the concept cells for those concepts.\\nWe test this hypothesis by considering whether the output\\nof a model, for a given sample, that ought to change had\\na given concept been identi\\ufb01ed, indeed changes when that\\nconcept is injected. For instance, if we feed an image of a\\ntrain having a passenger car and no reinforced cars to MA,\\nwhich was trained to identify TypeA trains, we would expect\\nit to output that this is not a TypeA train. However, if we\\nidentify the concept neurons for 9has:ReinforcedCar \\u2013 hav-\\ning a reinforced car \\u2013 and modify their outputs to indicate\\nthe presence of a reinforced car, i.e., we inject the conceptSpearman Accuracy Intersection\\nEmptyTrain 98.9% 99.1% 99.6%\\n9has:PassengerCar 90.7% 93.0% 92.9%\\n9has:ReinforcedCar 97.9% 98.3% 98.2%\\nWarTrain 66.6% 65.7% 99.0%\\nFigure 5: Correctly classi\\ufb01ed samples by sensitivity metric.\\n9has:ReinforcedCar , we expect the model to change its out-\\nput to identifying a TypeA train.\\nTo test our hypothesis in the setting of the XTRAINS\\ndataset, we subsample the dataset into four different 1000\\nsample sets, for each of the four relevant concepts considered\\nin Section 2.1, according to the following criteria:\\n\\u2022S1- contains:TypeA samples where the presence of\\nthat concept should change MA\\u2019s output.\\n\\u2022S2- contains:TypeA samples where the presence of\\nthat concept should not change MA\\u2019s output.\\n\\u2022S3- contains TypeA samples where the absence of that\\nconcept should change MA\\u2019s output.\\n\\u2022S4- contains TypeA samples where the absence of that that concept should not change MA\\u2019s output.\\n\\u2022S3- contains TypeA samples where the absence of that\\nconcept should change MA\\u2019s output.\\n\\u2022S4- contains TypeA samples where the absence of that\\nconcept should not change MA\\u2019s output.\\nTo determine whether the output of MAshould change, we\\nreason with the ontology provided with the dataset.\\nAs the \\ufb01rst step in our method is to compute a neurons\\u2019\\nsensitivity, we compare the results obtained by the three met-\\nrics described in Section 2.1 when applying our method to\\neach of the four described sets. Figure 5, shows the average\\nnumber of samples where the MA\\u2019s output behaved as ex-\\npected, when computing the concept neurons\\u2019 activation val-\\nuescM\\nias the median value of the neurons\\u2019 activations for\\nbothPCandNC, withjPCj=jNCj= 1000 . These results\\nseem to indicate that it is possible to manipulate a neural net-\\nworks\\u2019 perception regarding different concepts, by modify-\\ning the activations of speci\\ufb01c neurons which seem to identify\\nthose concepts. This is evidenced by the high percentage of\\nsamples where the models\\u2019 output changed as expected. Fur-\\nthermore, these results also indicate that the sensitivity metric\\nbased on the intersection of the probability density functions\\nof a neuron for positive and negative samples seems to pro-\\nvide more consistent and better results on average. For this\\nreason, all remaining experiments use this metric to compute\\nthe sensitivity of a neuron with regards to a concept ( sC).\\nRegarding the method to compute the neurons\\u2019 activations,\\ncM\\ni, besides using the median value of the neurons\\u2019 activa-\\ntions as in the previous experiments, we considered two al-\\nternatives: computing the mode of the neurons\\u2019 activations,\\nand computing the values through the method described in\\n[Tucker et al. , 2021 ]. Figure 6 shows the average results ob-\\ntained by each method. Using the median value achieved the\\nbest results, having the highest number of correctly classi\\ufb01ed\\nsamples among the three considered metrics for all concepts.\\nWe attribute the inferior results obtained by the method in\\n[Tucker et al. , 2021 ]to the fact that the information of each\\nconcept neuron is quite redundant, and thus the probe used in\\nthe method learns to perform its task from just a small subset\\nof those neurons, which might lead to most neurons having Median Mode [Tucker et al. , 2021 ]\\nEmptyTrain 99.6% 99.5% 72.4%\\n9has:PassengerCar 92.9% 92.8% 86.5%\\n9has:ReinforcedCar 98.2% 97.7% 73.9%\\nWarTrain 99.0% 98.6% 65.3%\\nFigure 6: Correctly classi\\ufb01ed samples by method to compute neuron\\nactivation.\\nS1 S2 S3 S4\\nEmptyTrain 99.7% - 99.5% 99.5%\\n9has:PassengerCar 99.9% 98.6% 98.7% 74.5%\\n9has:ReinforcedCar 99.2% 98.5% 96.2% 99.1%\\nWarTrain 98.4% - 100.0% 98.7%\\nFigure 7: Percentage of correctly classi\\ufb01ed samples in each set.\\ntheir activations unchanged. In all remaining experiments,\\nwe compute cM\\nias the median activation value of a neuron.\\nFigure 7 shows the percentage of samples where injecting\\na concept resulted in the expected MAoutput for each of the\\nsets described above. Overall the results seem to be quite pos-\\nitive, indicating that typically the model outputs the expected\\nresult given the concept being injected. The result of injecting\\n:9has:PassengerCar in set S4seems to be somewhat inferior\\nto the remaining results, which we believe to be an indication\\nof some spurious correlation learned by MA.\\nThe high percentage of samples where the injection of a\\ngiven sample leads to the expected change in the model\\u2019s out-\\nput constitutes strong evidence that it is possible to modify\\nthe perception of a neural network regarding speci\\ufb01c human-\\nde\\ufb01ned concepts, by manipulating the activations of the neu-\\nrons responsible for the identi\\ufb01cation of those concepts.\\n2.3 Importance of Selected Neurons\\nThe \\ufb01rst two steps in the proposed method have the goal of\\nidentifying which neurons in a model act as concept cells for a\\ngiven concept of interest. These neurons are then used to ma-\\nnipulate a neural networks\\u2019 perception regarding that concept.\\nThus, one might wonder about how important the speci\\ufb01c set\\nof selected concept neurons is to the overall performance of\\nthe method. This Section addresses the effects of varying the\\nthreshold value of function sC.\\nIn order to understand how dependent the methods\\u2019 per-\\nformance is on the amount of concept neurons, we measure,\\nfor each concept, the ratio of samples where the output of the\\nmodel is consistent with the manipulation performed. Figure\\n8, shows the average result obtained for all sets described in\\nSection 2.2 for each amount of considered concept neurons.\\nOur results indicate that if function sCis either too restric-\\ntive, excluding many neurons which act as concept cells, then\\nthe few considered ones might be unable to affect the model\\nand consistently produce the desired effects in the model \\u2013\\nthis is shown by the signi\\ufb01cant increase in performance as\\nmore concept neurons are initially added. However, if func-\\ntionsCtoo tolerant and starts to include neurons that are\\nnot effectively concept cells for a given concept performance\\nstarts to degrade \\u2013 as shown by the sharp decrease in perfor-\\nmance for higher number of considered concept neurons.\\nAs expected, the performance of the proposed method is\\nheavily dictated by the set of selected concept neurons. These\\n0 5 10 15 20 25 30\\n# Concept Neurons0.40.60.81.0CorrectEmptyTrain\\n\\u2203has.PassengerCar\\n\\u2203has.ReinforcedCar\\nWarTrainFigure 8: Correct samples by amount of selected concept neurons.\\n10 20 40 80 160 320 800\\n# Samples0.60.70.80.91.0CorrectEmptyTrain\\n\\u2203has.PassengerCar\\n\\u2203has.ReinforcedCar\\nWarTrain\\nFigure 9: Correct samples by amount of available labeled data.\\nresults shows how important it is to adjust the threshold value\\nof function sCfor each concept of interest, which might be\\ndone using a simple grid search with cross-validation.\\n2.4 Counterfactual\\u2019s Cost\\nThe results so far suggest that it is possible to manipulate a\\nneural network\\u2019s perception of speci\\ufb01c human-de\\ufb01ned con-\\ncepts by modifying the activation of neurons which seem to\\nbe identifying those concepts. However, they have assumed\\nthat there exists plenty of labeled data available, using a total\\nof2000 samples to compute neurons\\u2019 sensitivity values and\\nthe activation values to be injected for each concept. In this that there exists plenty of labeled data available, using a total\\nof2000 samples to compute neurons\\u2019 sensitivity values and\\nthe activation values to be injected for each concept. In this\\nsection, we assess whether the proposed method is practical\\nwhen taking into account the amount of necessary data.\\nTo assess the cost of applying this method, and how the\\namount of available labeled data impacts our capacity to prop-\\nerly modify a networks\\u2019 perception, we compare the average\\nresults obtained in the 4sample sets de\\ufb01ned in Section 2.2, for\\neach injected concept, when using different amounts of avail-\\nable data. We keep the amount of samples used to validate the\\nthreshold value of sCunchanged throughout this comparison,\\nusing 100samples as validation.\\nFigure 9, shows the average ratio of correctly classi\\ufb01ed\\nsamples after injection of a given concept, for each consid-\\nered concept. It is observable that even with as few as 40\\nlabeled samples \\u2013 20samples where the concept is present\\nand20where it is absent \\u2013 it is possible to manipulate the\\nneural networks perception for each considered concept with\\na high degree of success.\\nThe method\\u2019s performance starts to signi\\ufb01cantly drop\\nwhen the amount of labeled data is insuf\\ufb01cient to be able\\nto properly identify the concept neurons for a given concept.\\nThis seems to indicate that as long as there is enough labeled\\ndata to identify which neurons in a model are sensitive to the\\nconcepts we want to inject, we should be able to manipulate\\na neural network\\u2019s perception with regards to that concept. 3 Interpreting Neural Networks\\nSo far, we described a method to generate counterfactuals for\\na neural network model by modifying the activations of neu-\\nrons which act as concept cells for human-de\\ufb01ned concepts.\\nThrough this method, it is possible to understand how differ-\\nent concepts in\\ufb02uence the output of a model. What if one\\nwants to be able to understand how a model relates different\\nconcepts which are not represented in the model\\u2019s output?\\nFor example, one might be interested in understanding\\nwhetherMAhas learned that EmptyTrain s do not have any\\npassenger cars, and that if 9has:PassengerCar then that train\\nis not an EmptyTrain . Although both of these concepts are\\nnot represented inMA\\u2019s output they are both relevant for its\\ntask and one might be interested in assuring that the model is\\ncapable of understanding the relationship between both.\\nIn order to understand whether a given concept which is\\nnot represented in the output of a model is being identi\\ufb01ed\\nby it, we make use of mapping networks [de Sousa Ribeiro\\nand Leite, 2021 ]. These are small neural networks trained to\\nidentify in the activations of a neural network model whether\\na given human-de\\ufb01ned concept was identi\\ufb01ed.\\nTo verify whetherMAhas learned the described relations\\nbetween both concepts, we train two mapping networks for\\nthe concepts of EmptyTrain and9has:PassengerCar \\u2013 both\\nachieving an accuracy of more than 99% on a 1000 bal-\\nanced test set. To test whether MAhas learned that empty\\ntrains do not have any passenger cars, we inject the con-\\ncept EmptyTrain on1000 samples of trains with passenger\\ncars. Through the mapping network for 9has:PassengerCar ,\\nwe verify that in 95:6%of the samples, after injection, the\\nconcept9has:PassengerCar goes from being present to being\\nabsent. This is strong indication that this model has learned\\nthat empty trains do not have passenger cars.\\nSimilarly, we test whether MAhas learned that if a train\\nhas a passenger car, then it is not an empty train. We inject the\\nconcept9has:PassengerCar on1000 samples of empty trains,\\nand observe that in only 2%of them the concept EmptyTrain\\nturns absent. This indicates that MAhas not learned that if a\\ntrain has a passenger car, it is not an empty train.\\nThis example illustrates a possible application of our\\nmethod, namely to investigate whether certain relations be-\\ntween user-de\\ufb01ned concepts are encoded in the model. But\\nmore importantly, the model seems to naturally integrate the\\ninjected information, impacting the related concepts, as if the\\ninjected concept was truly being perceived by the model.\\n4 Correcting a Neural Network\\u2019s\\nMisunderstandings\\nWhen a neural network model provides an incorrect result, it\\nis dif\\ufb01cult to interpret the cause of the error. It might often\\nbe because the model was unable to correctly perceive part of\\nwhat is represented in its input, in which case the proposed\\nmethod might be used to \\u201ccorrect\\u201d a model\\u2019s wrong results.\\nTo test our hypothesis, we select all XTRAINS sam-\\nples whereMAprovides an incorrect result. We train a\\nmapping networks for the concepts 9has:ReinforcedCar and\\n9has:PassengerCar \\u2013 with an accuracy of more than 99% on\\na1000 balanced test set \\u2013 and use them to identify whether\\nFigure 10: Images of \\u2018Rhodesian ridgeback dog face\\u2019.\\nMAprovided a wrong result because either of these concepts\\nwere not perceived by the model when they should have.\\nWe select all samples where MAprovides a false nega-\\ntive result, indicating that a sample input was not of TypeA\\nwhen it was. From these samples, we observe that in those\\nwhere the concept of 9has:ReinforcedCar was not identi\\ufb01ed,\\nbut present in the input, injecting 9has:ReinforcedCar led to\\nthe output being corrected in 96:1%of the samples. Similarly,\\nwhen considering the samples where 9has:PassengerCar was\\nnot identi\\ufb01ed, but present in the input, injecting it led to the\\noutput being corrected in 98:7%of the samples.\\nThese results provide evidence that our method is applica- not identi\\ufb01ed, but present in the input, injecting it led to the\\noutput being corrected in 98:7%of the samples.\\nThese results provide evidence that our method is applica-\\nble even when a model provides incorrect results, allowing\\none to test whether different concepts might be the responsi-\\nble for the provided result. This enables users to further un-\\nderstand why their models achieved an incorrect output, and\\nidentify potential \\ufb02aws in the model or in its training.\\n5 Validation with Real World Data\\nSo far, we have only considered a synthetic dataset. We now\\nconsider whether we can replicate similar results with real\\ndata. To perform this validation, we consider the setting of\\nthe ImageNet dataset [Russakovsky et al. , 2015 ], and exam-\\nine the pre-trained MobileNetV2 model from [Sandler et al. ,\\n2018 ].\\nThe proposed method is based on the assumption that by\\nidentifying which neurons in a model are responsible for\\nidentifying a given human-de\\ufb01ned concept, we are able to\\nmodify the model\\u2019s perception of that concept by modifying\\nthose neuron\\u2019s activations. Thus, we focus our validation on\\nwhether we can identify the neurons responsible for a given\\nhuman-de\\ufb01ned concept, and whether we are able to modify a\\nmodel\\u2019s outputs by injecting that concept.\\nIn order to test our method in this setting, we selected a ran-\\ndom output class: \\u2018Rhodesian ridgeback\\u2019, which is a speci\\ufb01c\\ndog\\u2019s breed, and de\\ufb01ne the concept of \\u2018Rhodesian ridgeback\\ndog face\\u2019 by selecting a set of 98images from ImagenetNet\\nwhere a Rhodesian ridgeback dog is observable and its face is\\ncentered and completely visible, as illustrated in the samples\\nshown in Figure 10. We select this concept based on the as-\\nsumption that it is a useful concept for the model to classify\\nimages of Rhodesian ridgeback dogs.\\nTo test whether by injecting this concept we were able to\\nmodify the model\\u2019s perception, we used all 329 samples \\u2013\\nfrom training and validation sets \\u2013 where the model outputs a\\nfalse negative for the Rhodesian ridgeback class considering\\nits top-1 result. The injection of the concept of \\u2018Rhodesian\\nridgeback dog face\\u2019 led 48% of these samples to output the\\nclass of Rhodesian ridgeback. If we consider the 38samples\\nwhere the model outputs a false negative considering its top-5 Figure 11: Images of censored \\u2018Rhodesian ridgeback dog face\\u2019.\\nNot Censored Censored Censored + Injection\\nTop-1 70% 36% 64%\\nTop-5 100% 64% 88%\\nFigure 12: Model accuracy regarding \\u2018Rhodesian ridgeback\\u2019 class.\\noutputs, we were able to modify the model\\u2019s output in 71:1%\\nto Rhodesian ridgeback class.\\nSince the concept of \\u2018Rhodesian ridgeback dog face\\u2019 is not\\nsuf\\ufb01cient by itself to lead the model to output the class of\\n\\u2018Rhodesian ridgeback\\u2019, these results constitute evidence that\\nit is being used by the model, and that by manipulating its\\nconcept neurons we are able to modify the model\\u2019s output.\\nWe further test the assumptions that the concept of \\u2018Rhode-\\nsian ridgeback dog face\\u2019 is important for the model to be able\\nto recognize that breed, and that we are able to generate coun-\\nterfactuals by injecting that concept, by \\ufb01rst censoring every\\n\\u2018Rhodesian ridgeback dog face\\u2019 in ImageNet\\u2019s validation set,\\nas shown in Figure 11. We then measure the accuracy of the\\nmodel before censoring the dog\\u2019s faces, after censoring them,\\nand after injecting the concept of \\u2018Rhodesian ridgeback dog\\nface\\u2019 on the censored ones. The results, shown in Figure 12,\\nindicate that censoring the dog\\u2019s face leads to a signi\\ufb01cant\\ndrop in accuracy, which seems to be evidence of its impor-\\ntance for the model. Then, when the concept is injected even\\nafter the faces are censored, we observe a big increase in ac-\\ncuracy, indicating that the injected concept was successful in\\nproviding some of the information removed by the censoring\\nand helping the model provide the correct classi\\ufb01cation.\\nThese results show us that even in a setting with real data, a\\nmoderately sized model ( \\u00197\\u0002106neurons), and few labeled\\ndata, it is possible to identify which neurons in a model act\\nas concept cells for a given concept and, more importantly,\\nmanipulate the neural network\\u2019s perception of that concept\\nby manipulating the activations of those neurons.\\n6 Related Work\\nThe last few years has seen an increase in the develop-\\nment of methods for interpreting arti\\ufb01cial neural networks,\\nwith proxy-based methods being one of the most popular ap-\\nproaches (c.f., [Gilpin et al. , 2018 ]). These methods aim to\\nsubstitute the model being explained for one that is inher-\\nently interpretable and which exhibits similar behavior. In\\ncontrast, our approach to generate counterfactual scenarios\\ndoes not require changing or substituting the original model,\\nwhich might not always be feasible.\\nAnother popular approach is that of saliency and attribution\\nmethods (c.f., [Liet al. , 2021 ]), where a model\\u2019s behavior\\nis explained by attributing a contribution value to each input\\nfeature representing its contribution to a given prediction. Al-\\nthough these methods might provide some insights into whichfeatures contributed most for a given prediction, they do not\\nprovide clari\\ufb01cation regarding why those contributions jus-\\ntify the output, leaving the burden of understanding how those\\ncontributions are related with the speci\\ufb01c output to the user.\\nSome counterfactual-based methods (c.f., [Guidotti,\\n2022 ]) suffer from similar issues, providing a counterfactual\\nsample but lacking clari\\ufb01cation of why that sample leads to\\na different result. We believe that counterfactual methods\\nwould bene\\ufb01t from abstracting away from only generating\\nspeci\\ufb01c counterfactual samples, to describing in a human-\\nunderstandable way why those counterfactuals lead to some\\nother particular output based on how they affect the model.\\nRecently, neuro-symbolic approaches have aimed at un-\\nderstanding whether models are sensitive to speci\\ufb01c human-\\nde\\ufb01ned concepts [Kim et al. , 2018 ], and how we can lever-\\nage the representations of those concepts in a model to pro-\\nvide explanations for its outputs [de Sousa Ribeiro and Leite,\\n2021 ], as well as how to produce human-understandable the-\\nories representing the classi\\ufb01cation process of a model [Fer-\\nreira et al. , 2022 ]. In this paper, we link both approaches 2021 ], as well as how to produce human-understandable the-\\nories representing the classi\\ufb01cation process of a model [Fer-\\nreira et al. , 2022 ]. In this paper, we link both approaches\\nthrough a method that focuses on what a model is perceiving\\nand enables the generation of counterfactuals via manipula-\\ntion of what is being perceived wrt. human-de\\ufb01ned concepts.\\n7 Conclusions\\nIn this paper, we proposed a method to generate counterfactu-\\nals regarding what a neural network model is perceiving about\\nhuman-de\\ufb01ned concepts, enabling users to inquire the model,\\nand understand how its outputs are dependent on those con-\\ncepts. To this end, we explored how to identify which neurons\\nin a model are sensitive to a given concept, and how to lever-\\nage such neurons to manipulate a model into perceiving that\\nconcept. Through experimental evaluation, we show that it is\\npossible manipulate a neural network\\u2019s perception regarding\\ndifferent concepts, requiring few labeled data to do so, and\\nwithout needing to change or retrain the original model.\\nWe provide a formalization of the proposed method, and\\nillustrate how it can be applied to generate counterfactuals,\\nbut also how to use the it to better understand how different\\nconcepts are related within a model, and how to inspect and\\ncorrect a model\\u2019s misclassi\\ufb01cations.\\nWe conclude that for concepts that are related to the task\\nof a neural network, it is often the case that there exist a few\\nspeci\\ufb01c neurons which encode that concept and are respon-\\nsible for its identi\\ufb01cation within the model. Modifying the\\nactivations of these neurons, similarly to stimulating concept\\ncells in the human brain, allows one to trick the model that\\nit is perceiving that concept. This simple method seems ef-\\nfective in allowing sophisticated manipulations of high-level\\nabstract concepts, enabling users to explore, with little effort,\\nhow the model would respond in different scenarios.\\nIn the future, we are interested in exploring how to leverage\\nthis method to search for learned biases, and identify missing\\nor undesired associations between concepts.\\nAcknowledgments\\nThe authors would like to acknowledge the support provided\\nby FCT through PhD grant (UI\\/BD\\/151266\\/2021) and strate- gic project NOV A LINCS (UIDB\\/04516\\/2020).\\nReferences\\n[Abiodun et al. , 2018 ]Oludare Isaac Abiodun, Aman Jan-\\ntan, Abiodun Esther Omolara, Kemi Victoria Dada,\\nNachaat AbdElatif Mohamed, and Humaira Arshad. State-\\nof-the-art in arti\\ufb01cial neural network applications: A sur-\\nvey. Heliyon , 4(11):e00938, 2018.\\n[Adebayo et al. , 2020 ]Julius Adebayo, Michael Muelly,\\nIlaria Liccardi, and Been Kim. Debugging Tests for Model\\nExplanations. In Hugo Larochelle, Marc\\u2019Aurelio Ran-\\nzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien\\nLin, editors, Advances in Neural Information Processing\\nSystems 33: Annual Conference on Neural Information\\nProcessing Systems 2020, NeurIPS 2020, December 6-12,\\n2020, virtual , 2020.\\n[Chu et al. , 2020 ]Eric Chu, Deb Roy, and Jacob Andreas.\\nAre Visual Explanations Useful? A Case Study in Model-\\nin-the-Loop Prediction. CoRR , abs\\/2007.12248, 2020.\\n[de Sousa Ribeiro and Leite, 2021 ]Manuel\\nde Sousa Ribeiro and Jo \\u02dcao Leite. Aligning Arti\\ufb01cial\\nNeural Networks and Ontologies towards Explainable AI.\\nInProcs. of AAAI\\u201921 , pages 4932\\u20134940. AAAI Press,\\n2021.\\n[de Sousa Ribeiro et al. , 2020 ]Manuel de Sousa Ribeiro,\\nLudwig Krippahl, and Jo \\u02dcao Leite. Explainable Abstract\\nTrains Dataset. CoRR , abs\\/2012.12115, 2020.\\n[Ferreira et al. , 2022 ]Jo\\u02dcao Ferreira, Manuel\\nde Sousa Ribeiro, Ricardo Gonc \\u00b8alves, and Jo \\u02dcao Leite.\\nLooking Inside the Black-Box: Logic-based Explanations\\nfor Neural Networks. In Gabriele Kern-Isberner, Gerhard\\nLakemeyer, and Thomas Meyer, editors, Proceedings\\nof the 19th International Conference on Principles of\\nKnowledge Representation and Reasoning, KR 2022,\\nHaifa, Israel. July 31 - August 5, 2022 , 2022.\\n[Gastaldi et al. , 2022 ]Chiara Gastaldi, Tilo Schwalger,\\nEmanuela De Falco, Rodrigo Quian Quiroga, and Wulfram\\nGerstner. When shared concept cells support associations:\\nTheory of overlapping memory engrams. PLOS Compu-\\ntational Biology , 17(12):1\\u201344, 12 2022.\\n[Ghoddusi et al. , 2019 ]Hamed Ghoddusi, Germ \\u00b4an G.\\nCreamer, and Nima Ra\\ufb01zadeh. Machine learning in\\nenergy economics and \\ufb01nance: A review. Energy\\nEconomics , 81:709\\u2013727, 2019.\\n[Gilpin et al. , 2018 ]Leilani H. Gilpin, David Bau, Ben Z.\\nYuan, Ayesha Bajwa, Michael A. Specter, and Lalana Ka-\\ngal. Explaining Explanations: An Overview of Inter-\\npretability of Machine Learning. In Francesco Bonchi,\\nFoster J. Provost, Tina Eliassi-Rad, Wei Wang, Ciro Cat-\\ntuto, and Rayid Ghani, editors, 5th IEEE International\\nConference on Data Science and Advanced Analytics,\\nDSAA 2018, Turin, Italy, October 1-3, 2018 , pages 80\\u201389.\\nIEEE, 2018.\\n[Goh et al. , 2021 ]Gabriel Goh, Nick Cammarata, Chelsea\\nV oss, Shan Carter, Michael Petrov, Ludwig Schubert,Alec Radford, and Chris Olah. Multimodal Neu-\\nrons in Arti\\ufb01cial Neural Networks. Distill , 2021.\\nhttps:\\/\\/distill.pub\\/2021\\/multimodal-neurons.\\n[Guidotti, 2022 ]Riccardo Guidotti. Counterfactual explana-\\ntions and how to \\ufb01nd them: literature review and bench-\\nmarking. Data Mining and Knowledge Discovery , April\\n2022.\\n[Hitzler et al. , 2020 ]Pascal Hitzler, Federico Bianchi,\\nMonireh Ebrahimi, and Md. Kamruzzaman Sarker.\\nNeural-symbolic Integration and the Semantic Web.\\nSemantic Web , 11(1):3\\u201311, 2020.\\n[Hvilsh\\u00f8j et al. , 2021 ]Frederik Hvilsh\\u00f8j, Alexandros Iosi-\\n\\ufb01dis, and Ira Assent. ECINN: Ef\\ufb01cient Counterfactuals\\nfrom Invertible Neural Networks. In 32nd British Machine\\nVision Conference 2021, BMVC 2021, Online, November\\n22-25, 2021 , page 43. BMV A Press, 2021.\\n[Khan et al. , 2020 ]Asifullah Khan, Anabia Sohail, Umme\\nZahoora, and Aqsa Saeed Qureshi. A survey of the re-\\ncent architectures of deep convolutional neural networks.\\nArtif. Intell. Rev. , 53(8):5455\\u20135516, 2020.\\n[Kim et al. , 2018 ]Been Kim, Martin Wattenberg, Justin\\nGilmer, Carrie J. Cai, James Wexler, Fernanda B. Vi \\u00b4egas,\\nand Rory Sayres. Interpretability Beyond Feature Attribu-\\ntion: Quantitative Testing with Concept Activation Vectors\\n(TCA V). In Jennifer G. Dy and Andreas Krause, editors,\\nProceedings of the 35th International Conference on Ma- tion: Quantitative Testing with Concept Activation Vectors\\n(TCA V). In Jennifer G. Dy and Andreas Krause, editors,\\nProceedings of the 35th International Conference on Ma-\\nchine Learning, ICML 2018, Stockholmsm \\u00a8assan, Stock-\\nholm, Sweden, July 10-15, 2018 , volume 80 of Proceed-\\nings of Machine Learning Research , pages 2673\\u20132682.\\nPMLR, 2018.\\n[Liet al. , 2021 ]Xiao-Hui Li, Yuhan Shi, Haoyang Li, Wei\\nBai, Caleb Chen Cao, and Lei Chen. An Experimental\\nStudy of Quantitative Evaluations on Saliency Methods.\\nIn Feida Zhu, Beng Chin Ooi, and Chunyan Miao, editors,\\nKDD \\u201921: The 27th ACM SIGKDD Conference on Knowl-\\nedge Discovery and Data Mining, Virtual Event, Singa-\\npore, August 14-18, 2021 , pages 3200\\u20133208. ACM, 2021.\\n[Miller, 2019 ]Tim Miller. Explanation in Arti\\ufb01cial Intelli-\\ngence: Insights from the Social Sciences. Artif. Intell. ,\\n267:1\\u201338, 2019.\\n[Okuyama, 2018 ]Teruhiro Okuyama. Social memory en-\\ngram in the hippocampus. Neuroscience Research ,\\n129:17\\u201323, 2018. Achievements of Japan neuroscience\\naward winners.\\n[Quiroga et al. , 2005 ]R Quian Quiroga, Leila Reddy,\\nGabriel Kreiman, Christof Koch, and Itzhak Fried. Invari-\\nant visual representation by single neurons in the human\\nbrain. Nature , 435(7045):1102\\u20131107, 2005.\\n[Quiroga, 2012 ]Rodrigo Quian Quiroga. Concept cells: the\\nbuilding blocks of declarative memory functions. Nature\\nReviews Neuroscience , 13(8):587\\u2013597, 2012.\\n[Reddy and Thorpe, 2014 ]Leila Reddy and Simon J.\\nThorpe. Concept Cells through Associative Learning of\\nHigh-Level Representations. Neuron , 84(2):248\\u2013251,\\n2014. [Russakovsky et al. , 2015 ]Olga Russakovsky, Jia Deng,\\nHao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma,\\nZhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael\\nBernstein, Alexander C. Berg, and Li Fei-Fei. Ima-\\ngeNet Large Scale Visual Recognition Challenge. Inter-\\nnational Journal of Computer Vision (IJCV) , 115(3):211\\u2013\\n252, 2015.\\n[Sandler et al. , 2018 ]Mark Sandler, Andrew G. Howard,\\nMenglong Zhu, Andrey Zhmoginov, and Liang-Chieh\\nChen. MobileNetV2: Inverted Residuals and Linear Bot-\\ntlenecks. In 2018 IEEE Conference on Computer Vision\\nand Pattern Recognition, CVPR 2018, Salt Lake City, UT,\\nUSA, June 18-22, 2018 , pages 4510\\u20134520. Computer Vi-\\nsion Foundation \\/ IEEE Computer Society, 2018.\\n[Shahid et al. , 2019 ]Nida Shahid, Tim Rappon, and Whit-\\nney Berta. Applications of arti\\ufb01cial neural networks in\\nhealth care organizational decision-making: A scoping re-\\nview. PLOS ONE , 14(2):1\\u201322, 02 2019.\\n[Shen and Huang, 2020 ]Hua Shen and Ting-Hao (Kenneth)\\nHuang. How Useful Are the Machine-Generated Interpre-\\ntations to General Users? A Human Evaluation on Guess-\\ning the Incorrectly Predicted Labels. In Lora Aroyo and\\nElena Simperl, editors, Proceedings of the Eighth AAAI\\nConference on Human Computation and Crowdsourcing,\\nHCOMP 2020, Hilversum, The Netherlands (virtual), Oc-\\ntober 25-29, 2020 , pages 168\\u2013172. AAAI Press, 2020.\\n[Stepin et al. , 2021 ]Ilia Stepin, Jos \\u00b4e Maria Alonso, Alejan-\\ndro Catal \\u00b4a, and Martin Pereira-Fari \\u02dcna. A Survey of Con-\\ntrastive and Counterfactual Explanation Generation Meth-\\nods for Explainable Arti\\ufb01cial Intelligence. IEEE Access ,\\n9:11974\\u201312001, 2021.\\n[Tucker et al. , 2021 ]Mycal Tucker, Peng Qian, and Roger\\nLevy. What if This Modi\\ufb01ed That? Syntactic Inter-\\nventions with Counterfactual Embeddings. In Chengqing\\nZong, Fei Xia, Wenjie Li, and Roberto Navigli, editors,\\nFindings of the Association for Computational Linguistics:\\nACL\\/IJCNLP 2021, Online Event, August 1-6, 2021 , vol-\\nume ACL\\/IJCNLP 2021 of Findings of ACL , pages 862\\u2013\\n875. Association for Computational Linguistics, 2021.\\n[Yang et al. , 2021 ]Fan Yang, Ninghao Liu, Mengnan Du,\\nand Xia Hu. Generative Counterfactuals for Neural Net-\\nworks via Attribute-Informed Perturbation. SIGKDD Ex-\\nplor., 23(1):59\\u201368, 2021.\",\"21\":\" Notice\\nThis work has been submitted to the IEEE for possible publication.\\nCopyright may be transferred without notice, after which this version\\nmay no longer be accessible.arXiv:2303.02640v1  [cs.LG]  5 Mar 2023 Swim: A General-Purpose, High-Performing, and Ef\\ufb01cient Activation\\nFunction for Locomotion Control Tasks\\nMaryam Abdool1and Tony Dear2\\nAbstract \\u2014 Activation functions play a signi\\ufb01cant role in\\nthe performance of deep learning algorithms. In particular,\\nthe Swish activation function tends to outperform ReLU on\\ndeeper models, including deep reinforcement learning models,\\nacross challenging tasks. Despite this progress, ReLU is the\\npreferred function partly because it is more ef\\ufb01cient than Swish.\\nFurthermore, in contrast to the \\ufb01elds of computer vision and\\nnatural language processing, the deep reinforcement learning\\nand robotics domains have seen less inclination to adopt new\\nactivation functions, such as Swish, and instead continue to\\nuse more traditional functions, like ReLU. To tackle those\\nissues, we propose Swim, a general-purpose, ef\\ufb01cient, and high-\\nperforming alternative to Swish, and then provide an analysis of\\nits properties as well as an explanation for its high-performance\\nrelative to Swish, in terms of both reward-achievement and\\nef\\ufb01ciency. We focus on testing Swim on MuJoCo\\u2019s locomotion\\ncontinuous control tasks since they exhibit more complex\\ndynamics and would therefore bene\\ufb01t most from a high-\\nperforming and ef\\ufb01cient activation function. We also use the\\nTD3 algorithm in conjunction with Swim and explain this choice\\nin the context of the robot locomotion domain. We then conclude\\nthat Swim is a state-of-the-art activation function for continuous\\ncontrol locomotion tasks and recommend using it with TD3 as\\na working framework.\\nI. INTRODUCTION AND RELATED WORK\\nDeep reinforcement learning can automate the design of\\ncomplex controllers for locomotion tasks [9]. Before the\\nemergence of deep reinforcement learning, a common tech-\\nnique was to manually design controllers for each locomotion\\ntask; this process requires an accurate dynamic model of\\nthe robot that may be dif\\ufb01cult to achieve [9]. In addition\\nto exhibiting complex dynamics, locomotion problems also\\nfeature high degrees of freedom, complicating the task of\\nengineering controllers [5]. Therefore, we utilize deep rein-\\nforcement learning to automate the design of our controllers\\nfor MuJoCo\\u2019s locomotion continuous control tasks presented\\nin this paper.\\nIn a neural network, each neuron performs a transforma-\\ntion on the inputs using the incoming weights and biases, but\\nsuch a simple stand-alone model fails to learn complex tasks\\n[17]. Therefore, non-linearity is introduced so that neural\\nnetworks can act as non-linear function approximators in\\nvarious settings, including reinforcement learning [19]. This\\nis achieved by using non-linear activation functions, such as\\nthe Recti\\ufb01ed Linear Unit (ReLU), in the hidden layers of the\\n1Maryam Abdool is with Department of Computer Sci-\\nence, Columbia University, New York, NY 10027, USA\\nmaryam.r.abdool@gmail.com\\n2Tony Dear is with Department of Computer Science, Columbia Univer-\\nsity, New York, NY 10027, USA tony.dear@columbia.eduneural network [15]. Non-linearity is a powerful characteris-\\ntic of neural networks because of the universal approximation\\ntheorem\\u2019s implication that non-linear activation functions\\ncan approximate any continuous function arbitrarily well\\n[10]. This has vital implications for the application of deep\\nreinforcement learning algorithms to continuous control and\\nlocomotion tasks, as it guarantees that neural networks have\\nthe potential to learn complex controllers.\\nTD3 is one such deep reinforcement learning algorithm\\nthat is designed to learn complex locomotion controllers [7].\\nAs a deep learning algorithm, TD3 follows the standard\\napproach of using ReLU, a non-linear activation function,\\nin the hidden layers of the neural network to model complex\\ntasks [7]. What distinguishes TD3 from other algorithms are\\nthree aspects: an actor-critic approach, sample ef\\ufb01ciency, and\\nthe twin critic networks. For these reasons, we select TD3\\nover other algorithms.\\nFirst, in TD3\\u2019s actor-critic neural network architecture, the twin critic networks. For these reasons, we select TD3\\nover other algorithms.\\nFirst, in TD3\\u2019s actor-critic neural network architecture,\\nthe policy network is called the actor because it is used\\nto select actions, and the estimated value function network\\nis called the critic because it criticizes the actions made\\nby the actor [19]. An actor-critic method is advantageous\\nbecause it can be applied to problems with continuous\\naction spaces, namely MuJoCo\\u2019s continuous control tasks,\\nwhere Q-learning methods cannot be directly applied; for\\ncontinuous control tasks with an in\\ufb01nite set of actions, Q-\\nlearning methods must search through the in\\ufb01nite set to\\nselect the action, while the separation of the policy from\\nthe value function in the actor-critic method reduces the\\nextensive computation needed for each action selection [19].\\nSecond, TD3 is sample-ef\\ufb01cient because it is based on a\\ndeterministic policy gradient method [18]. The deterministic\\npolicy gradient has the simple form of the expected gradient\\nof the action-value function [18]. Because of this simple\\nform, the deterministic policy gradient can be estimated\\nmuch more ef\\ufb01ciently than its stochastic counterpart [18].\\nFinally, TD3 uses two critic networks, instead of one, to\\naddress the overestimation bias problem present in DDPG\\nand other Q-learning algorithms that occurs when the noisy\\nvalue estimate is maximized [7], [20].\\nAlthough designing new algorithms is an important prob-\\nlem, such as the TD3 algorithm that we employ to model\\nour controllers, research in creating new non-linear acti-\\nvation functions is neglected in favor of designing more\\ncomplex deep reinforcement learning algorithms. Therefore,\\nwe pursue the \\ufb01eld of activation functions as research shows\\nthat not all non-linear activation functions have the same\\nperformance. For example, it is believed that ReLU is advan-\\nManuscript submitted for review to the IEEE\\/RSJ International Conference on Intelligent Robots and Systems (IROS) tageous over Sigmoid because it reduces the likelihood of the\\nvanishing gradient problem and it is a simpler, more ef\\ufb01cient\\nactivation function [8]. Similarly, [16] speculates that Swish\\ntends to outperform common baseline functions, including\\nReLU, because it is non-monotonic. However, Swish is\\nless ef\\ufb01cient than ReLU due to the required exponential\\ncomputations, which is one of the reasons ReLU is a widely\\nused function, in addition to being more well-known; as\\nmentioned earlier, the choice of using ReLU in the original\\nTD3 implementation is one notable example [11], [7].\\nBecause of those limitations, we invent Swim, a non-\\nmonotonic, smooth activation function mathematically de-\\n\\ufb01ned as f(x) =x\\n2(kxp\\n1+k2x2+1). Swim is more ef\\ufb01cient and\\nmore likely to be high-performing than Swish, and is conse-\\nquently also more likely to be high-performing than ReLU;\\nas mentioned earlier, [16] and [6] have shown that Swish\\ntends to outperform ReLU on deeper models, including deep\\nreinforcement learning models, across challenging tasks. We\\nachieve ef\\ufb01ciency and high performance without compro-\\nmising any of the desirable properties, such as smoothness,\\nthat make Swish outperform baseline functions. This is in\\ncontrast to previous work, such as [11]\\u2019s design of H-Swish,\\nan ef\\ufb01cient but non-smooth version of Swish. Ultimately, we\\nstrive to achieve high performance and ef\\ufb01ciency as better\\nactivation functions correspond to better locomotion con-\\ntrollers. High performance and ef\\ufb01ciency are important met-\\nrics for the locomotion domain due to the complex dynamics\\nand time-intensive characteristics mentioned earlier. Finally,\\nwe also hypothesize that there are speci\\ufb01c properties intrinsic\\nto Swim\\u2019s success in the deep reinforcement learning and\\nlocomotion control domain.\\nII. SWIM: AN ANALYSIS\\nThe newly proposed activation function, Swim, and its \\ufb01rst\\nderivative are de\\ufb01ned as:\\nf(x) =x\\n2\\u0012kxp\\n1+k2x2+1\\u0013\\n;\\nf0(x) =1\\n2\\u0012kx(k2x2+2)\\n(p\\n1+k2x2)3+1\\u0013 (1)\\nwhere kis a constant that can be tuned or de\\ufb01ned before\\ntraining. We pick a ksuch that it can support our analysis that\\nSwim outperforms Swish because of the properties intrinsic\\nto it, and not because of a constant rescaling of the function\\nthat could be analogously applied to Swish by changing b,\\nwhich is also a learnable parameter. For this reason, we set k\\n= 0.5 to approximate the Swish function ( x1\\n1+e\\u0000bx) atb= 1,\\nwhich is the parameter that the original Swish paper runs and\\nreports the results on [16]. This also means that this may not\\nbe the most optimal setting of the kconstant, but we make\\nthis trade-off to support our analysis. The difference between\\nthe two functions, alongside ReLU ( max(0;x)) [15], can be\\nseen in Figure 1.\\nSimilarly, the plots of the \\ufb01rst derivatives of Swim, Swish,\\nand ReLU are shown in Figure 2.\\n(a)\\n(b)\\nFig. 1: (a) Graphs of Swim, Swish, and ReLU. (b) Zoomed-in graphs of\\nSwim, Swish, and ReLU at the negative values.\\n(a)\\n(b)\\nFig. 2: (a) First derivative graphs of Swim, Swish, and ReLU. (b) Zoomed-in\\n\\ufb01rst derivative graphs of Swim, Swish, and ReLU at the negative values.\\nIn the next subsection, we incorporate the behaviors of\\nSwim, Swish, and ReLU and their \\ufb01rst derivatives in our analysis. We focus our analysis on comparing Swim (for\\nk= 0.5) against Swish as [16] and [6] have shown that\\nSwish\\/SiLU outperforms ReLU and other baseline functions\\nacross various tasks.\\nA. Properties of Swim\\nLike Swish, Swim is unbounded above, bounded below,\\nnon-monotonic, and smooth. Out of these four properties,\\nnon-monotonicity and smoothness are the properties that\\ndistinguish Swish and Swim from ReLU and other baseline\\nactivations. As shown in Figure 1, the non-monotonicity\\nproperty allows Swim and Swish to produce negative values\\nfor small negative inputs. The impact on the gradient is\\nshown in Figure 2, where the \\ufb01rst derivative of Swim and\\nSwish are larger than zero for those small negative inputs.\\nThis has the effect of increasing expressivity and improving\\ngradient \\ufb02ow [16]. Similarly, the smoothness of Swim and\\nSwish correspond to a smooth loss landscape that is easier\\nto optimize [16].\\nAs shown in Figure 1, negative inputs of Swim saturate\\nat a slower rate than Swish. For k= 0.5 (Swim) and b= 1\\n(Swish), Swim maintains this property while bounding the\\nSwish function from below for all negative inputs, noting that\\nboth functions have an approximately equal local minimum\\nvalue of 0.3 in the negative region. Essentially, this allows\\nSwim to produce more and larger negative values for negative\\ninputs without causing the gradient to explode, and without\\nforgoing the regularization effect that is induced for functions\\nthat approach zero [8]. Although the value of Swish\\u2019s bpa-\\nrameter could be decreased to match Swim\\u2019s slow saturation\\nrate, it cannot be decoupled from the consequence of making\\nSwish approach a linear function, which depending on some\\ntasks, may be undesirable as well as disadvantageous due to\\nlosing the properties of a non-linear activation function.\\nIn the context of deep off-policy temporal difference al-\\ngorithms, namely TD3, Swim\\u2019s smoothly-saturating property\\nbecomes relevant as the neural networks do not incorporate\\nfeature normalization techniques. This results in a non-zero-\\ncentered (unnormalized) activation function that becomes\\nsusceptible to experiencing the vanishing gradient problem\\nat points that saturate quickly, which is the case for Swish.\\nIn other words, Swim mitigates the lack of normalization\\nissue in TD algorithms by slowly saturating the negative\\ninputs. As [1] has suggested, normalization techniques are\\nnot used partly because the action-value function is evaluated\\ntwo times (Q(s;a)and Q(s0;p(s0)))during the training of\\nthe critic network, thereby producing actions that come from\\ndifferent distributions. Furthermore, the observation space is\\nalso not necessarily normalized, which is the case for the\\nMuJoCo locomotion continuous control benchmark tasks on\\nwhich we run our tests.\\nUnlike Swish, Swim is less likely to over\\ufb02ow due to\\nthe absence of the exponentials [2]. As shown in Equation\\n1, Swim and its \\ufb01rst derivative are instead composed of\\nsquare roots and a quadratic term. Furthermore, Swim is\\nalso less computationally expensive than Swish. Although\\nthey are asymptotically equivalent functions, in practice, thesquare root and quadratic functions are faster to compute\\nthan exponentials, and the inverse square roots as a whole,\\nwhich the Swim function and its derivative are composed\\nof, are even faster to calculate than exponentials [3]. John\\nCarmack is regarded to have written a fast implementation\\nof the inverse square root function [13], and on an x86\\narchitecture, [3] has shown that his inverse square root-based\\nactivation function is faster than [4]\\u2019s exponential-based acti-\\nvation function named ELU. Furthermore, Intel publishes the\\nCPU performance of vector functions, including the Inverse\\nSquare Root and the Exponential\\/Exp function, which [3]\\nhas aggregated as shown in Table I. Depending on the x86\\narchitecture, Table I shows that the Inverse Square Root is\\napproximately 1.2-2.2x faster than the Exp function.\\nTABLE I: CPU Performance of the Vector Inverse Square Root and architecture, Table I shows that the Inverse Square Root is\\napproximately 1.2-2.2x faster than the Exp function.\\nTABLE I: CPU Performance of the Vector Inverse Square Root and\\nExponential Functions (Clocks per Element)\\nVector\\nFunction\\nSingle\\nPrecision\\n(EP)Intel Xeon\\nE5-2699 v3\\n(Haswell\\nA VX2)Intel Xeon\\nE5-2699 v4\\n(Broadwell\\nA VX2)Intel Xeon\\nPlatinum\\n8180 (Skylake\\nA VX-512)\\nInvSqrt 0.66 0.64 0.24\\nExp 0.81 0.89 0.52\\nExp\\/InvSqrt 1.2\\u00d7 1.4\\u00d7 2.2\\u00d7\\nIII. METHODS\\n(a)\\n (b)\\n(c)\\n (d)\\nFig. 3: (a) Walker2d-v2. (b) Hopper-v2. (c) HalfCheetah-v2. (d) Swimmer-\\nv2. Shown in Figure 3, we run our experiments on four\\nof MuJoCo\\u2019s benchmark locomotion control environments:\\n(a) Walker2d-v2, (b) Ant-v2, (c) Hopper-v2, and (d)\\nHalfCheetah-v2. We use the TD3 algorithm and the Swim\\nactivation function to solve those environments. We also\\nreport the performance and ef\\ufb01ciency of those experiments.\\nA. Preliminaries\\nWe model each environment as a Markov Decision Process\\n(MDP), where at each discrete time step tand a given state\\ns2S, an agent selects an action a2Aaccording to its policy\\np:S!A. The agent then receives a reward rand the new\\nstate of the environment s0. The objective is to then \\ufb01nd the\\noptimal policy pf, with parameters f.\\nWe select the Twin Delayed Deep Deterministic Policy\\nGradient algorithm (TD3) [7] to learn the optimal policy.\\nBased on the Deep Deterministic Policy Gradient (DDPG)\\nalgorithm [12], TD3 maintains a single actor and two critic\\nnetworks, with the action-value functions, Qf1(s;a)and\\nQf2(s;a), acting as the two critic neural network approxi-\\nmators. TD3 learns the two Q-functions, Qf1and Qf2, by\\nmean square Bellman error minimization.\\nAt each time step, the targets in the Bellman error loss\\nfunctions are updated according to the following equation:\\ny=r+gmin\\ni=1;2Qq0\\ni(s0;pf0(s0) +e);\\ne\\u0018clip(N(0;s);\\u0000c;c)(2)\\nwhere gis the discount factor, q0\\niis the parameter of the\\ntarget Q networks, f0is the parameter of the target actor net-\\nwork, eis the clipped noise, and (\\u0000c;c)are hyperparameter\\nvalues that are set to clip target policy noise. That is, the\\nsmaller of the two target values given by the Q-functions is\\nused to update the target.\\nWe use the authors\\u2019 original implementation of TD3 in\\nPyTorch because the original paper also focuses on running\\nTD3 on MuJoCo control tasks, including the ones we use\\nin this paper. Therefore, we also use the same optimal\\nhyperparameters and only change the activation functions in\\nour experiments.\\nB. Applying Activation Functions\\nWe write our custom implementation of the Swim acti-\\nvation function in PyTorch and use it in all of the hidden\\nlayers of the actor and critic networks. We follow the\\nsame approach when applying the Swish activation function,\\nalthough Swish uses the underlying numerically stable Sig-\\nmoid torch function in PyTorch. We do not implement a\\nnumerically stable version but write a direct translation of\\nthe Swim function, as written in Equation 1, using the torch\\nfunctions. A numerically stable function could potentially\\nimprove the performance of Swim, although we leave that for\\nfuture work. We also use double-precision instead of single-\\nprecision to improve accuracy and run our experiments on a\\nmachine with an ARM architecture.\\nAs mentioned in Section II, we only evaluate Swim against\\nSwish as [16] and [6] have shown that Swish\\/SiLU out-\\nperforms ReLU and other baseline functions across varioustasks. We also select Swim as an activation function for this\\nproblem for reasons explained in Section I and Section II.A,\\nand also because simple kinematic controller models, such as\\ntanh (x)[21], would not suf\\ufb01ce as an activation function for\\nthis problem since locomotion control tasks exhibit complex\\ndynamics.\\nC. Evaluation\\nWe use performance curves to compare the performance\\nof Swim with Swish, following standard practices in deep\\nreinforcement learning. We also measure inference time of\\nthe actor network to compare the ef\\ufb01ciency of the two\\nactivation functions, effectively isolating the evaluation of\\nef\\ufb01ciency from performance.\\nIV. RESULTS\\nFor the performance metric, we run TD3 with the se-\\nlected activation function for 1 million time steps on each\\nenvironment, where we record the average reward over 10\\nepisodes with no exploration noise every 5000 time steps.\\nWe then report the performance curves over 5 random seeds\\nfor each environment. We also report the max average return\\nover those 5 seeds to compare the results of the activation\\nfunctions.\\nFor the ef\\ufb01ciency metric, we calculate the inference time for each environment. We also report the max average return\\nover those 5 seeds to compare the results of the activation\\nfunctions.\\nFor the ef\\ufb01ciency metric, we calculate the inference time\\nover the \\ufb01rst 100 iterations on each environment and report\\nthe results.\\nA. Performance\\nTABLE II: Max Average Return over 5 trials of 1 million time steps. \\u00b1\\ncorresponds to a single standard deviation over trials.\\nEnvironment Swish Swim Improvement\\nWalker2d-v2 3774.668 \\u00b1\\n149.7794387.442 \\u00b1\\n453.52616%\\nHopper-v2 3450.055 \\u00b1\\n129.2033431.081 \\u00b1\\n110.3940%\\nHalfCheetah-\\nv210519.091 \\u00b1\\n872.69311130.364 \\u00b1\\n652.7165%\\nSwimmer-v2 132.838 \\u00b1\\n8.146134.503 \\u00b1\\n6.8061%\\nThe learning curves are presented in Figure 4 with the\\nmax average return reported in Table II. The results show\\nthat Swim matches or outperforms Swish in both \\ufb01nal\\nperformance and learning speed across all tasks. Based on\\nthe curve and the max average return of the two activation\\nfunctions, Swim outperforms Swish in 3 out of 4 environ-\\nments: the Walker2d-v2, HalfCheetah-v2, and Swimmer-v2;\\nhowever, improvement is more signi\\ufb01cant in the Walker2d-\\nv2 and HalfCheetah-v2 environments, with a 16% and 5%\\nimprovement rate, respectively.\\nThe results support the hypothesis and analysis presented\\nin Section II; for k= 0.5, it is expected that Swim will\\nnot always outperform Swish, as the constant has been\\nselected such that Swim will approximate Swish at b= 1.\\nAn exploration of a different value for kcould potentially\\nresult in higher performance. (a)\\n (b)\\n(c)\\n (d)\\nFig. 4: Learning curves for MuJoCo\\u2019s continuous control tasks: (a)\\nWalker2d-v2, (b) Hopper-v2, (c) HalfCheetah-v2, and (d) Swimmer-v2. The\\nshaded region represents half a standard deviation of the average evaluation\\nover 5 trials.\\nHowever, Swim still outperforms Swish in some environ-\\nments. As explained in Section II, we theorize that this is\\nbecause the negative values of Swim saturate at a slower\\nrate than Swish, which mitigates the issue of the lack of\\nfeature normalization techniques in TD3 and other deep\\nreinforcement learning algorithms, in addition to the fact that\\nthe observation space of our environments is not normalized.\\nB. Ef\\ufb01ciency\\nTABLE III: Inference time over the \\ufb01rst 100 iterations for each environment.\\n\\u00b1 corresponds to a single standard deviation over the iterations.\\nEnvironment Swish (ms) Swim (ms) Improvement\\nWalker2d-v2 0.810 \\u00b1 0.066 0.687\\u00b1 0.077 17.9%\\nHopper-v2 0.720 \\u00b1 0.055 0.666 \\u00b1 0.056 8.1%\\nHalfCheetah-\\nv20.826 \\u00b1 0.057 0.733 \\u00b1 0.064 12.7%\\nSwimmer-v2 0.737 \\u00b1 0.060 0.644\\u00b1 0.045 14.4%\\nInference time for each of the four environments is re-\\nported in Table III. The table shows that Swim consistently\\noutperforms Swish in terms of ef\\ufb01ciency across all environ-\\nments, with improvement ranging from 8.1% to 17.9%. In\\ncontrast to the results of the max average return, improve-\\nment in ef\\ufb01ciency is signi\\ufb01cant across all environments.\\nThe results support our analysis in Section II, where we\\nhypothesize that Swim is more ef\\ufb01cient than Swish becauseinverse square roots are computationally less expensive than\\nexponentials. Since we did not implement a custom backward\\npass that leverages the properties of Swim, we do not\\nmeasure backpropagation time, although the inverse square\\nroot is de\\ufb01ned in the \\ufb01rst derivative of the Swim function, as\\nshown in Equation 1, which could be leveraged to optimize\\nthe ef\\ufb01ciency of the backward pass as [3] has done in his\\npaper.\\nV. DISCUSSION AND FUTURE WORK\\nBased on our analysis and results, the primary advantage\\nof Swim is its reduced computational complexity compared\\nwith Swish, as it signi\\ufb01cantly outperforms Swish in ef\\ufb01-\\nciency across all tasks, although it also achieves state-of-the-\\nart performance and sometimes exceeds Swish in learning\\nspeed and reward-achievement. Swim\\u2019s low computational\\ncomplexity allows it to be a suitable general-purpose acti-\\nvation function, which is why we urge future researchers\\nto test and optimize Swim on other models beyond deep\\nreinforcement learning and the robotics domain. For the\\ndeep reinforcement learning and robotics domain, Swim is\\nespecially applicable because of its both high-performing\\nand high-ef\\ufb01ciency properties, where locomotion continuous\\ncontrol tasks bene\\ufb01t from deep off-policy temporal differ-\\nence learning algorithms that use activation functions with\\nnegatively saturating inputs to mitigate the lack of feature\\nnormalization techniques in addition to the lack of nor-\\nmalization of the observation space; locomotion continuous\\ncontrol tasks also bene\\ufb01t from an ef\\ufb01cient activation function\\nthat accelerates running time because they exhibit complex\\ndynamics and high degrees of freedom, requiring them to\\ntake a long time in training. Those results also give rise\\nto the idea of domain-speci\\ufb01c activation functions, which\\ncurrent research is not focused on; current research trends\\ninstead focus on designing generalizable activation functions\\nand then testing them on various tasks, without investigating\\nthe properties that would bene\\ufb01t certain areas more than\\nothers. We hope to see future work in the area of domain-\\nspeci\\ufb01c activation functions, especially in the \\ufb01elds of deep\\nreinforcement learning and robotics that see less work with\\nactivation functions.\\nFurther work with Swim remains. In terms of testing\\nand optimization, we only test our function on the CPU,\\nbut not on the GPU. This also means that Swim could be\\noptimized using a CUDA-based implementation to further and optimization, we only test our function on the CPU,\\nbut not on the GPU. This also means that Swim could be\\noptimized using a CUDA-based implementation to further\\nimprove ef\\ufb01ciency as [14] has done in his paper. We also\\nonly test our results on an ARM architecture and report our\\nef\\ufb01ciency results, although [3] has previously reported the\\nfast results of the inverse square root on an x86 architecture,\\nas well as Intel has published the fast performance of the\\ninverse square root, which we have reported in this paper.\\nFurthermore, the backward pass of Swim could be imple-\\nmented such that its ef\\ufb01ciency is improved using the inverse\\nsquare root as [3] has done in his paper. In terms of the\\nactivation function itself, we do not implement a numerically\\nstable version, although the performance we report shows\\nthat Swish matches or outperforms Swish in learning speed and reward-achievement without doing so. We speculate\\nthat a numerically stable version would further improve the\\nperformance of Swim. We also speculate that a different\\nvalue of kcould potentially improve the performance of\\nSwim. Based on these \\ufb01ndings, we plan on exploring and\\nimproving the performance and ef\\ufb01ciency of Swim by testing\\nit on a GPU and an ARM architecture, writing a CUDA-\\nbased implementation, designing a custom backward pass for\\nSwim, inventing a numerically stable version, and exploring\\ndifferent values of k.\\nAPPENDIX\\nThe code and other supplementary materials used for this\\nresearch paper, including our own implementation of Swim,\\nare available at https:\\/\\/github.com\\/maryam-abdool\\/Swim-\\nControl\\nREFERENCES\\n[1] Aditya Bhatt, Max Argus, Artemij Amiranashvili, and Thomas Brox.\\nCrossnorm: Normalization for off-policy td reinforcement learning.\\narXiv preprint arXiv:1902.05605 , 2019.\\n[2] Pierre Blanchard, Desmond J Higham, and Nicholas J Higham.\\nAccurately computing the log-sum-exp and softmax functions. IMA\\nJournal of Numerical Analysis , 41(4):2311\\u20132330, 2021.\\n[3] Brad Carlile, Guy Delamarter, Paul Kinney, Akiko Marti, and Brian\\nWhitney. Improving deep learning by inverse square root linear units\\n(isrlus). arXiv preprint arXiv:1710.09967 , 2017.\\n[4] Djork-Arn \\u00b4e Clevert, Thomas Unterthiner, and Sepp Hochreiter. Fast\\nand accurate deep network learning by exponential linear units (elus).\\narXiv preprint arXiv:1511.07289 , 2015.\\n[5] Yan Duan, Xi Chen, Rein Houthooft, John Schulman, and Pieter\\nAbbeel. Benchmarking deep reinforcement learning for continuous\\ncontrol. In International conference on machine learning , pages 1329\\u2013\\n1338. PMLR, 2016.\\n[6] Stefan Elfwing, Eiji Uchibe, and Kenji Doya. Sigmoid-weighted linear\\nunits for neural network function approximation in reinforcement\\nlearning. Neural Networks , 107:3\\u201311, 2018.\\n[7] Scott Fujimoto, Herke Hoof, and David Meger. Addressing function\\napproximation error in actor-critic methods. In International confer-\\nence on machine learning , pages 1587\\u20131596. PMLR, 2018.\\n[8] Xavier Glorot, Antoine Bordes, and Yoshua Bengio. Deep sparse\\nrecti\\ufb01er neural networks. In Proceedings of the fourteenth interna-\\ntional conference on arti\\ufb01cial intelligence and statistics , pages 315\\u2013\\n323. JMLR Workshop and Conference Proceedings, 2011.\\n[9] Tuomas Haarnoja, Sehoon Ha, Aurick Zhou, Jie Tan, George Tucker,\\nand Sergey Levine. Learning to walk via deep reinforcement learning.\\narXiv preprint arXiv:1812.11103 , 2018.\\n[10] Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer\\nfeedforward networks are universal approximators. Neural networks ,\\n2(5):359\\u2013366, 1989.\\n[11] Andrew Howard, Mark Sandler, Grace Chu, Liang-Chieh Chen,\\nBo Chen, Mingxing Tan, Weijun Wang, Yukun Zhu, Ruoming Pang,\\nVijay Vasudevan, et al. Searching for mobilenetv3. In Proceedings\\nof the IEEE\\/CVF international conference on computer vision , pages\\n1314\\u20131324, 2019.\\n[12] Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas\\nHeess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra.\\nContinuous control with deep reinforcement learning. arXiv preprint\\narXiv:1509.02971 , 2015.\\n[13] Chris Lomont. Fast inverse square root. Tech-315 nical Report , 32,\\n2003.\\n[14] Diganta Misra. Mish: A self regularized non-monotonic neural\\nactivation function. arXiv preprint arXiv:1908.08681 , 4(2):10\\u201348550,\\n2019.\\n[15] Vinod Nair and Geoffrey E Hinton. Recti\\ufb01ed linear units improve\\nrestricted boltzmann machines. In Icml, 2010.\\n[16] Prajit Ramachandran, Barret Zoph, and Quoc V Le. Searching for\\nactivation functions. arXiv preprint arXiv:1710.05941 , 2017.\\n[17] J \\u00a8urgen Schmidhuber. Deep learning in neural networks: An overview.\\nNeural networks , 61:85\\u2013117, 2015.[18] David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wier-\\nstra, and Martin Riedmiller. Deterministic policy gradient algorithms.\\nInInternational conference on machine learning , pages 387\\u2013395.\\nPMLR, 2014. stra, and Martin Riedmiller. Deterministic policy gradient algorithms.\\nInInternational conference on machine learning , pages 387\\u2013395.\\nPMLR, 2014.\\n[19] Richard S Sutton and Andrew G Barto. Reinforcement learning: An\\nintroduction . MIT press, 2018.\\n[20] Sebastian Thrun and Anton Schwartz. Issues in using function\\napproximation for reinforcement learning. In Proceedings of the\\n1993 Connectionist Models Summer School Hillsdale, NJ. Lawrence\\nErlbaum , volume 6, pages 1\\u20139, 1993.\\n[21] Dongdong Xie, Shenquan Wang, and Yuenan Wang. Trajectory\\ntracking control of differential drive mobile robot based on im-\\nproved kinematics controller algorithm. In 2018 Chinese Automation\\nCongress (CAC) , pages 2675\\u20132680. IEEE, 2018.\",\"22\":\" Ensemble Reinforcement Learning: A Survey\\nYanjie Song1, P. N. Suganthan2,3, Witold Pedrycz4, Junwei Ou1, Yongming He1, Yingwu\\nChen1\\nAbstract\\nReinforcement learning (RL) has achieved state-of-the-art performance in many scienti\\fc\\nand applied problems. However, some complex tasks still are di\\u000ecult to handle using a\\nsingle model and algorithm. The highly popular ensemble reinforcement learning (ERL)\\nhas become an important method to handle complex tasks with the advantage of combining\\nreinforcement learning and ensemble learning (EL). ERL combines several models or training\\nalgorithms to fully explore the problem space and has strong generalization characteristics.\\nThis study presents a comprehensive survey on ERL to provide the readers with an overview\\nof the recent advances and challenges. The background is introduced \\frst. The strategies\\nsuccessfully applied in ERL are analyzed in detail. Finally, we outline some open questions\\nand conclude by discussing some future research directions of ERL. This survey contributes\\nto ERL development by providing a guide for future scienti\\fc research and engineering\\napplications.\\nKeywords: ensemble reinforcement learning, reinforcement learning, ensemble learning,\\narti\\fcial neural network, ensemble strategy\\nEmail addresses: songyj_2017@163.com (Yanjie Song),\\np.n.suganthan@qu.edu.qa,EPNSugan@ntu.edu.sg (P. N. Suganthan ), wpedrycz@ualberta.ca (Witold\\nPedrycz ), junweiou@163.com (Junwei Ou), heyongming10@hotmail.com (Yongming He),\\nywchen@nudt.edu.cn (Yingwu Chen)\\n1College of Systems Engineering, National University of Defense Technology, Changsha, China\\n2KINDI Center for Computing Research, College of Engineering, Qatar University, Doha, Qatar\\n3School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore\\n4Department of Electrical & Computer Engineering, University of Alberta, Edmonton AB, Canada\\nPreprint submitted to Review March 7, 2023arXiv:2303.02618v1  [cs.LG]  5 Mar 2023 1. Introduction\\nIn the last decades, reinforcement learning (RL) methods have demonstrated the advan-\\ntages of solving complex problems in many \\felds such as gaming, robotics, and computer\\nvision. Along with a series of breakthroughs like deep Q neural networks [1], AlphaGo\\n[2], video game [3, 4] and robotic control tasks [5], we have witnessed the revitalization\\nof reinforcement learning outperforming humans. The success of the approach stems from\\nthe agent's ability to automate the acquisition of problem features and complete end-to-end\\nlearning. Arti\\fcial neural networks (ANN) and gradient descent further enhance RL's explo-\\nration and exploitation capabilities. This makes it easier for RL to handle time-consuming\\nmanual work or challenging tasks.\\nEach type of RL has unique advantages and inevitably has some limitations. For exam-\\nple, deep reinforcement learning (DRL) requires extensive training to obtain a policy [4].\\nAdditional problems often occur during training, including over\\ftting [6], error propagation\\n[7], and imbalance between exploration and exploitation [8]. These problems motivate re-\\nsearchers to design their models or training algorithms. Implementing ensemble learning\\nto the RL framework is a new way to enhance the learning and representation of algo-\\nrithms. This method is called ensemble reinforcement learning (ERL) and has excellent\\nperformance. Marquis de Condorcet [9] \\frst demonstrated that average voting outperforms\\nindividual model decisions. Krogh and Vedelsby [10], Breiman [11] and a series of other\\nstudies have theoretically demonstrated the great advantages of ensemble methods from\\nother perspectives. The ensemble idea has been able to achieve such great success in the\\n\\feld of deep learning and reinforcement learning due to the decomposition of datasets [12],\\npowerful learning capabilities [13] and diverse ensemble methods [11].\\nThe classi\\fcation of ERL is diverse according to di\\u000berent criteria. According to the\\nconstituent elements in the ensemble framework, ERL can be classi\\fed as a high-level en-\\nsemble [14] and a low-level ensemble [15]. ERL can also be classi\\fed as single-agent ERL\\n2 [16] and multi-agent ERL [17] according to the number of agents. Centralized ERL [18]\\nand distributed ERL [19], on the other hand, are classi\\fcations of ERLs based on how the\\nagents work. In this paper, we present a detailed description of ERL methods according to\\nthe improvement strategies used and discuss their applications to guide the design of new\\nmethods.\\nExisting related work of ERL on the use of training algorithms, ensemble strategies, and\\napplication areas are diverse. The motivation of this paper is to provide readers with a\\nsystematic overview of the existing related research and the current research progress and\\nthe valuable conclusions achieved. Readers can quickly \\fnd the corresponding studies and\\nfollow up on the research according to their concerns. To the best of our knowledge,\\nthis is the \\frst survey focusing solely on ensemble reinforcement learning. We\\npresent the strategies used in ERL and related applications, discuss several open questions\\nand give a guide for future exploration in the ERL area.\\nThe remainder of this paper is structured as follows. Section 2 presents the background\\nof ensemble reinforcement learning methods. Section 3 introduces implementation strategies\\nin ERL. Section 4 discusses the application of ERL to di\\u000berent domains. Section 5 discusses\\nseveral open questions and possible future research directions. Section 6 gives the conclusion\\nof this paper (see Figure 1).\\n2. Background\\nTo facilitate the readers' better understanding of ensemble reinforcement learning meth-\\nods, this section brie\\ry introduces reinforcement learning and ensemble learning.\\n2.1. Reinforcement Learning\\nThe reinforcement learning method is an arti\\fcial intelligence method in which the agent\\ninteracts with the environment and makes decisions to continuously correct errors to obtain\\n3 Figure 1: Structure of the paper\\n4 optimal decisions. Markov Decision Process (MDP) is the basis for using RL to solve prob-\\nlems [33]. RL can be used when the agent decision is only related to the current state and\\nnot to the previous state. Figure 2 illustrates the agent-environment interaction process. A\\ntuplehS;A;P;R;\\rican be used to represent the MDP, where Sdenotes the state, Adenotes\\nthe action, P:S\\u0002A!P(S) denotes the state transfer matrix with the probability value\\np(s0js) =p(St+1=s0jSt=s),R:S\\u0002A!Rdenotes the reward function, and \\r2[0;1]\\ndenotes the discount factor. The state of the agent at time step tisst, and it will take action\\nat. The combination of all states and actions de\\fnes a policy \\u0019. Here, Q-value is used to\\nevaluate the expected return obtained by the agent following policy \\u0019.\\nQ\\u0019(s;a) =E\\u0019\\\"1X\\nt=0\\rtR(st;at)js0=s;a 0=a#\\n(1)\\nThe goal of solving a problem using RL methods is to \\fnd an optimal policy \\u0019that\\nmaximizes Q\\u0019. For \\fnite-state MDPs, the most typical RL is the Q-learning method [20],\\nwhich uses Q-table records the combinations of hstate,actioni. Subsequently, a series of RL\\nmethods using arti\\fcial neural networks were proposed to cope with the in\\fnite state space.\\nFigure 2: Interaction process between agent and environment\\nTraining algorithms can be divided into model-based RL and model-free RL according\\nto whether the environment model in RL is given in advance or can be obtained through\\nlearning, and the training algorithms can be divided according to state-based or policy-\\nbased, or state-policy combination based. Detailed research progress on RL can be found in\\n5 ref. [21].\\n2.2. Ensemble Learning\\nEnsemble learning (EL) methods occupy an essential position in the \\feld of machine\\nlearning (ML). The core idea of EL methods is to train multiple predictors, combine them,\\nand make a decision from all predictions as the \\fnal result of an ensemble model. This EL\\nmethod can harness the characteristics of various types of models compared to individual\\nbasic models, and the prediction performance of EL models will be improved and the results\\nobtained will be more robust. The main types of ensemble learning methods include bagging\\n[22], boosting [23], and stacking [24]. Figure 3 gives a schematic diagram of these three types\\nof EL methods, where Ddenotes the dataset, D1toDndenote the sample selection from\\nthe dataset, M1toMndenote the model, and FRdenotes the \\fnal result. The dotted line\\nin Figure 3-(b) indicates that the weights of samples in the dataset are changed as the next\\nround of the dataset. And the dotted line in Figure 3-(c) indicates that all datasets are\\nused for model prediction from level 2to levelL. The main di\\u000berence between these three\\ntypes of methods is the way of sample selection. These original and improved EL methods\\nhave been applied in various areas. And domain knowledge implemented in the improved\\nEL method achieves outstanding performance. To sum up, the EL method has been proven\\nto be advantageous in three ways.\\n\\u000fBias{variance Decomposition\\nBias-variance decomposition has successfully demonstrated the superiority of EL meth-\\nods over individual learning methods. The bagging method mentioned above reduces vari-\\nance among base learners, while the other two methods reduce both bias and variance.\\nKrogh and Vedelsby \\frst demonstrated that EL is e\\u000bective for problems with a single data\\nset using the idea of ambiguity decomposition to reduce the variance [10]. Subsequently,\\nBrown et al. [25], Geman et al. [26] veri\\fed the e\\u000bectiveness of EL methods for problems\\n6 (a) Bagging [22]\\n (b) Boosting [23]\\n(c) Stacking [24]\\nFigure 3: Schematic diagram of three types of EL methods\\n7 with multiple data sets and is given as [12]:\\nE[s\\u0000t]2=bias2+1\\nNvar+ (1\\u00001\\nN)covar (2)\\nbias=1\\nNX\\ni(E[si]\\u0000t) (3)\\nvar=1\\nNX\\niE[si\\u0000E[si]]2(4)\\ncovar =1\\nN(N\\u00001)X\\niX\\nj6=iE[si\\u0000E[si]][sj\\u0000E[sj]] (5)\\nwhereidenotes the i-th model of EL, sdenotes the solution of the problem, and Ndenotes\\nthe number of models in EL. Here, both biasandvarare obtained using the average of the\\ndi\\u000berences between multiple models, and covar measures the pairwise di\\u000berence between\\nmodels in the EL method.\\nFor a single model, a decrease in bias leads to an increase in variance. As seen in Eq. 2,\\nif the ensemble model is used to predict, it can reduce the variance value without increasing\\nbias.\\n\\u000fStatistical Perspective\\nThe statistical perspective was proposed by Dietterich to support the great advantages\\nof the EL approach [13]. A machine learning problem is interpreted from a statistical point\\nof view. The search space exists multiple hypotheses, and the model prediction \\fnds the\\noptimal hypothesis from these hypotheses. In general, the size of the data used for training\\nis only a fraction of the size within the search space and there is a risk of making wrong\\ninferences. If an EL method is used, these hypotheses can be combined reasonably to obtain\\na better understanding of the search space features and reduce the chance of incorrect\\n8 classi\\fcation or invalid prediction.\\n\\u000fDiversity Perspective\\nThe diversity perspective is used to justify the success of ERL methods from another\\npoint of view. Dietterich points out that the combination of di\\u000berent base models can\\nenhance diversity [13]. Some typical methods, such as adaboost, and random forest, show\\nthe importance of diversity from the perspective of training data. The use of random noise\\ncan enhance the richness of the output. Diversity allows decision-makers to combine the\\nmodel output with usage requirements to get a more reasonable \\fnal result.\\n3. Strategies for Ensemble Reinforcement Learning\\nFrom the experiment results of related studies, it can be seen that the ERL method has\\nbetter average performance and sampling e\\u000eciency compared to the RL methods in some\\npublic RL test sets as well as some practical tasks [15, 27]. The performance improvement\\ncan reach about 20% [28, 29, 30]. For the classi\\fcation task, the ERL method achieves the\\nbest accuracy for multiple benchmarks in the UCI online data repository [31, 32].\\nThe strategies employed in the performance methods of ERL to achieve superiority over\\nother solution methods in numerous problems are closely related. The strategies used in\\nensemble reinforcement learning are diverse and can be classi\\fed according to the improve-\\nments made to the composition of ERL. Strategies for ERL mainly include ensemble strate-\\ngies for the Q-function, ensemble strategies for reward, ensemble strategies for the loss\\nfunction, combination of models, combination training algorithms, and decision strategies.\\nThese strategies are introduced separately in this section.\\n3.1. Ensemble Strategies for Q-functions\\nQ-function re\\rects how good the agent is in any given state in most of RL methods\\n[33]. \\\"Good state\\\" here means that the agent can obtain a high expected return, which\\n9 depends on what action is taken by the agent. The background section gives the most clas-\\nsical Q-function formula, which is also applicable to ERL methods. In addition, designing\\nQ-functions speci\\fcally for the ERL method can further improve the search performance of\\nthe algorithm [15, 36, 37, 38]. Lan et al. proposed a Maxmin Q-learning algorithm that uses\\nmultiple Q-functions to evaluate performance [39]. The maxmin mechanism integrates mul-\\ntiple predicted values as a reference for agent decision-making. Speci\\fcally, the prediction\\nterm in the original Q-value calculation formula is determined by the smallest of the multi-\\nple Q-functions. The performance of this algorithm is demonstrated by using the Mountain\\nCar environment. As can be seen from the results, the improvement of the Q-function has\\na positive e\\u000bect on both the convergence performance and the search performance of the\\nalgorithm.\\nBayesian optimization can also be used to improve the Q-function. Chen et al. applied\\nthis idea in the design of the ERL method to update Q\\u0003using Bayesian optimization. This\\nmethod is mainly suitable for solving high-dimensional ERL problems and is particularly\\nuseful when the solution space is ultra large [8]. In addition, an upper-con\\fdence bounds\\n(UCB) based solution space exploration strategy is used for the agent's action selection.\\nAtari game is used to test the performance of the proposed method. The experimental\\nresults verify the e\\u000bectiveness of the proposed strategy.\\nSome Q-value approximation methods from related work of RL can also be used in ERL\\nmethods. Ghosh et al. used an ERL method based on a multi-agent framework to solve the\\nair tra\\u000ec control problem [40]. A kernel-based Q-value approximation via sample transitions\\nis used to speed up the convergence [41].\\n3.2. Ensemble Strategies for Reward\\nReward re\\rects the agent's performance in actions taken based on the state. A good\\ndecision generally corresponds to a high reward, while a problematic decision should prompt\\nthe agent to \\fnd and correct the error through the reward. Yao et al. designed an averaging\\n10 reward calculation method for the ERL method, which allows the ERL method to take into\\naccount the relationship between exploration and exploitation [42]. Soft actor-critic is used\\nto train ANN models. This ERL method is well suited to solve the problem of exploring\\nunknown regions.\\nThe combination of reward functions in ERL can also be used with weight aggregation.\\nLin et al. proposed an adaptive adjustment method for reward function weights combining\\nUCB and error [43]. The weight update strategy allows the ERL method to evaluate the\\ncorrectness of previous policies and improve generalizability. Qi et al. also used an ERL\\nmethod with weighted reward aggregated functions to solve the tra\\u000ec signal control problem\\n[44].\\nA fuzzy set can a\\u000bect the reward value obtained by agents by measuring dissimilarity.\\nPan et al. proposed a dissimilarity evaluation metric for deciding the weight value of each\\nagent's reward in ERL [45]. In this way, ERL can achieve a good training e\\u000bect with fewer\\niterations.\\n3.3. Ensemble Strategies for Loss Function\\nThe loss function is an essential basis for ERL to update the network parameters and\\nimprove the performance of agent decisions. A smaller loss value means that the predicted\\nvalue of the ERL model is closer to the actual value. However, gradient explosion and\\ngradient disappearance are two fatal problems that often occur in the training process of\\nRL models. Some studies of ERL have attempted to improve the accuracy of RL model\\ndecisions by improving the loss function [15, 50, 51, 52]. Kumar et al. theoretically analyzed\\nthe bootstrapping error and proposed an error accumulation reduction method to enhance\\nthe stability of ensemble Q-learning algorithms [7].\\nDesigning a global loss function for all models used is an approach speci\\fc to ERL.\\nAdebola et al. proposed an improved global loss function with each member model included\\nin the function [46]. In addition, an interpolation method is used to control the di\\u000berence\\n11 between policies that the algorithm needs to train. This ERL approach can also optimize\\nthe agent's policy selection using \\fne-tuning techniques. Jiang et al. add the training data\\nerror between models to the overall loss calculation formula [19]. This method is well applied\\nin mobile edge computing (MEC) systems for rational resource scheduling.\\nImproved loss functions can also take uncertainty into account. Sun et al. used the\\nuncertainty reduction technique to design an ensemble loss function [48], which e\\u000bectively\\nprevents the RL model from falling into a local optimum. A distillation method is used to\\nselect the training data for the model. And the performance of the ERL method is veri\\fed\\nby the Atari game.\\n3.4. Combination of Models\\nThe ensemble of di\\u000berent types of models is a common and simple strategy in ERL. These\\nmodels can be either ML models or ANN models. The structure of the model combination\\nis determined according to the speci\\fc problem and the solution target. A single type or a\\ncombination of multiple types of models are both popular. For using only one type of model,\\nANNs with di\\u000berent depths can be considered, and di\\u000berent random initializations [16] or\\ndi\\u000berent training stages of ANN [53] can be used. Table 1 gives brief information about the\\nrelated work implementing model combination strategies. The model types of relevant work\\nare mainly around three. Such an ERL framework is less di\\u000ecult to implement and achieves\\nbetter performance than individual RL methods. Amal Saadallah et al. [54], Guohui Li\\net al. [55], Gitika Sharma et al. [56], on the other hand, attempted a large-scale ensemble\\nof numerous machine learning models and neural network models implemented in the ERL\\nframework.\\nERL can be divided into parallel ERL and sequential ERL according to the relation-\\nship between base learners in ERL. Figures 4{5 give schematic diagrams of these two ERL\\nmethods. In most ERL studies, such as Liu et al. [32], Schubert et al. [66], and Shen et al.\\n[67], base learners are constructed in a parallel framework. These ML or ANN models are\\n12 Table 1: Combination of models\\nYear Author Combination of Models\\n2019 Dong et al. [57] long short-term memory network, gated recurrent unit\\nnetwork\\n2019 Goyal et al. [52] convolution neural network, gated recursive unit\\n2020 Liu et al. [58] long short-term memory network, deep belief network,\\necho state network\\n2020 Perepu et al. [59] a linear regression model, long short-term memory model,\\narti\\fcial neural network, random forest\\n2021 Liu et al. [60] graph convolutional network, long short-term memory\\nnetworks, gated recursive unit\\n2021 Saadallah et al. [54] autoregressive integrated moving average, exponential\\nsmoothing, gradient boosting machines, gaussian pro-\\ncesses, support vector regression, random forest, projec-\\ntion pursuit regression, MARS, principal component re-\\ngression, decision tree regression, partial least squares re-\\ngression, multilayer perceptron, long short-term memory\\nnetwork (LSTM), Bi-LSTM: Bidirectional LSTM, CNN-\\nbased LSTM, convolutional LSTM\\n2021 Daniel L. Elliott and\\nCharles Anderson [61]convolution neural network, gated recursive unit, arti\\f-\\ncial neural network\\n2022 Shang et al. [29] gated recursive unit, graph convolutional network, graph\\nattention network\\n2022 Tan et al. [30] graph attention network, long short-term memory net-\\nworks, temporal convolutional network\\n2022 Li et al. [62] gated recurrent unit, deep belief network, temporal con-\\nvolutional network\\n2022 Zijie Cao and Hui Liu\\n[63]temporal convolutional network, Bidirectional long\\nshort-term memory network, kernel extreme learning ma-\\nchine\\n2022 Birman et al. [64] machine learning models, arti\\fcial neural network\\n2022 Li et al. [55] naive bayes, support vector machine with stochastic gra-\\ndient descent, FastText, Bi-directional long short-term\\nmemory\\n2022 Sharma et al. [56] support vector regressor (SVR), eXtreme gradient boost-\\ning (XGBoost), random Forest (RF), arti\\fcial neural net-\\nwork (ANN), long short-term memory (LSTM), convo-\\nlution neural network (CNN), CNN-LSTM, CNN-XGB,\\nCNN-SVR, and CNN-RF\\n2022 Shi Yin and Hui Liu\\n[65]group method of data handling, echo state network, ex-\\ntreme learning machine\\n2023 Yu et al. [28] graph attention network, gated recursive unit, temporal\\nconvolutional network 13 Figure 4: Parallel ensemble reinforcement learning\\nFigure 5: Sequential ensemble reinforcement learning\\nresponsible for the same task, and the output of each model generates the \\fnal prediction\\naccording to a certain strategy. There are also some studies, such as Qin et al. and Ferreira\\net al. that try to construct the ERL method in the sequential framework [68, 69]. In this\\nframework, the base learner completes the \\fnal prediction step by step in a certain order\\n[69].\\n3.5. Combination of Training Algorithms\\nTable 2: Combination of training algorithms\\nYear Author Combination of Training Algorithms\\n2008 Marco A. Wiering and\\nHado van Hasselt [70]Q-learning, Sarsa, actor-critic, QV-learning, ACLA\\n2018 Chen et al. [35] deep Q-networks, deep Sarsa networks, double deep Q-\\nnetworks\\n2020 Yang et al. [14] proximal policy optimization, advantage actor-critic,\\ndeep deterministic policy gradient\\n2020 Saphal et al. [71] advantage actor-critic, sample e\\u000ecient actor-critic with\\nexperience replay, actor-critic using Kronecker-factored\\ntrust region, deep deterministic policy gradient, soft\\nactor-critic, trust region policy optimization\\n2021 Smit et al. [27] double deep Q-Learning, soft actor-critic\\n2022 Eriksson et al. [72] residual gradient, TD, TD( \\u0015)\\n2022 N\\u0013 emeth, Marcell and\\nSz} ucs, G\\u0013 abor [73]deep deterministic policy gradient, advantage actor-\\ncritic, proximal policy optimization\\n14 ERL can use model combinations to obtain diverse prediction results, and can also\\nuse di\\u000berent training algorithms to achieve full exploration of the solution space. Training\\nalgorithms can be classi\\fed into three categories: state-based, policy-based, and state-policy\\ncombination based. Each of these training algorithms has its unique sampling strategy\\nand output data characteristics. Similar to the EL method, researchers can quickly use\\nthe training algorithms according to the application scenarios without focusing on data\\nsampling technology. Table 2 gives information about the studies using the combination\\nstrategy of training algorithms. Combining online and o\\u000fine training algorithms or using\\ntraining algorithms based on di\\u000berent optimization strategies can take advantage of their\\nrespective strengths to handle complex tasks. Accordingly, the complexity of ERL methods\\nusing such improved strategies increases and the training stage takes more time. Moreover,\\nthe ensemble model obtained also requires the design of a decision strategy to select the\\nprediction results that are closest to the actual situation.\\nCurrently, the research related to the combination of training algorithms is not deep\\nenough and simply combines multiple typical algorithms. There are some similarities be-\\ntween training algorithms, and the use of migration learning to transfer sampled data can be\\nconsidered to reduce the training time. In addition, the termination conditions of multiple\\ntraining algorithms also deserve in-depth analysis. Using the same number of iterations is\\nlikely to result in some models \\fnding the best policy long ago, while some models still need\\nfurther training.\\n3.6. Decision Strategies\\nMultiple base learners are implemented to obtain one result each. There may be some dif-\\nferences between multiple results. So, the algorithm of ERL needs to adopt certain decision\\nstrategies to determine the \\fnal model and output. In the existing ERL-based research, the\\ndecision strategies used include voting, optimal combination, binning, aggregation, weighted\\naggregation, stacking, and Boltzmann multiplication (see \\fgure 6).\\n15 Figure 6: Decision strategies\\nVoting : Voting is a common ERL decision strategy [74]. First, the number of occur-\\nrences of each prediction is recorded. Then, the \\fnal prediction can be selected from the\\nresults according to the principle of majority orranking .\\nOptimal Combination : For classi\\fcation problems, this is a commonly used decision\\nstrategy [75]. Multiple base classi\\fers are trained separately, from which the optimal subset\\nof models is selected to form an ensemble to classify the test set.\\nBinning : This is a majority voting decision strategy with continuous action space [71].\\nFirst, the action space is discretized into multiple intervals. After that, the number of\\noccurrences of the actions in each interval is recorded. Finally, the average value of the\\naction within the interval with the highest number of occurrences is selected as the \\fnal\\nprediction result.\\nAggregation : The prediction results of all the models in ERL are summed to produce\\n16 an overall evaluation value and take it as the \\fnal result [55]. In the aggregation method,\\neach model is considered to be equally reliable.\\nWeighted Aggregation : The prediction results obtained from di\\u000berent models are\\nsummed according to their weights [59]. A high value of weight is used for aggregation\\nmodels with high prediction accuracy.\\nStacking : An additional machine learning model is used to further predict the results of\\nbase learners, and the output of this machine learning model is used as the \\fnal prediction\\nresult [64].\\nBoltzmann Multiplication : Boltzman distribution is the basis for decision making\\n[70]. The probability value of each action can be calculated according to the Boltzman\\ndistribution. The outcome with the highest probability value is selected and will not be\\nchanged.\\n3.7. Discussions\\nStrategies are particularly critical for designing ERL methods, and overall strategies can\\nbe divided into two categories: improvement of ERL elements and model use. Among the\\nstrategies for improving the ERL element, the model combination is the easiest to implement.\\nThe use of model combination allows the ERL method to have multiple machine learning\\nmodels or ANN search strategies at the same time, which is su\\u000ecient for some practical\\napplication problems. The training algorithm combination strategy is slightly more compli-\\ncated than the model combination, and the researcher needs to understand the conditions of\\nuse of each training algorithm and design the combination of algorithm and model accord-\\ningly. When in a multi-agent or distributed framework, some new framework-related issues\\narise such as how base learners interact with the environment, and whether training is done\\nindependently or jointly. Improving Q-functions, reward, and loss functions are considered\\nfor the generalization of ERL methods. This is more demanding for researchers to propose\\n17 new methods. Accordingly, such method improvements are highly generalizable and can be\\napplied to various types of practical application problems.\\nThe decision strategy is another aspect to improve the performance of ERL. The outputs\\nfrom multiple base learners may be similar or can di\\u000ber signi\\fcantly. These outputs can be\\nadopted in their entirety, or only one or more of them can be selected as the result. There\\nis little research related to the combined use of decision strategies, and using this approach\\nallows the ERL to search the solution space comprehensively. The design of which strategy\\nto use under which conditions becomes an important factor for the strategy combination to\\nwork. The idea of integration can also be used in decision strategies, where a new model\\nevaluating the performance of the results generated using various decision strategies can\\nselect the optimal one.\\n4. Applications of Ensemble Reinforcement Learning\\nA part of the existing ERL-based research takes discrete\\/continuous control actions\\n[43, 76, 77, 78], the environment of the game [47, 79, 80] to verify the e\\u000bectiveness of\\nthe proposed algorithm. A part of the research also attempts to use ERL methods to solve\\napplication problems. The following, this section discusses the application of the ERL.\\n4.1. Energy and Environment Area\\nAs the world economy continues to grow, energy and environmental issues are receiving\\na great deal of attention worldwide. The use of neural network methods to predict future\\nconditions using historical data has become the basis for policy formulation. There is a\\nspatiotemporal relationship between data in this type of prediction problem. Therefore,\\nrecurrent neural network (RNN) is preferred for this situation. ERL methods combine\\nseveral classical RNN models to obtain more reliable conclusions. The application in energy\\nand environment area using ERL methods are shown in Table 3. Wind power and PM\\n18 2.5 prediction are hot research topics in this area. Sixty-four percent of ERL-based studies\\nused Q-learning as the training algorithm in ERL, while the remaining studies used Sarsa,\\ndeep Q-network, and deep deterministic policy gradient algorithms. Overall, the use of\\nERL methods in the energy and environment area focuses on an ensemble model, while\\nreinforcement learning algorithms are mainly used directly. Studies not using RL mainly use\\nML or ANN prediction methods [81, 82]. There is a signi\\fcant gap between the prediction\\nresults obtained by these methods and ERL.\\nTable 3: Application in energy and environment area\\nYear Authors Problem Training Algorithm\\n2020 Liu et al. [58] wind speed short term fore-\\ncastingQ-learning\\n2021 Jalali et al. [83] solar irradiance forecasting Q-learning\\n2021 Liu et al. [60] PM2.5 forecasting Q-learning\\n2021 Li et al. [84] PM2.5 forecasting Q-learning\\n2021 Chao Chen and Hui\\nLiu [85]wind speed prediction deep Q-Network\\n2021 Jalali et al. [86] wind power forecasting Q-learning\\n2022 Tan et al. [30] PM2.5 prediction Sarsa\\n2022 Qin et al. [87] unit commitment problem deep Q-Network\\n2022 Sogabe et al. [88] smart energy optimization\\nand risk evaluationQ-learning\\n2022 Sharma et al. [56] estimating reference evapo-\\ntranspirationQ-learning\\n2022 He et al. [89] wind farm control deep deterministic policy\\ngradient\\n2022 Jalali et al. [90] solar irradiance forecasting Q-learning\\n2022 Shi Yin and Hui Liu\\n[65]wind power prediction Q-learning\\n2023 Yu et al. [28] wind power prediction deep deterministic policy\\ngradient\\n4.2. Internet of Things and Cloud Computing Area\\nERL can be well applied in the Internet of Things (IoT) and cloud computing area and is\\noften used to optimize the working state of a system or business processing capabilities. The\\n19 IoT connects sensors, smart terminals, and industrial systems with the Internet into a global\\nsystem. Optimizing the e\\u000eciency of devices and facilities has a positive e\\u000bect on improving\\nthe overall performance of IoT systems. While cloud computing is a technology closely\\nrelated to the IoT. Speci\\fcally, cloud computing is a model in which users in a distributed\\nsystem can access computing resources or services provided by a cloud platform through the\\nnetwork on demand. Resource allocation and optimization have been the focus of research\\nin the IoT and cloud computing area. Table 4 gives the applications of ERL methods in this\\narea. The o\\u000fine algorithm is used in most of the studies except Polyzos et al. [91] which\\nuses an online algorithm. The performance of the ERL method is veri\\fed on the simulation\\nplatform [92]. The experimental results show that the use of the ensemble model makes the\\nERL method schedule signi\\fcantly better than compared RL methods. How to match the\\napplication requirements of multi-agent and distributed architecture is a core point when\\napplying ERL in this area. Such a system architecture allows the ensemble models in ERL\\nto handle the same or di\\u000berent tasks.\\nTable 4: Application in IoT and cloud computing area\\nYear Authors Problem Training Algorithm\\n2020 Ashiquzzaman et al.\\n[93]IoT sensor calibration deep Q-Network\\n2021 Polyzos et al. [91] resource allocation Sarsa\\n2021 Jiang et al. [19] large-scale MEC systems deep Q-Network\\n2021 Gu et al. [94] online cloud task scheduler deep deterministic policy\\ngradient\\n2021 Liu et al. [95] deep reinforcement learn-\\ning training on GPU cloud\\nplatformactor-critic network\\n2022 Mahmud et al. [96] non orthogonal multiple ac-\\ncess unmanned aerial net-\\nworkdeep Q-Network\\n20 4.3. Financial Area\\nIn the \\fnancial area, problems such as pricing \\fnancial products and portfolio optimiza-\\ntion often face complex decisions. Single models can make predictions on a speci\\fc problem,\\nbut the generalization is a\\u000bected by the problem scenario. Ensemble reinforcement learning\\nmethods, on the other hand, o\\u000ber new options for solving problems in \\fnance. The appli-\\ncation of ERL methods in the \\fnance area is shown in Table 5. The percentage of studies\\nusing only one training algorithm is 67%, while the rest of the studies use multiple training\\nalgorithms in an ERL method. Three algorithms, proximal policy optimization, advantage\\nactor-critic, and deep deterministic policy gradient, can be well implemented in the training\\nalgorithm.\\nTable 5: Application in \\fnancial area\\nYear Authors Problem Training Algorithm\\n2020 Yang et al. [14] stock trading proximal policy optimiza-\\ntion, advantage actor-\\ncritic, deep deterministic\\npolicy gradient\\n2020 Xu et al. [97] fuel economy improvement Q-learning\\n2021 Carta et al. [53] stock market forecasting deep Q-Network\\n2022 Li et al. [62] regional GDP prediction deep Q-Network\\n2022 Zijie Cao and Hui Liu\\n[63]carbon price forecasting Q-learning\\n2022 N\\u0013 emeth, Marcell and\\nSz} ucs, G\\u0013 abor [73]algorithmic trading proximal policy optimiza-\\ntion, advantage actor-\\ncritic, deep deterministic\\npolicy gradient\\n4.4. Other Areas\\nIn addition to the above three classic application areas, ERL has also had successful\\nattempts in other areas such as transportation, medical, and security. In this part, the\\napplications of ERL in other areas are discussed, and Table 6 gives the applications of ERL\\n21 in other areas. Predictions are the main types of problems tackled by ERL methods, and\\nthere are also a few classi\\fcation problems such as diagnosis and recognition that try to\\nsolve using ensemble ideas. Eriksson et al. [72] used ERL methods on autonomous driving\\nproblems is well worth further attention. If the ERL method can be well applied to the\\nautomatic assisted driving of small cars, it can be believed that it will trigger a new research\\nand application boom.\\nTable 6: Application in other areas\\nYear Authors Problem Area\\n2021 Ghosh et al. [40] air tra\\u000ec control tra\\u000ec\\n2021 Dong et al. [57] tra\\u000ec speed forecasting tra\\u000ec\\n2022 Shang et al. [29] tra\\u000ec volume forecasting tra\\u000ec\\n2022 Qi et al. [44] tra\\u000ec signal control tra\\u000ec\\n2022 Eriksson et al. [72] autonomous driving tra\\u000ec\\n2016 Tang et al. [98] symptom checker medicine\\n2021 Jalali et al. [75] COVID-19 diagnosis medicine\\n2022 Birman et al. [64] malware detection security\\n2022 Li et al. [55] rumor tracking security\\n2023 Henna et al. [99] FSO\\/RF communication\\nsystemsoptics\\n2019 Cuay\\u0013 ahuitl et al. [100] chatbots dialogue system\\n2010 Alexander Hans and\\nSte\\u000ben Udluft [36]pole balancing engineering control\\n2018 Ferreira et al. [69] cognitive satellite commu-\\nnicationaerospace\\nIn the future, we believe that there will be more areas using ERL methods for complex\\ntasks. The existing research results can be used as a reference for subsequent research,\\nto improve the existing algorithms to overcome their limitations or to extend the problem\\nto obtain valuable conclusions. In the next section, we will discuss some directions where\\nfurther research can be conducted on ERL methods.\\n22 5. Open Questions and Future Research Directions\\n5.1. Open Questions\\nIn this section, we outline three open questions in ERL-based research. The discussion\\ncan contribute to the future development of ERL (see Figure 7).\\nFigure 7: Open Questions\\n5.1.1. Which Models to Choose?\\nModels are the basis for constructing ERL methods and have a direct impact on the \\fnal\\noutput. The most critical aspect of model selection is the ability to feature selection and\\nlearning. If a model cannot learn valid information, it is meaningless. Ensemble methods\\nthat use multiple same or di\\u000berent models can reduce the possibility of incorrect inference\\nand improve the overall predictive performance. Models implemented should be bene\\fcial\\nto the overall predictive performance ERL method.\\nThe models implemented in ERL mainly include ML and ANN models, which have\\na simple structure and strong generalization ability and are suitable for some practical\\napplications. If the data scale is large and there are many features, ML models may be\\ndi\\u000ecult to handle. In this case, it is easier for the ANN model to get features from the\\ndataset and get the prediction results close to the actual value. However, some tasks that\\nneed to be understood are di\\u000ecult for ANN models. Some studies such as [54], [64] have\\nconsidered implementing ML models and ANN models together into ERL methods.\\n23 Using models with di\\u000berent training stages simultaneously is much easier for ERL design\\n[53]. There are di\\u000berences in the parameters within these models, and the predicted bias and\\nvariance are also distinctive. If such an ensemble model is used, it is worthwhile to investigate\\nin depth what conditions are chosen to save each ensemble element in the training process.\\n5.1.2. Which Training Algorithms to Use?\\nModel training is a well-known challenge in the ERL method. For model-free based ERL,\\nthe agent updates model parameters based on state transfer or trajectory after a certain\\nnumber of iterations. But there is no guarantee that all the data obtained by sampling are\\nuseful. The classic strategy is to use experience delay technology, which can improve the\\nsampling e\\u000eciency. While Schaul et al. [101] found that the sampling ine\\u000eciency problem\\noccurs when the sample in the relay bu\\u000ber is useless. Selected experience relay makes the\\nalgorithm training more e\\u000ecient by selectively choosing the adopted data into the replay\\nbu\\u000ber [102]. It is worth noting that the experience relay bu\\u000ber is only applicable to the\\no\\u000b-policy RL approach. On the other hand, the model-based RL approach can guarantee\\nsampling e\\u000eciency by learning the environment model. However, such a training algorithm\\nhas to face a huge action space. In this case, approximating the environment model becomes\\nan extremely di\\u000ecult task.\\nGood training algorithms should balance exploration and exploitation, while it is di\\u000ecult\\nto achieve by using only one. Therefore, using multiple types of training algorithms simul-\\ntaneously should be the general trend for future development. This ERL method requires\\ndesigning the sampling technique of the solution space according to the training strategy\\nseparately. How to use sampled data from the same type of strategy for model training pro-\\ncesses is also worth considering. In addition, such algorithm training will have high demands\\non the machine's CPU and GPU computing ability.\\n24 5.1.3. Computing Overhead\\nComputing overhead is another issue that is closely related to the \\frst two problems and\\nhas to be considered for ERL. Compared to a single model, multiple ML or ANN models\\nimplemented in ERL make the number of parameters ultra-large. Especially when each\\nANN model has a complex structure, memory consumption becomes a factor that cannot\\nbe ignored. Similarly, multiple training algorithms can complicate the training process.\\nEven when using techniques related to computational acceleration, the time consumed by\\na large number of computations can be signi\\fcantly longer than that of the individual\\ntraining. Many studies have found that ERL methods can complete sampling e\\u000eciently,\\nbut are also accompanied by an increase in computation time [15, 91]. In the testing stage,\\ncomplex decisions then easily lead to longer computation time than other methods [103].\\nTargeted design can reduce the computational overhead to some extent. An et al. achieved\\na reduction in model training time along with a reduction in memory consumption by taking\\nuncertainty into account in the ERL method [38]. Pan et al. reduced the time consumed by\\nthe algorithm for each iteration of training by fuzzifying the reward [45]. While the scale of\\nthese successfully computationally cost-reducing research models is still small, it is di\\u000ecult\\nto show that the improvement strategy is still applicable to large-scale models. In addition,\\nthe cost of data interaction needs extra attention when the models are deployed on multiple\\nmachines, which will a\\u000bect the e\\u000eciency of the system.\\nThe cost-e\\u000bectiveness of ERL using complex structures and training processes, i.e., the\\nadditional computational resources consumed by ERL methods compared to RL methods\\nfor problem processing ability improvement, is the basis for measuring the method design\\nand algorithm training. The increase in the number of models leads to an improvement\\nin the ensemble prediction performance, which is also accompanied by an increase in the\\ncomputing overhead. After a certain number of models are implemented, the computing\\noverhead of using more models can be signi\\fcantly more than the improvement in method\\n25 performance. When this happens, it is not wise to increase the size of the ensemble model.\\nIn some practical application problems, some scenarios allow the feasible solutions ob-\\ntained by ERL methods to be used as the results of problem-solving. Appropriate control of\\nthe number of iterations of the training process is an alternative if the computing overhead\\nof searching for the optimal strategy is much greater than the contribution it can make.\\n5.2. Future Research Directions\\nThere has been a lot of valuable research that uses ERL well to solve problems in science\\nand multiple application areas. An analysis of the year of publication of the available relevant\\nliterature shows that most of the research is concentrated within the last decade. There are\\nstill many directions waiting to be explored by researchers. Here, some potential research\\ndirections are given.\\n1.Hierarchical ensemble : Hierarchical reinforcement learning methods can be used\\nto solve some challenging problems. Qin et al. and Ferreira et al. respectively tried to use\\nmultiple RL models to complete di\\u000berent tasks separately in order [30, 68, 69]. The current\\nmodel structure is designed from experience and lacks systematic theoretical validation.\\nThe performance of the single RL model alone and the ensemble model should be weighed\\nin hierarchical ensemble reinforcement learning. The role played by each element in the\\nhierarchical framework also deserves a detailed design according to speci\\fc problems.\\n2.Large-scale ensemble : The number of base learners used in existing ERL ap-\\nproaches is generally about three [59, 61]. From the diversity perspective, a larger number\\nof models constituting a new ERL method can explore a large amount of feature informa-\\ntion and make accurate predictions. And from the statistical point of view, the increase of\\nensemble components can make more hypotheses and increase the chances of \\fnding the op-\\ntimal hypothesis. The use of large-scale ERL can design the information-sharing mechanism\\nbetween base learners and reduce the total training cost of the model.\\n26 3.Distributed approach : Ensemble reinforcement learning can also be trained or\\nused in a distributed manner. Existing distributed ERL-based research only uses ERL\\nin a distributed framework and lacks methodological improvements [19, 72]. How to take\\nadvantage of both ERL and distributed reinforcement learning deserves further analysis.\\nThe implementation of ERL into a distributed framework will inevitably lead to increased\\nmodel training and communication costs. In a distributed framework, focusing on low-cost\\ntraining methods and controlled training time is necessary to ensure that ERL is applied to\\npractical scenarios.\\n4.Online model training : Currently, ERL adopts o\\u000fine training and direct online\\nuse. If the model needs to update, it has to be trained o\\u000fine again. Such a model training\\napproach will make it di\\u000ecult for the model to grasp the latest situation. Accordingly,\\nthe strategy adopted by the agent is not optimal. If online or near-online model training\\nmethods can be used, new information will be added to the training dataset in time to ensure\\nthat the model can respond to the latest situation. Online training needs to focus on the\\ntriggering mechanism of model training, Too much or too little training can have a negative\\nimpact on the performance of the ERL method. Moreover, forgetting history memory can\\nalso help ERL \\fnd new optimal strategies.\\n5.E\\u000ecient training : The sampling e\\u000eciency of DRL also deserves attention, as this\\nproblem is still prevalent in ERL methods. This raises many related problems, such as how\\nto split the data set for training, initialize the model parameters, set the hyperparameters,\\nand update the strategy. Models belonging to di\\u000berent training stages can also be used\\ntogether to \\fnd the optimal combination of model con\\fgurations [53].\\nFuture research can be carried out in the above but not limited to these aspects. Several\\nnew situations will also be encountered when designing methods and solving problems. No\\nfree lunch theorem applies to any ERL method. It means the complexity of the ensemble\\nmodel and the training time needs to be taken into account in the method design [33].\\n27 6. Conclusion\\nThis paper reviews the research progress on ensemble reinforcement learning methods\\nfrom several aspects. The description of reinforcement learning methods and ensemble\\nlearning methods can enhance the understanding of ERL. Various strategies such as Q-\\nfunction ensemble, model combination, and decision strategies have helped ERL to be more\\ne\\u000bective. In addition, we discuss future research directions. ERL's powerful predictive or\\nclassi\\fcation capability makes it a promising framework for solving complex problems. We\\nbelieve that ERL can achieve satisfactory performance in more application areas in the\\nfuture.\\nReferences\\n[1] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller,\\nA. K. Fidjeland, G. Ostrovski, et al., Human-level control through deep reinforcement learning, nature\\n518 (7540) (2015) 529{533.\\n[2] D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser,\\nI. Antonoglou, V. Panneershelvam, M. Lanctot, et al., Mastering the game of go with deep neural\\nnetworks and tree search, nature 529 (7587) (2016) 484{489.\\n[3] O. Vinyals, I. Babuschkin, W. M. Czarnecki, M. Mathieu, A. Dudzik, J. Chung, D. H. Choi, R. Powell,\\nT. Ewalds, P. Georgiev, et al., Grandmaster level in starcraft ii using multi-agent reinforcement\\nlearning, Nature 575 (7782) (2019) 350{354.\\n[4]  L. Kaiser, M. Babaeizadeh, P. Mi los, B. Osi\\u0013 nski, R. H. Campbell, K. Czechowski, D. Erhan, C. Finn,\\nP. Kozakowski, S. Levine, et al., Model based reinforcement learning for atari, in: International\\nConference on Learning Representations, 2020.\\n[5] R. Liu, F. Nageotte, P. Zanne, M. de Mathelin, B. Dresp-Langley, Deep reinforcement learning for\\nthe control of robotic manipulation: a focussed mini-review, Robotics 10 (1) (2021) 22.\\n[6] S. Fujimoto, H. Hoof, D. Meger, Addressing function approximation error in actor-critic methods, in:\\nInternational conference on machine learning, PMLR, 2018, pp. 1587{1596.\\n[7] A. Kumar, J. Fu, M. Soh, G. Tucker, S. Levine, Stabilizing o\\u000b-policy q-learning via bootstrapping\\nerror reduction, Advances in Neural Information Processing Systems 32 (2019).\\n28 [8] R. Y. Chen, S. Sidor, P. Abbeel, J. Schulman, Ucb exploration via q-ensembles, arXiv preprint\\narXiv:1706.01502 (2017).\\n[9] M. d. Condorcet, Essay on the application of analysis to the probability of majority decisions, Paris:\\nImprimerie Royale (1785).\\n[10] A. Krogh, J. Vedelsby, Neural network ensembles, cross validation, and active learning, Advances in\\nneural information processing systems 7 (1995) 231{238.\\n[11] L. Breiman, Random forests, Machine learning 45 (2001) 5{32.\\n[12] G. Brown, J. Wyatt, R. Harris, X. Yao, Diversity creation methods: a survey and categorisation,\\nInformation fusion 6 (1) (2005) 5{20.\\n[13] T. G. Dietterich, Ensemble methods in machine learning, in: Multiple Classi\\fer Systems: First In-\\nternational Workshop, MCS 2000 Cagliari, Italy, June 21{23, 2000 Proceedings 1, Springer, 2000, pp.\\n1{15.\\n[14] H. Yang, X.-Y. Liu, S. Zhong, A. Walid, Deep reinforcement learning for automated stock trading: An\\nensemble strategy, in: Proceedings of the \\frst ACM international conference on AI in \\fnance, 2020,\\npp. 1{8.\\n[15] H. Sheikh, M. Phielipp, L. Boloni, Maximizing ensemble diversity in deep reinforcement learning, in:\\nInternational Conference on Learning Representations, 2022.\\n[16] X. Chen, C. Wang, Z. Zhou, K. W. Ross, Randomized ensembled double q-learning: Learning fast\\nwithout a model, in: International Conference on Learning Representations, 2021.\\n[17] S. Fau\\u0019er, F. Schwenker, Ensemble methods for reinforcement learning with function approximation,\\nin: Multiple Classi\\fer Systems: 10th International Workshop, MCS 2011, Naples, Italy, June 15-17,\\n2011. Proceedings 10, Springer, 2011, pp. 56{65.\\n[18] O. Anschel, N. Baram, N. Shimkin, Averaged-dqn: Variance reduction and stabilization for deep\\nreinforcement learning, in: International conference on machine learning, PMLR, 2017, pp. 176{185.\\n[19] F. Jiang, L. Dong, K. Wang, K. Yang, C. Pan, Distributed resource scheduling for large-scale mec sys-\\ntems: A multiagent ensemble deep reinforcement learning with imitation acceleration, IEEE Internet\\nof Things Journal 9 (9) (2021) 6597{6610.\\n[20] C. J. Watkins, P. Dayan, Q-learning, Machine learning 8 (1992) 279{292.\\n[21] K. Arulkumaran, M. P. Deisenroth, M. Brundage, A. A. Bharath, Deep reinforcement learning: A\\nbrief survey, IEEE Signal Processing Magazine 34 (6) (2017) 26{38.\\n[22] L. Breiman, Bagging predictors, Machine learning 24 (1996) 123{140.\\n29 [23] R. E. Schapire, The boosting approach to machine learning: An overview, Nonlinear estimation and\\nclassi\\fcation (2003) 149{171.\\n[24] D. H. Wolpert, Stacked generalization, Neural networks 5 (2) (1992) 241{259.\\n[25] G. Brown, J. L. Wyatt, P. Tino, Y. Bengio, Managing diversity in regression ensembles., Journal of\\nmachine learning research 6 (9) (2005).\\n[26] S. Geman, E. Bienenstock, R. Doursat, Neural networks and the bias\\/variance dilemma, Neural\\ncomputation 4 (1) (1992) 1{58.\\n[27] J. Smit, C. T. Ponnambalam, M. T. Spaan, F. A. Oliehoek, Pebl: Pessimistic ensembles for o\\u000fine\\ndeep reinforcement learning, in: Robust and Reliable Autonomy in the Wild Workshop at the 30th\\nInternational Joint Conference of Arti\\fcial Intelligence, 2021.\\n[28] Y. Chengqing, Y. Guangxi, Y. Chengming, Z. Yu, M. Xiwei, A multi-factor driven spatiotemporal\\nwind power prediction model based on ensemble deep graph attention reinforcement learning networks,\\nEnergy 263 (2023) 126034.\\n[29] P. Shang, X. Liu, C. Yu, G. Yan, Q. Xiang, X. Mi, A new ensemble deep graph reinforcement learning\\nnetwork for spatio-temporal tra\\u000ec volume forecasting in a freeway network, Digital Signal Processing\\n123 (2022) 103419.\\n[30] J. Tan, H. Liu, Y. Li, S. Yin, C. Yu, A new ensemble spatio-temporal pm2.5 prediction method based\\non graph attention recursive networks and reinforcement learning, Chaos, Solitons & Fractals 162\\n(2022) 112405.\\n[31] I. Partalas, G. Tsoumakas, I. Katakis, I. Vlahavas, Ensemble pruning using reinforcement learning,\\nin: Advances in Arti\\fcial Intelligence: 4th Helenic Conference on AI, SETN 2006, Heraklion, Crete,\\nGreece, May 18-20, 2006. Proceedings 4, Springer, 2006, pp. 301{310.\\n[32] Z. Liu, K. Ramamohanarao, Instance-based ensemble selection using deep reinforcement learning, in:\\n2020 International Joint Conference on Neural Networks (IJCNN), IEEE, 2020, pp. 1{7.\\n[33] R. S. Sutton, A. G. Barto, Reinforcement learning: An introduction, MIT press, 2018.\\n[34] I. Osband, C. Blundell, A. Pritzel, B. Van Roy, Deep exploration via bootstrapped dqn, Advances in\\nneural information processing systems 29 (2016).\\n[35] X.-l. Chen, L. Cao, C.-x. Li, Z.-x. Xu, J. Lai, Ensemble network architecture for deep reinforcement\\nlearning, Mathematical Problems in Engineering 2018 (2018).\\n[36] A. Hans, S. Udluft, Ensembles of neural networks for robust reinforcement learning, in: 2010 Ninth\\nInternational Conference on Machine Learning and Applications, IEEE, 2010, pp. 401{406.\\n30 [37] Q. He, H. Su, C. Gong, X. Hou, Mepg: A minimalist ensemble policy gradient framework for deep\\nreinforcement learning, arXiv preprint arXiv:2109.10552 (2021).\\n[38] G. An, S. Moon, J.-H. Kim, H. O. Song, Uncertainty-based o\\u000fine reinforcement learning with diver-\\nsi\\fed q-ensemble, Advances in neural information processing systems 34 (2021) 7436{7447.\\n[39] Q. Lan, Y. Pan, A. Fyshe, M. White, Maxmin q-learning: Controlling the estimation bias of q-learning,\\narXiv preprint arXiv:2002.06487 (2020).\\n[40] S. Ghosh, S. Laguna, S. H. Lim, L. Wynter, H. Poonawala, A deep ensemble method for multi-\\nagent reinforcement learning: A case study on air tra\\u000ec control, in: Proceedings of the International\\nConference on Automated Planning and Scheduling, Vol. 31, 2021, pp. 468{476.\\n[41] D. Ormoneit, A. Sen, Kernel-based reinforcement learning, Machine learning 49 (2-3) (2002) 161.\\n[42] Y. Yao, L. Xiao, Z. An, W. Zhang, D. Luo, Sample e\\u000ecient reinforcement learning via model-ensemble\\nexploration and exploitation, in: 2021 IEEE International Conference on Robotics and Automation\\n(ICRA), IEEE, 2021, pp. 4202{4208.\\n[43] J.-L. Lin, K.-S. Hwang, H. Shi, W. Pan, An ensemble method for inverse reinforcement learning,\\nInformation Sciences 512 (2020) 518{532.\\n[44] R. Qi, J. Huang, H. Li, Q. Tan, L. Huang, J. Cui, Random ensemble reinforcement learning for tra\\u000ec\\nsignal control, arXiv preprint arXiv:2203.05961 (2022).\\n[45] W. Pan, R. Qu, K.-S. Hwang, H.-S. Lin, An ensemble fuzzy approach for inverse reinforcement learn-\\ning, International Journal of Fuzzy Systems 21 (2019) 95{103.\\n[46] S. Adebola, S. Sharma, K. Shivakumar, Deft: Diverse ensembles for fast transfer in reinforcement\\nlearning, arXiv preprint arXiv:2209.12412 (2022).\\n[47] G. Chen, Y. Peng, M. Zhang, E\\u000bective exploration for deep reinforcement learning via bootstrapped\\nq-ensembles under tsallis entropy regularization, arXiv preprint arXiv:1809.00403 (2018).\\n[48] Y. Sun, P. Fazli, Ensemble policy distillation in deep reinforcement learning, in: Workshop on Rein-\\nforcement Learning in Games, 2020, pp. 1{9.\\n[49] J. Wang, J. Hu, J. Mills, G. Min, M. Xia, Federated ensemble model-based reinforcement learning in\\nedge computing, arXiv preprint arXiv:2109.05549 (2021).\\n[50] S. Lee, Y. Seo, K. Lee, P. Abbeel, J. Shin, O\\u000fine-to-online reinforcement learning via balanced replay\\nand pessimistic q-ensemble, in: Conference on Robot Learning, PMLR, 2022, pp. 1702{1712.\\n[51] Z. Yang, K. Ren, X. Luo, M. Liu, W. Liu, J. Bian, W. Zhang, D. Li, Towards applicable reinforcement\\nlearning: Improving the generalization and sample e\\u000eciency with policy ensemble, arXiv preprint\\n31 arXiv:2205.09284 (2022).\\n[52] A. Goyal, S. Sodhani, J. Binas, X. B. Peng, S. Levine, Y. Bengio, Reinforcement learning with\\ncompetitive ensembles of information-constrained primitives, arXiv preprint arXiv:1906.10667 (2019).\\n[53] S. Carta, A. Ferreira, A. S. Podda, D. R. Recupero, A. Sanna, Multi-dqn: An ensemble of deep\\nq-learning agents for stock market forecasting, Expert systems with applications 164 (2021) 113820.\\n[54] A. Saadallah, K. Morik, Online ensemble aggregation using deep reinforcement learning for time series\\nforecasting, in: 2021 IEEE 8th International Conference on Data Science and Advanced Analytics\\n(DSAA), IEEE, 2021, pp. 1{8.\\n[55] G. Li, M. Dong, L. Ming, C. Luo, H. Yu, X. Hu, B. Zheng, Deep reinforcement learning based ensemble\\nmodel for rumor tracking, Information Systems 103 (2022) 101772.\\n[56] G. Sharma, A. Singh, S. Jain, Deepevap: Deep reinforcement learning based ensemble approach for\\nestimating reference evapotranspiration, Applied Soft Computing 125 (2022) 109113.\\n[57] S. Dong, C. Yu, G. Yan, J. Zhu, H. Hu, A novel ensemble reinforcement learning gated recursive\\nnetwork for tra\\u000ec speed forecasting, in: 2021 Workshop on Algorithm and Big Data, 2021, pp. 55{60.\\n[58] H. Liu, C. Yu, H. Wu, Z. Duan, G. Yan, A new hybrid ensemble deep reinforcement learning model\\nfor wind speed short term forecasting, Energy 202 (2020) 117794.\\n[59] S. K. Perepu, B. S. Balaji, H. K. Tanneru, S. Kathari, V. S. Pinnamaraju, Reinforcement\\nlearning based dynamic weighing of ensemble models for time series forecasting, arXiv preprint\\narXiv:2008.08878 (2020).\\n[60] X. Liu, M. Qin, Y. He, X. Mi, C. Yu, A new multi-data-driven spatiotemporal pm2. 5 forecasting model\\nbased on an ensemble graph reinforcement learning convolutional network, Atmospheric Pollution\\nResearch 12 (10) (2021) 101197.\\n[61] D. L. Elliott, C. Anderson, The wisdom of the crowd: Reliable deep reinforcement learning through\\nensembles of q-functions, IEEE transactions on neural networks and learning systems (2021).\\n[62] Q. Li, C. Yu, G. Yan, A new multipredictor ensemble decision framework based on deep reinforcement\\nlearning for regional gdp prediction, IEEE Access 10 (2022) 45266{45279.\\n[63] Z. Cao, H. Liu, A novel carbon price forecasting method based on model matching, adaptive decompo-\\nsition, and reinforcement learning ensemble strategy, Environmental Science and Pollution Research\\n(2022) 1{24.\\n[64] Y. Birman, S. Hindi, G. Katz, A. Shabtai, Cost-e\\u000bective ensemble models selection using deep rein-\\nforcement learning, Information Fusion 77 (2022) 133{148.\\n32 [65] S. Yin, H. Liu, Wind power prediction based on outlier correction, ensemble reinforcement learning,\\nand residual correction, Energy 250 (2022) 123857.\\n[66] F. Schubert, C. Benjamins, S. D\u007f ohler, B. Rosenhahn, M. Lindauer, Polter: Policy trajectory ensemble\\nregularization for unsupervised reinforcement learning, arXiv preprint arXiv:2205.11357 (2022).\\n[67] M. Shen, J. P. How, Robust opponent modeling via adversarial ensemble reinforcement learning in\\nasymmetric imperfect-information games, arXiv preprint arXiv:1909.08735 (2019).\\n[68] Y. Qin, Z. Wang, C. Chen, Hrl2e: Hierarchical reinforcement learning with low-level ensemble, in:\\n2022 International Joint Conference on Neural Networks (IJCNN), IEEE, 2022, pp. 1{7.\\n[69] P. V. R. Ferreira, R. Pa\\u000benroth, A. M. Wyglinski, T. M. Hackett, S. G. Bil\\u0013 en, R. C. Reinhart,\\nD. J. Mortensen, Multiobjective reinforcement learning for cognitive satellite communications using\\ndeep neural network ensembles, IEEE Journal on Selected Areas in Communications 36 (5) (2018)\\n1030{1041.\\n[70] M. A. Wiering, H. Van Hasselt, Ensemble algorithms in reinforcement learning, IEEE Transactions\\non Systems, Man, and Cybernetics, Part B (Cybernetics) 38 (4) (2008) 930{936.\\n[71] R. Saphal, B. Ravindran, D. Mudigere, S. Avancha, B. Kaul, Seerl: Sample e\\u000ecient ensemble rein-\\nforcement learning, arXiv preprint arXiv:2001.05209 (2020).\\n[72] H. Eriksson, D. Basu, M. Alibeigi, C. Dimitrakakis, Sentinel: taming uncertainty with ensemble based\\ndistributional reinforcement learning, in: Uncertainty in Arti\\fcial Intelligence, PMLR, 2022, pp. 631{\\n640.\\n[73] M. N\\u0013 emeth, G. Sz} ucs, Split feature space ensemble method using deep reinforcement learning for\\nalgorithmic trading, in: Proceedings of the 2022 8th International Conference on Computer Technology\\nApplications, 2022, pp. 188{194.\\n[74] S. Fau\\u0019er, F. Schwenker, Neural network ensembles in reinforcement learning, Neural Processing\\nLetters 41 (2015) 55{69.\\n[75] S. M. J. Jalali, M. Ahmadian, S. Ahmadian, A. Khosravi, M. Alazab, S. Nahavandi, An oppositional-\\ncauchy based gsk evolutionary algorithm with a novel deep ensemble reinforcement learning strategy\\nfor covid-19 diagnosis, Applied Soft Computing 111 (2021) 107675.\\n[76] J. Wu, H. Li, Deep ensemble reinforcement learning with multiple deep deterministic policy gradient\\nalgorithm, Mathematical Problems in Engineering 2020 (2020) 1{12.\\n[77] H. Sheikh, K. Frisbee, M. Phielipp, Dns: Determinantal point process based neural network sampler\\nfor ensemble reinforcement learning, in: International Conference on Machine Learning, PMLR, 2022,\\n33 pp. 19731{19746.\\n[78] J. Buckman, D. Hafner, G. Tucker, E. Brevdo, H. Lee, Sample-e\\u000ecient reinforcement learning with\\nstochastic ensemble value expansion, Advances in neural information processing systems 31 (2018).\\n[79] O. Peer, C. Tessler, N. Merlis, R. Meir, Ensemble bootstrapping for q-learning, in: International\\nConference on Machine Learning, PMLR, 2021, pp. 8454{8463.\\n[80] A. Brown, M. Petrik, Interpretable reinforcement learning with ensemble methods, arXiv preprint\\narXiv:1809.06995 (2018).\\n[81] U. Pak, J. Ma, U. Ryu, K. Ryom, U. Juhyok, K. Pak, C. Pak, Deep learning-based pm2. 5 prediction\\nconsidering the spatiotemporal correlations: A case study of beijing, china, Science of The Total\\nEnvironment 699 (2020) 133561.\\n[82] J. Ma, Z. Yu, Y. Qu, J. Xu, Y. Cao, et al., Application of the xgboost machine learning method in\\npm2. 5 prediction: A case study of shanghai, Aerosol and Air Quality Research 20 (1) (2020) 128{138.\\n[83] S. M. J. Jalali, M. Khodayar, S. Ahmadian, M. Sha\\fe-Khah, A. Khosravi, S. M. S. Islam, S. Nahavandi,\\nJ. P. Catal~ ao, A new ensemble reinforcement learning strategy for solar irradiance forecasting using\\ndeep optimized convolutional neural network models, in: 2021 International Conference on Smart\\nEnergy Systems and Technologies (SEST), IEEE, 2021, pp. 1{6.\\n[84] Y. Li, Z. Liu, H. Liu, A novel ensemble reinforcement learning gated unit model for daily pm2. 5\\nforecasting, Air Quality, Atmosphere & Health 14 (2021) 443{453.\\n[85] C. Chen, H. Liu, Dynamic ensemble wind speed prediction model based on hybrid deep reinforcement\\nlearning, Advanced Engineering Informatics 48 (2021) 101290.\\n[86] S. M. J. Jalali, G. J. Os\\u0013 orio, S. Ahmadian, M. Lot\\f, V. M. Campos, M. Sha\\fe-khah, A. Khosravi, J. P.\\nCatal~ ao, New hybrid deep neural architectural search-based ensemble reinforcement learning strategy\\nfor wind power forecasting, IEEE Transactions on Industry Applications 58 (1) (2021) 15{27.\\n[87] J. Qin, Y. Gao, M. Bragin, N. Yu, An optimization method-assisted ensemble deep reinforcement\\nlearning algorithm to solve unit commitment problems, arXiv preprint arXiv:2206.04249 (2022).\\n[88] T. Sogabe, D. B. Malla, C.-C. Chen, K. Sakamoto, Attention and masking embedded ensemble rein-\\nforcement learning for smart energy optimization and risk evaluation under uncertainties, Journal of\\nRenewable and Sustainable Energy 14 (4) (2022) 045501.\\n[89] B. He, H. Zhao, G. Liang, J. Zhao, J. Qiu, Z. Y. Dong, Ensemble-based deep reinforcement learning\\nfor robust cooperative wind farm control, International Journal of Electrical Power & Energy Systems\\n143 (2022) 108406.\\n34 [90] S. M. J. Jalali, S. Ahmadian, B. Nakisa, M. Khodayar, A. Khosravi, S. Nahavandi, S. M. S. Is-\\nlam, M. Sha\\fe-khah, J. P. Catal~ ao, Solar irradiance forecasting using a novel hybrid deep ensemble\\nreinforcement learning algorithm, Sustainable Energy, Grids and Networks 32 (2022) 100903.\\n[91] K. D. Polyzos, Q. Lu, A. Sadeghi, G. B. Giannakis, On-policy reinforcement learning via ensemble\\ngaussian processes with application to resource allocation, in: 2021 55th Asilomar Conference on\\nSignals, Systems, and Computers, IEEE, 2021, pp. 1018{1022.\\n[92] A. Sadeghi, F. Sheikholeslami, G. B. Giannakis, Optimal and scalable caching for 5g using reinforce-\\nment learning of space-time popularities, IEEE Journal of Selected Topics in Signal Processing 12 (1)\\n(2017) 180{190.\\n[93] A. Ashiquzzaman, H. Lee, T.-W. Um, J. Kim, Energy-e\\u000ecient iot sensor calibration with deep rein-\\nforcement learning, IEEE Access 8 (2020) 97045{97055.\\n[94] D. Gu, J. Chen, X. Shi, L. Ran, Y. Zhang, M. Shang, Heterogeneous-aware online cloud task scheduler\\nbased on clustering and deep reinforcement learning ensemble, in: Advances in Natural Computation,\\nFuzzy Systems and Knowledge Discovery, Springer, 2021, pp. 152{159.\\n[95] X.-Y. Liu, Z. Li, Z. Yang, J. Zheng, Z. Wang, A. Walid, J. Guo, M. I. Jordan, Elegantrl-podracer: Scal-\\nable and elastic library for cloud-native deep reinforcement learning, arXiv preprint arXiv:2112.05923\\n(2021).\\n[96] S. K. Mahmud, Y. Chen, K. K. Chai, Ensemble reinforcement learning framework for sum rate op-\\ntimization in noma-uav network, in: 2022 IEEE World AI IoT Congress (AIIoT), IEEE, 2022, pp.\\n032{038.\\n[97] B. Xu, X. Hu, X. Tang, X. Lin, H. Li, D. Rathod, Z. Filipi, Ensemble reinforcement learning-based\\nsupervisory control of hybrid electric vehicle for fuel economy improvement, IEEE Transactions on\\nTransportation Electri\\fcation 6 (2) (2020) 717{727.\\n[98] K.-F. Tang, H.-C. Kao, C.-N. Chou, E. Y. Chang, Inquire and diagnose: Neural symptom checking\\nensemble using deep reinforcement learning, in: NIPS workshop on deep reinforcement learning, 2016.\\n[99] S. Henna, A. A. Minhas, M. S. Khan, M. S. Iqbal, Ensemble consensus representation deep reinforce-\\nment learning for hybrid fso\\/rf communication systems, Optics Communications 530 (2023) 129186.\\n[100] H. Cuay\\u0013 ahuitl, D. Lee, S. Ryu, Y. Cho, S. Choi, S. Indurthi, S. Yu, H. Choi, I. Hwang, J. Kim,\\nEnsemble-based deep reinforcement learning for chatbots, Neurocomputing 366 (2019) 118{130.\\n[101] T. Schaul, J. Quan, I. Antonoglou, D. Silver, Prioritized experience replay, arXiv preprint\\narXiv:1511.05952 (2015).\\n35 [102] D. Isele, A. Cosgun, Selective experience replay for lifelong learning, in: Proceedings of the AAAI\\nConference on Arti\\fcial Intelligence, Vol. 32, 2018.\\n[103] I. Partalas, G. Tsoumakas, I. Vlahavas, Pruning an ensemble of classi\\fers via reinforcement learning,\\nNeurocomputing 72 (7-9) (2009) 1900{1909.\\n36\",\"23\":\" Diffusion Models Generate Images Like Painters:\\nan Analytical Theory of Outline First, Details Later\\nBinxu Wang1 2 3 *John J. Vastola1 *\\nAbstract\\nHow do diffusion generative models convert pure\\nnoise into meaningful images? We argue that gen-\\neration involves \\ufb01rst committing to an outline, and\\nthen to \\ufb01ner and \\ufb01ner details. The corresponding\\nreverse diffusion process can be modeled by dy-\\nnamics on a (time-dependent) high-dimensional\\nlandscape full of Gaussian-like modes, which\\nmakes the following predictions: (i) individual\\ntrajectories tend to be very low-dimensional; (ii)\\nscene elements that vary more within training data\\ntend to emerge earlier; and (iii) early perturba-\\ntions substantially change image content more\\noften than late perturbations. We show that the\\nbehavior of a variety of trained unconditional and\\nconditional diffusion models like Stable Diffu-\\nsion is consistent with these predictions. Finally,\\nwe use our theory to search for the latent image\\nmanifold of diffusion models, and propose a new\\nway to generate interpretable image variations.\\nOur viewpoint suggests generation by GANs and\\ndiffusion models have unexpected similarities.\\n1. Introduction\\nImagine an artist painting a picture of a natural landscape.\\nWe generally expect higher-level scene elements to appear\\n\\ufb01rst, and lower-level details later: the borders of the land and\\nsky might be drawn, then the largest objects (like mountains\\nand trees) might be placed, then minor objects (like rocks\\nand small animals) might be placed, and \\ufb01nally \\ufb01ne details\\n(like textures and shading) might be \\ufb01lled in. Do diffusion\\ngenerative models (Sohl-Dickstein et al., 2015; Song &\\nErmon, 2019; Song et al., 2021), which can generate natural\\nlandscapes like those of an artist, also construct images like\\n*Equal contribution1Department of Neurobiology, Har-\\nvard Medical School, Boston, MA, USA2Department of\\nNeuroscience, Washington University in St Louis, St Louis,\\nMO, USA3Arti\\ufb01cio, Cambridge, USA. Correspondence to:\\nBinxu Wang <binxu wang@hms.harvard.edu >, John Vastola\\n<John Vastola@hms.harvard.edu >.\\nPreprint. Work in Progress.this? If not, how do they work?\\nBy visualizing images throughout a reverse diffusion\\ntrajectory\\u2014from pure noise to the \\ufb01nal image\\u2014the naive\\nanswer appears to be no. One gets the impression of an\\nimage emerging fully-formed from the noise; one is \\u2018un-\\ncovering\\u2019 the image, or \\u2018opening one\\u2019s eyes\\u2019 to reveal an\\nimage that was always there. But this naive answer is highly\\nmisleading, and we will later argue that it is entirely wrong.\\nOne signi\\ufb01cant step towards understanding this question\\nwas provided by Hertz et al., who found that models\\u2019\\ncross-attention maps substantially control scene composi-\\ntion (2022). They noted that different parts of an image\\nmay be \\u2018attended to\\u2019 at different times in a reverse diffusion\\ntrajectory. But it is unclear if attention mechanisms are nec-\\nessary for image generation to be like painting, or if this is\\na generic property of diffusion models.\\nUnderstanding image generation is important for a few rea-\\nsons. As Hertz et al. show, improving understanding can\\nmake the process more controllable, and hence more precise,\\nef\\ufb01cient, and useful. These improvements may be crucial\\nfor generalizing image generation to stable video generation,\\nas some early work has attempted (Ho et al., 2022). Beyond\\nAI, these details could have some bearing on the brain; in\\nthe same way convolutional neural networks (CNNs) are\\nused to test hypotheses about how early visual areas in the\\nbrain might behave, diffusion models may also be useful as\\na toy model of human visual imagination. Some research\\nin psychology suggests that humans imagine visual scenes\\nhierarchically, with important scene elements imagined \\ufb01rst\\nand details imagined later (Kosslyn & Shwartz, 1977).\\nMotivated by a variety of salient observations, in this pa-\\nper we propose a simple but insightful theory of reverse\\ndiffusion based on approximating image distributions as Motivated by a variety of salient observations, in this pa-\\nper we propose a simple but insightful theory of reverse\\ndiffusion based on approximating image distributions as\\nGaussian mixtures. It suggests an outline-\\ufb01rst, details-later\\nview of image generation that begins with a prototype and\\ngradually re\\ufb01nes it. This theoretical view yields several\\ninteresting predictions for how real generation proceeds,\\nincluding that: (i) individual trajectories tend to be very\\nlow-dimensional; (ii) scene elements that vary more within\\ntraining data tend to emerge earlier; and (iii) early pertur-\\nbations substantially change image content more often thanarXiv:2303.02490v1  [cs.CV]  4 Mar 2023 Diffusion Models Generate Images Like Painters\\nFigure 1. Characteristics of image generation by diffusion models .A. Tracking latent states G(xt)(top row), differences between\\nnearby time steps G(k(xt\\u00001\\u0000xt))(middle row), and \\ufb01nal image estimates G(^x0(xt))(bottom row) suggests different measures of\\nprogress. B. Individual trajectories are effectively two-dimensional, with the transition from xTtox0being rotation-like.\\nlate perturbations.\\nThis view also sheds light on an open problem: what is the\\nnature of the low-dimensional latent manifold that, at least\\nin principle, smoothly parameterizes diffusion-generated im-\\nages? For generative adversarial networks (GANs) (Good-\\nfellow et al., 2014), the geometry of this manifold is fairly\\nwell-understood (Wang & Ponce, 2021); for diffusion mod-\\nels, it is unclear \\u2018where\\u2019 it even is. Recently, Wenliang and\\nMoran searched for it by dissecting the Jacobian of the score\\nvector \\ufb01eld (2022). Here, we claim on-manifold directions\\nthat yield meaningful image perturbations can be identi\\ufb01ed\\nvia principal component analysis (PCA) on trajectories.\\nThe paper is organized as follows. In Section 2, we provide\\nnecessary background on diffusion models. In Section 3,\\nwe collect salient observations about image generation, e.g.\\nabout trajectory geometry and feature emergence order. In\\nSection 4, we explain these observations using an analytical\\nmodel of reverse diffusion, and explore some of the model\\u2019s\\nconsequences. In Section 5, we test these predictions in a\\nvariety of trained unconditional diffusion models. In Sec-\\ntion 6, we apply our theoretical view to ef\\ufb01ciently generate\\ninterpretable variations of Stable Diffusion images.\\n2. Diffusion generative modeling basics\\nThere are several complementary theoretical viewpoints on\\ndiffusion generative modeling (Sohl-Dickstein et al., 2015;\\nSong & Ermon, 2019; Yang et al., 2022); here, we review\\nthe details of Song et al.\\u2019s unifying view (2021).\\nDiffusion generative models involve mapping one proba-\\nbility distribution p(x)to another via a stochastic process\\u2014\\ntypically pure diffusion or an Ornstein-Uhlenbeck (OU)\\nprocess. This so-called \\u2018forward\\u2019 process can be inverted by\\nconsidering its \\u2018reverse\\u2019 process, which is mathematically\\nguaranteed to exist for reasonable choices of initial distri-\\nbution and forward process (Anderson, 1982). If p(x)is\\nmapped to a distribution that is easy to sample from, like\\na normal distribution, by sampling from it and running thereverse process one can obtain samples from p(x). (Details\\nfollow; Appendix D relates our notation to others\\u2019 notation.)\\nForward\\/reverse diffusion. A fairly general forward pro-\\ncess is given by the stochastic differential equation (SDE)\\n_x=f(x) +g(t)\\u0011(t) (1)\\nwhereg(t)is a time-dependent noise amplitude, \\u0011(t)is a\\nvector of independent Gaussian white noise terms, and time\\nruns fromt= 0tot=T= 1. The reverse process is\\n_x=f(x)\\u0000g(t)2s(x;t) +g(t)\\u0011(t) (2)\\nwhere s(x;t) :=rxlogp(x;t)is the score function, and\\nwhere we use the standard convention that times runs back-\\nward , i.e. fromt= 1tot= 0. In this paper, we focus on\\none common forward process: the variance-preserving SDE\\n_x=\\u0000\\f(t)x+g(t)\\u0011(t) (3)\\nwhere we enforce the constraint that \\f(t) =1\\n2g2(t). The\\nmarginal probabilities of this process are\\np(xtjx0) =N(xtj\\u000btx0;\\u001b2\\ntI) (4)\\nwhere\\u000btand\\u001b2\\ntsatisfy\\u000b2\\nt+\\u001b2\\nt= 1and are de\\ufb01ned to be\\n\\u000bt:=e\\u0000Rt\\n0\\f(t0)dt0\\u001b2\\nt:= 1\\u0000e\\u00002Rt\\n0\\f(t0)dt0:(5)\\nIn practice, to reverse an SDE like this, we do not simulate\\nthe corresponding reverse process; instead, we simulate a\\nprobability \\ufb02ow ODE with the same marginal probabilities\\n(Song et al., 2021). To reverse Eq. 3, we would use\\n_x=\\u0000\\f(t)x\\u00001\\n2g2(t)s(x;t) (6)\\nwhere time again runs backward from t= 1tot= 0. Diffusion Models Generate Images Like Painters\\nLearning the score function. The score function, which\\nis required to reverse the forward process, can be learned\\nvia gradient descent on a score-matching objective\\nEx0\\u0018p(x0);\\u000f\\u0018N (0;I)Z1\\n0\\rtk\\u000f\\u0012(\\u000btx0+\\u001bt\\u000f;t)\\u0000\\u000fk2\\n2dt(7)\\nwhere \\u000f\\u0012(xt;t)\\u0019 \\u0000\\u001btrxlogp(xt)is a neural network\\napproximation of the negative scaled score function, and \\rt\\nis a positive weighting function (Song et al., 2021).\\nLatent diffusion models. Pixel space representations of\\nimages are extremely high-dimensional, and movement\\nthrough this space often corresponds to changing imper-\\nceptible high-frequency details, making diffusion inef\\ufb01cient.\\nOne alternative uses an auto-encoder ( x$G(x)) to map\\nimages into an expressive latent space, and trains a diffu-\\nsion model in latent rather than pixel space. Models like\\nStable Diffusion leverage this ef\\ufb01ciency increase to scale to\\nmassive data sets at a modest cost (Rombach et al., 2022).\\nDDIM\\/PNDM samplers. Sampling the reverse process is\\nsomewhat independent of score function learning, enabling\\nresearchers to separately study its ef\\ufb01ciency. Ho et al. use\\nthe DDPM sampler (2020), which is effectively the same as\\ndiscretizing a reverse SDE (Eq. 2). Song et al. propose the\\nDDIM sampler (2020), a deterministic sampler equivalent\\nto solving the probability \\ufb02ow ODE (c.f. Eq. 6), which\\ndramatically reduced the required number of steps. The idea\\nof sampling via solving an ODE enables researchers to ex-\\nploit various numerical methods for integrating ODEs; one\\npowerful example is Liu et al.\\u2019s PNDM sampler (2022), the\\ncurrent default sampler for Stable Diffusion. In this work,\\nwe focus on the DDIM\\/PNDM samplers, and comment on\\nhow other samplers affect our results in Section B.6.\\n3. Salient observations about image\\ngeneration\\n3.1. How should we measure generation progress?\\nA common way to monitor image generation progress is\\nto observe how xt(or, in the case of latent diffusion, the\\ndecoded image G(xt)that corresponds to it) changes over\\ntime. As previously mentioned, this approach tends to show\\na fully-formed image unveiled from noise (Fig. 1A, top\\nrow). But is this what is \\u2018actually\\u2019 happening?\\nA simple but useful alternative is to observe (appropriately\\nscaled) differencesk(xt\\u0000k\\u0000xt)between close time points\\u2014\\nthe next \\u2018layer of paint\\u2019 that has been added to the canvas.\\nThese often appear to be like a sketch of the \\ufb01nal outcome\\nearly in generation, and are increasingly contaminated by\\nnoise towards the end (Fig. 1A, middle row).\\nA more principled alternative, \\ufb01rst proposed by Ho et al.(2020), is to consider the sequence of endpoint estimates\\n^x0of the reverse diffusion trajectory. In particular, the\\nweighted combination of the latent state and score given by\\n^x0(x(t)) :=x(t)\\u0000\\u001bt\\u000f\\u0012(x(t);t)\\n\\u000bt(8)\\nprovides an endpoint estimate that improves over time. We\\nfound tracking this statistic throughout reverse diffusion\\nprovides substantial insight into the generation process: as\\nearly as the \\ufb01rst time step, a rough outline is often visible.\\nAs time goes on, one tends to see progressively \\ufb01ner de-\\ntails \\ufb01lled in (Fig. 1A, bottom). Intriguingly, the endpoint\\nestimate ^x0(xt)is initially similar to the scaled state dif-\\nferencek(xt\\u0000k\\u0000xt). While Fig. 1 depicts an example\\nof conditional generation in a latent diffusion model, we\\nobserved similar results for many unconditional diffusion\\nmodels. For MNIST, the estimate ^x0is initially ambiguous,\\nthen starts to resemble the digit eventually generated (Fig.\\n9). For the relatively unimodal face data set CelebA-HQ,\\nthe estimate appears to be interpretable even at the \\ufb01rst time\\nstep (Fig.7)\\u2014speci\\ufb01cally, it is an average face on a generic\\nbackground (Langlois & Roggman, 1990).\\n3.2. When do different image features tend to emerge?\\nWhat is the order of feature emergence according to the\\nendpoint estimate ^x0(xt)? As Ho et al. (2020) note, high-\\nlevel features tend to emerge before low-level ones. For\\nexample, when generating \\u201ca portrait of an aristocrat\\u201d (Fig. endpoint estimate ^x0(xt)? As Ho et al. (2020) note, high-\\nlevel features tend to emerge before low-level ones. For\\nexample, when generating \\u201ca portrait of an aristocrat\\u201d (Fig.\\n1A), a generic face-like shape appears, and then is re\\ufb01ned to\\ninclude blobs that correspond to hat\\/hair and body. Coarse\\nfacial features and various image colors emerge, facial hair\\nappears, and the blob above the face is \\u2018reinterpreted\\u2019 into a\\nhat. Finally, high-frequency details of the face and clothes\\nare added. Similar \\ufb01ndings are observed in unconditional\\nmodels (e.g. of faces, churches, etc.; Fig. 7,8).\\n3.3. What is the shape of individual trajectories?\\nWe also studied the geometry of reverse diffusion trajec-\\ntories. Although the dimensionality of image\\/latent space\\nis quite large, individual trajectories are effectively two-\\ndimensional: the average variance explained by the top\\ntwo principal components (PCs) is 99.98% for our CelebA\\nmodel, and 99.54% for Stable Diffusion (see Table B.2 and\\nFig. 11). To good approximation, xtalways remains in the\\nplane de\\ufb01ned by the initial noise xTand the \\ufb01nal state x0:\\nthe variance explained by a projection onto this 2D plane\\nis higher than 99.2% for all models. Furthermore, the re-\\nverse diffusion trajectory is well-approximated by a rotation\\nwithin this plane (Fig. 1B), i.e.\\nxt\\u0019Ktx0+q\\n1\\u0000K2\\ntxT (9)\\nwhere 0\\u0014Kt\\u00141,KT= 0, andK0= 1. WhenKt=\\u000bt,\\nthis equation explains almost all trajectory variance: 97.51% Diffusion Models Generate Images Like Painters\\nfor Stable Diffusion and 98.93% for CelebA diffusion.\\n4. Theoretical analysis of reverse trajectories\\nHow should we understand the previous observations? The\\nexplicit form of the probability \\ufb02ow ODE (Eq. 6) suggests\\nan obvious but surprisingly powerful geometric view of\\nimage generation: a particle climbs down the changing\\nlandscape de\\ufb01ned by the minus log-likelihood function and\\nforward process. Here, we describe a simpli\\ufb01ed but useful\\nversion of this picture that both informs our intuition, and\\nmakes quantitative predictions about reverse diffusion.\\nFigure 2. Conceptual view of unimodal landscape dynamics .\\n4.1. Summary of landscape dynamics view of\\ngeneration\\nAssume that the training set of images is well-approximated\\nby a vast collection of Gaussian-like modes. The forward\\nprocess smears these modes out, repeatedly grouping nearby\\nmodes until only one large and nondescript mode remains.\\nThe reverse process does exactly the opposite: one mode\\nsplits into several, those modes split into more, and so on,\\nuntil one recovers a Gaussian mixture like the training set.\\nImage generation can be characterized by two related pro-\\ncesses: the dynamics of a particle driven by the constantly\\nchanging shape of this landscape, and the dynamics of the\\ncontinuously updated estimate of that particle\\u2019s endpoint.\\nParticle dynamics correspond to the conversion of noise into\\nan image, while estimate dynamics represent the layer-by-\\nlayer (outline-\\ufb01rst, details-later) painting of that image.\\nTo understand these processes, we must know: (i) their\\ndynamics on short time scales, when the number of modes\\nis \\ufb01xed; (ii) the frequency and effects of mode-splitting.\\nBoth aspects cause higher-level features to appear earlier.\\n4.2. One Gaussian mode: dynamics of particle\\nWe will begin by studying a simpler problem: reverse diffu-\\nsion given a training set well-described by a single Gaussianmode. In this setting, we can be quantitatively precise. In\\nSections 4.4 and 4.5, we will show how to apply this theory\\nto the general multi-mode setting.\\nSuppose that the mean and covariance of this mode are \\u0016\\nand\\u0006. The score function at time t >0is the score of a\\nGaussianN(\\u000bt\\u0016;\\u001b2\\ntI+\\u000b2\\nt\\u0006), which is\\ns(x;t) =rxp(x;t) = (\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001(\\u000bt\\u0016\\u0000x):(10)\\nThe corresponding probability \\ufb02ow ODE\\n_x=\\u0000\\f(t)x\\u0000\\f(t)(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001(\\u000bt\\u0016\\u0000x)(11)\\ncan be solved exactly (Appendix G). y:=x\\u0000\\u000bt\\u0016satis\\ufb01es\\n\\u0000_y=\\f(t) =\\u0000\\u0002\\n(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001\\u0000I\\u0003\\ny (12)\\nThe covariance matrix has a compact singular value decom-\\nposition (SVD) \\u0006=U\\u0003UT, where U= [u1;:::;ur]is a\\nD\\u0002rsemi-orthogonal matrix. The columns of Uare the\\nprincipal axes along which the mode varies, and together\\ncomprise an \\u2018image manifold\\u2019. If we de\\ufb01ne the matrix\\n[~\\u0003t]ij:=\\u000eij\\u000b2\\nt\\u0015i\\n\\u001b2\\nt+\\u000b2\\nt\\u0015i; (13)\\nwhere\\u0015i:=\\u0003ii, we can rewrite Eq. 12 as\\n\\u0000_y=\\f(t) =1\\n\\u001b2\\nth\\n\\u0000\\u000b2\\ntI+U~\\u0003tUTi\\ny: (14)\\nEq. 12 is linear, and the dynamics of ytalong each principal\\naxis are independent of dynamics along the others. We can\\nwriteytas a sum of on- and off-manifold components:\\nyt= (t;0)y?\\nT+rX\\nk=1 (t;\\u0015k)ck(T)uk\\nck(T) =uT\\nk(xT\\u0000\\u000bT\\u0016)\\ny?\\nT= (I\\u0000UUT)(xT\\u0000\\u000bT\\u0016)\\n (t;\\u0015) :=s\\n1 + (\\u0015\\u00001)\\u000b2\\nt\\n1 + (\\u0015\\u00001)\\u000b2\\nT(15)\\nwhereT > 0marks the beginning of dynamics. Since\\n\\u000btt!0\\u0000\\u0000\\u0000! 1, the components of yorthogonal to the im-\\nage manifold decay toward zero. The coef\\ufb01cients of on-\\nmanifold components grow if \\u0015k<1, remain \\ufb01xed if\\n\\u0015k= 1, and shrink if \\u0015k>1; in all cases,  (t;\\u0015k)ap-\\nproachesp\\u0015k, the standard deviation along uk, ast!0.\\nEq. 15 implies high variance image manifold features (i.e.\\ndirections with high \\u0015k) appear \\ufb01rst, and that low variance\\nfeatures appear later. It also implies an increasing com-\\nmitment to image features as time goes on. Suppose at\\ntimet<t0<T the off-manifold directions are perturbed Diffusion Models Generate Images Like Painters\\nby\\u000ey?, and the on-manifold direction coef\\ufb01cients are per-\\nturbed by amounts \\u000eck; the total differences \\u0001y?(t)and\\n\\u0001ck(t)in these components at time tare\\n\\u0001y?\\nt=\\u000ey? (t;0)\\n (t0;0)=\\u000ey?s\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nt0\\n\\u0001ck(t) =\\u000eck (t;\\u0015k)\\n (t0;\\u0015k)=\\u000ecks\\n1 + (\\u0015k\\u00001)\\u000b2\\nt\\n1 + (\\u0015k\\u00001)\\u000b2\\nt0:(16)\\nBoth on- and off-manifold perturbations have smaller effects\\nast0gets closer to t, since\\u000bt0is closer to\\u000bt. Off-manifold\\nperturbations decay, while on-manifold perturbations have a\\nlasting effect (and perturbations in high variance directions\\nhave larger effects). Eq. 15 also implies (see Appendix H)\\nxt\\u0019\\u000btx0+q\\n1\\u0000\\u000b2\\ntxT\\n+rX\\nk=1\\u001aq\\n\\u001b2\\nt+\\u0015k\\u000b2\\nt\\u0000\\u000btp\\n\\u0015k\\u0000\\u001bt\\u001b\\nck(T)uk;\\nwhich can be used to explain our observation that xtdy-\\nnamics tend to look like a rotation within a 2D plane (Fig.\\n1B). It isa rotation when the sum term vanishes, which\\nhappens when \\u000bt\\u00190and\\u000bt\\u00191. The sum term approxi-\\nmately vanishes when \\u0015kis suf\\ufb01ciently large or small\\u2014but\\nin general, it does not, meaning xtis only approximately a\\nrotation. Interestingly, since \\u000btdoes not depend on \\u0016or\\u0006,\\nthe rotation does not depend on any properties of the mode.\\nWe empirically \\ufb01nd, however, that it does depend on the\\nsampler (Section B.6.1).\\n4.3. One Gaussian mode: dynamics of endpoint\\nestimate\\nThe endpoint estimate ^x0(see Eq. 8 and Appendix F) is\\n^x0(xt) :=xt+\\u001b2\\nt(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001(\\u000bt\\u0016\\u0000xt)\\n\\u000bt:(17)\\nInitially, when \\u001b2\\nt\\u00181, we have ^x0\\u0019\\u0016, which we can\\ninterpret as something like the \\u2018average face\\u2019 (Langlois &\\nRoggman, 1990) if the training set consists of face images.\\nLater, this might look like the average of a more specialized\\nset of faces, e.g. the average older male face. Close to the\\nend of reverse diffusion, when \\u001b2\\nt\\u00180, we have ^x0\\u0019xt, i.e.\\nthe endpoint estimate and true trajectory endpoint match.\\nUsing Eq. 15, note that ^x0can be written\\n^x0(xt) =\\u0016+\\u0002\\nI\\u0000\\u001b2\\nt(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001\\u0003\\nyt=\\u000bt\\n=\\u0016+U~\\u0003tUTyt=\\u000bt\\n=\\u0016+rX\\nk=1\\u0018(t;\\u0015k)ck(T)uk\\n\\u0018(t;\\u0015) :=\\u000bt\\u0015p\\n(\\u001b2\\nt+\\u0015\\u000b2\\nt)(\\u001b2\\nT+\\u0015\\u000b2\\nT)(18)which quanti\\ufb01es the fraction of ^x0captured by the mean\\n\\u0016and each of the principal component. Interestingly, this\\nequation implies that ^x0always remains on the image mani-\\nfold, which explains why visualized estimates always look\\nlike images, rather than images contaminated with noise.\\nEq. 18 implies that image manifold features appear in ^x0\\nin order of descending variance: \\ufb01rst the highest variance\\nfeatures appear, then the next highest, and so on. There is\\nalso increasing commitment to features that appear earlier,\\nin the sense that perturbations are less likely to change them\\nlater in reverse diffusion. To quantify this, we can derive an\\nequation analogous to Eq. 16:\\n\\u0001^x0=rX\\nk=1\\u000bt\\u0015kp\\n[\\u001b2\\nt+\\u000b2\\nt\\u0015k] [\\u001b2\\nt0+\\u000b2\\nt0\\u0015k]\\u000eckuk(19)\\nwhere \\u0001^x0is the change in ^x0at timetdue to perturbations\\n\\u000eckat timet0. Our conceptual view of particle and estimate\\ndynamics in a unimodal landscape is depicted in Fig. 2.\\n4.4. Short time scale dynamics in general\\nMoving beyond the unimodal case, on short time scales the\\nlandscape looks like a Gaussian mixture whose modes are\\nshrinking in size. A priori, we might expect that multiple\\nmodes in\\ufb02uence the particle\\u2019s movement at any given time.\\nHowever, a quirk of high-dimensional Gaussians is that\\nalmost all of their probability is concentrated in a thin shell\\naround their mode (see Appendix E), and we can use this\\nfact to approximate the score. Note that if p(x;t)represents\\na Gaussian mixture whose components have weights \\u0019k,\\nmeans \\u0016k(t), and covariance matrices \\u0006k(t),\\ns(x;t) =P\\nk\\u0019k\\u0006\\u00001(t)(\\u0016(t)\\u0000x)N(x;\\u0016k(t);\\u0006k(t))P\\nk\\u0019kN(x;\\u0016k(t);\\u0006k(t)):\\nThis is essentially a softmax function, with each mode con-\\ntributing based on size and proximity to x. The sparseness\\nof high-dimensional Gaussians means that only the mode\\nclosest tox(k0, say) will contribute appreciably to the score,\\nso that s(x;t)\\u0019\\u0006k0(\\u0016k0\\u0000x)to very good approximation.\\nHence, we can use the theory from the previous two sections\\nto accurately describe short time scale dynamics.\\nxtreverse di\\ufb00usion timenoise\\nFigure 3. Conceptual view of mode-splitting. Diffusion Models Generate Images Like Painters\\nFigure 4. Analytical predictions of convergence speed along dimensions of different variances .A.The (t;\\u0015k)function, with\\ndifferent\\u0015k= 0:0;0:01;0:1;1;10;100. It governs the dynamics of ytalong eigendimension ukwith variance \\u0015k.B.Time derivative of\\nthe curves in A., i.e. the rate of change along each eigendimension. C.The\\u0018(t;\\u0015k)function governing the dynamics of ^x0(xt)\\u0000\\u0016\\nalong eigendimension uk, normalized by the standard deviationp\\u0015k, showing the course of convergence for the projected outcome. D.\\nThe time derivative of the curve in C., showing the relative rates of change of ^x0(xt)\\u0000\\u0016along each eigendimension, highlighting that\\nthe projected outcome \\ufb01rst moves along directions of larger variance. The speci\\ufb01c \\u000btschedule is copied from ddpm-CIFAR-10 (see\\nAppendix A). Other schedules yield similar results.\\n4.5. Frequency and effects of mode-splitting\\nOn longer time scales, the number of modes tends to in-\\ncrease exponentially: as the noise scale gets smaller, the\\nlandscape re\\ufb02ects \\ufb01ner and \\ufb01ner details of the training set,\\ncausing one mode to become two, two to become four, and\\nso on. From the particle\\u2019s point of view, the mean and co-\\nvariance matrix of the closest mode suddenly switches, then\\nsingle mode dynamics runs until the next splitting event.\\nTo make our intuition more precise, consider a training\\nset ofMclosely packed D-dimensional isotropic Gaus-\\nsian modes. Given a coarse asymptotic bound on high-\\ndimensional sphere packing (Ball, 1992), we can estimate\\nthat a given mode will have roughly 2Dnearest neighbors.\\nAt the beginning of reverse diffusion, one mode splits into\\n2Dmodes. Each of these splits into 2Dmore modes, each\\nof those splits, and so on; after ksplittings, there are 2kD\\nmodes. Hence, on the order of logMsplittings are required\\nto recover the resolution of the training set.\\nSplittings occur when the noise scale goes below the vari-\\nances of the current modes (equivalently: the distance be-\\ntween the underlying modes), uncovering modes with vari-\\nances that are a factor of roughly 2\\u0000Dsmaller. After k\\nsplittings, they are a factor of 2\\u0000kDsmaller. Because \\f(t)\\nis usually chosen to depend linearly on time, the noise scale\\n\\u001btchanges exponentially with time, which means splittings\\nshould occur at regular time intervals, and the characteristic\\namount of time between splittings should scale as 1=logM.\\nAccording to this model, image generation should look\\nhierarchical . High-level features, which correspond to a\\nsmeared out collection of alike modes, are committed to\\n\\ufb01rst. As these modes split, the particle ends up in the attrac-\\ntor basin of one of the component modes, and the image\\ncommits to lower-level features. This process continues\\nuntil only high-frequency details remain (Fig. 3).\\nThis commitment to a particular collection of modes yieldsanother reason that later perturbations have less of an effect:\\nthey tend to push the particle into nearby modes, which\\ncorrespond to similar images. The typical distances between\\nparticles that committed to different modes is expected to\\nincrease exponentially with divergence time; for that reason,\\nthe strength of the perturbation required to push them into\\nnearby basins of attraction also increases exponentially.\\n4.6. Numerical simulation\\nTo sharpen our intuition about our analytical results (espe-\\ncially Eq. 15 and 18), we used a common \\u000btschedule (see\\nAppendix A) and plotted the function  (t;\\u0015k)and (t;0)\\nwhich govern the xtdynamics, for different \\u0015k, i.e. the vari-\\nance of the distribution in the direction uk. The projection\\ncoef\\ufb01cients of xtalong high variance directions increase at\\n\\ufb01rst, while the coef\\ufb01cients along the low variance directions\\nremain the same or shrink later (Fig. 4A).\\nBy plotting the derivative of  (t;\\u0015)function, we can exam-\\nine_xti.e. the velocity of particle movement (Fig. 4B). It\\nis initially dominated by contributions from the large vari-\\nance dimensions. Close to the end, it features comparable ine_xti.e. the velocity of particle movement (Fig. 4B). It\\nis initially dominated by contributions from the large vari-\\nance dimensions. Close to the end, it features comparable\\nbut opposite-sign contributions from the high variance di-\\nmensions and the off-manifold (i.e. noise) directions. This\\nexplains the observation that state differences xt\\u0000k\\u0000xtlook\\ninterpretable at \\ufb01rst, and later look like noise-contaminated\\nimages (Fig. 1A, middle row).\\nFinally, we examined \\u0018(t;\\u0015k)which governs the dynamics\\nof^x0(xt)along each principal direction uk. These func-\\ntions look sigmoidal and converge to 1 when normalized\\nby the standard deviationp\\u0015k(Fig. 4C). The functions\\nconverge earlier along large variance dimensions, and later\\nalong small variance dimensions. Furthermore, their time\\nderivativesd\\ndt\\u0018(t;\\u0015k)are bump-like, and peak earlier for\\nhigh variance dimensions. In plain language, high variance\\nfeatures are added to the endpoint estimate \\ufb01rst, and low Diffusion Models Generate Images Like Painters\\nvariance features are added later. This partly explains our\\nearlier observations about the order of feature emergence\\n(Fig. 1A, bottom row).\\n5. Validating theory on unconditional models\\nEven if approximating the training set as a high-dimensional\\nGaussian mixture is valid, it is dif\\ufb01cult to make detailed\\nquantitative predictions about the effects of mode-splitting\\non reverse diffusion dynamics. As a \\ufb01rst step, we can instead\\napproximate the training set as a single Gaussian mode, and\\ntest our earlier predictions. To do this, for a few accessible\\nimage data sets\\u2014MNIST, CIFAR-10, and CelebA\\u2014we nu-\\nmerically computed the mean and covariance of all images\\nin pixel space. We then investigated three unconditional\\npixel space diffusion models trained on these data sets (see\\nAppendix A). Although crude, this approximation is not un-\\nreasonable, since e.g. face space may be well-approximated\\nby a multivariate Gaussian (Turk & Pentland, 1991).\\nCoarse agreement between theory and actual trajectory.\\nUnder the single mode approximation, we can predict how\\nxtand^x0(xt)will evolve. In many cases, we can accurately\\ndescribe the early part of the trajectory and predict the low-\\nfrequency layout of the \\ufb01nal image well (Fig. 5A-B). High-\\nfrequency details like edges tend not to be well-predicted.\\nWhen we compared the theory and actual trajectory through\\ntime, we found the deviation blows up around 20 steps\\n(Fig. 5C); we think this is a signature of when the single\\nstatic mode assumption breaks down, and the trajectory\\nstarts to be guided by the updated \\ufb01ne-grained mode. (See\\nFig. 12,13 for examples from MNIST and CelebA.) On the\\nother hand, this also showed that the single Gaussian mode\\nassumption is a good approximation in the initial phase.\\nSince the high-variance layout information is determined\\nin this initial phase, our theory illuminates the connection\\nbetween the initial noise pattern xTand the low frequency\\nfeatures in the \\ufb01nal sample x0.\\nOff-manifold dynamics well-predicted by theory. Eq.\\n15 indicates that off-manifold components of xt\\u0000\\u000bt\\u0016\\nshould shrink at a universal rate that depends only on \\u000bt.\\nTo test this, we picked a few PCs along which our data sets\\nvary the least, and found that dynamics in these directions\\nare well-predicted by our formula (Fig. 15). Even though\\nour theory does not account in detail for mode-splitting, its\\npredictions for off-manifold directions are still accurate.\\nPrediction errors happen along image manifold. Next,\\nwe examined the deviation between our predictions and the\\nreal trajectory. We found the deviation tends to lie within\\nthe subspace spanned by the top PCs (Fig. 5D, Fig.12E,\\nFig.13E). One interpretation is that mode-splitting majorly\\nhappens within the general image manifold: after mode-splitting, the new effective mean and covariance change\\nalong the image manifold, which induces error for the trajec-\\ntory predicted by using the old static mean and covariance.\\nFigure 5. Comparing analytical solution to actual trajectory\\nfor a CIFAR-10 diffusion model .A.Actual ^x0(xt)and ana-\\nlytical solution using same initial conditions. B.Collections of\\nactual and theory-predicted x0.C.Mean squared error between\\nthe actual and analytical solution of trajectory xt.D.Fraction of\\nsquared error as a function of PC, between the actual and analytical\\nx0.\\n6. Application: Stable Diffusion\\nIn this section, we study Stable Diffusion, a powerful la-\\ntent diffusion model that can generate images from natural\\nlanguage text prompts (Rombach et al., 2022). It is a con-\\nditional diffusion model: when we \\ufb01x a text prompt \\u001c, the\\nmodel samples from the distribution p(xj\\u001c). Properties\\nof this high-dimensional distribution like \\u0016and\\u0006are ex-\\ntremely hard to characterize (partly due to an inaccessible\\nground truth), but we can still show that most of our qualita-\\ntive predictions hold. We can also use our theory to guide\\nthe search for \\u2018on-manifold\\u2019 directions, which can be used ground truth), but we can still show that most of our qualita-\\ntive predictions hold. We can also use our theory to guide\\nthe search for \\u2018on-manifold\\u2019 directions, which can be used\\nfor editing and creating variations.\\n6.1. PCs of latent trajectory are on-manifold variations\\nWe observed that Stable Diffusion trajectories are higher-\\ndimensional than other models\\u2019 trajectories (Table B.2).\\nWhat do these additional dimensions represent? Our theory\\npredicts that the dimensions outside the x0;xTplane are\\nlying within the image manifold spanned by U(Eq.4.2).\\nIndeed, when we visualized the PC directions through the Diffusion Models Generate Images Like Painters\\nFigure 6. Effects of perturbation during reverse diffusion .A.\\nThe PCs of the state difference along the trajectory are on-manifold.\\nB.The projected outcome G(^x0(xt))for the perturbed trajec-\\ntory ~xtwhen the same perturbation is injected at different steps\\n5;10;15;20;35. Here the perturbation is aligned with PC3 of the\\nU-Net output along the original trajectory. C.The LPIPS distance\\n(Zhang et al., 2018) between the projected outcome in the original\\ntrajectory and perturbed trajectory, D(G(^x0(xt));G(^x0(~xt))).\\ndecoder, they appeared to be interpretable variations around\\nthe target image (Fig. 6A, 16). This gave us an ef\\ufb01cient way\\nto \\ufb01nd local on-manifold vectors using a single sampling\\ntrajectory.\\n6.2. Effects of perturbation on image generation\\nHo et al. (2020) use a noisy SDE solver, which implic-\\nitly includes perturbations at every time step; they notice\\nthat earlier perturbations tend to affect higher-level features,\\nwhile later perturbations tend to affect lower-level features.\\nBy perturbing the latent state at different time steps, we\\nnotice the same phenomenon (Fig. 6A). Speci\\ufb01cally, we\\nperturb the latent state once at time t0by an amount K\\u000ex,\\nthen continue generation. At any time point, the effect on\\nthe \\ufb01nal image scales with the strength of perturbation (Fig.\\nB.5). As predicted by single mode theory (Eq. 16), when\\nstrength and direction are \\ufb01xed, the earlier the perturba-\\ntion, the larger the effect on the image (Fig. 6A,B). Plainly\\nspeaking, the earlier perturbations have more time to be\\n\\u201campli\\ufb01ed\\u201d to the \\ufb01nal result. We also noticed that the effect\\nof perturbation is rapidly re\\ufb02ected by the projected outcome\\nG(^x0(xt)), which allows for fast testing and synthesis of\\nperturbations (Fig. 6B).\\nAt a \\ufb01xed time point, not all perturbations are created equal.\\nAt the same scale K, Gaussian noise perturbations often\\nhave much less perceptible effect on the \\ufb01nal image, while\\nperturbations along trajectory PC directions often have asigni\\ufb01cant effect. Quantitatively, the difference between\\nthe perturbed and original trajectory along the perturbed\\ngrows through time when the perturbation is along a PC\\ndirection; in contrast, the difference along a noise direction\\nwill decrease over time (Fig.B.5,B.5). This observation\\naligns with our theory (c.f. Eq. 16), which says off-manifold\\nnoise directions will shrink to zero and signal directions will\\nbe ampli\\ufb01ed to the target standard deviationp\\u0015k. Given\\na high-dimensional image space, it is highly likely that a\\nrandomly chosen perturbation has little overlap with the\\nsignal manifold. Can we design better perturbations?\\n6.3. Perturbation design guided by sampling trajectory\\nThe PCs of a sampling trajectory correspond to meaning-\\nful and performant \\u2018on-manifold\\u2019 perturbations. One nice\\nfeature of applying these perturbations is that on-manifold\\ndirections need not be perfectly known, since the \\u2018noise\\u2019\\ndirections will decay to zero by the end. Hence, as long as\\nthe perturbation direction has enough signal direction in it,\\none tends to obtain good results.\\nUsing this method, we can create manifolds of smoothly con-\\nnected images, answering a long-standing question about the\\nexistence of latent manifolds in diffusion models (Fig.B.5\\nB.5). One nice application of the method is to create local\\nvariations of an existing sample without changing the noise\\nseed.\\n7. Discussion\\nTo what extent do our main \\ufb01ndings\\u2014outline-\\ufb01rst and\\ndetails-later image generation, low-dimensional trajectories,\\nand increasing commitment to image elements\\u2014hold true\\nfor other diffusion model variants and other kinds of gener-\\native models? Simulating reverse SDEs (Eq. 2) instead of\\nODEs should not yield many qualitative differences; instead\\nof linear ODEs, one has similarly-behaved OU processes.\\nThe results of Antognini and Sohl-Dickstein (2018) show\\nOU trajectories are also very low-dimensional. Arguments\\nsimilar to ones we have made suggest other kinds of models The results of Antognini and Sohl-Dickstein (2018) show\\nOU trajectories are also very low-dimensional. Arguments\\nsimilar to ones we have made suggest other kinds of models\\nmay be e.g. hierarchical. Xu and Liu et al.\\u2019s Poisson \\ufb02ow\\ngenerative models (2022) share the qualitative splitting be-\\nhavior we described in Section 4.5, but feature smeared-out\\ncharge distributions instead of smeared out Gaussian modes.\\nOur theory provides a conceptual bridge between diffusion\\nmodels and GANs. In the GAN literature, the idea of gen-\\nerating images by modeling progressively greater levels of\\nresolution is well-established, e.g. by successful architec-\\ntures like Progressive Growing GAN and StyleGAN (Karras\\net al., 2017; 2020). In this paradigm, early layers synthesize\\nthe rough layout of the image from noise, while the last\\nfew layers add realistic details. At any layer, the image can\\nbe visualized by a decoding head, with the decoded image Diffusion Models Generate Images Like Painters\\ngradually adding details as it approaches the \\ufb01nal result.\\nWe showed that when looking at the projected outcome ^x0,\\ndiffusion models have an intriguingly similar generative\\nprocess: the \\ufb01rst few steps map the noise pattern to a rough\\noutline, and then \\ufb01ner details are gradually added in. The\\nonly signi\\ufb01cant difference is that GAN layers are replaced\\nby reverse diffusion sampling steps. This connection may\\ninspire future designs of diffusion models or GANs.\\nThere may also be a sense in which the latent manifold\\ngeometry of GANs and diffusion models is similar. Wang\\nand Ponce (2021) found that perturbing the large variance\\ndirections of a latent space metric has large and interpretable\\neffects on image generation in GANs. If we interpret the\\ncovariance matrix of a Gaussian mode as inducing a metric\\non the latent space of diffusion models, our perturbation-\\nrelated observations can be cast in a similar light.\\nIn psychology, the idea of generating mental images hier-\\narchically is related to the idea of non-commitment : not all\\ndetails of a scene may be imagined. Non-commitment is\\npervasive in human mental imagery (Bigelow et al., 2022).\\nWe found that visualizing endpoint estimates also yields a\\nsort of non-commitment (e.g. to \\ufb01ne face details) early in\\nreverse diffusion, and it would be interesting to see to what\\nextent this phenomenon resembles the psychological one.\\nReferences\\nAnderson, B. D. Reverse-time diffusion equation\\nmodels. Stochastic Processes and their Applica-\\ntions , 12(3):313\\u2013326, 1982. ISSN 0304-4149.\\ndoi: https:\\/\\/doi.org\\/10.1016\\/0304-4149(82)90051-5.\\nURL https:\\/\\/www.sciencedirect.com\\/\\nscience\\/article\\/pii\\/0304414982900515 .\\nAntognini, J. M. and Sohl-Dickstein, J. PCA of high dimen-\\nsional random walks with comparison to neural network\\ntraining. In NeurIPS , pp. 10328\\u201310337, 2018.\\nBall, K. A lower bound for the optimal density of lat-\\ntice packings. International Mathematics Research No-\\ntices, 1992(10):217\\u2013221, 05 1992. ISSN 1073-7928. doi:\\n10.1155\\/S1073792892000242. URL https:\\/\\/doi.\\norg\\/10.1155\\/S1073792892000242 .\\nBigelow, E. J., McCoy, J., and Ullman, T. D. Non-\\ncommitment in mental imagery, Oct 2022. URL\\npsyarxiv.com\\/pn4zd .\\nGoodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B.,\\nWarde-Farley, D., Ozair, S., Courville, A., and Bengio,\\nY . Generative adversarial nets. In Ghahramani, Z.,\\nWelling, M., Cortes, C., Lawrence, N., and Weinberger,\\nK. (eds.), Advances in Neural Information Pro-\\ncessing Systems , volume 27. Curran Associates,\\nInc., 2014. URL https:\\/\\/proceedings.neurips.cc\\/paper\\/2014\\/file\\/\\n5ca3e9b122f61f8f06494c97b1afccf3-Paper.\\npdf.\\nHertz, A., Mokady, R., Tenenbaum, J., Aberman, K.,\\nPritch, Y ., and Cohen-Or, D. Prompt-to-prompt im-\\nage editing with cross attention control. arXiv preprint\\narXiv:2208.01626 , 2022.\\nHo, J. and Salimans, T. Classi\\ufb01er-free diffusion guidance.\\narXiv preprint arXiv:2207.12598 , 2022.\\nHo, J., Jain, A., and Abbeel, P. Denoising diffusion proba-\\nbilistic models. Advances in Neural Information Process-\\ning Systems , 33:6840\\u20136851, 2020.\\nHo, J., Salimans, T., Gritsenko, A. A., Chan, W., Norouzi,\\nM., and Fleet, D. J. Video diffusion models. In\\nICLR Workshop on Deep Generative Models for Highly\\nStructured Data , 2022. URL https:\\/\\/openreview.\\nnet\\/forum?id=BBelR2NdDZ5 .\\nKarras, T., Aila, T., Laine, S., and Lehtinen, J. Progressive\\ngrowing of GANs for improved quality, stability, and\\nvariation. arXiv preprint arXiv:1710.10196 , 2017.\\nKarras, T., Laine, S., Aittala, M., Hellsten, J., Lehtinen, J.,\\nand Aila, T. Analyzing and improving the image quality\\nof Stylegan. In Proceedings of the IEEE\\/CVF conference\\non computer vision and pattern recognition , pp. 8110\\u2013\\n8119, 2020.\\nKosslyn, S. M. and Shwartz, S. P. A simula-\\ntion of visual imagery. Cognitive Science , 1\\n(3):265\\u2013295, 1977. ISSN 0364-0213. doi:\\nhttps:\\/\\/doi.org\\/10.1016\\/S0364-0213(77)80020-7.\\nURL https:\\/\\/www.sciencedirect.com\\/\\nscience\\/article\\/pii\\/S0364021377800207 . (3):265\\u2013295, 1977. ISSN 0364-0213. doi:\\nhttps:\\/\\/doi.org\\/10.1016\\/S0364-0213(77)80020-7.\\nURL https:\\/\\/www.sciencedirect.com\\/\\nscience\\/article\\/pii\\/S0364021377800207 .\\nLanglois, J. H. and Roggman, L. A. Attractive faces are only\\naverage. Psychological science , 1(2):115\\u2013121, 1990.\\nLiu, L., Ren, Y ., Lin, Z., and Zhao, Z. Pseudo numeri-\\ncal methods for diffusion models on manifolds. arXiv\\npreprint arXiv:2202.09778 , 2022.\\nRombach, R., Blattmann, A., Lorenz, D., Esser, P., and\\nOmmer, B. High-resolution image synthesis with latent\\ndiffusion models. In Proceedings of the IEEE\\/CVF Con-\\nference on Computer Vision and Pattern Recognition , pp.\\n10684\\u201310695, 2022.\\nRuderman, D. L. The statistics of natural images. Network:\\ncomputation in neural systems , 5(4):517, 1994.\\nSohl-Dickstein, J., Weiss, E., Maheswaranathan, N., and\\nGanguli, S. Deep unsupervised learning using nonequi-\\nlibrium thermodynamics. In Bach, F. and Blei, D. (eds.), Diffusion Models Generate Images Like Painters\\nProceedings of the 32nd International Conference on Ma-\\nchine Learning , volume 37 of Proceedings of Machine\\nLearning Research , pp. 2256\\u20132265, Lille, France, 07\\u2013\\n09 Jul 2015. PMLR. URL https:\\/\\/proceedings.\\nmlr.press\\/v37\\/sohl-dickstein15.html .\\nSong, J., Meng, C., and Ermon, S. Denoising diffusion\\nimplicit models. arXiv preprint arXiv:2010.02502 , 2020.\\nSong, Y . and Ermon, S. Generative modeling by estimating\\ngradients of the data distribution. In Wallach, H.,\\nLarochelle, H., Beygelzimer, A., d 'Alch \\u00b4e-Buc, F.,\\nFox, E., and Garnett, R. (eds.), Advances in Neural\\nInformation Processing Systems , volume 32. Curran As-\\nsociates, Inc., 2019. URL https:\\/\\/proceedings.\\nneurips.cc\\/paper\\/2019\\/file\\/\\n3001ef257407d5a371a96dcd947c7d93-Paper.\\npdf.\\nSong, Y ., Sohl-Dickstein, J., Kingma, D. P., Kumar, A.,\\nErmon, S., and Poole, B. Score-based generative mod-\\neling through stochastic differential equations. In In-\\nternational Conference on Learning Representations ,\\n2021. URL https:\\/\\/openreview.net\\/forum?\\nid=PxTIG12RRHS .\\nSorscher, B., Ganguli, S., and Sompolinsky, H. Neural\\nrepresentational geometry underlies few-shot concept\\nlearning. Proceedings of the National Academy of Sci-\\nences , 119(43):e2200800119, 2022. doi: 10.1073\\/pnas.\\n2200800119. URL https:\\/\\/www.pnas.org\\/doi\\/\\nabs\\/10.1073\\/pnas.2200800119 .\\nTurk, M. and Pentland, A. Eigenfaces for recognition. Jour-\\nnal of cognitive neuroscience , 3(1):71\\u201386, 1991.\\nWang, B. and Ponce, C. R. A geometric analysis of\\ndeep generative image models and its applications. In\\nInternational Conference on Learning Representations ,\\n2021. URL https:\\/\\/openreview.net\\/forum?\\nid=GH7QRzUDdXG .\\nWenliang, L. K. and Moran, B. Score-based generative\\nmodel learn manifold-like structures with constrained\\nmixing. In NeurIPS 2022 Workshop on Score-Based\\nMethods , 2022. URL https:\\/\\/openreview.net\\/\\nforum?id=eSZqaIrDLZR .\\nXu, Y ., Liu, Z., Tegmark, M., and Jaakkola, T. S. Pois-\\nson \\ufb02ow generative models. In Oh, A. H., Agarwal, A.,\\nBelgrave, D., and Cho, K. (eds.), Advances in Neural\\nInformation Processing Systems , 2022. URL https:\\n\\/\\/openreview.net\\/forum?id=voV_TRqcWh .\\nYang, L., Zhang, Z., Song, Y ., Hong, S., Xu, R., Zhao, Y .,\\nShao, Y ., Zhang, W., Cui, B., and Yang, M.-H. Diffu-\\nsion models: A comprehensive survey of methods and\\napplications. arXiv preprint arXiv:2209.00796 , 2022.Zhang, R., Isola, P., Efros, A. A., Shechtman, E., and Wang,\\nO. The unreasonable effectiveness of deep features as a\\nperceptual metric. In Proceedings of the IEEE conference\\non computer vision and pattern recognition , pp. 586\\u2013595,\\n2018. Diffusion Models Generate Images Like Painters\\nA. Diffusion models used in numerical experiments\\nTable 1. Diffusion models used for this paper\\u2019s numerical experiments .\\ny: The MNIST diffusion model uses the upsampled 3\\u000232\\u000232RGB pixel space as sample space, while the original MNIST data set\\nconsists of 28\\u000228single channel black and white images. Thus the effective dimensionality of these images is around 784.\\nDATA SET HUGGING FACE MODEL ID DIMENSIONALITY LATENTS ? C ONDITIONAL ?\\nMNIST DIMPO \\/DDPM -MNIST 3X32X32=3072y \\u0002 \\u0002\\nCIFAR-10 GOOGLE \\/DDPM -CIFAR 10-32 3 X32X32=3072 \\u0002 \\u0002\\nLSUN-CHURCH GOOGLE \\/DDPM -CHURCH -256 3 X256 X256 = 196,608 \\u0002 \\u0002\\nCELEB A-HQ GOOGLE \\/DDPM -CELEBAHQ -256 3 X256 X256 = 196,608 \\u0002 \\u0002\\nLAION-2B RUNWAYML \\/STABLE -DIFFUSION -V1-5 4 X64X64 = 16,384p p\\nB. Supplementary Results\\nB.1. Visualizing image generation progress for other diffusion models\\nFigure 7. Visualizing the image generation process for the DDPM-CelebA model . Same layout as Fig.1.\\nFigure 8. Visualizing the image generation process for the DDPM-Church model . Same layout as Fig.1. Diffusion Models Generate Images Like Painters\\nFigure 9. Visualizing the image generation process for the DDPM-MNIST model . Same layout as Fig.1.\\nFigure 10. Visualizing the image generation process for the DDPM-CIFAR-10 model . Same layout as Fig.1. Diffusion Models Generate Images Like Painters\\nB.2. Trajectory geometry statistics\\nTable 2. Trajectory geometry statistics for different diffusion models . Residual variance is here de\\ufb01ned to be the squared norm of\\nthe error vector divided by the squared norm kxtk2. Residual variance is computed for three approximations: 1) projecting a trajectory\\nonto the top 2 PCs, 2) projecting a trajectory onto the plane spanned by x0andxT, 3) approximating the trajectory by a rotation\\n\\u000btx0+p\\n1\\u0000\\u000b2\\ntxT. Effective dimensionality is de\\ufb01ned as the number of PCs needed to account for 99.9% of the variance. Shown are\\nthe effective dimensionalities of the trajectory xt, difference xt\\u00001\\u0000xt, and the U-Net output \\u000f\\u0012(xt). All sampling used 51 time steps.\\nRESIDUAL VARIANCE DIM.FOR 99.9 V AR.\\nSAMPLERTOP 2 PC\\nPROJ .x0;xT\\nPROJ .x0;xT\\nROTATIONxt\\u0001xt\\u000f\\u0012(xt)\\nDDPM -MNIST DDIM 0.08% 0.26% 1.70% 2 5 8\\nDDPM -CIFAR 10-32 DDIM 0.05% 0.30% 1.01% 2 4 7\\nDDPM -CHURCH -256 DDIM 0.05% 0.31% 1.09% 2 5 8\\nDDPM -CELEBAHQ -256 DDIM 0.02% 0.10% 1.07% 2 4 7\\nSTABLE -DIFFUSION -V1-5 PNDM 0.46% 0.81% 2.49% 5 34 28 Diffusion Models Generate Images Like Painters\\nFigure 11. Geometry of diffusion sampling trajectories for 5 different diffusion models. Left panel : the error fraction of the projection\\nonto the 2D plane de\\ufb01ned by x0andxT, and the error fraction under the rotation approximation. Right panel : the cumulative explained\\nvariance of PCs for the trajectory xt, state differences xt\\u00001\\u0000xt, and the output from the U-Net \\u000f\\u0012(xt). 2 PCs explained almost all\\nvariance for the xttrajectory, while the state difference xt\\u00001\\u0000xtand U-Net outputs are higher dimensional. For quanti\\ufb01cation see Tab.\\nB.2. Diffusion Models Generate Images Like Painters\\nB.3. Validation of the single mode theory on CIFAR, MNIST, and CelebA\\nFigure 12. Comparing analytical solution and actual diffusion process for the DDPM-MNIST model .A.True and predicted endpoint\\nestimate ^x0(xt)throughout reverse diffusion. B.Collection of samples of diffusion-generated images and the corresponding images\\npredicted by our analytical theory. C.Mean squared error of the trajectory xt.D.MSE of the endpoint estimate during diffusion. E.x0\\nprediction error along each eigendimension. Diffusion Models Generate Images Like Painters\\nFigure 13. Comparing analytical solution and actual diffusion process for the DDPM-CelebA model . Same layout as Fig.12. Note\\nthat in A.the general layout and shading around the face is consistent between the theory and actual diffusion trajectory. Note in B, the\\nzoomed-out version of the predicted and actual sample look highly similar. Diffusion Models Generate Images Like Painters\\nFigure 14. Comparing analytical solution and actual diffusion process for the DDPM-CIFAR-10 model . Same layout as Fig.12.\\nFigure 15. Trajectory along off-manifold directions well-predicted by theory . We computed the PC projection of actual trajectories\\nand compared them to the theoretical prediction given the same initial value; they aligned well. DDPM-MNIST model. Diffusion Models Generate Images Like Painters\\nB.4. Principal dimensions of Stable Diffusion sampling trajectories\\nFigure 16. Visualizing PCs of Stable Diffusion trajectories . We computed the principal components of the trajectory xtand the\\ntrajectory difference xt\\u00001\\u0000xt, and visualized the scaled version of PC vectors through the decoder. We can see that they represent an\\ninterpretable vector space around the actual sample x0. Diffusion Models Generate Images Like Painters\\nB.5. Perturbation experiments\\nFigure 17. Example perturbation experiment result. Different rows represent different perturbation times, from top to bottom\\n5;10;:::50. Different columns represent different perturbation scales, left to right: negative to positive, \\u000020;\\u000015;:::15;20. Per-\\nturbations are along PC6 of U-Net output. Stable Diffusion, PNDM sampler, seed 100. Prompt: \\u201ca portrait of an aristocrat\\u201d. Diffusion Models Generate Images Like Painters\\nFigure 18. Example perturbation experiment result. Different rows represent different perturbation times, from top to bottom\\n5;10;:::50. Different columns represent different perturbation scales, left to right: negative to positive, \\u000020;\\u000015;:::15;20. Per-\\nturbations are along PC5 of U-Net output. Stable Diffusion, PNDM sampler, seed 101. Prompt: \\u201ca portrait of an aristocrat\\u201d. Diffusion Models Generate Images Like Painters\\nFigure 19. Example perturbation experiment result. Different rows represent different perturbation times, from top to bottom\\n5;10;:::50. Different columns represent different perturbation scales, left to right: negative to positive, \\u000020;\\u000015;:::15;20. Per-\\nturbations are along a random noise direction, which is less effective than the previous PC perturbation. Stable Diffusion, PNDM sampler,\\nseed 101. Prompt: \\u201ca portrait of an aristocrat\\u201d. Diffusion Models Generate Images Like Painters\\nFigure 20. Geometry of trajectory perturbation .A.Quanti\\ufb01cation of sample deviation due to a perturbation at time t0and strength\\nK, heatmap encodes the perceptual image similarity per LPIPS. Left panel: perturbation along PC02, Right panel: perturbation along\\na random vector. The perturbation along the PC02 is much more effective than the random pattern in affecting the sample. B.The\\nunderlying geometry, projecting the difference between the perturbed trajectory and the original trajectory onto the unit perturbation\\nvector. As we predicted the perturbation along signal manifold gets ampli\\ufb01ed, while the random perturbation decays over time. (16)\\nB.6. Impact of design choices on the geometry of diffusion models\\nThe \\ufb01eld of diffusion generative modeling has greatly advanced since the DDIM paper (Song et al., 2020). Here, we consider\\nhow making slightly different design choices (e.g. using different samplers) affects the geometry and dynamics of sampling\\ntrajectories. Diffusion Models Generate Images Like Painters\\nB.6.1. D IFFERENTIAL EQUATION SOLVER\\nStable Diffusion by default uses the deterministic PNDM sampler, but one can use other solvers. Because the choice\\nof solver slightly modi\\ufb01es the size and direction of individual time steps, changing the solver is akin to doing an early\\nperturbation on the trajectory. Consistent with this expectation, we observe a variety of effects on image generation with a\\n\\ufb01xed random seed and different solvers: sometimes there is effectively no difference (e.g. DDIM and PNDM), sometimes\\nthere is a slight difference, and sometimes there is a major difference (e.g. PNDM vs LMSDiscrete Solver) (Fig.B.6.2). As\\nfor trajectory geometry, usually, DDIM, PNDM, and DPMSolverMultistep create rotation-like 2D trajectories well-predicted\\nby our theory, while LMSDiscrete and EulerDiscrete create more linear 1D trajectories.\\nB.6.2. C LASSIFIER -FREE GUIDANCE STRENGTH\\nClassi\\ufb01er-free guidance (Ho & Salimans, 2022) has been widely used in conditional diffusion models as a method to generate\\nsamples highly aligned with the conditional signal (e.g. prompt). We examined the effect of the strength of classi\\ufb01er-free\\nguidance on the geometry of trajectories. We found that, generally, a higher guidance scale generates trajectories xtand\\ntrajectory differences \\u0001xtwith higher dimensionality (Fig.22 top). Furthermore, when visualizing the top PC vectors,\\na larger number of interpretable PC dimensions can be found for a higher guidance scale (Fig. 22 bottom). We also\\nobserved that with a smaller guidance value, the trajectory is usually smooth; with a higher guidance value, it induced\\nstrong oscillatory movement in the trajectory at the early phase, when combined with certain higher-order schedulers e.g.\\nthe default solver PNDM (Liu et al., 2022) (Fig. 22 middle). This effect could be a feature for the sampler to explore the\\nlandscape more. It could also be an artifact, which could be \\ufb01xed by modifying the sampler.\\nFigure 21. Effect of classi\\ufb01er-free guidance (cfg) strength and diffusion sampler on the sample . Diffusion Models Generate Images Like Painters\\nFigure 22. Effect of classi\\ufb01er-free guidance strength and sampler on trajectory geometry .Top panel : Dimensionality of trajectory\\nxtand state difference \\u0001xt, measured by explained variance of PCs. Higher guidance induces higher dimensionality in xtand\\u0001xt.\\nMiddle panel : Trajectory difference \\u0001xtprojected onto the x0;xTplane. Higher guidance induced oscillation in the search trajectory,\\nesp. for PNDM sampler. Bottom panel : Comparing the top 16 PCs of state difference \\u0001xt, for high guidance (7.5) versus low guidance\\n(1.0) trajectory from the same noise seed. The samples are shown in the middle. Trajectories sampled with higher guidance have more\\n\\u2018on-manifold\\u2019 PC dimensions. Diffusion Models Generate Images Like Painters\\nC. Endpoint estimate trajectory examples\\nFigure 23. Example endpoint estimate trajectory. CelebA-HQ, DDIM sampler, seed 129.\\nFigure 24. Example endpoint estimate trajectory. CelebA-HQ, DDIM sampler, seed 152. Diffusion Models Generate Images Like Painters\\nFigure 25. Example endpoint estimate trajectory. Stable Diffusion, PNDM sampler, seed 101. Prompt: \\u201ca portrait of an aristocrat\\u201d.\\nFigure 26. Example endpoint estimate trajectory. Stable Diffusion, PNDM sampler, seed 107. Prompt: \\u201ca portrait of an aristocrat\\u201d. Diffusion Models Generate Images Like Painters\\nD. Notation correspondence\\nDiffusion models usually have forward processes whose conditional probabilities are\\np(xtjx0) =N(Atx0;BtI) (20)\\nfor allt2[0;T]. In the limit of small time steps, the transition probability distribution can be captured by the SDE\\n_xt=\\u0000Ctxt+Dt\\u0011(t) (21)\\nwhere \\u0011(t)is a vector of independent Gaussian white noise terms.\\nPapers discussing these models may use slightly different notation. In the table below, we brie\\ufb02y indicate how various\\nchoices of notation correspond to one another. To make comparing discrete and continuous models easier, we assume the\\ntime step size is \\u0001t= 1.\\nTable 3. Comparison of notation for diffusion model parameters .\\nPAPER CITATION At Bt Ct Dt\\nDDPM (H O ET AL ., 2020)p\\u0016\\u000bt 1\\u0000\\u0016\\u000bt 1\\u0000p1\\u0000\\ftp\\ft\\nDDIM (S ONG ET AL ., 2020)p\\u000bt 1\\u0000\\u000bt 1\\u0000p\\n\\u000bt=\\u000bt\\u00001p\\n1\\u0000\\u000bt=\\u000bt\\u00001\\nSTABLE DIFF. (R OMBACH ET AL ., 2022) \\u000bt \\u001b2\\nt 1\\u0000\\u000bt=\\u000bt\\u00001q\\n\\u001b2\\nt\\u0000(\\u000bt=\\u000bt\\u00001)2\\u001b2\\nt\\u00001\\nVP SDE (S ONG ET AL ., 2021) exph\\n\\u00001\\n2Rt\\n0\\f(s)dsi\\n1\\u0000exph\\n\\u0000Rt\\n0\\f(s)dsi\\n\\f(t)=2p\\n\\f(t)\\nOURS \\u000bt \\u001b2\\nt \\f(t) g(t)\\nIn the huggingface diffusers library implementation of reverse diffusion, the function alphas cumprod corresponds\\nto our\\u000b2\\nt. Diffusion Models Generate Images Like Painters\\nE. High-dimensional Gaussians are thin shells\\nUnderstanding the counterintuitive properties of Gaussians in high dimensions is important for understanding reverse\\ndiffusion dynamics in a landscape of high-dimensional Gaussian modes. Although this counterintuitive behavior is well-\\nknown (see e.g. (Song et al., 2021; Sorscher et al., 2022)), we present the relevant details here for the reader\\u2019s convenience.\\nConsider an isotropic Gaussian centered at the origin in Ddimensions, where D\\u001d1. Its probability density function is\\np(x) =1\\n(2\\u0019\\u001b2)D=2exp\\u001a\\n\\u0000x2\\n2\\u001b2\\u001b\\n: (22)\\nIn spherical coordinates with r:=p\\nx2, the probability of measuring a value near ris\\np(r) =2\\n\\u0000(D=2)(2\\u001b2)D=2rD\\u00001exp\\u001a\\n\\u0000r2\\n2\\u001b2\\u001b\\n(23)\\nwhere the factor of rD\\u00001comes from the Jacobian determinant associated with the change of coordinates. The peak of this\\ndistribution occurs at r\\u0003=p\\nD\\u00001\\u001band its mean occurs at hri=p\\nD\\u001b. Its variance is var(r) = 2\\u001b2, which suggests\\nthat almost all of the probability mass is contained in a thin shell, i.e.\\n(p\\nD\\u0000p\\n2)\\u001b\\u0014r\\u0014(p\\nD+p\\n2)\\u001b: (24)\\nAsDgets large, \\ufb02uctuations about this region become small compared to the typical radius, and essentially all probability\\nlies on the shell. By appropriately modifying this argument, it is also true for a non-isotropic Gaussian. In this more general\\ncase, the peak of p(r)occurs on (D\\u00001)-dimensional ellipsoid de\\ufb01ned by\\nxT\\u0006\\u00001x=D\\u00001: (25)\\nMost probability is concentrated on the surface of this ellipsoid, i.e.\\np\\nD\\u0000p\\n2\\u0014p\\nxT\\u0006\\u00001x\\u0014p\\nD+p\\n2: (26)\\nThe fact that D-dimensional Gaussians are essentially (D\\u00001)-dimensional ellipsoids has interesting consequences for\\nmean-reversion dynamics (like what we study in this paper) in high dimensions. As Antognini and Sohl-Dickstein (2018)\\npoint out, OU processes in high dimensions almost never reach the mode , or even get close to it, despite the de\\ufb01ning\\ncharacteristic of OU processes being mean reversion. When particles reach the ellipsoid described above, the driving force\\ntoward the mode is balanced by the noise in the process, causing them to be overwhelmingly likely to remain on the ellipsoid. Diffusion Models Generate Images Like Painters\\nF. Theory of the projected outcome ^x0\\nIn this section, we study the endpoint estimate ^x0in the case of a Gaussian distribution. We use this theory in the main text\\nto understand the short time dynamics of reverse diffusion.\\nF.1. Projected outcome for a Gaussian image distribution\\nIf the image distribution is a Gaussian, i.e. p(x) =N(\\u0016;\\u0006), then we have the explicit score function rxlogp(x) =\\n\\u0006\\u00001(\\u0016\\u0000x). The forward diffusion process yields the conditional distribution p(xtjx0) =N(\\u000btx0;\\u001b2\\ntI), whose marginal\\ndistribution is p(xt) =N(\\u000bt\\u0016;\\u000b2\\nt\\u0006+\\u001b2\\ntI). When time tincreases,\\u000btdecreases from 1 to 0, while \\u001btincreases.\\nThe time-dependent score is\\ns(x;t) =rxlogp(xt) = (\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001(\\u000bt\\u0016\\u0000x):\\nNote that in the most common implementation of diffusion models, people use a neural network to parameterize the scaled\\nnegative score function \\u000f\\u0012(x;t) =\\u0000\\u001bts(x;t). Then we can write the analytical form of the estimate ^x0(xt)as\\n^x0(xt) =xt\\u0000\\u001bt\\u000f\\u0012(xt;t)\\n\\u000bt(27)\\n=xt+\\u001b2\\nts(xt;t)\\n\\u000bt(28)\\n=xt+\\u001b2\\nt(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001(\\u000bt\\u0016\\u0000xt)\\n\\u000bt: (29)\\nLet\\u2019s examine the difference between ^x0(xt)and the mean \\u0016. We \\ufb01nd that\\n^x0(xt)\\u0000\\u0016=1\\n\\u000bt(xt\\u0000\\u000bt\\u0016+\\u001b2\\nt(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001(\\u000bt\\u0016\\u0000xt)) (30)\\n=\\u001b2\\nt\\n\\u000bt(1\\n\\u001b2\\ntI\\u0000(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001)(xt\\u0000\\u000bt\\u0016):\\nWhat can we learn from this formula? We explore several consequences of it below.\\nThe projected outcome is always exact for a delta function. If the initial distribution p(x0)is a delta function, \\u0006!0.\\nThen at any time t,p(xt) =N(\\u000bt\\u0016;\\u001b2\\ntI), and the projected outcome is ^x0(xt)\\u0000\\u0016\\u00110;8xt;8t. Hence, the projected\\noutcome is always exact, regardless of the position of xtand the time t. This can be regarded as one justi\\ufb01cation for this\\nstatistic ^x0(xt): it is exact and invariant in the isotropic score \\ufb01eld created by a point distribution.\\nThis point may be relevant for understanding the very end of reverse diffusion dynamics, when xtis likely to live within a\\nscore \\ufb01eld created by a single point. At such a time, ^x0(xt)scarcely changes.\\nProjected outcome for an isotropic Gaussian. When \\u0006= \\u0016\\u001b2I, Eq. 30 takes a particularly simple form:\\n^x0(xt)\\u0000\\u0016=1\\n\\u000bt\\u000b2\\nt\\u0016\\u001b2\\n\\u001b2\\nt+\\u000b2\\nt\\u0016\\u001b2(xt\\u0000\\u000bt\\u0016): (31)\\nF.2. Low rank \\u0006\\nImages are commonly recognized as residing on a low-dimensional manifold. To study analytically what this might imply\\nfor diffusion models that generate images, we consider the speci\\ufb01c case of a Gaussian whose covariance matrix \\u0006is\\ndegenerate and low rank. Let \\u0006=U\\u0003UTbe the eigendecomposition or compact SVD of the covariance matrix, where U\\nis aD\\u0002rsemi-orthogonal matrix whose columns are normalized (i.e. UTU=Ir), and \\u0003is ther\\u0002rdiagonal eigenvalue\\nmatrix. Denote the kth column of Ubyukand thekth diagonal element of \\u0003by\\u0015k. Diffusion Models Generate Images Like Painters\\nWe can use the Woodbury matrix inversion identity to write\\n(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001= (\\u001b2\\ntI+\\u000b2\\ntU\\u0003UT)\\u00001(32)\\n=1\\n\\u001b2\\ntI\\u00001\\n\\u001b4\\ntU(1\\n\\u000b2\\nt\\u0003\\u00001+1\\n\\u001b2\\ntUTU)\\u00001UT(33)\\n=1\\n\\u001b2\\ntI\\u00001\\n\\u001b4\\ntU(1\\n\\u000b2\\nt\\u0003\\u00001+1\\n\\u001b2\\ntIr)\\u00001UT: (34)\\nDe\\ufb01ne the diagonal matrix\\n~\\u0003t=1\\n\\u001b2\\nt(1\\n\\u000b2\\nt\\u0003\\u00001+1\\n\\u001b2\\ntIr)\\u00001(35)\\n=1\\n\\u001b2\\ntdiag[(\\u000b2\\nt\\u0015k)\\u00001+ (\\u001b2\\nt)\\u00001)\\u00001] (36)\\n=diag\\u0014\\u000b2\\nt\\u0015k\\n\\u000b2\\nt\\u0015k+\\u001b2\\nt\\u0015\\n: (37)\\nThe diagonal elements of this matrix have the same form as the function\\n\\u001e(\\u0015;t) :=\\u000b2\\nt\\u0015\\n\\u000b2\\nt\\u0015+\\u001b2\\nt: (38)\\nTo improve our intuition about the temporal behavior of these diagonal elements, we plotted this function (Fig.27).\\nWe can use the matrix ~\\u0003tto write (\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001as\\n(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001=1\\n\\u001b2\\nt(I\\u0000U~\\u0003tUT): (39)\\nUsing this result, we can write the endpoint estimate as\\n^x0(xt)\\u0000\\u0016=\\u001b2\\nt\\n\\u000bt(1\\n\\u001b2\\ntI\\u0000(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001)(xt\\u0000\\u000bt\\u0016) (40)\\n=1\\n\\u000btU~\\u0003tUT(xt\\u0000\\u000bt\\u0016): (41)\\nThis formula has a series of interesting implications.\\nProjected outcome stays on image manifold. Note that the deviation of the endpoint estimate from the distribution mean\\n^x0(xt)\\u0000\\u0016always remains in the subspace spanned by the columns of U, i.e.^x0(xt)\\u0000\\u00162span(U). This an interesting\\nresult: if the data distribution is a low-dimensional manifold (e.g. image manifold), the estimate ^x0will not deviate from\\nthis manifold. Even if the projected outcome does not exactly re\\ufb02ect the true outcome, it will not make errors out of the\\nimage manifold, i.e. orthogonal to high variance directions. Visually, such directions would correspond to random noise,\\nand hence perturbations in those directions would be \\u2018nonsense\\u2019 perturbations to images.\\nProjected outcome starts around the center of the distribution. At the start of the reverse diffusion, t\\u0019T,\\u000bt\\u00190,\\nand\\u001b2\\nt\\u001d\\u000b2\\nt. This means ~\\u0003t\\u00190and^x0(xt)\\u0000\\u0016\\u00190. Hence, the projected outcome ^x0initially corresponds to the\\ncenter of the distribution. For multi-class unconditional generation, this center will be relatively class-ambiguous. For the\\nunconditional generation of faces, we observed that ^x0(xT)points to a generic face, which is close to the \\u2018average face\\u2019.\\nProjected outcome ends in the real outcome. At the end of reverse diffusion, t= 0,\\u001bt= 0;and\\u000bt= 1. This means\\n^x0(x0) =x0, i.e. the projected outcome corresponds to the real result of reverse diffusion.\\nFeatures emerge in descending order of their variance. Consider the projection of this vector ^x0(xt)\\u0000\\u0016on an\\neigenvector ukof\\u0006:\\nuT\\nk(^x0(xt)\\u0000\\u0016) =1\\n\\u000bt\\u000b2\\nt\\u0015k\\n\\u000b2\\nt\\u0015k+\\u001b2\\ntuT\\nk(xt\\u0000\\u000bt\\u0016): (42) Diffusion Models Generate Images Like Painters\\nFigure 27. The function \\u001e(\\u0015;t)with different \\u0015values, using a common \\u000btschedule.\\nWhen\\u001b2\\nt\\u001d\\u000b2\\nt\\u0015k, i.e. when the noise scale is much larger than the signal scale, we have\\nuT\\nk(^x0(xt)\\u0000\\u0016)\\u001cuT\\nk(xt\\u0000\\u000bt\\u0016)\\n\\u000bt; (43)\\nso the projected outcome is approximately 0. At the other extreme, when \\u001b2\\nt\\u001c\\u000b2\\nt\\u0015k, the signal variance is much bigger\\nthan the noise variance, and we have\\nuT\\nk(^x0(xt)\\u0000\\u0016)\\u0019uT\\nk(xt\\u0000\\u000bt\\u0016)\\n\\u000bt: (44)\\nTo interpret this, note that if we regard ukas a feature direction, then \\u0015kis the variance along this feature direction. The\\nabove equation tells us that this feature stays around the mean value of the distribution when the noise variance is much\\nlarger than the scaled variance of this feature. In plain language, when the signal scale along a certain dimension is less than\\nthe noise scale, the projected outcome along that dimension will remain undetermined.\\nEmpirically, people have found that natural image spaces have spectra close to 1=f(Ruderman, 1994), which means more\\nimage variance exists in low-frequency features than in high-frequency features. Thus, we expect that in the generating\\nprocess, the projected outcome ^x0(xt)will specify the low-frequency layout \\ufb01rst, and then gradually add the high-frequency\\ndetails.\\nInvariance of projected outcome to off-manifold perturbations. Consider a perturbation of xtat timetby\\u000ex. The\\nprojected outcome will change by an amount\\n^x0(xt+\\u000ex)\\u0000^x0(xt) =1\\n\\u000btU~\\u0003tUT\\u000ex: (45)\\nNotice that if xtis perturbed in a direction orthogonal to the manifold spanned by the columns of U, the projected outcome\\n^x0will not change. Thus, perturbation in a random non-signal direction will not change the projected outcome of reverse\\ndiffusion (and it will also, as we will see, not affect the real result either). In contrast, if the perturbation is aligned with the\\nimage manifold, the perturbation willaffect the projected outcome. Diffusion Models Generate Images Like Painters\\nEffect of a perturbation decreases over time. As we can see from the formula above, the effect of the same perturbation\\nchanges over time. After \\u000b2\\nt\\u0015k\\u001d\\u001b2\\nt, the term\\u000b2\\nt\\u0015k\\n\\u000b2\\nt\\u0015k+\\u001b2\\ntis approximately 1, so1\\n\\u000bt\\u000b2\\nt\\u0015k\\n\\u000b2\\nt\\u0015k+\\u001b2\\ntwill decrease over time. We\\npredict on this basis that after certain features emerge, perturbations along those dimensions will have a limited effect.\\nG. Theory of state space dynamics xt\\nIn this section, we study the dynamics of the latent state xtand endpoint estimate ^x0. These results are used in Section 4 of\\nthe main text.\\nG.1. Alternative form of state dynamics\\nUsing the form of the projected outcome (Eq. 30), we can write the probability \\ufb02ow ODE (Eq. 6) as\\n_x=\\u0000\\f(t)x\\u00001\\n2g2(t)rxlogp(x;t) (46)\\n=\\u0000\\f(t)x\\u00001\\n2g2(t)\\n\\u001b2\\nt(\\u000bt^x0(x)\\u0000x): (47)\\nNotice that\\n\\u000b(t) = exp(\\u0000Zt\\n0\\f(\\u001c)d\\u001c)d\\ndt\\u000b(t) =\\u0000\\f(t)\\u000b(t); (48)\\nwhich allows us to write\\nd\\ndt(xt\\n\\u000bt) =_xt\\n\\u000bt\\u0000_\\u000btxt\\n\\u000b2\\nt(49)\\n=_xt\\n\\u000bt+\\ftxt\\n\\u000bt(50)\\nd\\ndt(xt\\n\\u000bt) =\\u00001\\n2g2(t)\\n\\u001b2\\nt(^x0(xt)\\u0000xt\\n\\u000bt): (51)\\nFrom this, we can see that the quantity xt=\\u000btis being isotropically attracted towards the moving target ^x0(xt)at a rate\\ndetermined by1\\n2g2(t)\\n\\u001b2\\nt.\\nG.2. Analytical solution of reverse diffusion dynamics for a Gaussian distribution\\nFor a multivariate Gaussian distribution, the exact ODE governing the reverse diffusion sampling process is\\n_x=\\u0000\\f(t)x\\u00001\\n2g2(t)(\\u001b2\\ntI+\\u000b2\\nt\\u0006)\\u00001(\\u000bt\\u0016\\u0000x): (52)\\nUsing the eigendecomposition of \\u0006described in the previous section,\\n_x=\\u0000\\f(t)x\\u00001\\n2g2(t)1\\n\\u001b2\\nt(I\\u0000U~\\u0003tUT)(\\u000bt\\u0016\\u0000x) (53)\\nwhere we recall that ~\\u0003is de\\ufb01ned to be the time-dependent diagonal matrix\\n~\\u0003t=diag\\u0014\\u000b2\\nt\\u0015k\\n\\u000b2\\nt\\u0015k+\\u001b2\\nt\\u0015\\n: (54)\\nConsider the dynamics of the quantity xt\\u0000\\u000bt\\u0016. Using the relationship between \\ftand\\u000bt, we have\\nd\\ndt(xt\\u0000\\u000bt\\u0016) =_xt\\u0000\\u0016_\\u000bt (55)\\n=_xt+\\ft\\u000bt\\u0016 (56)\\n=\\ft(\\u000bt\\u0016\\u0000x)\\u00001\\n2g2(t)1\\n\\u001b2\\nt(I\\u0000U~\\u0003tUT)(\\u000bt\\u0016\\u0000x) (57)\\n= (1\\n2g2(t)1\\n\\u001b2\\nt(I\\u0000U~\\u0003tUT)\\u0000\\ftI)(x\\u0000\\u000bt\\u0016): (58) Diffusion Models Generate Images Like Painters\\nIf we assume that the forward process is a variance-preserving SDE, then \\ft=1\\n2g2(t), which implies \\u000b2\\nt= 1\\u0000\\u001b2\\nt. Using\\nthis, we obtain\\nd\\ndt(xt\\u0000\\u000bt\\u0016) =\\ft(1\\n\\u001b2\\nt(I\\u0000U~\\u0003tUT)\\u0000I)(x\\u0000\\u000bt\\u0016) (59)\\n=\\ft((1\\n\\u001b2\\nt\\u00001)I\\u00001\\n\\u001b2\\ntU~\\u0003tUT))(x\\u0000\\u000bt\\u0016): (60)\\nDe\\ufb01ne the variable yt:=xt\\u0000\\u000bt\\u0016. We have just shown that its dynamics are fairly \\u2018nice\\u2019, in the sense that the above\\nequation is well-behaved separable linear ODE. As we are about to show, it is exactly solvable.\\nWrite ytin terms of the orthonormal columns of Uand a component that lies entirely in the orthogonal space U?:\\nyt=y?(t) +rX\\nk=1ck(t)uk;y?(t)2U?: (61)\\nThe dynamics of the coef\\ufb01cient ck(t)attached to the eigenvector ukare\\n_ck(t) =d\\ndt(uT\\nkyt) =\\ft((1\\n\\u001b2\\nt\\u00001)\\u00001\\n\\u001b2\\nt\\u000b2\\nt\\u0015k\\n\\u000b2\\nt\\u0015k+\\u001b2\\nt)(uT\\nkyt) (62)\\n=\\ft\\n\\u001b2\\nt(1\\u0000\\u001b2\\nt\\u0000\\u000b2\\nt\\u0015k\\n\\u000b2\\nt\\u0015k+\\u001b2\\nt)ck(t) (63)\\n=\\ft\\u000b2\\nt\\n\\u001b2\\nt(1\\u0000\\u0015k\\n\\u000b2\\nt\\u0015k+\\u001b2\\nt)ck(t): (64)\\nUsing the constraint that \\u000b2\\nt+\\u001b2\\nt= 1, this becomes\\n_ck(t) =\\ft\\u000b2\\nt(1\\u0000\\u0015k)\\n\\u000b2\\nt\\u0015k+\\u001b2\\ntck(t): (65)\\nFor the orthogonal space component y?(t), it will stay in the orthogonal space U?, and more speci\\ufb01cally the 1D space\\nspanned by the initial y?(t)\\u2014so, when going backward in time, its dynamics is simply a downscaling of y?(T).\\n_y?(t) =\\ft(1\\n\\u001b2\\nt\\u00001)y?(t) =\\ft1\\u0000\\u001b2\\nt\\n\\u001b2\\nty?(t) =\\ft\\u000b2\\nt\\n\\u001b2\\nty?(t): (66)\\nCombining these two results and solving the ODEs in the usual way, we have the trajectory solution\\nyt=d(t)y?(T) +rX\\nk=1ck(t)uk (67)\\nd(t) = exp\\u0012Zt\\nTd\\u001c\\f\\u001c\\u000b2\\n\\u001c\\n\\u001b2\\u001c\\u0013\\n(68)\\nck(t) =ck(T) exp\\u0012Zt\\nTd\\u001c\\f\\u001c\\u000b2\\n\\u001c(1\\u0000\\u0015k)\\n\\u000b2\\u001c\\u0015k+\\u001b2\\u001c)\\u0013\\n: (69)\\nThe initial conditions are\\nck(T) =uT\\nkyT (70)\\ny?(T) =yT\\u0000rX\\nk=1ck(T)uk;y?(T)2U?: (71)\\nTo solve the ODEs, it is helpful to use a particular reparameterization of time. In particular, consider a reparameterization in Diffusion Models Generate Images Like Painters\\nterms of\\u000btusing the relationship \\u0000\\ft\\u000btdt=d\\u000bt. The integral we must do is\\nZt\\nTd\\u001c\\f\\u001c\\u000b2\\n\\u001c(1\\u0000\\u0015k)\\n\\u000b2\\u001c\\u0015k+\\u001b2\\u001c=Zt\\nTd\\u001c\\f\\u001c\\u000b2\\n\\u001c(1\\u0000\\u0015k)\\n1 +\\u000b2\\u001c(\\u0015k\\u00001)(72)\\n=Z\\u000bt\\n\\u000bTd\\u000b\\u001c\\u000b\\u001c(\\u0015k\\u00001)\\n1 +\\u000b2\\u001c(\\u0015k\\u00001)(73)\\n=1\\n2log(1 +\\u000b2\\n\\u001c(\\u0015k\\u00001))\\f\\f\\f\\u000bt\\n\\u000bT(74)\\n=1\\n2log\\u00121 + (\\u0015k\\u00001)\\u000b2\\nt\\n1 + (\\u0015k\\u00001)\\u000b2\\nT\\u0013\\n: (75)\\nNote that taking \\u0015k= 0gives us the solution to dynamics in the directions orthogonal to the manifold. We have\\nck(t) =ck(T)s\\n1 + (\\u0015k\\u00001)\\u000b2\\nt\\n1 + (\\u0015k\\u00001)\\u000b2\\nT(76)\\nd(t) =s\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nT: (77)\\nThe time derivatives of these coef\\ufb01cients are\\n_ck(t) =ck(T)\\u0000(\\u0015k\\u00001)\\u000b2\\nt\\ftp\\n(1 + (\\u0015k\\u00001)\\u000b2\\nT)(1 + (\\u0015k\\u00001)\\u000b2\\nt)(78)\\n_d(t) =\\u000b2\\nt\\ftp\\n(1\\u0000\\u000b2\\nT)(1\\u0000\\u000b2\\nt): (79)\\nFinally, we can write out the explicit solution for the trajectory xt:\\nxt=\\u000bt\\u0016+d(t)y?(T) +rX\\nk=1ck(t)uk: (80)\\nWe can see that there are three terms: 1) \\u000bt\\u0016, an increasing term that scales up to the mean \\u0016of the distribution; 2)\\nd(t)y?(T), a decaying term downscaling the residual part of the initial noise vector, which is orthogonal to the data\\nmanifold; and 3) the ck(t)uksum, each term of which has independent dynamics.\\nWe also now have the analytical solution for the projected outcome:\\n^x0(xt)\\u0000\\u0016=1\\n\\u000btU~\\u0003tUT(xt\\u0000\\u000bt\\u0016) (81)\\n=rX\\nk=1ck(t)\\u000bt\\u0015k\\n\\u000b2\\nt\\u0015k+\\u001b2\\ntuk\\n=rX\\nk=1ck(T)\\u000bt\\u0015kp\\n(\\u000b2\\nt\\u0015k+\\u001b2\\nt)(\\u000b2\\nT\\u0015k+\\u001b2\\nT)uk:\\nSimilarly, we can write out the exact formula for the trajectory\\u2019s tangent vector _xt:\\n_xt= _\\u000bt\\u0016+_d(t)y?(T) +rX\\nk=1_ck(t)uk (82)\\n=\\u0000\\u000bt\\ft\\u0016+\\u000b2\\nt\\ftp\\n(1\\u0000\\u000b2\\nT)(1\\u0000\\u000b2\\nt)y?(T)\\u0000rX\\nk=1ck(T)(\\u0015k\\u00001)\\u000b2\\nt\\ftp\\n(1 + (\\u0015k\\u00001)\\u000b2\\nT)(1 + (\\u0015k\\u00001)\\u000b2\\nt)uk: Diffusion Models Generate Images Like Painters\\nTo simply notation, we de\\ufb01ne two functions\\n (t;\\u0015) :=s\\n\\u001b2\\nt+\\u000b2\\nt\\u0015\\n\\u001b2\\nT+\\u000b2\\nT\\u0015(83)\\n\\u0018(t;\\u0015) :=\\u000bt\\u0015p\\n(\\u000b2\\nt\\u0015+\\u001b2\\nt)(\\u000b2\\nT\\u0015+\\u001b2\\nT): (84)\\nIn terms of these functions, our solutions can be written as\\nxt=\\u000bt\\u0016+ (t;0)y?(T) +rX\\nk=1 (t;\\u0015k)ck(T)uk (85)\\n^x0(xt) =\\u0016+rX\\nk=1\\u0018(t;\\u0015k)ck(T)uk: (86)\\nThe dynamics of  (t;\\u0015k)and\\u0018(t;\\u0015k)are plotted in main text (Fig.4). Finally, we note that there are various special\\nrelationships between  (t;\\u0015),\\u001e(t;\\u0015), and\\u0018(t;\\u0015):\\n\\u0015d\\ndt (t;\\u0015) =\\u0000(\\u0015\\u00001)\\ft\\u000bt\\u0018(t;\\u0015) (87)\\n\\u0018(t;\\u0015) =1\\n\\u000bt (t;\\u0015)\\u001e(t;\\u0015): (88)\\nH. Reverse diffusion dynamics look approximately like a 2D rotation\\nIn this section, we derive Eq. 4.2 from the main text, which quanti\\ufb01es the extent to which the dynamics of xtlook like a\\nrotation within a 2D plane spanned by x0(the reverse diffusion endpoint) and xT(the initial noise). We will derive the\\nformula by assuming that the training set consists of a single Gaussian mode, but will comment later on the fact that this\\nassumption can be relaxed somewhat.\\nAssume that reverse diffusion begins at time Twith\\u000bT\\u00190, and ends at time t= 0with\\u000b0= 1. Using Eq. 15, at some\\nintermediate time twe have that\\nxt=\\u000bt\\u0016+s\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nTy?(T) +rX\\nk=1s\\n1 + (\\u0015k\\u00001)\\u000b2\\nt\\n1 + (\\u0015k\\u00001)\\u000b2\\nTck(T)uk: (89)\\nIt is also true, by substituting t= 0andt=T, that\\n\\u0016=x0\\u0000rX\\nk=1s\\n\\u0015k\\n1 + (\\u0015k\\u00001)\\u000b2\\nTck(T)uk\\ny?(T) =xT\\u0000\\u000bT\\u0016\\u0000rX\\nk=1ck(T)uk:(90)\\nUsing these two equations, we can rewrite Eq. 89 as\\nxt=\\u000bt\\u0016+s\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nT\\\"\\nxT\\u0000\\u000bT\\u0016\\u0000rX\\nk=1ck(T)uk#\\n+rX\\nk=1s\\n1 + (\\u0015k\\u00001)\\u000b2\\nt\\n1 + (\\u0015k\\u00001)\\u000b2\\nTck(T)uk\\n=\\\"\\n\\u000bt\\u0000\\u000bTs\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nT#\\n\\u0016+s\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nTxT+rX\\nk=1(s\\n1 + (\\u0015k\\u00001)\\u000b2\\nt\\n1 + (\\u0015k\\u00001)\\u000b2\\nT\\u0000s\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nT)\\nck(T)uk\\n=\\\"\\n\\u000bt\\u0000\\u000bTs\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nT#\\nx0+s\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nTxT+R(t)(91) Diffusion Models Generate Images Like Painters\\nwhere the remainder term R(t)is equal to\\nR(t) =rX\\nk=1(s\\n1 + (\\u0015k\\u00001)\\u000b2\\nt\\n1 + (\\u0015k\\u00001)\\u000b2\\nT\\u0000s\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nT\\u0000\\\"\\n\\u000bt\\u0000\\u000bTs\\n1\\u0000\\u000b2\\nt\\n1\\u0000\\u000b2\\nT#s\\n\\u0015k\\n1 + (\\u0015k\\u00001)\\u000b2\\nT)\\nck(T)uk: (92)\\nThe expression simpli\\ufb01es somewhat if we take \\u000bT= 0. Doing so, we obtain the equation seen in the main text:\\nxt\\u0019\\u000btx0+q\\n1\\u0000\\u000b2\\ntxT+rX\\nk=1\\u001aq\\n\\u001b2\\nt+\\u0015k\\u000b2\\nt\\u0000\\u000btp\\n\\u0015k\\u0000\\u001bt\\u001b\\nck(T)uk:\\nIf\\u000bTis not very close to zero, there may be a small \\ufb01rst-order correction to the rotation-like dynamics, even if the sum\\nterms are approximately zero.\\nAlthough we derived this formula by assuming a single Gaussian mode, its form does not actually depend on the details\\nof the nearest mode. In the landscape picture, although the mean and variance of the nearest mode regularly change in a\\ndiscontinuous way, the form of our rotation equation should stay the same.\",\"24\":\" Con\\ufb02ict-driven Structural Learning Towards Higher\\nCoverage Rate in ATPG\\nHui-Ling Zhen1, Naixing Wang2, Junhua Huang1, Xinyue Huang2, Mingxuan Yuan1and Yu Huang2\\n1. Noah\\u2019s Ark Lab, Huawei; 2. Hisilicon, Huawei\\nAbstract \\u2014Due to the increasing challenges posed by the\\nrelentless rise in the design complexity of integrated circuits,\\nBoolean Satis\\ufb01ability (SAT) has emerged as a robust alternative\\nto structural APTG techniques. However, the high cost of trans-\\nforming a circuit testing problem to a Conjunctive Normal Form\\n(CNF) limits the application of SAT in industrial ATPG scenarios,\\nresulting in a loss of test coverage. In Order to address this\\nproblem, this paper proposes a con\\ufb02ict-driven structural learning\\n(CDSL) ATPG algorithm \\ufb01rstly, in which the con\\ufb02ict-driven\\nheuristic methods in modern SAT solver are implemented on the\\nlogic cone of fault propagation and activation directly. The pro-\\nposed CDSL algorithm is composed of three parts: (1) According\\nto the implication graph, various con\\ufb02ict constraints have been\\nlearned to prune search space. (2) Con\\ufb02ict-driven implication\\nand justi\\ufb01cation have been applied to increase decision accuracy\\nand solving ef\\ufb01ciency. (3) A con\\ufb02ict-based diagnosis method is\\nfurther proposed in the case of low coverage debug, leading to\\nmaking the aborted faults testable by relaxing or modifying some\\nconstraints on primary inputs. Extensive experimental results on\\nindustrial circuits demonstrate the effectiveness and ef\\ufb01ciency\\nof the proposed CDSL algorithm. It is shown that compared\\nwith the SAT-based ATPG, the proposed CDSL can on average\\ndecrease 25:6%aborted faults with 94:51% less run time. With\\na two-stage computational \\ufb02ow, it has shown that the proposed\\nCDSL can lead to 46:37% less aborted faults than a one-stage\\nstructural algorithm, further with the 3:19% improvement on\\nfault coverage. In addition, the con\\ufb02ict diagnosis can lead to\\n8:89% less aborted faults on average, and 0:271% improvement\\nin fault coverage rate.\\nIndex Terms \\u2014Con\\ufb02ict-driven, ATPG, Con\\ufb02ict Diagnosis\\nI. I NTRODUCTION\\nContinuous progress in decreasing device sizes and in-\\ncreasing design complexity has brought increasing demand\\nfor high product quality and low defective parts-per-million\\n(DPPM) goals. Thus, scan-based structural testing has become\\neven more important than ever, and Automatic Test Pattern\\nGeneration (ATPG) has served as an essential procedure for\\ngenerating appropriate test patterns for testing logical faults\\nthat model physical defects.\\nGiven a targeted fault of the circuit-under-test, the goal of\\nATPG is to either generate a test pattern for the targeted\\nfault (i.e., \\ufb01nding the test vector that can differentiate the\\ngood and faulty machines and that such fault is detectable)\\nor prove that it is undetectable (i.e. there is no test vector\\nthat can differentiate the good and faulty machines). There\\nhave been several structural algorithms for ATPG, such as D-\\nalgorithm [1] and its advanced variants [2], [3].\\nThere are two core problems in ATPG. One is how to\\nimprove decision ef\\ufb01ciency under a given backtrack limit,\\nespecially considering a large number of hard-to-detect faults\\nin today\\u2019s complex designs. There mainly exist two methods\\nto solve this problem. One is to utilize Boolean Satis\\ufb01ability\\n(SAT) solver directly [4], [5]. Unlike structural ATPG working\\non a circuit network, SAT-based ATPG makes use of symboliccalculation techniques to implement ef\\ufb01cient con\\ufb02ict-driven\\nsearch on the Conjunctive Normal Form (CNF). Many SAT-\\nbased ATPG algorithms have been proposed, such as TG-\\nPro [6], TIGUAN [7], and PASSAT [8]. Similar SAT-based\\ntechniques have been applied, so as to insert test points for\\nlow-capture-power testing while maintaining the same fault\\ncoverage [9].\\nFig. 1. A hybrid computational \\ufb02ow\\nin ATPG, which begins at the struc-\\ntural ATPG and ends with the SAT.A hybrid computational\\n\\ufb02ow composed of struc-\\ntural ATPG and SAT-based\\nalgorithms has been pro-\\nposed, as shown in Fig- in ATPG, which begins at the struc-\\ntural ATPG and ends with the SAT.A hybrid computational\\n\\ufb02ow composed of struc-\\ntural ATPG and SAT-based\\nalgorithms has been pro-\\nposed, as shown in Fig-\\nure 1 [4]. Here, the struc-\\ntural ATPG algorithm is\\nadopted \\ufb01rstly under a\\ngiven backtrack limit and\\nit targets relatively easy-\\nto-detect faults, which can\\nbe detected via a test\\npattern or proved to be\\nundetectable. Then SAT\\ntargets the hard-to-detect\\nfaults which are aborted\\nby the structural ATPG.\\nUnlike structural ATPG,\\nwhich is performed directly on the circuit, SAT-based algo-\\nrithms rely on the CNF transformed from the logic cone of\\nfault propagation and activation. This transformation is an\\nextra step in SAT-based algorithms.\\nFig. 2. Comparison between the CNF generation time vs. solving\\ntime (in CPU microseconds). The horizontal axis is the fault index,\\nwhile the vertical axis is the respective runtime.\\nTake one circuit Stuck 4as an example (with additional\\ndetails provided in Section IV), we demonstrate a signi\\ufb01cant\\nchallenge for SAT in Figure 2. The \\ufb01gure examines the time\\nrequired for the transformation from the circuit to CNF in\\ncomparison to the related solving time. All targeted faults are\\nstuck-at, and the SAT-based framework follows TG-Pro [6].\\nThe chosen SAT Solver is Kissat [10], a reference SAT solverarXiv:2303.02290v1  [cs.AI]  4 Mar 2023 in SAT competition 2022. It is revealed that the transforma-\\ntion process requires more runtime than solving itself. This\\nindicates that despite the signi\\ufb01cant advancements made in\\nSAT solver, which have displayed considerable potential in\\nsolving ATPG problems [5], the additional overhead required\\nfor CNF transformation limits SAT\\u2019s applications in industrial\\nATPG. Several works have been done to alleviate this problem.\\nAn incremental SAT-based framework has been proposed\\nwhich aims to generate CNF incrementally and \\ufb01nd the \\ufb01nal\\nsolutions, or prove UNSAT, via partial CNF, hence decreasing\\nthe transformation time as well as solving time on average [4].\\nPreprocessing methods have been utilized to simplify the logic\\ncone of fault propagation and activation, leading to a decrease\\nin the generation and solving time by solving an equivalent\\nsubstitute [11].\\nNevertheless, the CNF transformation on large-scale circuits\\nremains a big bottleneck, resulting in utilizing SAT solver\\nbeing directly limited. Thus, the second method is to attempt\\nto utilize SAT\\u2019s heuristics on the circuit. A con\\ufb02ict-driven\\nrecursive learning which combines with a dynamic decision\\nordering technique has been proposed to resolve hard-to-\\nresolve faults [12]. A con\\ufb02ict-driven implication method has\\nbeen proposed to improve the justi\\ufb01cation ef\\ufb01ciency and avoid\\nthe over-speci\\ufb01cations of test vectors [13]. An untestable\\ndebug algorithm has also been utilized for low test coverage\\nanalysis [14]. However, the method of constructing learning\\ncon\\ufb02icts in modern SAT solvers, like the unique implication\\npoint (UIP), has not been considered.\\nThe other problem is that the ATPG constraints are usually\\nconservative during the early stage of the design [14]. The\\nconservatism often results in the implementation not being\\nsuf\\ufb01ciently mature in practice. Therefore, in the early stages,\\nthe DFT engineers have some degree of freedom to relax or\\nmodify certain constraints, making that some of the aborted\\nfaults as well as untestable faults which are not led by the\\ncircuit structure can be potentially resolved. To address this\\nissue, we employ a con\\ufb02ict diagnosis approach after running\\nATPG engine to resolve low test coverage. Take one aborted\\nfault as an example. We consider that the reason for abortion\\nis due to the encountered con\\ufb02icts exceeding the backtrack\\nlimit. Finally, the statistical analysis for the learnt con\\ufb02icts\\nwould provide meaningful suggestions to the DFT engineer,\\nleading to a decrease in the number of aborted or untestable\\nfaults and improving the coverage rate.\\nMotivated by the aforementioned discussions, this paper\\nproposes a con\\ufb02ict-driven structural learning (CDSL) ATPG\\nalgorithm, which aims to utilize the advantages brought by\\nthe structural ATPG and SAT-based algorithms. To summarize,\\nour contributions include:\\n(i)We \\ufb01rst build learnt con\\ufb02ict-based constraints di-\\nrectly on circuit, aiming to prune the searching space by using\\nthe optimization process data. According to the implication\\ngraph which is directly related to the decision-making process,\\nwe construct two kinds of con\\ufb02ict constraints, i.e., decision-\\nvariable-based constraint and UIP-based constraint, leading to\\navoiding meaningless searching in subsequent iterations.\\n(ii)We adopt the con\\ufb02ict-driven decision rules to im-\\nprove the decision accuracy. After accumulating the learnt\\ncon\\ufb02icts, we construct new implications and justi\\ufb01cation\\napproaches based on those con\\ufb02icts. Extensive experiments\\ndemonstrate the effectiveness of con\\ufb02ict constraints on impli-cation ef\\ufb01ciency with fewer backtracks and aborted faults.\\n(iii)We further construct the con\\ufb02ict diagnosis accord-\\ning to the learnt con\\ufb02icts in the case of low coverage debug.\\nIn this method, we utilize the learnt con\\ufb02icts to analyze the\\nreason from PIs\\u2019 constraints and relax or modify certain of\\nthem, aiming at further improving the test coverage rate.\\nThe remainder of this paper is organized as follows. After reason from PIs\\u2019 constraints and relax or modify certain of\\nthem, aiming at further improving the test coverage rate.\\nThe remainder of this paper is organized as follows. After\\nsome preliminaries in Section II, Section III presents our new\\nSAT-based ATPG approach. Experimental results are demon-\\nstrated in Section IV, in which we show the effectiveness of\\nthe proposed framework both on solution quality and runtime.\\nFinally, we conclude this work in Section V.\\nII. P RELIMINARIES\\nA. Con\\ufb02ict-Driven-Clause-Learning (CDCL) in SAT\\nSAT-based ATPG makes use of ef\\ufb01cient SAT solvers to\\nsolve APTG problems. It begins with building a CNF format\\nSAT model which represents the fault condition and prop-\\nagation between the PIs and the POs. In general, a CNF\\nformula\\u001econsists of a conjunction of clauses !, each of\\nwhich denotes a disjunction of literals. A literal is either a\\nvariablexior its complement. Each variable can be assigned\\na logic value, either 0or1. Any general Boolean problems\\ncan be represented as a CNF formula model. A SAT solver\\neither \\ufb01nds an assignment such that \\u001eis satis\\ufb01ed, or proves\\nthat no such assignment exists, i.e., UNSAT. A key heuristics\\nin modern SAT solver is Con\\ufb02ict-Driven-Clause-Learning\\n(CDCL) algorithm [5]. In general, CDCL is a Branch-and-\\nBound (BB) search framework, in which each step a literal and\\na propositional value (either 0 or 1) are selected for branching\\npurposes. A key characteristic of CDCL is to learn new clauses\\nfrom con\\ufb02icts during backtrack searches.\\nB. Structural ATPG Algorithm\\nDifferent from SAT-based algorithms, the structural ATPG\\nalgorithm is performed on the circuit directly. Until now,\\nseveral kinds of algorithms like D-algorithm, PODEM, and\\nFAN have been proposed. In practice, D-algorithm tries to\\npropagate the stuck-at-fault value denoted by D(for Stuck-\\nat-0) orD(for Stuck-at-1) to a primary output (PO) [1].\\nThe conventional D-algorithm generates a decision structure\\nto evaluate the value of every node in the circuit to obtain\\nthe test vectors. PODEM and FAN are the advanced variants\\nby limiting the searching space and accelerating backtracing,\\nwhile PODEM limits the searching space only to Primary\\nInputs (PIs) [15], and FAN limits the searching space to\\nheadlines [16].\\nC. Comparison between Structural ATPG and CDCL\\nThere exists a certain difference between CDCL and\\nstructural ATPG algorithm. The \\ufb01rst difference must root\\nin the branching rules. The structural ATPG algorithm is\\nrequirement-driven [1], which denotes that the decision or-\\nder accords with the fault propagation and circuit structural\\ncharacteristics. Unlike this, the initial decision order in CDCL\\naccords to the input literal order which is random, and this\\norder is modi\\ufb01ed based on the literal\\u2019s frequency in learnt\\ncon\\ufb02ict constraints after some backtracks. The second differ-\\nence roots the backtrack rules after con\\ufb02ict occurs. We take an\\nexample to discuss other differences, as shown in Figure 3. All\\nthe decision variables ( x0,x2,x3, andx4) are in square boxes, while all the implicated variables are in oval boxes. Each\\ndecision variable is assigned with a decision level according\\nto the decision order. The direction of the arrow is consistent\\nwith the direction of the implication.\\nFig. 3. An example of a decision-\\nmaking process. All decision vari-\\nables are in square boxes, and\\nimplications in are in oval boxes.\\nThe related decision level is also\\nlabeled.Figure 3 shows that, af-\\nter the fourth decision vari-\\nable, a con\\ufb02ict occurs (i.e.,\\nx8cannot be 0and1at the\\nsame time). In the structural\\nATPG algorithm, the deci-\\nsion pointer will backtrack\\nto the last decision variable\\n(i.e.,x3), but without analy-\\nsis of the reason for the oc-\\ncurrence of con\\ufb02icts. In the\\ngiven con\\ufb02ict-driven meth-\\nods [12]\\u2013[14], there will be\\nadded one learnt con\\ufb02ict\\nconstraintx4 6= 1 , which\\nlimits the following impli-\\ncations under new searching\\nrules. Apparently, a better\\nsearching strategy must combine both advantages of struc-\\ntural ATPG and CDCL, i.e., the branching rules follow the\\nstructural ATPG algorithm which aims to decrease the cost of\\nwrong decisions, while once con\\ufb02ict occurs, the reasons for\\ncon\\ufb02ict should be considered like CDCL to avoid same wrong\\nsearching path.\\nIII. P ROPOSED CDSL A LGORITHM\\nFig. 4. New proposed CDSL algorithm. Different from the conven-\\ntional structural ATPG algorithm, we incorporate SAT\\u2019s heuristics\\nsuch as learnt con\\ufb02ict constraints, con\\ufb02ict-driven implication, and\\ncon\\ufb02ict-driven branch\\/decision, aiming to prune the searching space\\nbased on data from the optimization process and \\ufb01nd solutions\\nor prove UNSAT, with fewer backtracks. After the new ATPG\\ncomputation, we propose to add the con\\ufb02ict diagnosis in case of\\nlow coverage.\\nConsidering the above, we propose a con\\ufb02ict-driven struc-\\ntural learning (CDSL) ATPG algorithm which combines two\\nmethods, as shown in Figure 4. Compared with the con-\\nventional structural ATPG and SAT-based ATPG algorithms,\\nthe CDSL algorithm has two advantages: (1) It accumulatescon\\ufb02ict constraints after backtracks, with the aim of avoiding\\nthe same wrong decisions and \\ufb01nding solutions with fewer\\nbacktracks. (2) It employs con\\ufb02ict-driven implications to prune\\nthe searching space and con\\ufb02ict-driven branching rules, with\\na score heuristics, to improve decision accuracy.\\nGiven a fault site, we \\ufb01rst trace the circuit to get the logic\\ncone related to fault propagation and activation. The decision\\nrules begin at the fault site and follow the conventional struc-\\ntural ATPG algorithm until one con\\ufb02ict occurs. In the process,\\nall structural ATPG algorithms like D-algorithm, PODEM, and\\nFAN can be used.\\nA. Implication Graph\\nFirstly, we construct an implication graph according to the\\ndecision-making process:\\n(1) We construct a directed acyclic graph in which each\\nvertex represents a variable\\u2019s assignment, and each incident\\nedge to a vertex represents the reason leading to that assign-\\nment. If one implication is inferred via other implications,\\nthere also exists an edge among different implications. Thus,\\ndecision variables have no incident edges in contrast to implied\\nvariables that have assignments forced during propagation.\\n(2) Each decision variable is assigned a decision level ac-\\ncording to the related decision-making order, while its related\\nimplications have the same decision level.\\nNote that each variable in CDSL\\u2019s implication graph denotes\\na logic gate. Once a con\\ufb02ict occurs, the proposed CDSL\\nalgorithm would trace the implication graph to \\ufb01nd all the his-\\ntorical assignments which result in the con\\ufb02ict and construct\\nlearnt con\\ufb02ict constraint.\\nB. Learnt Con\\ufb02ict Constraints\\nTake Figure 3 as an example, in which a con\\ufb02ict occurs\\nthroughx8, we construct two kinds of learnt con\\ufb02ict con-\\nstraints in the proposed CDSL algorithm.\\n(1) Decision Variable-based Con\\ufb02ict. The basic principle\\nis that the current con\\ufb02ict, at least, is caused by all historical\\ndecision variables. As shown in Figure 3, before the con\\ufb02ict (1) Decision Variable-based Con\\ufb02ict. The basic principle\\nis that the current con\\ufb02ict, at least, is caused by all historical\\ndecision variables. As shown in Figure 3, before the con\\ufb02ict\\noccurs, there are four decision variables, i.e., x0= 1,x1= 1\\nx2= 1 ,x3= 1 andx4= 1 , thereby we can add a\\nlearnt con\\ufb02ict constraint as x0+x1+x2+x3+x4that\\nis constructed via the decision variables. It denotes that in\\nthe following decision-making process, even though the four\\nvariables can serve as decision variables, they cannot repeat\\nthe same assignments, in other words, when it is found that\\nthree of these variables repeat the historical assignments, the\\nfourth variable must take the opposite assignment.\\n(2) Unique Implication Point (UIP)-based Con\\ufb02ict. A\\nUIP is a special node that any node at the current decision\\nlevel such that any path from the decision variable to the\\ncon\\ufb02ict node must pass through it [17]. As shown in Figure 3,\\nthe con\\ufb02ict occurs in node x8whose decision level is 4. The\\ninference of UIP-based learnt con\\ufb02ict constraints can be given\\nas follows:\\n(i) We \\ufb01rst \\ufb01nd the direct reason for the con\\ufb02ict node.\\nFigure 3 exhibits that one x8\\u2019s direct reasons are x4andx7,\\nand the other x8\\u2019s direct reason is x0andx2. Hereby, both\\nx0,x2, andx4are decision variables and their decision level\\nis0,2, and 4, respectively. x7is implications from x4,x5,\\nandx9. Thus, the direct learnt con\\ufb02ict constraint can be given\\nasx0+x2+x4+x7. (ii) Check the decision level, and we should decide whether\\nsome of the reason nodes are replaced by the corresponding\\nparents. The evaluation rule is that in the \\ufb01nal learnt con\\ufb02ict\\nconstraint, there exists only one variable whose decision level\\nis the same as the con\\ufb02ict node, and this variable is UIP.\\n(ii-a) Consider x0+x2+x4+x7, since both x7,x9, and\\nx4are in decision level 4andx4is a decision variable, we\\nutilizex7\\u2019s parent nodes (i.e., x4,x5andx9) to replace it.\\nAfter deduplication, the learnt con\\ufb02ict constraint is updated\\nasx0+x2+x4+x5+x9, in which the decision levels of x5\\nandx9are3and4, respectively.\\n(ii-b) Since x9andx4are in the same decision level, we\\nutilizex9\\u2019s parents (i.e., z1,x3andx4) to replace it, and then\\nthe learnt con\\ufb02ict is updated as x0+x2+x4+x5+z1+x3.\\nFinally, we can obtain the UIP-based learnt con\\ufb02ict con-\\nstraint asx0+x2+x4+x5+z1+x3. Considering that the\\nonly variable whose decision level is the same as the con\\ufb02ict\\nnode isx4, thus,x4serves as the UIP node. Note that we\\nonly show the learnt relationship among different variables,\\nnot including the logic values. After accumulating different\\nlearnt con\\ufb02ict constraints, the proposed CDSL algorithm will\\nutilize those in the following three aspects:\\nC.Con\\ufb02ict-driven Implications\\nAll learnt con\\ufb02ict constraints are applied for the implication\\nphase, aiming to avoid repeating the wrong searching paths.\\nTake the UIP-based learnt con\\ufb02ict constraint x0+x2+x4+\\nx5+z1+x3of Figure 3 as an example, if we \\ufb01nd that \\ufb01ve of\\nthe related variables (i.e., x0,x2,x4,x5andz1) have the same\\nassignments with historical ones, the sixth must be assigned as\\nthe opposite value. To avoid the extra computational overhead\\nwhen too many learnt con\\ufb02ict constraints are accumulated,\\nwe also add a forgotten rule in the implication phase: if one\\nlearnt con\\ufb02ict constraint is not utilized in recent Nloops,\\nthis constraint is considered to be no longer relevant and\\nit would be deleted in the following loops. Hereby, Nis a\\nhyperparameter.\\nD.Con\\ufb02ict-driven Branch Heuristics\\nThe learnt con\\ufb02ict constraints can also be applied through\\nVariable State Independent Decaying Sum (VSIDS) heuristic,\\naiming to improve the decision accuracy in the following\\ndecision phase. There are three steps in the VSIDS strategy:\\na) We start by assigning each variable a \\ufb02oating point\\nscore. When a con\\ufb02ict occurs, the activity of some variables\\nis increased by 1. In general, the initial score is set to 0.\\nb) After each con\\ufb02ict, the variable activity is decayed\\nperiodically, aiming to trade off the historical decisions and\\nfollowing ones. Such decay factor is set [0;1].\\nc) To balance VSIDS and structural strategies, we would\\ncheck each variable\\u2019s score during branching. The variable\\nwith the highest score is selected under a given probability.\\nFurther, different from the structural ATPG algorithm which\\nrequires backtracking to the last decision variable, we adopt\\nanon-chronological backtrack rule in the proposed CDSL\\nalgorithm. This rule accords with the UIP-based con\\ufb02ict con-\\nstraint, and the backtrack point is the variable that is with the\\nlargest decision level except for the UIP node. Take Figure 3\\nas an example, the scores of x0,x5,x3andx4are higher\\nthan others\\u2019 after both decision-variable-based and UIP-based\\ncon\\ufb02ict constraints are accumulated, and once one con\\ufb02ict\\noccurs, the backtrack point is chosen as x3.E. Con\\ufb02ict Diagnosis for Low Coverage Debug\\nExcept for the implications and branching, we also explore\\nadopting the con\\ufb02ict diagnosis to beat the low test coverage\\nin the initial phase of design:\\n(i) Compute each logic gate\\u2019s score according to the fre-\\nquency in the learnt con\\ufb02ict constraints.\\n(ii) Choose the top-k gates according to the score\\u2019s rank.\\nThen trace the circuit to \\ufb01nd the related external constraints.\\nUsually, those constraints are put on either primary inputs or\\nthe fan-in gates of decision level 0.\\nIn con\\ufb02ict diagnosis, we choose to relax or modify the Usually, those constraints are put on either primary inputs or\\nthe fan-in gates of decision level 0.\\nIn con\\ufb02ict diagnosis, we choose to relax or modify the\\nidenti\\ufb01ed external ATPG constraints, which would provide an\\nopportunity to make the aborted or untestable fault testable.\\nIV. E XPERIMENTAL RESULTS\\nA. Experiments Setup\\nIn this section, we aim to evaluate the proposed CDSL\\nalgorithm from the following three aspects:\\nRQ1 : Can it have a performance advantage over the traditional\\nSAT-based algorithms?\\nRQ2 : Can it be bene\\ufb01cial for improving test coverage compared\\nto the structural algorithm?\\nRQ3 : Can the con\\ufb02ict diagnosis be exploited to debug the\\naborted or untestable faults?\\nIn the following, the CDSL framework is implemented on the\\nstructural D-algorithm. and its performance is evaluated from\\ntwo perspectives, one is the number of aborted faults (unob-\\nserved faults, abbreviated as UO) under the set aborted limit,\\nthe other one is fault coverage rate, i.e., Fault Coverage =\\nNTestable\\nNTotal, whereNTotal andNTestable are the number of\\ntotal faults and testable faults, respectively. All experiments\\nare carried out for industrial circuits, and their designs are\\nshown in Table I.\\nTABLE I\\nDESIGN CHARACTERISTICS\\nCircuit Fault Type #gates #State Circuit Fault Type #gates #State\\nStuck 1 Stuck-at 246078 14979 Tran 1 Transition 139871 9644\\nStuck 2 Stuck-at 246078 14979 Tran 2 Transition 785559 26288\\nStuck 3 Stuck-at 221004 18190 Tran 3 Transition 785559 383963\\nStuck 4 Stuck-at 78600 12047 Tran 4 Transition 785559 357483\\nStuck 5 Stuck-at 221004 18190 Tran 5 Transition 221004 357483\\nStuck 6 Stuck-at 206221 15772 Tran 6 Transition 221004 331291\\nStuck 7 Stuck-at 56586 8194 Tran 7 Transition 221004 374009\\nStuck 8 Stuck-at 221004 357483 Tran 8 Transition 206221 331291\\nStuck 9 Stuck-at 246078 331291 Tran 9 Transition 206221 331291\\nStuck 10 Stuck-at 785559 26288 Tran 10 Transition 221004 331291\\nB. Evaluation on Run Time\\nTo answer RQ1 , we choose stuck-at faults to compare\\nthe proposed CDSL with SAT-based methods, as shown in\\nTable II. The \\ufb01rst column is the circuit name. The second\\nand third columns show the number of aborted faults led by\\nthe proposed CDSL algorithm and related run time (in CPU\\nseconds), respectively. Hereby, the aborted limit is set as 100.\\nThen from left to right, there are four different baselines to\\nevaluate the CDSL algorithm:\\ni) A basic SAT-based framework, TG-Pro [6]. It is also the\\nlatest open-source framework. The SAT solver is chosen as\\nKissat2022 [10]. ii) The basic D-algorithm. It is also a module of the\\nproposed CDSL algorithm.\\niii) An incremental SAT-based ATPG method with prepro-\\ncessing procedure [4].\\niv) A SAT-based ATPG method with a fault analysis mod-\\nule [18], which is a trained neural network and predicts the\\nfault classi\\ufb01cation for appropriate algorithm selection.\\nIt is shown that compared with the conventional SAT-\\nbased ATPG and structural D-algorithm, the proposed CDSL\\nalgorithm can decrease the aborted faults by 25:6% and\\n49:88% on average, while the run time is decreased by 94:51%\\nand25:88%, respectively. Although the two new variants, i.e.,\\nthe SAT-based ATPG with preprocessing or with the learnt\\nnetwork-based fault analysis can lead to fewer aborted faults\\nand better run time, the proposed CDSL can also decrease the\\nUO by 45:23% and12:35%, respectively, and the related run\\ntime can be decreased 58:79% and93:09%.\\nIt is worth mentioning that when the backtrack limit is the\\nsame, both the conventional structural ATPG and the proposed\\nCDSL algorithm can lead to fewer aborted faults than SAT-\\nbased methods. It is because the SAT\\u2019s heuristics, such as\\nbranching, restart, and local search, totally rely on the score\\nbased on accumulated con\\ufb02icts. It denotes that the limited\\ncon\\ufb02ict constraints may affect the performance of heuristics.\\nTABLE II\\nPERFORMANCE OF CDSL ONUO AND RUNTIME\\nCircuitCDSL TG-Pro Structural Incre Neural\\nUO time UO time UO time UO time UO time\\nStuck 1 147 229 174 10952 226 814 162 1528 162 9125\\nStuck 2 352 167 559 1722 793 128 638 218 475 1522\\nStuck 3 253 33 195 780 271 58 139 678 175 672\\nStuck 4 1 53 7 1103 8 101 12 206 7 856\\nStuck 5 144 18 119 393 158 36 105 79 110 326\\nStuck 6 1343 365 1318 5165 1949 1307 2125 806 986 4238\\nStuck 7 236 97 485 1389 453 92 383 234 429 1109\\nStuck 8 601 550 518 10543 664 498 836 631 492 7692\\nStuck 9 514 75 987 977 1303 812 1189 235 836 901\\nStuck 10 545 878 1197 11931 1028 984 1963 1368 975 9312\\nAverage 414 247 556 4496 825 333 755 598 465 3569\\nImprovement \\/ \\/ 25.6%94.51%49.88%25.88%45.23%58.79%12.35%93.09%\\nC. Evaluation on Coverage Rate\\nTo further compare the proposed CDSL with the structural\\nalgorithm, we construct a two-stage ATPG framework on\\ntransition faults. (i) In the \\ufb01rst stage, we set a relatively\\nsmall backtrack limit and close the con\\ufb02ict-driven modules.\\nWe aim at handling the easy-to-detect faults with a relatively\\nsmall aborted limit (The aborted limit is set 20). (ii) In the\\nsecond stage, we set a relatively large aborted limit and the\\nproposed CDSL algorithm targets the aborted faults (The\\naborted limit is set at 100). There are two baselines in the\\nfollowing experiments: (1) The \\ufb01rst baseline is the one-stage\\nconventional D-algorithm. (2) The second is also a two-stage\\nalgorithm, but the con\\ufb02ict-driven modules are closed in both\\ntwo stages. The results are shown in Table III.It is found that the one-stage conventional D-algorithm\\nresults in 8702 aborted faults on average, and the fault cov-\\nerage rate is 92:95%. However, when the same D-algorithm\\nis armed with a two-stage setting, the aborted fault can be\\ndecreased to 5975 and the fault coverage rate can reach\\n95:21%. Further, when the proposed CDSL is implemented\\nwith a two-stage setting, aborted faults can be decreased to\\n4667 , and the fault coverage rate can be increased to 96:14%.\\nIn other words, compared with the D-algorithm, the aborted\\nfaults can be decreased via 46:37% and the fault coverage\\nrate can be increased via 3:19%, while compared with the\\ntwo-stage algorithm which is without con\\ufb02ict-driven modules,\\nthe aborted faults can be decreased via 21:89% and the fault\\ncoverage rate is increased via 0:93%.\\nTABLE III\\nEVALUATION IN A TWO-STAGE FRAMEWORK\\nCircuitOne-Stage without Con\\ufb02ict Prop Model\\nUO coverage UO coverage UO coverage\\nTran 1 505 95.57% 402 96.785% 353 97.149%\\nTran 2 32319 98.71% 22710 99.109% 17154 99.325%\\nTran 3 105 97.86% 119 98.867% 98 99.029%\\nTran 4 604 97.59% 320 98.611% 214 98.928%\\nTran 5 5414 91.71% 3769 94.678% 2943 95.795% Tran 2 32319 98.71% 22710 99.109% 17154 99.325%\\nTran 3 105 97.86% 119 98.867% 98 99.029%\\nTran 4 604 97.59% 320 98.611% 214 98.928%\\nTran 5 5414 91.71% 3769 94.678% 2943 95.795%\\nTran 6 13211 90.55% 9110 93.548% 7339 94.777%\\nTran 7 14037 90.15% 9462 93.383% 7615 94.634%\\nTran 8 13436 90.50% 9152 93.603% 7364 94.819%\\nTran 9 1641 88.34% 671 91.342% 526 93.011%\\nTran 10 5757 88.53% 4043 92.25% 3067 93.97%\\nAverage 8702 92.95% 5975 95.21% 4667 96.14 %\\nImprovement 46.37% 3.19% 21.89% 0.93% \\/ \\/\\nD. Evaluation on Con\\ufb02ict Diagnosis\\nFinally, we evaluate the con\\ufb02ict diagnosis in the case of low\\ncoverage analysis. As described in Section III-E, according to\\nthe accumulated learnt con\\ufb02icts, we \\ufb01rst mark the top 5logic\\ngates. After tracing the circuits from the labeled logic gates,\\nthe con\\ufb02ict-related PI nodes are found, and the corresponding\\nlogic value is marked as \\u000bN(supposing that there are N\\nrelated PI nodes). If there exist constraints on the found PI\\nnodes, we would relax such constraints. Otherwise, if there are\\nnot any constraints on one of the found PI nodes, we prefer\\nto add a constraint on this node and the logic value is the\\nopposite of\\u000b. Finally, we recall the ATPG engine to generate\\nthe test pattern or prove the untestability. The results are given\\nin Table IV. It is shown that after the con\\ufb02ict diagnosis, the\\naborted faults decrease 8:89% on average, while the fault\\ncoverage rates increase by 0:271% .\\nV. C ONCLUSIONS\\nAiming at addressing the ef\\ufb01ciency problem brought by\\nthe SAT-based framework but exploiting ef\\ufb01cient heuristics\\nof modern SAT solver, we have proposed con\\ufb02ict-driven\\nstructural learning (CDSL) ATPG algorithm in this paper,\\nwhich allows the structural ATPG to bene\\ufb01t from the SAT\\u2019s TABLE IV\\nEVALUATION ON CONFLICT DIAGNOSIS\\nCircuit UO Coverage Circuit UO Coverage\\nStuck 1 554 99.120% Tran 1 306 97.337%\\nStuck 2 522 99.010% Tran 2 14928 99.505%\\nStuck 3 920 98.606% Tran 3 82 99.210%\\nStuck 4 8 99.803% Tran 4 126 98.600%\\nStuck 5 852 97.679% Tran 5 2812 96.004%\\nStuck 6 35 99.786% Tran 6 7002 95.232%\\nStuck 7 392 98.938% Tran 7 7213 94.887%\\nStuck 8 2356 96.022% Tran 8 6579 94.872%\\nStuck 9 5910 95.931% Tran 9 442 93.859%\\nStuck 10 3827 99.873% Tran 10 2913 93.953%\\ntechniques such as con\\ufb02ict management and con\\ufb02ict-driven\\nbranching. The proposed CDSL algorithm is composed of\\nthree parts: (1) Learnt con\\ufb02ict constraints before each back-\\ntrack has been constructed, aiming to learn from the mistakes\\nand utilize the optimization process data to prune search\\nspace. (2) Con\\ufb02ict-driven implication and justi\\ufb01cation have\\nbeen applied for decisions and implications, aiming to further\\nincrease the solving ef\\ufb01ciency and decision effectiveness. (3)\\nCon\\ufb02ict diagnosis based on the analysis of the learnt con\\ufb02icts\\nhas been attempted to improve test and fault coverage rate\\nby relaxing some of the external ATPG constraints. Extensive\\nexperimental results on industrial circuits have demonstrated\\nthe advantage of the proposed CDSL ATPG algorithm in three\\naspects: (i) Comparing with the conventional SAT-based ATPG\\nand structural D-algorithm, the proposed CDSL algorithm\\nhas decreased the aborted faults by 25:6%and49:88% on\\naverage, while the run time is decreased by 94:51% and\\n25:88%, respectively. (ii) With a two-stage setting, compared\\nwith the D-algorithm, the aborted faults can be decreased via\\n46:37% and the fault coverage rate can be increased via 3:19%,\\nwhile compared with the two-stage algorithm which is without\\ncon\\ufb02ict-driven modules, the aborted faults can be decreased\\nvia21:89% and fault coverage rate is increased via 0:93%.\\n(iii) Con\\ufb02ict diagnosis has been shown to decrease the aborted\\nfaults via 8:89% on average while increasing the fault coverage\\nrate0:271% . Future work includes the development of more\\nSAT heuristics on structural ATPG heuristics.\\nREFERENCES\\n[1] J. P. Roth, \\u201cDiagnosis of automata failures: A calculus\\nand a method,\\u201d IBM J. Res. Develop. , vol. 10, pp. 278\\u2013\\n291, 1966.\\n[2] N. Wang, C. Wang, K.-H. Tsai, W.-T. Cheng, X. Lin,\\nM. Kassab, and I. Pomeranz, \\u201cTea: A test generation\\nalgorithm for designs with timing exceptions,\\u201d Asian Test\\nSymposium , pp. 19\\u2013195, 2019.\\n[3] M. Schulz, E. Trischler, and T. Sarfert, \\u201cSocrates: A\\nhighly ef\\ufb01cient automatic test pattern generation system,\\u201d\\nInternational Test Conference , pp. 1016\\u20131026, 1987.\\n[4] J. Huang, H. L. Zhen, N. Wang, M. Yuan, H. Mao,\\nY . Huang, and J. Tao, \\u201cAccelerate sat-based atpg via\\npreprocessing and new con\\ufb02ict management heuristics,\\u201d\\n27th Asia and South Paci\\ufb01c Design Automation Confer-\\nence (ASP-DAC) , pp. 365\\u2013370, 2022.[5] B. Becker, R. Drechsler, and M. Sauer, \\u201cRecent advances\\nin sat-based atpg: Non-standard fault models, multi\\nconstraints and optimization,\\u201d International Conference\\non Design and Technology of Integrated Systems in\\nNanoscale Era , pp. 1\\u201310, 2014.\\n[6] H. Chen and J. Marques-silva, \\u201cTg-pro: A sat-based\\natpg system system description,\\u201d Journal on Satis\\ufb01ability,\\nBoolean Modeling and Computation , vol. 8, no. 1-2, pp.\\n83\\u201388, 2011.\\n[7] A. Czutro, I. Polian, M. Lewis, P. Engelke, S. M. Reddy,\\nand B. Becker, \\u201cTiguan: Thread-parallel integrated test\\npattern generator utilizing satis\\ufb01ability analysis,\\u201d Inter-\\nnational Conference on VLSI Design , pp. 227\\u2013232, 2009.\\n[8] S. Eggersgl \\u00a8u\\u00df, K. Schmitz, R. Krenz-B \\u02daa\\u02daath, and\\nR. Drechsler, \\u201cOn optimization-based atpg and its appli-\\ncation for highly compacted test sets.\\u201d IEEE Transactions\\non Computer-Aided Design of Integrated Circuits and\\nSystems. , pp. 2104\\u20132117, 2016.\\n[9] S. Eggersgl \\u00a8u\\u00df, S. Holst, D. Tille, K. Miyase, and\\nX. Wen., \\u201cFormal test point insertion for region-based\\nlow-capture-power compact at-speed scan test.\\u201d IEEE\\nAsian Test Symposium (ATS) , pp. 173\\u2013178, 2016. X. Wen., \\u201cFormal test point insertion for region-based\\nlow-capture-power compact at-speed scan test.\\u201d IEEE\\nAsian Test Symposium (ATS) , pp. 173\\u2013178, 2016.\\n[10] M. S. Cherif, D. Habet, and C. Terrioux, \\u201cKissat mab:\\nUpper con\\ufb01dence bound strategies to combine vsids and\\nchb,\\u201d SAT COMPETITION , 2022.\\n[11] D. Tille, S. Eggersgluss, and R. Drechsler, \\u201cIncremental\\nsolving techniques for sat-based atpg,\\u201d IEEE Transac-\\ntions on Computer-Aided Design of Integrated Circuits\\nand Systems , vol. 29, no. 7, pp. 1125\\u20131130, 2010.\\n[12] C. Wang, S. M. Reddy, I. Pomeranz, X. Lin, and J. Ra-\\njski, \\u201cCon\\ufb02ict driven techniques for improving determin-\\nistic test pattern generation.\\u201d IEEE\\/ACM international\\nconference on Computer-aided design , pp. 87\\u201393, 2002.\\n[13] S. Bommu, K. Chandrasekar, R. Kundu, and S. Sengupta,\\n\\u201cConcat: Con\\ufb02ict driven learning in atpg for industrial\\ndesigns.\\u201d IEEE International Test Conference (ITC) , pp.\\n1\\u201310, 2008.\\n[14] C. Kameshwar, S. Bommu, and S. Sengupta., \\u201cLow\\ncoverage analysis using dynamic un-testability debug in\\natpg.\\u201d IEEE VLSI Test Symposium (VTS) , pp. 291\\u2013296,\\n2011.\\n[15] P. Goel, \\u201cAn implicit enumeration algorithm to gener-\\nate tests for combinational logic circuits,\\u201d IEEE Trans.\\nComput. , vol. C-30, pp. 215\\u2013222, 1981.\\n[16] K. T. and M. R. Mercer, \\u201cA topological search algorithm\\nfor atpg.\\u201d In 24th ACM\\/IEEE Design Automation Con-\\nference , pp. 502\\u2013508, 1987.\\n[17] M.-S. Joao, I. Lynce, and S. Malik., \\u201cCon\\ufb02ict-driven\\nclause learning sat solvers. handbook of satis\\ufb01ability.\\u201d\\nIOS Press , pp. 133\\u2013182, 2021.\\n[18] J. Huang, H. L. Zhen, N. Wang, M. Yuan, H. Mao,\\nand Y . Huang, \\u201cNeural fault analysis for sat-based atpg.\\u201d\\nIEEE International Test Conference (ITC) , pp. 36\\u201345,\\n2022.\",\"25\":\" TopSpark: A Timestep Optimization Methodology for Energy-Ef\\ufb01cient\\nSpiking Neural Networks on Autonomous Mobile Agents\\nRachmad Vidya Wicaksana Putra\\u0003and Muhammad Sha\\ufb01quey\\nAbstract \\u2014 Autonomous mobile agents (e.g., mobile ground robots\\nand UA Vs) typically require low-power\\/energy-ef\\ufb01cient machine learn-\\ning (ML) algorithms to complete their ML-based tasks (e.g., object\\nrecognition) while adapting to diverse environments, as mobile agents\\nare usually powered by batteries. These requirements can be ful\\ufb01lled\\nby Spiking Neural Networks (SNNs) as they offer low power\\/energy\\nprocessing due to their sparse computations and ef\\ufb01cient online learning\\nwith bio-inspired learning mechanisms for adapting to different envi-\\nronments. Recent works studied that the energy consumption of SNNs\\ncan be optimized by reducing the computation time of each neuron\\nfor processing a sequence of spikes (i.e., timestep). However, state-of-\\nthe-art techniques rely on intensive design searches to determine \\ufb01xed\\ntimestep settings for only the inference phase, thereby hindering the\\nSNN systems from achieving further energy ef\\ufb01ciency gains in both the\\ntraining and inference phases. These techniques also restrict the SNN\\nsystems from performing ef\\ufb01cient online learning at run time. Toward\\nthis, we propose TopSpark, a novel methodology that leverages adaptive\\ntimestep reduction to enable energy-ef\\ufb01cient SNN processing in both\\nthe training and inference phases, while keeping its accuracy close to\\nthe accuracy of SNNs without timestep reduction. The key ideas of\\nour TopSpark include: (1) analyzing the impact of different timestep\\nsettings on the accuracy; (2) identifying neuron parameters that have\\na signi\\ufb01cant impact on accuracy in different timesteps; (3) employing\\nparameter enhancements that make SNNs effectively perform learning\\nand inference using less spiking activity due to reduced timesteps; and\\n(4) developing a strategy to trade-off accuracy, latency, and energy to\\nmeet the design requirements. The experimental results show that, our\\nTopSpark saves the SNN latency by 3.9x as well as energy consumption\\nby 3.5x for training and 3.3x for inference on average, across different\\nnetwork sizes, learning rules, and workloads, while maintaining the\\naccuracy within 2% of that of SNNs without timestep reduction. In this\\nmanner, TopSpark enables low-power\\/energy-ef\\ufb01cient SNN processing\\nfor autonomous mobile agents.\\nI. I NTRODUCTION\\nAutonomous mobile agents (e.g., UGVs and UA Vs) usually\\nrequire low-power ML algorithms to complete their ML-based tasks\\n(e.g., object recognition through images\\/videos), since these agents\\nare typically powered by batteries [1]; see Fig. 1(a). Furthermore,\\nthese mobile robots also require to continuously adapt to different\\noperational environments (so-called dynamic environments ) since\\nthe of\\ufb02ine-trained knowledge is typically learnt from limited sam-\\nples and may be obsolete at run time, hence leading to accuracy\\ndegradation; see Fig. 1(a). These requirements can be ful\\ufb01lled\\nby Spiking Neural Networks (SNNs) because of the following\\ntwo reasons. First, advancements in neuromorphic computing have\\nled SNNs to achieve ultra-low power\\/energy processing and high\\naccuracy by leveraging their bio-inspired spike-based operations\\n[2]\\u2013[4]. Second, SNNs employ bio-inspired learning rules locally in\\neach synapse, such as Spike-Timing-Dependent Plasticity (STDP),\\nwhich are suitable for ef\\ufb01cient online learning (i.e., training at run\\ntime for updating the knowledge of systems using unlabeled data1),\\nthereby adapting to dynamic environments [5]\\u2013[7]. Furthermore,\\n\\u0003Rachmad Vidya Wicaksana Putra is with the Institute of Computer\\nEngineering, Technische Universit \\u00a8at Wien (TU Wien), Vienna, Austria\\nrachmad.putra@tuwien.ac.atyMuhammad Sha\\ufb01que is with the Division of Engineering, New York\\nUniversity Abu Dhabi (NYUAD), Abu Dhabi, United Arab Emirates (UAE)\\nmuhammad.shafique@nyu.edu\\n1Unlabeled data from environments can be leveraged for ef\\ufb01cient SNN University Abu Dhabi (NYUAD), Abu Dhabi, United Arab Emirates (UAE)\\nmuhammad.shafique@nyu.edu\\n1Unlabeled data from environments can be leveraged for ef\\ufb01cient SNN\\ntraining at run time using STDP-based unsupervised learning.\\n(b)\\ninput\\n\\u2026excitatory \\nneurons\\nweights learned \\nusing STDP inhibition\\noutput5 neurons \\ncan recognize \\n5 classesinput: 51\\n5\\n3\\n2\\n41\\n5\\n33 neurons \\ncan recognize \\n3 classesinput: 5An autonomous mobile agent with \\na spiking neural network model\\n(e.g., mobile robot and UGV)\\nVolcanoes\\n(Environment 3)\\nThe agent moves across different environments\\n\\u2026\\nAdaptive to different operational environments\\nDeserts\\n(Environment 1)Forests\\n(Environment 2)(a)\\nLow -power \\/\\nenergy -efficient  \\nprocessingrequires requires\\nFig. 1. (a) Autonomous mobile agents typically require low-power pro-\\ncessing and capabilities for adapting to different operational environments to\\ncomplete ML-based tasks. (b) SNN architecture that supports STDP-based\\nunsupervised learning, i.e., fully-connected network. A larger SNN model\\nhas a higher number of excitatory neurons to recognize more features than\\na smaller one.\\nSNNs are also expected to provide high accuracy that meets the\\napplications\\u2019 requirements. To achieve higher accuracy, larger-sized\\nSNN models are usually employed as they have a larger number\\nof neurons to recognize more input features than the smaller-sized\\nmodels; see Fig. 1(b). For instance, the work of [8] studied that\\na network with 3600 excitatory neurons achieves \\u001890% accuracy\\nfor the MNIST, while a network with 100 excitatory neurons only\\nachieves \\u001875%. However, SNN hardware platforms for embedded\\napplications (e.g., mobile robots) typically have limited memory\\nand compute capabilities [9], thereby making it challenging to\\nachieve both high energy ef\\ufb01ciency and high accuracy for SNNs\\nat the same time. To address this, previous works have explored\\ndifferent optimization methodologies [8] [10]\\u2013[13]. However, most\\nof them have not optimized the computational time of a neuron\\nfor processing input spikes (i.e., timestep) , which also has the\\npotential to substantially reduce the energy consumption of SNN\\nprocessing, while preserving the excitatory neurons to recognize\\ndiverse features.\\nTargeted Research Problem: How can we improve the energy\\nef\\ufb01ciency of SNN processing in both the training and inference\\nphases through timestep optimizations, while maintaining the ac-\\ncuracy high. An ef\\ufb01cient solution to this problem will enable low-\\nlatency and energy-ef\\ufb01cient autonomous mobile agents\\/robots that\\ncan adapt to different environments.\\nA. State-of-the-Art and Their Limitations\\nState-of-the-art works have employed different timestep reduction\\ntechniques to optimize the energy consumption of SNN inference,\\nwhile achieving acceptable accuracy [14]\\u2013[17]. For instance, thearXiv:2303.01826v1  [cs.NE]  3 Mar 2023 020406080100\\n0 50 100 150 200 250 300 350Accuracy [%]\\nTimestepBaseline \\naccuracyAccuracy is comparable to the baseline accuracy\\nNotable accuracy degradation(a)\\n(b)\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350Latency Energy\\n0.00.20.40.60.81.010\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350\\nTimestepTraining InferenceLatency\\n(Normalized to \\nTimestep 350)\\nEnergy \\n(Normalized to \\nTimestep 350)Latency & energy are saved as the timestep is reduced Fig. 2. Experimental results considering an SNN with 400 excitatory\\nneurons with a fully-connected architecture in Fig. 1(a), rate coding, and\\npair-based STDP [10] under different timesteps in training and inference:\\n(a) accuracy pro\\ufb01les; (b) latency and energy consumption normalized to\\ntimestep 350.\\nwork of [14] trained binary-weighted SNNs and then studied the\\nimpact of different timesteps on accuracy. Another work performed\\na gradient descent-based retraining method to learn the parameter\\nvalues while considering reduced timesteps [15]. Similarly, the\\nwork of [16] gradually reduced the timesteps while retraining the\\nnetwork. Meanwhile, a recent work employed search algorithms to\\n\\ufb01nd the appropriate timestep reduction [17].\\nLimitations: State-of-the-art works only employed \\ufb01xed\\ntimesteps for SNN inference, through costly (re)training [14]\\u2013[16]\\nand costly intensive searches [17]. Therefore, the existing tech-\\nniques limit the SNN systems from further energy ef\\ufb01ciency gains\\nin both the training and inference phases. These techniques also hin-\\nder the SNN systems from performing ef\\ufb01cient online learning\\/\\ufb01ne-\\ntuning through training at run-time. Furthermore, smart AI-based\\nsystems usually need to adjust their accuracy for better battery\\nlife at run time [18], thereby requiring adaptive trade-offs between\\naccuracy, latency, and energy, which cannot be accommodated by\\nthe existing techniques.\\nTo address these limitations, a timestep optimization technique\\nfor both the training and inference phases is required . This solution\\nwill enable ef\\ufb01cient SNN training and inference with smaller\\ntimesteps (i.e., lower latency) and lower energy consumption.\\nMoreover, this ef\\ufb01cient training can also be leveraged at run time\\nto enable ef\\ufb01cient online learning mechanisms.\\nTo highlight the potential of timestep reduction in SNN pro-\\ncessing, we present an experimental case study in the following\\nSection I-B.\\nB. Motivational Case Study and Key Challenges\\nWe study the accuracy, latency, and energy consumption pro\\ufb01les\\nof an SNN model considering different timesteps. To do this, we\\nperform experiments employing a fully-connected SNN with 400\\nneurons, rate coding, and pair-wise STDP in unsupervised learning\\nsettings [10]. For each timestep setting, we perform training and\\ninference using the MNIST dataset2. Detailed information on the\\nexperimental setup will be discussed in Section IV. The experimen-\\ntal results are shown in Fig. 2, from which we draw the following\\nkey observations .\\n\\u000fAccuracy scores are typically proportional to timestep settings\\ndue to the spiking activity, i.e., smaller timestep leads to lower\\n2The MNIST dataset is commonly used in SNN community for evaluating\\nSNNs with unsupervised learning settings [2] [8] [10].accuracy, and larger timestep leads to higher accuracy; see\\nFig. 2(a).\\n\\u000fAccuracy pro\\ufb01le of an SNN with timestep reduction may have\\ntwo regions, compared to the baseline accuracy (i.e., an SNN\\nwithout timestep reduction): (1) a region with comparable ac-\\ncuracy, and (2) a region with notable accuracy degradation; see\\nFig. 2(a).\\n\\u000fTimestep reduction can effectively save the latency and energy\\nconsumption of SNN processing in both the training and infer-\\nence phases due to reduced neuron and learning operations; see\\nFig. 2(b).\\nAlthough reducing timesteps can effectively curtail the latency\\nand energy consumption of SNN processing, aggressive reductions\\nin the timestep may signi\\ufb01cantly decrease the accuracy , thereby\\nlimiting the applicability of timestep reduction techniques for and energy consumption of SNN processing, aggressive reductions\\nin the timestep may signi\\ufb01cantly decrease the accuracy , thereby\\nlimiting the applicability of timestep reduction techniques for\\nimproving the ef\\ufb01ciency gains of SNNs.\\nResearch Challenges: Our experiments and observations expose\\nseveral design challenges that should be solved to address the\\ntargeted problem, as discussed in the following.\\n\\u000fThe solution should maximally reduce the timestep in the train-\\ning and the inference to signi\\ufb01cantly optimize the computation\\nlatency and energy consumption of SNN processing, but without\\nnoticeably degrading the accuracy.\\n\\u000fThe solution should effectively learn from reduced spiking activity\\nto maintain the learning quality as compared to the original SNNs\\n(i.e., without timestep reduction).\\n\\u000fThe optimization techniques should incur minimum overheads\\n(e.g., energy) to accommodate ef\\ufb01cient online learning and better\\nbattery life management.\\nC. Our Novel Contributions\\nTo address the research challenges, we propose TopSpark ,\\na novel T imestep op timization methodology for energy ef\\ufb01cient\\nSpiking neura l network s in training and inference on autonomous\\nmobile agents . To the best of our knowledge, our TopSpark is the\\n\\ufb01rst work that optimizes timesteps of SNNs in both the training\\nand inference phases. Following are the novel steps performed in\\nthe TopSpark methodology (see the overview in Fig. 3).\\n1)Analyzing the accuracy under different timesteps to investi-\\ngate the impact of timestep reduction in the SNN training and\\ninference on the accuracy pro\\ufb01les.\\n2)Analyzing the impact of neuron parameters to investigate the\\nrole of neuron parameters and the in\\ufb02uence of their values on\\nthe accuracy pro\\ufb01les under different timesteps.\\n3)Devising parameter enhancement rules to maintain accuracy\\nby leveraging the knowledge from previous analysis steps,\\nthereby providing effective yet simple solutions with minimum\\noverheads.\\n4)Devising a strategy to trade-off accuracy, latency, and energy\\nto meet the constraints (i.e., target accuracy, target latency, and\\nenergy budget), hence enabling a better battery life management.\\nKey Results: We evaluate TopSpark methodology using Python\\nsimulations [19] on GPU machines to get the accuracy pro\\ufb01les and\\nleverage the computation time and power to estimate the latency and\\nenergy consumption of SNN training and inference. Experimental\\nresults with different workloads and STDP-based learning rules\\nshow that, our TopSpark saves the latency by 3.9x and energy\\nconsumption by 3.5x for training and 3.3x for inference on average,\\nwhile keeping the accuracy close to the SNNs without timestep\\nreduction.\\nII. P RELIMINARIES OF SNN S\\nSpiking neural networks (SNNs) are the neural networks\\u2019 class\\nthat employs bio-plausible computation models based on action \\u2026\\nEnhanced SNN\\nOptimized timestepOur Novel Contributions\\nTopSpark Methodology (Section III)\\nAnalysis of Accuracy Profiles (Section III -A)\\nAnalysis of the Impact of Neuron \\nParameters on Accuracy (Section III -B)\\nParameter Enhancements (Section III -C)\\nDesign Trade -off Strategy (Section III -D)\\u2026\\nBaseline SNN\\nBaseline timestep\\nConstraints (accuracy, \\nlatency, and energy)Fig. 3. An overview of our novel contributions (shown in blue boxes).\\npotentials (i.e., spikes) [2]. An SNN model has a speci\\ufb01c neuron\\nmodel, network architecture, learning rule, and spike coding [8].\\nNeuron model de\\ufb01nes how a neuron operates, updates its internal\\ndynamics (e.g., membrane potential), and generates output spikes\\nover time. The operational time of a neuron to process a sequence\\nof input spikes from a single input (e.g., an image pixel) is\\nde\\ufb01ned as timestep [17]. Here, we consider the Leaky Integrate-\\nand-Fire (LIF) neuron model because it has been widely used\\nin the SNN community because of its simple yet highly bio-\\nplausible operations [8]; see an overview in Fig. 4. The LIF neuron\\nupdates its membrane potential ( Vmem ) each timestep. Vmem is\\nincreased each time an input spike comes, and otherwise, Vmem\\nis decreased. If Vmem reaches a de\\ufb01ned threshold voltage ( Vth),\\nthe neuron generates an output spike. Then, Vmem goes to the\\nreset potential ( Vmem ) and the neuron cannot generate spikes in\\na de\\ufb01ned refractory period ( Tref).\\ntimestep\\ntimestepVmem\\nVresetVth\\nincoming \\/input spikestimestepgenerated output spikes\\nVth+ \\u03b8\\nTrefneuronal dynamics\\nFig. 4. The neuronal dynamics of a LIF neuron model.\\nNetwork architecture determines the connections among neu-\\nrons, synapses, as well as inputs and outputs. In this work, we\\nconsider a fully-connected network in Fig. 1(b) since it supports\\nunsupervised learning, which is required for enabling ef\\ufb01cient\\nonline learning mechanisms. In such a network, each input (e.g.,\\nan image pixel) is connected to all excitatory neurons. Each\\nexcitatory neuron is expected to recognize a speci\\ufb01c class. Hence,\\nthe connecting synapses are trained to learn the corresponding\\nfeatures. For the learning rules , we consider the bio-plausible STDP\\nrules (i.e., pair-based weight-dependent STDP [10] and adaptive\\nlearning rate STDP [8]) since they support unsupervised learning\\nscenarios to train the synaptic weights using unlabeled data from\\noperational environments, thus enabling ef\\ufb01cient online learning\\nmechanisms [5] [12]. To perform SNN processing, the input data\\nis converted into a sequence of spikes (i.e., spike train) using a\\nspeci\\ufb01c neural\\/spike coding . Here, we consider the rate coding\\nas it can be coupled with different STDP-based learning rules\\nfor achieving high accuracy [10]. This rate coding uses Poisson\\ndistribution to generate the spike train of each input (e.g., a pixel)\\nwhose probability mass function ( Ppmf) is given by Eq. 1, with \\u0015\\ndenotes the rate parameter, kdenotes the number of occurrences,\\nandedenotes the Eulers\\u2019 number.\\nPpmf =\\u0015k\\u0001e\\u0000\\u0015\\nk!(1)\\nTo avoid the domination of some excitatory neurons in the\\ntraining phase, the adaptation potential ( \\u0012) is employed [10]. Here,\\nVthis increased by \\u0012each time the corresponding neuron generates\\na spike that triggers the learning process for a speci\\ufb01c inputfeature\\/pattern, thereby making it harder to stimulate the same\\nneuron to generate spikes for other input patterns; see Fig. 4. In this\\nmanner, a certain neuron is expected to produce spikes only when\\nstimulated with a speci\\ufb01c pattern (i.e., recognizing a speci\\ufb01c class),\\nand other neurons can produce spikes when stimulated with other\\ninput patterns (different classes), thereby maximizing the accuracy.\\nIII. T OPSPARK METHODOLOGY\\nOur TopSpark methodology employs several key novel steps, as\\nshown in the overview in Fig. 5. Detailed descriptions of these steps\\nare provided in the subsequent sections.\\nA. Analysis of Accuracy Pro\\ufb01les in Different Timesteps\\nWe \\ufb01rst investigate and analyze the impact of timestep re- are provided in the subsequent sections.\\nA. Analysis of Accuracy Pro\\ufb01les in Different Timesteps\\nWe \\ufb01rst investigate and analyze the impact of timestep re-\\nduction in training and inference on accuracy. This study aims\\nat understanding the characteristics of the accuracy pro\\ufb01les con-\\nsidering a given SNN model, dataset, and timestep. To do this,\\nwe perform experimental studies with a 400-neuron SNN while\\nconsidering different timesteps, datasets (i.e., MNIST and Fashion\\nMNIST), and learning rules, i.e., a pair-based weight-dependent\\nSTDP (STDP1) [10] and an adaptive learning rate-based STDP\\n(STDP2) [8]3. The experimental results are shown in Fig. 6, from\\nwhich we draw the following key observations.\\n\\u000fTrends of the accuracy: In general, different models with\\ndifferent combinations of timesteps, learning rules, and datasets\\nhave similar trends, i.e., the accuracy pro\\ufb01les are proportional\\nto the timestep as a larger timestep leads to higher accuracy,\\nwhile a smaller one leads to lower accuracy. It indicates that,\\naccuracy degradation in different SNN models may be solved\\nusing a similar approach, which is bene\\ufb01cial for developing a\\nsimple yet effective solution.\\n\\u000fAdvantages: Most of the timesteps lead to accuracy scores that\\nare comparable to the baseline accuracy (i.e., SNNs without\\ntimestep reduction) despite employing different SNN models\\nwith different learning rules and datasets; see label- 1. It shows\\nthat, the timestep may be reduced signi\\ufb01cantly without facing\\nnoticeable accuracy degradation as compared to the baseline.\\n\\u000fPotentials of small timesteps: The low accuracy in small\\ntimesteps (shown by label- 2) should be improved so that these\\ntimesteps can be used at run time for offering good trade-offs\\namong accuracy, latency, and energy consumption. For instance,\\nif the SNN-based systems need to reduce their operational\\npower\\/energy for better battery life, they may reduce the timestep\\nwithout accuracy loss or with acceptable accuracy degradation as\\ncompared to the baseline.\\nOur observations expose that, a smaller timestep typically has\\nless spiking activity (i.e., a smaller number of pre- and post-synaptic\\nspikes), hence leading to relatively less learning activity and lower\\naccuracy. This indicates that, a small number of spikes in a reduced\\ntimestep should be effectively used for learning the input features .\\nB. Identifying the Roles of Neuron Parameters in Different\\nTimesteps\\nTo maintain the learning quality of SNNs in a reduced timestep,\\nthe learning rules should bene\\ufb01t from the available spikes during the\\ntraining. Hence, the neurons should effectively make use of the pre-\\nsynaptic spikes for generating the post-synaptic spikes, which are\\nthen used by the STDP learning rules to recognize different classes.\\nOtherwise, the learning rules will not bene\\ufb01t from the spikes.\\nTo address this, we investigate the roles of neuron parameters and\\ntheir impact on the accuracy . We perform experimental case studies\\nwith a 400-neuron network with different timesteps, datasets, and\\n3Detailed information on the experimental setup is provided in Section IV. TopSpark Methodology (Section III) Analysis of Accuracy \\nin Different Timesteps \\n(Section III -A)Original \\nSNN Model\\nTimestepsAnalysis of the Impact of Neuron Parameters (Section III -B)input\\n020406080100\\n050100150 200250300350Accuracy [%]\\nTimestep020406080100\\n050100150200250300350Accuracy [%]\\nTimestep020406080100\\n050100 150 200 250 300 350Accuracy [%]\\nTimestep020406080100\\n050100150200250300350Accuracy [%]\\nTimestepAdaptation Potential ( \\u03b8) Refractory Period ( Tref) Threshold Potential ( Vth)\\nNeuron Parameter \\nEnhancements (Section III -C)Threshold \\nPotential ( Vth)\\nRefractory \\nPeriod ( Tref)Adaptation \\nPotential ( \\u03b8)Trade -off Strategy\\n(Section III -D)Optimized SNN Model\\nOptimized Timestep\\ninput\\nConstraints \\n(Accuracy, \\nLatency, \\nEnergy)\\nAccuracy\\nLatency EnergyAutonomous mobile agents\\nB2B37075808590\\n1040\\n150\\n25085-90\\n80-85\\n75-80\\n70-75\\nFig. 5. The overview of our TopSpark methodology, where the novel steps are highlighted in blue boxes. We \\ufb01rst analyze the impact of timestep reduction\\non the accuracy pro\\ufb01les (Section III-A). We also identify the roles of neuron parameters in different timesteps (Section III-B). Then, we leverage previous\\nobservations to enhance the neuron parameters (Section III-C) as well as develop a strategy to trade-off accuracy, latency, and energy (Section III-D).\\nOutput of the TopSpark methodology is an optimized SNN model with optimized timestep, which is employed on autonomous mobile agents. Furthermore,\\nthe mobile agents can adjust the timestep of the SNN processing at run time to adaptively meet the power\\/energy requirements (e.g., to save battery life).\\n020406080100\\n0 50 100 150 200 250 300 350Accuracy [%]\\nTimestepSTDP1\\nSTDP2(a) MNIST\\n020406080\\n0 50 100 150 200 250 300 350\\nTimestepSTDP1\\nSTDP2(b) Fashion MNIST\\nSignificant \\naccuracy \\ndegradationSignificant \\naccuracy \\ndegradationBaseline accuracy\\nBaseline accuracy\\n1\\n2 12\\nFig. 6. The accuracy pro\\ufb01les of a 400-neuron network with reduced\\ntimesteps during training and inference phases, while considering different\\nlearning rules and datasets: (a) MNIST and (b) Fashion MNIST.\\nlearning rules as employed in Section III-A, while considering dif-\\nferent values of neuron parameters. Here, we investigate threshold\\npotential (Vth), refractory period ( Tref), and adaptation potential\\n(\\u0012) since we employ the LIF neuron model. We reduce the values\\nof these parameters with the following settings: (1) V1\\nth=V0\\nth\\u00001;\\n(2)Tref= 1; and (3)\\u0012= 0. Index- 0denotes a parameter with an\\noriginal value, while index- 1denotes a parameter with an adjusted\\nvalue. The experimental results are presented in Fig. 7, from which\\nwe derive the following key observations.\\n\\u000fReducedVth:We observe that V1\\nth=V0\\nth\\u00001leads to better\\naccuracy than the direct timestep reduction in most of the\\ntimesteps, and leads to competitive accuracy compared to the\\nbaseline in small timesteps, as shown by 3and 4. The rea-\\nson is that, a smaller Vthcan make the corresponding neuron\\nproduces post-synaptic spikes easier than the original Vth, thus\\nincreasing the post-synaptic spike and learning activities that lead\\nto better accuracy. Therefore, the idea of threshold potential ( Vth)\\nreduction can be exploited for devising an effective solution .\\n\\u000fReducedTref:We observe that Tref= 1 leads to comparable\\naccuracy as the baseline in small timesteps (see 3and 4) since\\nthis setting makes all neurons responsive to any input spikes,\\nwhich leads to higher learning activities and relatively good\\naccuracy. In high timesteps, this setting leads to highly responsive\\nneurons that may encounter dif\\ufb01culties in distinguishing a speci\\ufb01c\\nclass, hence leading to accuracy lower than the baseline and\\nthe direct timestep reduction; see 5. Therefore, the idea of\\nrefractory period ( Tref) reduction should be exploited carefully\\nif considered for developing an effective solution .\\n\\u000fReduced\\u0012:We observe that, in general, \\u0012= 0 leads to a\\nsigni\\ufb01cant accuracy drop as compared to the baseline and the if considered for developing an effective solution .\\n\\u000fReduced\\u0012:We observe that, in general, \\u0012= 0 leads to a\\nsigni\\ufb01cant accuracy drop as compared to the baseline and the\\ndirect timestep reduction, as shown by 6. The reason is that,\\nthis setting may make some neurons dominate the spiking activity\\n(i.e., spike generation), hence restricting the other neurons and\\ntheir connecting synapses from learning and recognizing diverse\\n0102030405060\\n0 50 100 150 200 250 300 350\\nSTDP1 STDP1 (Reduced Vth)\\nSTDP1 (Reduced Tref) STDP1 (Reduced \\u03b8)020406080100\\n0 50 100 150 200 250 300 350Accuracy [%]\\nTimestep40455055\\n10 20 30 40 5060708090\\n10 20 30 40 503 4\\n5\\n6\\n020406080100\\n0 50 100 150 200 250 300 350Accuracy [%]\\nTimestep\\nSTDP2 STDP2 (Reduced Vth)\\nSTDP2 (Reduced Tref) STDP2 (Reduced \\u03b8)020406080\\n0 50 100 150 200 250 300 350(a.1) \\nSTDP1\\n&\\nMNIST(b.1) \\nSTDP1\\n&\\nFashion\\nMNIST\\n60708090\\n10 20 30 40 505560657075\\n25 75 125 175(a.2) \\nSTDP2\\n&\\nMNIST(b.2) \\nSTDP1\\n&\\nFashion\\nMNIST34\\n6Baseline accuracy\\nBaseline \\naccuracy\\nBaseline accuracy Baseline \\naccuracy(Direct)\\n(Direct)TrefVth\\nTrefVthFig. 7. The accuracy pro\\ufb01les of a 400-neuron network with reduced\\ntimesteps during training and inference, while considering different values\\nof neuron parameters, learning rules, and datasets: (a.1) STDP1 - MNIST,\\n(a.2) STDP2 - MNIST, (b.1) STDP1 - Fashion MNIST, and (b.2) STDP2 -\\nFashion MNIST.\\ninput features, which in turn causes accuracy degradation [10].\\nTherefore, the adaption potential ( \\u0012) should be considered when\\ndeveloping an effective solution .\\nC. Parameter Enhancements for Maintaining Accuracy\\nSection III-B suggests that neuron parameters can be exploited to\\nmaintain accuracy in reduced timesteps. However, \\ufb01nding appropri-\\nate values for these parameters requires intensive searches, which\\nrestrict the SNN systems from ef\\ufb01cient online learning\\/\\ufb01ne-tuning.\\nToward this, we propose a set of simple policies that can enhance\\nthe neuron parameters with minimum overheads , thereby enabling\\nthe SNN-based systems to employ adaptive timestep reduction for\\nef\\ufb01cient online learning through training at run time. Threshold potential ( Vth):We leverage Vthreduction to main-\\ntain accuracy for most of the timesteps by adjusting the gap between\\nVthandVreset , so that the neurons can have proportional and\\nsuf\\ufb01cient spiking (i.e., pre- and post-synaptic spikes) and learning\\nactivities for distinguishing different classes. To do this, we propose\\nto linearly scale down Vthfrom its original value considering the\\nreduced timestep, as stated in Eq. 2 .V0\\nthandV1\\nthare the threshold\\npotentials for the original and the adjusted ones, respectively. Vreset\\nis the reset potential, while T0andT1are the timesteps for the\\noriginal and the adjusted ones, respectively.\\nV1\\nth=Vreset +\\u0018T1\\nT0\\u0001(V0\\nth\\u0000Vreset )\\u0019\\n(2)\\nRefractory period ( Tref):We leverage Trefreduction to de\\ufb01ne\\nthe effective duration for a neuron to be unresponsive after gener-\\nating spikes, so that the other neurons have a chance to process\\nthe input spikes and trigger the learning process. To do this, we\\npropose to proportionally decrease Treffrom its original value\\nconsidering the reduced timestep, as stated in Eq. 3 .T0\\nrefandT1\\nref\\ndenote the refractory period for the original and the adjusted ones,\\nrespectively. In this manner, Trefis set to a small value when the\\ntimestep is small, and vice versa. Furthermore, we also employ a\\nceiling function to discretize the refractory period into a timestep\\nform and ensure that the minimum value of Tref= 1.\\nT1\\nref=\\u0018T1\\nT0\\u0001T0\\nref\\u0019\\n(3)\\nAdaptation potential ( \\u0012):We keep\\u0012in the neural dynamics\\nand carefully adjust its value so that the spiking and learning\\nactivities are increased while avoiding the domination of some\\nneurons, thereby maintaining good accuracy. To do this, we propose\\nto linearly reduce \\u0012from its original value, as stated in Eq. 4 . Here,\\n\\u00120and\\u00121are the adaptation potentials for the original and adjusted\\nones, respectively.\\n\\u00121=T1\\nT0\\u0001\\u00120 (4)\\nD. Trade-Off Strategy for Accuracy, Latency, and Energy\\nSections III-A and III-B show that timestep reduction improves\\nthe latency and energy ef\\ufb01ciency of SNN systems. However, at the\\nsame time, their accuracy may be degraded below the acceptable\\nthreshold. Hence, the accuracy level, latency, and energy consump-\\ntion of SNN systems should meet the design constraints. It is espe-\\ncially important for applications that need adaptive adjustments at\\nrun time for better battery life, such as smart mobile agents\\/robots.\\nToward this, we propose a strategy to trade-off accuracy, latency,\\nand energy consumption to meet the constraints (i.e., acceptable\\naccuracy, acceptable latency, and energy budget ). Our strategy is\\nto quantify the trade-off bene\\ufb01t for a given model using our pro-\\nposed multi-objective trade-off function in Eq. 5, which considers\\naccuracy, latency, and energy consumption. Sis the score of trade-\\noff bene\\ufb01t, which is useful for design space exploration considering\\ndifferent trade-offs between accuracy, latency, and energy consump-\\ntion.Ais the accuracy and Lnis the normalized latency, i.e., the\\nratio between the latency after timestep reduction and the original\\nlatency. Meanwhile, Enis the normalized energy, i.e., the ratio\\nbetween the energy consumption after timestep reduction and the\\noriginal one. \\u001cand\\\"are the adjustment factors for latency and\\nenergy consumption, respectively. The adjustment factor should be\\nset higher than the others if the respective metric is more important\\nthan the others, and vice versa. Here, \\u001cand\\\"are the non-negative\\nreal numbers.\\nS=A\\u0000(\\u001c\\u0001Ln+\\\"\\u0001En)withLn=L1\\nL0andEn=E1\\nE0(5)The use of our TopSpark methodology in autonomous mobile\\nagents: The output of TopSpark is an optimized SNN model\\nwith enhanced parameters and optimized timestep, which can be\\nemployed directly for performing energy-ef\\ufb01cient SNN inference on\\nmobile agents\\/robots. If the mobile agents consider online learning\\nfor adapting to different operational environments, they can also\\nemploy the TopSparks\\u2019 output model, since this model can be\\ntrained under reduced timestep settings. Furthermore, the mobile for adapting to different operational environments, they can also\\nemploy the TopSparks\\u2019 output model, since this model can be\\ntrained under reduced timestep settings. Furthermore, the mobile\\nagents can adjust the timestep of SNN processing at run time to\\nadaptively meet the power\\/energy requirements through TopSparks\\u2019\\nparameter enhancements. In this manner, mobile agents\\/robots can\\nsave their battery life on-the-\\ufb02y without signi\\ufb01cantly\\/noticeably\\nsacri\\ufb01cing accuracy.\\nIV. E VALUATION METHODOLOGY\\nTo evaluate our TopSpark methodology, we use the same evalua-\\ntion scenarios that are employed widely in the SNN community with\\nthe experimental setup shown in Fig. 8. We use a Poisson-based\\nrate coding for converting data into spikes. We employ the fully-\\nconnected network architecture (as shown in Fig. 1) with a different\\nnumber of neurons. For brevity, we consider the term M nfor an\\nSNN model with n-number of neurons. We also employ different\\nSTDP-based learning rules, i.e., a pair-based weight-dependent\\nSTDP (STDP1) [10] and an adaptive learning rate-based STDP\\n(STDP2) [8]. We consider the MNIST and Fashion MNIST datasets\\nas the workloads. For each timestep setting, we perform both\\nthe training and inference phases. These experimental scenarios\\naim at highlighting the generality of TopSpark methodology. For\\ncomparison partners, we consider the original SNN models without\\ntimestep reduction as the baseline , and the SNNs with direct\\ntimestep reduction technique. Furthermore, for both comparison\\npartners, we consider the original parameter values from work\\nof [10], as shown in Table I.\\nEvaluation: We employ Python-based simulations [19] which\\nrun on GPU machines (i.e., Nvidia GeForce RTX 2080 Ti) to\\nevaluate the accuracy. Afterward, we leverage the simulation time to\\nevaluate the latency. We also obtain the power consumption using\\nthenvidia-smi utility, following the approach used in work [20].\\nThese recorded simulation time and operational power are then\\nleveraged to estimate energy consumption in the training and\\ninference phases.\\nAccuracy \\n(.txt)Latency\\n(.txt)DatasetSNN model \\nconfiguration \\nTimestepsAn SNN model generationPython -based Simulation\\nConstraints \\n\\u2022Accuracy\\n\\u2022Latency \\n\\u2022EnergyTopSpark Methodology\\nEnergy \\nestimator\\nRunning \\non GPU\\nEnhanced SNN model \\nPower \\n(.txt)Training\\nTestingParameter \\ncontroller\\nTrained \\nSNN \\nmodel Simulation \\ncontroller\\nEnergy \\n(.txt)Optimized \\ntimestepOptimized \\nSNN model\\nTrade -off \\ncalculatorSNN model \\n\\u2026\\n\\u2026\\nFig. 8. Experimental setup for evaluating TopSpark methodology.\\nTABLE I\\nTHE ORIGINAL PARAMETER SETTINGS FOR SNN S.\\nTimestepTrefVresetVth\\u0012\\n350 5 -60mV -52mV 1mV\\nV. R ESULTS AND DISCUSSION\\nA. Maintaining Accuracy\\nExperimental results for the accuracy are provided in Fig. 9.\\nIn general, the direct timestep reduction technique achieves com-\\nparable accuracy pro\\ufb01les in large timesteps, but suffers from\\na signi\\ufb01cant degradation in small timesteps due to the loss of\\ninformation (i.e., spikes) for maintaining the learning quality; M3600\\n020406080\\n050100 150 200 250 300 350020406080100\\n050100 150 200 250 300 350\\n020406080\\n050 100 150 200 250 300 350020406080100\\n050100 150 200 250 300 350020406080100\\n050100 150 200 250 300 350020406080100\\n0 50 100 150 200 250 300 350020406080100\\n050100 150 200 250 300 350Accuracy [%]STDP1 - Baseline STDP1 - TopSpark STDP2 - Baseline STDP2 - TopSpark\\n020406080\\n0 50 100 150 200 250 300 350020406080\\n0 50 100 150 200 250 300 350(a) MNIST (b) Fashion MNIST020406080\\n0 50 100 150 200 250 300 350Accuracy [%]406080100\\n10 20 30 40 50\\nTimestep(Direct)  (Direct)  Baseline accuracy with STDP1 Baseline accuracy with STDP2\\n406080100\\n10 20 30 40 50\\nM400 M900 M1600 M2500M400 M900 M1600 M2500 M36003 3\\n2 2\\n1(TopSpark)  (TopSpark)  Fig. 9. Accuracy pro\\ufb01les of SNN models across different timestep settings, learning rules, and workloads: (a) MNIST and (b) Fashion MNIST.\\n0.00.20.40.60.81.0\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350Latency and energy savings\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350Baseline (no timestep reduction)\\nM2500 M3600\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350\\n0.00.20.40.60.81.0\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n3500.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n3500.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350M2500M1600\\nM16000.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n3500.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350Latency (Direct) Latency (TopSpark) Energy (Direct) Energy (TopSpark)\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350\\n0.00.20.40.60.81.0\\n0.00.20.40.60.81.0\\n10203040501001501752002503500.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350M2500\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350M3600 M1600\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n3500.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350M400 M900\\n4\\n0.00.20.40.60.81.0\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350M3600\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350M2500\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350M1600\\nTimestep0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350\\nEnergy\\n(Normalized to Baseline)(i) STDP1 (ii) STDP2\\nLatency\\n(Normalized to Baseline)M400 M900\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n3504(b) Inference(a) Training\\n0.00.20.40.60.81.0\\n10\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350(i) STDP1 (ii) STDP2\\nLatency\\n(Normalized to Baseline)M900\\nM900M400\\nM400\\nEnergy\\n(Normalized to Baseline)M3600\\n44\\nFig. 10. Latency and energy consumption of SNNs during (a) training and (b) inference, considering different learning rules (STDP1 and STDP2), network\\nsizes, timestep settings, and workloads. The experimental results for MNIST and Fashion MNIST are similar due to the same number and size of samples.\\nsee 1. Meanwhile, our TopSpark achieves competitive accuracy\\npro\\ufb01les for all timestep settings across different learning rules and\\nworkloads, as shown in 1and 2. For instance, if we consider\\nan acceptable 2% accuracy loss from the baseline (i.e., original\\nSNN without timestep reduction) for M400 with the MNIST, our\\nTopSpark achieves the accuracy of 86% in timestep 30 (STDP1)\\nand 90% in timestep 100 (STDP2). Meanwhile, the direct reduction\\ntechnique achieves the accuracy of 85% in timestep 50 (STDP1)\\nand 90% in timestep 150 (STDP2). If we need to save more\\nbattery life, we can further reduce the timestep while relaxing the\\naccuracy constraint. For instance, TopSpark can achieve accuracy\\nof 77% (STDP1) and 75% (STDP2) in timestep 10 for M400\\nwith the MNIST, while the direct reduction technique achieves\\nonly accuracy of 52% (STDP1) and 14% (STDP2); see 3. The\\nreason is that, our TopSpark employs parameter enhancements that\\nproportionally scale down the values of neuron parameters to (1)\\nmake the neurons preserve the spiking and learning activities in\\nlarge timesteps for maintaining high accuracy, and (2) increase the proportionally scale down the values of neuron parameters to (1)\\nmake the neurons preserve the spiking and learning activities in\\nlarge timesteps for maintaining high accuracy, and (2) increase the\\nspiking and learning activities to compensate the loss of spikes for\\nimproving the learning quality in small timesteps and the accuracy.\\nB. Latency Improvements\\nFig. 10 shows the experimental results for latency. In general, the\\ndirect reduction technique and our TopSpark effectively reduce theprocessing latency as compared to the baseline, since they employ\\nsmaller timesteps. For instance, if we consider an acceptable 2%\\naccuracy loss from the baseline for M400 with the MNIST, the\\ndirect reduction technique improves latency by 3.9x in training\\nand 4.2x in inference for STDP1, and by 2.3x in training and\\ninference for STDP2, as compared to the baseline. Meanwhile,\\nour TopSpark can further improve latency by 10.8x in training and\\n10.9x in inference for STDP1, and by 2.5x in training and 3.5x in\\ninference for STDP2, as compared to the baseline; see 4. In all\\nexperimental scenarios, our TopSpark improves latency by 3.9x on\\naverage across different network sizes, learning rules, workloads,\\nand processing phases (i.e., training and inference). The reason is\\nthat, parameter enhancements in our TopSpark enable more timestep\\nreduction while preserving the learning quality through appropriate\\nspiking and learning activities.\\nC. Energy Ef\\ufb01ciency Improvements\\nThe experimental results for energy consumption are provided in\\nFig. 10. The direct reduction technique and our TopSpark effectively\\nimprove the energy ef\\ufb01ciency as compared to the baseline, since\\nthey have smaller latency and operational power. For instance, if we\\nconsider an acceptable 2% accuracy loss from the baseline for M400\\nwith the MNIST, the direct reduction technique improves energy\\nef\\ufb01ciency by 4x (STDP1) and by 2.3x (STDP2) in both the training B2B37075808590\\n10\\n30\\n50\\n150\\n200\\n35085-90\\n80-85\\n75-80\\n70-75\\nB2B37075808590\\n10\\n30\\n50\\n150\\n200\\n35085-90\\n80-85\\n75-80\\n70-75\\nB1B1: \\u03c4= 0, \\u03b5= 0\\nB2: \\u03c4= 10, \\u03b5= 0\\nB3 : \\u03c4= 0, \\u03b5= 1010\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n35010\\n20\\n30\\n40\\n50\\n100\\n150\\n175\\n200\\n250\\n350Trade -Off Benefit\\nB1Timestep\\nrepresents the latency\\nAdjustment factor settings\\nTimestep\\nrepresents the latency\\nAdjustment factor settings(a) STDP1\\n(b) STDP25\\nBaseline \\naccuracy\\nBaseline \\naccuracy567Trade -Off BenefitB1: \\u03c4= 0, \\u03b5= 0\\nB2: \\u03c4= 10, \\u03b5= 0\\nB3 : \\u03c4= 0, \\u03b5= 10Fig. 11. Design trade-offs on the accuracy, latency, and energy consumption\\nfor the M400 with MNIST workload across different learning rules. These\\nresults are also representative for other SNN models with different network\\nsizes, learning rules, and workloads, since they have similar trends.\\nand inference phases, as compared to the baseline. Meanwhile, our\\nTopSpark further improves energy ef\\ufb01ciency by 10x (STDP1) and\\nby 2.5x-3.5x (STDP2) in both the training and inference phases, as\\ncompared to the baseline; see 4. In all experimental scenarios, our\\nTopSpark improves energy ef\\ufb01ciency by 3.5x (training) and by 3.3x\\n(inference) on average across different network sizes, learning rules,\\nand workloads. The reason is that, our TopSpark employs effective\\nparameter enhancements that preserve the learning quality across\\ndifferent network sizes, timesteps, learning rules, workloads, and\\nprocessing phases. Therefore, the reduced timesteps lead to reduced\\nlatency and operational power, and hence the energy consumption.\\nD. Design Trade-Offs\\nExperimental results for design trade-offs are provided in Fig. 11.\\nThe results show that, we can set the adjustment factors to meet\\nthe design constraints. If we need to prioritize accuracy over the\\nother metrics, then we may set \\u001c= 0 and\\\"= 0. As a result, the\\ntrade-off bene\\ufb01t leads to design points that achieve high accuracy as\\npotential solutions, as pointed out by 5. If we set a higher priority\\nto latency over the other metrics (e.g., \\u001c= 10 and\\\"= 0), the trade-\\noff bene\\ufb01t is shifted to a design point that has a signi\\ufb01cant timestep\\nreduction; see 6. Similar results are observed if we set higher\\npriority to energy consumption, as a signi\\ufb01cant timestep reduction\\nalso effectively improves energy ef\\ufb01ciency; see 7. We also observe\\nthat, there may be a change of trade-off bene\\ufb01t when we change the\\nadjustment factors ( \\u001cand\\\"); see 5-7. The reason is that, when\\nwe prioritize one metric above the others, the bene\\ufb01t of the other\\nmetrics will decrease, and vice versa. These results also show that\\nthe SNN-based systems can employ our strategy to trade-off their\\naccuracy, latency, and energy consumption.\\nThe above results and discussion show that the TopSpark method-\\nology is applicable for enabling energy-ef\\ufb01cient SNNs with STDP-\\nbased learning rules in training and inference phases, across dif-\\nferent timesteps, network sizes, and workloads, hence making it\\namenable to autonomous mobile agents\\/robots.VI. C ONCLUSION\\nWe propose a novel TopSpark methodology that leverages adap-\\ntive timestep optimizations to enable energy-ef\\ufb01cient SNNs via\\nanalysis of accuracy in different timesteps, parameter enhance-\\nments, and trade-offs for accuracy-latency-energy. Our TopSpark\\nsaves latency by 3.9x and energy consumption by 3.5x (training)\\nand 3.3x (inference) on average, while keeping accuracy within\\n2% of SNNs without timestep reduction. Therefore, our work\\nmay enable low-latency and energy-ef\\ufb01cient SNN training and\\ninference for autonomous mobile agents\\/robots, including their\\nef\\ufb01cient online learning process.\\nREFERENCES\\n[1] R. Bonnevie, D. Duberg, and P. Jensfelt, \\u201cLong-term exploration in un-\\nknown dynamic environments,\\u201d in 2021 7th Int. Conf. on Automation,\\nRobotics and Applications (ICARA) , 2021, pp. 32\\u201337.\\n[2] A. Tavanaei et al. , \\u201cDeep learning in spiking neural networks,\\u201d Neural\\nNetworks , vol. 111, pp. 47\\u201363, 2019.\\n[3] F. Akopyan et al. , \\u201cTruenorth: Design and tool \\ufb02ow of a 65 mw 1 [2] A. Tavanaei et al. , \\u201cDeep learning in spiking neural networks,\\u201d Neural\\nNetworks , vol. 111, pp. 47\\u201363, 2019.\\n[3] F. Akopyan et al. , \\u201cTruenorth: Design and tool \\ufb02ow of a 65 mw 1\\nmillion neuron programmable neurosynaptic chip,\\u201d IEEE Trans. on\\nComputer-Aided Design of Integrated Circuits and Systems , 2015.\\n[4] M. Davies et al. , \\u201cLoihi: A neuromorphic manycore processor with\\non-chip learning,\\u201d IEEE Micro , vol. 38, no. 1, pp. 82\\u201399, Jan 2018.\\n[5] R. V . W. Putra and M. Sha\\ufb01que, \\u201clpspikecon: Enabling low-precision\\nspiking neural network processing for ef\\ufb01cient unsupervised continual\\nlearning on autonomous agents,\\u201d in 2022 Int. Joint Conf. on Neural\\nNetworks (IJCNN) , 2022, pp. 1\\u20138.\\n[6] J. M. Allred and K. Roy, \\u201cControlled forgetting: Targeted stimula-\\ntion and dopaminergic plasticity modulation for unsupervised life-\\nlong learning in spiking neural networks,\\u201d Frontiers in Neuroscience ,\\nvol. 14, p. 7, 2020.\\n[7] P. Panda et al. , \\u201cAsp: Learning to forget with adaptive synaptic\\nplasticity in spiking neural networks,\\u201d IEEE Journal on Emerging and\\nSelected Topics in Circuits and Systems , vol. 8, no. 1, 2018.\\n[8] R. V . W. Putra and M. Sha\\ufb01que, \\u201cFspinn: An optimization framework\\nfor memory-ef\\ufb01cient and energy-ef\\ufb01cient spiking neural networks,\\u201d\\nIEEE Trans. on Computer-Aided Design of Integrated Circuits and\\nSystems , vol. 39, no. 11, pp. 3601\\u20133613, 2020.\\n[9] A. Basu et al. , \\u201cSpiking neural network integrated circuits: A review of\\ntrends and future directions,\\u201d in 2022 IEEE Custom Integrated Circuits\\nConf. (CICC) , 2022, pp. 1\\u20138.\\n[10] P. Diehl and M. Cook, \\u201cUnsupervised learning of digit recognition\\nusing spike-timing-dependent plasticity,\\u201d Frontiers in Computational\\nNeuroscience , vol. 9, p. 99, 2015.\\n[11] S. Sen et al. , \\u201cApproximate computing for spiking neural networks,\\u201d\\ninDesign, Automation Test in Europe Conf. Exhibition (DATE) , March\\n2017, pp. 193\\u2013198.\\n[12] N. Rathi et al. , \\u201cStdp-based pruning of connections and weight quan-\\ntization in spiking neural networks for energy-ef\\ufb01cient recognition,\\u201d\\nIEEE Trans. on Computer-Aided Design of Integrated Circuits and\\nSystems , vol. 38, no. 4, pp. 668\\u2013677, April 2019.\\n[13] S. Krithivasan et al. , \\u201cDynamic spike bundling for energy-ef\\ufb01cient\\nspiking neural networks,\\u201d in 2019 IEEE\\/ACM Int. Symp. on Low Power\\nElectronics and Design (ISLPED) , July 2019, pp. 1\\u20136.\\n[14] S. Lu and A. Sengupta, \\u201cExploring the connection between binary and\\nspiking neural networks,\\u201d Frontiers in Neuroscience , vol. 14, 2020.\\n[15] N. Rathi and K. Roy, \\u201cDiet-snn: A low-latency spiking neural network\\nwith direct input encoding and leakage and threshold optimization,\\u201d\\nIEEE Trans. on Neural Networks and Learning Systems , pp. 1\\u20139, 2021.\\n[16] S. S. Chowdhury, N. Rathi, and K. Roy, \\u201cOne timestep is all you\\nneed: Training spiking neural networks with ultra low latency,\\u201d arXiv\\npreprint arXiv:2110.05929 , 2021.\\n[17] A. Nallathambi et al. , \\u201cLayerwise disaggregated evaluation of spiking\\nneural networks,\\u201d in ACM\\/IEEE Int. Symp. on Low Power Electronics\\nand Design (ISLPED) , 2022.\\n[18] J.-L. Yin et al. , \\u201cDeep battery saver: End-to-end learning for\\npower constrained contrast enhancement,\\u201d IEEE Trans. on Multimedia\\n(TMM) , vol. 23, 2021.\\n[19] H. Hazan et al. , \\u201cBindsnet: A machine learning-oriented spiking neural\\nnetworks library in python,\\u201d Frontiers in Neuroinformatics , 2018.\\n[20] S. Han et al. , \\u201cDeep compression: Compressing deep neural networks\\nwith pruning, trained quantization and huffman coding,\\u201d arXiv preprint\\narXiv:1510.00149 , 2015.\",\"26\":\" arXiv:2303.01695v1  [cs.NE]  3 Mar 2023EVOLUTIONARY MULTI -OBJECTIVE ALGORITHMS FOR THE\\nKNAPSACK PROBLEMS WITH STOCHASTIC PROFITS\\nA P REPRINT\\nKokila Perera\\nOptimisation and Logistics\\nSchool of Computer and Mathematical Sciences\\nThe University of Adelaide\\nAdelaide, AustraliaAneta Neumann\\nOptimisation and Logistics\\nSchool of Computer and Mathematical Sciences\\nThe University of Adelaide\\nAdelaide, Australia\\nFrank Neumann\\nOptimisation and Logistics\\nSchool of Computer and Mathematical Sciences\\nThe University of Adelaide\\nAdelaide, Australia\\nABSTRACT\\nEvolutionary multi-objective algorithms have been widely shown to be successful when utilized\\nfor a variety of stochastic combinatorial optimization pro blems. Chance constrained optimization\\nplays an important role in complex real-world scenarios, as it allows decision makers to take into\\naccount the uncertainty of the environment. We consider a ve rsion of the knapsack problem with\\nstochastic pro\\ufb01ts to guarantee a certain level of con\\ufb01dence in the pro\\ufb01t of the solutions. We in-\\ntroduce the multi-objective formulations of the pro\\ufb01t chan ce constrained knapsack problem and\\ndesign three bi-objective \\ufb01tness evaluation methods that w ork independently of the speci\\ufb01c con\\ufb01-\\ndence level required. We evaluate our approaches using well -known multi-objective evolutionary\\nalgorithms GSEMO and NSGA-II. In addition, we introduce a \\ufb01l tering method for GSEMO that\\nimproves the quality of the \\ufb01nal population by periodically removing certain solutions from the in-\\nterim populations based on their con\\ufb01dence level. We show th e effectiveness of our approaches on\\nseveral benchmarks for both settings where the knapsack ite ms have \\ufb01xed uniform uncertainties and\\nuncertainties that are positively correlated with the expe cted pro\\ufb01t of an item.\\nKeywords Multi-objective optimization, stochastic knapsack probl em, chance constrained problems\\n1 Introduction\\nReal world optimization problems involve often uncertain p roperties imposed by some stochastic components in the\\nproblem or noise in its environment Peng (2019); He and Shao ( 2009). Such uncertainties must be considered in the\\noptimization to \\ufb01nd reliable and useful solutions to proble ms. Chance constraints are a natural way to model uncer-\\ntainties in problems. They can capture the effects of uncert ain variables on the inequality constraints and formulate\\nthe optimization problems under uncertainties Peng (2019) ; Neumann et al. (2020); Doerr et al. (2020). A chance\\nconstraint is usually an inequality constraint on some stoc hastic variable of the problem, which can be violated by a\\nslight chance (a small probability) when optimizing the problem Neumann a nd Sutton (2019); Neumann and Neumann\\n(2020). Chance constraints enable one to set the desired con \\ufb01dence level of getting the maximal pro\\ufb01t when implement-\\ning a solution, unlike using a solution from a deterministic approach. The study of problems with chance constraints\\nleads to developing ef\\ufb01cient applications to solve complex optimization problems and is crucial in developing real\\nworld solutions for complex problems, such as in the mining i ndustry Xie et al. (2021a); Mohtasham et al. (2021),\\npower systems Geng and Xie (2019), communication systems Ab e et al. (2020) and transportation Kepaptsoglou et al.\\n(2015). Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nThis work considers a variation of the classical knapsack pr oblem (KP), where the items have stochastic pro\\ufb01ts and\\ndeterministic weights. As the weights of items are determin istic, the constraint on the weights remains the same\\nas in the classical KP. An additional constraint is introduc ed to this problem to capture the uncertainty of the pro\\ufb01t\\nvariable, which is a chance constraint on the pro\\ufb01t Neumann e t al. (2022). This constraint guarantees that the solution\\nhas maximal pro\\ufb01t P and only drops below P for a small probabil ity (\\u03b1). In summary, the optimization process for\\nthis problem is to \\ufb01nd the solution(s) that maximize the pro\\ufb01 t P, subjected to the weight constraint and pro\\ufb01t chance\\nconstraint, which guarantees that the pro\\ufb01t P is obtained wi th probability at least \\u03b1.\\nIn the literature, the KP with a chance constraint on the weig ht (as elements have deterministic pro\\ufb01ts and stochastic\\nweights) is more prevalent Xie et al. (2019, 2020). On the con trary, the pro\\ufb01t chance constrained KP is explored\\nonly in one study, which appears in recent literature Neuman n et al. (2022). This work considers single-objective\\nevolutionary approaches to address this problem Neumann et al. (2022). This scenario makes it possible to identify\\nthe risk of not achieving the expected pro\\ufb01t by implementing a particular solution for the problem and making better\\ndecisions in planning. This problem re\\ufb02ects a bene\\ufb01cial and valid real-world problem scenario like mine planning as\\nmentioned in Neumann et al. (2022).\\nEvolutionary algorithms (EAs) perform well in addressing s tochastic optimization problems Doerr and Neumann\\n(2021); Singh and Branke (2022). Also, they can produce opti mal or feasible solutions in a reasonable amount of\\ntime for complex combinatorial optimization problems like chance constrained knapsack and job shop scheduling\\nproblems. EAs may consider a different number of objectives depending on how the de\\ufb01nition of the underlying\\nproblem. Usually, when an EA considers a single objective, i t generates a single solution that optimizes the objective\\nvalue. In contrast, a multi-objective EA generates a set of s olutions that gives a trade-off between given objectives.\\nSuch a solution set makes it possible to have more insights in to improving the algorithms and search space than having\\na single solution as the outcome Coello et al. (2013); Deb (20 01). Therefore, multi-objective algorithms help one to\\nmake informed decisions on selecting a solution to implemen t.\\nIn this work, we explore the use of the multi-objective evolu tionary approaches for the chance constrained KP with\\nstochastic pro\\ufb01ts introduced in Neumann et al. (2022). Here we introduce multi-objective \\ufb01tness evaluation for EAs\\nto address this problem and methods to enhance their perform ance.\\n1.1 Related Work\\nThe use of evolutionary computation for chance constrained problems appears in the early literature He and Shao\\n(2009); Loughlin and Ranjithan (1999); Liu et al. (2013); Ma sutomi et al. (2013). Those works consider computation-\\nally expensive methods like simulations and sampling to cat er for chance constraints. More recent studies have looked\\ninto tail-bound inequalities, which more ef\\ufb01ciently deal w ith chance constraints Neumann et al. (2022); Xie et al.\\n(2019); Assimi et al. (2020).\\nThe chance constrained KP where the pro\\ufb01ts are deterministi c and weights are stochastic is considered in several\\npapers Xie et al. (2019, 2020). Xie et al. (2019) presents how to use well-known deviation inequalities: Chebyshev\\u2019s\\ninequality and Hoeffding bound to estimate the probability of constraint violation. In Xie et al. (2020), where the same\\nKP variation is considered, they introduce problem-speci\\ufb01 c operators for EAs with both single- and multi-objective\\nformulations. In the study Assimi et al. (2020), dynamic cha nce constrained KP with stochastic pro\\ufb01ts is studied. formulations. In the study Assimi et al. (2020), dynamic cha nce constrained KP with stochastic pro\\ufb01ts is studied.\\nIn addition to the objective function on the pro\\ufb01t of a given s tochastic solution, a second objective is introduced to\\naddress the dynamic capacity constraint. It captures the mi nimal capacity bound for the solution that meets the chance\\nconstraints.\\nRun-time analysis is an essential topic in studying problem s with chance constraints. The \\ufb01rst paper on run time\\nanalysis for chance constraint problems considers the KP wi th stochastic weights Neumann and Sutton (2019). This\\nwork considers different cases of that problem and studies t he run time of (1+1) EA for them. In Xie et al. (2021b),\\nthey perform the run time analysis of simple EAs for chance co nstrained KPs with uniform weights. The papers\\nNeumann and Witt (2022) and Shi et al. (2022) study the run tim e of simple EAs for different chance constrained\\nproblems. InNeumann and Witt (2022), the authors consider s ingle- and multi-objective EAs for chance constrained\\nproblems with normally distributed random problem variabl es. They also show how to use the proposed evolutionary\\napproaches for chance constrained minimum spanning tree pr oblems Neumann and Witt (2022). In Shi et al. (2022),\\nthey analyze the run time of random local search and (1+1) EA f or the chance constrained makespan problem.\\nIn the study Neumann et al. (2022), the authors simple EAs for the pro\\ufb01t chance constrained KP. Those algorithms\\ninclude (1+1) EA with standard bit-\\ufb02ip mutation and heavy-t ail mutation operators and population based ( \\u00b5+1) EA,\\nwhich uses a speci\\ufb01c crossover operator for the KP. This stud y evaluates the performance of all these algorithms\\nusing the single objective \\ufb01tness evaluation. The overall r esults show that (1+1) EA with heavy tail mutation operator\\nsigni\\ufb01cantly improved over other algorithms.\\n2 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nThis work is motivated by the recent study on evolutionary mu lti-objective algorithms that compute trade-offs with\\nrespect to the expected value and the variance of solutions p resented in Neumann and Witt (2022). Here we explore the\\nmulti-objective formulations for the pro\\ufb01t chance constra ined KP based on that work. In addition to the variance of the\\nsolutions, we consider the standard deviation and also the c ount of elements in the solutions (under certain conditions )\\nto introduce \\ufb01tness evaluations for the problem. The signi\\ufb01 cance of these \\ufb01tness functions is that they can evaluate the\\n\\ufb01tness of a solution independent of any speci\\ufb01c con\\ufb01dence le vel of its pro\\ufb01t (i.e. independent of speci\\ufb01c value for \\u03b1).\\nSince this generates a set of solutions that gives a trade-of f of speci\\ufb01c objectives for each \\ufb01tness formulation with eac h\\nalgorithm, it allows one to make more informed decisions whe n selecting a solution to implement. For example, to\\nidentify the solution that gives the best pro\\ufb01t with a partic ular\\u03b1value, we can calculate the pro\\ufb01t of all the solutions\\nfor that value and select the solution that gives the best pro \\ufb01t among the \\ufb01nal population.\\nThis study considers two well-known multi-objective EAs: G SEMO Giel (2003) and NSGA-II Deb et al. (2002).\\nFurthermore, we introduce a \\ufb01ltering method for GSEMO which aims to improve the quality of the \\ufb01nal population\\nresulting from the algorithm. This \\ufb01ltering method is appli ed regularly after a \\ufb01xed number of evaluations in the\\nevolutionary process. It considers whether a solution can b e the best solution for any \\u03b1\\u2208[0.0,1\\/2]considering the\\ninterim population at the time and, otherwise, removes it fr om the population. For all experimental settings, in additi on\\nto the two algorithms, GSEMO and NSGA-II, we consider this \\ufb01l tering method combined with GSEMO as a separate\\nalgorithm.\\nThe structure of this paper is as follows. Section 2 introduc es the problem formulation, and Section 2 discusses the\\nalgorithms we consider in this study. Section 3 discusses th e multi-objective formulation, including the objective\\nfunction on pro\\ufb01t, \\ufb01tness functions and how to use the probab ility bounds to estimate the con\\ufb01dence level of the\\nsolutions. Section 4 and 5 discuss the EAs considered in this paper and the new Filtering method introduced in this\\nwork. Section 6 presents the details of the experimental set tings and the results. Finally, Section 7 gives the conclusi ons\\nof this study.\\n2 Problem De\\ufb01nition\\nIn the classical KP, the pro\\ufb01t and weight values of the items a re deterministic. Let the KP has n items\\n{x1,...,x i,...,x n}with pro\\ufb01t piand weight wiand weight bound B. A solution to the problem can be represent ed as\\n{0,1}nsuch that xi= 1only when xiis selected in the solution. Given the pro\\ufb01t of solution xasp(x) =\\/summationtextn\\ni=1pi.xi\\nand weight w(x) =\\/summationtextn\\ni=1wi.xi, the classical KP is to \\ufb01nd the solution x\\u2217that maximize p(x\\u2217)subjected to the\\nweight constraint w(x\\u2217)\\u2264B.\\nIn this work, we consider a stochastic version of the classic al KP where the pro\\ufb01ts of the knapsack items are stochastic\\nwhile weights remain deterministic. Therefore the pro\\ufb01t of a solution will be uncertain and may vary from the expected\\nmaximal pro\\ufb01t value. A chance constraint on the pro\\ufb01t is used to capture the stochastic behaviour of the problem. This\\nconstraint ensures that for each feasible solution x, the maximum probability that the pro\\ufb01t will drop below pro\\ufb01 t value\\nP is only a small probability 0< \\u03b1 <1\\/2.\\nWe can formally present this problem as follows:\\nmaxP (1)\\ns. t.Pr(p(x)< P)\\u2264\\u03b1 (2)\\nand w (x)\\u2264B (3)\\nwhere\\u03b1is a parameter determining the allowed failure probability which is a small value <= 1\\/2. Equation 2\\nspeci\\ufb01es the chance constraint on the pro\\ufb01t.\\n2.1 Estimating Pro\\ufb01t of a Solution where\\u03b1is a parameter determining the allowed failure probability which is a small value <= 1\\/2. Equation 2\\nspeci\\ufb01es the chance constraint on the pro\\ufb01t.\\n2.1 Estimating Pro\\ufb01t of a Solution\\nComputing the probability of violating a given chance const raint is intractable in general Doerr and Neumann (2021).\\nTherefore, it is not straightforward to estimate the pro\\ufb01t o f a solution under uncertainties. However, tail bounds can\\nbe used to upper bound the probability of chance constraint v iolationDoerr and Neumann (2021). In Neumann et al.\\n(2022), the authors present pro\\ufb01t estimates for the problem based on the tail bounds: Chebyshev\\u2019s inequality and\\nHoeffding bound. Each of these applies to the problem under c ertain conditions.\\nIf the expectation and variance of the solutions\\u2019 pro\\ufb01ts are known for a given problem instance, we can use Cheby-\\nshev\\u2019s inequality to calculate the pro\\ufb01t of a solution for it Doerr (2020). Neumann et al. (2022) present a pro\\ufb01t estimate\\nin Equation 4, that ensure the pro\\ufb01t chance constraint Pr(p(x)< P)\\u2264\\u03b1is held. We can guarantee that the solution\\n3 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nxwill give the pro\\ufb01t \\u02c6pCheb(x,\\u03b1)except for a small probability \\u03b1as follows:\\n\\u02c6pCheb(x,\\u03b1) =\\u00b5(x)\\u2212\\/radicalbig\\n(1\\u2212\\u03b1)\\/\\u03b1\\u00b7\\/radicalbig\\nv(x). (4)\\nThe above equation gives a very general setting for the pro\\ufb01t of a solution that can be used in any scenario where the\\nexpected value and the variance are known. For example, we ca n consider that the pro\\ufb01ts pitake a uniform distribution\\nsuch that pi\\u2208{\\u00b5i\\u2212\\u03b4,\\u00b5i+\\u03b4}which gives the expected value \\u00b5iand variance\\u03b42\\n3.|x|1.\\nThe Hoeffding bound can be used to formulate a pro\\ufb01t estimate if the pro\\ufb01ts are taken randomly from a uniform\\nrandom distribution independent of each other Doerr (2020) . From Neumann et al. (2022), we get the formulation for\\nthe pro\\ufb01t of solution xusing the Hoeffding bound ( \\u02c6pHoef(x,\\u03b1)) as follows:\\n\\u02c6pHoef(x,\\u03b1) =\\u00b5(x)\\u2212\\u03b4\\u00b7\\/radicalbig\\nln(1\\/\\u03b1)\\u00b72\\u00b7|x|1. (5)\\n3 Multi-Objective Formulation\\nIn this section, we introduce the multi-objective formulat ions of the pro\\ufb01t chance constrained KP. As presented in\\nSection 2, the optimal solution for the KP maximizes the pro\\ufb01 t subjected to the weight bound constraint. For the pro\\ufb01t\\nchance constrained KP with deterministic weights, the mult i-objective formulation needs to consider the uncertainty\\nof the pro\\ufb01ts of knapsack items. In the following subsection s, we present the multi-objective formulations and the\\nfunctions to estimate pro\\ufb01ts.\\n3.1 Fitness Functions\\nIn general, we need to ensure that the expectation of pro\\ufb01t is maximized while the variance is minimized to maximize\\nthe pro\\ufb01t of the solution. Considering this, we introduce th e \\ufb01tness function g(x) = (\\u00b5(x),v(x))that will produce\\na set of solutions that gives a trade-off between the two obje ctives, irrespective of \\u03b1the guarantee of the pro\\ufb01t of the\\nindividual solution\\u2019s pro\\ufb01t value. The formula for the two o bjectives is given in Equation 6 and 7 where vmax=\\/summationtextn\\ni=1vi.\\n\\u00b5(x) =\\/braceleftbigg\\/summationtextn\\ni=1\\u00b5ixiw(x)\\u2264B\\nB\\u2212w(x)w(x)> B(6)\\nv(x) =\\/braceleftbigg\\/summationtextn\\ni=1\\u03c32\\nixi w(x)\\u2264B\\nvmax+(w(x)\\u2212B)w(x)> B(7)\\nWhen evaluating the \\ufb01tness of solutions, we need to determin e their dominance concerning the different objectives.\\nFor two solutions A and B, we say that Adominates B(denoted as A\\/{ollowsequalB)iff\\u00b5(A)\\u2265\\u00b5(B)\\u2227v(A)\\u2264v(B), andA\\nstrongly dominates B(denoted as A\\u227bB)iffA\\/{ollowsequalBand\\u00b5(A)> \\u00b5(B)\\u2228v(A)< v(B). For an infeasible solution\\nthat violates the weight capacity constraint ( w(A)> B ), the above formula penalises the two objective values (see\\nEquation 6 and 7). This formulation ensures that any feasibl e solution dominates the infeasible solution instances.\\nNext, we consider the \\ufb01tness function g\\u2032(x) = (\\u00b5(x),s(x)). Only the second objective of this \\ufb01tness function differs\\nfromg\\u2032(x)while the \\ufb01rst objective \\u00b5remains the same and can be de\\ufb01ned as given in Equation 6. We de note the\\nmaximal standard deviation as smax=\\/summationtextn\\ni=1\\u03c3iin Equation 8 which de\\ufb01nes the second objective of this \\ufb01tnes s\\nfunctiong\\u2032(x).\\ns(x) =\\/braceleftbigg\\/radicalbig\\/summationtextn\\ni=1\\u03c32\\nixi w(x)\\u2264B\\nsmax+(w(x)\\u2212B)w(x)> B.(8)\\nFor simple EAs like GSEMO, the \\ufb01nal population when optimizi ngg(x)is equivalent to when optimizing g\\u2032(x)as\\nthe difference between the two is that the second objective o f the latter is the square root of the second objective of the\\nformer. Therefore it does not change the order of search poin ts of such algorithms. However, it should be noted that\\nsome popular EMO algorithms like NSGA-II would encounter a d ifference in the diversity mechanism when working\\nwithv(x)instead of s(x). Therefore, we investigate both g(x)andg\\u2032(x)in this work.\\n4 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\n3.2 Fitness Function for Pro\\ufb01ts with Same Dispersion\\nWhen the value for \\u03b4is the same for all elements, we can consider the number of ite ms selected in the solution ( |x|1)\\nas the second objective. This objective enables the de\\ufb01niti on of a \\ufb01tness function that performs independent of both\\nthe con\\ufb01dence level of the solution ( \\u03b1) and the uncertainty level of the individual pro\\ufb01t values ( \\u03b4). We denote the\\n\\ufb01tness function for this scenario as g\\u2032\\u2032(x) = (\\u00b5(x),c(x)). The \\ufb01rst objective of the new \\ufb01tness function is based on\\nthe expectation of the pro\\ufb01t of x\\u00b5(x), similar to previous functions and calculated as given in Eq uation 6. The second\\nobjective c(x)is calculated as follows,\\nc(x) =\\/braceleftbigg\\/summationtextn\\ni=1xi w(x)\\u2264B\\nn+(w(x)\\u2212B)w(x)> B.(9)\\n3.3 Estimating the Best Pro\\ufb01t Value\\nAs the \\ufb01nal output of the multi-objective evolutionary appr oaches, we get a set of solutions that gives a trade-off of\\nthe objectives independent of the con\\ufb01dence level of the pro \\ufb01ts. We need to identify the solution with the best pro\\ufb01t\\nfor a given con\\ufb01dence level \\u03b1using these \\ufb01nal solutions. We can use Equations 4 and 5 from N eumann et al. (2022)\\nand calculate the pro\\ufb01t of all the solutions in the \\ufb01nal popul ation for the required con\\ufb01dence levels ( \\u03b1). The solution\\ngiving the highest pro\\ufb01t value for each \\u03b1is the best solution for that particular setting.\\n3.4 Con\\ufb01dence Level of a Solution\\u2019s Pro\\ufb01t\\nDifferent solutions in the \\ufb01nal population produced by mult i-objective approaches become the best solution for differ -\\nent con\\ufb01dence levels. We can identify the con\\ufb01dence level of each solution in the \\ufb01nal population for which it gives\\nthe highest pro\\ufb01t value. First, we obtain the \\u03b1threshold for a pair of solutions such that one becomes bette r than the\\nother by comparing the pro\\ufb01t estimates mentioned in Subsect ion 2.1. Based on the threshold \\u03b1value between solution\\npairs in the \\ufb01nal population, we de\\ufb01ne a con\\ufb01dence level inte rval for each solution such that they give the best pro\\ufb01t\\nvalue for any \\u03b1in that interval.\\n3.4.1 Con\\ufb01dence Level Threshold using Chebyshev\\u2019s Inequal ity\\nLet the solutions xandysuch that \\u00b5(x)> \\u00b5(y)andv(x)> v(y), then we can say that the pro\\ufb01t of the solution xis\\nbetter than ywhen the minimum required con\\ufb01dence level( \\u03b1x,y) is as follows:\\n\\u03b1Cheb(x,y)=1\\n1+(R(x,y))2s.t. R(x,y) =\\u00b5(x)\\u2212\\u00b5(y)\\/radicalbig\\nv(x)\\u2212\\/radicalbig\\nv(y)(10)\\nFor the con\\ufb01dence level \\u03b1\\u22651\\/(1+(R(x,y))2), the solution Xwill have a better pro\\ufb01t value than the solution Y.\\nTheorem 3.1. Let0< \\u03b1 <1,and x andybe two feasible solutions such that \\u00b5(x)> \\u00b5(y)andv(x)> v(y), holds.\\nIf\\u03b1\\u22651\\n1+(R(x,y))2holds such that R(x,y) =\\u00b5(x)\\u2212\\u00b5(y)\\u221a\\nv(x)\\u2212\\u221a\\nv(y)then\\u02c6pCheb(x,\\u03b1)\\u2265\\u02c6pCheb(y,\\u03b1).\\nProof. We have\\n\\u03b1\\u22651\\n1+(R(x,y))2\\n\\u21d0\\u21d2 \\u03b1+\\u03b1(R(x,y))2\\u22651\\n\\u21d0\\u21d2 (R(x,y))2\\u2265(1\\u2212\\u03b1)\\/\\u03b1\\n5 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nAs we assume 0< \\u03b1 <1,\\u00b5(x)> \\u00b5(y)andv(x)> v(y), we have R(x,y)>0and1\\u2212\\u03b1\\n\\u03b1>0.\\nThis implies,\\nR(x,y)\\u2265\\/radicalbig\\n(1\\u2212\\u03b1)\\/\\u03b1\\n\\u21d0\\u21d2\\u00b5(x)\\u2212\\u00b5(y)\\/radicalbig\\nv(x)\\u2212\\/radicalbig\\nv(y)\\u2265\\/radicalbig\\n(1\\u2212\\u03b1)\\/\\u03b1\\n\\u21d0\\u21d2 \\u00b5(x)\\u2212\\u00b5(y)\\u2265\\/radicalbigg\\n1\\u2212\\u03b1\\n\\u03b1\\u00b7\\/parenleftBig\\/radicalbig\\nv(x)\\u2212\\/radicalbig\\nv(y)\\/parenrightBig\\n\\u21d0\\u21d2 \\u00b5(x)\\u2212\\/radicalbigg\\n1\\u2212\\u03b1\\n\\u03b1\\u00b7\\/radicalbig\\nv(x)\\u2265\\u00b5(y)\\u2212\\/radicalbigg\\n1\\u2212\\u03b1\\n\\u03b1\\u00b7\\/radicalbig\\nv(y)\\n\\u21d0\\u21d2 \\u02c6pCheb(x,\\u03b1)\\u2265\\u02c6pCheb(y,\\u03b1)\\n3.4.2 Con\\ufb01dence Level Threshold using Hoeffding Bound\\nConsider the solutions xandysuch that \\u00b5(x)> \\u00b5(y)andv(x)> v(y). From the \\u02c6pHoef we can derive the minimum\\ncon\\ufb01dence level \\u03b1x,yfor which solution xgives a better pro\\ufb01t than yw.r.t.\\u02c6pHoef as follows:\\n\\u03b1Hoef(x,y)=e\\u2212(S(x,y))2s.t. S(x,y) =\\u00b5(x)\\u2212\\u00b5(y)\\n\\u03b4\\/parenleftBig\\/radicalbig\\n2|x|1\\u2212\\/radicalbig\\n2|y|1\\/parenrightBig (11)\\nTheorem 3.2. Let0< \\u03b1 <1,and x andybe two feasible solutions such that \\u00b5(x)> \\u00b5(y)andv(x)> v(y), holds.\\nIf\\u03b1\\u2265e\\u2212(S(x,y))2holds then \\u02c6pHoef(x,\\u03b1)\\u2265\\u02c6pHoef(y,\\u03b1).\\nProof. We have,\\n\\u03b1\\u2265e\\u2212(S(x,y))2\\n\\u21d0\\u21d2 e(S(x,y))2\\u22651\\/\\u03b1\\n\\u21d0\\u21d2 (S(x,y))2\\u2265ln(1\\/\\u03b1)\\nAs we assume 0< \\u03b1 <1,\\u00b5(x)> \\u00b5(y)andv(x)> v(y), we have S(x,y)>0andln1\\n\\u03b1>0.\\nThis implies,\\nS(x,y)\\u2265\\/radicalbig\\nln(1\\/\\u03b1)\\n\\u21d0\\u21d2\\u00b5(x)\\u2212\\u00b5(y)\\n\\u03b4\\/parenleftBig\\/radicalbig\\n2|x|1\\u2212\\/radicalbig\\n2|y|1\\/parenrightBig\\u2265\\/radicalbig\\nln(1\\/\\u03b1)\\n\\u21d0\\u21d2 \\u00b5(x)\\u2212\\u00b5(y)\\u2265\\u03b4\\u00b7\\/radicalBigg\\n2ln\\/parenleftbigg1\\n\\u03b1\\/parenrightbigg\\/parenleftBig\\/radicalbig\\n|x|1\\u2212\\/radicalbig\\n|y|1\\/parenrightBig\\n\\u21d0\\u21d2 \\u00b5(x)\\u2212\\u03b4\\u00b7\\/radicalBigg\\nln\\/parenleftbigg1\\n\\u03b1\\/parenrightbigg\\n\\u00b72\\u00b7|x|1\\u2265\\u00b5(y)\\u2212\\u03b4\\u00b7\\/radicalBigg\\nln\\/parenleftbigg1\\n\\u03b1\\/parenrightbigg\\n\\u00b72\\u00b7|y|1\\n\\u21d0\\u21d2 \\u02c6pHoef(x,\\u03b1)\\u2265\\u02c6pHoef(y,\\u03b1)\\n3.4.3 Con\\ufb01dence Level Interval for Solutions in the Final Po pulation\\nWe can use the \\u03b1threshold value in the previous subsections, to introduce t he con\\ufb01dence level range for solutions in\\na population as follows. As the \\u03b1threshold we can use either \\u03b1Cheb(x,y)or\\u03b1Hoef(x,y). Despite the speci\\ufb01c equation\\nto estimate the \\u03b1threshold, we can introduce the \\u03b1interval for a solution.\\nFirst, we sort all the solutions in the given population P as {x1,...xn}such that \\u00b5(x1)\\u2265\\u00b5(x2)\\u2265...\\u2265\\u00b5(xn). Then,\\nwe can de\\ufb01ne a (n+1)\\u00d7(n+1) symmetric matrix of con\\ufb01dence level thresholds as:\\n6 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nAlgorithm 1 GSEMO\\n1:Choosex\\u2208{0,1}nuniformly at random ;\\n2:S\\u2190{x};\\n3:while stopping criterion not met do\\n4: choosex\\u2208Suniformly at random;\\n5:y\\u2190\\ufb02ip each bit of xindependently with probability of1\\nn;\\n6: if(\\/\\\\e}atio\\\\slash\\u2203w\\u2208S:w\\u227by)then\\n7:S\\u2190(S\\u222a{y})\\\\{z\\u2208S|y\\/{ollowsequalz};\\n8: end if\\n9:end while\\n\\u03b1(i,j)=\\/braceleftbigg\\n1 i= 0orj= 0\\n\\u03b1Cheb(i,j)or\\u03b1Hoef(i,j)i= 1,...,n(12)\\nFinally, using the con\\ufb01dence level threshold values, we can get the con\\ufb01dence level range for solution k as given\\nin Equation 13. If there exists a valid \\u03b1interval for a particular solution, then for the \\u03b1values in that interval, that\\nsolution will give the best pro\\ufb01t value. If the \\u03b1interval is empty for a particular solution, then that solut ion won\\u2019t be\\nthe best solution for any \\u03b1value.\\nk\\u22121\\nmin\\nj=0\\u03b1i,k\\u2264\\u03b1k\\u2264n\\u22121max\\nj=k\\u03b1i,j (13)\\n4 Algorithms\\nIn this study, we consider two widely used multi-objective E As: GSEMO and NSGA-II. GSEMO is the most basic EA\\nthat addresses multi-objective optimization problems. It has been proven to be effective in solving chance constraine d\\nmulti-objective optimization problems in many studies Neu mann et al. (2022). The steps of GSEMO are given in\\nAlgorithm 1.\\nThe population S initially contains a solution that is gener ated randomly. Over the iterations, a parent solution xis\\nselected uniformly at random from S and an offspring solutio nyis generated by \\ufb02ipping each bit of xwith a probability\\nof1\\nn. Ifyis not dominated by any of the existing solutions in S, it is ad ded to S replacing all the existing solutions in S\\nthat are dominated by y. This guarantees at the end of any iter ation, population S will contain a set of non-dominating\\nsolutions that are equally ef\\ufb01cient with respect to the give n objective functions.\\nNSGA-II is the most prominent multi-objective EA that focus es on diversity by \\ufb01nding near optimal solutions\\nDeb et al. (2002). If we consider one iteration of the evoluti onary process of NSGA-II, it creates an offspring pop-\\nulation from the parent population. First, we use the binary tournament for the selection of two parent solutions and\\napply single point crossover to generate two offspring solu tions which are mutated by \\ufb02ipping each bit with a proba-\\nbility of1\\nn. When the offspring population is full, all the solutions fr om parent and offspring solutions are considered\\ntogether and divided into non-dominating solution fronts ( i.e., the solutions in a front do not dominate other solution s\\nin it). Starting with the front with the lowest rank, these ar e added to the new population until it reaches capacity. If\\nonly a part of a front is to be taken into the new population, cr owding distance-based sorting is used to decide the\\nselection. This improves the better spread (diversity) amo ng the solutions. In one iteration of this algorithm, a new\\n\\ufb01tness evaluation equal to the size of the offspring populat ion is considered. The number of iterations of the algorithm s\\ndepends on the offspring population size and the maximum num ber of evaluations as considered in the experimental\\nsetup.\\n5 Filtering for Problems with Chance Constraints\\nHere we propose a \\ufb01ltering method for GSEMO to improve its out come by removing solutions that do not have a valid\\n\\u03b1interval from the interim populations. The \\ufb01nal population of GSEMO contains solutions that do not give the best\\npro\\ufb01t value for any probability value for \\u03b1. Such solutions do not add value to the optimization goal of \\ufb01 nding the\\nbest solutions with given con\\ufb01dence levels. For instance, F igure 1 presents a \\ufb01nal population from GSEMO using 10\\nmillion \\ufb01tness evaluation of g(x)on uncorr-100: an instance used in the experiments. The plot shows\\u00b5(x)versus\\nv(x)for all solutions in the \\ufb01nal population. Blue star markers i ndicate that the solution has a valid con\\ufb01dence level\\ninterval. These solutions\\u2019 \\u03b1intervals compose the complete probability range [0.0,1.0 ]. At the end of the optimization interval. These solutions\\u2019 \\u03b1intervals compose the complete probability range [0.0,1.0 ]. At the end of the optimization\\n7 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\n0.0 2.0 4.0 6.0 8.0\\n\\u00b5(x)[in 1000\\u2019s]0.0 1.0 2.0 3.0v(x) [in 1000\\u2019s]\\u03b11\\n\\u03b12\\n\\u03b13\\n\\u03b14\\n\\u03b18\\n\\u03b110\\n\\u03b112\\n\\u03b118\\n\\u03b119\\nFigure 1: A sample of a \\ufb01nal population from GSEMO\\nAlgorithm 2 Filtering Method\\n0:Input: Population P0=x1,...,xnordered by the decreasing order of \\u00b5(xi);\\n1:setk\\u21901;\\n2:whilek\\u2264ndo\\n3: setupper\\u2190mink\\u22121\\nj=0\\u03b1i,k\\n4: setlower\\u2190maxn\\u22121\\nj=k\\u03b1i,j\\n5: ifupper\\u2265lower then\\n6:P1\\u222a{xk}\\n7: end if\\n8: setk\\u2190k+1\\n9:end while\\n10:returnP1\\nprocess, interest is in solutions with these blue markers. I t is one of these solutions that become the best solution for\\na given\\u03b1value. On the contrary, other solutions marked in black do no t become the best solution for any con\\ufb01dence\\nlevel.\\nThe \\ufb01ltering method removes these solutions that do not beco me the best solution for any con\\ufb01dence level. It is\\napplied to the interim populations of GSEMO regularly after a certain amount of \\ufb01tness evaluations. This process\\nremoves the solutions from interim populations, consideri ng the\\u03b1intervals they represent according to Equation 13.\\nAs the \\ufb01ltering method keeps only the solutions with valid \\u03b1interval, it increases the change of the evolutionary\\noptimization to improve on these solutions. Therefore, thi s method helps to improve the quality of the solutions in the\\n\\ufb01nal population.\\nThe steps of the \\ufb01ltering method are given in Algorithm 2. It t akes the population P0as the input, which can be either\\nthe \\ufb01nal population or an interim population created during the execution of GSEMO. Population P0needs solutions\\nin the decreasing order of the \\u00b5(x). For each solution xk, we consider \\u03b1i,kand select the interval for con\\ufb01dence level,\\nusing the inequality given in Equation 13. The solution xkis added to the resulting population P1iff the interval for\\n\\u03b1kis non-empty.\\n6 Experiments\\nIn this work, we evaluate the \\ufb01tness functions introduced pr eviously in different chance constraint settings using mul ti-\\nobjective EAs on different benchmarks; and we use multiple e xperiments for that. Here we present the experimental\\nsettings and the results of these experiments.\\n8 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINTTable 1: Results for same dispersion using Chebyshev\\u2019s ineq uality\\nB\\u03b1\\u03b4GSEMO with g(x) (1) GSEMO+Filtering with g(x) (2) NSGA-II with g(x) (3) NSGA-II with g\\u2032(x) (4) NSGA-II with g\\u2032\\u2032(x) (5)\\nmean std stat mean std stat mean std stat mean std stat mean std statuncorr-100\\n24070.12511029.6826 76.3191 2\\u22125+11085.6072 0.00001+3+4+5+11007.6251 88.8419 2\\u221210998.7574 50.6310 2\\u221211007.7624 52.5740 1\\u22122\\u2212\\n5010862.4750 61.4080 2\\u22125+10907.0000 0.00001+3+4+5+10837.2841 53.1386 2\\u221210811.4832 80.3604 2\\u221210832.9581 50.1309 1\\u22122\\u2212\\n0.012510620.5264 71.0446 2\\u22125+10672.3379 0.00001+3+4+5+10602.1969 81.4896 2\\u221210594.9482 45.5447 2\\u221210602.8065 46.9209 1\\u22122\\u2212\\n5010044.4941 49.7668 2\\u22125+10079.9649 0.00001+3+4+5+10025.8881 41.6418 2\\u221210004.2550 67.4364 2\\u221210023.0463 38.8802 1\\u22122\\u2212\\n0.00125 9345.9245 55.2560 2\\u22125+9384.5150 0.00001+3+4+5+9338.8078 59.0630 2\\u22125+9336.6045 30.1889 2\\u22129340.8894 29.5604 1\\u22122\\u22123\\u2212\\n50 7495.5153 17.9630 2\\u22125+7502.7716 0.00001+3+4+5+7498.7722 13.1282 2\\u22125+7488.8368 31.6085 2\\u22127499.2121 9.5267 1\\u22122\\u22123\\u2212strong-100\\n41870.125 8507.0584 130.1420 2\\u22128606.3413 87.2952 1+3+4+5+8525.5561 125.6131 2\\u22128500.0805 149.4958 2\\u22128499.3632 124.5506 2\\u2212\\n50 8368.0306 94.0118 2\\u22128422.6322 67.2273 1+3+5+8326.9385 114.3897 2\\u22128364.0549 124.7364 8319.3263 115.3753 2\\u2212\\n0.0125 8083.9678 106.1983 2\\u22128170.3360 69.5792 1+4+5+8102.4388 104.2026 8082.3479 124.2002 2\\u22128082.2853 103.3765 2\\u2212\\n50 7502.9361 59.8544 2\\u22127549.4937 41.2749 1+3+5+7489.7223 73.5765 2\\u22127513.5174 78.6964 7485.1707 73.8490 2\\u2212\\n0.00125 6770.3193 41.1347 2\\u22126814.1197 20.7669 1+4+5+6787.5739 42.8885 6782.2797 48.8989 2\\u22126784.1694 42.0052 2\\u2212\\n50 4957.5449 36.7770 2+3\\u22124\\u22124894.0039 60.2107 1\\u22123\\u22124\\u22125\\u22124990.7637 17.9608 1+2+5+4989.4145 13.2248 1+2+5+4991.4701 11.0454 2+3\\u22124\\u2212uncorr-300\\n68530.12533935.4067 205.6247 2\\u221234286.1802 147.5309 1+3+4+5+33681.6883 534.3217 2\\u221233671.7620 555.4450 2\\u221233615.8492 489.0223 2\\u2212\\n5033571.9980 260.8593 2\\u221233967.3813 159.2433 1+3+4+5+33418.4882 512.5693 2\\u221233284.5989 450.6255 2\\u221233319.4651 483.5459 2\\u2212\\n0.012533237.8865 200.4641 2\\u221233577.9421 141.7536 1+3+4+5+32992.6756 521.4969 2\\u221232984.2543 541.8390 2\\u221232929.2384 476.3437 2\\u2212\\n5032180.0106 245.8773 2\\u221232551.5342 144.8963 1+3+4+5+32039.6829 487.7705 2\\u221231913.5405 429.4896 2\\u221231946.2435 458.2449 2\\u2212\\n0.0012531066.6084 186.3571 2\\u221231372.5619 122.8614 1+3+4+5+30846.1243 482.1829 2\\u221230841.9917 499.6822 2\\u221230789.6329 437.0386 2\\u2212\\n5027843.2948 203.7335 2\\u221228141.7188 105.9385 1+3+4+5+27745.1612 411.5637 2\\u221227641.0702 364.7302 2\\u221227668.0494 380.2576 2\\u2212strong-300\\n138210.12523809.6581 433.2506 2\\u22123\\u22125\\u221224369.6211 216.9574 1+3+4+24099.5570 327.0870 1+2\\u221223986.4018 344.2409 2\\u221224176.0891 232.7994 1+\\n5023594.2993 335.6481 2\\u22123\\u22125\\u221224135.2769 220.4491 1+3+4+5+23867.9086 293.6342 1+2\\u221223695.2127 304.2994 2\\u221223899.3116 223.8869 1+2\\u2212\\n0.012523176.9548 406.2664 2\\u22123\\u22125\\u221223703.0401 197.3436 1+3+4+23464.5667 299.1984 1+2\\u221223360.1013 318.6357 2\\u221223534.8995 212.1891 1+\\n5022322.7651 282.0126 2\\u22123\\u22125\\u221222797.4912 177.4246 1+3+4+5+22588.5433 240.6771 1+2\\u221222446.1950 255.5942 2\\u221222616.9323 182.9564 1+2\\u2212\\n0.0012521208.9163 322.7908 2\\u22123\\u22125\\u221221626.7053 138.6421 1+4+21486.9475 214.2326 1+21411.6675 240.6127 2\\u221221536.8345 149.2069 1+\\n5018388.5805 159.4458 2\\u22123\\u22124\\u22125\\u221218647.0894 70.1852 1+4+18623.6627 98.1567 1+18569.9473 110.1685 1+2\\u221218638.7270 63.3870 1+uncorr-500\\n112430.12557076.8361 748.9305 2\\u22123+4+58431.4168 311.5788 1+3+4+5+55850.4588 1249.6235 1\\u22122\\u221255869.0104 1408.3737 1\\u22122\\u221256037.4884 1287.2936 2\\u2212\\n5056690.8982 859.2445 2\\u22123+4+5+58120.8249 314.3063 1+3+4+5+55563.0878 1044.9051 1\\u22122\\u221254981.1206 1223.7431 1\\u22122\\u221255667.0434 1278.2837 1\\u22122\\u2212\\n0.012556197.0249 738.3354 2\\u22123+4+57528.4355 304.4306 1+3+4+5+54995.4516 1230.9769 1\\u22122\\u221255013.2733 1387.3025 1\\u22122\\u221255179.3063 1266.4267 2\\u2212\\n5054931.1821 829.5866 2\\u22123+4+5+56312.8274 298.8610 1+3+4+5+53851.9927 1009.2563 1\\u22122\\u221253287.7928 1187.2402 1\\u22122\\u221253950.6793 1236.5882 1\\u22122\\u2212\\n0.0012553456.0312 705.5039 2\\u22123+4+54715.2628 282.5314 1+3+4+5+52331.0919 1172.9610 1\\u22122\\u221252346.6392 1321.7096 1\\u22122\\u221252505.0533 1201.5375 2\\u2212 0.0012553456.0312 705.5039 2\\u22123+4+54715.2628 282.5314 1+3+4+5+52331.0919 1172.9610 1\\u22122\\u221252346.6392 1321.7096 1\\u22122\\u221252505.0533 1201.5375 2\\u2212\\n5049451.2762 737.5644 2\\u22123+4+50683.8705 252.3483 1+3+4+5+48521.4630 900.6466 1\\u22122\\u221248012.1505 1072.2455 1\\u22122\\u221248602.1733 1107.2660 2\\u2212strong-500\\n222230.12538822.1695 692.1198 2\\u22123\\u22124\\u22125\\u221240391.0362 449.8195 1+3+4+5+39792.1607 415.5621 1+2\\u221239754.0904 424.9780 1+2\\u221239769.7011 392.0549 1+2\\u2212\\n5038444.0651 620.4975 2\\u22123\\u22124\\u22125\\u221240078.0983 348.7030 1+3+4+5+39442.9758 605.5909 1+2\\u221239485.7055 483.2892 1+2\\u221239416.6356 382.3801 1+2\\u2212\\n0.012538026.4154 657.3621 2\\u22123\\u22124\\u22125\\u221239525.2027 425.9579 1+3+4+5+38973.8435 390.8087 1+2\\u221238936.9771 399.8035 1+2\\u221238951.8432 369.6260 1+2\\u2212\\n5036864.1232 555.3952 2\\u22123\\u22124\\u22125\\u221238332.2087 312.3534 1+3+4+5+37800.5870 535.4997 1+2\\u221237839.6026 424.3667 1+2\\u221237781.3920 337.2743 1+2\\u2212\\n0.0012535546.6995 551.6446 2\\u22123\\u22124\\u22125\\u221236827.5418 352.7810 1+3+4+5+36424.1740 315.3111 1+2\\u221236391.1536 322.1788 1+2\\u221236404.2919 299.3640 1+2\\u2212\\n5031947.2385 360.7027 2\\u22123\\u22124\\u22125\\u221232899.2694 206.7434 1+32685.3625 321.9032 1+32727.4059 242.3192 1+32688.8779 201.6241 1+\\n9 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\n6.1 Experimental Setup\\nFor experimental investigations, we use the same benchmark s as the ones that are used in Neumann et al. (2022).\\nThe set of benchmarks includes three correlated instances a nd three bounded strongly correlated instances with the\\nnumbers of knapsack items n\\u2208{100,300,500}. We consider that the pro\\ufb01ts of the knapsack items have a unif orm\\ndistribution, such that the pro\\ufb01t of element iis chosen uniformly at random as pi\\u2208{\\u00b5i\\u2212\\u03b4,\\u00b5i+\\u03b4}. This allows to\\nuse both\\u02c6pCheb and\\u02c6pHoef when the pro\\ufb01ts have the same uncertainty level ( \\u03b4). The experimental investigation covers\\ntwo uncertainty levels for each benchmark, \\u03b4\\u2208{25,50}. Additionally, we consider the scenario where the pro\\ufb01ts\\nhave different dispersion such that each item ihas an uncertainty level \\u03b4i, which is chosen uniformly at random as\\n\\u03b4i\\u2208[0.0,\\u00b5i]. The benchmarks with different uncertainties are consider ed only with \\u02c6pCheb since\\u02c6pHoef requires the\\nsame uncertainty level for all elements.\\nThis study mainly considers three algorithms: two well-kno wn multi-objective evolutionary algorithms GSEMO and\\nNSGA-II and the third is GSEMO with the \\ufb01ltering method intro duced in Section 4. The latter is referred to as\\nGSEMO+Filtering hereafter in this paper. These algorithms are combined with \\ufb01tness functions as appropriate, to\\nuse in the experiments. For the benchmarks with \\ufb01xed uncerta inties, using GSEMO or GSEMO+Filtering with any\\n\\ufb01tness function ( g(x)org\\u2032(x)org\\u2032\\u2032(x)) will produce equivalent results as the \\ufb01nal populations. T herefore, GSEMO\\nand GSEMO+Filtering are only considered with the \\ufb01tness eva luationg(x). The \\ufb01tness function g\\u2032\\u2032(x)considers the\\nnumber of items selected in the solution for the scenario whe re pro\\ufb01ts have the same dispersion. Therefore, we do\\nnot consider algorithms with that \\ufb01tness function for the be nchmarks with different uncertainties for pro\\ufb01ts. Only\\nGSEMO and GSEMO+Filtering with g(x), and NSGA-II with g(x)andg\\u2032(x)are considered for those benchmarks.\\nEvery algorithm considers the 10 million \\ufb01tness evaluation and produces a population of solutions that gives a trade-\\noff concerning the objectives used for \\ufb01tness evaluation. T he quality of the output of these methods is evaluated\\nby analyzing the best pro\\ufb01t value for different con\\ufb01dence le vels considering \\u03b1\\u2208{0.1,0.01,0.001}. Each method\\ngenerates a \\ufb01nal population independent of \\u03b1, and we select the best solution from that population for dif ferent\\u03b1\\nvalues using pro\\ufb01t estimations \\u02c6pCheb or\\u02c6pHoef as applicable. The results summarise the best pro\\ufb01t value gi ven by 30\\nexperimental results for each \\u03b1. This summary requires running the method 30 times for each \\u03b4for benchmarks with\\nthe same pro\\ufb01t dispersion and 30 times for benchmarks with di fferent pro\\ufb01t dispersion. However, when using g\\u2032\\u2032(x),\\nas algorithms run independent of \\u03b4value, it is possible to get the best pro\\ufb01t values for differe nt\\u03b4from the same \\ufb01nal\\npopulation.\\nFinally, we test for the statistical signi\\ufb01cance validity o f the results using the Kruskal-Wallis test with 95% con\\ufb01den ce\\nwith the Bonferroni post-hoc statistical procedure. The st atistical comparison is indicated as X+orX\\u2212to indicate that\\nthe method in the column outperforms Xor vice versa. If there is no signi\\ufb01cant difference between t he two methods,\\nrespective numbers do not appear. For each method, the summa ry of the best pro\\ufb01t values is given as mean, std and\\nstat, which represent the mean and standard deviation of the results and statistical comparison with the corresponding\\nresults from other methods, respectively.\\n6.2 Results\\nTable 1 and 2 present the results for the benchmarks with \\ufb01xed uncertainty levels pro\\ufb01ts of elements. According to the\\nmean values GSEMO with the \\ufb01ltering method outperforms othe r methods in most of the settings in both tables. The mean values GSEMO with the \\ufb01ltering method outperforms othe r methods in most of the settings in both tables. The\\nstatistical comparisons give more insights into the perfor mance of the methods when applied to each instance. Results\\nfor uncorr-100 in Table 1 show that GSEMO+Filtering perform s the better than other four methods, and GSEMO\\nperforms better than NSGA-II with g\\u2032\\u2032(x). For smaller con\\ufb01dence levels \\u03b1= 0.001, NSGA-II with g(x)outperforms\\nthat withg\\u2032\\u2032(x). Strong-100 instance gets very different results for \\u03b1= 0.001and\\u03b4= 50 , than other cases of the same\\ninstance. There, GSEMO with g(x)and NSGA-II with g(x)andg\\u2032(x)perform well and GSEMO+Filtering gives the\\nlowest result. However, the second method performs well in o ther\\u03b1and\\u03b4settings and NSGA-II with g(x)andg\\u2032(x)\\nalso perform similarly in certain settings.\\nFor all settings of uncorr-300, GSEMO+Filtering outperfor ms the other four methods while performing similarly to\\neach other. For strong-300 instance, GSEMO gives lower resu lts than GSEMO+Filtering and NSGA-II with g(x)\\nandg\\u2032\\u2032(x). Also, NSGA-II with g(x)produces results as good as GSEMO+Filtering for \\u03b1= 0.001. For uncorr-\\n500, GSEMO+Filtering gives the best results and GSEMO outpe rforms most of the NSGA-II results. However,\\nwhen using g\\u2032\\u2032(x)with NSGA-II on this instance, results show equal performan ce except for \\u03b1={0.1,0.01}con-\\n\\ufb01dence levels when considering uncertainty level \\u03b4= 50 . Experiments on strong-500 also get the best results from\\nGSEMO+Filtering. However, GSEMO is outperformed by other m ethods for all settings considered for this instance.\\nIn comparison, the NSGA-II gives the second best results acr oss settings when \\u03b4= 50 and\\u03b1= 0.001NSGA-II\\nmethods also perform as well as GSEMO+Filtering.\\n10 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nTable 2: Results for same dispersion using Hoeffding bound\\nB\\u03b1\\u03b4GSEMO with g(x) (1) GSEMO+Filtering with g(x) (2) NSGA-II with g(x) (3) NSGA-II with g\\u2032(x) (4) NSGA-II with g\\u2032\\u2032(x) (5)\\nmean std stat mean std stat mean std stat mean std stat mean std statuncorr-100\\n24070.12510987.3004 75.7683 2\\u221211042.7989 0.00001+3+4+5+10965.6291 88.0780 2\\u221210956.9290 50.1020 2\\u221210965.9887 51.9898 2\\u2212\\n5010778.0078 60.1974 2\\u221210821.5978 0.00001+3+4+5+10753.4968 51.9386 2\\u221210728.1263 79.0134 2\\u221210749.4108 48.9647 2\\u2212\\n0.012510896.5878 74.5928 2\\u221210951.1744 0.00001+3+4+5+10875.7430 86.4446 2\\u221210867.4019 48.9713 2\\u221210876.2792 50.7360 2\\u2212\\n5010596.7649 57.6057 2\\u221210638.3488 0.00001+3+4+5+10573.7130 49.3723 2\\u221210549.2659 76.1316 2\\u221210569.9917 46.4640 2\\u2212\\n0.0012510826.9815 73.6938 2\\u221210880.8684 0.00001+3+4+5+10806.7709 85.1930 2\\u221210798.7053 48.1051 2\\u221210807.4426 49.7746 2\\u2212\\n5010457.6924 55.6226 2\\u221210497.7369 0.00001+3+4+5+10435.7601 47.4116 2\\u221210412.0216 73.9290 2\\u221210432.3187 44.5489 2\\u2212strong-100\\n41870.125 8463.1467 127.6172 2\\u22128561.1573 85.4504 1+3+4+5+8481.7278 123.3848 2\\u22128456.8099 146.8670 2\\u22128456.3390 122.3538 2\\u2212\\n50 8278.6327 90.2553 2\\u22128332.4692 64.4783 1+3+5+8240.4850 110.0787 2\\u22128276.2257 119.9424 8233.2780 111.0076 2\\u2212\\n0.0125 8369.3409 122.2698 2\\u22128464.4621 81.5096 1+3+4+5+8387.9199 118.6225 2\\u22128364.1958 141.2465 2\\u22128363.9441 117.6448 2\\u2212\\n50 8086.8101 82.3211 2\\u22128139.0049 58.6173 1+3+5+8054.9801 100.8829 2\\u22128087.7692 109.6779 8048.4883 101.6750 2\\u2212\\n0.00125 8297.3868 118.1961 2\\u22128390.2653 78.4972 1+3+4+5+8315.9386 114.9756 2\\u22128293.1306 136.9396 2\\u22128293.0470 114.0400 2\\u2212\\n50 7939.6195 76.3766 2\\u22127990.5545 54.1630 1+3+5+7912.6372 93.8879 2\\u22127943.1615 101.8268 7906.6941 94.5672 2\\u2212uncorr-300\\n68530.12533863.1544 205.0835 2\\u221234212.8177 146.9244 1+3+4+5+33610.2955 532.9802 2\\u221233600.5469 554.0346 2\\u221233545.0211 487.7131 2\\u2212\\n5033428.2570 259.2905 2\\u221233821.1723 157.7349 1+3+4+5+33276.1084 510.0035 2\\u221233143.0191 448.4365 2\\u221233177.8089 480.9303 2\\u2212\\n0.012533708.5096 203.9304 2\\u221234055.7967 145.6324 1+3+4+5+33457.5385 530.1364 2\\u221233448.1219 551.0166 2\\u221233392.9168 484.9026 2\\u2212\\n5033119.8295 255.9406 2\\u221233507.4493 154.5170 1+3+4+5+32970.6017 504.5017 2\\u221232839.2289 443.7443 2\\u221232873.6003 475.3176 2\\u2212\\n0.0012533589.8465 203.0502 2\\u221233935.3102 144.6467 1+3+4+5+33340.3278 527.9571 2\\u221233331.1621 548.7016 2\\u221233276.2031 482.7469 2\\u2212\\n5032883.1649 253.3855 2\\u221233266.7212 152.0660 1+3+4+5+32736.1783 500.2836 2\\u221232606.1226 440.1485 2\\u221232640.1729 471.0147 2\\u2212strong-300\\n138210.12523744.0892 430.4747 2\\u22123\\u22125\\u221224300.5479 214.9589 1+3+4+24033.7421 324.1660 1+2\\u221223921.4833 341.5510 2\\u221224109.9465 230.6676 1+\\n5023462.9510 329.9513 2\\u22123\\u22125\\u221223997.1125 215.9394 1+3+4+5+23735.7246 288.1238 1+2\\u221223566.2352 299.2535 2\\u221223767.0263 219.6354 1+2\\u2212\\n0.012523603.7492 424.5461 2\\u22123\\u22125\\u221224152.7138 210.6726 1+3+4+23892.9269 317.9729 1+2\\u221223782.5354 335.8005 2\\u221223967.9043 226.0936 1+\\n5023181.2241 317.9412 2\\u22123\\u22125\\u221223700.6927 206.3054 1+3+4+5+23452.1877 276.3622 1+2\\u221223289.4860 288.4378 2\\u221223482.9419 210.5250 1+2\\u2212\\n0.0012523496.0973 420.0184 2\\u22123\\u22125\\u221224039.3181 207.2892 1+3+4+23784.9190 313.2317 1+2\\u221223675.9269 331.3992 2\\u221223858.9115 222.5879 1+\\n5022965.0474 308.7926 2\\u22123\\u22125\\u221223473.2418 198.9571 1+3+4+5+23234.6394 267.3758 1+2\\u221223077.1290 280.1499 2\\u221223264.9563 203.5545 1+2\\u2212uncorr-500\\n112430.12556985.6975 747.8288 2\\u22123+4+58337.8820 310.8352 1+3+4+5+55761.8933 1247.6913 1\\u22122\\u221255780.3693 1406.1906 1\\u22122\\u221255948.9616 1285.1401 2\\u2212\\n5056509.1689 856.1817 2\\u22123+4+5+57934.0703 312.7435 1+3+4+5+55386.3718 1041.2515 1\\u22122\\u221254806.2623 1219.9708 1\\u22122\\u221255489.9899 1273.9787 1\\u22122\\u2212\\n0.012556790.6294 745.4714 2\\u22123+4+58137.6851 309.2459 1+3+4+5+55572.3326 1243.5564 1\\u22122\\u221255590.6468 1401.5183 1\\u22122\\u221255758.8493 1280.5162 2\\u2212\\n5056119.2292 849.6132 2\\u22123+4+5+57533.3999 309.3150 1+3+4+5+55007.2442 1033.3396 1\\u22122\\u221254431.0658 1211.8786 1\\u22122\\u221255109.7653 1264.7366 1\\u22122\\u2212\\n0.0012556640.9485 743.6632 2\\u22123+4+57984.0686 308.0287 1+3+4+5+55426.8776 1240.3841 1\\u22122\\u221255445.0676 1397.9335 1\\u22122\\u221255612.9710 1276.9688 2\\u2212 0.0012556640.9485 743.6632 2\\u22123+4+57984.0686 308.0287 1+3+4+5+55426.8776 1240.3841 1\\u22122\\u221255445.0676 1397.9335 1\\u22122\\u221255612.9710 1276.9688 2\\u2212\\n5055820.0179 844.5763 2\\u22123+4+5+57226.0199 306.6676 1+3+4+5+54716.3294 1027.2713 1\\u22122\\u221254143.1674 1205.6714 1\\u22122\\u221254818.0086 1257.6477 1\\u22122\\u2212strong-500\\n222230.12538739.7417 688.5044 2\\u22123\\u22124\\u22125\\u221240301.3374 447.3446 1+3+4+5+39707.3872 412.9829 1+2\\u221239669.4439 422.3664 1+2\\u221239685.3277 389.7418 1+2\\u2212\\n5038280.9153 613.7020 2\\u22123\\u22124\\u22125\\u221239897.8123 344.8981 1+3+4+5+39273.3775 598.3347 1+2\\u221239315.7031 477.1865 1+2\\u221239247.8886 377.7601 1+2\\u2212\\n0.012538563.3178 680.7776 2\\u22123\\u22124\\u22125\\u221240109.3511 442.0526 1+3+4+5+39525.9425 407.4683 1+2\\u221239488.2712 416.7806 1+2\\u221239504.1345 384.7767 1+2\\u2212\\n5037930.8422 599.1733 2\\u22123\\u22124\\u22125\\u221239510.9695 336.7709 1+3+4+5+38909.4678 582.7780 1+2\\u221238950.9755 464.0994 1+2\\u221238885.5815 367.8049 1+2\\u2212\\n0.0012538427.9430 674.8592 2\\u22123\\u22124\\u22125\\u221239962.0501 437.9920 1+3+4+5+39386.7151 403.2421 1+2\\u221239349.2525 412.4983 1+2\\u221239365.1001 380.9688 1+2\\u2212\\n5037662.2216 588.0761 2\\u22123\\u22124\\u22125\\u221239214.1346 330.5707 1+3+4+5+38630.2301 570.8539 1+2\\u221238671.1102 454.0722 1+2\\u221238607.6080 360.1434 1+2\\u2212\\nTable 3: Results for different dispersion using Chebyshev\\u2019 s inequality\\nInstance B\\u03b1GSEMO with g(x) (1) GSEMO+Filtering with g(x) (2) NSGA-II with g(x) (3) NSGA-II with g\\u2032(x) (4)\\nmean std stat mean std stat mean std stat mean std stat\\nuncorr-100 24070.1 8546.7015 322.7956 8562.9620 323.1475 8565.8942 324.9223 8564.2549 323.8494\\n0.01 4682.2710 799.1875 4675.9295 795.5029 4674.8735 797.5334 4677.1350 799.3174\\n0.001 1967.4965 880.0104 1962.3962 886.0726 1919.2445 917.6798 1953.6975 894.4411\\nstrong-100 41870.1 7100.1631 245.6990 7099.8604 250.2445 7122.7698 249.2182 7123.3386 250.5080\\n0.01 5315.4258 326.6578 5308.5248 343.0757 5329.4856 326.9290 5333.3263 325.5847\\n0.001 3746.9429 732.1072 3746.1213 731.9397 3734.2093 731.9080 3743.5859 731.6689\\nuncorr-300 68530.1 29089.9230 479.0749 2\\u22123\\u22124\\u221229580.8035 355.6388 1+29735.0392 358.7213 1+29676.9763 388.6763 1+\\n0.01 19396.8836 1128.3967 19431.8036 1126.0478 19589.0418 1131.0819 19580.3092 1131.0342\\n0.001 8910.0087 1370.4531 8850.8254 1381.8863 8899.5025 1378.4832 8971.3050 1383.6696\\nstrong-300 138210.1 21789.9295 335.7672 2\\u22123\\u22124\\u221222171.9138 358.9991 1+22345.3397 309.3574 1+22297.4300 307.0689 1+\\n0.01 18172.8378 560.3248 18195.2974 615.1645 18338.0359 588.6787 18342.4421 576.7652\\n0.001 14629.7944 809.3377 14617.5558 794.6289 14643.0814 808.6424 14667.4349 812.9751\\nuncorr-500 112430.1 50266.4398 709.0211 2\\u22123\\u22124\\u221252494.0984 556.8082 1+52468.0194 532.9634 1+52149.3408 700.4027 1+\\n0.01 37753.4240 1566.1944 38510.4882 1564.4777 38746.1887 1539.8167 38686.7230 1555.8618\\n0.001 18969.0800 2144.1783 18880.7433 2144.3506 19153.0886 2137.5579 19190.4696 2134.7447\\nstrong-500 222230.1 35919.5415 631.8822 2\\u22123\\u22124\\u221237833.8138 352.1352 1+37832.1320 332.8651 1+37690.0363 317.8082 1+\\n0.01 30977.9111 679.1163 2\\u22123\\u22124\\u221231554.3119 682.8664 1+31822.0362 649.7576 1+31805.6899 637.4521 1+\\n0.001 25041.2112 721.5121 25018.0126 704.1720 25131.2311 723.9024 25193.7178 741.8816\\nTable 2 gives the results from the experiments that use \\u02c6pCheb to estimate the pro\\ufb01t values of solutions. uncorr-100,\\nstrong-100, and uncorr-300 results are highest when using G SEMO+Filtering. It outperforms all the other methods\\nexcept NSGA-II with g\\u2032(x)on strong-100 instance for uncertainty level \\u03b4= 50 . Experiments on strong-300 instance\\nshow that NSGA-II with g\\u2032\\u2032(x)performs equally as the GSEMO+Filtering when considering a lower uncertainty level\\n(25) for the knapsack items. GSEMO and NSGA-II with g\\u2032(x)methods give the lowest results for this instance. On the\\ncontrary, GSEMO performs better than most NSGA-II methods f or uncorr-500. When using g\\u2032\\u2032(x), NSGA-II performs\\nequally as GSEMO for lower uncertainty level value 25. For al l\\u03b1and\\u03b4values, NSGA-II methods are outperformed\\nby GSEMO+Filtering in the experiments on strong-500 in Tabl e 2. However, NSGA-II methods perform better than\\nGSEMO on that benchmark.\\n11 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nGSEMO+Filtering performs signi\\ufb01cantly well when applied t o benchmarks with the same dispersion for pro\\ufb01ts. The\\n\\ufb01ltering method allows the interim populations to contain m ore solutions that yield a valid con\\ufb01dence level interval.\\nTherefore, improving upon these solutions eventually give s better results in the \\ufb01nal outcome. Considering NSGA-II\\nresults,g(x)andg\\u2032(x)tends to produce better results than g\\u2032\\u2032(x). This can be due to the fact the crowding distance\\nassignment is better when considering the variance or the st andard deviation of solutions\\u2019 pro\\ufb01ts.\\nWe can compare the results for benchmarks with the same dispe rsion for pro\\ufb01ts (given in Table 1 and 2) with the\\nprevious work in Neumann et al. (2022) as it also considers th e same benchmarks and similar experimental setup. The\\nexperimental settings are the same in both works except for t he number of \\ufb01tness evaluations the algorithms consider.\\nFor each \\u03b1and\\u03b4value, methods in Neumann et al. (2022) run for one million \\ufb01t ness evaluations. On the contrary,\\nwe run the multi-objective methods on benchmarks for each \\u03b4value for 10 million \\ufb01tness evaluations which yield\\nresults for all \\u03b1values from the same algorithm output. The results show that for Chebyshev results 14 out of the\\n36 settings, the highest mean pro\\ufb01t values given in Table 1 ou tperform all the three methods used in Neumann et al.\\n(2022) and for Hoeffding results in 15 out of the 36 settings g ets better pro\\ufb01ts according to Table 2. Generally, most\\nof these cases are from experiments on uncorrelated benchma rks. For the cases where the new methods perform better\\non bounded strongly correlated instances, it is for higher u ncertainty level \\u03b4= 50 and lower con\\ufb01dence values like\\n\\u03b1={0.01,0.01}.\\nTable 3 presents results for benchmarks with different disp ersion for pro\\ufb01ts of elements. The highest mean values\\nreported for each case show, for smaller instances, GSEMO an d NSGA-II give the highest mean pro\\ufb01ts, and for in-\\nstances with 300 or 500 elements, GSEMO+Filtering with g\\u2032(x)and NSGA-II give the highest mean pro\\ufb01t. Although\\nthe highest mean values can be identi\\ufb01ed from different meth ods, the results are similar in most of the settings. Based\\non the statistical comparisons, we can see that smaller inst ances: strong-100 and uncorr-100, do not signi\\ufb01cantly diff er\\nbetween each method\\u2019s results. Results from other instance s show that for \\u03b1= 0.1GSEMO+Filtering and NSGA-II\\nmethods outperform the GSEMO. In addition, strong-500 inst ance shows better results from GSEMO+Filtering with\\ng\\u2032(x)and NSGA-II with both \\ufb01tness functions for \\u03b1= 0.01. However, in other settings with \\u03b1={0.01,0.001}, all\\nmethods show similar results for the instances with 300 and 5 00 items.\\nCompared to the benchmarks with the same dispersion for pro\\ufb01 ts, the ones with different dispersion can give higher\\nvariance for the pro\\ufb01t of some elements. Therefore, it is cru cial to have a good spread of the solutions in the \\ufb01nal\\npopulation as more solutions tend to give negative pro\\ufb01t val ues when considering certain con\\ufb01dence levels. NSGA-\\nII using crowding distance based sorting appears to achieve this when use with selected objectives in the \\ufb01tness\\nevaluations. In comparison, GSEMO+Filtering is also able t o achieve similarly good results by ignoring the solutions\\nwithout a valid \\u03b1interval and eventually improving the \\ufb01nal population.\\n7 Conclusion\\nThis paper explores multi-objective evolutionary approac hes to solve the pro\\ufb01t chance constrained KP. We introduce\\n\\ufb01tness evaluations for EAs to cater for this problem. These \\ufb01 tness functions can evaluate the solutions irrespective\\nof the required con\\ufb01dence level of the solutions. Therefore , the outcome of EAs gives us a population that includes\\nsolutions giving the best pro\\ufb01t value for different con\\ufb01den ce levels. So it is unnecessary to decide on the required solutions giving the best pro\\ufb01t value for different con\\ufb01den ce levels. So it is unnecessary to decide on the required\\ncon\\ufb01dence level before executing the algorithms, and no nee d to have multiple executions to investigate solutions\\nfor different con\\ufb01dence levels. After considering the avai lable solutions and risks associated with their pro\\ufb01ts, it i s\\npossible to make more informed decisions on what solutions t o implement for the problem instance. Furthermore, we\\nintroduce a \\ufb01ltering method, which is applied at regular int ervals of \\ufb01tness evaluations. It keeps only the solutions wi th\\na valid\\u03b1interval in the interim populations, enabling the new offsp ring solutions in the next generations to improve\\nupon these solutions. The performance of these methods is ev ident in the experimental investigations.\\nAcknowledgements\\nThis work has been supported by the Australian Research Coun cil (ARC) through grant FT200100536, and by the\\nSouth Australian Government through the Research Consorti um \\\"Unlocking Complex Resources through Lean Pro-\\ncessing\\\". This work was also supported with supercomputing resources provided by the Phoenix HPC service at the\\nUniversity of Adelaide.\\nReferences\\nYuma Abe, Masaki Ogura, Hiroyuki Tsuji, Amane Miura, and Shu ichi Adachi. 2020. Resource and Network Manage-\\nment for Satellite Communications Systems: A Chance-Const rained Approach. IFAC 53 (2020), 3304\\u20133309. Issue\\n12 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\n2.\\nHirad Assimi, Oscar Harper, Yue Xie, Aneta Neumann, and Fran k Neumann. 2020. Evolutionary Bi-Objective Opti-\\nmization for the Dynamic Chance-Constrained Knapsack Prob lem Based on Tail Bound Objectives. In ECAI 2020 ,\\nV ol. 325. IOS Press, 307\\u2013314.\\nC.C. Coello, D.A. Van Veldhuizen, and G.B. Lamont. 2013. Evolutionary Algorithms for Solving Multi-Objective\\nProblems . Springer US.\\nKalyanmoy Deb. 2001. Multi-Objective Optimization using Evolutionary Algorit hms. John Wiley & Sons.\\nKalyanmoy Deb, Samir Agrawal, Amrit Pratap, and T. Meyariva n. 2002. A fast and elitist multiobjective genetic\\nalgorithm: NSGA-II. IEEE Trans. Evol. Comput. 6, 2 (2002), 182\\u2013197.\\nBenjamin Doerr. 2020. Probabilistic Tools for the Analysis of Randomized Optimiz ation Heuristics . Springer, Chap-\\nter 1, 1\\u201387.\\nBenjamin Doerr, Carola Doerr, Aneta Neumann, Frank Neumann , and Andrew M. Sutton. 2020. Optimization of\\nChance-Constrained Submodular Functions. In The Thirty-Fourth AAAI Conference on Arti\\ufb01cial Intelligen ce, AAAI\\n2020 . AAAI Press, 1460\\u20131467.\\nBenjamin Doerr and Frank Neumann. 2021. A Survey on Recent Pr ogress in the Theory of Evolutionary Algorithms\\nfor Discrete Optimization. ACM Trans. Evol. Learn. Optim. 1, 4, Article 16 (oct 2021), 43 pages.\\nXinbo Geng and Le Xie. 2019. Data-driven decision making in p ower systems with probabilistic guarantees: Theory\\nand applications of chance-constrained optimization. Annual Reviews in Control 47 (2019), 341\\u2013363.\\nO. Giel. 2003. Expected runtimes of a simple multi-objectiv e evolutionary algorithm. In The 2003 Congress on\\nEvolutionary Computation, 2003. CEC \\u201903. , V ol. 3. 1918\\u20131925 V ol.3.\\nFangguo He and Guiming Shao. 2009. An Evolutionary Algorith m for Uncertain Optimization Problems. In 2009\\nInternational Conference on Information Engineering and C omputer Science . IEEE, 1\\u20134.\\nKonstantinos Kepaptsoglou, Grigorios Fountas, and Matthe w G. Karlaftis. 2015. Weather impact on containership\\nrouting in closed seas. Transportation Research Part C: Emerging Technologies 55 (2015), 139\\u2013155.\\nBo Liu, Qingfu Zhang, Francisco V . Fern\\u00e1ndez, and Georges G. E. Gielen. 2013. An Ef\\ufb01cient Evolutionary Algorithm\\nfor Chance-Constrained Bi-Objective Stochastic Optimiza tion. IEEE Trans. Evol. Comput. 17, 6 (2013), 786\\u2013796.\\nDaniel H. Loughlin and S. Ranji Ranjithan. 1999. Chance-Con strained Genetic Algorithms. In GECCO \\u201999 . Morgan\\nKaufmann Publishers Inc., 369\\u2013376.\\nKazuyuki Masutomi, Yuichi Nagata, , and Isao Ono. 2013. An Ev olutionary Algorithm for Black-Box Chance-\\nConstrained Function Optimization. Journal of Advanced Computational Intelligence and Intell igent Informatics\\n17, 2 (2013), 272\\u2013282.\\nMehrnaz Mohtasham, Hossein Mirzaei-Nasirabad, and Behroo z Alizadeh. 2021. Optimization of truck-shovel alloca-\\ntion in open-pit mines under uncertainty: a chance-constra ined goal programming approach. Mining Technology\\n130 (2021), 81\\u2013100.\\nAneta Neumann and Frank Neumann. 2020. Optimising Monotone Chance-Constrained Submodular Functions Using\\nEvolutionary Multi-objective Algorithms. In Parallel Problem Solving from Nature - PPSN XVI - 16th Intern ational\\nConference, PPSN 2020, Proceedings, Part I (Lecture Notes i n Computer Science, Vol. 12269) . Springer, 404\\u2013417.\\nAneta Neumann, Yue Xie, and Frank Neumann. 2022. Evolutiona ry Algorithms for Limiting the Effect of Uncertainty\\nfor the Knapsack Problem with Stochastic Pro\\ufb01ts. In PPSN XVII . Springer, Cham, 294\\u2013307.\\nFrank Neumann, Mojgan Pourhassan, and Vahid Roostapour. 20 20.Analysis of Evolutionary Algorithms in Dynamic\\nand Stochastic Environments . Springer, Chapter 7, 323\\u2013357.\\nFrank Neumann and Andrew M. Sutton. 2019. Runtime Analysis o f the (1 + 1) Evolutionary Algorithm for the\\nChance-Constrained Knapsack Problem. In FOGA \\u201919 (FOGA \\u201919) . ACM, 147\\u2013153.\\nFrank Neumann and Carsten Witt. 2022. Runtime Analysis of Si ngle- and Multi-Objective Evolutionary Algorithms Chance-Constrained Knapsack Problem. In FOGA \\u201919 (FOGA \\u201919) . ACM, 147\\u2013153.\\nFrank Neumann and Carsten Witt. 2022. Runtime Analysis of Si ngle- and Multi-Objective Evolutionary Algorithms\\nfor Chance Constrained Optimization Problems with Normall y Distributed Random Variables. In IJCAI-22 . 4800\\u2013\\n4806.\\nShen Peng. 2019. Chance constrained problem and its applications . Theses. Universit\\u00e9 Paris Saclay (COmUE) ; Xi\\u2019an\\nJiaotong University.\\nFeng Shi, Xiankun Yan, and Frank Neumann. 2022. Runtime Anal ysis of Simple Evolutionary Algorithms for the\\nChance-Constrained Makespan Scheduling Problem. In PPSN XVII . Springer, 526\\u2013541.\\n13 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nHemant Kumar Singh and J\\u00fcrgen Branke. 2022. Identifying Sto chastically Non-dominated Solutions Using Evolu-\\ntionary Computation. In PPSN (2) (Lecture Notes in Computer Science, Vol. 13399) . Springer, 193\\u2013206.\\nYue Xie, Oscar Harper, Hirad Assimi, Aneta Neumann, and Fran k Neumann. 2019. Evolutionary Algorithms for the\\nChance-Constrained Knapsack Problem. In GECCO \\u201919 . ACM, 338\\u2013346.\\nYue Xie, Aneta Neumann, and Frank Neumann. 2020. Speci\\ufb01c Sin gle- and Multi-Objective Evolutionary Algorithms\\nfor the Chance-Constrained Knapsack Problem. In GECCO \\u201920 . ACM, 271\\u2013279.\\nYue Xie, Aneta Neumann, and Frank Neumann. 2021a. Heuristic Strategies for Solving Complex Interacting Stockpile\\nBlending Problem with Chance Constraints. In GECCO \\u201921 . ACM, 1079\\u20131087.\\nYue Xie, Aneta Neumann, Frank Neumann, and Andrew M. Sutton. 2021b. Runtime Analysis of RLS and the\\n(1+1) EA for the Chance-Constrained Knapsack Problem with C orrelated Uniform Weights. In GECCO \\u201921 . ACM,\\n1187\\u20131194.\\n14\",\"27\":\" Linear CNNs discover the statistical structure of the dataset\\nusing only the most dominant frequencies\\nHannah Pinson1Joerie Lenaerts1Vincent Ginis1 2\\nAbstract\\nOur theoretical understanding of the inner work-\\nings of general convolutional neural networks\\n(CNN) is limited. We here present a new stepping\\nstone towards such understanding in the form of a\\ntheory of learning in linear CNNs. By analyzing\\nthe gradient descent equations, we discover that\\nusing convolutions leads to a mismatch between\\nthe dataset structure and the network structure.\\nWe show that linear CNNs discover the statisti-\\ncal structure of the dataset with non-linear, stage-\\nlike transitions, and that the speed of discovery\\nchanges depending on this structural mismatch.\\nMoreover, we \\ufb01nd that the mismatch lies at the\\nheart of what we call the \\u201cdominant frequency\\nbias\\u201d, where linear CNNs arrive at these discover-\\nies using only the dominant frequencies of the dif-\\nferent structural parts present in the dataset. Our\\n\\ufb01ndings can help explain several characteristics\\nof general CNNs, such as their shortcut learning\\nand their tendency to rely on texture instead of\\nshape.\\nIn addition to a neural network\\u2019s pre-de\\ufb01ned architecture,\\nthe parameters of the network obtain an implicit structure\\nduring training. For example, it has been shown that weight\\nmatrices can exhibit structural patterns, such as clusters and\\nbranches (V oss et al., 2021; Casper et al., 2022). On the\\nother hand, the input dataset also has an implicit structure\\narising from patterns and relationships between the samples.\\nE.g., in a classi\\ufb01cation task, dogs are more visually similar\\nto cats than to cars. A general theory on how the implicit\\nstructure in the network arises and how it depends on the\\nstructure of the dataset has yet to be developed. Here we\\nderive such a theory for the speci\\ufb01c case of two-layer, linear\\nCNN. Our approach is inspired by previous work on the\\n1Data Lab\\/Applied Physics, Vrije Universiteit Brussel, Plein-\\nlaan 2, 1050 Brussel, Belgium2Harvard John A. Paulson School of\\nEngineering and Applied Sciences, Harvard University, 29 Oxford\\nStreet, Cambridge, Massachusetts 02138, USA. Correspondence\\nto: Hannah Pinson <hannah.pinson@vub.be >.learning dynamics in linear fully connected neural networks\\n(FCNN) (Saxe et al., 2014; 2019). We study the different\\ninvolved structures in terms of singular value decomposi-\\ntions (SVD): of the input-output correlation matrix of the\\ndataset on the one hand (sec. 2.1), and of the SVD of the\\nconstrained network structure induced by the convolutions\\non the other hand (sec. 2.2). We subsequently consider the\\nequations of gradient descent in the slow learning regime,\\nyielding a set of differential equations (sec. 2.3). These\\nequations describe the evolution of the implicit network\\nstructure, given the statistical dataset structure. Our analysis\\nreveals that the constraints in the network structure of a\\nCNN give rise to additional factors in those gradient descent\\nequations: these factors represent a mismatch between the\\nsingular vectors of the data and those of the network (sec.\\n2.4). This mismatch in\\ufb02uences the dynamics of learning,\\nchanging the speed of discovery of the different parts of the\\ndataset structure (sec. 3) with respect to the speed of dis-\\ncovery in a FCNN. Internally, this mismatch also leads to a\\ndominant frequency bias: only the dominant frequency com-\\nponents of each singular vector of the dataset are used by the\\nCNN (sec. 4). Experiments con\\ufb01rm the overall dynamics of\\nlearning and the existence of the dominant frequency bias\\n(sec. 5).\\nOur results can be put in the context of the implicit reg-\\nularisation resulting from training with gradient descent.\\nNeural networks used in practice contain much more train-\\nable weights than there are input samples. Therefore, the\\ntarget mapping between input and output could potentially\\nbe realised by a lot of different weight settings. However,\\nusing gradient descent yields speci\\ufb01c outcomes for the (evo- target mapping between input and output could potentially\\nbe realised by a lot of different weight settings. However,\\nusing gradient descent yields speci\\ufb01c outcomes for the (evo-\\nlution of the) network weights (Du et al., 2019; Arora et al.,\\n2019; Gidel et al., 2019; Advani et al., 2020; Satpathi &\\nSrikant, 2021). In (Saxe et al., 2014; 2019) the authors show\\nthat the structure of the \\ufb01nal weights of linear FCNN re\\ufb02ect\\nthe dataset structure, at least when starting from random ini-\\ntial conditions and when using gradient descent. They \\ufb01nd\\nanalytical solutions for the learning dynamics in a two-layer,\\nlinear FCNN. These solutions indicate the stage-like discov-\\nery of different structural parts of the dataset. The authors\\nassume a small learning rate, and the analytical solutions\\nare derived from a continuous time approximation to the\\nupdate equations of gradient descent. In (Gidel et al., 2019),arXiv:2303.02034v1  [cs.CV]  3 Mar 2023 Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nthe discrete dynamics are studied and in (Braun et al., 2022;\\nAtanasov et al., 2022), the authors develop the theory for dif-\\nferent initialisation regimes. The relationship with a.o. early\\nstopping and learning order constancy for both CNN and\\nFCNN is further studied in (Hacohen & Weinshall, 2022)\\nas the principal component bias or PC-bias. Furthermore,\\nsimilar theories exist for linear and shallow non-linear auto-\\nencoders (Bourlard & Kamp, 1988; Pretorius et al., 2018;\\nRe\\ufb01netti & Goldt, 2022). Finally, it has been shown that\\nlinear CNN trained with gradient descent exhibit an implicit\\nbias in the form of sparsity in the frequency domain (Gu-\\nnasekar et al., 2018). We here show which mechanism gives\\nrise to this sparsity in the frequency domain, and we \\ufb01nd\\nwhich frequencies are developed over time.\\n1. Prerequisites\\nNotation The input consists of n\\u0002nimages denoted Xs\\nwheresis the sample index. We will omit this index when\\nthe context is clear. Often, we will need a vectorized (or\\n\\u2018\\ufb02attened\\u2019) representation of Xand all other n\\u0002nmatrices:\\nwe turn those matrices in n2\\u00021vectors through stacking\\nall rows one after the other, and transposing the result. The\\nresulting column vector is denoted with a lower case letter,\\ne.g.,vec(X) =x. We reverse this operation to show the\\nvectors as 2D images in \\ufb01gures. An index linto the vec-\\ntor which results from vectorizing a 2D matrix, denoted a\\n\\u2018vec-2D\\u2019 index, runs from 0ton2\\u00001. The corresponding\\nindices in the 2D matrix are given by i=div(l;n)and\\nj=mod(l;n), withdivinteger division and mod the mod-\\nulo operator. The row iof a matrixBis denotedBi;:; the\\ncolumnjis denotedB:;j.\\u0003denotes the complex conjugate.\\nArchitecture and assumptions We consider a CNN with a\\nsingle convolutional layer, a \\ufb02atten layer, and a single fully\\nconnected layer. There are only linear activation functions.\\nThe convolutional layer consists of a single kernel Kwith\\nthe same dimensions as the input images ( n\\u0002n). When\\nthe kernel reaches the boundaries of the image, it \\u2018wraps\\u2019\\naround, such that the convolution is circular. This simpli\\ufb01es\\nthe math while still being a good approximation to the zero-\\npadding often used in practice (Gray, 2006). The task is\\nimage classi\\ufb01cation with one-hot encoded labels, we assume\\na slow learning rate, an MSE loss, and small, random initial\\nconditions.\\nThe singular value decomposition (SVD) The SVD of\\nanp\\u0002n2(p < n2) real-valued matrix Bis given by:\\nB=USVT, whereUis an orthogonal p\\u0002pma-\\ntrix (UT=U\\u00001) containing the left singular vectors as\\ncolumns,Vis an orthogonal n2\\u0002n2matrix containing\\nthe right singular vectors as columns, and Sis ap\\u0002n2\\nrectangular diagonal matrix containing the preal and posi-\\ntive singular values, denoted s\\u000b. The singular vectors and\\nvalues are by de\\ufb01nition sorted from highest to lowest sin-gular value. The modes M(\\u000b),\\u000b2f0;:::;pgform the\\ndecomposition of the original matrix: B=P\\n\\u000bM(\\u000b), with\\nM(\\u000b)=s\\u000bU:;\\u000bVT\\n\\u000b;:.\\nThe vectorized 2D discrete Fourier transform The\\nFourier transform is another type of decomposition, used\\nto decompose a signal in its frequency components. The\\nresulting amplitudes in the frequency domain, the \\u2018Fourier\\ncoef\\ufb01cients\\u2019, describe the relative importance of each fre-\\nquency in the original signal. For an n\\u0002nmatrixB,\\nthe conventional 2D discrete Fourier transform (denoted\\nF2D) is given by F2D(B)\\u0016;\\u0017= \\u0006n\\ns;t(!n)\\u0016s(!n)\\u0017tBs;t\\nwith!n= exp(\\u00002\\u0019i=n), withi2=\\u00001. The result is\\na 2Dn\\u0002ncomplex-valued matrix, whose values tell us the\\nrelative importance of the n2different spatial frequencies:\\nnfrequencies in the vertical direction, combined with nfre-\\nquencies in the horizontal direction. We will make use of an\\nequivalent transform, but instead of applied to a 2D matrix,\\napplied to the 1D vector that results from vectorizing the\\nn\\u0002nmatrix \\ufb01rst. This transform is given by multiplication\\nwith then2\\u0002n2matrixQ, i.e.:\\nQx=1\\nnvec(F2D(X) ): (1) applied to the 1D vector that results from vectorizing the\\nn\\u0002nmatrix \\ufb01rst. This transform is given by multiplication\\nwith then2\\u0002n2matrixQ, i.e.:\\nQx=1\\nnvec(F2D(X) ): (1)\\nQis illustrated in Fig.1; for its exact de\\ufb01nition, see SI. Note\\nthat this operation is different from the 1D discrete Fourier\\ntransform of a vector: we will call it the vectorized 2D\\ndiscrete Fourier transform, or vec-2D DFT in short.\\nFigure 1. Illustration of the matrix Q, where each row\\/column is\\na vectorized Fourier eigenmatrix (4 examples given).\\nIf the original vector is real-valued, the Fourier spectrum of\\nthis vector will exhibit symmetries. Let a vec-2D frequency\\nindexjcorrespond to a pair of horizontal and vertical fre-\\nquency indices (\\u0016;\\u0017), then the symmetric vec-2D frequency\\nindexjsymm corresponds to the pair (n\\u0000\\u0016;n\\u0000\\u0017), thus\\nj= (n\\u00001)\\u0016+\\u0017andjsymm = (n\\u00001)(n\\u0000\\u0016) + (n\\u0000\\u0017).\\nThe symmetry in the spectrum is then given by:\\n(Qx)jsymm = (Qx)\\u0003\\nj: (2)\\n2. Differential Equations of Gradient Descent\\nWe want to study the relationship between the dataset struc-\\nture and the network structure during training with gradient\\ndescent. To this end, we \\ufb01rst discuss the SVD that capture\\nthese structures. We will then show how the equations of\\ngradient descent can be reformulated in terms of those SVD. Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nFigure 2. Illustration of the relationship between statistical structure and SVD. a)Example dataset consisting of geometric shapes: a\\ncircle, an octagon, a square and a star. There is only one sample per class. b)Visualisation of VTand its \\ufb01rst p rows, the singular vectors\\ndenoted\\u001e\\u000b(here p=4), as n\\u0002nmatrices. The other rows carry no meaning. c)Visualisation of the summation of the modes M(\\u000b)of\\n\\u0006yx, given by the SVD (see prerequisites). Each mode is \\ufb01rst reshaped to p n\\u0002nmatrices.d)Interpretation of the implicit hierarchical\\nstructure in \\u0006yx, which can be derived from the visualisations in sub\\ufb01gures b) and c). e)Vec-2D Fourier coef\\ufb01cients of the 4 singular\\nvectors, i.e.,jQ\\u001e\\u000bjjin function of the vec-2D indices j. The values for j > 2048 are given by the symmetry properties of Fourier\\ntransforms. When comparing with sub\\ufb01gure b), we see that broader distinctions in shape correspond to spectra with limited peaks at low\\nfrequencies, while \\ufb01ne distinctions also contain higher frequencies.\\n2.1. Statistical Structure of the Dataset\\nThe dataset has an implicit structure, e.g., some images con-\\ntain the same backgrounds, and images of cats are similar\\nto images of dogs. Given a general dataset, it is unclear a\\npriori which patterns might be relevant during the training\\nof a network. For linear CNN, however, we \\ufb01nd that the\\nrelevant structure is given by the statistical structure: the\\nstructure that can be derived from the correlations in the\\ndataset. In Fig.2, we illustrate the statistical structure of a\\ndataset consisting of geometric shapes. Given ground-truth\\nlabelsywhen there are pclasses, we can de\\ufb01ne the p\\u0002n2\\ninput-output correlation matrix \\u0006yxas:\\n\\u0006yx=hyxTi; (3)\\nwherexis ann2\\u00021vectorized sample, and hidenotes the\\naverage over all samples. Note that yxTis an outer product,\\ni.e.,\\u0006yx\\nl;m=hylxmi. Given one-hot encoded labels, each\\nrow of \\u0006yxcontains the vectorized average image of each\\nclass. The structure relevant when training a (linear) neural\\nnetwork is captured through the singular value decomposi-\\ntions (SVD) of this matrix (see (Saxe et al., 2019) and Fig.\\n2(b-d)). The SVD of \\u0006yxis given by:\\n\\u0006yx=USVT: (4)\\nIn this particular case, the \\ufb01rst prows ofVT, which we\\ndenote\\u001e(\\u000b),\\u000b2f0;:::;p\\u00001g, are the principal compo-\\nnents of the averaged class images (Pearson, 1901; Jolliffe,\\n2002). The p\\u0002pmatrixUthen contains the coef\\ufb01cients\\nto reconstruct the pvectorized, averaged class images as\\nlinear combinations of the pright singular vectors \\u001e\\u000b. By\\nvisualizing the singular vectors \\u001e\\u000band the corresponding\\nmodesM\\u000b, we can see the SVD here embodies broader to\\ufb01ner distinctions between the averaged class images (see\\nFig.2(b-d)). As such, it captures our intuitive understanding\\nof an implicit hierarchical structure between the shapes in\\nthe dataset.\\nWe can also consider the singular value decomposition of the\\ninput-input correlation matrix \\u0006xx=hxxTi. In general,\\nthis matrix could have n2different singular values and cor-\\nresponding singular vectors, and its exact shape in\\ufb02uences\\nthe dynamics of learning as well. However, to focus on the\\neffect of a convolutional architecture only, we consider the\\ncase where \\u0006xxis diagonalizable in the basis given by V:\\n\\u0006xx=V\\u0006xxVT; (5)\\nwhere \\u0006xxis a diagonal matrix with only the \\ufb01rst pentries\\non the diagonal non-zero. In our case of classi\\ufb01cation with\\none-hot encoded labels, this is the case if all images in a\\nclass are the same (see SI). The less images deviate from\\nthe average image for their class, the better this assumption\\nholds. Finally, given any set of predictions ^yfor a set of\\ninput samples x, we can also compute the input-predicted\\noutput correlation matrix \\u0006^yx(cfr. Eq. 3):\\n\\u0006^yx=h^yxTi; (6)\\nwhere \\u0006^yxhas dimensions p\\u0002n2. To relate the evolution\\nof this matrix to the structure of the dataset, we study \\u0006^yx\\nin the SVD basis associated to \\u0006yx(see Eq. 4):\\n\\u0006^yx=UAVT: (7) \\u0006^yx=h^yxTi; (6)\\nwhere \\u0006^yxhas dimensions p\\u0002n2. To relate the evolution\\nof this matrix to the structure of the dataset, we study \\u0006^yx\\nin the SVD basis associated to \\u0006yx(see Eq. 4):\\n\\u0006^yx=UAVT: (7)\\nThep\\u0002n2matrixAis not necessarily rectangular diagonal;\\nif it would be, and if the diagonal elements would be positive\\nreal numbers, Eq. 7 would be an SVD with singular values\\ngiven by the values on the diagonal of A. In that case, we\\ncould say that \\u0006^yxhas the same structural elements as \\u0006yx,\\nonly with different strengths. Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\n2.2. Structure of the Network\\nWe also want to study the implicit structure of the convolu-\\ntional network through an SVD. To this end, we replace the\\nconvolution operations with convolution-equivalent weight\\nmatrices, such that we can study the SVD of these matrices\\n(Sedghi et al., 2019). A convolutional layer can be seen as a\\nconstrained fully connected layer: the sliding of the kernel\\nover the image is actually a repeated application of the same\\nweights (Goodfellow et al., 2016). A convolution-equivalent\\nweight matrix is then a matrix where elements are repeated\\nin a \\ufb01xed, circulant pattern. Such a matrix is called a doubly\\nblock circulant matrix (see (Jain, 1989; Sedghi et al., 2019)),\\ndenoteddbc (de\\ufb01nition, see SI). We can also de\\ufb01ne such a\\nmatrix through its eigendecomposition: the n2\\u0002n2doubly\\nblock circulant matrix of the kernel Kis given by\\ndbc(K) =nQ\\u00001diag(Qk)Q (8)\\nand it thus acts as a convolution-equivalent weight matrix:\\nvec(X~K) =dbc(K)x (9)\\nwhere~denotes circular convolution. Here kis the vec-\\ntorized kernel, and Qis the matrix corresponding to the\\nvec-2D DFT (see prerequisites and Fig. 1). diag(Qk)is an\\nn2\\u0002n2diagonal matrix with the n2elements of the vec-2D\\nDFT ofk, thusQk, on the diagonal. Readers familiar with\\nthe convolution theorem might recognize it in Eq. 8: to\\napply a convolution, transform the input to the Fourier do-\\nmain, multiply with the transformed kernel, and transform\\nthe result back to the original domain.\\nIn short, for our theoretical derivations, we consider the\\nnetwork with predictions ^ygiven by:\\n^y=Wdbc (K)x; (10)\\nwithWthep\\u0002n2weight matrix of the fully connected\\nlayer. We can again formalize the notion of implicit struc-\\nture through considering the singular value decomposi-\\ntions\\/eigenvalue decompositions of the network\\u2019s weight\\nmatricesWanddbc(K). While a general weight matrix\\nWcould in principle have any singular value decomposi-\\ntion, re\\ufb02ecting its unconstrained structure, the eigendecom-\\nposition ofdbc(K)is\\ufb01xed and always given by Eq. 8: the\\neigenvectors stay the same. Only the eigenvalues, given by\\nnQk, can change. This re\\ufb02ects the fact that of the n2\\u0002n2\\nmatrixdbc(K), onlyn2values can change independently.\\nThese values are given by the n\\u0002nkernel. The different\\nsingular values of dbc(K)are actually given by nj(Qk)ij.\\nFor a discussion on the role of the phases of the complex\\neigenvalues, see SI.\\n2.3. Differential Equations of Gradient Descent\\nWe can now focus on what happens during training with\\ngradient descent. We start from a MSE loss function L=1\\n2Pp\\nl=0(yl\\u0000^yl)2, and its gradient is used to update the\\nnetwork parameters at every discrete timestep:\\n\\u0001k=\\u0000\\u0015@L\\n@kT; \\u0001W=\\u0000\\u0015@L\\n@WT; (11)\\nwith\\u0015the learning rate. For the convolutional layer, the\\ngradient of the loss with respect to the kernel is in itself\\ngiven by a convolution, but with a \\ufb02ipped image (see, e.g.,\\n(Goodfellow et al., 2016), Ch. 9). The gradients are then\\ngiven by:\\n@L\\n@kT=dbc(Xflip)pX\\nl=0(yl\\u0000^yl)(WT):;l (12)\\n\\u0000@L\\n@WT\\u0001\\nl;:= (yl\\u0000^yl)xTdbc(k)T: (13)\\nIn the slow learning regime, the weights and kernel change\\nminimally with each update. The updates can then be ap-\\nproximated through an averaged update over samples ((Saxe\\net al., 2019), SI p1.). This allows us to introduce the matri-\\nces\\u0006yx(cfr. Eq. 3) and \\u0006^yx(cfr. Eq. 7) in the equations.\\nGiven the slow learning rate, the discrete equations can also\\nbe reformulated as differential equations. We can subse-\\nquently transform the different variables using the SVD\\nbasis of \\u0006yx:\\nW=UTWR (14)\\ndbc(K)=R\\u00001dbc(K)V (15)\\n=nR\\u00001Q\\u00001diag(Qk)QV (16)\\nhereRis an arbitrary n2\\u0002n2invertible matrix. This matrix\\nre\\ufb02ects the freedom in the actual shape of the weights,\\nas long as their product remains the same. Now we have\\n\\u0006^yx=Wdbc (K)\\u0006xx=UAVT=UWdbc(K)\\u0006xxVT\\n(see also Eq. 5), and we arrive at the differential equations\\nin the following, somewhat complicated shape (for full\\nderivation, see SI) :\\n1 \\u0006^yx=Wdbc (K)\\u0006xx=UAVT=UWdbc(K)\\u0006xxVT\\n(see also Eq. 5), and we arrive at the differential equations\\nin the following, somewhat complicated shape (for full\\nderivation, see SI) :\\n1\\nn\\u0015\\u0000ddiag (Q\\u00001k)\\ndt\\u0001\\nj;j=Q\\u00001\\nj;:R\\u0000\\u0003TW\\u0003T\\u0000\\nS\\u0000A)(QV)T\\n:;j;(17)\\n1\\nn\\u0015\\u0000dW\\ndt\\u0001\\n=\\u0000\\nS\\u0000A)(QV)Tdiag(Qk)Q\\u00001R; (18)\\nwherediag(Q\\u00001k)j;j= (Q\\u00001k)j= (Q\\u0003Tk)j, and all\\nderivatives of off-diagonal elements of diag(Q\\u00001k)are\\nzero. There are a couple of key observation to make about\\nthis set of differential equations (Eqs. 17 and 18 ): \\ufb01rstly,\\nthey are coupled and non-linear, making them dif\\ufb01cult or\\nimpossible to solve analytically but for very speci\\ufb01c cases.\\nWe can also see that at every step Ais brought closer to\\nS(cfr. Eq. 4), that Athus becomes diagonal, and that\\nthe updates become zero if these matrices are equal: i.e.,\\nthe predictions perfectly match the labels. Furthermore, the\\ndynamics of learning are in\\ufb02uenced by the vectors QV :;\\u000b=\\nQ\\u001e(\\u000b), the vec-2D Fourier transforms of the right singular Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nvectors\\u001e(\\u000b). This turns out to be a crucial point: this\\nembodies the \\u2018mismatch\\u2019 between the dataset structure ( V)\\nand the constrained structure of the convolutional layer ( Q);\\nin the equivalent equations for fully connected networks, no\\nsuch extra factors are present (see Eqs. 21 and 22 below).\\n2.4. Comparison with Fully Connected networks\\nThe dynamics of learning during gradient descent in linear\\nFCNN are discussed in (Saxe et al., 2014; 2019). In the case\\nof a two-layer linear FCNN, we have\\n\\u0006^yxFC=<^yFCxT>=W2W1\\u0006xx=UAFCVT;(19)\\nwhereW2andW1are the weight matrices for the second\\nand \\ufb01rst layer of the FCNN network, respectively. The\\nmatricesUandVare the same as before (Eq. 4). However,\\nthe evolution of the matrices \\u0006^yxFCandAFCwill be\\ndifferent from the evolution of their equivalents in the CNN\\ncase. Using transformed weight matrices:\\nW1=R0\\u00001W1V;W2=UTW2R0; (20)\\nwithR0an arbitrary invertible matrix, the authors in (Saxe\\net al., 2014; 2019) then arrive at the system of differential\\nequations:\\n1\\n\\u0015dW1\\ndt=W2T(S\\u0000AFC) (21)\\n1\\n\\u0015dW2\\ndt= (S\\u0000AFC)W1T\\n: (22)\\nwhich can be compared to the equations we derived for the\\nCNN (Eqs. 17 and 18). The authors subsequently show\\nthat, starting from small, random initial conditions, W2and\\nW1each quickly become diagonal. Comparing this to the\\ncase of the CNN, we see that dbc(K)cannot be diagonal\\n(Eq. 15). This indicates that the constrained structure of the\\nconvolutional layer in itself cannot fully re\\ufb02ect the dataset\\nstructure, while a fully connected layer can.\\n3. Non-linear learning dynamics\\nWe now explore how the mismatch between VandQin-\\n\\ufb02uences the dynamics of learning. In particular, we study\\nthe evolution of the predictions: we can study this evolution\\nthrough analysing the evolution of the matrix A(see Eq. 7),\\nwithA=Wdbc(K)\\u0006xx. We will \\ufb01rst derive analyti-\\ncal solutions for this evolution for a particular, illuminating\\ndataset. Later we will show that the characteristics of the\\nfound trajectories roughly hold for more general datasets as\\nwell. Consider the dataset where the image for each class is\\ngiven by:\\nX(c)\\nl;m=b(c)cos\\u0012\\n2\\u0019\\u0016l\\nn+ 2\\u0019\\u0017m\\nn\\u0013\\n: (23)Herecis the class index, and for each class we pick a\\ndifferent pair of \\u0016and\\u00172f0;\\u0001\\u0001\\u0001;n\\u00001g, indexing the\\nnpossible horizontal and vertical frequencies. landm\\nindex the pixels. b(c)is the amplitude of the frequency\\n(one value per class). Examples of this type of input are\\nshown in Fig. 1, since the vectorized input images each\\ncorrespond to the real part of a column of Q(apart from the\\namplitude). Since the class vectors are already orthogonal,\\nnormalisation yields the psingular vectors V:;\\u000b=\\u001e\\u000b.\\nThis type of input decouples the differential equations (Eqs.\\n17 and 18): the intuition behind this is that the structural\\n\\u2018mismatch\\u2019 partly vanishes, because we pick the vectors\\n\\u001e\\u000bto be orthogonal to the real part of the columns of Q.\\nHowever,Qis complex-valued and the singular vectors\\nare real-valued; therefore an artefact of the mismatch will\\nremain in the form of a factor1p\\n2. In the SI, we \\ufb01rst de\\ufb01ne\\na set of speci\\ufb01c initial conditions such that Astarts out\\nrectangular diagonal. Subsequently we show that given\\nthe Eqs. 17 and 18, each a\\u000b=A\\u000b;\\u000bexactly follows a\\nsigmoidal trajectory:\\na\\u000b(t) =s\\u000be2n\\u0015d\\u000bs\\u000bt\\ne2\\u0015nd\\u000bs\\u000bt\\u00001 +s\\u000b=a(0)\\n\\u000b; (24)\\nwithd\\u000b=1p\\n2(unless\\u0016= 0 and\\u0017= 0, thend\\u000b= 1),\\nanda(0)\\n\\u000b=a\\u000b(t= 0):The time, in number of samples,\\nto grow from an initial value a(0)\\n\\u000b=\\u000fwith\\u000f\\u001c1, to a\\nvaluea\\u000b=s\\u000b\\u0000\\u000f, is approximately given byd\\u000bn\\ns\\u000b\\u0015(see\\n(Saxe et al., 2019)). Under analogous initial conditions and\\nassumptions, the values a\\u000b(t)in a linear FCNN follow a\\nsimilar sigmoidal trajectory (see (Saxe et al., 2019)). How-\\never, there are no factors d\\u000bandn: the number of samples\\nneeded to reach convergence for each value a\\u000bis approxi-\\nmately given by1\\ns\\u000b\\u0015for FCNN.\\nGiven these analytical results, we can draw the following\\nconclusions: \\ufb01rst, given the sigmoidal trajectories, the pre- mately given by1\\ns\\u000b\\u0015for FCNN.\\nGiven these analytical results, we can draw the following\\nconclusions: \\ufb01rst, given the sigmoidal trajectories, the pre-\\ndictions of the network change in such a way that the dif-\\nferent structural modes of \\u0006yxare discovered with rapid,\\nstage-like transitions by both types of networks. This discov-\\nery is ordered in time (through the factor1\\ns\\u000b), from highest\\nsingular value to lowest singular value, corresponding to\\nthe discovery from broader to \\ufb01ner distinctions between\\nthe classes. However, the CNN exhibits a different effec-\\ntive learning rate \\u0015eff=nd\\u000b\\u0015for each mode \\u000bw.r.t. the\\nFCNN. The factor nis a speed-up resulting from the con-\\nvolution with a kernel of dimension n; the factord\\u000b<= 1\\nre\\ufb02ects a delay stemming from the mismatch between the\\ndataset structure and the constrained network structure. In\\nFig. 3, we show the results of experiments with a dataset\\nof \\u2018pure cosines\\u2019 as described by Eq. 23 (for details, see\\nSI). The \\ufb01rst class\\/singular vector \\u001e0is a constant image,\\ni.e.,\\u0016=\\u0017= 0; the subsequent classes have different ran-\\ndomly selected frequencies with decreasing amplitudes. The\\nFCNN is trained with a higher learning rate \\u0015FC=n\\u0015CNN , Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nsuch that the graphs of both networks can be compared on\\nthe same plot. d0= 1, such that the trajectories of a0for the\\nCNN and FCNN overlap. d\\u000b= 1=p\\n2\\u00190:71for the other\\nmodes, resulting in a shift between the trajectories. The\\ntheoretical predictions, made using the same set of initial\\nconditions, exactly match the experimental trajectories.\\nWe derived the analytical solutions using aligned, balanced\\ninitial conditions (see SI). These conditions render Arectan-\\ngular diagonal from the beginning. Several previous studies\\nshow that when starting from fully random, small initial\\nconditions,Avery quickly becomes diagonal as well (Saxe\\net al., 2014; 2019; Hacohen & Weinshall, 2022; Braun et al.,\\n2022). We discuss this further in the SI. Apart from a small\\nadditional delay related to this alignment phase, the tra-\\njectories ofAwhen starting from small, random initial\\nconditions are thus described by the same sigmoidal curves.\\nFigure 3. Experimental evolution and theoretical predictions (left)\\nofa\\u000b, and experimental evolution of the MSE loss (right), for\\nlinear FCNN (round markers) and linear CNN (square markers),\\ntrained on the same dataset. The black, dashed lines indicate the\\npredictions. All plotted lines are averaged over trials, shadowed\\nregions indicate the standard deviations. Dashed horizontal lines\\nleft indicate the values s\\u000b.\\n4. Dominant Frequency Bias\\nSo far, we have studied the evolution of the predictions for\\ntwo-layer linear CNN and FCNN. It turns out that the way\\nthese networks arrive at those predictions internally is very\\ndifferent. We will show that during training the kernel of\\nthe linear CNN becomes an implicit regularizer: it \\ufb01lters\\nout a small fraction of the frequency components present\\nin the dataset. The whole network arrives at its predictions\\nusing only those frequencies. Driving this implicit regulari-\\nsation is a soft winner-takes-all dynamics (sWTA) (Lazzaro\\net al., 1988; Fang et al., 1996; Fukai & Tanaka, 1997), where\\nduring training different vec-2D Fourier coef\\ufb01cients of the\\nkerneljQkjj\\u2018compete\\u2019 with each other to be part of the\\n\\ufb01nal kernel. Whether they \\u2018win\\u2019 depends on their initial\\nvalues and the corresponding coef\\ufb01cients of the singular\\nvectors of the dataset jQ\\u001e\\u000bjj. The word \\u2018soft\\u2019 denotes\\nthat there is not a single winning frequency. This sWTA\\ndynamics\\u2014and thus the implicit regularisation\\u2014can be de-\\nrived from the given differential equations.To see this, we \\ufb01rst consider a slightly more general type\\nof dataset: a dataset for which the modes now consist of\\nsums of cosines, where each cosine possibly has a different\\namplitude (see SI). However, we still assume frequencies\\nare not shared between modes, such that modes remain de-\\ncoupled. Formally, given an index j,jQ\\u001e\\u000bjjis non-zero for\\nonly one mode \\u000b.\\u001b(\\u000b)then denotes the set of all vec-2D\\nindicesjof the frequencies associated to a mode \\u000b. Ifj\\nis the vec-2D index corresponding to the pair of indices\\n(\\u0016;\\u0017), then withjsymm we denote the vec-2D index that\\ncorresponds to (n\\u0000\\u0016;n\\u0000\\u0017). Since the singular vectors are\\nreal-valued, their Fourier spectra exhibit symmetries, and\\njQ\\u001e\\u000bjj=jQ\\u001e\\u000bjjsymm .\\u001b(\\u000b)\\nsymm includes all the symmetric\\nindicesjsymm as well. In the SI, we show that when start-\\ning from small, random initial conditions, the development\\nof the vec-2D Fourier coef\\ufb01cients of the kernel jQkjjis\\napproximately given by:\\n1\\n2n\\u0015dj(Qk)jj2\\ndt=j(Qk)jj2j(Q\\u001e\\u000b)jj(s\\u000b\\u0000a\\u000b);(25)\\nwhenj2\\u001b(\\u000b)\\nsymm , with\\na\\u000b=n\\u0006xx\\n\\u000b;\\u000b\\u0010\\n2j(Qk)jj2j(Q\\u001e\\u000b)jj\\n+X\\nj02\\u001b(\\u000b)\\nsymmnfj;jsymmgj(Qk)j0j2j(Q\\u001e\\u000b)j0j\\u0011\\n:(26)\\nWhiles\\u000bcan be considered as the input that drives\\nthe system,\\u0000j(Qk)jj2is a self-inhibition term, and\\n\\u0000P\\nj02\\u001b(\\u000b)\\nsymmnfj;jsymmgj(Qk)j0j2is a term that captures\\nthe lateral inhibition coming from the other 2D-vec ker-\\nnel Fourier coef\\ufb01cients associated to the same mode. A\\nformulation and analysis of sWTA dynamics close to the\\nequations we consider is given in (Fukai & Tanaka, 1997). nel Fourier coef\\ufb01cients associated to the same mode. A\\nformulation and analysis of sWTA dynamics close to the\\nequations we consider is given in (Fukai & Tanaka, 1997).\\nNote that unlike the common description of WTA dynamics\\nas a system of competing neurons, we here have a system\\nof \\u2018competing\\u2019 kernel Fourier coef\\ufb01cients. The overall dy-\\nnamics are as follows: if we initialize the kernel with small\\ninitial values, its vec-2D Fourier coef\\ufb01cients j(Qk)ijwill\\nalso be small. For each input mode \\u000b, we have a number of\\nassociated frequency indices j, and the corresponding terms\\nj(Qk)jjeach contribute to the respective effective singular\\nvaluea\\u000b(Eq. 26). Initially, a\\u000bis much smaller than s\\u000b.\\nTherefore, the values j(Qk)jj2initially grow exponentially\\n(Eq. 25 when a\\u000b\\u00190). However, they do so with different\\nexponents; within the set of frequencies associated to the\\nmode\\u000b, the difference lies in the factors j(Q\\u001e\\u000b)jj. As soon\\nas the valuesj(Qk)jj2start growing, the self-inhibition as\\nwell as the mutual inhibition between the coef\\ufb01cients as-\\nsociated to the same mode starts to take off. In practice,\\nthe coef\\ufb01cientsj(Qk)jjwith the largest factors j(Q\\u001e\\u000b)jj\\nvery strongly inhibit the growth of the other coef\\ufb01cients,\\nsuch that only the former coef\\ufb01cients grow and signi\\ufb01cantly\\ncontribute to the effective singular value a\\u000b. Eventually, a Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nFigure 4. Illustration of the dominant frequency bias. a)Illustration of the class images as sums of pure cosines. Since the class vectors\\nxcare orthogonal, the singular vectors \\u001e\\u000bare normalised versions of the vectors xc.b)Evolution of the effective singular values. c)\\nEvolution of the squared Fourier coef\\ufb01cients of the kernel (middle), evolution of the kernel Kat selected timepoints (below), frequencies\\ncorresponding to highest Fourier coef\\ufb01cients of \\ufb01nal kernel (right). Compare the development of the kernel over time with the discovery\\nof the modes, shown in b), and the values of jQ\\u001e\\u000bjj, shown in a).b),c): solid lines: averages over experimental runs. Shaded regions:\\nstandard deviations. Vertical dashed lines: selected timepoints. Horizontal dashed lines: singular values s\\u000b.\\nsaturates to its \\ufb01nal value s\\u000b, and all derivatives become\\nzero. Depending on the factors j(Q\\u001e\\u000b)jj, we thus end up\\nwith one or more coef\\ufb01cients j(Qk)jjthat have \\u2018won\\u2019 the\\ncompetition; this means the \\ufb01nal kernel consists of a sum of\\nthose frequencies only.\\nWhile in a linear, fully connected network, the weight ma-\\ntrices take on the inherent structure of the dataset (see\\nSec. 2.4); the kernel in our linear CNN only picks up the\\nmost dominant frequencies associated to each mode. It thus\\nacts as a \\ufb01lter; not necessarily a low-pass \\ufb01lter, but a \\ufb01lter\\nof the most dominant frequencies. The results of an exper-\\niment with two classes, each a sum of pure cosines with\\ndifferent amplitudes, is illustrated in Fig. 4.\\nFor more general datasets, the singular vectors can in gen-\\neral share the same frequency. Formally, jQ\\u001e\\u000bjjcan be\\nsigni\\ufb01cant for different modes \\u000b. In this case, Eq. 25 does\\nnot hold and we have to revert back to the more general\\nequation Eq. 17. The experiments discussed in the next\\nsection, show that in those more general cases a form of the\\ndominant frequency \\ufb01ltering is still present. However, the\\ndominancy of a frequency is now determined through an-\\nother factor as well: assume two values jQ\\u001e\\fjjandjQ\\u001e\\rjj,\\nwith\\f <\\r , are signi\\ufb01cant. During the discovery of mode \\f\\nthe kernel Fourier coef\\ufb01cient j(Qk)jjdevelops. Later, at the\\nstart of the discovery of the mode \\r, the coef\\ufb01cientj(Qk)jj\\nstarts from a higher value than coef\\ufb01cients j(Qk)j0jthat\\nwere not developed before, and that still have their small,\\nrandom initial value. Loosely speaking, this gives the coef-\\n\\ufb01cientj(Qk)jja \\u2018competitive advantage\\u2019. This also means\\nthat frequencies that are dominant in the \\ufb01rst few modes\\n(thus highjQ\\u001e\\u000bjjfor the \\ufb01rst modes \\u000b) are the frequencies\\nthat are more likely to be important for the \\ufb01nal kernel.5. Experiments with More General Datasets\\nFinally, we report the results of training a linear CNN\\nfrom small, random initial conditions on two more gen-\\neral datasets: the dataset of geometric shapes (see Fig. 2)\\nand on the dataset consisting of the \\ufb01rst 4 classes of CIFAR-\\n10 (\\u2018airplane\\u2019, \\u2018automobile\\u2019, \\u2018bird\\u2019 & \\u2018cat\\u2019), which we will\\ncall CIFAR-4. We subtract the \\ufb01rst mode, i.e. the average\\nover images, from the CIFAR-4 dataset. Therefore only the\\n3 modes needed to distinguish between 4 classes remain (SI\\nFig. S.4). CIFAR-4 has multiple samples per class: \\u0006xxis\\nno longer diagonal in the basis given by V( Eq. 5 does not\\nhold), which in\\ufb02uences the dynamics through a coupling\\nof the modes. Moreover, we can hold out a separate test\\nset to track the test loss for CIFAR-4. We train the CNN\\nwith learning rate \\u0015, and the FCNN with a learning rate\\n\\u0015FC=n\\u0015. The results are shown in Fig. 5; experimental\\ndetails see SI.\\nWe can conclude that the general insights we derived before\\nare still valid. First of all, in both linear CNN and linear\\nFCNN the modes of \\u0006yxare discovered with rapid, suc-\\ncessive transitions (Fig. 5 (a) and (b)). However, the CNN\\nexhibits a different effective learning rate for each mode (see\\nSec. 3). Since the \\ufb01rst mode of the geometric shape dataset cessive transitions (Fig. 5 (a) and (b)). However, the CNN\\nexhibits a different effective learning rate for each mode (see\\nSec. 3). Since the \\ufb01rst mode of the geometric shape dataset\\nis essentially the average over the shapes (see Fig. 2), and an\\naverage corresponds to the zero-frequency, the CNN does\\nnot show an additional delay for this mode ( d0\\u00191). All\\nother mode discoveries have an additional delay with respect\\nto the trajectories for the FCNN. Fig. 5 (c) and (d) show the\\nevolution of the loss. The loss for CIFAR-4 does not go to\\nzero, meaning there are samples that cannot be classi\\ufb01ed by\\na linear CNN or FCNN. The FCNN has discovered all the\\nmodes after around 18000 samples, and at that point reaches\\na train and test loss of around 0.21. After that, it starts to\\nover\\ufb01t. The CNN needs a lot more samples to discover\\nall the modes, but doesn\\u2019t start to over\\ufb01t. The latter might Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nFigure 5. Top row: geometric shapes. Bottom row: CIFAR-4. a;b)Evolution of the effective singular values for the CNN (square\\nmarkers) and the FCNN (round markers). c;d): MSE loss for the CNN (square markers) and the FCNN (round markers). For CIFAR-4,\\nthe test loss is plotted in blue. The inset in d) shows the loss up to 106timesteps.e;g)Evolution of the weights connected to 3 randomly\\nselected nodes in the hidden layer of the FCNN, reshaped as n\\u0002nmatrices. Shown timepoints (in number of samples) roughly correspond\\nto the moments each mode is fully discovered (compare to sub\\ufb01gures a) and b) ). f;h)Evolution of the kernel at analogous timepoints\\nfor the CNN.i;j)Evolution of the coef\\ufb01cients j(Qk)jj2(middle), frequencies corresponding to highest j(Qk)jj2of \\ufb01nal kernel (right).\\nOnly the kernel of the blue channel is shown for CIFAR-4. a\\u0000f)Solid lines: averages over experimental runs. Shaded regions:\\nstandard deviations. Vertical dashed lines: selected timepoints. Horizontal dashed lines: singular values s\\u000b. The kernels and partial\\nweight matrices shown, are randomly selected across experiment runs.\\nbe due to the implicit regularisation given by the dominant\\nfrequency bias: Fig. 5 (e) and (g) show that the rows of the\\n\\ufb01rst weight matrix in FCNN are mixtures of the discovered\\nsingular vectors, as given by Eq. 20. Fig. 5 (f) and (h), on\\nthe other hand, show that, at any point during training, the\\nkernel is a sparse mixture of the dominant frequencies of\\nthe singular vectors discovered so far. Fig. 5 (i) and (j) show\\nthe evolution of the kernel Fourier coef\\ufb01cients j(Qk)jjin\\ndetail. In the plot for the geometric shapes dataset, we have\\nhighlighted a trajectory in blue. This cascaded trajectory\\nis an example of how a frequency initially becomes domi-\\nnant through a high value jQ\\u001e\\u000bjjduring the discovery of\\nmode\\u000b, and subsequently can remain dominant due to its\\nhigher value at the start of the discovery of a subsequent\\nmode. That the kernel frequencies are really the dominant\\nfrequencies is further illustrated in the SI.\\n6. Discussion\\nThe above results shed new light on the use of convolu-\\ntions in neural networks. Our theoretical analysis uses a\\nsimple, linear CNN and a number of assumptions (listed\\nin Sec. 1). We \\ufb01rst discussed how the relevant structure\\nof the dataset is related to its principal components. Wesubsequently showed that linear CNN discover this statisti-\\ncal structure with rapid, stage-like transitions. They do so\\nat an effective learning rate that depends on the mismatch\\nbetween the dataset structure and their constrained network\\nstructure. Moreover, this mismatch yields a dominant fre-\\nquency bias. Loosely speaking, we could summarize this\\ndominant frequency bias as \\u2018a linear CNN uses only the\\nprincipal frequencies of the principal components\\u2019. This\\nimplicit regularisation is seemingly at odds with the \\u2018spec-\\ntral bias\\u2019, which states that in neural networks trained with\\ngradient descent, lower frequencies are learned \\ufb01rst (Xu,\\n2018; Rahaman et al., 2019; Xu et al., 2019; Ronen et al.,\\n2019; Basri et al., 2020; Cao et al., 2021). We claim on\\nthe other hand that the bias is towards frequencies that are\\ndominant in the SVD decomposition of the dataset struc-\\nture, irrespective of whether these are actually high or low\\nfrequencies. One important thing to note is that broader\\ndistinctions in shape correspond to lower frequencies (see\\nFig. 2). In our experiments with more general datasets we\\n\\ufb01nd that broader distinctions are uncovered \\ufb01rst, and that\\ntherefore lower frequencies are learned \\ufb01rst. But the \\u2018spec-\\ntral bias\\u2019 is also speci\\ufb01cally derived for non-linear networks,\\nthus it is currently unclear how the two phenomena actually\\nrelate. Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nThe structure we consider is the statistical structure given\\nby the SVD; the hierarchy in this structure is discovered\\nthrough time. Deeper, non-linear CNN are known to pro-\\ncess higher-order visual relationships in later layers as well:\\ni.e., there is an additional notion of hierarchy through depth.\\nUncovering how our new insights relate to this, forms the\\ngoal of future work. However, our results already provide\\nsome understanding on why general CNN are sensitive to ad-\\nversial perturbations in the form of Fourier eigenfunctions\\n(Tsuzuku & Sato, 2019), and why they exhibit shortcut\\nlearning (Geirhos et al., 2020) through relying on textures\\n(images with a sparse, peaked frequency spectrum) instead\\nof shapes (Geirhos et al., 2019), and why they have the\\ntendency to learn surface statistical regularities as opposed\\nto higher level abstractions (Jo & Bengio, 2017). Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nReferences\\nAdvani, M. S., Saxe, A. M., and Sompolinsky, H. High-\\ndimensional dynamics of generalization error in neural\\nnetworks. Neural Networks , 132:428\\u2013446, 2020.\\nArora, S., Du, S., Hu, W., Li, Z., and Wang, R. Fine-grained\\nanalysis of optimization and generalization for overpa-\\nrameterized two-layer neural networks. In International\\nConference on Machine Learning , pp. 322\\u2013332. PMLR,\\n2019.\\nAtanasov, A., Bordelon, B., and Pehlevan, C. Neural net-\\nworks as kernel learners: The silent alignment effect. In\\nInternational Conference on Learning Representations ,\\n2022.\\nBach, F. Learning theory from \\ufb01rst principles,\\n2023. URL https:\\/\\/www.di.ens.fr\\/ \\u02dcfbach\\/\\nltfp_book.pdf .\\nBasri, R., Galun, M., Geifman, A., Jacobs, D., Kasten, Y .,\\nand Kritchman, S. Frequency bias in neural networks for\\ninput of non-uniform density. In International Conference\\non Machine Learning , pp. 685\\u2013694. PMLR, 2020.\\nBourlard, H. and Kamp, Y . Auto-association by multilayer\\nperceptrons and singular value decomposition. Biological\\ncybernetics , 59(4):291\\u2013294, 1988.\\nBraun, L., Domin \\u00b4e, C. C. J., Fitzgerald, J. E., and Saxe,\\nA. M. Exact learning dynamics of deep linear networks\\nwith prior knowledge. In Advances in Neural Information\\nProcessing Systems , 2022.\\nCao, Y ., Fang, Z., Wu, Y ., Zhou, D. X., and Gu, Q. Towards\\nunderstanding the spectral bias of deep learning. In Pro-\\nceedings of the Thirtieth International Joint Conference\\non Arti\\ufb01cial Intelligence , pp. 2205\\u20132211, 2021.\\nCasper, S., Hod, S., Filan, D., Wild, C., Critch, A., and\\nRussell, S. Graphical clusterability and local specializa-\\ntion in deep neural networks. In ICLR 2022 Workshop\\non PAIR2Struct: Privacy, Accountability, Interpretability,\\nRobustness, Reasoning on Structured Data , 2022.\\nDu, S. S., Zhai, X., P \\u00b4oczos, B., and Singh, A. Gradient\\ndescent provably optimizes over-parameterized neural\\nnetworks. In 7th International Conference on Learning\\nRepresentations , 2019.\\nFang, Y ., Cohen, M. A., and Kincaid, T. G. Dynamics of a\\nwinner-take-all neural network. Neural Networks , 9(7):\\n1141\\u20131154, 1996.\\nFukai, T. and Tanaka, S. A simple neural network exhibiting\\nselective activation of neuronal ensembles: from winner-\\ntake-all to winners-share-all. Neural computation , 9(1):\\n77\\u201397, 1997.Geirhos, R., Rubisch, P., Michaelis, C., Bethge, M., Wich-\\nmann, F. A., and Brendel, W. Imagenet-trained CNNs are\\nbiased towards texture; increasing shape bias improves\\naccuracy and robustness. In International Conference on\\nLearning Representations , 2019.\\nGeirhos, R., Jacobsen, J.-H., Michaelis, C., Zemel, R., Bren-\\ndel, W., Bethge, M., and Wichmann, F. A. Shortcut learn-\\ning in deep neural networks. Nature Machine Intelligence ,\\n2(11):665\\u2013673, 2020.\\nGidel, G., Bach, F., and Lacoste-Julien, S. Implicit regu-\\nlarization of discrete gradient dynamics in linear neural\\nnetworks. Advances in Neural Information Processing\\nSystems 32 , pp. 3196\\u20133206, 2019.\\nGoodfellow, I., Bengio, Y ., and Courville, A. Deep\\nLearning . MIT Press, 2016. http:\\/\\/www.\\ndeeplearningbook.org .\\nGray, R. M. Toeplitz and circulant matrices: A review. Foun-\\ndations and Trends \\u00aein Communications and Information\\nTheory , 2(3):155\\u2013239, 2006. ISSN 1567-2190.\\nGunasekar, S., Lee, J. D., Soudry, D., and Srebro, N. Im-\\nplicit bias of gradient descent on linear convolutional\\nnetworks. Advances in Neural Information Processing\\nSystems , 31, 2018.\\nGunning, R. and Rossi, H. Analytic Functions of Several\\nComplex Variables . Prentice-Hall, 1965.\\nHacohen, G. and Weinshall, D. Principal components bias in\\nover-parameterized linear models, and its manifestation\\nin deep neural networks. Journal of Machine Learning\\nResearch , 23(155):1\\u201346, 2022.\\nJain, A. K. Fundamentals of digital image processing .\\nPrentice-Hall, Inc., 1989.\\nJo, J. and Bengio, Y . Measuring the tendency of cnns\\nto learn surface statistical regularities. arXiv preprint\\narXiv:1711.11561 , 2017. Prentice-Hall, Inc., 1989.\\nJo, J. and Bengio, Y . Measuring the tendency of cnns\\nto learn surface statistical regularities. arXiv preprint\\narXiv:1711.11561 , 2017.\\nJolliffe, I. Principal Component Analysis . Springer New\\nYork, 2002.\\nLazzaro, J., Ryckebusch, S., Mahowald, M. A., and Mead,\\nC. A. Winner-take-all networks of o (n) complexity. Ad-\\nvances in neural information processing systems , 1, 1988.\\nPearson, K. Liii. on lines and planes of closest \\ufb01t to systems\\nof points in space. The London, Edinburgh, and Dublin\\nphilosophical magazine and journal of science , 2(11):\\n559\\u2013572, 1901.\\nPretorius, A., Kroon, S., and Kamper, H. Learning dynamics\\nof linear denoising autoencoders. In International Con-\\nference on Machine Learning , pp. 4141\\u20134150. PMLR,\\n2018. Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nRahaman, N., Baratin, A., Arpit, D., Draxler, F., Lin, M.,\\nHamprecht, F., Bengio, Y ., and Courville, A. On the spec-\\ntral bias of neural networks. In International Conference\\non Machine Learning , pp. 5301\\u20135310. PMLR, 2019.\\nRe\\ufb01netti, M. and Goldt, S. The dynamics of representation\\nlearning in shallow, non-linear autoencoders. In Inter-\\nnational Conference on Machine Learning , pp. 18499\\u2013\\n18519. PMLR, 2022.\\nRonen, B., Jacobs, D., Kasten, Y ., and Kritchman, S. The\\nconvergence rate of neural networks for learned functions\\nof different frequencies. Advances in Neural Information\\nProcessing Systems , 32, 2019.\\nSatpathi, S. and Srikant, R. The dynamics of gradient de-\\nscent for overparametrized neural networks. In Learning\\nfor Dynamics and Control , pp. 373\\u2013384. PMLR, 2021.\\nSaxe, A. M., McClelland, J. L., and Ganguli, S. Exact\\nsolutions to the nonlinear dynamics of learning in deep\\nlinear neural networks. In 2nd International Conference\\non Learning Representations , 2014.\\nSaxe, A. M., McClelland, J. L., and Ganguli, S. A mathe-\\nmatical theory of semantic development in deep neural\\nnetworks. Proceedings of the National Academy of Sci-\\nences , 116(23):11537\\u201311546, 2019.\\nSedghi, H., Gupta, V ., and Long, P. M. The singular values\\nof convolutional layers. In 7th International Conference\\non Learning Representations , 2019.\\nTsuzuku, Y . and Sato, I. On the structural sensitivity of deep\\nconvolutional networks to the directions of fourier basis\\nfunctions. In Proceedings of the IEEE\\/CVF Conference\\non Computer Vision and Pattern Recognition , pp. 51\\u201360,\\n2019.\\nV oss, C., Goh, G., Cammarata, N., Petrov, M.,\\nSchubert, L., and Olah, C. Branch specializa-\\ntion. Distill , 2021. doi: 10.23915\\/distill.00024.008.\\nhttps:\\/\\/distill.pub\\/2020\\/circuits\\/branch-specialization.\\nXu, Z. J. Understanding training and generalization\\nin deep learning by fourier analysis. arXiv preprint\\narXiv:1808.04295 , 2018.\\nXu, Z.-Q. J., Zhang, Y ., and Xiao, Y . Training behavior of\\ndeep neural network in frequency domain. In Interna-\\ntional Conference on Neural Information Processing , pp.\\n264\\u2013274. Springer, 2019. Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nA. Diagonality of \\u0006xxin the Basis Given by Vand Relationship to Class Coherence\\nWe here discuss the diagonality of \\u0006xxin the basis given by V, and its relationship to class coherence. We start from the\\nassumption that the label vectors are orthonormal, yT\\ncpycq=\\u000ecpcq, wherecpandcqare class indices. This is the case when\\nthe classes are one-hot encoded. Then, a property of the singular value decomposition of a general, real matrix Bis that the\\nright singular vectors of this matrix are the eigenvectors of the matrix BTB. We can make use of the orthonormality of the\\nlabel vectors and general properties of the kronecker product (denoted \\n) to derive:\\n\\u0006yxT\\u0006yx=hy\\nxTiThy\\nxTi (S.1)\\n=hx\\nyTihy\\nxTi (S.2)\\n=C2\\nN2m\\u00001X\\ncr=0m\\u00001X\\ncq=0(hxicr\\nyT\\ncr)(ycq\\nhxTicq) (S.3)\\n=C2\\nN2m\\u00001X\\ncr=0m\\u00001X\\ncq=0yT\\ncrycq(hxicr\\nhxTicq) (S.4)\\n=C2\\nN2m\\u00001X\\ncr=0hxicr\\nhxTicr (S.5)\\nwherehicrdenotes the average over all samples belonging to the same class cr,pis the number of classes, Nis the\\ntotal number of samples, and Cis the number of samples per class (we assume a balanced dataset). If we de\\ufb01ne\\n\\bxx=C2\\nN2Pm\\u00001\\ncr=0hxicr\\nhxicr, we thus have that VT\\bxxVis diagonal: the right singular vectors of \\u0006yxare the\\neigenvectors of \\bxx=\\u0006yxT\\u0006yx. The corresponding eigenvalues are equal to the square of the singular values, thus the\\ndiagonal values of STS.\\n\\u0006xx=hx\\nxTiand\\bxx(eq. S.5) are different in general, but they are close to equal if samples within a class are very\\nsimilar (at least in pixel-space):\\nx(i)=hxicr+\\u000f(i);k\\u000f(i)k<<khxicrk (S.6)\\nwherex(i)is the sample with index ibelonging to a class with index cr. We have:\\n\\u0006xx=hx\\nxTi (S.7)\\n=\\n(hxicr+\\u000f(i))\\n(hxicr+\\u000f(i))T\\u000b\\n(S.8)\\n=\\n(hxicr\\nhxiT\\ncr) + (hxicr\\n\\u000f(i)) +:::\\u000b\\n(S.9)\\nThus, ifx(i)\\u0019hxicr(eq. S.6), then:\\n\\u0006xx\\u0019\\n(hxicr\\nhxiT\\ncr)\\u000b\\n(S.10)\\n=C\\nNm\\u00001X\\ncp=0hxicr\\nhxiT\\ncr(S.11)\\n=N\\nC\\bxx(S.12)\\n=p\\bxx: (S.13)\\nIn other words, if samples within a class are very similar (=high class coherence), we can use the same eigen-\\/right singular\\nvectors to decompose \\u0006xxand\\u0006yxinto a sum of their respective modes. Thus the higher the class coherence, the more the\\ntwo matrices represent the same inherent structure. If \\u0006xxis exactly equal to p\\bxx, thepeigenvalues of \\u0006xxare given by\\nthepnon-zero diagonal elements of pSTS. Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nB. De\\ufb01nition of Q\\nWe \\ufb01rst recall the de\\ufb01nition of the one-dimensional discrete Fourier transform (DFT) matrix, denoted by the n\\u0002nmatrix\\nF(n):\\n(F(n))p;q= (!n)pq; !n= exp(\\u00002\\u0019i=n); (S.14)\\nwhere the superscript pqdenotes the product of indices pandqapplied as an exponent, and i2=\\u00001. Then then2\\u0002n2\\ncomplex-valued matrix Qis given by:\\nQ=1\\nnF(n)\\nF(n)(S.15)\\nwith\\nthe kronecker product. Qis unitary (Q\\u0003T=Q\\u00001) and symmetric.\\nC. Doubly Block Circulant Matrices\\nC.1. Doubly Block Circulant Matrix De\\ufb01nition\\ncirc(b): circulant matrix of the vector bwith lengthq:\\ncirc([b0;b1;\\u0001\\u0001\\u0001;bq\\u00002;bq\\u00001]T) =2\\n666664b0bq\\u00001\\u0001\\u0001\\u0001b2b1\\nb1b0bq\\u00001\\u0001\\u0001\\u0001b2\\nb2b1b0bq\\u00001\\u0001\\u0001\\u0001\\n......\\nbq\\u00001::: b 03\\n777775(S.16)\\nthendbc(B), the doubly block circulant matrix of the matrix B, is given by (Sedghi et al., 2019):\\ndbc(B) =2\\n6664circ(B0;:)circ(Bu\\u00001;:)::: circ (B1;:)\\ncirc(B1;:)circ(B0;:)::: circ (B2;:)\\n......\\ncirc(Bu\\u00001;:) ::: circ (B0;:)3\\n7775(S.17)\\nwhereuis the number of rows of B, andBi;:is theithrow. Note that this is the de\\ufb01nition for an actual convolution, see\\nbelow.\\nC.2. Convolution and correlation: notes on \\ufb02ipping the kernel or the image\\nGiven a kernel Kand an imageX, the convolution operation is de\\ufb01ned as:\\n(X\\u0003K)i;j=X\\nmX\\nlXm;lKi\\u0000m;j\\u0000l (S.18)\\nor, equivalently (since convolution is commutative):\\n(K\\u0003X)i;j=X\\nmX\\nlXi\\u0000m;j\\u0000lKm;l: (S.19)\\nThis implies the kernel (or the image) is applied in a \\u2018\\ufb02ipped\\u2019 manner, i.e., when we increase mandl, we decrease the\\ncorresponding indices. This amounts to \\ufb02ipping the kernel (or image) over the x- and y-axis, or equivalently, rotating it 180\\ndegrees, before applying the convolution.\\nIn practice, however, convolutional layers are usually implemented as correlations (here denoted ?):\\n(X?K)i;j=X\\nmX\\nlXm;lKi+m;j+l (S.20)\\nthus without \\ufb02ipping the kernel beforehand. This makes the operation more intuitive (but it\\u2019s not commutative). Whether the\\nkernel is \\ufb02ipped beforehand or not, the learning algorithm will learn the appropriate values of the kernel in the appropriate Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nplace: the result of learning with correlations instead of convolutions is just a \\ufb02ipped kernel (Goodfellow et al., 2016). Most\\nliterature therefore uses the term \\u201cconvolution\\u201d in lieu of \\u201ccorrelation\\u201d. The actual operation used in\\ufb02uences the de\\ufb01nition\\nof the doubly block circulant matrices and the gradient descent equations, however, and so it\\u2019s important to keep track of the\\ndetails when comparing different sources.\\nAs it turns out, we can de\\ufb01ne a very similar circulant and doubly block circulant matrix to replace a (circular) correlation\\n(see the de\\ufb01nitions used in (Sedghi et al., 2019)):\\ncirccorr([b0;b1;\\u0001\\u0001\\u0001;bq\\u00002;bq\\u00001]T) =2\\n666664b0b1\\u0001\\u0001\\u0001bq\\u00002bq\\u00001\\nbq\\u00001b0b1\\u0001\\u0001\\u0001bq\\u00002\\nbq\\u00002bq\\u00001b0b1\\u0001\\u0001\\u0001\\n......\\nb1::: b 03\\n777775(S.21)\\nand\\ndbccorr(B) =2\\n666664circ(B0;:)circ(B1;:)::: circ (Bu\\u00001;:)\\ncirc(Bu\\u00001;:)circ(B0;:)::: circ (Bu\\u00002;:)\\n......\\ncirc(B1;:) ::: circ (B0;:)\\n:3\\n777775(S.22)\\nWhen we compare those de\\ufb01nitions to the earlier de\\ufb01ned matrices for circular convolutions, we see that the difference lies in\\na permutation of the indices equal to a reversion and a shift, e.g., for a (column) vector b:\\nrev(bT) = [bq\\u00001;bq\\u00002;\\u0001\\u0001\\u0001;b0] (S.23)\\nand\\nshift!1(rev(bT)) = [b0;bq\\u00001;\\u0001\\u0001\\u0001;b1]; (S.24)\\nyielding\\ncirccorr(b) =circ(shift!1(rev(b))); (S.25)\\nand a similar permutation of the rows for the doubly block circulant matrices. Due to the properties of Fourier transforms,\\nwe can also \\ufb02ip the kernel through applying the Fourier transform twice:\\nQQk =shift!1(rev(kT)) (S.26)\\nThis allows us to translate all the derivations from implementations with convolutions to implementations with correlations:\\nwhere\\ndbc(K) =nQ\\u00001diag(Qk)Q=nQdiag(Q\\u00001k)Q\\u00001(S.27)\\n(since the dbc matrix is real valued, thus dbc(K) =dbc(K)\\u0003andQ\\u00001=Q\\u0003), we can use eq. S.26 to write:\\ndbccorr(K) =dbc(Kflip) =nQdiag(Q\\u00001(QQk))Q\\u00001(S.28)\\nthus\\ndbccorr(K) =dbc(Kflip) =nQdiag(Qk)Q\\u00001: (S.29)\\nD. Derivation of the Differential Equations of Gradient Descent\\nD.1. De\\ufb01nition of the network and learning algorithm\\nAs described in the main paper, we consider a convolutional network where the predictions ^yare given by:\\n^y=Wdbc (k)x (S.30) Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nherex=vec(X)is the vectorized input image, k=vec(K)is the vectorized kernel, and Wis the weight matrix of the\\nfully connected layer.\\nThe loss is given by the MSE loss, i.e.,\\nL=1\\n2pX\\nl=0(yl\\u0000^yl)2: (S.31)\\nThe kernel and the weights are updated according to the gradients of this loss:\\n\\u0001k=\\u0000\\u0015@L\\n@kT(S.32)\\n\\u0001W=\\u0000\\u0015@L\\n@WT(S.33)\\nwhere\\u0015is the learning rate.\\nD.2. Gradient descent for the convolutional layer\\nFor the convolutional layer, the gradient of the loss with respect to the kernel is in itself given by a convolution, but with a\\n\\ufb02ipped image (or, equivalently, an image rotated 180 degrees). We can write this equation using doubly block circulant\\nmatrices:\\n@L\\n@kT=vec\\u0000\\nXflip~@L\\n@HT\\u0001\\n(S.34)\\n=dbc(Xflip)@L\\n@hT; (S.35)\\nwith\\n@L\\n@hT=\\u0000pX\\nl=0(yl\\u0000^yl)(WT):;l: (S.36)\\nHereHdenotes the activity of the hidden layer, and h=vec(H).(WT):;ldenotes column lofWT.\\nWe will now rewrite this equation in a form that will turn out to be useful in the next steps:\\n@L\\n@kT=nmX\\nl=0Qdiag(Qx)Q\\u00001(yl\\u0000^yl)(WT):;l (S.37)\\n()\\u00001\\nnQ\\u00001@L\\n@kT\\u0001\\nj=pX\\nl=0diag(Qx)j;:(yl\\u0000^yl)(Q\\u00001WT):;l (S.38)\\n()\\u00001\\nnQ\\u00001@L\\n@kT\\u0001\\nj=pX\\nl=0(Qx)j(yl\\u0000^yl)(Q\\u00001WT)j;l (S.39)\\n()\\u00001\\nnQ\\u00001@L\\n@kT\\u0001\\nj=pX\\nl=0\\u0000\\n(y\\u0000^y)\\n(Qx)T\\u0001\\nl;i(Q\\u00001WT)j;l (S.40)\\n()1\\nn\\u0000\\nQ\\u00001@L\\n@kT\\u0001\\nj= (Q\\u00001WT)j;:\\u0000\\n(y\\u0000^y)\\nxT\\u0001\\nQT\\n:;j (S.41)\\nD.3. Gradient descent for the fully connected layer\\nFor the fully connected layer with weights W, we can write down the gradient in the conventional form:\\n\\u0000@L\\n@WT\\u0001\\nl;:= (yl\\u0000^yl)xTdbc(k)T(S.42) Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nand, for future use, rewrite this as:\\n\\u0000@L\\n@WT\\u0001\\nl;:= (yl\\u0000^yl)xTnQTdiag(Qk)TQ\\u0000T(S.43)\\n()1\\nn\\u0000@L\\n@WT\\u0001\\nl;:=\\u0000\\n(y\\u0000^y)\\nxT\\u0001\\nl;:Qdiag(Qk)Q\\u00001(S.44)\\nwhere we used the fact Q\\u00001=Q\\u0000TandQ=QT.\\nD.4. From discrete gradient descent to differential equations\\nThe equations S.41 and S.44, together with the update rules given by eq. S.32 and S.33 , tell us how to update the kernel and\\nthe weight matrix at each discrete time step \\u0001t. We will proceed to make a continuous approximation of these discrete\\nupdates. This is a valid approximation as long as the learning rate is small enough. In this slow learning regime, we can\\nassume the weights only change minimally over a number of updates with different samples. The factors\\u0000\\n(y\\u0000^y)\\nxT\\u0001\\ncan then be replaced by their average over those samples (see (Saxe et al., 2019)). In the main body, we de\\ufb01ned the matrices\\n\\u0006xx=hx\\nxTiand\\u0006yx=hy\\nxTi, which can be computed from the dataset alone. We also de\\ufb01ned \\u0006^yx=h^y\\nxTi.\\nThese de\\ufb01nition allow us to rewrite the difference eqs. S.41, S.44, S.32 and S.33, as:\\n1\\nn\\u0015\\u0000ddiag (Q\\u00001k)\\ndt\\u0001\\nj;j=Q\\u00001\\nj;:WT\\u0000\\n\\u0006yx\\u0000\\u0006^yx)Q:;j (S.45)\\n1\\nn\\u0015\\u0000dW\\ndt\\u0001\\n=\\u0000\\n\\u0006yx\\u0000\\u0006^yx)Qdiag(Qk)Q\\u00001(S.46)\\nwherediag(Q\\u00001k)is then2\\u0002n2diagonal matrix with the elements of Q\\u00001kplaced on the diagonal. The derivatives of\\nthe off-diagonal elements are zero (\\u0000ddiag (Q\\u00001k)\\ndt\\u0001\\ni;j= 0ifi6=j).\\nD.5. Interpreting the gradient descent equations in terms of the dataset structure\\nWe can now \\ufb01nally link the process of gradient descent with the structure present in the dataset. We can describe both the\\ninput-input as the input-output structure of the dataset with singular value decompositions. We can use these singular value\\ndecompositions to perform a change of variables. Starting from:\\n\\u0006yx=USVT(S.47)\\n\\u0006^yx=UAVT(S.48)\\n\\u0006xx=V\\u0006xxVT(S.49)\\nwe transformWtoWanddbc(K)todbc(K):\\nW=UWR\\u00001(S.50)\\n()W=UTWR (S.51)\\nwhereRis an arbitrary invertible matrix, and\\ndbc(K)=nQ\\u00001diag(Qk)Q=Rdbc(K)VT(S.52)\\n()dbc(K)=R\\u00001dbc(K)V=nR\\u00001Q\\u00001diag(Qk)QV (S.53)\\n()dbc(K)=nR\\u00001Qdiag(Q\\u00001k)Q\\u00001V (S.54)\\nwhere in the last step we used that dbc(K)is a real valued matrix, thus dbc(K)=dbc(K)\\u0003, and thatQand\\nQ\\u00001(=Q\\u0003T)are unitary and symmetric.\\nWe perform this change of variables such that we can describe the network, given by the product Wdbc (K), in terms of\\nthe singular vectors of the dataset given by UandV. With this change of variables, we indeed have (compare to eq.S.47) :\\nWdbc (K)=UWR\\u00001Rdbc(K)VT=UWdbc(K)VT: (S.55) Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nThe matricesWanddbc(K)are thus de\\ufb01ned up to an invertible matrix; but their product remains the same. If the p\\u0002n2\\nproductWdbc(K)would be rectangular diagonal with preal, positive entries, eq. would represent a singular value\\ndecomposition. Moreover, this singular value decomposition would be the same as that of the input-output relationship in\\nthe dataset (eq.S.47), only with different singular values.\\nWe can also use this change of variables to describe the complete input-predicted output relation \\u0006^yx(eq. 6):\\n\\u0006^yx=Wdbc (K)\\u0006xx(S.56)\\n=UWdbc(K)\\u0006xxVT(S.57)\\n=UAVT(S.58)\\nwithA:\\nA=Wdbc(K)\\u0006xx=nWR\\u00001Qdiag(Q\\u00001k)Q\\u00001V\\u0006xx (S.59)\\nwhere again, if Awould be rectangular diagonal with preal, positive entries, this would be an SVD with the same singular\\nvectors as the input-(ground truth) output relationship of the dataset \\u0006yx. In that case, we will call the matrix Athe matrix\\nof effective singular values. Note that in practice, we can compute Airrespective of the internal details of the network: we\\nonly need predictions ^yfor samplesxto computeA=UT\\u0006^yxV. The evolution of Aduring training thus essentially\\ncoincides with the evolution of the predictions of the network.\\nWe are now ready to rewrite our system of differential equations of gradient descent using the new variables:\\n1\\nn\\u0015\\u0000ddiag (Q\\u00001k)\\ndt\\u0001\\nj;j=Q\\u00001\\nj;:WT\\u0000\\n\\u0006yx\\u0000\\u0006^yx)QT\\n:;j (S.60)\\n()1\\nn\\u0015\\u0000ddiag (Q\\u00001k)\\ndt\\u0001\\nj;j=Q\\u00001\\nj;:R\\u0000\\u0003TW\\u0003TUTU\\u0000\\nS\\u0000A)VTQT\\n:;j (S.61)\\n1\\nn\\u0015UT\\u0000dW\\ndt\\u0001\\nR=UT\\u0000\\n\\u0006yx\\u0000\\u0006^yx)QTdiag(Qk)Q\\u00001R (S.62)\\n()1\\nn\\u0015UT\\u0000dW\\ndt\\u0001\\nR=UTU\\u0000\\nS\\u0000A)VTQTdiag(Qk)Q\\u00001R (S.63)\\nWe thus arrive at the coupled system of equations:\\n1\\nn\\u0015\\u0000ddiag (Q\\u00001k)\\ndt\\u0001\\nj;j=Q\\u00001\\nj;:R\\u0000\\u0003TW\\u0003T\\u0000\\nS\\u0000A)(QV)T\\n:;j (S.64)\\n1\\nn\\u0015\\u0000dW\\ndt\\u0001\\n=\\u0000\\nS\\u0000A)(QV)Tdiag(Qk)Q\\u00001R (S.65)\\nWhich are the equations mentioned in the main paper, Eqs. 17 and 18.\\nE. Sums of Cosines Datasets\\nIn the theoretical derivations, we will often use a \\u2019sums of cosines dataset\\u2019. This dataset consists of a sum of pure 2D cosines,\\nwhere the frequencies of those cosines are not shared between classes:\\nX(c)\\nl;m=X\\nj2\\u001b(c)bjcos\\u0012\\n2\\u0019\\u0016l\\nn+ 2\\u0019\\u0017m\\nn+\\u000ej\\u0013\\n; (S.66)\\nwhere\\u001b(c)is a set, belonging to class c, of vec-2D indices jthat map to pairs of horizontal ( \\u0016) and vertical ( \\u0017) frequency\\nindices.bjis the corresponding amplitude, and \\u000ejis the corresponding phase. We here thus consider the case where the sets\\n\\u001b(c)are disjoint. Note that this set only differs from a general image dataset on this point: all images can be decomposed as Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nsums of cosines, but they could possibly contain the same frequencies. However, in the special case where frequencies are\\nnot shared (between modes, see next) we can derive analytical results.\\nIn this case the right singular vectors \\u001e(\\u000b)are given by:\\n\\u001e(\\u000b)\\ni=1qP\\nj2\\u001b(\\u000b)b2\\njnp\\n2X\\nj2\\u001b(\\u000b)bjcos\\u0012\\n2\\u0019\\u0016l\\nn+ 2\\u0019\\u0017m\\nn+ (\\u000e\\u001e)j\\u0013\\n; (S.67)\\ni.e., again the class vectors normalised and sorted according to singular value: the mode indices \\u000bform a permutation\\nof the class indices cbased on the singular values. Here \\u0016=div(j;n); \\u0017=mod(j;n)are the frequency indices and\\nl=div(i;n); m=mod(i;n)are the pixel indices. There is a subtlety regarding the phase (\\u000e\\u001e)j: the SVD is de\\ufb01ned up to\\na minus sign, i.e., if both the left and right singular vector obtain a minus sign, the total mode remains unaltered. Therefore\\nthe phase (\\u000e\\u001e)jcould be the same as the original phase (\\u000e\\u001e)j=\\u000ej, or a possible minus sign could be absorbed in the phase\\nas(\\u000e\\u001e)j=\\u000ej+\\u0019. We can furthermore note that the sets \\u001b\\u000b, where each set corresponds to a set \\u001b(c), are also disjoint.\\nFormally, this means that given an index j,j(Q\\u001e\\u000b)jjis non-zero for only one mode \\u000b. The latter property will be crucial in\\nthe derivations below. We will also still assume that \\u0006xx=V\\u0006xxVTwith\\u0006xxdiagonal.\\nF. Minimal Norm Weights\\nIt is well-known that, when gradient descent is used to train a linear neural network from near-zero initial conditions in\\nthe over-parametrized regime, the \\ufb01nal network parameters will have minimal norm (for a derivation with MSE loss, see\\ne.g., (Bach, 2023), chapter 11). I.e., among all the possible combinations of network weights that implement the desired\\ninput-output map, gradient descent leads to the network weights that have the smallest norm. This is called the implicit\\nregularization of gradient descent. The minimal norm solution for the two-layer linear CNN with a single kernel is given by\\nthe following constrained minimization problem:\\nminW;dbc(K)\\u0010\\njjWjj2\\nF+jjdbc(K)jj2\\nF\\u0011\\n(S.68)\\nsubject to:\\nWdbc (K)\\u0006xx=USVT(S.69)\\nwherejjBjjFdenotes the Frobenius norm of the l\\u0002mmatrixB, in essencejjBjjF=qPl\\niPm\\njjBijj.\\nF.1. Exact Solutions for Sum of Cosines Datasets\\nWe here derive what those \\ufb01nal solutions look for a sums of cosines dataset. We \\ufb01rst de\\ufb01ne three new n2\\u00021vectors,\\u000e\\u001e,\\nand\\u000ekand\\u000ew. These vectors contain the phases (or angles) associated to the complex-valued vec-2D DFT of the singular\\nvalues, of the kernel and of the weights, respectively. The de\\ufb01nition of \\u000ekis straightforward:\\n\\u000ek=angle (Qk): (S.70)\\nThe de\\ufb01nition of \\u000e\\u001eand\\u000ewis more subtle. The p\\u0002n2matrixWhasprows, and there are also psingular vectors \\u001e\\u000b\\n(\\u000b2f0;\\u0001\\u0001\\u0001;p\\u00001g) of dimension n2\\u00021. For every index j, there are thus pcomplex numbers, e.g., (Q\\u001e0)j,(Q\\u001e1)j,\\netc. However, when we use a dataset with disjoint sets of frequencies, we have by design a non-zero value (Q\\u001e\\u000b)jfor only\\none single mode index \\u000bper frequency index j. We select the phase of this value to be the jthelement of the vector \\u000e\\u001e. We\\nthus have:\\n(\\n(\\u000e\\u001e)j=angle (Q\\u001e\\u000b)j; j!\\u000b;\\n(\\u000e\\u001e)j= 0; freq.jnot present in singular vectors.(S.71)\\nwhere again j!\\u000bdenotes that the frequency jis used in the sum of cosines making up the singular vector \\u001e\\u000b. We de\\ufb01ne\\n\\u000ewin a similar way:\\n(\\n(\\u000ew)j=angle ((QWT)j;\\u000b); j!\\u000b;\\n(\\u000ew)j= 0; freq.jnot present in singular vectors.(S.72) Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nNote that because the kernel, the weights and the singular vectors are all real-valued, we have (\\u000e\\u001e)jsymm =\\u0000(\\u000e\\u001e)j,\\n(\\u000ek)jsymm =\\u0000(\\u000ek)j, and (\\u000ew)jsymm =\\u0000(\\u000ew)j.\\nWe are now ready to state what the \\ufb01nal, minimal norm solutions look like. A kernel Kand the weight matrix Wcorrespond\\nto a minimal norm solution, if and only if:\\nS\\u000b;\\u000b=nX\\nj2\\u001b\\u000bsymmj(Q\\u001e\\u000b)jjj(Qk)jj2\\u0006\\u000b;\\u000b; (S.73)\\nwith\\u001b\\u000b\\nsymm is the original set of indices jin\\u001b\\u000btogether with all their corresponding symmetric indices jsymm , and\\nW=U\\n\\u0002wQ; (S.74)\\nwhere \\nis ap\\u0002n2matrix de\\ufb01ned by:\\n(\\n\\n\\u000b;j=j(Qk)jj;\\n\\u000b;jsymm =j(Qk)jjj!\\u000b;\\n\\n\\u000b;j= 0 else;(S.75)\\nand\\n\\u0002w=diag(ei\\u000ew): (S.76)\\nWith, for all j2\\u001b\\u000b\\nsymm :\\n(\\u000ek)j+ (\\u000ew)j=\\u0000(\\u000e\\u001e)j: (S.77)\\nFrom these minimal norm solutions, we can then also conclude (cfr. Eq. 14):\\nR=Q\\u00001; (S.78)\\nW=\\n\\u0002w: (S.79)\\nNote that the only constraint on the Fourier spectrum of the kernel is given by Eq. S.73; the kernel is not uniquely de\\ufb01ned.\\nWe show in the main paper that the \\ufb01nal coef\\ufb01cients j(Qk)jjare in\\ufb02uenced or determined through a winner-takes-all process\\nthat takes place during training with gradient descent. This effect can only be derived from the gradient equations themselves,\\nhowever.\\nProof overview:\\nWe start fromdbc(K)=Q\\u00001diag(Qk)Q, withk=vec(K)an arbitrary, real-valued kernel, and from W=UP where\\nPis an arbitrary n\\u0002nreal-valued matrix, such that Wis an arbitrary real-valued weight matrix. We then use the method of\\nLagrange multipliers on the constrained optimization problem given by Eq. S.68 and Eq. S.69 to \\ufb01nd a system of equations\\ninvolvingk,P, and the given matrices U,V,S,Qand\\u0006xx. This system of equations captures the relationships between\\nthese matrices when dbc(K)andWform a minimal norm solution. When frequencies are not shared between modes, i.e.,\\nwhen for every frequency index jonly one value Qj;:V:;\\u000bis signi\\ufb01cant, the solutions to these equations are as described\\nabove.\\nF.2. Detailed Proof\\nLetdbc(K)=Q\\u00001diag(Qk)Q, withk=vec(K)an arbitrary, real-valued kernel, and let W=UP whereP\\nis an arbitrary n\\u0002nreal-valued matrix, such that Wis an arbitrary real-valued weight matrix. We \\ufb01rst reformulate\\nthe constrained minimization problem Eq. S.68 using properties of the Frobenius norm. The Frobenius norm can be\\nexpressed in function of singular values: jjBjjF=qPmin(l;m)\\ni\\u001bi(B)2where\\u001bi(B)are the singular values of B. We can\\nmake use of the latter property of the Frobenius norm to explicitly include the constraint on the network structure, given\\nby the eigendecomposition dbc(K)=nQ\\u00001diag(Qk)Q. The singular values of dbc(K)are the square roots of the\\neigenvalues of dbc(K)Tdbc(K)=n2Q\\u00001diag(Q\\u0003k)diag(Qk)Q. The singular values of dbc(K)are thus given by\\nnj(Qk)jj. The Frobenius norm can also be expressed as a trace, jjBjjF=q\\ntr(B\\u0003TB). ThereforejjWjj2\\nFcan be expressed as Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\ntr\\u0000\\nPTUTUP\\u0001\\n=tr\\u0000\\nPTP\\u0001\\n, given thatPis real andUis real and orthogonal.\\nWe thus reformulate the constrained minimization problem (Eq. S.68 and S.69) as:\\nminP;Qk\\u0010\\ntr\\u0000\\nP\\u0003TP\\u0001\\n+n2X\\nij(Qk)ij2\\u0011\\n; (S.80)\\nsubject to\\nnPQ\\u00001diag(Qk)QV\\u0006xx=S: (S.81)\\nWe can make use of Lagrange multipliers to solve the constrained optimization problem given by Eqs. S.80 and S.81. We\\ndenote the multipliers by the p\\u0002n2matrix \\u0003, and the Lagrangian Lis then given by:\\nL=tr\\u0010\\nPTP\\u0011\\n+n2X\\njj(Qk)jj2+tr\\u0010\\n\\u0003T\\u0000\\nnPQ\\u00001diag(Qk)QV\\u0006xx\\u0000S\\u0001\\u0011\\n(S.82)\\nTo \\ufb01nd the minimal norm solution, we compute the derivatives with respect to the matrix P, the values (Qk)jand the\\nmatrix \\u0003. These derivatives are zero at the minimum.\\ndL\\ndP:\\nTo compute the derivative with respect to the matrix P, we make use of the cyclic property of the trace function, and the\\nfact thatdtr(BC)\\ndB=CT. We obtain:\\ndL\\ndP=d\\ndP\\u0010\\ntr\\u0000\\nPPT\\u0001\\n+tr\\u0000\\nPQ\\u00001diag(Qk)QV\\u0006xx\\u0003T\\u0001\\u0011\\n(S.83)\\n=P+\\u0003\\u0006xxTVTQTdiag(Qk)Q\\u0000T(S.84)\\ndL\\nd(Qk)j:\\nTo calculate this derivative, we will make use of the following properties:\\n\\u0000\\ndiag(Qk)Q\\u0001\\nl;m=X\\nidiag(Qk)l;iQi;m=diag(Qk)l;lQl;m (S.85)\\nyielding\\n(Q\\u00001diag(Qk)Q)i;m=X\\nl(Q\\u00001\\ni;l\\u0000\\ndiag(Qk)Q\\u0001\\nl;m=X\\nlQ\\u00001\\ni;ldiag(Qk)l;lQl;m (S.86)\\nand, for any matrix BandC,\\ntr(BC) =X\\niBi;:C:;i (S.87)\\nsince the trace operation is the sum of diagonal elements. Moreover, we can compute the derivative of a function fwith\\nrespect to a complex number z=zR+izIasdf\\ndz=1\\n2(df\\ndzR\\u0000idf\\ndzI)(see, e.g., (Gunning & Rossi, 1965)). The derivative of Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nthe third term is then given by:\\nd\\nddiag (Qk)j;j\\u0010\\ntr\\u0000\\nPQ\\u00001diag(Qk)QV\\u0006xx\\u0003T\\u0001\\u0011\\n(S.88)\\n=d\\nddiag (Qk)j;j\\u0010\\ntr\\u0000\\nPX\\nlQ\\u00001\\n:;ldiag(Qk)l;lQl;:V\\u0006xx\\u0003T\\u0001\\u0011\\n(S.89)\\n=d\\nddiag (Qk)j;j\\u0010X\\nmPm;:X\\nlQ\\u00001\\n:;ldiag(Qk)l;lQl;:V\\u0006xx\\u0003T\\n:;m\\u0011\\n(S.90)\\n=1\\n2(X\\nmPm;:Q\\u00001\\n:;jQj;:V\\u0003T\\n:;m\\u0000i2X\\nmPm;:Q\\u00001\\n:;jQj;:V\\u0003T\\n:;m) (S.91)\\n=X\\nmPm;:Q\\u00001\\n:;jQj;:V\\u0006xx\\u0003T\\n:;m (S.92)\\n=tr(PQ\\u00001\\n:;jQj;:V\\u0006xx\\u0003T) (S.93)\\nThis yields, for the total derivativedL\\nd(Qk)j:\\ndL\\nd(Qk)j=1\\n2\\u0000\\n2(Qk)Rj\\u00002i(Qk)Ij\\u0001\\n+tr(PQ\\u00001\\n:;jQj;:V\\u0006xx\\u0003T) (S.94)\\n= (Qk)\\u0003\\nj+tr(PQ\\u00001\\n:;jQj;:V\\u0006xx\\u0003T) (S.95)\\ndL\\nd\\u0003:\\nFinally, we computedL\\nd\\u0003:\\ndL\\nd\\u0003=1\\n2\\u0000dL\\nd\\u0003R\\u0000idL\\nd\\u0003I\\u0001\\n(S.96)\\n=1\\n2\\u0010\\u0000\\nnPQ\\u00001diag(Qk)QV\\u0006xx\\u0000S\\u0001\\n\\u0000i2\\u0000\\nnPQ\\u00001diag(Qk)QV\\u0006xx\\u0000S\\u0001\\u0011\\n(S.97)\\n=nPQ\\u00001diag(Qk)QV\\u0006xx\\u0000S (S.98)\\nThe complete system of equations is then given by:\\n8\\n><\\n>:dL\\ndP= 0\\ndL\\nd(Qk)j= 0\\ndL\\nd\\u0003= 0()8\\n><\\n>:P+\\u0003\\u0006xxTVTQTdiag(Qk)Q\\u0000T= 0\\n(Qk)\\u0003\\nj+tr(PQ\\u00001\\n:;jQj;:V\\u0006xx\\u0003T) = 0\\nnPQ\\u00001diag(Qk)QV\\u0006xx\\u0000S= 0(S.99)\\nFrom the \\ufb01rst equation of eqs. S.99 we obtain:\\nP=\\u0000\\u0003\\u0006xxTVTQTdiag(Qk)Q\\u0000T(S.100)\\n()P=\\u0000\\u0003\\u0006xxTVTQ\\u0000Tdiag(Qk)\\u0003QT(S.101)\\nwhere we used that dbc(K)is real-valued. We can plug this in the second equation of the system of eqs. S.99 :\\n(Qk)\\u0003\\nj+tr(PQ\\u00001\\n:;jQj;:V\\u0006xx\\u0003T) = 0 (S.102)\\n() (Qk)\\u0003\\nj+tr(\\u0000\\u0003\\u0006xxTVTQ\\u0000Tdiag(Qk)\\u0003QTQ\\u00001\\n:;jQj;:V\\u0006xx\\u0003T) = 0 (S.103)\\n() (Qk)\\u0003\\nj+tr(\\u0000\\u0003\\u0006xxTVTQ\\u0000T\\n:;jdiag(Qk)\\u0003\\nj;jQj;:V\\u0006xx\\u0003T) = 0 (S.104) Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nIf the frequency indexed by jis not present in any mode, i.e., (QV)j;\\u000b= 0for all mode indices \\u000b, then we obtain:\\n(Qk)\\u0003\\nj= (Qk)j= 0 (S.105)\\nIf the frequency indexed by jis present in mode \\u000b, then (QV)j;\\u000b6= 0for a single mode \\u000b(by assumption). We then obtain:\\n(Qk)\\u0003\\nj+tr(\\u0000\\u0003\\u0006xxT\\n:;\\u000b(VTQ\\u0000T)\\u000b;jdiag(Qk)\\u0003\\nj;j(QV)j;\\u000b\\u0006xx\\u000b;:\\u0003T) = 0 (S.106)\\nand because \\u0006xxis diagonal (by assumption):\\n(Qk)\\u0003\\nj+tr(\\u0000\\u0003:;\\u000b\\u0006xxT\\n\\u000b;\\u000b(VTQ\\u0000T)\\u000b;jdiag(Qk)\\u0003\\nj;j(QV)j;\\u000b\\u0006xx\\u000b;\\u000b\\u0003T\\n\\u000b;:) = 0 (S.107)\\n() (Qk)\\u0003\\nj+tr(\\u0000\\u0003:;\\u000b\\u0006xxT\\n\\u000b;\\u000b(QV)\\u0003T\\n\\u000b;jdiag(Qk)\\u0003\\nj;j(QV)j;\\u000b\\u0006xx\\u000b;\\u000b\\u0003T\\n\\u000b;:) = 0 (S.108)\\n() (Qk)\\u0003\\nj+tr(\\u0000\\u0003:;\\u000b\\u0006xxT\\n\\u000b;\\u000bdiag(Qk)\\u0003\\nj;jj(QV)j;\\u000bj2\\u0006xx\\u000b;\\u000b\\u0003T\\n\\u000b;:) = 0 (S.109)\\n() (Qk)\\u0003\\nj\\u0000(Qk)\\u0003\\nj\\u0006xx2\\n\\u000b;\\u000bj(QV)j;\\u000bj2tr(\\u0003:;\\u000b\\u0003T\\n\\u000b;:) = 0 (S.110)\\n()\\u0003T\\n\\u000b;:\\u0003:;\\u000b=\\u0006xx\\u00002\\n\\u000b;\\u000bj(QV)j;\\u000bj\\u00002(S.111)\\nThis yields:\\n\\u0003:;\\u000b=~R:;\\u000b\\u0006xx\\u00001\\n\\u000b;\\u000bj(QV)j;\\u000bj\\u00001(S.112)\\nwith~Rap\\u0002porthogonal matrix.\\nWe now propose to decompose PasP=\\n\\u0002wQ. We can use the obtained equations to determine the shape of \\nand\\u0002w\\ncorresponding to a minimal norm solution. From the \\ufb01rst equation:\\nP=\\u0000\\u0003\\u0006xxTVTQ\\u0000Tdiag(Qk)\\u0003QT(S.113)\\n()\\n\\u0002w=\\u0000\\u0003\\u0006xxTVTQ\\u0000Tdiag(Qk)\\u0003(S.114)\\n() (\\n\\u0002w):;j=\\u0000\\u0003:;\\u000b\\u0006xxT\\n\\u000b;\\u000b(QV)\\u0003T\\n\\u000b;jdiag(Qk)\\u0003\\nj;j (S.115)\\n() (\\n\\u0002w):;j=\\u0000~R:;\\u000b\\u0006xx\\u00001\\n\\u000b;\\u000bj(QV)j;\\u000bj\\u00001\\u0006xxT\\n\\u000b;\\u000b(QV)\\u0003T\\n\\u000b;jdiag(Qk)\\u0003\\nj;j (S.116)\\n() (\\n\\u0002w):;j=\\u0000~R:;\\u000bj(QV)j;\\u000bj\\u00001e\\u0000i(\\u000e\\u001e)jj(QV)\\u000b;jje\\u0000i(\\u000ek)jj(Qk)jj (S.117)\\n()\\n:;j(\\u0002w)j;j=\\u0000~R:;\\u000be\\u0000i(\\u000e\\u001e)je\\u0000i(\\u000ek)jj(Qk)jj (S.118)\\nThus we obtain, for the real-valued matrix \\n:\\n\\n:;j=\\u0000~R:;\\u000bj(Qk)jj (S.119)\\nand for the diagonal matrix with associated phases:\\n(\\u0002w)j;j=ei(\\u000ew)j=ei\\u0000\\n\\u0000(\\u000e\\u001e)j\\u0000(\\u000ek)j\\u0001\\n; (S.120)\\nor also\\n(\\u000ew)j=\\u0000(\\u000e\\u001e)j\\u0000(\\u000ek)j: (S.121)\\nFinally, we can use the third equation of the system of eqs. S.99, involving the p\\u0002n2rectangular diagonal matrix Swith Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nreal, positive values on the diagonal of S0:p;0:p, and zero elsewhere. Starting with the elements of S0:p;0:p:\\nS\\f;\\u000b=n\\n\\f;:\\u0002wdiag(Qk)QV\\u0006xx:;\\u000b (S.122)\\n=X\\njn\\n\\f;j(\\u0002w)j;jdiag(Qk)j;j(QV)j;\\u000b\\u0006xx\\u000b;\\u000b (S.123)\\n=X\\njn\\n\\f;j(\\u0002w)j;j(\\u0002k)j;j(\\u0002\\u001e)j;jjdiag(Qk)j;jjj(QV)j;\\u000bj\\u0006xx\\u000b;\\u000b (S.124)\\n=X\\njn\\n\\f;jjdiag(Qk)j;jjj(QV)j;\\u000bj\\u0006xx\\u000b;\\u000b (S.125)\\n=X\\nj\\u0000n~R\\f;\\u000bj(Qk)jj2j(QV)j;\\u000bj\\u0006xx\\u000b;\\u000b (S.126)\\nIf\\f6=\\u000b, thenS\\f;\\u000b= 0. Therefore ~R\\f;\\u000b= 0if\\f6=\\u000b. Given that ~Ris orthogonal, and S\\u000b;\\u000bis positive and real, we have\\n~R\\u000b;\\u000b=\\u00001. The other elements of S,S\\u000b;iwithi=p, are indeed zero, since \\u0006xx\\u000b;i= 0. This completes the proof.\\nFor completeness, we also show that Pis a real-valued matrix:\\nPl;m= (\\n\\u0002wQ)l;m (S.127)\\n=X\\nj\\nl;jdiag(ei\\u000ew)j;jQj;m (S.128)\\n=X\\nj2\\u001b\\u000b\\u0010\\n\\nl;jdiag(ei\\u000ew)j;jQj;m+\\nl;jsymmdiag(ei\\u000ew)jsymm;jsymmQjsymm;m\\u0011\\n(S.129)\\nGiven the vec-2D indices and their corresponding pair of horizontal and vertical indices, j!(\\u0016;\\u0017),m!(a;b), we \\ufb01nd:\\nnQj;m= (F\\nF)(\\u0016\\u00001)n+\\u0017;(a\\u00001)n+b (S.130)\\n=F\\u0016;aF\\u0017;b (S.131)\\n=e\\u00002\\u0019i\\nn(\\u0016a+\\u0017b)(S.132)\\nand forjsymm!(n\\u0000\\u0016;n\\u0000\\u0017):\\nnQjsymm;m=Fn\\u0000\\u0016;aFn\\u0000\\u0017;b (S.133)\\n=e\\u00002\\u0019i\\nn((n\\u0000\\u0016)a+(n\\u0000\\u0017)b)(S.134)\\n=e\\u00002\\u0019in\\nnae\\u00002\\u0019in\\nnbe+2\\u0019i\\nn(\\u0016a+\\u0017b)(S.135)\\n=e+2\\u0019i\\nn(\\u0016a+\\u0017b)(S.136)\\n=nQ\\u0003\\nj;m (S.137)\\nThe vector\\u000ewhas the property that (\\u000ew)j=\\u0000(\\u000ew)jsymm . This yields diag(ei\\u000ew)jsymm;jsymm =diag(ei\\u000ew)\\u0003\\nj;j. Since \\nis real-valued, each term in the summation in Eq.S.129 is real, and therefore Pis real.\\nG. Silent Alignment, Balanced Weights and WTA Dynamics\\nIn the previous section, we derived the shape of the \\ufb01nal weights and kernel for datasets consisting of sums of cosines. These\\nwere the minimal norm solutions\\u2014the kind of network parameters the linear CNN converges to when starting from small,\\nrandom initial conditions. In the next section, we derive analytic solutions to the evolution of the predictions of the network,\\nfor speci\\ufb01c datasets and assuming aligned andbalanced initial conditions. With aligned initial conditions, we mean initial\\nconditions that render the matrix Arectangular diagonal, with positive real, entries from the start. The alignment can be Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\ninterpreted as follows: if Ais such a rectangular diagonal matrix, the network parameters are aligned with the statistical\\ndataset structure, in the sense that \\u0006^yxand\\u0006yxhave the same singular vectors. With balanced we mean that the (Fourier\\ntransformed) weight matrix consists of (Fourier transformed) elements of the kernel; implying some balancedness between\\nthe two types of layers. We borrowed this term from the theory of fully connected networks (Saxe et al., 2019), but the\\nshapes of the matrix involved for CNN are more complicated, which makes the concept more subtle.\\nThe assumption of aligned and balanced initial conditions turns out to be a good approximation to small, random initial\\nconditions and a sums of cosines dataset: when starting from small, random initial conditions, we see that the network\\nparameters quickly become aligned and balanced. This also happens in a stage-like fashion, and the alignment associated to\\na mode occurs before that mode is learned and the loss appreciably decreases. Therefore the dynamics of learning can be\\napproximately described by assuming aligned and balanced conditions from the start. This phenomenon has been rigorously\\nanalysed for the case of fully connected networks (Atanasov et al., 2022; Braun et al., 2022). It has also been shown that\\nthe error of this approximation decreases with initialization scale, e.g., it decreases as the standard deviation \\u001bused in a\\ngaussian initializer decreases.\\nFigure S.1. Evolution of the matrix A0:2;0:2during training: diagonal elements (left) and off-diagonal elements (right). Solid lines:\\naverage over trials. Shaded regions: standard deviation over trials. Horizontal, dashed lines: singular values s\\u000b.\\nFigure S.2. Plot of the evolution of the different fractionsjjW\\u000b;jj\\u0000\\n\\u000b;jj\\njW\\u000b;jj, a measure of how quickly the network parameters become\\nbalanced.\\nIn Fig. S.1, we have plotted, for the sums of cosines dataset, both the evolution of the two diagonal and the two off-diagonal\\nelements ofA0:2;0:2, averaged over 10 trials using small, random, initial conditions ( \\u001b= 0:00001 ). As discussed in the\\nmain paper, the diagonal values of Afollow sigmoidal, stage-like trajectories: after some time, they suddenly appear to\\ngrow very fast, eventually saturating on their corresponding singular value. I.e., A0;0saturates at the value s0andA1;1\\nsaturates at the value s1. The off-diagonal elements, on the other hand, remain relatively small (compare the y-axis between\\nthe left and right plot) and exhibit transient peaks, before becoming \\u00190.\\nFig.S.2 shows how quickly the magnitudes of the elements W\\u000b;jbecome equal to \\n\\u000b;j=j(Qk)jj, i.e., how quickly the Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nsolution becomes balanced. We plotted the evolution of the fractionjjW\\u000b;jj\\u0000\\n\\u000b;jj\\njW\\u000b;jjduring training, for all the different\\nfrequencies (indexed by j) involved in the two modes \\u000b= 0and\\u000b= 1of the sums of cosines dataset. Wis computed\\nfromWusingR=Q\\u00001. We see that the balancedness also happens in a stage-like or cascaded fashion: very quickly, the\\nelements corresponding to the \\ufb01rst mode become balanced; in a next phase and after the \\ufb01rst mode has been learned, the\\nelements corresponding to the second mode also become balanced.\\nFor the sums of cosines dataset, we thus quickly arrive at the following shape for the fully connected weight matrix W:\\nW(t)=U\\n(t)\\u0002w(t)Q (S.138)\\nwhere \\n(t)is ap\\u0002n2matrix de\\ufb01ned by:\\n(\\n\\n\\u000b;j=j(Qk(t))jj;\\n\\u000b;jsymm =j(Qk(t))jjj!\\u000b\\n\\n\\u000b;j= 0 else(S.139)\\nand\\n\\u0002w(t) =diag(ei\\u000ew(t)) (S.140)\\nWith, for all j2\\u001b\\u000b\\nsymm :\\n(\\u000ek)j(t) + (\\u000ew)j(t) =\\u0000(\\u000e\\u001e)j(t) (S.141)\\n.\\nNaturally, the product Wdbc(K)\\u0006xx=Athen becomes:\\n8\\n<\\n:A\\u000b;\\u000b(t) =nP\\nj2\\u001b\\u000bsymmj(Q\\u001e\\u000b)jjj(Qk(t))jj2\\u0006\\u000b;\\u000b\\u000b2f0;\\u0001\\u0001\\u0001;p\\u00001g\\nA\\u000b;\\f(t) = 0 else(S.142)\\nAnd we also have (cfr. Eq. 14):\\nR=Q\\u00001(S.143)\\nW(t) =\\n(t)\\u0002w(t) (S.144)\\nW=UW(t)Q (S.145)\\nas soon as the parameters become aligned and balanced. Given that this happens fast, the dynamics can be approximated by\\nassuming this shape for Wfrom the start. Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nWe can plug this in the differential equation given by Eq. 17 to obtain:\\n1\\nn\\u0015\\u0000ddiag (Q\\u00001k)\\ndt\\u0001\\nj;j=Q\\u00001\\nj;:R\\u0000\\u0003TW\\u0003T\\u0000\\nS\\u0000A)(QV)T\\n:;j (S.146)\\n()1\\nn\\u0015\\u0000ddiag (Qk)\\ndt\\u0001\\nj;j=Qj;:R\\u0000\\u0003TW\\u0003T\\u0000\\nS\\u0000A)(Q\\u00001V)T\\n:;j (S.147)\\n()1\\nn\\u0015\\u0000ddiag (Qk)\\ndt\\u0001\\nj;j=Qj;:Q\\u00001(\\n\\u0002w)\\u0003T\\u0000\\nS\\u0000A)(Q\\u00001V)T\\n:;j (S.148)\\n()1\\nn\\u0015\\u0000ddiag (Qk)\\ndt\\u0001\\nj;j=Qj;:Q\\u00001(\\n\\u0002w)\\u0003T\\u0000\\nS\\u0000A)(Q\\u00001V)T\\n:;j (S.149)\\n()1\\nn\\u0015\\u0000ddiag (Qk)\\ndt\\u0001\\nj;j=Ij;:\\u0002w\\u0003(\\n)T\\u0000\\nS\\u0000A)(Q\\u00001V)T\\n:;j (S.150)\\n()1\\nn\\u0015\\u0000ddiag (Qk)\\ndt\\u0001\\nj;j=\\u0002w\\u0003\\nj;j(\\n)T\\nj;\\u000b\\u0000\\nS\\u0000A)\\u000b;\\u000b(Q\\u00001V)T\\n\\u000b;j (S.151)\\n()1\\nn\\u0015d\\u0000\\nj(Qk)jjei(\\u000ek)j\\u0001\\ndt=e\\u0000i(\\u000ew)jj(Qk)jj\\u0000\\nS\\u0000A)\\u000b;\\u000be\\u0000i(\\u000e\\u001e)jj(Q\\u001e\\u000b)jj (S.152)\\n()1\\nn\\u0015d\\u0000\\nj(Qk)jjei(\\u000ek)j\\u0001\\ndtj(Qk)jje\\u0000i(\\u000ek)j=e\\u0000i((\\u000ew)j+(\\u000ek)j+(\\u000e\\u001e)j)j(Qk)jj2\\u0000\\nS\\u0000A)\\u000b;\\u000bj(Q\\u001e\\u000b)jj (S.153)\\n()1\\n2n\\u0015\\u0000dj(Qk)jj2\\ndt\\u0001\\nj;j=j(Qk)jj2(S\\u0000A)\\u000b;\\u000bj(Q\\u001e\\u000b)jj: (S.154)\\nThis is Eq. 25 of the main paper, describing the winner-takes-all dynamics.\\nH. Analytical Solutions to the Dynamics of Learning for Pure Cosines Datasets\\nWe derive exact, analytical solutions to the evolution of the effective singular values given the following input:\\nX(c)\\nl;m=b(c)cos\\u0012\\n2\\u0019\\u0016l\\nn+ 2\\u0019\\u0017m\\nn\\u0013\\n(S.155)\\nas discussed in the main paper. We thus use a single cosine per class image, and we set phase \\u000ej= 0(cfr. Eq. S.66). The\\nchoice of the phase does not in\\ufb02uence the conclusions, but makes the discussion in the main paper less complicated.\\nIn this case the right singular vectors \\u001e\\u000b=\\u001e(\\u000b)\\u0003are given by:\\n\\u001e\\u000b\\ni=1\\nd\\u000bncos\\u0012\\n2\\u0019\\u0016l\\nn+ 2\\u0019\\u0017m\\nn+ (\\u000e\\u001e)j\\u0013\\n; (S.156)\\nwith(\\u000e\\u001e)j= 0or(\\u000e\\u001e)j=\\u0019, introducing a possible minus sign (see Eq. S.67 and discussion following the equation). Here\\n\\u0016=div(j;n); \\u0017=mod(j;n)are the frequency indices and l=div(i;n); m=mod(i;n)are the pixel indices. For ease\\nof notation, we have introduced d\\u000b:\\n(\\nd\\u000b= 1j= 0;j!\\u000b\\nd\\u000b=1p\\n2j6= 0;j!\\u000b(S.157)\\nwhere with j!\\u000bwe indicate that jis the vec-2D index that indexes the horizontal (index \\u0016) and vertical (index \\u0017)\\nfrequencies that are used to construct the singular vector indexed by \\u000b. We then \\ufb01nd:\\n8\\n><\\n>:(Q\\u001e\\u000b)j=d\\u000bei(\\u000e\\u001e)j j!\\u000b\\n(Q\\u001e\\u000b)jsymm =d\\u000bei(\\u000e\\u001e)jsymm =d\\u000be\\u0000i(\\u000e\\u001e)jj!\\u000b\\n(Q\\u001e\\u000b)j= 0 else(S.158)\\nwhere we denote with jsymm the frequency index that corresponds to the pair of frequency indices (n\\u0000\\u0017;n\\u0000\\u0016). Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nWe initialize the kernel with small, random weights, and set the initial weights of the fully connected layer according to Eq.\\nS.138. This makes the network balanced and aligned. We get (see Eq. S.142):\\n8\\n><\\n>:A\\u000b;\\u000b(t) = 2nd\\u000bj(Qk(t))jj2\\u0006\\u000b;\\u000bj!\\u000b;j6= 0;\\u000b2f0;\\u0001\\u0001\\u0001;p\\u00001g\\nA\\u000b;\\u000b(t) =nd\\u000bj(Qk(t))jj2\\u0006\\u000b;\\u000bj!\\u000b;j= 0;\\u000b2f0;\\u0001\\u0001\\u0001;p\\u00001g\\nA\\u000b;\\f(t) = 0 else(S.159)\\nWhich can be succinctly written as:\\nA\\u000b;\\u000b=n\\nd\\u000bj(Qk)jj2\\u0006xx\\n\\u000b;\\u000b (S.160)\\nandA\\u000b;\\f= 0if\\u000b6=\\f.\\nThese initial conditions decouple the general system of equations (eqs. 17 and 18); we can use Eq. S.154 for the speci\\ufb01c\\npairs of modes and frequencies (\\u000b;j)used in the single cosines dataset; all other derivatives are zero. From this equation\\nand the fact that we only have one frequency jper mode\\u000b, we can derive the full evolution of the effective singular values\\nA\\u000b;\\u000b:\\ndA\\u000b;\\u000b\\ndt=n\\nd\\u000bdj(Qk)jj2\\ndt\\u0006xx\\n\\u000b;\\u000b (S.161)\\n= 2n\\u0015n\\nd\\u000b\\u0006xx\\n\\u000b;\\u000bj(Qk)jj2(S\\u0000A)\\u000b;\\u000bd\\u000b (S.162)\\n= 2nd\\u000b\\u0015A\\u000b;\\u000b(S\\u000b;\\u000b\\u0000A\\u000b;\\u000b) (S.163)\\nThis is a non-linear, separable differential equation that we can solve for time t(see also (Saxe et al., 2019)). To simplify\\nnotation, we will use a\\u000b=A\\u000b;\\u000bands\\u000b=S\\u000b;\\u000b. Using\\nt=1\\n2nd\\u000b\\u0015Za(f)\\n\\u000b\\na(0)\\n\\u000bda\\u000b\\na\\u000b(s\\u000b\\u0000a\\u000b)=1\\n2nd\\u000b\\u0015s\\u000bln\\u0010a(f)\\n\\u000b(s\\u000b\\u0000a(0)\\n\\u000b)\\na(0)\\n\\u000b(s\\u000b\\u0000a(f)\\n\\u000b)\\u0011\\n(S.164)\\nwheretis the training time (in number of samples) required to arrive at value a(f)\\n\\u000bat time t from initial value a(0)\\n\\u000b. The\\nevolution during training of the effective singular value, a\\u000b(t), is then given by:\\na\\u000b(t) =s\\u000be2\\u0015nd\\u000bs\\u000bt\\ne2\\u0015d\\u000bns\\u000bt\\u00001 +s\\u000b=a(0)\\n\\u000b: (S.165)\\nI. Details of experiments\\nAll experiments are implemented with tensor\\ufb02ow\\/keras. A factor 1\\/p is used in the implementation of the MSE loss, with p\\nthe number of classes. All experiments can be completed in a couple of hours on a basic CPU.\\nWhen only one image per class is used, this means in practice that for each update, a sample\\/class is randomly selected. For\\nthe CIFAR-4 dataset, we used the more conventional setup with a number of epochs, wherein the samples of the train set are\\ngiven as input in a random order.\\nWe use CNNs with a single convolutional layer, a \\ufb02atten layer, and a single fully connected layer. There are only linear\\nactivation functions. The convolutional layer consists of a single kernel Kwith the same dimensions as the input images\\n(n\\u0002n) for grayscale images, and 3 such kernels for colour images. Images are circularly padded beforehand: we extend the\\nimages with n-1 rows\\/columns of the pixel values of the opposite side. When the kernel of n\\u0002nis applied, the padding is\\nthen in practice equivalent to wrapping the kernel around the image when it slides over the original image boundary. The\\nresult of this convolution is again an n\\u0002nmatrix.\\nFCNNs consist of two fully connected layers; the hidden one with n2nodes, the output one with p nodes. The \\ufb01rst weight\\nmatrix therefore has dimensions n2\\u0002n, the second p\\u0002n2. The choice of nodes in the hidden layer corresponds to the\\narchitecture of the CNN, which performs a convolution from an n\\u0002nto ann\\u0002nimage, and therefore has an associated\\ndoubly block circulant matrix of size n2\\u0002n2. For the CIFAR-4 dataset, we use 3n2nodes in the hidden layer. Images are\\n\\ufb02attened ton2\\u00021or3n2\\u00021vectors beforehand. Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nI.1. Pure cosines experiment\\nFigure S.3. Cosines used in the \\u2018pure cosines\\u2019 experiment.\\nImage dimension: 16\\u000216; horizontal and vertical freq. pairs for each class: (0,0), (5,2), (1,7), (0,4); amplitudes for each\\nclass: 1.5, 1, 0.5, 0.2.\\n\\u2022\\u0015CNN = 1=2000 ,\\u0015FCNN = 16=2000\\n\\u2022init CNN: random normal with \\u0016= 0and\\u001b= 0:00001 for the kernel K, weightsWbalanced toKusingR=Q\\n(eq. S.157). This makes Ainitrectangular diagonal.\\n\\u2022init FCNN: using Ainitfrom CNN, setting the \\ufb01rst p diagonal elements of W1andW2to the valueqA(init )\\u000b;\\u000b\\n\\u0006xx\\u000b,\\nother elements to zero.\\n\\u2022 number of updates: 8000\\n\\u2022 number of repeated experiments: 10\\nI.2. Dominant frequency bias experiment (sums of cosines)\\nSums of cosines dataset as shown in \\ufb01g. 4, image dimensions 64\\u000264.\\n\\u2022\\u0015CNN = 1=10000 ,\\u0015FCNN = 64=10000\\n\\u2022 init CNN: random normal with \\u0016= 0and\\u001b= 0:00001 for the kernel and weights.\\n\\u2022 init FCNN: random normal with \\u0016= 0and\\u001b= 0:00001 for all weights.\\n\\u2022 number of updates: 600\\n\\u2022 number of repeated experiments: 10\\nI.3. Geometric shapes experiment\\nDataset as shown in \\ufb01g. 2. Image dimensions 64\\u000264.\\n\\u2022\\u0015CNN = 1=20000 ,\\u0015FCNN = 64=20000\\n\\u2022 init CNN: random normal with \\u0016= 0and\\u001b= 0:00001 for the kernel and weights.\\n\\u2022 init FCNN: random normal with \\u0016= 0and\\u001b= 0:00001 for all weights.\\n\\u2022 number of updates: 60000\\n\\u2022 number of repeated experiments: 10 Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nFigure S.4. Modes of the CIFAR-4 dataset with removed \\ufb01rst mode (i.e., subtracted mean over samples). Three modes remain to\\ndistinguish between the four classes. The top row is a visualisation of the \\ufb01rst mode, the second row of the second mode, etc. 1\\u00023n2\\nvectors are visualised as n\\u0002ncolor images. Each image is normalised to values between 0 and 1. In this way, we can see that the\\n\\ufb01rst mode distinguishes central objects in a blue background, which is especially relevant for the \\ufb01rst class \\u2018airplane\\u2019; the second mode\\ndiscovers the class \\u2018automobile\\u2019 based on a reddish color and a vague frontal view of a car, and the third mode distinguishes \\u2018bird\\u2019 based\\nin a central object in a predominantly green background.\\nI.4. CIFAR-4 experiment\\nFirst four classes of the CIFAR-10 dataset. Train set consists of the 5000 samples for each class in the CIFAR-10 train set\\n(thus 20000 samples in total) with mean subtracted, test set consist of the 1000 samples for each class in the CIFAR-10 test\\nset (4000 samples in total) with mean subtracted. Image dimension 32\\u000232.\\n\\u2022\\u0015CNN = 1=10000 ,\\u0015FCNN = 32=10000\\n\\u2022 init CNN: random normal with \\u0016= 0and\\u001b= 0:0001 for the kernel and weights.\\n\\u2022 init FCNN: random normal with \\u0016= 0and\\u001b= 0:0001 for all weights.\\n\\u2022 number of updates: 100000\\n\\u2022 number of repeated experiments: 10\\nJ. Comparison of the 2D-vec Fourier Spectra of the Kernel and the Singular Vectors\\nIn Fig. S.5 (next page), we visualize that the frequencies that make up the kernel are the dominant frequencies of the singular\\nvectors. The data shown are from the experiment with the geometric shapes dataset (cfr. Fig. 5).\\nEach row of the \\ufb01gure corresonds to a timepoint during training with the CNN:\\n\\u2022 row 0, timepoint = 2000 samples, mode 0 discovered;\\n\\u2022 row 1, timepoint = 9000 samples, mode 1 discovered;\\n\\u2022 row 2, timepoint = 16000 samples, mode 2 discovered;\\n\\u2022 row 3, timepoint = 55000 samples, mode 3 discovered\\n(cfr. Fig. 5 (a)). At each row we plot the 2D-vec Fourier spectra of the singular vectors, i.e., the values j(Q\\u001e\\u000b)jj, that\\ncorrespond to the modes discovered at that point (cfr. Fig. 2 (e)). These spectra are unaltered during training, and are thus\\nrepeated across rows. The kernel, and therefore the values j(Qk)jj, do change during training. We consider the average of\\nthe valuesj(Qk)jjover the number of experiment runs. We overlay these averaged values j(Qk)jjfor the given timepoints\\nat the corresponding values j(Q\\u001e\\u000b)jj. This overlay is indicated with black crosses. The size of these crosses codes for Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nthe magnitude of the values j(Qk)jj. In this way, we can \\ufb01rst note that the Fourier spectrum of kis much sparser than the\\ndifferent Fourier spectra of the singular vectors (in the spectrum of each singular vectors, the number of crosses is much\\nsmaller than the total number of other markers). Moreover, large values j(Qk)jj(large crosses) mainly correspond to large\\nvalues ofj(Q\\u001e\\u000b)jj. Thus the kernel is, at any point during training, a sparse mixture of the dominant frequencies of the\\nsingular vectors discovered so far. Linear CNNs discover the statistical structure of the dataset using only the most dominant frequencies\\nFigure S.5. Comparison of the 2D-vec Fourier Spectra of the Kernel and the Singular Vectors at Selected Timepoints.\",\"28\":\" Magnetic Stochastic Synapses\\nMachine learning using magnetic stochastic synapses\\nMatthew O. A. Ellis,1,a)Alexander Welbourne,2,a)Stephan J. Kyle,2Paul W. Fry,3Dan A. Allwood,2Thomas J.\\nHayward,2and Eleni Vasilaki1\\n1)Department of Computer Science, University of She\\u000eeld, She\\u000eeld, S1 4DP,\\nUnited Kingdom\\n2)Department of Materials Science and Engineering, University of She\\u000eeld, She\\u000eeld, S1 3JD,\\nUnited Kingdom\\n3)Department of Electronic and Electrical Engineering, University of She\\u000eeld, She\\u000eeld, S1 3JD,\\nUnited Kingdom\\n(Dated: 3rd March 2023)\\nThe impressive performance of arti\\fcial neural networks has come at the cost of high energy usage and CO 2\\nemissions. Unconventional computing architectures, with magnetic systems as a candidate, have potential as\\nalternative energy-e\\u000ecient hardware, but, still face challenges, such as stochastic behaviour, in implementa-\\ntion. Here, we present a methodology for exploiting the traditionally detrimental stochastic e\\u000bects in magnetic\\ndomain-wall motion in nanowires. We demonstrate functional binary stochastic synapses alongside a gradient\\nlearning rule that allows their training with applicability to a range of stochastic systems. The rule, utilising\\nthe mean and variance of the neuronal output distribution, \\fnds a trade-o\\u000b between synaptic stochasticity and\\nenergy e\\u000eciency depending on the number of measurements of each synapse. For single measurements, the\\nrule results in binary synapses with minimal stochasticity, sacri\\fcing potential performance for robustness.\\nFor multiple measurements, synaptic distributions are broad, approximating better-performing continuous\\nsynapses. This observation allows us to choose design principles depending on the desired performance and\\nthe device's operational speed and energy cost. We verify performance on physical hardware, showing it is\\ncomparable to a standard neural network.\\nINTRODUCTION\\nThe meteoric rise of arti\\fcial intelligence (AI) as a part\\nof modern life has brought many advantages. However, as\\nAI programs become increasingly more complex, their en-\\nergy footprint becomes larger1,2, with the training of one\\nof today's state-of-the-art natural language processing\\nmodels now requiring similar energy consumption to the\\nchildhood of an average American citizen3. Several non-\\ntraditional computing architectures aim to reduce this\\nenergy cost, including non-CMOS technologies4{7. How-\\never, competitive performance with non-CMOS technolo-\\ngies requires overcoming the latent advantage of years of\\ndevelopment in CMOS.\\nIn biological neural networks, synapses are considered\\nall-or-none or graded and non-deterministic, unlike the\\nfully analogue synapses modelled in arti\\fcial networks8.\\nInspired by biology, several approaches have considered\\nnetworks with binary synapses and neurons, with the\\nview that binary operations are simpler to compute and\\nthus lower energy9{12. However, while these binarised\\nneural networks are more robust to noise, they su\\u000ber\\nfrom lower performance than analogue versions. In con-\\ntrast, networks with stochastic synapses provide sam-\\npling mechanisms for probabilistic models13and can ri-\\nval analogue networks at the expense of long sampling\\ntimes14{19. Adapted training methods are required to\\nprovide higher performance for a lower number of sam-\\nples, while implementations require hardware that can\\na)These authors contributed equally to this work.natively (with low energy cost) provide the stochastic-\\nity required. Magnetic architectures are one possible\\nroute for unconventional computing. They have long\\npromised a role in computing logic following the strong\\ninterest in the \\feld stemming from the data storage\\nmarket6,7,20{26. The non-volatility of magnetic elements\\nnaturally allows for the data storage, while ultra-low-\\npower control mechanisms, such as spin-polarised cur-\\nrents or applied strain27,28o\\u000ber routes towards energy-\\ne\\u000ecient logic-in-memory computing. Ongoing develop-\\nments have shown how to manipulate magnetic domains rents or applied strain27,28o\\u000ber routes towards energy-\\ne\\u000ecient logic-in-memory computing. Ongoing develop-\\nments have shown how to manipulate magnetic domains\\nto both move data and process it22,29{31. However, mag-\\nnetic domain wall logic is limited by stochastic e\\u000bects,\\nparticularly when compared to the low error tolerance\\nenvironment of CMOS computing32,33.\\nHere we propose a methodology where, rather than\\nseeking to eliminate stochastic e\\u000bects, they become a\\ncrucial part of our computing architecture. As a proof of\\nconcept, we demonstrate how a nanowire is usable as a\\nstochastic magnetic synapse able to perform handwritten\\ndigit recognition using multiplexing of one of the hard-\\nware synapses.\\nWe have developed a learning rule that can e\\u000bectively\\ntrain arti\\fcial neural networks made of such \\\\noisy\\\"\\nsynapses by considering the synaptic distribution. Sup-\\npose we allow a single measurement to identify the state\\nof the synapse. In that case, the learning rule will ad-\\njust its parameter, i.e. the \\feld at which the wall is\\npropagated, to reduce the synaptic stochasticity. If we\\nallow multiple measurements, the gradient rule will \\fnd\\nparameters that allow for a broad synaptic distribution,\\nmimicking a continuous synapse and improving perfor-\\nmance. Without the stochasticity, the operation wouldarXiv:2303.01886v1  [cs.ET]  3 Mar 2023 Magnetic Stochastic Synapses 2\\nbe limited to binary operations, which lack the resolu-\\ntion power of analogue synapses. With stochasticity,\\nwe have a \\rexible system tunable between quick-run-\\ntime approximation and long-run-time performance. Our\\nlearning rule provides e\\u000ecient network training despite\\nthe high or variable noise environment and di\\u000bers from\\nother stochastic neural network computing schemes that\\nemploy mean-\\feld-based learning rules14,16,19. Here, the\\ninclusion of the network variance allows the training to\\n\\fnd better solutions in low sampling regimes, providing a\\ntrade-o\\u000b between operational speed\\/energy cost and test\\naccuracy.\\nWe have veri\\fed the model performance experimen-\\ntally by transferring the trained weights to a network\\nutilising such a hardware synapse, with excellent agree-\\nment between the experimental performance and that of\\na simulated network. Our observations allow for a de-\\nsign framework where we can identify the number of re-\\nquired measurements (and hence energy requirements)\\nfor a given desired accuracy and vice versa.\\nThis work opens up the prospect of utilising the\\nlow-energy-cost bene\\fts of spintronic-based logic5{7,34.\\nIn particular, it enables the use of domain wall-based\\nnanowire devices24,31,35,36whilst transforming the hith-\\nerto hindrance of noisy operation32,33into the basis of a\\nhigh-performance stochastic machine learning paradigm.\\nRESULTS\\nHardware stochastic synapse\\nOur proposed elementary computation unit is a binary\\nstochastic synapse based on a ferromagnetic nanowire\\nwith two favourable magnetic orientations. The tran-\\nsitions between regions of di\\u000bering magnetisation orien-\\ntation are known as domain walls (DWs). While di\\u000ber-\\nent forms of DWs exist, here they form a `vortex' pat-\\ntern with a cyclical magnetisation texture. Our synapse\\nwas a 400 nm wide, 54 nm thick permalloy nanowire with\\nnotches patterned halfway along its length to create an\\narti\\fcial defect site. Figure 1.a shows an SEM image of\\nthe system, with the inset enlarging the notch. DWs were\\nnucleated at the left-hand side of the wire (false-coloured\\nblue) by applying a voltage pulse across a gold current\\nline (false-coloured orange).\\nThe operation of this system as a stochastic synapse is\\ndescribed schematically in \\fgure 1.b. A vortex DW37can\\nbe injected into the wire by applying a current pulse in\\nthe line. This corresponds to presenting the synapse with\\nan input of 1, while no DW injection corresponds to an\\ninput of 0. An applied magnetic \\feld is used to propagate\\nthe DW along the length of the wire. If the propagation\\n\\feld is su\\u000eciently high, the DW does not pin at the defect\\nsite and can pass to the end of the wire, resulting in an\\noutput of 1. If the propagation \\feld is low, the DW is\\npinned at the notch, resulting in an output of 0. For\\nintermediate values of the \\feld, the behaviour becomesstochastic but with a well de\\fned pinning probability.\\nWe can consider the \\feld control as controlling the weight\\nin a binary synapse with detecting a DW on the right\\nhand side of the nanowire as the output of the synapse.\\nAs the propagation \\feld is tuned, the probability of the\\nDW passing changes. Figure 1.c shows this passing prob-\\nability, as measured using the focused Magneto-Optical\\nKerr e\\u000bect (FMOKE), as a function of the propagation\\n\\feld. The probability of passing behaves in a sigmoid-\\nlike manner, and the orange dashed line shows a \\ft using\\na logistic sigmoid function f(hij) (see methods).\\nTherefore, a binary stochastic synapse is determined\\nby\\nwij=(\\n1 with probability f(hij)\\n0 otherwise,(1)\\nwheref(hij) is the DW passing probability function, hij\\nis the propagation \\feld for the synapse connecting input\\nneuronjwith output neuron i. Through this de\\fnition\\nour synapses are purely excitatory, which corresponds\\nto the physical representation of a magnetic DW being\\npinned or not, rather than the complementary binary\\nscheme with values f\\u00001;1g, which is not naturally rep-\\nresented by the physical system. to the physical representation of a magnetic DW being\\npinned or not, rather than the complementary binary\\nscheme with values f\\u00001;1g, which is not naturally rep-\\nresented by the physical system.\\nCompared to binary synapses, neural networks with\\nanalogue or graded synapses tend to perform better due\\nto the wider range of states38,39. Here, we adopt a scheme\\nsimilar to that of stochastic computing, where the aver-\\nage of a series of binary measurements or samples are\\nused to represent a value. Thus, we allow for K\\u00151\\nmeasurements to identify the state of a synapse and de-\\nnote the equivalent mean weight as\\n\\u0016wij=1\\nKX\\nkw(k)\\nij; (2)\\nwhereKis the total number of samples taken and the\\nsuperscript ( k) indicates the individual sampling of the\\nsynaptic weights as per eqn. 1. The mean synapse has\\n1 +Kstates, e.g. for K= 1 the two states will be 0\\nand 1, while for K= 2 the states will be 0, 0.5, and\\n1. It follows that for K!1 , \\u0016wijwill be equivalent to\\na sigmoidally-shaped continuous synapse, bounded be-\\ntween 0 and 1. An example demonstrating the average\\nweight as a function of the number of samples can be\\nseen in \\fgure 1.d, where we plot eq. 2 for K= 1 (pur-\\nple squares), 4 (blue diamonds) and 128 (green circles).\\nEach example is calculated by sampling wijthe desired\\nnumber of times with a \\fxed hijthat was selected ran-\\ndomly. In each case only discrete levels are available but\\nwhenK= 128 the sampling is su\\u000ecient to provide an\\nalmost continuous representation.\\nIn this way, our proposed binary stochastic synapse can\\nbe used to construct neural networks that will approach\\na bounded analogue network when multiple samples are\\ntaken. Physically, this is achieved by repeated operation\\nof the hardware devices to accumulate the average values. Magnetic Stochastic Synapses 3\\na b\\nc\\n d\\n1 \\u03bcm\\n10 \\u03bcm\\nField axisCurrent axisInput, xjWeight\\nControl, hij\\nHigh\\nLow\\nHigh\\nLowSynaptic\\nOutput, wijxj\\n1\\n0\\n0\\n01\\n0\\nFIG. 1. Characterisation of notched permalloy nanowire (NW) as a stochastic synapse. a, False-coloured SEM\\nimage of the permalloy NW (blue) and current injection line (orange). The inset shows detail of the arti\\fcial notch. The \\feld\\n(green) and current (white) axes are marked. b,Schematic of the operating principle of the stochastic synapse. The current\\nline allows input ( xj) of 1 (current pulse, DW injected) or 0 (no pulse, no DW). Field inline with the NW drives (if present)\\nthe DW through the system: high \\felds pass the DW through the notch and produce an output of 1, low \\felds result in the\\nnotch blocking the DW and an output of 0. Intermediary \\felds (not shown) provide intermediate probabilities of passing the\\nnotch. c,Experimentally measured probability of an injected domain wall passing the notch. Tuning the propagation \\feld can\\ncontrol this probability across the whole range in a logistic sigmoid-like fashion. Points are averages of 1000 samples, x error\\nbars represent precision in choice of propagation \\feld, and y error bars are given p(1\\u0000p)=p\\n1000. The logistic sigmoid \\ft is\\ngiven in methods. The nucleation \\feld with no input (no injection, xj= 0) is (10 :74\\u00060:07) mT. Therefore, below 10 mT the\\npassing probability for no input is zero. d,Average synaptic weight, as de\\fned in eq. 2. Depending on the number of samples,\\ni.e. repetitions of the operation in b, the e\\u000bective synaptic weight varies from purely binary (one repetition\\/sample, K= 1) to\\nalmost continuous ( K= 128 samples).\\nStochastic network\\nWe embed these synapses in an arti\\fcial neural net-\\nwork where the output of neuron iis given by\\nyi=X\\nj\\u0016wijxj; (3)\\nwherejis an index over the input dimension.We trained the network as a classi\\fer for a problem\\nofCclasses with Cindependent neurons (perceptrons),\\nwhere each neuron represented one class. This task was\\nbased on the well-known MNIST dataset but with each\\nimage downsampled to give images with a shape of 14 by\\n14 pixels instead of the standard 28 by 28. This was nec-\\nessary to reduce the time of the operation when running\\non the prototype experimental hardware (see methods).\\nIn \\fgure 2.a we depict the perceptron that corresponds Magnetic Stochastic Synapses 4\\na\\nbMNIST Subsampled \\nInputw3\\nw4w2x1\\nx2\\nx3\\nx4yw1K synapse copies\\n\\u00afwij=1\\nK\\/summationdisplay\\nkw(k)\\nijStochastic perceptron\\nyi=\\/summationdisplay\\nj\\u00afwijxj\\n}\\nOne perceptron for each yi (i.e. numbers 0-9)\\n?\\n0\\nComputer holds\\nnetworkRequests result of synapse \\nfrom hardware\\n1 or 0 returnedRepeat for \\neach synapse\\nSummations \\nperformed to \\ncalculate final answerHardware network architectureSynapse operation for neuron \\u201c0\\u201d c\\nHigh\\nPropagation\\nfield, h0j\\nPassed = \\u201c1\\u201d\\nLow\\nPropagation\\nfield, h0jPinned = \\u201c0\\u201d\\n1\\n0input zero,\\n bypass synapse\\nImage Input, xjweight \\ncontrol, h0jsynapse result, w0jxj\\u201cHigh\\u201d 1\\n0\\n \\u201cLow\\u201drequest result \\nfrom hardware\\n10 \\u201cHigh\\u201d\\n0 \\u201cLow\\u201drequest result \\nfrom hardware0input zero,\\n bypass synapse\\nFIG. 2. Stochastic network operation. a , Sketch of the stochastic perceptron. Each input value from an image is fed via\\na mean weight to the neurons for each class. Here, the weighted inputs are summed to give the neuron's activity (as in eq. 3).\\nIn the case of the MNIST task, there is a neuron for each of the 10 classes (numbers \\\\0\\\" to \\\\9\\\"; y0toy9). When trained, the\\nneuron for the class corresponding to the correct input, here y0, should have the highest activity. Each mean weight in our\\nnetwork ( \\u0016 wij) is the average of multiple measurements ( K\\u00151) of the output of a synapse with individual weight wijset by its\\ntrained propagation \\feld (see eqs. 1 & 2). A clear distinction should be made here from traditional neural networks that these\\nweights are stochastic and will vary for each run of the network. The individual weights take the value \\\\1\\\" with the probability\\nf(hij) (DW passing probability, as characterised in \\fgure 1.c) or \\\\0\\\" otherwise. The mean weights, therefore, take values from\\nthe distributions shown in \\fgure 1.d. b, The architecture of the hardware network. For the purpose of demonstrating successful\\nperformance, only the stochastic synapses are run directly on the hardware. The perceptrons are stored on a computer, which\\nrequests results (\\\\1\\\" or \\\\0\\\") from the magnetic stochastic synapses for a given synaptic parameter (trained propagation \\feld,\\nhij). After this is repeated for each synapse, summations are performed to predict the correct class of the input. c, Idealised\\noperation of single synapses in materia for the neuron y0. The data path is shown for two inputs, or pixels, for the case of\\na correct image for the class (\\\\0\\\") and an incorrect image (\\\\5\\\"). The value of the weight control, the propagation \\feld hij,\\nis expected to be correlated with pixels in images from the correct class: where the pixels are \\\\on\\\" for correct images, high\\nvalues of the weight control are expected; when \\\\o\\u000b\\\", low values. If the input pixel value is \\\\0\\\", the synapse is bypassed as\\nthe result is \\\\0\\\" by construction. However, if it is \\\\1\\\", a result is requested from the hardware using the corresponding weight\\ncontrol. As shown in the top graph, high propagation \\felds result in the DW directly passing the notch (only a single step is\\nseen) which is interpreted as an output of \\\\1\\\". As in the lower graph, low propagation \\felds result in a two step procedure\\nwhere the DW initially pins at the notch before depinning at a higher driving \\feld. This is interpreted as a \\\\0\\\". In practice,\\nthe results from the synapses will vary stochastically re\\recting the passing probability f(hij). Magnetic Stochastic Synapses 5\\nto class \\\\0\\\". If we present to the neuron a representative\\nof its corresponding class (in this case an image of the\\ndigit \\\\0\\\"), the neuron should produce a high activity for\\nrecognising the input as zero.\\nThe experimental process is shown in \\fgure 2.b. For\\nease of demonstration, only a single hardware synapse is\\nused, with operations serialise in time. Potential devices\\nwould have multiple synapses running in parallel with\\na summation performed during the measurement. The\\nperceptron parameters are stored on a computer, which\\nsends the input and synaptic parameter to the external\\nhardware synapse and requests the result. The process\\nis repeated until Ksamples per synapse (see eq. 2) are\\ncollected. Summation of the results takes place on the\\ncomputer with an additional bias term applied. To avoid\\nredundant measurements, pixels corresponding to inputs\\nof \\\\0\\\" (white pixels in our example image) were omitted,\\nsince the output is deterministically \\\\0\\\" by design. A\\nsynapse receiving a black pixel ( xj= 1) will produce \\\\1\\\"\\nif the \\feld is set at a high value or \\\\0\\\" if the \\feld is set\\nat a low value, see \\fgure 2.c. Intermediate \\feld values\\nwill produce outputs that vary scholastically, re\\recting\\nthe passing probability f(hij).\\nAnalysis of the stochastic learning rule\\nWe now sketch the derivation of the learning rule that\\nwe apply to the synapses of the neural network. Each\\nsynapsew(k)\\nijis an independent sample from a Bernoulli\\ndistribution, and therefore the sum of these samples will\\nfollow a Poisson-Binomial distribution. The mean, \\u0016i,\\nand variance, \\u001b2\\ni, for each output neuron (calculated by\\neq. 3) are given by:\\n\\u0016i=X\\njf(hij)xj; (4)\\n\\u001b2\\ni=1\\nKX\\njf(hij)xj[1\\u0000f(hij)xj]: (5)\\nFor a detailed calculation of these values see the supple-\\nmentary material.\\nSince the number of inputs and the sampling process\\nmeans this sum will be over a large number of events,\\nthe Poisson-Binomial distribution can be approximated\\nas a Gaussian40. Using this approximation, the neuronal\\noutput can be re-parameterised so that the stochasticity\\nis only in a term with no dependence on the trainable\\nparameters. In this way, we write\\n~yi=\\u0016i+\\u001bi\\u0018i; (6)\\nwhere ~yidenotes the approximation of neuronal output\\nyiand\\u0018iis a sample from a Gaussian distribution with\\nzero mean and unit variance.\\nIf we assume that we are in a supervised learning\\nframework and that Eis the error function we would liketo minimise (e.g. square mean error or cross-entropy),\\nthenEis a function of the pattern pwe present to the\\nnetwork, which de\\fnes the desirable output target. E\\nis also a function of the output neurons, represented by\\nvectory, which also depends on p. The learning rule will\\nupdate the values of the applied \\feld to each synapse\\nhijby \\u0001hijaccording to the following \\\\online\\\" gradient\\nrule:\\n\\u0001hij=\\u0000\\u0011@E(~y)\\n@~yi@~yi\\n@hij(7)\\n=\\u0000\\u0011@E(~y)\\n@~yi\\u0012@\\u0016i\\n@hij+@\\u001bi\\n@hij\\u0018i\\u0013\\n; (8)\\nwhere\\u0011is a small positive number representing the learn-\\ning rate. We calculate the derivative of@~yi\\n@hijfrom eq. 6,\\n4, 5. We also calculate the value \\u0018iusing eq. 6, comput-\\ning\\u0016iand\\u001bifrom eq. 4 and 5 and ~ yifrom eq. 3 (setting\\n~yi=yi). It follows that for K!1 ,\\u001b!0 and we ob-\\ntain a \\\\mean-\\feld\\\" gradient rule that takes into account\\nthe mean but not the variance of the output neurons.\\nWe have tested the performance of this rule on the\\ndownsampled MNIST dataset. During training, the num-\\nber of repeats (samples) Kis set as a parameter of the\\nnetwork, which we de\\fne as Ktrain, and as such modi\\fes\\nhow the training progresses. The variance of the output\\nhas an important e\\u000bect on the classi\\fcation procedure; if\\nthe variance is high then mis-classi\\fcation will be more\\nlikely, especially in classes that have similar mean val-\\nues for each neuron. Therefore, during supervised train-\\ning the network aims to minimise this variance. When\\nK is low, this happens through changing the weights,\\ncontrolled through the magnetic \\felds hij, so that the\\nprobabilities are close to either 1 or 0 (high or low ap- K is low, this happens through changing the weights,\\ncontrolled through the magnetic \\felds hij, so that the\\nprobabilities are close to either 1 or 0 (high or low ap-\\nplied \\feld), as this minimises the single sample variance\\nin eqn. 5. This leads to a solution that is almost a deter-\\nministic binary network. However, if K is large then the\\nvariance is reduced by the factor 1 =Kand therefore the\\nsystem can tolerate higher synaptic variance than in the\\ncase ofK= 1. Thus, a pseudo-analogue solution can be\\nfound.\\nFigure 3 describes the e\\u000bect of the learning rule on the\\nnetwork synapses. We plot the distribution of the propa-\\ngation \\felds, hij, over all the neurons from 5 independent\\nmodels before training (\\fgure 3.a), after training with\\nKtrain = 1 sample (\\fgure 3.b) and after training with\\nKtrain= 128 samples (\\fgure 3.c). The \\fnal distributions\\ncon\\frms the theoretical expectation that Ktrain= 1 leads\\nto a binary network (low variance) while Ktrain = 128\\napproximates a standard perceptron with a continuous\\ndistribution of synaptic weight (high variance).\\nIn \\fgure 3.e-f we show the distributions of the neuronal\\noutput when presented with the same image repeatedly\\nfor the three training cases above and \\fnd that the neu-\\nronal distribution re\\rects the synaptic distribution. We\\nnow consider the case where during testing a di\\u000berent\\nnumber of samples are drawn when calculating eqn. 2, Magnetic Stochastic Synapses 6\\nfea b c\\nd\\ngh\\nh\\nKtrain  = \\u221e\\nFIG. 3. Analysis of the stochastic learning rule. a-c , Probability density histograms of synaptic magnetic \\feld parameters\\nover 5 independent models. ashows the distribution before training, where all the \\felds are initialised so that the passing\\nprobability (shown in orange, right hand axis) is 0.5. bandcshow the distributions when trained using 1 or 128 samples\\nrespectively. With 1 sample, the distribution is bimodal with peaks at \\felds with probabilities close to 0 or 1. While when\\ntraining with 128 samples, the distribution is focused on the central region of the passing probability function. d-f, Distribution\\nof the neuron values ywhen an image of a zero is shown 10,000 times independently for neurons either identifying the correct\\n(output 0 in this example) or incorrect (outputs 1-9) classi\\fcation. In dthe model is untrained so all outputs have the same\\ndistribution, while in eandfthe distribution is split into the correct output neuron and the incorrect output neurons when\\ntraining with 1 and 128 samples respectively. The top row shows Ktest= 1 while the bottom row shows Ktest= 128. Using more\\nsamples during testing reduces the variance and therefore the chance of mis-classi\\fcation. g, The standard deviation averaged\\nover all the neurons when increasing number of samples are used in training, with Ktest= 1 (circles) and Ktest=Ktrain\\n(squares). This summarises the conclusions from the distribution plots in d-f. More samples during training allows the\\nstandard deviation for a single sample to increase as the standard deviation over all samples is reduced. However, testing\\nwith Ktest< K train results in an increased overall standard deviation. h, Accuracy on the test set against number of samples\\nduring training when using the stochastic (dark green circles) or the mean-\\feld (light green squares) learning rules. The points\\nshow the accuracy averaged over 5 independently trained models, while the shaded region indicates 1 standard deviation. The\\nstochastic learning rule maintains a higher test accuracy when the number of samples is low. Magnetic Stochastic Synapses 7\\nwhich we de\\fne as Ktest. The top row shows the dis-\\ntribution when Ktest= 1, while the bottom row shows\\nKtest= 128. The untrained neuron values exhibit a\\nGaussian distribution across all data samples, with \\felds\\ninitialised to give the largest possible variance; see \\fg-\\nure 3.d. After training, with Ktrain= 1 orKtrain= 128\\nandKtest=Ktrain the distributions of the correct and\\nincorrect class neurons minimally overlap. However, if\\nwe test the network with Ktest= 1 after we train it\\nwithKtrain = 128 there is a rather signi\\fcant overlap\\n(3.f, upper panel) suggesting a high probability of miss-\\nclassi\\fcation. In all cases, when testing with Ktest= 128\\n(bottom row) the variance is reduced 1 =128 as given in\\neqn. 5 and allows for better resolution of the mean values.\\nIn the case Ktrain= 128, the learning rule has exploited\\nthis additional sampling and variance reduction by better\\nutilising a continuous range of weights to boost perfor-\\nmance. However, when Ktest<128 (as in 3.f, upper\\npanel), the increase in variance decreases the probabil-\\nity of correct classi\\fcation. In the other training case\\n(Ktrain = 1, 3.e, upper panel), the learning rule adapts\\nthe weights to \\fnd a low variance, almost deterministic\\nbinary, solution. Further sampling during testing (3.e,\\nlower panel) reduces this variance further, as expected,\\nbut doesn't signi\\fcantly change the overlap as it has al-\\nready been optimised for the lower sampling regime. As\\nwe will show, this leads to higher performance when test\\nsampling (Ktest) is small, but capped high performance\\nwhen test sampling is allowed to rise, in contrast to the\\nlargeKtraincase.\\nFigure 3.g compares the average variance during test-\\ning withKtest= 1 samples (circles) and Ktest=Ktrain\\nsamples (squares) as a function of the number of samples\\nused during training, Ktrain. As discussed before, when\\ntraining with 1 sample the variance is kept low by having\\npassing probabilities close to 0 or 1. However, when more\\nsamples are used during training, the variances for a sin-\\ngle sample can increase as the variance of the averaged\\nsamples decreases.\\nThis behaviour of minimising the variance to reduce\\nmiss-classi\\fcation arises due to the variance term in the\\n\\\\stochastic\\\" learning rule. Other rules that only consider\\nthe mean term14,19cannot \\fnd these deterministic solu-\\ntions when using a stochastic network. Figure 3.h shows\\nthe test accuracy with Ktest=Ktrain as a function of\\nKtrain samples for our stochastic learning rule (squares)\\nvs the mean \\feld rule (circles) averaged over \\fve inde-\\npendently trained models. For both rules, increasing the\\nnumber of samples leads to an improvement in the test\\naccuracy as more levels are possible for the synapse av-\\nerages (see \\fg. 1.d). However, in all cases, the stochastic\\nlearning rule out performs the mean-\\feld rule, with con-\\nvergence when a large number of samples ( K\\u00158) is used\\nfor training and testing. The dashed line shows the per-\\nformance for a fully mean-\\feld network, where e\\u000bectively\\nan in\\fnite number of samples are taken (i.e continuous\\nbut bounded synapses), and represents the best possible\\naccuracy for such a network given the task.Hardware and operational principles.\\nWe now proceed to demonstrate our neural networks\\nworking on physical hardware and not only within sim-\\nulation. Figures 4.a and b shows the test accuracy\\ncomputed when the synaptic operation has been simu-\\nlated (lines) and processed using the hardware (points)\\nfor models trained with either a, Ktrain = 1 or b,\\nKtrain= 128 and tested with increasing numbers of sam-\\nples (Ktest). Due to the throughput of our prototype\\ndevice, we only demonstrate experimental results up to\\nKtest= 8.\\nThe simulation and hardware results show excellent\\nagreement and highlight di\\u000berent behaviours in models\\ntrained with di\\u000berent sampling levels. In the case trained\\non one repeat, the network is deterministic and as such\\nthe accuracy does not signi\\fcantly improve when we av- trained with di\\u000berent sampling levels. In the case trained\\non one repeat, the network is deterministic and as such\\nthe accuracy does not signi\\fcantly improve when we av-\\nerage over more samples during testing. On the other\\nhand, while the model trained with 128 repeats shows a\\nlower performance with only one testing repeat, the ac-\\ncuracy improves as we increase the number of samples\\nduring testing. This arises from the increased stochastic-\\nity at low sampling levels and resultant increased preci-\\nsion at high sampling levels. This behaviour is corrob-\\norated by the corresponding neuronal distributions (3.e)\\nand (3.f), which show that the neuronal variance when\\ntraining with one sample and testing with one sample is\\nmuch lower than in the case of training with K= 128\\nsamples and testing with one sample. It is akin to major-\\nity voting, where classi\\fers have to be diverse to improve\\nperformance (see41and references therein). Here, perfor-\\nmance increase increases with increasing Ktest(number\\nof voters) when the neuronal distribution has a high vari-\\nance.\\nFigure 4. callows further interrogation of the major-\\nity voting behaviour. It presents (using the now veri-\\n\\fed simulation model) a colour plot of the test accuracy\\nas a function of the number of training and test sam-\\nples. This variation in performance when testing using\\na di\\u000berent number of repeats raises an essential trade-\\no\\u000b in speed vs accuracy. To a \\frst approach, the re-\\nsults follow the behaviour of stochastic computing: fast\\napproximation with increasing accuracy over time if re-\\nquired. This trend is matched on average with the ex-\\ntra repeats, implying an extra time and energy cost to\\naccumulate the samples, but providing a boost in accu-\\nracy. However by utilising our learning rule's ability to\\nenable low sampling deterministic solutions we can out-\\nperform the naive stochastic computing reasoning in the\\nlow sampling limit (as also seen in \\fgure 3.h). If \\fxing\\nKtrain, this leads to a competition between low repeat\\nperformance and ultimate high repeat accuracy, i.e if a\\nmodel is trained on a high number of repeats, but uses\\na low number of repeats during testing (inference) time,\\nthen the accuracy will be sub-optimal. Similarly, if the\\nmodel is trained on a low number of repeats, but tested\\non many, the ultimate accuracy su\\u000bers. One possibility\\nis to always tie Ktest=Ktrain, but this requires multiple Magnetic Stochastic Synapses 8\\na b\\nMean-field binary stochasticStandard neural network\\nc\\nFIG. 4. Hardware veri\\fcation and choice of sampling. a,b Comparison of the testing accuracy computed using either\\nthe physical hardware (points) or from simulation (curves). The test data set is restricted to the \\frst 600 images with an\\napproximately equal balance of digits. In both, training was done using the model, and hardware testing was limited to eight\\nsamples due to throughput limitations of the prototype device. ashows the accuracy when the network was trained with only\\n1 sample while bwas trained with 128 samples. As before, training with 1 sample reaches an almost deterministic solution,\\nso repeated sampling during testing does not improve the accuracy. Training with 128 shows an increase in accuracy as more\\nsamples are used during testing, reaching higher peak performance (albeit with a lower initial base). The dashed line shows the\\nperformance on a standard neural network and the dotted-dashed is for a full mean-\\feld binary stochastic network. In both\\ncases the hardware performance shows excellent agreement with the model calculations. The model accuracy is averaged over\\n5 independent tests with the same trained weights, with the shaded area showing 1 standard deviation. This can be taken to\\nrepresent the variability in performance for a given task due to the inherent stochasticity of the network. Naturally, it decreases\\nas the number of test samples increases and is lower for the, more deterministic, Ktrain = 1 case. The hardware accuracy is\\nfrom a single run over the 600 images, so the error bars show the standard error of the estimation of the accuracy over the\\nmini-batches. cTest accuracy (as measured with the model) over di\\u000berent combinations of training and testing sampling for\\nthe sub-sampled MNIST task. The data is bi-linearly interpolated, which can be considered as averaging over fractions of\\nthe data set with di\\u000berent sampling rates. In general, testing with more samples increases accuracy, but, this is limited when\\nKtest> K train. In particular, in the Ktrain = 1 case, further sampling provides little improvement due to the deterministic\\nweight distributions. Training with 2 samples is better in all test cases than when training with 1, but best overall accuracy\\nis when 128 samples are used in both training and testing. Data such as this provide a guide to choosing training and testing\\nsamples depending on desired accuracy and operation times for a given task.\\ntrained weights. It is, therefore, constructive to utilise\\ndata such as Figure 4. cas a guide on training and test-\\ning the synapse depending on the desired accuracies and\\noperational times. Whilst maintaining the simplicity of\\na single set of trained weights (\\fxing Ktrain), a horizon-\\ntal range of testing values can be chosen to achieve the\\ndesired accuracies and energy cost envelope.\\nAnalysing the performance over the space of training\\nvs test repeats for the MNIST task, we \\fnd that in most\\ncases, testing with a similar number of repeats to the\\ntraining performs well. A signi\\fcant outlier to this was\\nthat training with two samples consistently outperformed\\ntraining with one across all levels of testing, including\\ntesting with one sample. We attribute this to the smaller\\nstep sizes in the parameter space with two samples com-\\npared to one, which allows for a better solution while the\\nvariance is still very low and remains small when testing\\nwith one sample.DISCUSSION\\nNeuromorphic devices are a promising route to devel-\\noping low-energy-cost machine learning systems, seeking\\nto overcome one of the chief drawbacks of traditional neu-\\nral networks. Stochastic, binary neural networks have\\nshown promise in this regard due to their reduced en-\\nergy cost and simple implementation9{13. Multiple sam-\\npling of these networks allows their performance to rival\\nanalogue networks14{19. Outstanding problems, however, ergy cost and simple implementation9{13. Multiple sam-\\npling of these networks allows their performance to rival\\nanalogue networks14{19. Outstanding problems, however,\\nhave been providing training rules to achieve high perfor-\\nmance even at low sampling rates (where calculations can\\nbe performed faster and at less energy cost) and identify-\\ning hardware implementations that can natively provide\\nthe stochasticity required. We have developed a learning\\nmethodology for stochastic binary neural networks that\\nwe verify experimentally, using the behaviour of mag-\\nnetic domain walls in nanowires as stochastic synapses.\\nStochasticity has traditionally been considered a limit- Magnetic Stochastic Synapses 9\\ning factor in nanomagnetic logic devices32,33, but here is\\na functional aspect that drives learning. We have shown\\nperformance of the hardware network comparable to a\\nstandard neural network and demonstrated high perfor-\\nmance at low sampling thanks to the novel learning rule.\\nExperimentally, we have observed that a DW injected\\ninto a nanowire with an arti\\fcial pinning site can be\\nstochastically pinned and tuned by using an applied mag-\\nnetic \\feld. We have then demonstrated that this tunable\\nstochastic pinning can create synapses for a neural net-\\nwork device. Due to the nature of the physical system,\\nthese synapses behave as binary stochastic synapses. Our\\nfundamental ingredient for training such a network is a\\nlearning rule that considers the variance of the stochas-\\ntic output of the network. This training method con-\\nsiders taking multiple samples ( Ktrain\\/Ktest) of the net-\\nwork output to compute a sample average and deviation.\\nA low number of samples leads toward a predominantly\\ndeterministic binary solution and is fast to compute but\\nhas lower performance than a high number of samples\\nthat approximates a standard \\\\analogue\\\" network and\\nrequire more time (and energy). This trade-o\\u000b allows\\n\\rexibility in designing the network based on the required\\nperformance or operating speed.\\nKey is that the learning rule developed here has al-\\nlowed us to \\fnd a range of operating regimes because\\nthe stochastic part of the output is considered. Other bi-\\nnary stochastic computing approaches, such as Hirtzlin\\net al.19, train using the expectation of the network (which\\nwe call mean-\\feld and is equivalent to K!1 ) and leads\\nto a reduced accuracy when fewer samples are used dur-\\ning inference (testing). The Gaussian approximation was\\nalso used by Esser et al.42to train a network with binary\\nstochastic synapses on the IBM TrueNorth neurosynap-\\ntic system but the contribution from the variance term\\nis considered to be negligible. The contribution from the\\nvariance term in our rule allows for weights to be trained\\nthat operate better in the low sampling regime compared\\nto the mean-\\feld versions.\\nOther learning methods where the variance is taken in\\naccount stem from the Likelihood-Ratio framework43{45,\\nwhich is related to policy gradient methods in rein-\\nforcement learning46. While these methods consider the\\nstochasticity of the neurons and synapse, they depend\\nheavily on the choice of baseline values for the loss which\\nrequire complex approximation methods. Additionally,\\nthe reparameterisation method applied here allows for a\\ndirect feedback of the error signal to the synaptic \\feld pa-\\nrameters and \\fts within existing backpropagation-based\\nlearning methodologies.\\nOverall, the stochastic learning rule presented in\\nthis paper has shown tunability in both high and\\nlow sampling regimes and can be implemented simply\\nwithin backpropagation-style codes. The ability, due\\nto consideration of the variance of the output, to tune\\nbetween low-sampling deterministic binary and high-\\nsampling stochastic \\\\analogue like\\\" behaviour lends it-\\nself to the \\rexibility of our system between operationalspeed\\/energy cost and test accuracy.\\nThe magnetic DW synapse that we have demonstrated\\nhere is a proof of principle component and as such it\\nimportant to look towards changes that would be nec-\\nessary for a more \\\\production ready\\\" neuromorphic de-\\nvice. Optimised devices would likely look towards spin-\\ntorque driven domain wall motion31alongside the use\\nof local nanomagnetic elements to encode the weights.\\nIt is also possible to envisage our learning methodology\\napplied to networks built of alternative magnetic ele-\\nments with similar stochastic properties, such as mag-\\nnetic tunnel junctions, amongst others12,14,47{49. Else-\\nwhere, DW devices have been used as neurons50or acti-\\nvation functions51and magnetic elements in general have\\nbeen demonstrated in a range of alternative low energy where, DW devices have been used as neurons50or acti-\\nvation functions51and magnetic elements in general have\\nbeen demonstrated in a range of alternative low energy\\ncomputation schemes16,47,52{54that exploit the stochas-\\nticity of magnetic devices. Our fundamental element,\\nthe magnetic stochastic synapse, could \\ft within such\\nparadigms where e\\u000ecient production of random bits is\\nkey. It is important, however, to state that the key re-\\nsult here is demonstrating performance as run on experi-\\nmental hardware, enabled by our stochastic learning rule.\\nFurther optimisation is a matter of future research and\\nengineering development.\\nWhilst the single layer network demonstrated here can\\nonly solve linearly separable problems, it can be extended\\nin a number of ways. Retaining the single layer simplicity\\nand looking towards an all magnetic architecture, it has\\npotential applications in the \\feld of reservoir comput-\\ning. In reservoir computing, a \\fxed reservoir performs\\na non-linear spatial-temporal transformation of an input\\nsequence such that the output representation is linearly\\nseparable. The advantage of RC is that the reservoir\\ntransform can be o\\u000foaded to a physical system with ap-\\npropriate properties and there has been considerable re-\\ncent interest in developing magnetic (spintronics) based\\nphysical reservoir computing7,55{62. There is potential\\nto connect our magnetic DW based neural network to\\nthese reservoirs to create a complete hardware reservoir\\ncomputing system. There is also the more traditional\\nroute of scaling our current approach towards multi-layer\\nnetworks as the learning rule is compatible with back-\\npropagation. An open research question in this avenue is\\nwhether the sampling procedure should apply at a local\\nor global scale of the network. One approach is imple-\\nmentation of multi-layers using nanowire interconnects\\nand logic gates, but if we look away from the limita-\\ntion of all magnetic architectures, it is also possible to\\nenvisage hybrid magnetic-CMOS application speci\\fc in-\\ntegrated circuits (as in Ref. 63) that might provide a\\nroute to larger scale network hardware. However, details\\nof these implementations are beyond the scope of this\\ncurrent work.\\nIn conclusion, we have developed a training method-\\nology for binary stochastic synapses that considers the\\nnetwork's stochasticity during learning and resampling\\nof the stochastic output allows for a trade-o\\u000b between\\ndevice run time and desired accuracy. This approach has Magnetic Stochastic Synapses 10\\nbeen demonstrated on a proof of concept magnetic do-\\nmain wall-based stochastic synapse with excellent agree-\\nment between hardware and model during inference.\\nMETHODS\\nDevice Fabrication\\nThe devices were fabricated using two-stage electron\\nbeam lithography with the CSAR-62 resist. Nanowires\\nwere deposited in the \\frst stage using thermal evap-\\noration of permalloy (Ni 81Fe19) to a thickness of\\n54 nm (base pressure, 7 \\u000210\\u00007mbar; process pressure,\\n\\u00185\\u000210\\u00005mbar; rate, 0 :5\\/RingA s\\u00001). Current lines and\\nconnection pads were deposited in the second stage as\\nTi\\/Au (Nominally 10 nm\\/ 200 nm via thermal evapora-\\ntion). Samples were electrically connected to PCB de-\\nvices using silver DAG.\\nDevice operation\\nThe device operation procedes as in \\fgure 2. An\\nAVTECH pulse generator was used to apply 30 volt,\\n100-nanosecond pulses along the current line (resistance\\n290 \\n). An electromagnet was used to apply \\felds along\\nthe wire lengths. A National Instruments DAQ card was\\nused to control timing between these two, with pulses\\nbeing triggered at particular times during repeated si-\\nnusoidal \\feld sequences. The \\feld at which the pulse\\nis triggered is the propagation \\feld. On the \\ry calibra-\\ntion of timing enabled correction of any drift between the\\ntrigger and \\feld sequence (due to heating) to .0:1 mT.\\nA focused-MOKE magnetometer (spot size \\u00185 mi-\\ncrometer) was used to measure the NW response. Hys-\\nteresis loops were obtained with the laser spot positioned\\nover the notch. Single steps in the hysteresis loop indicate\\nthe domain wall passing the notch (an output of 1). Dou-\\nble steps indicate a two-stage pinning\\/depinning process\\n(an output of 0). An algorithmic method allowed auto-\\nmated evaluation of each hysteresis loop. The number of\\npeaks were calculated in the di\\u000berentiated Kerr signal; if\\ntwo peaks were present then the DW had been pinned.\\nTo eliminate false positives, the steps in the raw signal\\ncorresponding to the peaks were required to be greater\\nthan 24 % of the total signal change. This was optimised\\nexperimentally to allow for peak detection even with a\\nslightly o\\u000b centre laser spot (unequal step sizes), but to\\nminimises erroneous detections arising from noise.\\nDomain passing probability\\nThe probability of a domain wall not being pinned by\\nthe arti\\fcial defect site was observed to have a sigmoid-\\nlike behaviour. A functional form of this probability wasused to simulate magnetic stochastic synapses for com-\\nputational training of the networks. We \\ftted this prob-\\nability using\\nf(h) =d+1\\u0000d\\n1 + exp(\\u0000\\u0001(h\\u0000h0)); (9)\\nwhered= 0:0219 is a \\fnite passing probability at low\\n\\feld,h0= 4:63 mT is the \\feld centre and \\u0001 = 2 :73 mT\\u00001\\nis the sigmoid width. We note that this exact form of\\nthe \\ftting function is not necessary for the stochastic\\nlearning rule used to train the network.\\nStochastic learning rule\\nFor a network comprised of binary stochastic synapses\\nthe value of each neuron can be approximated by a Gaus-\\nsian given by equation (6), where the mean and variance\\nof each neurons is de\\fned by equations (4) and (5) re-\\nspectively. Using this approximation, a gradient based\\nlearning rule can be derived as the random variable no\\nlonger has a dependence on the model parameters. The\\nparameters of this network are the magnetic \\felds which\\ndetermine the passing probability of the synapse so a\\ngradient descent update is given by\\n\\u0001hij=\\u0000\\u0011@E(~y)\\n@~yi@~yi\\n@hij(10)\\n=\\u0000\\u0011@E(~y)\\n@~yi\\u0012@\\u0016i\\n@hij+@\\u001bi\\n@hij\\u0018i\\u0013\\n: (11)\\nThe gradient of the mean and variance with respect to\\nthe magnetic \\felds are\\n@\\u0016i\\n@hij=f0(hij)xj (12)\\n@\\u001bi\\n@hij=(1\\u00002f(hij)xj)\\n2\\u001bif0(hij)xj; (13)\\nwheref0(h) =@f(h)=@his the derivative of the passing\\nprobability function. Combining this result into equation\\n(10) gives the update rule\\n\\u0001hij=\\u0000\\u0011@E(~y)\\n@~yif0(hij)xj\\n\\u0002\\u0012\\n1 +1\\u00002f(hij)xj\\n2\\u001bi\\u0018i\\u0013\\n: (14)\\nIn this form the rule contains the mean \\feld compo-\\nnent multiplied by a factor that depends on the variance.\\nWhile for the derivation of the rule we have speci\\fed that \\u0002\\u0012\\n1 +1\\u00002f(hij)xj\\n2\\u001bi\\u0018i\\u0013\\n: (14)\\nIn this form the rule contains the mean \\feld compo-\\nnent multiplied by a factor that depends on the variance.\\nWhile for the derivation of the rule we have speci\\fed that\\n\\u0018iis a Guassian random variable with zero mean and unit\\nvariance, during training it is calculated exactly from the\\nforward phase using \\u0018i= (yi\\u0000\\u0016i)=\\u001bi, so if the neuron\\noutput is higher than the mean it will be positive while\\nif it is lower it will be negative. This combines with the\\n1\\u00002f(hij)xjto determine whether the factor increases\\nthe weight update or reduces it. Magnetic Stochastic Synapses 11\\nModel training details\\nAs a benchmark we use the MNIST dataset64but to\\nreduce the number of synaptic operations for the experi-\\nmental hardware it was downsampled by using the Max-\\nPool operation with a \\flter size of 2x2. This created a set\\nof 14 x 14 pixel images which were mapped to a binary\\ninput by thresholding the pixel intensity at 0.5.\\nThe training part of the dataset was randomly split\\ninto a 50,000 training and 10,000 validation subsets. A\\nreal valued bias was applied output of the simulated\\nbinary synapses and these values were converted into\\na probability using the Softmax function with the loss\\nagainst the image labels measured using Cross-Entropy\\nloss. Training was performed using mini-batches of 50\\nimages, and iterated until the validation loss did not de-\\ncrease over 20 epochs. The model with the lowest vali-\\ndation error before the end of training was returned as\\nthe trained model. The Adam optimiser was used with\\na learning rate \\u0011= 0:001 forK\\u00152 and\\u0011= 0:01 for\\nK= 1, determined based on the lower validation error.\\nOn device machine learning testing\\nFor the demonstration of our stochastic network in ma-\\nteria we have used an automated control system to inject\\na domain wall into the magnetic nanowire at the desired\\nmagnetic \\feld given by the synaptic weights. We \\frst\\noptimised the synaptic magnetic \\felds for our network\\nmodels in simulation for the cases of Ktrain= 1 and 128,\\nusing the method detailed below. For each Ktrain, we\\ntrained 5 models before selecting the model that had the\\nlowest error on the validation dataset. We then trans-\\nferred these to the hardware with the control software\\nloading the pixel binary values ( xj) from the test dataset\\nand using the simulation trained magnetic \\felds ( hij) to\\ncontrol the magnetic synapses. As detailed in \\fgure 2, if\\nthe pixel value was 1 the control system would determine\\nwhether the domain wall has pinned or passed the defect\\nsite and return a 0 or 1 respectively. The result of this\\nsynaptic operation was then passed back to the program\\nrunning the neural network inference, which computed\\nthe neuron values to predict the correct class of the test\\ndata.\\nACKNOWLEDGMENTS\\nThe authors gratefully acknowledge funding from EP-\\nSRC (Grant: EP\\/S009647\\/1) and Leverhulme Trust (Re-\\nsearch Grant: RPG-2019-097). The authors would like to\\nthank Luca Manneschi and Ian Vidamour for their help\\nand feedback on this work.CODE AVAILABILITY\\nCode related to this paper is available at\\nhttps:\\/\\/github.com\\/mattoaellis\\/binary_\\nstochastic_synapses .\\nAppendix A: Mean and variance of the Poisson-Binomial\\ndistribution\\nFor a network of binary stochastic synapses, the out-\\nput of each synapses is assumed to be an independent\\nrandom binary event (Bernoulli trial). If these had the\\nsame probability then the sum of these events would re-\\nsult in the Binomial distribution but as each synapse has\\na di\\u000berent input and synaptic probability the value of the\\nneuron will follow a Poisson-Binomial distribution. Since\\nthis distribution can be complex to calculate in full we\\napproximate the distribution by a Gaussian40. The mean\\nof the neuron output, yi, for a given number of samples\\nKis\\n\\u0016i=E[yi] (A1)\\n=E2\\n41\\nKX\\nkX\\njw(k)\\nijxj3\\n5 (A2)\\n=1\\nKX\\nkX\\njE[w(k)\\nijxj] (A3)\\n=1\\nKX\\nkX\\nj(0f(hij)xj+ 1f(hij)xj) (A4)\\n=X\\njf(hij)xj: (A5)\\nwhere in the \\fnal step here we note that the synaptic\\npassing probability f(hij) is independent of k. The vari-\\nance of the distribution is\\n\\u001b2\\ni= var[yi] (A6)\\n= var2\\n41\\nKX\\nkX\\njw(k)\\nijxj3\\n5: (A7)\\nWe now use the fact that the variance of a sum of inde-\\npendent random events is the sum of the variances, and\\nthat var [y=K] = var[y]=K2such that\\n\\u001b2\\ni=1\\nK2X\\nkX\\njvarh\\nw(k)\\nijxji\\n: (A8)\\nThe variance of each synapse as a Bernoulli event is\\nvarh\\nw(k)\\nijxji\\n=f(hij)xj(1\\u0000f(hij)xj); (A9) Magnetic Stochastic Synapses 12\\nand again since this is independent of kthen the variance\\nof the neuron output is\\n\\u001b2\\ni=K\\nK2X\\njf(hij)xj(1\\u0000f(hij)xj) (A10)\\n=1\\nKX\\njf(hij)xj(1\\u0000f(hij)xj): (A11)\\n1N. C. Thompson, K. Greenewald, K. Lee, and G. F. Manso,\\n\\\\Deep learning's diminishing returns: The cost of improvement\\nis becoming unsustainable,\\\" IEEE Spectrum 58, 50{55 (2021).\\n2M. M. Sabry Aly, T. F. Wu, A. Bartolo, Y. H. Malviya,\\nW. Hwang, G. Hills, I. Markov, M. Wootters, M. M. Shulaker,\\nH. Philip Wong, and S. Mitra, \\\\The n3xt approach to energy-\\ne\\u000ecient abundant-data computing,\\\" Proceedings of the IEEE\\n107, 19{48 (2019).\\n3E. Strubell, A. Ganesh, and A. McCallum, \\\\Energy and policy\\nconsiderations for deep learning in nlp,\\\" (2019).\\n4M. Ziegler, \\\\Novel hardware and concepts for unconventional\\ncomputing,\\\" Scienti\\fc reports 10, 1{3 (2020).\\n5M. T. Niemier, G. H. Bernstein, G. Csaba, A. Dingler, X. S. Hu,\\nS. Kurtz, S. Liu, J. Nahas, W. Porod, M. Siddiq, and E. Varga,\\n\\\\Nanomagnet logic: Progress toward system-level integration,\\\"\\nJournal of Physics: Condensed Matter 23, 493202 (2011).\\n6G. Finocchio, M. Di Ventra, K. Y. Camsari, K. Everschor-Sitte,\\nP. K. Amiri, and Z. Zeng, \\\\The promise of spintronics for un-\\nconventional computing,\\\" Journal of Magnetism and Magnetic\\nMaterials 521, 167506 (2021).\\n7J. Grollier, D. Querlioz, K. Y. Camsari, K. Everschor-Sitte,\\nS. Fukami, and M. D. Stiles, \\\\Neuromorphic spintronics,\\\" Na-\\nture Electronics 3, 360{370 (2020).\\n8C. C. Petersen, R. C. Malenka, R. A. Nicoll, and J. J. Hop\\feld,\\n\\\\All-or-none potentiation at ca3-ca1 synapses,\\\" Proceedings of\\nthe National Academy of Sciences 95, 4732{4737 (1998).\\n9T. Simons and D.-J. Lee, \\\\A Review of Binarized Neural Net-\\nworks,\\\" Electronics 8, 661 (2019).\\n10A. Laborieux, M. Bocquet, T. Hirtzlin, J.-O. Klein, L. H. Diez,\\nE. Nowak, E. Vianello, J.-M. Portal, and D. Querlioz, \\\\Low\\nPower In-Memory Implementation of Ternary Neural Networks\\nwith Resistive RAM-Based Synapse,\\\" in 2020 2nd IEEE Interna-\\ntional Conference on Arti\\fcial Intelligence Circuits and Systems\\n(AICAS) (2020) pp. 136{140.\\n11S. Yu, B. Gao, Z. Fang, H. Yu, J. Kang, and H.-S. P. Wong,\\n\\\\Stochastic learning in oxide binary synaptic device for neuro-\\nmorphic computing,\\\" Frontiers in neuroscience 7, 186 (2013).\\n12B. Penkovsky, M. Bocquet, T. Hirtzlin, J.-O. Klein, E. Nowak,\\nE. Vianello, J.-M. Portal, and D. Querlioz, \\\\In-Memory Resistive\\nRAM Implementation of Binarized Neural Networks for Medical\\nApplications,\\\" in 2020 Design, Automation & Test in Europe\\nConference & Exhibition (2020) pp. 690{695.\\n13E. O. Neftci, B. U. Pedroni, S. Joshi, M. Al-Shedivat, and\\nG. Cauwenberghs, \\\\Stochastic synapses enable e\\u000ecient brain-\\ninspired learning machines,\\\" Frontiers in neuroscience 10, 241\\n(2016).\\n14M. W. Daniels, A. Madhavan, P. Talatchian, A. Mizrahi, and\\nM. D. Stiles, \\\\Energy-e\\u000ecient stochastic computing with super-\\nparamagnetic tunnel junctions,\\\" Phys. Rev. Appl. 13, 034016\\n(2020).\\n15Z. Li, J. Li, A. Ren, R. Cai, C. Ding, X. Qian, J. Draper, B. Yuan,\\nJ. Tang, Q. Qiu, and Y. Wang, \\\\HEIF: Highly E\\u000ecient Stochas-\\ntic Computing-Based Inference Framework for Deep Neural Net-\\nworks,\\\" IEEE Transactions on Computer-Aided Design of Inte-\\ngrated Circuits and Systems 38, 1543{1556 (2019).\\n16Y. Shao, S. L. Sinaga, I. O. Sunmola, A. S. Borland, M. J. Carey,\\nJ. A. Katine, V. Lopez-Dominguez, and P. K. Amiri, \\\\Imple-\\nmentation of arti\\fcial neural networks using magnetoresistive\\nrandom-access memory-based stochastic computing units,\\\" IEEE\\nMagnetics Letters 12, 1{5 (2021).17P. K. Muthappa, F. Neugebauer, I. Polian, and J. P.\\nHayes, \\\\Hardware-based Fast Real-time Image Classi\\fcation\\nwith Stochastic Computing,\\\" in 2020 IEEE 38th International\\nConference on Computer Design (ICCD) (2020) pp. 340{347.\\n18A. Nisar, F. A. Khanday, and B. K. Kaushik, \\\\Implementation\\nof an e\\u000ecient magnetic tunnel junction-based stochastic neural\\nnetwork with application to iris data classi\\fcation,\\\" Nanotech-\\nnology 31, 504001 (2020). of an e\\u000ecient magnetic tunnel junction-based stochastic neural\\nnetwork with application to iris data classi\\fcation,\\\" Nanotech-\\nnology 31, 504001 (2020).\\n19T. Hirtzlin, B. Penkovsky, M. Bocquet, J.-O. Klein, J.-M. Por-\\ntal, and D. Querlioz, \\\\Stochastic computing for hardware imple-\\nmentation of binarized neural networks,\\\" IEEE Access 7, 76394{\\n76403 (2019).\\n20M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petro\\u000b,\\nP. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, \\\\Giant\\nMagnetoresistance of (001)Fe\\/(001)Cr Magnetic Superlattices,\\\"\\nPhysical Review Letters 61, 2472{2475 (1988).\\n21G. Binasch, P. Gr\u007f unberg, F. Saurenbach, and W. Zinn, \\\\En-\\nhanced magnetoresistance in layered magnetic structures with\\nantiferromagnetic interlayer exchange,\\\" Physical Review B 39,\\n4828{4830 (1989).\\n22D. A. Allwood, G. Xiong, C. Faulkner, D. Atkinson, D. Petit,\\nand R. Cowburn, \\\\Magnetic domain-wall logic,\\\" Science 309,\\n1688{1692 (2005).\\n23W. P. Mccray, \\\\How spintronics went from the lab to the iPod,\\\"\\nNature Nanotechnology 4(2009).\\n24S. S. Parkin, M. Hayashi, and L. Thomas, \\\\Magnetic domain-\\nwall racetrack memory,\\\" Science 320, 190{194 (2008).\\n25R. Lavrijsen, J.-H. Lee, A. Fern\\u0013 andez-Pacheco, D. C. M. C. Petit,\\nR. Mansell, and R. P. Cowburn, \\\\Magnetic ratchet for three-\\ndimensional spintronic memory and logic,\\\" Nature 493, 647{650\\n(2013).\\n26A. Fern\\u0013 andez-Pacheco, N.-J. Steinke, D. Mahendru, A. Wel-\\nbourne, R. Mansell, S. Chin, D. Petit, J. Lee, R. Dalgliesh,\\nS. Langridge, and R. Cowburn, \\\\Magnetic State of Multilay-\\nered Synthetic Antiferromagnets during Soliton Nucleation and\\nPropagation for Vertical Data Transfer,\\\" Advanced Materials In-\\nterfaces (2016).\\n27S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. Beach,\\n\\\\Current-driven dynamics of chiral ferromagnetic domain walls,\\\"\\nNature materials 12, 611{616 (2013).\\n28J.-M. Hu, Z. Li, L.-Q. Chen, and C.-W. Nan, \\\\High-density\\nmagnetoresistive random access memory operating at ultralow\\nvoltage at room temperature,\\\" Nature communications 2, 1{8\\n(2011).\\n29S. S. Parkin, M. Hayashi, and L. Thomas, \\\\Magnetic domain-\\nwall racetrack memory,\\\" Science 320, 190{194 (2008).\\n30D. Sanz-Hern\\u0013 andez, R. Hamans, J.-W. Liao, A. Welbourne,\\nR. Lavrijsen, and A. Fern\\u0013 andez-Pacheco, \\\\Fabrication, Detec-\\ntion, and Operation of a Three-Dimensional Nanomagnetic Con-\\nduit,\\\" ACS Nano 11(2017).\\n31Z. Luo, A. Hrabec, T. P. Dao, G. Sala, S. Finizio, J. Feng,\\nS. Mayr, J. Raabe, P. Gambardella, and L. J. Heyderman,\\n\\\\Current-driven magnetic domain-wall logic,\\\" Nature 579, 214{\\n218 (2020).\\n32T. Hayward, \\\\Intrinsic nature of stochastic domain wall pinning\\nphenomena in magnetic nanowire devices,\\\" Scienti\\fc reports 5,\\n1{12 (2015).\\n33D. Kumar, T. Jin, S. Al Risi, R. Sbiaa, W. S. Lew, and S. N.\\nPiramanayagam, \\\\Domain Wall Motion Control for Racetrack\\nMemory Applications,\\\" IEEE Transactions on Magnetics 55, 1{\\n8 (2019).\\n34B. Lambson, D. Carlton, and J. Bokor, \\\\Exploring the Ther-\\nmodynamic Limits of Computation in Integrated Systems: Mag-\\nnetic Memory, Nanomagnetic Logic, and the Landauer Limit,\\\"\\nPhysical Review Letters 107, 010604 (2011).\\n35M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. P.\\nParkin, \\\\Current-Controlled Magnetic Domain-Wall Nanowire\\nShift Register,\\\" Science 320, 209{211 (2008).\\n36S.-H. Yang, K.-S. Ryu, and S. Parkin, \\\\Domain-wall velocities of\\nup to 750 m s-1 driven by exchange-coupling torque in synthetic Magnetic Stochastic Synapses 13\\nantiferromagnets,\\\" Nature Nanotechnology 10, 221{226 (2015).\\n37Y. Nakatani, A. Thiaville, and J. Miltat, \\\\Head-to-head do-\\nmain walls in soft nano-strips: a re\\fned phase diagram,\\\" Journal\\nof Magnetism and Magnetic Materials Proceedings of the Joint\\nEuropean Magnetic Symposia (JEMS' 04), 290-291 , 750{753\\n(2005).\\n38J. Satel, T. Trappenberg, and A. Fine, \\\\Are binary synapses\\nsuperior to graded weight representations in stochastic attractor\\nnetworks?\\\" Cognitive neurodynamics 3, 243{250 (2009).\\n39A. M. Dubreuil, Y. Amit, and N. Brunel, \\\\Memory capacity of\\nnetworks with stochastic binary synapses,\\\" PLoS computational\\nbiology 10, e1003727 (2014).\\n40Y. Hong, \\\\On computing the distribution function for the poisson\\nbinomial distribution,\\\" Computational Statistics & Data Analy-\\nsis59, 41{51 (2013).\\n41A. Vouros, T. V. Gehring, K. Szydlowska, A. Janusz, Z. Tu,\\nM. Croucher, K. Lukasiuk, W. Konopka, C. Sandi, and E. Vasi-\\nlaki, \\\\A generalised framework for detailed classi\\fcation of swim-\\nming paths inside the morris water maze,\\\" Scienti\\fc reports 8,\\n1{15 (2018).\\n42S. K. Esser, R. Appuswamy, P. Merolla, J. V. Arthur, and\\nD. S. Modha, \\\\Backpropagation for energy-e\\u000ecient neuromor-\\nphic computing,\\\" Advances in neural information processing sys-\\ntems 28(2015).\\n43R. J. Williams, \\\\Simple statistical gradient-following algorithms\\nfor connectionist reinforcement learning,\\\" Machine learning 8,\\n229{256 (1992).\\n44S. Gu, S. Levine, I. Sutskever, and A. Mnih, \\\\Muprop: Unbiased\\nbackpropagation for stochastic neural networks,\\\" arXiv preprint\\narXiv:1511.05176 (2015).\\n45P. Parmas and M. Sugiyama, \\\\A uni\\fed view of likelihood ratio\\nand reparameterization gradients,\\\" in International Conference\\non Arti\\fcial Intelligence and Statistics (PMLR, 2021) pp. 4078{\\n4086.\\n46E. Vasilaki, N. Fr\\u0013 emaux, R. Urbanczik, W. Senn, and W. Gerst-\\nner, \\\\Spike-based reinforcement learning in continuous state and\\naction space: when policy gradient methods fail,\\\" PLoS compu-\\ntational biology 5, e1000586 (2009).\\n47M. A. Azam, D. Bhattacharya, D. Querlioz, C. A. Ross, and\\nJ. Atulasimha, \\\\Voltage control of domain walls in magnetic\\nnanowires for energy-e\\u000ecient neuromorphic devices,\\\" Nanotech-\\nnology 31, 145201 (2020).\\n48D. Sanz-Hern\\u0013 andez, M. Massouras, N. Reyren, N. Rougemaille,\\nV. Sch\\u0013 anilec, K. Bouzehouane, M. Hehn, B. Canals, D. Querlioz,\\nJ. Grollier, F. Montaigne, and D. Lacour, \\\\Tunable Stochasticity\\nin an Arti\\fcial Spin Network,\\\" Advanced Materials 33, 2008135\\n(2021).\\n49W. A. Misba, M. Lozano, D. Querlioz, and J. Atulasimha,\\n\\\\Energy E\\u000ecient Learning With Low Resolution Stochastic Do-\\nmain Wall Synapse for Deep Neural Networks,\\\" IEEE Access 10,\\n84946{84959 (2022).\\n50N. Hassan, X. Hu, L. Jiang-Wei, W. H. Brigner, O. G. Akinola,\\nF. Garcia-Sanchez, M. Pasquale, C. H. Bennett, J. A. C. Incorvia,\\nand J. S. Friedman, \\\\Magnetic domain wall neuron with lateral\\ninhibition,\\\" Journal of Applied Physics 124, 152127 (2018).\\n51W. H. Brigner, N. Hassan, X. Hu, C. H. Bennett, F. Garcia-\\nSanchez, C. Cui, A. Velasquez, M. J. Marinella, J. A. C. Incorvia,\\nand J. S. Friedman, \\\\Domain wall leaky integrate-and-\\fre neu-\\nrons with shape-based con\\fgurable activation functions,\\\" IEEE\\nTransactions on Electron Devices 69, 2353{2359 (2022).\\n52W. A. Borders, A. Z. Pervaiz, S. Fukami, K. Y. Camsari,\\nH. Ohno, and S. Datta, \\\\Integer factorization using stochastic\\nmagnetic tunnel junctions,\\\" Nature 573, 390{393 (2019).\\n53W. Al Misba, M. Lozano, D. Querlioz, and J. Atulasimha, \\\\En-\\nergy e\\u000ecient learning with low resolution stochastic domain wall\\nsynapse for deep neural networks,\\\" IEEE Access 10, 84946{84959\\n(2022).\\n54M. Koo, G. Srinivasan, Y. Shim, and K. Roy, \\\\Sbsnn:\\nStochastic-bits enabled binary spiking neural network with on-\\nchip learning for energy e\\u000ecient neuromorphic computing at theedge,\\\" IEEE Transactions on Circuits and Systems I: Regular\\nPapers 67, 2546{2555 (2020).\\n55R. W. Dawidek, T. J. Hayward, I. T. Vidamour, T. J. Broomhall, Papers 67, 2546{2555 (2020).\\n55R. W. Dawidek, T. J. Hayward, I. T. Vidamour, T. J. Broomhall,\\nG. Venkat, M. A. Mamoori, A. Mullen, S. J. Kyle, P. W. Fry,\\nN.-J. Steinke, J. F. K. Cooper, F. Maccherozzi, S. S. Dhesi,\\nL. Aballe, M. Foerster, J. Prat, E. Vasilaki, M. O. A. Ellis,\\nand D. A. Allwood, \\\\Dynamically-driven emergence in a nano-\\nmagnetic system,\\\" Advanced Functional Materials 31, 2008389\\n(2021).\\n56R. V. Ababei, M. O. A. Ellis, I. T. Vidamour, D. S. Devadasan,\\nD. A. Allwood, E. Vasilaki, and T. J. Hayward, \\\\Neuromorphic\\ncomputation with a single magnetic domain wall,\\\" Scienti\\fc Re-\\nports 11, 15587 (2021).\\n57A. Welbourne, A. L. R. Levy, M. O. A. Ellis, H. Chen, M. J.\\nThompson, E. Vasilaki, D. A. Allwood, and T. J. Hay-\\nward, \\\\Voltage-controlled superparamagnetic ensembles for low-\\npower reservoir computing,\\\" Applied Physics Letters 118, 202402\\n(2021).\\n58J. C. Gartside, K. D. Stenning, A. Vanstone, H. H. Holder, D. M.\\nArroo, T. Dion, F. Caravelli, H. Kurebayashi, and W. R. Bran-\\nford, \\\\Recon\\fgurable training and reservoir computing in an ar-\\nti\\fcial spin-vortex ice via spin-wave \\fngerprinting,\\\" Nature Nan-\\notechnology 17, 460{469 (2022).\\n59I. T. Vidamour, M. O. A. Ellis, D. Gri\\u000en, G. Venkat,\\nC. Swindells, R. W. S. Dawidek, T. J. Broomhall, N. J. Steinke,\\nJ. F. K. Cooper, F. Maccherozzi, S. S. Dhesi, S. Stepney, E. Vasi-\\nlaki, D. A. Allwood, and T. J. Hayward, \\\\Quantifying the\\ncomputational capability of a nanomagnetic reservoir computing\\nplatform with emergent magnetisation dynamics,\\\" Nanotechnol-\\nogy33, 485203 (2022).\\n60I. Vidamour, C. Swindells, G. Venkat, P. Fry, A. Welbourne,\\nR. Rowan-Robinson, D. Backes, F. Maccherozzi, S. Dhesi,\\nE. Vasilaki, D. Allwood, and T. Hayward, \\\\Reservoir Computing\\nwith Emergent Dynamics in a Magnetic Metamaterial,\\\" (2022),\\narXiv:2206.04446 [cond-mat].\\n61D. A. Allwood, M. O. A. Ellis, D. Gri\\u000en, T. J. Hayward,\\nL. Manneschi, M. F. K. Musameh, S. O'Keefe, S. Stepney,\\nC. Swindells, M. A. Trefzer, E. Vasilaki, G. Venkat, I. Vidamour,\\nand C. Wringe, \\\\A perspective on physical reservoir computing\\nwith nanomagnetic devices,\\\" (2022), arXiv:2212.04851 [physics].\\n62K. D. Stenning, J. C. Gartside, L. Manneschi, C. T. S. Che-\\nung, T. Chen, A. Vanstone, J. Love, H. H. Holder, F. Caravelli,\\nK. Everschor-Sitte, E. Vasilaki, and W. R. Branford, \\\\Adaptive\\nProgrammable Networks for In Materia Neuromorphic Comput-\\ning,\\\" (2022), arXiv:2211.06373 [cond-mat].\\n63T. Hirtzlin, M. Bocquet, B. Penkovsky, J.-O. Klein, E. Nowak,\\nE. Vianello, J.-M. Portal, and D. Querlioz, \\\\Digital biologically\\nplausible implementation of binarized neural networks with dif-\\nferential hafnium oxide resistive memory arrays,\\\" Frontiers in\\nneuroscience 13, 1383 (2020).\\n64Y. LeCun, \\\\The mnist database of handwritten digits,\\\"\\nhttp:\\/\\/yann. lecun. com\\/exdb\\/mnist\\/ (1998).\\n1N. C. Thompson, K. Greenewald, K. Lee, and G. F. Manso,\\n\\\\Deep learning's diminishing returns: The cost of improvement\\nis becoming unsustainable,\\\" IEEE Spectrum 58, 50{55 (2021).\\n2M. M. Sabry Aly, T. F. Wu, A. Bartolo, Y. H. Malviya,\\nW. Hwang, G. Hills, I. Markov, M. Wootters, M. M. Shulaker,\\nH. Philip Wong, and S. Mitra, \\\\The n3xt approach to energy-\\ne\\u000ecient abundant-data computing,\\\" Proceedings of the IEEE\\n107, 19{48 (2019).\\n3E. Strubell, A. Ganesh, and A. McCallum, \\\\Energy and policy\\nconsiderations for deep learning in nlp,\\\" (2019).\\n4M. Ziegler, \\\\Novel hardware and concepts for unconventional\\ncomputing,\\\" Scienti\\fc reports 10, 1{3 (2020).\\n5M. T. Niemier, G. H. Bernstein, G. Csaba, A. Dingler, X. S. Hu,\\nS. Kurtz, S. Liu, J. Nahas, W. Porod, M. Siddiq, and E. Varga,\\n\\\\Nanomagnet logic: Progress toward system-level integration,\\\"\\nJournal of Physics: Condensed Matter 23, 493202 (2011). Magnetic Stochastic Synapses 14\\n6G. Finocchio, M. Di Ventra, K. Y. Camsari, K. Everschor-Sitte,\\nP. K. Amiri, and Z. Zeng, \\\\The promise of spintronics for un-\\nconventional computing,\\\" Journal of Magnetism and Magnetic\\nMaterials 521, 167506 (2021).\\n7J. Grollier, D. Querlioz, K. Y. Camsari, K. Everschor-Sitte,\\nS. Fukami, and M. D. Stiles, \\\\Neuromorphic spintronics,\\\" Na-\\nture Electronics 3, 360{370 (2020).\\n8C. C. Petersen, R. C. Malenka, R. A. Nicoll, and J. J. Hop\\feld,\\n\\\\All-or-none potentiation at ca3-ca1 synapses,\\\" Proceedings of\\nthe National Academy of Sciences 95, 4732{4737 (1998).\\n9T. Simons and D.-J. Lee, \\\\A Review of Binarized Neural Net-\\nworks,\\\" Electronics 8, 661 (2019).\\n10A. Laborieux, M. Bocquet, T. Hirtzlin, J.-O. Klein, L. H. Diez,\\nE. Nowak, E. Vianello, J.-M. Portal, and D. Querlioz, \\\\Low\\nPower In-Memory Implementation of Ternary Neural Networks\\nwith Resistive RAM-Based Synapse,\\\" in 2020 2nd IEEE Interna-\\ntional Conference on Arti\\fcial Intelligence Circuits and Systems\\n(AICAS) (2020) pp. 136{140.\\n11S. Yu, B. Gao, Z. Fang, H. Yu, J. Kang, and H.-S. P. Wong,\\n\\\\Stochastic learning in oxide binary synaptic device for neuro-\\nmorphic computing,\\\" Frontiers in neuroscience 7, 186 (2013).\\n12B. Penkovsky, M. Bocquet, T. Hirtzlin, J.-O. Klein, E. Nowak,\\nE. Vianello, J.-M. Portal, and D. Querlioz, \\\\In-Memory Resistive\\nRAM Implementation of Binarized Neural Networks for Medical\\nApplications,\\\" in 2020 Design, Automation & Test in Europe\\nConference & Exhibition (2020) pp. 690{695.\\n13E. O. Neftci, B. U. Pedroni, S. Joshi, M. Al-Shedivat, and\\nG. Cauwenberghs, \\\\Stochastic synapses enable e\\u000ecient brain-\\ninspired learning machines,\\\" Frontiers in neuroscience 10, 241\\n(2016).\\n14M. W. Daniels, A. Madhavan, P. Talatchian, A. Mizrahi, and\\nM. D. Stiles, \\\\Energy-e\\u000ecient stochastic computing with super-\\nparamagnetic tunnel junctions,\\\" Phys. Rev. Appl. 13, 034016\\n(2020).\\n15Z. Li, J. Li, A. Ren, R. Cai, C. Ding, X. Qian, J. Draper, B. Yuan,\\nJ. Tang, Q. Qiu, and Y. Wang, \\\\HEIF: Highly E\\u000ecient Stochas-\\ntic Computing-Based Inference Framework for Deep Neural Net-\\nworks,\\\" IEEE Transactions on Computer-Aided Design of Inte-\\ngrated Circuits and Systems 38, 1543{1556 (2019).\\n16Y. Shao, S. L. Sinaga, I. O. Sunmola, A. S. Borland, M. J. Carey,\\nJ. A. Katine, V. Lopez-Dominguez, and P. K. Amiri, \\\\Imple-\\nmentation of arti\\fcial neural networks using magnetoresistive\\nrandom-access memory-based stochastic computing units,\\\" IEEE\\nMagnetics Letters 12, 1{5 (2021).\\n17P. K. Muthappa, F. Neugebauer, I. Polian, and J. P.\\nHayes, \\\\Hardware-based Fast Real-time Image Classi\\fcation\\nwith Stochastic Computing,\\\" in 2020 IEEE 38th International\\nConference on Computer Design (ICCD) (2020) pp. 340{347.\\n18A. Nisar, F. A. Khanday, and B. K. Kaushik, \\\\Implementation\\nof an e\\u000ecient magnetic tunnel junction-based stochastic neural\\nnetwork with application to iris data classi\\fcation,\\\" Nanotech-\\nnology 31, 504001 (2020).\\n19T. Hirtzlin, B. Penkovsky, M. Bocquet, J.-O. Klein, J.-M. Por-\\ntal, and D. Querlioz, \\\\Stochastic computing for hardware imple-\\nmentation of binarized neural networks,\\\" IEEE Access 7, 76394{\\n76403 (2019).\\n20M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petro\\u000b,\\nP. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, \\\\Giant\\nMagnetoresistance of (001)Fe\\/(001)Cr Magnetic Superlattices,\\\"\\nPhysical Review Letters 61, 2472{2475 (1988).\\n21G. Binasch, P. Gr\u007f unberg, F. Saurenbach, and W. Zinn, \\\\En-\\nhanced magnetoresistance in layered magnetic structures with\\nantiferromagnetic interlayer exchange,\\\" Physical Review B 39,\\n4828{4830 (1989).\\n22D. A. Allwood, G. Xiong, C. Faulkner, D. Atkinson, D. Petit,\\nand R. Cowburn, \\\\Magnetic domain-wall logic,\\\" Science 309,\\n1688{1692 (2005).\\n23W. P. Mccray, \\\\How spintronics went from the lab to the iPod,\\\"\\nNature Nanotechnology 4(2009).24S. S. Parkin, M. Hayashi, and L. Thomas, \\\\Magnetic domain-\\nwall racetrack memory,\\\" Science 320, 190{194 (2008).\\n25R. Lavrijsen, J.-H. Lee, A. Fern\\u0013 andez-Pacheco, D. C. M. C. Petit, wall racetrack memory,\\\" Science 320, 190{194 (2008).\\n25R. Lavrijsen, J.-H. Lee, A. Fern\\u0013 andez-Pacheco, D. C. M. C. Petit,\\nR. Mansell, and R. P. Cowburn, \\\\Magnetic ratchet for three-\\ndimensional spintronic memory and logic,\\\" Nature 493, 647{650\\n(2013).\\n26A. Fern\\u0013 andez-Pacheco, N.-J. Steinke, D. Mahendru, A. Wel-\\nbourne, R. Mansell, S. Chin, D. Petit, J. Lee, R. Dalgliesh,\\nS. Langridge, and R. Cowburn, \\\\Magnetic State of Multilay-\\nered Synthetic Antiferromagnets during Soliton Nucleation and\\nPropagation for Vertical Data Transfer,\\\" Advanced Materials In-\\nterfaces (2016).\\n27S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. Beach,\\n\\\\Current-driven dynamics of chiral ferromagnetic domain walls,\\\"\\nNature materials 12, 611{616 (2013).\\n28J.-M. Hu, Z. Li, L.-Q. Chen, and C.-W. Nan, \\\\High-density\\nmagnetoresistive random access memory operating at ultralow\\nvoltage at room temperature,\\\" Nature communications 2, 1{8\\n(2011).\\n29S. S. Parkin, M. Hayashi, and L. Thomas, \\\\Magnetic domain-\\nwall racetrack memory,\\\" Science 320, 190{194 (2008).\\n30D. Sanz-Hern\\u0013 andez, R. Hamans, J.-W. Liao, A. Welbourne,\\nR. Lavrijsen, and A. Fern\\u0013 andez-Pacheco, \\\\Fabrication, Detec-\\ntion, and Operation of a Three-Dimensional Nanomagnetic Con-\\nduit,\\\" ACS Nano 11(2017).\\n31Z. Luo, A. Hrabec, T. P. Dao, G. Sala, S. Finizio, J. Feng,\\nS. Mayr, J. Raabe, P. Gambardella, and L. J. Heyderman,\\n\\\\Current-driven magnetic domain-wall logic,\\\" Nature 579, 214{\\n218 (2020).\\n32T. Hayward, \\\\Intrinsic nature of stochastic domain wall pinning\\nphenomena in magnetic nanowire devices,\\\" Scienti\\fc reports 5,\\n1{12 (2015).\\n33D. Kumar, T. Jin, S. Al Risi, R. Sbiaa, W. S. Lew, and S. N.\\nPiramanayagam, \\\\Domain Wall Motion Control for Racetrack\\nMemory Applications,\\\" IEEE Transactions on Magnetics 55, 1{\\n8 (2019).\\n34B. Lambson, D. Carlton, and J. Bokor, \\\\Exploring the Ther-\\nmodynamic Limits of Computation in Integrated Systems: Mag-\\nnetic Memory, Nanomagnetic Logic, and the Landauer Limit,\\\"\\nPhysical Review Letters 107, 010604 (2011).\\n35M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. P.\\nParkin, \\\\Current-Controlled Magnetic Domain-Wall Nanowire\\nShift Register,\\\" Science 320, 209{211 (2008).\\n36S.-H. Yang, K.-S. Ryu, and S. Parkin, \\\\Domain-wall velocities of\\nup to 750 m s-1 driven by exchange-coupling torque in synthetic\\nantiferromagnets,\\\" Nature Nanotechnology 10, 221{226 (2015).\\n37Y. Nakatani, A. Thiaville, and J. Miltat, \\\\Head-to-head do-\\nmain walls in soft nano-strips: a re\\fned phase diagram,\\\" Journal\\nof Magnetism and Magnetic Materials Proceedings of the Joint\\nEuropean Magnetic Symposia (JEMS' 04), 290-291 , 750{753\\n(2005).\\n38J. Satel, T. Trappenberg, and A. Fine, \\\\Are binary synapses\\nsuperior to graded weight representations in stochastic attractor\\nnetworks?\\\" Cognitive neurodynamics 3, 243{250 (2009).\\n39A. M. Dubreuil, Y. Amit, and N. Brunel, \\\\Memory capacity of\\nnetworks with stochastic binary synapses,\\\" PLoS computational\\nbiology 10, e1003727 (2014).\\n40Y. Hong, \\\\On computing the distribution function for the poisson\\nbinomial distribution,\\\" Computational Statistics & Data Analy-\\nsis59, 41{51 (2013).\\n41A. Vouros, T. V. Gehring, K. Szydlowska, A. Janusz, Z. Tu,\\nM. Croucher, K. Lukasiuk, W. Konopka, C. Sandi, and E. Vasi-\\nlaki, \\\\A generalised framework for detailed classi\\fcation of swim-\\nming paths inside the morris water maze,\\\" Scienti\\fc reports 8,\\n1{15 (2018).\\n42S. K. Esser, R. Appuswamy, P. Merolla, J. V. Arthur, and\\nD. S. Modha, \\\\Backpropagation for energy-e\\u000ecient neuromor-\\nphic computing,\\\" Advances in neural information processing sys-\\ntems 28(2015). Magnetic Stochastic Synapses 15\\n43R. J. Williams, \\\\Simple statistical gradient-following algorithms\\nfor connectionist reinforcement learning,\\\" Machine learning 8,\\n229{256 (1992).\\n44S. Gu, S. Levine, I. Sutskever, and A. Mnih, \\\\Muprop: Unbiased\\nbackpropagation for stochastic neural networks,\\\" arXiv preprint\\narXiv:1511.05176 (2015).\\n45P. Parmas and M. Sugiyama, \\\\A uni\\fed view of likelihood ratio\\nand reparameterization gradients,\\\" in International Conference\\non Arti\\fcial Intelligence and Statistics (PMLR, 2021) pp. 4078{\\n4086.\\n46E. Vasilaki, N. Fr\\u0013 emaux, R. Urbanczik, W. Senn, and W. Gerst-\\nner, \\\\Spike-based reinforcement learning in continuous state and\\naction space: when policy gradient methods fail,\\\" PLoS compu-\\ntational biology 5, e1000586 (2009).\\n47M. A. Azam, D. Bhattacharya, D. Querlioz, C. A. Ross, and\\nJ. Atulasimha, \\\\Voltage control of domain walls in magnetic\\nnanowires for energy-e\\u000ecient neuromorphic devices,\\\" Nanotech-\\nnology 31, 145201 (2020).\\n48D. Sanz-Hern\\u0013 andez, M. Massouras, N. Reyren, N. Rougemaille,\\nV. Sch\\u0013 anilec, K. Bouzehouane, M. Hehn, B. Canals, D. Querlioz,\\nJ. Grollier, F. Montaigne, and D. Lacour, \\\\Tunable Stochasticity\\nin an Arti\\fcial Spin Network,\\\" Advanced Materials 33, 2008135\\n(2021).\\n49W. A. Misba, M. Lozano, D. Querlioz, and J. Atulasimha,\\n\\\\Energy E\\u000ecient Learning With Low Resolution Stochastic Do-\\nmain Wall Synapse for Deep Neural Networks,\\\" IEEE Access 10,\\n84946{84959 (2022).\\n50N. Hassan, X. Hu, L. Jiang-Wei, W. H. Brigner, O. G. Akinola,\\nF. Garcia-Sanchez, M. Pasquale, C. H. Bennett, J. A. C. Incorvia,\\nand J. S. Friedman, \\\\Magnetic domain wall neuron with lateral\\ninhibition,\\\" Journal of Applied Physics 124, 152127 (2018).\\n51W. H. Brigner, N. Hassan, X. Hu, C. H. Bennett, F. Garcia-\\nSanchez, C. Cui, A. Velasquez, M. J. Marinella, J. A. C. Incorvia,\\nand J. S. Friedman, \\\\Domain wall leaky integrate-and-\\fre neu-\\nrons with shape-based con\\fgurable activation functions,\\\" IEEE\\nTransactions on Electron Devices 69, 2353{2359 (2022).\\n52W. A. Borders, A. Z. Pervaiz, S. Fukami, K. Y. Camsari,\\nH. Ohno, and S. Datta, \\\\Integer factorization using stochastic\\nmagnetic tunnel junctions,\\\" Nature 573, 390{393 (2019).\\n53W. Al Misba, M. Lozano, D. Querlioz, and J. Atulasimha, \\\\En-\\nergy e\\u000ecient learning with low resolution stochastic domain wall\\nsynapse for deep neural networks,\\\" IEEE Access 10, 84946{84959\\n(2022).\\n54M. Koo, G. Srinivasan, Y. Shim, and K. Roy, \\\\Sbsnn:\\nStochastic-bits enabled binary spiking neural network with on-\\nchip learning for energy e\\u000ecient neuromorphic computing at the\\nedge,\\\" IEEE Transactions on Circuits and Systems I: Regular\\nPapers 67, 2546{2555 (2020).\\n55R. W. Dawidek, T. J. Hayward, I. T. Vidamour, T. J. Broomhall,\\nG. Venkat, M. A. Mamoori, A. Mullen, S. J. Kyle, P. W. Fry,N.-J. Steinke, J. F. K. Cooper, F. Maccherozzi, S. S. Dhesi,\\nL. Aballe, M. Foerster, J. Prat, E. Vasilaki, M. O. A. Ellis,\\nand D. A. Allwood, \\\\Dynamically-driven emergence in a nano-\\nmagnetic system,\\\" Advanced Functional Materials 31, 2008389\\n(2021).\\n56R. V. Ababei, M. O. A. Ellis, I. T. Vidamour, D. S. Devadasan,\\nD. A. Allwood, E. Vasilaki, and T. J. Hayward, \\\\Neuromorphic\\ncomputation with a single magnetic domain wall,\\\" Scienti\\fc Re-\\nports 11, 15587 (2021).\\n57A. Welbourne, A. L. R. Levy, M. O. A. Ellis, H. Chen, M. J.\\nThompson, E. Vasilaki, D. A. Allwood, and T. J. Hay-\\nward, \\\\Voltage-controlled superparamagnetic ensembles for low-\\npower reservoir computing,\\\" Applied Physics Letters 118, 202402\\n(2021).\\n58J. C. Gartside, K. D. Stenning, A. Vanstone, H. H. Holder, D. M.\\nArroo, T. Dion, F. Caravelli, H. Kurebayashi, and W. R. Bran-\\nford, \\\\Recon\\fgurable training and reservoir computing in an ar-\\nti\\fcial spin-vortex ice via spin-wave \\fngerprinting,\\\" Nature Nan-\\notechnology 17, 460{469 (2022).\\n59I. T. Vidamour, M. O. A. Ellis, D. Gri\\u000en, G. Venkat,\\nC. Swindells, R. W. S. Dawidek, T. J. Broomhall, N. J. Steinke,\\nJ. F. K. Cooper, F. Maccherozzi, S. S. Dhesi, S. Stepney, E. Vasi- 59I. T. Vidamour, M. O. A. Ellis, D. Gri\\u000en, G. Venkat,\\nC. Swindells, R. W. S. Dawidek, T. J. Broomhall, N. J. Steinke,\\nJ. F. K. Cooper, F. Maccherozzi, S. S. Dhesi, S. Stepney, E. Vasi-\\nlaki, D. A. Allwood, and T. J. Hayward, \\\\Quantifying the\\ncomputational capability of a nanomagnetic reservoir computing\\nplatform with emergent magnetisation dynamics,\\\" Nanotechnol-\\nogy33, 485203 (2022).\\n60I. Vidamour, C. Swindells, G. Venkat, P. Fry, A. Welbourne,\\nR. Rowan-Robinson, D. Backes, F. Maccherozzi, S. Dhesi,\\nE. Vasilaki, D. Allwood, and T. Hayward, \\\\Reservoir Computing\\nwith Emergent Dynamics in a Magnetic Metamaterial,\\\" (2022),\\narXiv:2206.04446 [cond-mat].\\n61D. A. Allwood, M. O. A. Ellis, D. Gri\\u000en, T. J. Hayward,\\nL. Manneschi, M. F. K. Musameh, S. O'Keefe, S. Stepney,\\nC. Swindells, M. A. Trefzer, E. Vasilaki, G. Venkat, I. Vidamour,\\nand C. Wringe, \\\\A perspective on physical reservoir computing\\nwith nanomagnetic devices,\\\" (2022), arXiv:2212.04851 [physics].\\n62K. D. Stenning, J. C. Gartside, L. Manneschi, C. T. S. Che-\\nung, T. Chen, A. Vanstone, J. Love, H. H. Holder, F. Caravelli,\\nK. Everschor-Sitte, E. Vasilaki, and W. R. Branford, \\\\Adaptive\\nProgrammable Networks for In Materia Neuromorphic Comput-\\ning,\\\" (2022), arXiv:2211.06373 [cond-mat].\\n63T. Hirtzlin, M. Bocquet, B. Penkovsky, J.-O. Klein, E. Nowak,\\nE. Vianello, J.-M. Portal, and D. Querlioz, \\\\Digital biologically\\nplausible implementation of binarized neural networks with dif-\\nferential hafnium oxide resistive memory arrays,\\\" Frontiers in\\nneuroscience 13, 1383 (2020).\\n64Y. LeCun, \\\\The mnist database of handwritten digits,\\\"\\nhttp:\\/\\/yann. lecun. com\\/exdb\\/mnist\\/ (1998).\",\"29\":\" Reservoir computing based on solitary-like waves dynamics of \\flm\\n\\rows: a proof of concept\\nIvan S. Maksymov1andAndrey Pototsky2\\n1Arti\\fcial Intelligence and Cyber Futures Institute, Charles Sturt University, Bathurst, NSW 2795, Australia\\n2Department of Mathematics, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia\\nAbstract { Several theoretical works have shown that solitons|waves that self-maintain constant\\nshape and velocity as they propagate|can be used as a physical computational reservoir, a concept\\nwhere machine learning algorithms designed for digital computers are replaced by analog physical\\nsystems that exhibit nonlinear dynamical behaviour. Here we propose and experimentally validate\\na novel reservoir computing (RC) system that for the \\frst time employs solitary-like (SL) waves\\npropagating on the surface of a liquid \\flm \\rowing over an inclined surface. We demonstrate the\\nability of the SL wave RC system (SLRC) to forecast chaotic time series and to successfully pass\\nessential benchmark tests, including a memory capacity test and a Mackey-Glass model test.\\nIntroduction. { Reservoir computing (RC) [1] and\\nits foundational concepts of context of reverberated input\\nhistories [2], Liquid State Machine (LSM) [3] and Echo\\nState Network (ESN) [4] underpin an important class of\\nmachine learning (ML) algorithms that can perform cer-\\ntain functions of a biological brain [3] and that can be\\ntrained to recognise patterns and to forecast the response\\nof nonlinear dynamical systems that exhibit chaotic be-\\nhaviour [5]. This functionality enables using RC systems\\nin many areas of science and technology, including physics,\\npsychology and \\fnance [6,7].\\nThe ability of an RC system to perform complex fore-\\ncasting tasks originates from its structural organisation\\n[Fig. 1(a)] that separates a fast linear readout layer from\\na much slower arti\\fcial neural network called the reservoir\\n[4]. The reservoir consists of a network of randomly con-\\nnected arti\\fcial neurons, and it converts a time-varying\\ninput into a spatio-temporal pattern of neural activations\\nthat is then transformed into practical outputs using the\\nreadout layer. While the readout layer is task-speci\\fc and\\nmemory-less, the universal reservoir is based on the princi-\\nples of nonlinear dynamical systems and has certain mem-\\nory [8]. Such a structure enables a reservoir consisting of a\\nrelatively small number of arti\\fcial neurons to be more ef-\\n\\fcient in resolving certain practical problems than a com-\\nputer program run on a supercomputer. Thus, one can\\ndraw an analogy between a reservoir and some biological\\nbrains [3], including the brains of fruit \\ries and mosquitos\\nthat, on average, have just about 200,000 neurons (in con-\\ntrast to mammalian brains that have tens of billions neu-rons) but enable the insects to perform complicated tasks\\nwhile they navigate and \\fnd food [9].\\nAlthough in the framework of LSM the analogy between\\na reservoir and a liquid was made rather at an abstract\\nlevel [3], follow-up research works have demonstrated that\\nphysical liquids inherently possess all characteristics re-\\nquired for the creation of a reservoir [10{16]. There have\\nalso been theoretical proposals [17{20] of RC systems\\nbased on optical solitons [21]. However, there have been\\nno experimental demonstrations of soliton-based RC sys-\\ntem yet.\\nIn this paper, we experimentally demonstrate a physical\\nRC system that exploits the dynamics of solitary-like (SL)\\nwaves [Fig. 1(b)] that originate from the spatio-temporal\\nevolution of liquid \\flms \\rowing over an inclined surface\\n[22{24]. We con\\frm the plausibility of the SL waves based\\nRC (SLRC) system using several standard test protocols\\ndesigned to benchmark computer programs that imple-\\nment the concept of RC. SL waves are ubiquitous in nature\\nand can be readily observed in many everyday-life situa-\\ntions [Fig. 1(c)]. Subsequently, the principles of SLRC\\ndemonstrated in this work should be applicable to nu- and can be readily observed in many everyday-life situa-\\ntions [Fig. 1(c)]. Subsequently, the principles of SLRC\\ndemonstrated in this work should be applicable to nu-\\nmerous physical system that support roll waves [24, 25],\\nincluding micro\\ruidic devices [26].\\nAlgorithmic reservoir computing. { A standard\\nESN algorithm [Fig. 1(a)] that we employ as a reference\\nuses the following nonlinear update equation:\\nxn= (1\\u0000\\u000b)xn\\u00001+\\u000btanh( Winun+Wxn\\u00001); (1)\\np-1arXiv:2303.01801v1  [physics.flu-dyn]  3 Mar 2023 I. S. Maksymov and A. Pototsky\\nFig. 1: (a) ESN-based RC algorithm.(b) Sketch of the proposed SLRC. (c) Photograph of SL waves of natural origin seen on\\nthe bowl of a drinking fountain that spouts water.\\nwherenis the index denoting equally-spaced discrete time\\ninstancestn,unis the vector of Nuinput values, xnis a\\nvector ofNxneural activations of the reservoir, the oper-\\nator tanh(\\u0001) is a sigmoid activation function of a neuron,\\nWinis the input matrix containing Nx\\u0002Nuelements,\\nWis the recurrent weight matrix consisting of Nx\\u0002Nx\\nelements and \\u000b2(0;1] is a parameter that controls the\\nreservoir's temporal dynamics.\\nOne calculates the output weights Woutby solving a\\nsystem of linear equations Ytarget=WoutX, where X\\nandYtargetare the state matrix and the target matrix\\nthat are constructed using, respectively, xnand the vec-\\ntor of target outputs ytarget\\nn as columns for each dis-\\ncrete time instant tn. The solution is obtained in the\\nform Wout=YtargetX>(XX>+\\fI)\\u00001, where Iis the\\nidentity matrix, \\f\\u00150 is a regularisation coe\\u000ecient and\\nX>is the transpose of X[4]. Then, one solves Eq. (1)\\nfor new input data unand computes the output vector\\nyn=Wout[1;un;xn] using a constant bias and the con-\\ncatenation [ un;xn].\\nSolitary-like wave physical reservoir system. {\\nFigure 1(b) shows a sketch of the experimental setup,\\nwhere SL waves propagating over a metal plate inclined\\nwith respect to the ground by the angle \\u0012= 3o[27] are\\nused as a computational reservoir. The electronic modula-\\ntion of the pump \\row rate is used to control the thickness\\nof the liquid \\flm and to create SL waves on its surface.\\nAdding an organic \\ruorescent dye to the liquid (tap wa-\\nter) and irradiating it with UV light, we generate green\\n\\ruorescence light and measure variations of its intensity\\ncaused by the SL waves. The optical intensity pro\\fles are\\nrecorded using an overhead high-speed (210 frames per sec-\\nond) digital camera and then processed using customised\\nsoftware [27].To validate SLRC, we adopt the approach used to test\\nthe ability of ESN computer programs to predict chaotic\\ntime series (TS) [5, 8, 15]. First, the response of the SL\\nwaves to a su\\u000eciently long data set is recorded in one\\nsingle measurement. Then, the \\frst half of the input data\\nset is used for training of the RC system but its second half\\nis reserved for tests. Thus, using the \\frst half of the input\\ndata we calculate Wout, but the second half the data set,\\nun, produces the neural activations xnneeded to compute\\nthe output as yn=Wout[1;un;xn].\\nAs the \\frst TS that is intended to test the memory\\ncapacity of SLRC, we use a randomly generated vector\\nof binary '0' or '1' values. This data vector is converted\\ninto an electric signal using a digital function generator,\\nwhere 0.25-second-long square pulses correspond to the\\n'1' states but 0.25-second-long spaces between the pulses\\nrepresent the '0' states. This electric signal modulates the\\npump \\row [Fig. 1(b)]: the '1' states of the TS are im-\\nplemented as a brief change in the \\row rate that results\\nin the generation of one SL wave but the '0' states cor-\\nrespond to no \\row modulation. For example, an input\\ndata subset '10101' is represented by three 0.25-second-\\nlong pulses separated by two 0.25-second-long periods of\\nno \\row modulations, but '111001111' corresponds to one\\n0.75-second-long pulse separated by 0.5 seconds of no \\row\\nmodulation from the following 1-second-long pulse. The\\nchosen duration of both the pulses and the gaps between\\nthem is based on the optimal SL wave generation param-\\neters established in [27].\\nFigure 2 shows the SL waves produced by the train-\\ning portion of the random TS. While a periodic square\\npulse signal corresponding to a regular '101010...' binary\\nsequence produces a pattern of equally spaced SL waves\\n(not shown here, see [27] for a relevant discussion), the\\np-2 Reservoir computing based on solitary-like waves dynamics\\nFig. 2: SL waves produced by a short training section of the randomly generated binary input signal, where each '1' and '0'\\nvalue is represented as 0.25-second-long square pulses and 0.25-second-long gaps between them, respectively.\\nSL wave pro\\fle in Fig. 2 follows the random pattern of\\nthe input TS. The SL waves do not precisely reproduce\\nthe training pulses, which is a typical picture seen at the\\ntraining stage of physical RC systems [15,28]. As also es-\\ntablished in algorithmic RC systems [8], the response of\\nthe reservoir must not be a copy of the training signal since\\nin that case the trained RC system would not be able to\\nproduce an output that resembles the target TS.\\nThe SL wave signal in Fig. 2 corresponds to a single neu-\\nral activation of the reservoir. To generate more neural ac-\\ntivations, we sample this signal with a small discrete time\\nstep following the procedure described in [28], thus pro-\\nducing 40 neural activations. We arrange the so-obtained\\nactivations into a state matrix Xand use it to calculate\\nWoutfollowing the standard ESN algorithm [Fig. 1(a)].\\nAs the second test, using the same training approach\\nas above, we task the SLRC to predict a Mackey-Glass\\nTS (MGTS). MGTS is often employed to test RC systems\\ndue to its nonlinear chaotic behaviour [15,29,30] and it is\\nproduced by the delay di\\u000berential equation [31]\\n_xMG(t) =\\fMGxMG(\\u001cMG\\u0000t)\\n1 +xq\\nMG(\\u001cMG\\u0000t)\\u0000\\rMGxMG(t);(2)\\nwhere we choose \\u001cMG= 17 and set q= 10,\\fMG= 0:2 and\\n\\rMG= 0:1. The resulting TS modulates liquid \\row in the\\nexperiment. However, in contract to the square pulses,\\nthe MGTS modulation is continuous. We note that the\\nchosen set of MGTS parameters is often used to test the\\nperformance of standard ML algorithms, including autore-\\ngressive moving average (ARMA) [29]. Algorithmic RC\\nsystems are known to outperform ARMA in forecasting\\nchaotic TS [32]. Thus, we also use the MGTS test to ver-\\nify whether SLRC could outperform ARMA.\\nBenchmark test. { Memory capacity. A reservoir\\nsuitable for applications in RC should have an internal\\nmemory [8]. We test the memory capacity of SLRC using\\nthek-delay task applied to the random binary TS, where\\nwe calculate the correlation coe\\u000ecient r(ytarget\\nn\\u0000k;yk;n) be-\\ntween a delayed target signal ytarget\\nn\\u0000kand the predicted\\noutput yk;nproduced by the RC system trained using thetraining data uk;nwith a discrete time delay k[8]. The\\nfunctionr2(ytarget\\nn\\u0000k;yk;n) can accept any value from 0 to\\n1, being 0 an indicator of the full loss of correlation and\\n1 corresponding a 100% correlation between the correct\\ntarget data and the prediction made by the RC system.\\nCalculating r2as a function of the discrete delay time k,\\nwe determine the memory capacity (MC) as\\nMC =kmaxX\\nk=1r2(ytarget\\nn\\u0000k;yk;n); (3)\\nbeingkmaxthe maximum delay considered in the analysis.\\nFigure 3(a) plots r2forkmax= 20 and produces MC =\\n13:4 bits. We observe that the last non-zero value of r2\\ncorresponds to k= 40 thus showing that SLRC, which\\nhasN= 40 neurons, retains some memory of the past\\ninputs up to the theoretical fundamental limit MC =N\\nof algorithmic RC systems [8]. Yet, in practice, in algo-\\nrithmic RC systems MC\\u00190:3Ndue to an accumulated\\ne\\u000bect of rounding errors in the network update equations\\n[8]. Obtaining N=MC\\u00190:35 for SLRC, we conclude that\\nimperfections of the experimental setup and of the image\\nprocessing technique used to detect SL waves play the role\\nof rounding error in an algorithmic RC system.\\nThe value of MC reported in Fig. 3(a) is an order of\\nmagnitude higher than that reported in [18], where an op-\\ntical soliton RC system was theoretically suggested. We\\nhypothesise that the application of SL waves in the \\feld of\\nRC since may be advantageous since they possess unique\\nnonlinear properties that are not available with solitary\\nwave of other physical nature. Indeed, SL waves can merge\\n[22] instead of passing one through another, with the lat- nonlinear properties that are not available with solitary\\nwave of other physical nature. Indeed, SL waves can merge\\n[22] instead of passing one through another, with the lat-\\nter being the case of well-known optical [21] and matter\\n[33] soliton waves, which we speculate may increase the\\nmemory of SLRC to the past inputs.\\nTo support our arguments, we \\frst refer to the liter-\\nature on the memory capacity of RC systems and we\\ncon\\frm that the behaviour of r2observed in Fig. 3(a)\\nclosely resembles that of reservoirs consisting of arti\\f-\\ncial neurons with highly nonlinear activation functions\\n[34, 35]. We also task the SLRC with a parity check test\\np-3 I. S. Maksymov and A. Pototsky\\nFig. 3: Memory capacity of the SLRC determined using (a) k-\\ndelay and (b) parity check tasks.\\nused to evaluate the performance of recurrent neural net-\\nwork that operate in a nearly chaotic dynamic regime\\n[36]. In this test, the RC system predicts a TS de\\fned\\nasPARITY (un\\u0000k;un\\u0000k\\u00001;un\\u0000k\\u00002) for increasing delays\\nk. The so-generated TS is a binary sequence itself and\\nits forecast is a challenging task for any RC system since\\nthe parity function is not linearly separable and its predic-\\ntion requires the reservoir to have memory. Similarly to\\nEq. (3), we calculate PC memory capacity by summation\\nof the parity check values up to the maximum delay kmax.\\nFigure 3(b) shows the outcome of the parity check test\\nthat produces PC= 12:4 bits. In the literature on physi-\\ncal RC systems, the value of PCis usually slightly smaller\\nthan the value of MC and, typically, it reaches 2 bits\\n[16,28,37,38]. The value of PCfor algorithmic RC systems\\nis larger than that of physical RC systems [36]. While for\\nthe SLRC the value of PCis also slightly smaller than\\nthat ofMC, it is six times larger than the typical value\\nfor physical RC systems used in this work as a reference.\\nA plausible explanation of this di\\u000berence may be the\\nfact that the phase space of a standard algorithmic reser-\\nvoir is chosen so that is does not contain multiple periodic\\nor chaotic attractors or \\fxed points, the presence of which\\nmay disrupt the echo state property [30]. Physical RC sys-\\ntems are also often designed according to this rule. How-\\never, in practice, many physical RC systems with complex\\ndynamics not only can reproduce the behaviour of algo-\\nrithmic RC systems having an echo state property but\\nthey can also resolve complex nonlinear problems with a\\nhigher e\\u000eciency [15, 36]. Since SL waves exhibit a com-\\nplex nonlinear behaviour [22], similarly to the RC systems\\ndiscussed in [15, 36] SLRC should be capable of resolving\\nsome complex nonlinear problems with high e\\u000eciency.\\nWe also note that in Fig. 3(b) at the zero delay the par-\\nity check produces the memory of approximately 0 :96 bit\\ninstead of the theoretically expected 1 bit corresponding\\nto 100% correlation. The exact origin of this discrepancy\\nin our proof-of-principle experiment is unknown. How-\\never, the following explanations may be plausible. Firstly,\\nit may be an artefact since it is not observed in the MC\\ntest. Secondly, in algorithmic RC systems, this situation isknown to originate from long-term reverberations in the\\nreservoir that deteriorate the recall of immediately past\\ninputs [2, 8]. In the SLRC, reverberations may be caused\\nby technical imperfections of the inlet \\fxture and by small\\nre\\rections of SL from the downstream edge of the inclined\\nplate, which both are well-established facts [27].\\nMackey-Glass time series. We test the ability of SLRC\\nto predict MGTS using a one-step ahead prediction tasks\\nthat is often employed to test the accuracy of neural net-\\nwork models [6,39,40]. In this task, the value of a TS at a\\ntime steptnis mapped to a future time step tn+1and the\\nresulting prediction is compared with the expected target\\nsignal to calculate the accuracy of the reservoir.\\nUsing the approach outlined in [15], we resample the\\nMGTS signal so that the frequency of the principal peak\\nin its Fourier spectrum equals 2 Hz, which is the frequency\\nof SL waves investigated in the relevant experimental work\\n[27]. In Fig. 4(a) we plot a section of the training MGTS\\nsignal and the traces of the SL waves created by it. As with\\nthe square pulses in Fig. 2, we observe an SL wave pat-\\ntern that follows but not precisely \\ft the training MGTS,\\nwhich indicates that the trained RC system will be able\\nto generalise the learnt input signal patterns [8].\\nIn Fig. 4(b), we can see that the SLRC correctly pre-\\ndicts the target MGTS. We calculate the normalised mean\\nsquare error (NMSE), which is a standard measure of the\\naccuracy of algorithmic RC systems [4], as\\nNMSE (y;ytarget) =D\\nkyn\\u0000ytarget dicts the target MGTS. We calculate the normalised mean\\nsquare error (NMSE), which is a standard measure of the\\naccuracy of algorithmic RC systems [4], as\\nNMSE (y;ytarget) =D\\nkyn\\u0000ytarget\\nnk2E\\nD\\r\\ryn\\u0000hytarget\\nni\\r\\r2E; (4)\\nwherek\\u0001kis the norm andh\\u0001idenotes the averaging over\\ntime. AnNMSE = 0 means that the RC system produces\\na 100% correct prediction but NMSE = 1 means that the\\nRC system has fully failed to predict the target data. We\\nobtainNMSE SLRC\\u00196:6\\u000210\\u00004. Then, we task software\\nthat implements the ESN model Eq. (1) [30] to forecast\\nMGTS and obtain NMSE ESN\\u00191:75\\u000210\\u00007. A three or-\\nders of magnitude di\\u000berence between NMSE of SLRC and\\nESN is inconsequential for our analysis and is attributed\\nto a noise present in the SLRC measurements. We also\\nuse software that implements an ARMA model [41] and\\npredicts MGTS in a one-step-ahead regime. We obtain\\nNMSE ARMA\\u00191:5\\u000210\\u00003, which shows that the accu-\\nracy of ARMA is lower than that of RC systems, which is\\nan established fact [32].\\nTheoretical analysis. { It has been shown that con-\\nsiderable memory capacity of physical RC systems can be\\nachieved under a condition, when the system's response\\nto one pulse of a chaotic TS does not fully decay before\\nit is presented with the following pulses, which enables\\nthe system to retain some traces of its response to several\\nconsecutive pulses of the input TS [28]. SLRC has a sim-\\nilar memory mechanism that is based on the property of\\nSL waves of higher amplitude to overtake and absorb SL\\np-4 Reservoir computing based on solitary-like waves dynamics\\n(b)\\nFig. 4: MGTS test of SLRC. (a) SL waves produced by a short training section of the MGTS input signal. (b) One-timestep\\nahead prediction made by SLRC (dots) compared with the expected target signal (line). Note the di\\u000berent timescales use to\\nplot the training and prediction stage data.\\nwaves of lower amplitude [22]. We theoretically investi-\\ngate this mechanism using a model, where, for the sake of\\nillustration, we excite SL waves using three square pulses.\\nWe employ a simpli\\fed hydrodynamic Shkadov model\\nthat is based on the boundary layer approximation of\\nthe Navier-Stokes equation under the assumption of the\\nparabolic longitudinal velocity pro\\fle [42, 43] and that is\\nformulated in terms of two coupled equations for the local\\n\\flm thickness h(x;t) and the \\ruid \\rux across the layer\\nq(x;t)\\n\\u001a\\u0014\\n@tq+6\\n5@x\\u0012q2\\nh\\u0013\\u0015\\n=\\u00003\\u0016q\\nh2+\\u001bh@3\\nxh\\n\\u0000\\u001agcos(\\u0012)h@xh+\\u001agsin(\\u0012)h;\\n@th+@xq=f(x;t); (5)\\nwhere\\u0012is the inclination angle, \\u0016the dynamic viscosity\\nand\\u001bthe liquid-air surface tension. The term f(x;t) in\\nthe second equation in Eqs. (5) is introduced to mimic the\\ntime-dependent in\\rux of the \\ruid induced by the inlet.\\nWe use a Gaussian f(x;t) = exp (\\u0000a(x\\u0000x0)2)s(t) cen-\\ntred around the location of the inlet x0with parameter\\nachosen su\\u000eciently large to ensure that the width of the\\nGaussian is signi\\fcantly smaller than the typical length of\\nthe surface waves. The binary function s(t) =faf0;1gis\\nintroduced to describe the time modulation of the \\ruid in-\\n\\rux andfastands for the amplitude of the in\\rux. We nu-\\nmerically integrate Eqs. (5) in the ( \\u0000L\\u0014x\\u0014L) domain\\nof total length of 2 L= 400 cm using periodic boundaries.The inlet is located at x0= 100 cm. The relatively large\\ndomain length enables us to observe the dynamics of the\\nwaves over a time period of up to 15 seconds, before the\\nwaves travel over the distance of 2 L.\\nTo demonstrate the interaction and collision of waves\\nwith di\\u000berent amplitudes, we excite three SL waves on\\na 0.7-mm-thick water \\flm using three \\row modulation\\nphases with a gradually increasing amplitude. All three\\nphases have a duration of 0.5 seconds and they are sep-\\narated one from another by 0.5 seconds of no \\row mod-\\nulation [Fig. 5(a)]. The space-time plot of the excited\\nwaves is shown in Fig. 5(b), where the arrows labelled as\\n(1;2;3) highlight the propagation direction of the three SL\\nwaves. As anticipated, the \\frst smaller wave propagates\\nslower than the second larger wave. These two waves even-\\ntually collide approximately 100 cm further downstream\\nand form a new wave with a complex pro\\fle. Such an in-\\nteraction scenario is known as an inelastic wave collision\\n[22]. All three waves (1 ;2;3) eventually form one com-\\npact structure that consists of many smaller waves around\\n150 cm further downstream from the inlet location.\\nThus, we can see that, when the SL waves are detected\\nvery close to the inlet, they do not have enough time to de-\\nvelop and interact, which, in turn, means that the memory\\ncapacity of SLRC is low. The memory capacity will be low\\nalso when SL waves are detected far downstream, where\\na signi\\fcant transformation of the waveform implies that\\nthe ability of the reservoir to memorise the input signal is\\np-5 I. S. Maksymov and A. Pototsky\\n100 200 300 x [cm]time\\nfa(1)(2)(3)secondary wave(a) (b)\\nFig. 5: (a) Modulation function s(t) used to excite three SL waves with di\\u000berent amplitudes. (b) Space-time plot of the temporal\\nevolution of the three SL waves, labelled by the arrows (1 ;2;3), excited by the modulation function shown in panel (a). The\\n\\frst smaller wave (1) propagates slower than the second and the third larger waves (2 ;3). The second wave inelastically collides\\nwith the \\frst wave about 100 cm further downstream.\\ncompromised. Subsequently, it is plausible that the opti-\\nmal memory capacity will be achieved when SL waves are\\ndetected midway the inlet and the far downstream region\\nwhere the SL waves merge, which is the approach adopted\\nin the experiments conducted in this work.\\nDiscussion. { Before we conclude, we discuss sev-\\neral potential improvements of SLRC that can be done in\\nthe future. Firstly, the experimental setup used in this\\nwork holds the potential to be extended to enable mea-\\nsurements, where the RC system could generate a new\\nTS using its own predictions as the input data. Such an\\noperating regime is called the generative mode and its im-\\nplementation is a challenging task for both physical and\\nalgorithmic RC systems [8, 15, 30]. In particular, the op-\\neration in a generative mode requires introducing a feed-\\nback circuit that converts the output of the readout layer\\ninto new input data at every discrete time step. However,\\ntransfer of data from the readout layer, which, typically,\\nis a computer program itself, to the input layer of the RC\\nsystem introduces a time delay that can be several orders\\nof magnitude larger than the timescale of the reservoir dy-\\nnamics. For instance, if the reservoir's natural frequency is\\nof order of several GHz [28], all readout calculations must\\nbe done at the GHz scale or above. In the case of SLRC,\\nwhere the frequencies of SL waves are of order of several\\nHz, this problem can be resolved using a laser vibrometry\\ntechnique to detect the SL waves [44] and then process-\\ning the optical signal using ultrafast opto-electronic andphotonics systems that have already been adopted in the\\n\\feld of ML [45,46]. The time delay introduced by such a\\nprocessing scheme will be negligibly small compared with\\nthe temporal dynamics of SL waves.\\nApart from the operation in the generative mode, the\\nintroduction of a time delay between the output and input\\nof the reservoir may help increase the memory capacity of\\nthe RC system [28, 47, 48], also enabling the reservoir to\\noperate using just a single arti\\fcial neuron [49]. Yet, in\\nthe \\feld of \\ruid dynamics, the introduction of a feedback\\ncontrol has been used to stabilise liquid \\flm \\rows [50],\\nwhich may also help improve the performance of SLRC.\\nConclusions. { We have experimentally demon-\\nstrated a physical computational reservoir system enabled\\nby solitary-like waves propagating on the surface of a liq-\\nuid thin \\flm \\rowing over an inclined surface. While com-\\nputational reservoirs based on optical and matter-wave\\nsolitons have been theoretically predicted but not experi-\\nmentally demonstrated yet, solitary wave dynamics of liq-\\nuid \\flms has unique features that we have employed in\\nthe \\feld of machine learning for the \\frst time. We trained\\nand exploited the solitary-like wave reservoir to perform\\nseveral essential benchmark tests used to evaluate the per-\\nformance of machine learning algorithms designed for digi-\\ntal computer, showing that the solitary-like wave reservoir\\nhas considerable memory capacity and can predict chaotic\\ntime series more e\\u000eciently than certain standard algorith-\\nmic machine learning models.\\np-6 Reservoir computing based on solitary-like waves dynamics\\nREFERENCES\\n[1]Nakajima K. and Fisher I. ,Reservoir Computing\\n(Springer, Berlin) 2021.\\n[2]Kirby K. G. ,Proc. Florida AI Research Symposium\\n(FLAIRS) , (1991) 66.\\n[3]Maass W., Natschl \u007fager T. andMarkram H. ,Neural\\nComput. ,14(2002) 2531.\\n[4]Luko \\u0014sevi\\u0014cius M. andJaeger H. ,Comput. Sci. Rev. ,3\\n(2009) 127.\\n[5]Gauthier D. J., Bollt E., Griffith A. andBarbosa\\nW. A. S. ,Nat. Commun. ,12(2021) 5564.\\n[6]Marcellino M., Stock J. H. andWatson M. W. ,\\nJ. Econom. ,135(2006) 499.\\n[7]Small M. ,Applied Nonlinear Time Series Analysis: Ap-\\nplications in Physics, Physiology and Finance (World Sci-\\nenti\\fc, Singapore) 2005.\\n[8]Jaeger H. ,A tutorial on training recurrent neural net-\\nworks, covering BPPT, RTRL, EKF and the \\\\echo state\\nnetwork\\\" approach (GMD Report 159, German National\\nResearch Center for Information Technology) 2005.\\n[9]Raji J. I. andPotter C. J. ,PLOS ONE ,16(2021) 1.\\n[10]Fernando C. andSojakka S. ,Pattern recognition in a\\nbucket in proc. of Advances in Arti\\fcial Life , edited by\\nBanzhaf W., Ziegler J., Christaller T., Dittrich\\nP.andKim J. T. , (Springer, Berlin) 2003 pp. 588{597.\\n[11]Jones B., Stekel D., Rowe J. andFernando C. ,Is\\nthere a liquid state machine in the bacterium escherichia\\ncoli? in proc. of 2007 IEEE Symposium on Arti\\fcial Life\\n2007 pp. 187{191.\\n[12]Nakajima K. andAoyagi T. ,IEICE Tech. Rep. ,115\\n(2015) 109.\\n[13]Nakajima K. ,Jpn. J. Appl. Phys. ,59(2020) 060501.\\n[14]Goto K., Nakajima K. andNotsu H. ,N. J. Phys. ,23\\n(2021) 063051.\\n[15]Maksymov I. S., Pototsky A. andSuslov S. A. ,\\nPhys. Rev. E ,105(2021) 044206.\\n[16]Matsuo T., Sato D., Koh S.-G., Shima H., Naitoh\\nY., Akinaga H., Itoh T., Nokami T., Kobayashi\\nM.andKinoshita K. ,ACS Appl. Mater. Interfaces ,14\\n(2022) 36890.\\n[17]Jiang J. andLai Y.-C. ,Phys. Rev. Research ,1(2019)\\n033056.\\n[18]Zeng Q., Wu Z., Yue D., Tan X., Tao J. andXia G. ,\\nAppl. Opt. ,59(2020) 6932.\\n[19]Marcucci G., Pierangeli D. and Conti C. ,\\nPhys. Rev. Lett. ,125(2020) 093901.\\n[20]Silva N. A., Ferreira T. D. andGuerreiro A. ,New\\nJ. Phys. ,23(2021) 023013.\\n[21]Kivshar Y. S. andAgrawal G. P. ,Optical Soli-\\ntons: From Fibers to Photonic Crystals (Academic Press,\\nNew York) 2003.\\n[22]Liu J. andGollub J. P. ,Phys. Fluids ,6(1994) 1702.\\n[23]Chang H.-C. ,Annu. Rev. Fluid Mech. ,26(1994) 103.\\n[24]Kalliadasis S., Ruyer-Quil C., Scheid B. andVe-\\nlarde M. G. ,Falling Liquid Films (Springer-Verlag, Lon-\\ndon) 2012.\\n[25]Balmforth N. J. andMandre S. ,J. Fluid Mech. ,514\\n(2004) 1.\\n[26]Hu X. andCubaud T. ,Phys. Rev. Lett. ,21(2018)\\n044502.\\n[27]Maksymov I. S. andPototsky A. ,Appl. Sci. ,13(2022)1888.\\n[28]Watt S. andKostylev M. ,Phys. Rev. Appl. ,13(2020)\\n034057.\\n[29]Thissen U., van Brakel R., de Weijer A. P.,\\nMelssen W. J. andBuydens L. M. C. ,Chemom. In-\\ntell. Lab. Syst. ,69(2003) 35.\\n[30]Luko \\u0014sevi\\u0014cius M. ,A practical guide to applying echo\\nstate networks inNeural Networks: Tricks of the Trade,\\nReloaded , edited by Montavon G., Orr G. B. and\\nM\u007fuller K.-R. , (Springer, Berlin) 2012 pp. 659{686.\\n[31]Mackey M. C. andGlass L. ,Science ,197(1977) 287.\\n[32]Maciel L., Gomide F., Santos D. andBallini R. ,\\nExchange rate forecasting using echo state networks for\\ntrading strategies in proc. of 2014 IEEE Conference on\\nComputational Intelligence for Financial Engineering and\\nEconomics (CIFEr) 2014 pp. 40{47.\\n[33]Remoissenet M. ,Waves Called Solitons: Concepts and\\nExperiments (Springer) 1994.\\n[34]Ganguli S., Huh D. andSompolinsky H. ,PNAS ,105\\n(2008) 18970.\\n[35]Verzelli P., Alippi C. andLivi L. ,Sci. Reps. ,9(2019)\\n13887.\\n[36]Bertschinger N. andNatschl \u007fager T. ,Neural Com-\\nput.,16(2004) 1413.\\n[37]Furuta T., Fujii K., Nakajima K., Tsunegi S., Kub-\\nota H., Suzuki Y. andMiwa S. ,Phys. Rev. Appl. ,10\\n(2018) 034063.\\n[38]Liao Z., Wang Z., Yamahara H. andTabata H. ,\\nChaos Solitons Fract. ,153(2021) 111503.\\n[39]Rodan A. andTino P. ,IEEE Trans. Neural Netw. ,22\\n(2011) 131.\\n[40]Morales G. B., Mirasso C. R. andSoriano M. C. ,\\nNeurocomputing ,461(2021) 705. Chaos Solitons Fract. ,153(2021) 111503.\\n[39]Rodan A. andTino P. ,IEEE Trans. Neural Netw. ,22\\n(2011) 131.\\n[40]Morales G. B., Mirasso C. R. andSoriano M. C. ,\\nNeurocomputing ,461(2021) 705.\\n[41]Khan S. ,Mackey Glass Time Series Prediction Using\\nLeast Mean Square (Matlab Central File Exchange) 2023.\\n[42]Shkadov V. Y. , Izv. Akad. Nauk SSSR,\\nMekh. Zhidk. Gaza ,1(1967) 43.\\n[43]Shkadov V. Y. , Izv. Akad. Nauk SSSR,\\nMekh. Zhidk. Gaza ,2(1968) 20.\\n[44]Maksymov I. andPototsky A. ,Phys. Rev. E ,100\\n(2019) .\\n[45]Sorokina M. ,J. Phys. Photonics ,2(2020) 044006.\\n[46]Chembo Y. K. ,Chaos ,30(2020) 013111.\\n[47]Riou M., Torrejon J., Garitaine B., Araujo F. A.,\\nBortolotti P., Cros V., Tsunegi S., Yakushiji K.,\\nFukushima A., Kubota H., Yuasa S., Querlioz D.,\\nStiles M. D. andGrollier J. ,Phys. Rep. Appl. ,12\\n(2019) 024049.\\n[48]Kondrashov A. V., Nikitin A. A., Nikitin\\nA. A., Kostylev M. and Ustinov A. B. ,\\nJ. Magn. Magn. Mater. ,563(2022) 169968.\\n[49]Appeltant L., Soriano M. C., der Sande G. V.,\\nDanckaert J., Massar S., Dambre J., Schrauwen\\nB., Mirasso C. R. andFischer I. ,Nat. Commun. ,2\\n(2011) 468.\\n[50]Thompson A. B., Gomes S. N., Pavliotis G. A. and\\nPapageorgiou D. T. ,Phys. Fluids ,28(2016) 012107.\\np-7\",\"30\":\" EVOLUTIONARY AUGMENTATION POLICY OPTIMIZATION FOR\\nSELF-SUPERVISED LEARNING\\nA P REPRINT\\nNoah Barrett1,2,\\u2020, Zahra Sadeghi1,2,\\u2020,*, and Stan Matwin1,2\\n1Facutly of Computer Science, Dalhousie University, Halifax, Canada\\n2Institute for Big Data Analytics, Halifax, Canada\\n*Correspondence: zahras@dal.ca\\nABSTRACT\\nSelf-supervised learning (SSL) is a Machine Learning algorithm for pretraining Deep Neural Networks (DNNs)\\nwithout requiring manually labeled data. The central idea of this learning technique is based on an auxiliary\\nstage aka pretext task in which labeled data are created automatically through data augmentation and exploited\\nfor pretraining the DNN. However, the effect of each pretext task is not well studied or compared in the\\nliterature. In this paper, we study the contribution of augmentation operators on the performance of self supervised\\nlearning algorithms in a constrained settings. We propose an evolutionary search method for optimization of\\ndata augmentation pipeline in pretext tasks and measure the impact of augmentation operators in several SOTA\\nSSL algorithms. By encoding different combination of augmentation operators in chromosomes we seek the\\noptimal augmentation policies through an evolutionary optimization mechanism. We further introduce methods for\\nanalyzing and explaining the performance of optimized SSL algorithms. Our results indicate that our proposed\\nmethod can \\ufb01nd solutions that outperform the accuracy of classi\\ufb01cation of SSL algorithms which con\\ufb01rms the\\nin\\ufb02uence of augmentation policy choice on the overall performance of SSL algorithms. We also compare optimal\\nSSL solutions found by our evolutionary search mechanism and show the effect of batch size in the pretext task on\\ntwo visual datasets.\\nKeywords Self-supervised learning, Data Augmentation, Evolutionary Algorithms, Deep Learning, Deep Neural\\nNetworks, Optimization, Explainability, Arti\\ufb01cial Intelligence\\n1 Introduction\\nThe supreme power of Machine Learning algorithms is founded on supervised learning techniques. However, while\\nthese algorithms have shown to be tremendously successful in solving classi\\ufb01cation problems, they remain entirely\\ndependent on a large corpus of manually annotated data. As a result, the whole process of learning is not autonomous\\nand tends to be in\\ufb02uenced by the error of annotation. Self-supervised learning (SSL) has been introduced in response\\nto tackling these limits by making the possibility of training DNNs without relying on labeled data. The core of SSL\\nis based on an auxiliary phase aka pretext task which focuses on pretraining networks using automatically generated\\nlabeled data. One crucial aspect of pretext tasks is transforming unlabeled images using augmentation operators in\\norder to produce labeled samples. Although many different SSL paradigms have been developed, there is no speci\\ufb01c\\ninvestigation about their level of effectiveness and usefulness. Moreover, there is no speci\\ufb01c study to investigate\\nthe effect of augmentation operators on the performance of each of the proposed self supervised algorithms. In this\\nyThese authors contributed equally to this workarXiv:2303.01584v1  [cs.CV]  2 Mar 2023 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\nstudy, we show that the choice of augmentation operators can bear an impact on the achievable performance of the\\ndownstream problem. Through a series of experiments, we seek to measure the in\\ufb02uence of different augmentation\\npolicies. We formulate an evolutionary search algorithm in order to optimize the transformation settings that lead to\\nbest outputs. To this end, we focus on four state-of-the-art SSL algorithms that have indicated high performance on\\nvarious datasets. This paper encompasses the following original contributions. We \\ufb01rst investigate the best combination\\nof augmentation operators in four state-of-the-art SSL algorithms through an evolutionary learning algorithm. Then,\\nwe compare the performance of these SSL methods before and after the optimization process. We \\ufb01nd and evaluate\\nthe best augmentation operators for each SSL method across two different datasets. In the end, we run explanatory\\nanalysis over the optimized chromosomes to understand the impact and importance of augmentation operators for\\neach SSL algorithm and dataset. This paper is organized as follows: We provide an overview about Self-Supervised\\nLearning, Auto Augmentation and Search Optimization Algorithms in section 2. In section 3, implementation details\\nare mentioned. Then, in section 4, our method is described. The experiments and results are provided in section 5. The\\nexplainability methods are presented in section 6. Finally, section 7 concludes the paper.\\n2 Related work\\nIn this section, we provide a literature survey about related prior work in three subsections. First, we introduce the\\nparadigm of self-supervised learning algorithms and review ongoing research in this \\ufb01eld. Secondly, we go over recent\\nadvances in Auto Augmentation. Finally, we provide an overview about Intelligent Search algorithms.\\n2.1 Self-Supervised Learning\\nSelf-Supervised Learning has increased attention in Machine Learning due to its ability to handle unlabeled data more\\nef\\ufb01ciently than the traditional unsupervised learning. This is achieved by designing a new phase of learning known as\\npretext task, which is an attempt for solving an auxiliary classi\\ufb01cation task. The auxiliary task is focused on classifying\\nthe data into self-labeled categories in order to obtain a pretrained model with rich feature representation about the\\nunderlying structure of data. Various approaches have been proposed for designing the pretext tasks, which can be\\nclassi\\ufb01ed into patch-based, instance-based, and video-based methods. Patch-based methods aim to learn the relative\\nposition between patches of an image Doersch et al. (2015), while instance-based methods utilize the whole image and\\ntake advantage of an augmentation task such as an af\\ufb01ne transformation Doersch et al. (2015) or colorization Isola\\nand Efros (2016). In contrast, video-based methods design a task that can take advantage of the sequential ordering of\\nframes in a video clip Misra et al. (2016). Self-supervised learning can also be categorized into generative or contrastive\\nmodels. Generative modeling approaches aim to reconstruct the input through the deployment of autoencoders Pathak\\net al. (2016) and GANs Donahue et al. (2016) while the contrastive approach includes the attempts to increase the\\ndiscrimination between different images and decrease it for similare ones by applying contrastive learning Oord et al.\\n(2018). Self-supervised learning is highly associated to contrastive models, while the generative approach is often\\nconsidered as an unsupervised learning scheme. A variety of different approaches have been proposed in order to solve\\nthe pretext task which are highly focused on different forms of augmentation, such as spatial location and relative\\npositions of patches Noroozi and Favaro (2016)Dosovitskiy et al. (2014), image transformation including rotation positions of patches Noroozi and Favaro (2016)Dosovitskiy et al. (2014), image transformation including rotation\\nGidaris et al. (2018), altering global statistics while preserving local statistics Jenni et al. (2020), data and color jittering,\\npainting and coloring Isola and Efros (2016). Given the fact that such pretext tasks are intrinsically different, it is\\nvery intriguing to investigate the effect of pretraining process of pretext task in quest of discovering performance\\ndifferences in downstream tasks. In this paper, we focus on comparing four state-of-the-art SSL algorithms, namely,\\nBYOL gri (2020), simsiam Chen and He (2021), NNCLR Dwibedi et al. (2021) and SWA V Caron et al. (2020). All of\\nthese SSL algorithms share fundamental similarities in their architecture and underlying methodology. Speci\\ufb01cally, all\\nalgorithms employ a Siamese architecture at its core. Siamese architectures consist of a shared input network which\\nfeeds into two separate neural networks that share weights, and an architecture that is analogous to siamese twins in\\nnature. Conventionally, Siamese networks are used for images, where the inputs are different images, and the actual\\n2 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\ncomparability of images are determined in a supervised manner Chen and He (2021). BYOL, SimSiam, NNCLR, and\\nSwA V share the general notion of contrastive learning using a siamese architecture, however the speci\\ufb01c details of\\nimplementation and theoretical motivation creates signi\\ufb01cant differences in the four algorithms. It is of great importance\\nto note that the authors of these four algorithms highlight, to varying degrees, their invariability to augmentation. The\\nproposed work looks into the margins of these \\ufb01ndings and expose that while there is an astounding invariance to\\naugmentation strategies in the proposed algorithms overall, there still is room for improvement for understanding the\\neffect of augmentation in contrastive learning. The key idea of BYOL is that from a given representation, called the\\ntarget representation, it is possible to train an enhanced representation,i.e., the online representation. BYOL works\\niteratively, to improve the learned representation by using a slow moving average of the online network as the target\\nnetwork gri (2020). BYOL does not require negative samples, and claims to be insensitive to batch sizes and the types of\\naugmentations used. BYOL\\u2019s ability of not requiring a large batch sizes, negative pairs, as well as having a novel level\\nof robustness to augmentations was what made it a breakthrough algorithm at the time of its release. BYOL\\u2019s robustness\\nto augmentation is in comparison to one of the earliest forms of contrastive SSL SimCLR Chen et al. (2020), it is shown\\nthat BYOL suffers a much smaller performance drop than SimCLR when only using random crops. SimSiam is an SSL\\nmethod that opts for simplicity and highlights the signi\\ufb01cance of the Siamese network architecture of SSL methods.\\nSimSiam shows that the shared weight con\\ufb01guration is key to many core SSL algorithms, and a simple stop gradient\\napproach can be used to prevent collapse Chen and He (2021) which is a common issue in many algorithms employing\\nSiamese networks. Collapse occurs when the model maps every image to the same point, maximizing similarity when\\ncomparing, but in turn not learning any important information about problem at hand. SimSiam provided surprising\\nevidence that meaningful representations can be learned without the use of negative sample pairs, large batches and\\nmomentum encoders. The \\ufb01rst two of these were already unnecessary in the earlier work shown in BYOL, however,\\nthe momentum encoder component was thought of as a crucial component to its avoidance of collapsed solutions.\\nPerhaps, the most remarkable discovery produced by SimSiam could be that a simple stop gradient is all that is needed\\nto successfully train a Siamese architecture. Compared to BYOL and SimSiam, SwA V provides a novel mechanism for\\nlearning. Following the vanilla approach of contrastive learning, BYOL and SimSiam learn by predicting the \\\"closeness\\\"\\nof two views of a sample, this is a core concept to Siamese Networks and their impactful discoveries in representation\\nlearning. Simply put, SwA V learns by computing a code from an augmented version of an image and then predicts\\nthis code from other augmented versions of the same image. The method simultaneously clusters data while enforcing\\nconsistent cluster assignments between different views Caron et al. (2021). Additionally, SwA V introduces a novel\\naugmentation multi-crop, which allows for more memory ef\\ufb01cient comparisons by utilizing a small set of large crop\\nwindows and a larger set of small crop windows. This means, despite employing a Siamese architecture, SwA V is not\\nactually a form of contrastive learning. However, SwA V at its core still aims to learn a strong latent representation of\\nthe problem at hand. The use of this alternate approach yields a new breakthrough unfound by BYOL and SimSiam, the problem at hand. The use of this alternate approach yields a new breakthrough unfound by BYOL and SimSiam,\\nwhich is the ability to signi\\ufb01cantly reduce the number of pairwise feature comparisons, reducing the computational\\nand memory requirements of the algorithm. Compared to the other discussed SSL methods, NNCLR introduces a very\\nnovel concept with respect to the selection of positive pairs. Rather than deriving positive samples for an image via\\naugmentation, they show that it is possible to use the nearest neighbours of an image of interest in the given data space\\nas positive samples. Like BYOL and SimSiam, NNCLR is a contrastive learning method, which similar to SimSiam\\nemploys a very simple architecture. The novelty is in the selection of positive samples from the given dataset, rather\\nthan derivation from augmentation. NNCLR samples positive samples by computing the nearest neighbours of a given\\nsample in the learned latent space of the dataset. This approach provides more semantic class wise variations rather\\nthan pre-de\\ufb01ned transformations which tend to provide more geometric information Dwibedi et al. (2021). Alike all the\\nabove discussed approaches, NNCLR \\ufb01nds that it is not very reliant on data augmentation.\\nAll four discussed algorithms share the use of a Siamese architecture, SimSiam can be thought of as the simplest version\\nof these learning approaches using a Siamese architecture. BYOL adds a slight sort of complexity to the learning\\nmethod by employing a slow moving average approach for the target component of the Siamese network. Like SimSiam,\\nNNCLR also opts for simplicity in its architecture, however it refrains from employing only augmentation to generate\\n3 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\npositive samples, and instead, determines the nearest neighbours of a given image to select positive samples. SwA V\\ndiffers from all three methods in that it utilizes a swapped prediction mechanism for learning rather than the typical\\ncontrastive mechanism. All four algorithms have recently shown remarkable breakthroughs in the Self-Supervised\\nComputer vision world, and for this reason they have been chosen as focal points in our work. The four algorithms all\\nclaim to have a varying degree of invariance to data augmentation based on empirical \\ufb01ndings. However, the contrastive\\nmechanism of BYOL, SwA V and SimSiam, rely solely on data augmentation. Given the limited discussion on the\\neffect of data augmentation, the question of its impact to the different SSL algorithms still remains. In this work, we\\nhypothesize that by picking the best set of augmentation operators, we can boost the performance of SSL algorithms.\\n2.2 Auto Augmentation\\nData Augmentation is a regularization technique that has been mainly developed for dealing with over\\ufb01tting issue\\nin training DNNs by increasing the diversity of training examples Shorten and Khoshgoftaar (2019). Recently, this\\napproach has been adopted in contrastive self-supervised learning procedure for creating additional positive samples\\nwhile preserving the semantic content of observations V on K\\u00fcgelgen et al. (2021). As mentioned in previous section,\\nthe positive samples will be discriminated against negative samples in a training procedure to obtain a rich embedding\\nthat can be used for learning different tasks. While data augmentation is the core basis of self-supervised learning,\\nlittle attention has been placed over understanding its impact and there is not many investigations about the theoretical\\nsuccess of augmentation tasks. To the best of our knowledge there is no systematic study to assess the effect of\\nemploying different augmentation policies in Self-supervised Learning. Some authors have pointed out the important\\nrole of transformation operators in self-supervised feature learning performance. Jenni et al., argued that transformation\\nfunction selection is data-speci\\ufb01c and hence each dataset might bene\\ufb01t from a different set of transformations. In\\naddition, different tasks could require tailored augmentation techniques. For example, for anomaly detection a novel\\nproxy task has been proposed Li et al. (2021). Along the recent endeavors in AutoML which aim to automate the\\nMachine Leaning procedure, Auto Augmentation methods are developed in order to seek the optimal augmentation\\nstrategies for training Deep Neural Networks to improve the performance of DNNs. AutoAugment can be regarded as\\none of the pioneering approaches for designing an automatic data augmentation learning procedure Cubuk et al. (2019).\\nThis approach is based on employing reinforcement learning to explore search space of augmentation policies. In Lim\\net al. (2019) the auto augmentation procedure is accelerated by matching the density between training and validation\\ndata. PBA Ho et al. (2019) is another approach for that uses a hyperparameter search algorithm to learn a schedule of\\naugmentation polices.\\n2.3 Search optimization algorithms\\nSearch optimization algorithms are one of the core components for the tremendous success of Machine Learning\\ntechniques and they are increasingly exploited in fundamental problems of parameter and hyperparameter tuning and\\nmodel selection Brazdil et al. (2022). An optimization algorithm seeks to adjust the con\\ufb01guration of Machine Learning\\nalgorithms by minimizing a cost function such that it will lead to accurate estimation and prediction. Through iterative\\nsearch algorithms, the best con\\ufb01guration that best describes a distribution of observable data can be obtained. There\\nare a variety of search optimization techniques which are employed for Machine Learning tasks Yang and Shami are a variety of search optimization techniques which are employed for Machine Learning tasks Yang and Shami\\n(2020). The simplest one is known as exhaustive search which looks for \\ufb01nding the best parameters by checking\\nas many possible solution as possible based on the level of desirable performance or computational resources. Two\\nwell-known subcategory of this type are random search which randomly explores the search space, and grid search\\nwhich systematically discretizes the search space and checks the solutions that fall on in grid. Other more intelligent\\nmethods which are developed for a more effective search are including Gradient based methods, Reinforcement\\nLearning and Evolutionary Algorithms. Gradient based approaches are iterative search algorithms which attempt to\\n\\ufb01nd the optimal solutions according to gradient value at each step. These algorithms often gets stuck in local optima.\\nReinforcement learning algorithms, on the other hand, is a search paradigm that traverses solution space based on taking\\n4 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\nTable 1. All augmentation operators used and the associated intensity ranges. This set of operators is the same as\\nAutoAugment Cubuk et al. (2019).\\nAugmentation Intensity Range\\nHorizontalFlip 0.0, 1.0\\nVerticalFlip 0.0, 1.0\\nShearX 0.0, 0.3\\nShearY 0.0, 0.3\\nTranslateX 0, 14\\nTranslateY 0, 14\\nRotate -30, 30\\nColor 0.1, 1.9\\nSolarize 0.0, 1.0\\nContrast 0.1, 1.9\\nSharpness 0.1, 1.9\\nBrightness 0.1, 1.9\\nactions from a prede\\ufb01ned list. Each action changes the state of the agent and induces a reward. The goal is to maximize\\nthe return or the sum of discounted rewards. Evolutionary optimization is referred to bio-inspired methods in which a\\npopulation of solutions are created, evolved and evaluated iteratively. Three popular Evolutionary algorithm variants are\\nGenetic Algorithms (GA), Genetic Programming (GP) and Evolutionary Strategies (ES) O\\u2019Neill (2009). Evolutionary\\nalgorithms have attracted attention of researcher due to multiple unique properties. They can search the space in a\\nparallel and independent manner, they are not based on gradient calculation and hence do not suffer from exploding or\\nvanishing gradients Such et al. (2017). Due to these intriguing features, we have applied a Genetic Algorithm in this\\npaper which is described in the next section.\\n3 Implementation Details\\nThis work is interested in exploring a massive amount of different con\\ufb01gurations of augmentations in the pretext task of\\ncommon SSL algorithms, in order to do so, two standard benchmark which are commonly used for image classi\\ufb01cation\\nare used. We explore the performance of four SSL algorithms using two different batch sizes of 32 and 256. We carry\\nout the experiments on CIFAR-10 and SVHN datasets. In the following subsections, we provide more in-depth details\\nabout the outcome of each experiment.\\n3.1 Dataset\\nCIFAR-10 . The CIFAR-10 dataset Krizhevsky (2009) is a popular coloured image dataset consisting of 10 classes of\\nairplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck. This dataset consists of 60000 32x32 images,\\nwith 50000 train images and 10000 test images. The test set consists of 1000 randomly sampled instances from each\\nclass, and the remaining 50000 are used for the training set. The classes in this dataset are mutually exclusive and the\\ncreators took extra caution to ensure the similar classes of truck and automobile contain no overlap.\\nSVHN . The Street View House Numbers (SVHN) Dataset Netzer et al. (2011) is an image dataset consisting of real\\nworld images of house numbers taken from Google street view images. This dataset is presented as a more challenging\\nversion of MNIST Deng (2012) consisting of 600,000 32x32 labeled digits which are cropped from street view images.\\nThe classi\\ufb01cation problem posed in SVHN is signi\\ufb01cantly harder than MNIST as it contains real-world images of\\nnumbers.\\n5 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\n3.2 Network Architecture and Training Speci\\ufb01cations\\nFor all experiments, a small and simple convolutional neural network is used as the backbone for both pretraining\\nand downstream tasks. As illustrated in 1, this network consists of 3 convolutional blocks. Each block contains two\\nconvolutions with a kernel size of 3x3, followed by ReLU activation functions, a max pool and a batch normalization\\noperation. For each SSL algorithm, i.e., BYOL, NNCLR, SimSiam and SwA V , this backbone is applied in accordance\\nto the speci\\ufb01c algorithm. All the hyperparameter used for network training of each of the SSL algorithm are according\\nto the original papers. For the downstream classi\\ufb01cation task, all experiments use an Adam optimizer with a learning\\nrate of 0.001 and weight decay of 0.0005. Cross entropy is used as loss function. In addition, a three layer linear model\\nwith ReLU activation is used on top of the pretrained networks. We train both the pretext and downstream tasks for\\n10 epochs. The backbone is \\ufb01rst pretrained using one of the four SSL algorithms then the linear head is added to the\\nnetwork for \\ufb01ne-tuning on the downstream task. For each experiment, we roughly exploit 4GB of GPU and 12 CPUs.\\nFigure 1. Architecture of Network used for the downstream task.\\nThe implementation of the proposed work relies on several different libraries. For the deep learning components, such\\nas model architectures, dataset loading, and supervised training, PyTorch is used Paszke et al. (2019). Building on\\nPyTorch, the python library, Lightly Susmelj et al. (2020), implements cutting edge SSL algorithms, this library is used\\nfor implementing the four SSL algorithms in this research. To implement the evolutionary mechanism in this research,\\nthe python library Deap Fortin et al. (2012) is employed.\\n4 Method\\nWe propose two evolutionary mechanism for augmentation operator optimization through Genetic algorithm to \\ufb01nd the\\nbest performing chromosomes and corresponding models. The \\ufb01rst approach is a single optimization method, in which\\naugmentation policy of each one of the four SSL algorithm is optimized individually, whereas the second approach\\nis a multi optimization method through which not only we optimize the augmentation policy but also optimize the\\ndownstream task performance by \\ufb01nding the best performing algorithm.\\n4.1 Evolutionary Data Augmentation Optimization\\nThe proposed research aims to apply a Genetic algorithm to optimize augmentation policies for four different cutting-\\nedge self-supervised algorithms. From a genetic algorithm perspective, the key area that requires the most attention is\\nthe de\\ufb01nition of the \\ufb01tness function. The proposed \\ufb01tness function serves as a proxy function that aims to gauge the\\nimpact of changing the augmentation policies for a given self-supervised algorithm. Below, our formulation of the \\ufb01tness\\nfunction is elaborated. To represent the augmentation strategies as chromosomes we employ a value representation. We\\nexperiment with two different \\ufb02avours of evolutionary augmentation optimization, namely single optimization (SO)\\nand Multiple Optimization (MO) modes. With SO we choose one SSL algorithm and exclusively optimize it with GA,\\nwhile with MO all four SSL algorithms are evolved together.\\n6 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\n4.1.1 Chromosome De\\ufb01nitions and Augmentation Policies\\nTaking inspiration from AutoAugment and the preceding work in learning augmentation policies, the proposed value\\nencoding assumes a set of kaugmentations, A=fa1;a2;:::;a kg. For each augmentation aian intensity value i\\ncan be de\\ufb01ned. Authors of Cubuk et al. (2019) de\\ufb01ned a range of values for intensity of each augmentation operator.\\nFor the purpose of this work, these ranges have been borrowed. Table 1 shows all the different operators used in our\\nexperiments along with their possible ivalues. Letikbe the intensity value assigned to augmentation operator akand\\nKbe the maximum number of augmentation operators, in SO, we formulate an augmentation policy in a chromosome\\nasCi=f(a1;i1);(a2;i2);:::; (al;il)g, wherel\\u0014K. For the purpose of our experiments, we use a value of 3 for\\nl. In MO, we represent a chromosome as Ci= (algj;f((a1;i1);(a2;i2);:::; (al;il)g)wherealgjis one of the SSL\\nalgorithms infSwAV;NNCLR;Simsiam;Byol g.\\n4.1.2 Fitness Function\\nThe proposed \\ufb01tness function aims to evaluate augmentation policies for SSL algorithms based on downstream tasks.\\nMore speci\\ufb01cally, the test accuracy in the downstream supervised task serves as the \\ufb01tness for augmentation optimization.\\nIt is important to note that in this work we are focused on the augmentation policies for the pretext task, and a \\ufb01xed\\nset of augmentations are used for the supervised downstream task. The most crucial part of our \\ufb01tness function is the\\n\\ufb01xing of randomness. By \\ufb01xing every random component of the deep learning pipeline, the difference between two\\naugmentation strategies can be soundly compared in relative terms. In order to compensate for bias due to \\ufb01xing the\\nrandom components of the deep learning pipeline we evaluate the our method using Ndifferent random seeds for the\\nneural network initialization and data shuf\\ufb02ing. The seed values for each experiment are shown in Table 3.\\n4.1.3 Selection\\nThere exists a large swath of different selection methods such as Roulette, tournament and Rank ?. Our method opts for\\na simple and common approach of Roulette. In this approach, every chromosome has an equal chance of being selected\\nproportional to their \\ufb01tness.\\n4.1.4 Crossover\\nWhen selecting a crossover operation caution must be taken to prevent the children chromosomes from having duplicate\\naugmentations. To handle this issue, Partially Mapped Crossover (PMX) Reeves (2003) is employed. This method\\navoids creation of chromosomes with duplicate genes. When performing crossover between two or more chromosomes,\\nthe mechanism for preventing duplicate genes is based purely on the augmentation operator and not the intensity. When\\ntwo chromosomes are crossed over, the shared genes are completely copied over to maintain the same intensity as the\\nparent\\u2019s gene. When performing crossover for our MO approach, PMX is applied to the augmentation policy portion of\\nthe chromosome, but for the SSL gene, it is probabilistically swapped with the other chromosome.\\n4.1.5 Mutation\\nFor mutation, only the intensities of the augmentation operators are mutated. In order to perform this, a custom mutation\\nfunction is created. We call this function MutGaussianChoice . This mutation operator probabilistically mutates the\\ngenes intensity values within the acceptable range of intensity for that speci\\ufb01c operator. The intensity is increased or\\ndecreased incrementally by the range of the intensity values (as shown in Table 1) divided by the increment value which\\nis computed using equation 1. For MO, MutGaussianChoice is applied to the augmentation policy portion of the\\nchromosome, but for the SSL gene, the gene is randomly mutated to a different SSL algorithm.\\n7 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\nincrement =max(irange )\\u0000min(irange )\\n10(1)\\n1.Equation for computing increment value for MutGaussianChoice\\npc= (fmax\\u0000f0)=(fmax\\u0000\\u0016f);f0\\u0015\\u0016f\\npc= 1;f0<\\u0016f(2)\\npm= (fmax\\u0000f)=(fmax\\u0000\\u0016f);f0\\u0015\\u0016f\\npm= 0:5;f < \\u0016f(3)\\n4.1.6 Adaptive Mutation Rates\\nWhen applying Mutation and Crossover, whether to apply the operation to the chromosome(s) is probabilistically\\ndetermined using a mutation and crossover rate. Our method employs adaptive crossover and mutation rates (i.e. pcand\\npm) inspired by Srinivas and Patnaik (1994) as shown in Equations 2 and 3: where fmax and\\u0016fdenote the maximum\\n\\ufb01tness value and the average \\ufb01tness value of the population respectively, and f0denotes the larger \\ufb01tness of the two\\nchromosomes to be crossed or mutated.\\n5 Experiments\\nIn this work, we aim to understand the impact of augmentation operators used in the pretext task of four popular SSL\\nalgorithms. By using an evolutionary mechanism to evolve augmentation policies, hundreds of models are trained during\\none run of the algorithm. To provide baselines for comparison, we train models in both supervised and self-supervised\\nfashions with the augmentation operators according to the original papers. This provides us with an insight into how\\nour evolved augmentation pipelines compare with a purely supervised approach as well as SSL approaches using\\nthe original augmentation pipelines. Given that we train the networks under a constrained setting (i.e., using a small\\nnetwork and minimum number of training epochs), we conduct another experiment in which we study how the best\\nevolved augmentation pipelines perform when trained for more epochs in both the pretext task and downstream task.\\nFor this experiment, we exploit the optimal augmentation operators we found through our proposed evolutionary search\\nmechanism.\\nSupervised setting . For training the supervised baseline models, we conduct the supervised classi\\ufb01cation task using\\nall of the same hyperparameters as the downstream task in the SSL experiments with no pretraining, i.e., we train\\na randomly initialized architecture from scratch. This baseline gives us the ability to compare the representations\\nlearned with self-supervision and a randomly initialized representation. In all supervised baselines we use the same\\ncon\\ufb01guration described in the downstream task.\\nSelf-supervised baselines .To understand how the learned augmentation pipelines compare with the original SSL\\nalgorithm, we train models using the augmentations used in the original papers. Because our speci\\ufb01c training\\ncon\\ufb01guration is signi\\ufb01cantly constrained in terms of network architecture and trarining epochs compared to the original\\npapers, these baselines serve as a method for us to understand how the SSL algorithm behaves in our constrained\\nsetting. When training these baselines, the exact same hyperparameters such as batch size, and number of epochs as the\\nalgorithms trained with evolved augmentations are used. This allows us to understand how changing the augmentations\\nin our setting affect the SSL algorithm with respect to the original augmentation con\\ufb01guration.\\nEvolutionary SSL . The \\ufb01tness function in the proposed GA employs the downstream test accuracy as its metric. The\\nhyperparameterization of the SSL con\\ufb01guration in our evolutionary based search is the exact same as the self-supervised\\nbaseline, except for the augmentations in the pretext task. Precautions were taken to ensure that all random components\\n8 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\n(a) BS 32, Cifar10 (b) BS 256, Cifar10\\nFigure 2. Training models for more epochs in both the pretext and downstream task. X-axis representing the different\\nSSL algorithms, hue representing the number of epochs used for the pretext task (50, 100 or 1000), and the y-axis\\nrepresenting the improvement on the initial accuracy of the augmentation pipeline.\\nof the SSL pipeline were controlled with a \\ufb01xed seed, ensuring that the only component of the pipeline that changes is\\nthe augmentations itself. Both the default SSL con\\ufb01guration and Evolutionary Optimized Augmentation con\\ufb01guration\\nusing the evolved augmentation use the exact same downstream con\\ufb01guraiton as the supervised baseline. Because of\\nthis, with the basic SSL and supervised baselines, we are able to compare how the SSL pretraining using an evolved\\naugmentation policy improves the performances achieved by both the supervised and SSL baselines.\\nEffect of batch size We study the effect of batch size in the pretext task, and so batch sizes of 32 and 256 were\\nexperimented for all the supervised, self-supervised and evolutionary self-supervised modes.\\nEffect of number of epochs Given that in our experiments are conducted in a constrained setting, the evolutionary\\nalgorithm utilized a low number of epochs. To better evaluate the performance of SSL algorithms, we carry out an\\nexperiment, in which we use the best found augmentation policies by our evolutionary search mechanism and train SSL\\nmodels for 50, 100, and 1000 epochs in the pretext task and 50 epochs in the downstream task.\\n5.1 Results\\nAs explained in the previous sections, in order to understand the effect of changing the augmentation operators in the\\npretext task of the four mentioned SSL algorithms, the Evolutionary Algorithm is carried out. For each of the four\\nalgorithms, BYOL, SimSiam, SwA V and NNCLR, two datasets of SVHN and Cifar10 and pretext batch sizes of 32 and\\n256 were used. To mitigate the bias due to randomness, Nrandom seeds were used to control all randomness in the\\ndeep learning components of the algorithm, where Nfor each experiment is shown in table 3. This ensures that the\\ndifferences in training outcomes are not due to randomness in the training pipeline, such as model initialization, or\\nshuf\\ufb02ing of data. For each of the SSL algorithms, batch sizes and data sets, our Evolutionary Optimized Augmentation\\nAlgorithm was employed with a population size of 15. Running for 10 generations, with a population size of 15 and for\\nthe number of seeds shown in Table 3 resulted in a total of 15000 different augmentation con\\ufb01gurations in competition\\n9 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\nTable 2. Best found augmentation policies\\nDataset ssl algorithm aug1 op1 aug2 op2 aug3 op3\\nsvhn SwaV Color 1.48 TranslateX 3.00 ShearX 0.13\\nsvhn BYOL Sharpness 0.95 Contrast 1.28 Solarize 0.32\\nsvhn SimSiam Contrast 1.15 TranslateX 5.00 ShearY 0.12\\nsvhn NNCLR Color 0.90 ShearY 0.05 Solarize 0.32\\ncifar10 NNCLR Sharpness 0.88 Contrast 1.18 ShearX 0.10\\ncifar10 SwaV TranslateX 8.00 Brightness 0.69 Color 0.50\\ncifar10 BYOL Contrast 0.97 Sharpness 0.34 Rotate -1.00\\ncifar10 SimSiam TranslateX 8.00 VerticalFlip 0.64 Contrast 1.24\\nwith one another. Note that due to the nature of the GA, with individuals being carried over to the next generation, not\\nall 15000 con\\ufb01gurations are unique.\\nWe \\ufb01nd that we can consistently improve the SSL algorithms performance by applying our proposed evolutionary\\nmethod. In Figure 3, we show the average of the best \\ufb01tness over all the seeds for each of the four algorithms across\\ngeneration. More speci\\ufb01cally, for each random seed experiment, the best \\ufb01tness at each generation is extracted and\\nthen the average for all seeds is computed. For both Cifar10 and SVHN, an overall monotonic trend of optimization\\nis observed which con\\ufb01rms that our proposed evolutionary method improves the performance of SSL aglorithms.\\nWe observe that, mostly, NNCLR accounts for the smallest net improvement, whereas BYOL shows the largest\\nimprovement. Furthermore, we compare the distribution of classi\\ufb01cation accuracy of the best solutions of different\\nrandom initialization found in the \\ufb01nal generation of evolutionary process in the both datasets using two different batch\\nsizes in pretext task of SSL. Figure 4 suggests that the batch size of 256 has a larger impact on the improvement of\\nthe accuracy of downstream task classi\\ufb01cation. In addition, we observe that the results obtained from SVHN dataset\\npresents on par average accuracy for all the four algorithms. In contrast, in CIFAR-10 there is a noticeable variation in\\nthe average accuracy of the four algorithms in both batch sizes. With batch size 32, the highest performance is achieved\\nby SimSiam in CIFAR-10 and Byol in SVHN, while with batch size 256, all the four algorithms produce similar results.\\nFurthermore, the details of optimal performance obtained by each of the four algorithms are compared in Tables 4, 5, 6\\nand 7. The results indicate that mostly higher accuracies are achieved when using larger batch size in both datasets.\\nIn addition, the solutions found by both SO and MO, outperform those obtained by supervised and self supervised\\nlearning. In all cases, including both datasets, both batch sizes and for the four SSL algorithms, the SO algorithm \\ufb01nds\\nbetter solutions than MO. This is due to the fact that single optimization procedure is focused on solutions from a\\nspeci\\ufb01c SSL method and has a more limited search space. More interestingly, in all cases, when we deploy the best\\ncon\\ufb01gurations found by augmentation operators from SO and train for 50 in pretext and 100 in downstream tasks, we\\noutperform all the results when compared to the supervised training. These results are denoted by SSL (optimized)\\nin the Tables. As mentioned before, in all the experiments, we used 10 epochs for both the pretext and downstream\\ntasks. However, in order to better understand how well the proposed evolved augmentation pipelines behave for larger\\nnumber of epochs in our experiment setting, we conduct another experiment in which we measure the effect of number\\nof epochs on the results. For this experiment, the best augmentation pipeline for each SSL algorithm is used in their\\nrespective pretext tasks and are trained for 50, 100, and 1000 epochs. The obtained pretrained models were then \\ufb01ne\\ntuned in the downstream task for 50 epochs. With this experiment, we compare the models being trained for 10 epochs tuned in the downstream task for 50 epochs. With this experiment, we compare the models being trained for 10 epochs\\nin both the pretext and downstream phases with models that are trained for 50, 100, and 1000 epochs in the pretext task\\nand 50 epochs in the downstream task. As illustrated in Figure 2, we see that there is an increasing boost in performance\\nin downstream task when using 50 and 100 epochs in the pretext task, but this improvement degrades when using 1000\\nepochs. This \\ufb01nding indicates that training the pretext task for too long results in over\\ufb01tting and a decay in classi\\ufb01cation\\naccuracy. Furthermore, we summarize the best found policies in Table 2 and visualize the effect of these policies by a\\nfew examples in Figures 5 and 6 for CIFAR-10 and SVHN datsets respectively. Details about number of seeds for each\\nexperiment is listed in Table 3.\\n10 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\n(a) Cifar10, BS 32 (b) Cifar10, BS 256\\n(c) SVHN, BS 32 (d) SVHN, BS 256\\nFigure 3. The average of the best found \\ufb01tness at each generation for all seeds listed in Table 3\\n11 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\n(a) Cifar10, BS 32 (b) Cifar10, BS 256\\n(c) SVHN, BS 32 (d) SVHN, BS 256\\nFigure 4. The distribution of best results in the \\ufb01nal generation for all seeds as listed in Table 3\\n12 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\nFigure 5. Visualization of best found operators cifar10\\nTable 3. Number of seeds run for each experiment,\\nexp algo seed\\nbsize=256, data=svhn SwaV [0, 1, 2, 3, 4, 5, 6, 7, 8]\\nbsize=256, data=svhn BYOL [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\\nbsize=256, data=svhn SimSiam [0, 1, 2, 5, 6, 7, 8]\\nbsize=256, data=svhn NNCLR [0, 1, 2, 3, 4, 5, 6, 7, 8]\\nbsize=32, data=svhn SwaV [0, 1, 2]\\nbsize=32, data=svhn BYOL [0, 1, 2, 3]\\nbsize=32, data=svhn SimSiam [0, 1, 2]\\nbsize=32, data=svhn NNCLR [0, 1, 2]\\nbsize=32, data=cifar10 NNCLR [0, 1, 2, 3]\\nbsize=32, data=cifar10 SwaV [0, 1, 2, 3, 4, 5]\\nbsize=32, data=cifar10 BYOL [0, 1, 2, 3, 4, 8]\\nbsize=32, data=cifar10 SimSiam [0, 1, 2, 3]\\nbsize=256, data=cifar10 SwaV [0, 1, 2, 3, 4, 5, 6, 7]\\nbsize=256, data=cifar10 BYOL [0, 1, 2, 3, 4, 5, 6, 7]\\nbsize=256, data=cifar10 SimSiam [0, 1, 2, 3, 4, 5, 6, 7]\\nbsize=256, data=cifar10 NNCLR [0, 1, 2, 3, 4, 5, 6, 7]\\n13 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\nFigure 6. Visualization of best found operators svhn\\n6 Explainability\\nIn this section, we propose two algorithms for analyzing the best models encoded in the optimal performing\\nchromosomes and explain the effect of augmentation operators. In the following section, we introduce an augmentation\\nsensitivity and augmentation importance analysis in order to understand the most in\\ufb02uential operators for each dataset\\nand SSL algorithms.\\n6.1 Augmentation Sensitivity\\nIn order to measure the sensitivity of each algorithm to each augmentation operator, we design an ablation study in\\nwhich we measure the amount of difference in performance in absence of each operator. More speci\\ufb01cally, in order\\nto measure the sensitivity of aioperator, we remove this operator from all chromosomes that include it and replace\\nit with all possible operators from our augmentation set and obtain a new set of chromosomes. Then we obtain the\\ndifference between the average performance of this set and the performance of the original chromosome: OS( ai) =\\nperformance( ai) - average(performance( ak)) for all k The pseudocode is presented in Algorithm 1\\n6.2 Augmentation Importance\\nIn this analysis, for each operator and each SSL method, we calculate the number of times an operator has appeared in\\nthe top 50 chromosomes from all the generations (based on test accuracies). We normalized this value by number of\\n14 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\nAlgorithm 1 Computing Sensitivity\\nop Operator of interest\\nC Set of all Chromosomes\\nCop Subset of chromosomes in Cthat contain operator op\\nSensitivity 0\\nforciin Cop do\\nAvgSimAcc 0\\nTotalSimilarChromos  0\\nforcjin C do\\nNumEqualOps 0\\nforopiinci.operations do\\nNumEqualOps NumEqualOps + 1\\nend for\\nifopnot incj.operations and NumEqualOps\\u0011ci.length\\u00001then\\nAvgSimAcc AvgSimAcc +cj.accuracy\\nTotalSimilarChromos  TotalSimilarChromos + 1\\nend if\\nAvgSimAcc AvgSimAcc\\u0004TotalSimilarChromos\\nSensitivity Sensitivity +jci.accuracy\\u0000AvgSimAccj\\nend for\\nend for\\nSensitivity Sensitivity\\u0004Cop.length\\nTable 4. Best found accuracy for Cifar10 with a batch size of 32 in the pretext task.\\nalgo NNCLR SimSiam SwaV BYOL\\nSSL (SO) 84.26 84.48 84.30 84.27\\nSSL (MO) 83.94 84.18 84.18 83.92\\nSSL (default) 82.81 83.59 81.59 83.83\\nSupervised 83.24 83.24 83.24 83.24\\nSSL (optimized) 85.58 85.72 84.74 86.03\\nTable 5. Best found accuracy for Cifar10 with a batch size of 256 in the pretext task.\\nalgo NNCLR SimSiam SwaV BYOL\\nSSL (SO) 84.36 84.43 84.57 84.20\\nSSL (MO) 84.24 83.85 83.97 84.20\\nSSL (default) 83.86 83.23 83.22 84.01\\nSupervised 83.24 83.24 83.24 83.24\\nSSL (optimized) 85.90 86.06 85.93 85.81\\nTable 6. Best found accuracy for SVHN with a batch size of 32 in the pretext task.\\nalgo NNCLR SimSiam SwaV BYOL\\nSSL (SO) 93.35 93.41 93.29 93.39\\nSSL (MO) 93.07 93.07 93.07 92.70\\nSSL (default) 92.34 92.52 91.88 92.85\\nSupervised 91.28 91.28 91.28 91.28\\nSSL (optimized) 93.67 93.75 93.58 93.55\\nTable 7. Best found accuracy for SVHN with a batch size of 256 in the pretext task.\\nalgo NNCLR SimSiam SwaV BYOL\\nSSL (SO) 93.41 93.55 93.50 93.37\\nSSL (MO) 92.94 92.94 92.83 92.94\\nSSL (default) 92.75 91.87 91.95 92.34\\nSupervised 83.24 83.24 83.24 83.24\\nSSL (optimized) 93.70 93.68 93.63 93.67\\n15 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\noperators and number of chromosomes. The details of this procedure are illustrated in Algorithm 2 This method allows\\nus to \\ufb01nd the degree of utilization of augmentation operators for generating the best found chromosomes. This metric\\nallows us to understand how different augmentations are impacting the top chromosomes.\\nAlgorithm 2 Computing Operator Importance\\nop Operator of interest\\nC Set of all Chromosomes\\nN Number of chromosomes to consider\\nImportance 0\\nC sorted(C)\\nforciin C[:N]do\\nifopinci.operations then\\nImportance Importance + 1\\nend if\\nend for\\n6.3 Explainability Findings\\nFigure 7 demonstrates an overall view of augmentation importance and sensitivity for both batch sizes. From this Figure\\nwe can interestingly observe that both analyses follow nearly a similar trend and disclose similar information about\\neach operator. Note that importance values show the more frequently used operators in the best policies, whereas, the\\nsensitivity analysis accounts for the operators that make a bigger change in accuracy. Crucially, certain augmentation\\noperators are found to be globally prevalent whereas others do not. In addition, the results of the four algorithms BYOL,\\nNNCLR, SwA V and SimSiam, resembles a non-uniform distribution of sensitivity and importance values for the various\\noperators. This supports the idea that speci\\ufb01c augmentations lead to a stronger pretext task in the SSL algorithms. We\\nwould expect a more uniform distribution if the augmentations had no effect on the outcome. In our results we observe\\nthat in all cases, one or several augmentations dominate while others are very uncommon or are not even present in bar\\ncharts. Plots a and c of Figure 7 which are corresponding to the sensitivity analysis, illustrate that the algorithms are\\nmost sensitive to Contrast and Sharpness and least sensitive to translateY . Both \\ufb02ip operations have the second and third\\nlowest overall sensitivity value, meaning that none of the algorithms appear to be overly sensitive to the \\ufb02ip operation.\\nTranslateX, Color, Brightness, Rotate, Solarize and both Shear operators all have a medium sensitivity value. These\\n\\ufb01ndings suggest that all four SSL algorithms are more sensitive to the augmentations that are non-geometric transforms\\nsuch as contrast and sharpness and are less sensitive to the geometric transforms, such as Translate and Flip. From the\\nstacked bar charts, it is also observable that certain operators are more in\\ufb02uential in each dataset. It can be observed\\nthat SimSiam appears to be consistently sensitive to the Brightness operator in SVHN and to Contrast and ShearY in\\nCIFAR-10, SwA V has a relatively high sensitivity to Color and Rotate in SVHN and several operators such as Color and\\nSharpness in CIFAR-10. BYOL consistently has a varied range of sensitivities to the different augmentation operators\\nin SVHN, and in CIFAR-10, and the algorithm is most sensitive to Contrast and Sharpness operators. For NNCLR\\nwe see that the experiments using SVHN are most sensitive to Sharpness, and for CIFAR-10 there is not as clear of a\\npattern, however HorizontalFlip is relatively showing a high sensitivity.\\nThe augmentation importance bar charts in plots b and d of Figure 7 suggest that for both datasets ShearX and TranslateX\\nhave appeared dominantly in top chromosomes, while VerticalFlip, HorizontalFlip and TranslateY are the least applied\\noperators. For SVHN the remaining augmentaitons are more evenly distributed in terms of importance than Cifar10. We\\nsee overall the same augmentations for both datasets in between the lowest and high importance augmentations, yet the\\nordering is slightly different. It is observed that sharpness has a higher overall importance in SVHN than Cifar10, as\\ndoes color. Contrast, solarize, rotate, ShearY and brightness are all relatively similar in terms of overall importance does color. Contrast, solarize, rotate, ShearY and brightness are all relatively similar in terms of overall importance\\nfor both datasets. When considering the importance metrics for all four algorithms in Figure 7, it is clear that speci\\ufb01c\\naugmentations are most important. For BYOL it is found that Solarize is the most important for SVHN and TranslateX\\n16 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\n(a) Sensitivity SVHN (b) Importance SVHN\\n(c) Sensitivity Cifar10 (d) Importance Cifar10\\nFigure 7. Global View of augmentation sensitivity and importance\\nfor CIfar10. Additionally, it can be seen that both shear operations are consistently of relatively high importance for\\nboth datasets with BYOL. For NNCLR, color is most important with SVHN and ShearX is most important with Cifar10.\\nThe most important operators for SimSiam are TranslateX for SVHN and ShearX and ShearY for Cifar10. Lastly, for\\nSwA V contrast is the most important augmentation for both datasets.\\n6.4 Landscape Analysis\\nTo better compare the SSL algorithms and in order to measure the effect of batch size in our experiments we utilize a\\nstrategy for analyzing the loss of the resultant downstream models, known as loss landscape analysis. Loss Landscape\\nanalysis aims to provide a deeper understanding of the optimized model parameters and its context in its respective\\nparameter space by slightly perturbing the model parameters around the optimized parameterization. This provides a\\nlandscape of the loss function for the given parameter space, allowing for insights into how convex or chaotic the loss\\nspace around the solution is. Producing a loss landscape for a given model architecture and problem is not a trivial task\\nand many different approaches exist for doing so. This work opts to employ a \\ufb01lter normalization approach presented\\nby Li et al. Li et al. (2018) to visualize the loss landscape. The \\ufb01lter normalization loss landscape visualization was\\nchosen for this work as it was found to accurately capture the local sharpness and \\ufb02atness of minimizers. Understanding\\nthe local geometry of the loss landscape is an essential component for understanding the behaviour of the loss function\\nin the problems at hand. As discussed in Li et al. (2018), a key component of the \\ufb01lter wise normalization technique is a\\nrandom vector technique used by Goodfellow et al. (2014), Im et al. (2016). In this technique two random vectors \\u000eand\\n17 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\n(a) BYOL (b) NNCLR (c) SimSiam (d) SwA V\\nFigure 8. Comparing the loss landscape of the downstream models, using SSL methods with a batch size of 32 (top\\nrow) and batch size of 256 (bottom row). We see that the smaller batch size in the SSL pretraining leads to a less sharp\\nminimization in the downstream task.\\n\\u0011are sampled from a random Gaussian distribution, as well as a center point \\u0012\\u0003, where\\u0012\\u0003is the optimized parameter\\ncon\\ufb01guration. Then a 2d contour is generated by plotting the function4:\\nf(\\u000b;\\f) =L(\\u0012\\u0003+\\u000b\\u000e+\\f\\u0011) (4)\\nwhere\\u000b, and\\fare the respective lengths of random vectors \\u000eand\\u0011. The key component of this approach which differs\\nfrom earlier random vector approaches is that, the sampled directions are \\ufb01lter-wise normalized. That is the random\\nvectors are scaled to the \\ufb01lters within the CNN to ensure that the area covered by the random vectors are not too small\\nor too large for the set of parameters. Mainly, it was found that the results with Cifar-10 were more interesting in terms\\nof effect of batch size. For this reason we choose to further explore the loss landscapes of the experiments on Cifar-10\\ndataset. For all four SSL algorithms, NNCLR, BYOL, SimSiam and SwA V and the two batch sizes 32, and 256, we\\nuse the \\ufb01lter normlaization technique with a \\u000b, and\\fboth set to 10and a sample size of 50resulting in a 50x50loss\\nlandscape. Li et al. Li et al. (2018) found that with the \\ufb01lter normalization method it was possible to visualize how batch\\nsize impacts minima sharpness, large batches were found to produce visually sharper minima. We visualize the loss\\nlandscape of the best solutions found by our evolutionary search mechanism. Remarkably, in all four SSL algorithms\\nwe notice a difference in sharpness when training with a batch sizes of 32 and 256 in the pretext task and keeping a\\n\\ufb01xed batch size of 32 in the downstream task. As illustrated in Figure 8, we observe that BYOL, NNCLR and SimSiam\\nall have converged to sharp minima in the downstream task when using a batch size of 256 in the pretext task, and a\\nvisibly \\ufb02atter minima when using a batch size of 32 in the pretext task, the opposite result is observed in SwA V .\\n7 Conclusion\\nOur contribution in this paper is two-fold; On one hand, we focused on optimizing the performance of SSL algorithms\\nby \\ufb01nding the best augmentation operators. To this end, we proposed an approach based on evolutionary optimization\\nto automatically \\ufb01nd the optimal augmentation operators and their intensities in order to maximize the accuracy of\\n18 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\ndownstream task. In this regard, we optimized and compared the performance of four SOTA SSL algorithms. On the\\nother hand, we proposed algorithms for explaining the optimized solutions in order to analyze and \\ufb01nd the impact\\nof augmentation operators applied in the pretext task of SSL algorithms. Accordingly, we designed explainability\\nexperiments to elicit the most in\\ufb02uential augmentation operators using two standard visual datasets. We further analyzed\\nthe impact of different parameters including batch size and epoch number, and visualized loss landscapes to better\\nunderstand the best augmentation policies and best performing models. To the best of our knowledge this work is\\nthe \\ufb01rst attempt to optimize hyper-parameters used in the pretext task for self-supervised learning algorithms. Our\\nresults con\\ufb01rm that SSL algorithms are sensitive to the choice of augmentation policies. We also demonstrated that our\\nproposed evolutionary augmentation optimization method can \\ufb01nd the solutions that lead to performance enhancement\\nof SSL algorithms by ef\\ufb01ciently searching the augmentation policy space.\\nCon\\ufb02ict of Interest Statement\\nThe authors declare that the research was conducted in the absence of any commercial or \\ufb01nancial relationships that\\ncould be construed as a potential con\\ufb02ict of interest.\\nFunding\\nThis research has been supported by the Natural Sciences and Engineering Research Council of Canada.\\nReferences\\n[Dataset] (2020). Bootstrap your own latent: A new approach to self-supervised Learning.\\ndoi:10.48550\\/arXiv.2006.07733. ArXiv:2006.07733 [cs, stat]\\nBrazdil, P., van Rijn, J. N., Soares, C., and Vanschoren, J. (2022). Metalearning: Applications to Automated Machine\\nLearning and Data Mining (Springer Nature)\\nCaron, M., Misra, I., Mairal, J., Goyal, P., Bojanowski, P., and Joulin, A. (2020). Unsupervised learning of visual\\nfeatures by contrasting cluster assignments. Advances in Neural Information Processing Systems 33, 9912\\u20139924\\n[Dataset] Caron, M., Misra, I., Mairal, J., Goyal, P., Bojanowski, P., and Joulin, A. (2021). Unsupervised Learning of\\nVisual Features by Contrasting Cluster Assignments. doi:10.48550\\/arXiv.2006.09882. ArXiv:2006.09882 [cs]\\nChen, T., Kornblith, S., Norouzi, M., and Hinton, G. (2020). A Simple Framework for Contrastive Learning of Visual\\nRepresentations. arXiv:2002.05709 [cs, stat] ArXiv: 2002.05709\\nChen, X. and He, K. (2021). Exploring simple siamese representation learning. In Proceedings of the IEEE\\/CVF\\nConference on Computer Vision and Pattern Recognition . 15750\\u201315758\\nCubuk, E. D., Zoph, B., Mane, D., Vasudevan, V ., and Le, Q. V . (2019). Autoaugment: Learning augmentation strategies\\nfrom data. In Proceedings of the IEEE\\/CVF Conference on Computer Vision and Pattern Recognition . 113\\u2013123\\nDeng, L. (2012). The mnist database of handwritten digit images for machine learning research. IEEE Signal Processing\\nMagazine 29, 141\\u2013142\\nDoersch, C., Gupta, A., and Efros, A. A. (2015). Unsupervised visual representation learning by context prediction. In\\nProceedings of the IEEE international conference on computer vision . 1422\\u20131430\\nDonahue, J., Kr\\u00e4henb\\u00fchl, P., and Darrell, T. (2016). Adversarial feature learning. arXiv preprint arXiv:1605.09782\\nDosovitskiy, A., Springenberg, J. T., Riedmiller, M., and Brox, T. (2014). Discriminative unsupervised feature learning\\nwith convolutional neural networks. Advances in neural information processing systems 27\\n[Dataset] Dwibedi, D., Aytar, Y ., Tompson, J., Sermanet, P., and Zisserman, A. (2021). With a Little Help from\\nMy Friends: Nearest-Neighbor Contrastive Learning of Visual Representations. doi:10.48550\\/arXiv.2104.14548.\\nArXiv:2104.14548 [cs]\\n19 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\nFortin, F.-A., De Rainville, F.-M., Gardner, M.-A., Parizeau, M., and Gagn\\u00e9, C. (2012). DEAP: Evolutionary algorithms\\nmade easy. Journal of Machine Learning Research 13, 2171\\u20132175\\nGidaris, S., Singh, P., and Komodakis, N. (2018). Unsupervised representation learning by predicting image rotations.\\narXiv preprint arXiv:1803.07728\\n[Dataset] Goodfellow, I. J., Vinyals, O., and Saxe, A. M. (2014). Qualitatively characterizing neural network\\noptimization problems. doi:10.48550\\/ARXIV .1412.6544\\nHo, D., Liang, E., Chen, X., Stoica, I., and Abbeel, P. (2019). Population based augmentation: Ef\\ufb01cient learning of\\naugmentation policy schedules. In International Conference on Machine Learning (PMLR), 2731\\u20132741\\nIm, D. J., Tao, M., and Branson, K. (2016). An empirical analysis of deep network loss surfaces. CoRR abs\\/1612.04010\\nIsola, R. Z. P. and Efros, A. A. (2016). Colorful image colorization. ECCV . Richard Zhang Phillip Isola and Alexei A\\nEfros\\nJenni, S., Jin, H., and Favaro, P. (2020). Steering self-supervised feature learning beyond local pixel statistics. In\\nProceedings of the IEEE\\/CVF Conference on Computer Vision and Pattern Recognition . 6408\\u20136417\\nKrizhevsky, A. (2009). Learning multiple layers of features from tiny images . Tech. rep.\\nLi, C.-L., Sohn, K., Yoon, J., and P\\ufb01ster, T. (2021). Cutpaste: Self-supervised learning for anomaly detection and\\nlocalization. In Proceedings of the IEEE\\/CVF Conference on Computer Vision and Pattern Recognition . 9664\\u20139674\\nLi, H., Xu, Z., Taylor, G., Studer, C., and Goldstein, T. (2018). Visualizing the Loss Landscape of Neural Nets . Tech.\\nRep. arXiv:1712.09913, arXiv. ArXiv:1712.09913 [cs, stat] type: article\\nLim, S., Kim, I., Kim, T., Kim, C., and Kim, S. (2019). Fast autoaugment. Advances in Neural Information Processing\\nSystems 32\\nMisra, I., Zitnick, C. L., and Hebert, M. (2016). Shuf\\ufb02e and learn: unsupervised learning using temporal order\\nveri\\ufb01cation. In European conference on computer vision (Springer), 527\\u2013544\\nNetzer, Y ., Wang, T., Coates, A., Bissacco, A., Wu, B., and Ng, A. Y . (2011). Reading digits in natural images with\\nunsupervised feature learning. In NIPS Workshop on Deep Learning and Unsupervised Feature Learning 2011\\nNoroozi, M. and Favaro, P. (2016). Unsupervised learning of visual representations by solving jigsaw puzzles. In\\nEuropean conference on computer vision (Springer), 69\\u201384\\nO\\u2019Neill, M. (2009). Riccardo poli, william b. langdon, nicholas f. mcphee: a \\ufb01eld guide to genetic\\nOord, A. v. d., Li, Y ., and Vinyals, O. (2018). Representation learning with contrastive predictive coding. arXiv preprint\\narXiv:1807.03748\\nPaszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., et al. (2019). Pytorch: An imperative style, high-\\nperformance deep learning library. In Advances in Neural Information Processing Systems 32 (Curran Associates,\\nInc.). 8024\\u20138035\\nPathak, D., Krahenbuhl, P., Donahue, J., Darrell, T., and Efros, A. A. (2016). Context encoders: Feature learning by\\ninpainting. In Proceedings of the IEEE conference on computer vision and pattern recognition . 2536\\u20132544\\nReeves, C. (2003). Genetic Algorithms. In Handbook of Metaheuristics , eds. F. Glover and G. A. Kochenberger\\n(Boston, MA: Springer US). 55\\u201382. doi:10.1007\\/0-306-48056-5_3\\nShorten, C. and Khoshgoftaar, T. M. (2019). A survey on image data augmentation for deep learning. Journal of big\\ndata 6, 1\\u201348\\nSrinivas, M. and Patnaik, L. M. (1994). Adaptive probabilities of crossover and mutation in genetic algorithms. IEEE\\nTransactions on Systems, Man, and Cybernetics 24, 656\\u2013667\\nSuch, F. P., Madhavan, V ., Conti, E., Lehman, J., Stanley, K. O., and Clune, J. (2017). Deep neuroevolution: Genetic\\nalgorithms are a competitive alternative for training deep neural networks for reinforcement learning. arXiv preprint\\narXiv:1712.06567 algorithms are a competitive alternative for training deep neural networks for reinforcement learning. arXiv preprint\\narXiv:1712.06567\\nSusmelj, I., Heller, M., Wirth, P., Prescott, J., and et al., M. E. (2020). Lightly. GitHub. Note: https:\\/\\/github.com\\/lightly-\\nai\\/lightly\\n20 Evolutionary Augmentation Policy Optimization for Self-Supervised Learning A P REPRINT\\nV on K\\u00fcgelgen, J., Sharma, Y ., Gresele, L., Brendel, W., Sch\\u00f6lkopf, B., Besserve, M., et al. (2021). Self-supervised\\nlearning with data augmentations provably isolates content from style. Advances in neural information processing\\nsystems 34, 16451\\u201316467\\nYang, L. and Shami, A. (2020). On hyperparameter optimization of machine learning algorithms: Theory and practice.\\nNeurocomputing 415, 295\\u2013316\\n21\",\"31\":\" Hindsight States:\\nBlending Sim & Real Task Elements for\\nEf\\ufb01cient Reinforcement Learning\\nSimon Guist, Jan Schneider, Alexander Dittrich, Vincent Berenz, Bernhard Sch \\u00a8olkopf and Dieter B \\u00a8uchler\\nAbstract \\u2014Reinforcement learning has shown great potential\\nin solving complex tasks when large amounts of data can\\nbe generated with little effort. In robotics, one approach to\\ngenerate training data builds on simulations based on dynamics\\nmodels derived from \\ufb01rst principles. However, for tasks that, for\\ninstance, involve complex soft robots, devising such models is\\nsubstantially more challenging. Being able to train effectively in\\nincreasingly complicated scenarios with reinforcement learning\\nenables to take advantage of complex systems such as soft\\nrobots. Here, we leverage the imbalance in complexity of the\\ndynamics to learn more sample-ef\\ufb01ciently. We (i) abstract the\\ntask into distinct components, (ii) off-load the simple dynamics\\nparts into the simulation, and (iii) multiply these virtual parts\\nto generate more data in hindsight . Our new method, Hindsight\\nStates (HiS), uses this data and selects the most useful transitions\\nfor training. It can be used with an arbitrary off-policy algorithm.\\nWe validate our method on several challenging simulated tasks\\nand demonstrate that it improves learning both alone and\\nwhen combined with an existing hindsight algorithm, Hindsight\\nExperience Replay (HER). Finally, we evaluate HiS on a physical\\nsystem and show that it boosts performance on a complex table\\ntennis task with a muscular robot. Videos and code of the\\nexperiments can be found on webdav.tuebingen.mpg.de\\/his\\/.\\nI. I NTRODUCTION\\nReinforcement learning (RL) holds great potential for de-\\nvising optimal behavior for challenging robotics tasks. Dif-\\n\\ufb01culties in these tasks often arise from contact in object\\nmanipulation, the requirement to perform fast but accurate\\nmotions, or from hard-to-control systems like soft robots.\\nWhile it has been shown that RL can handle such problems,\\nthe major drawback of RL methods to require large amounts\\nof interactions with the environment remains. This fact makes\\nlearning on real robots challenging.\\nTraining in simulation and transferring the resulting policy\\nto the real system is one approach to alleviate this challenge.\\nHowever, even small inaccuracies in the simulation can cause\\nthe simulated and real system to behave differently, a problem\\nknown as reality gap . This problem is worse for complex\\nsystems like soft robots, which are hard to model accurately.\\nSim-to-real techniques attempt to mitigate the impact of the\\nreality gap. Many popular techniques accomplish more ef-\\nfective policy transfer by making the learned policies more\\nrobust to variations in the task dynamics through domain\\nThis work was supported by the Max Planck Institute for Intelligent\\nSystems.\\nAll authors are af\\ufb01liated with the MPI for Intelligent Systems, Max-Planck-\\nRing 4, 72076 T \\u00a8ubingen, Germany.randomization [26, 41]. However, these methods are gen-\\nerally computationally expensive and require precise \\ufb01ne-\\ntuning of the parameters, such as the amount of randomization\\non the dynamics parameters. These downsides of sim-to-\\nreal approaches accumulate as tasks advance in complexity,\\nthus, necessitating learning many complex tasks partly or\\ncompletely on the real system.\\nHybrid sim and real (HySR) [5] is a way to ease the training\\nof complex tasks on real systems. The method is based on the\\ninsight that for some tasks, certain parts of the environment\\nhave simpler dynamics than others. For instance, many robotic\\nball games consist of a robot, whose dynamics are more\\ncomplex compared to those of the ball. The mismatch in mod-\\neling complexity is exacerbated if the robot is equipped with\\nsoft components, which render modeling even more dif\\ufb01cult.\\nThe idea behind HySR is to keep the complicated parts of\\nthe task real, whereas the simpler parts are simulated. This\\nstrategy yields signi\\ufb01cant practical bene\\ufb01ts, while facilitating The idea behind HySR is to keep the complicated parts of\\nthe task real, whereas the simpler parts are simulated. This\\nstrategy yields signi\\ufb01cant practical bene\\ufb01ts, while facilitating\\nthe transfer to the entirely real system. First, the approach\\nsimpli\\ufb01es setting up the task since fewer real objects have\\nto be handled that, for instance, are subject to wear-and-\\ntear or require safety considerations. Second, it can greatly\\nsimplify the reset mechanism in episodic RL tasks. Robots are\\nactuated and can typically reset themselves, while environment\\nresets often require manually moving all objects to their initial\\npositions. Third, data-augmentation techniques can be added\\nto the virtual objects that would not be feasible on a fully\\nreal system. Lastly, the simulation provides information about,\\ne.g., the exact positions of virtual objects or about contacts\\nthat occur, which might not be easily attainable for real\\nobjects. This data can be used to generate labels or rewards\\nthat can aid the training process. While HySR can simplify\\npractical aspects of the training, the large number of real robot\\ninteractions required for training can still cause problems for\\nreal-world RL.\\nIn this work, we present an approach that can signi\\ufb01cantly\\nreduce the number of interactions with the real system required\\nfor learning. The key idea behind our method, Hindsight\\nStates (HiS), is to pair the data of a single real instance with\\nadditional data generated by concurrently simulating multiple\\ndistinct instances of the virtual part. In the example of robot\\nball games, our method simulates the effect of the robot\\u2019s\\nactions on several virtual balls simultaneously. We relabel the\\nvirtual part of each roll-out with this additional virtual data\\nin hindsight . Intuitively, the agent experiences what wouldarXiv:2303.02234v2  [cs.RO]  9 Mar 2023 (a) Pushing\\n (b) Sliding\\n (c) Simulated robot table tennis\\n (d) Real robot table\\ntennis\\nFig. 1: Visualization of HiS applied to the tasks considered in this work. Rather than training with a single object, HiS uses\\nmultiple virtual objects in parallel to generate more data and experience higher rewards with increased probability early on in\\nthe training.\\nhave happened if it had encountered a different virtual part\\nbut applied the same actions as in the original episode. This\\nrelabeling process enables the RL agent to generate extra\\nexperience without collecting further transitions on the real\\nsystem. The hindsight data can then be used in addition to the\\nregular training data by any off-policy RL algorithm, such as\\nSoft Actor-Critic (SAC) [12]. Particularly for tasks with sparse\\nrewards, the additional data is bene\\ufb01cial since it increases the\\nprobability that the agent experiences positive rewards early in\\nthe training. The contributions of this paper are the following:\\n1)Extending HySR for sample ef\\ufb01cient training\\nOur main contribution is to devise a sample ef\\ufb01cient way to\\ntrain in the Hybrid Sim and Real (HySR) setting. To that end,\\nwe formalize the HySR training and, based on it, develop a\\nnovel algorithm, called Hindsight States (HiS).\\n2)Evaluation of the interplay between HER and HiS\\nOur experiments demonstrate that the combination of HiS\\nand HER achieves higher sample ef\\ufb01ciency than each method\\nby itself. We argue that HiS and HER improve task perfor-\\nmance in distinct and complementary manners.\\n3)Thorough experimental validation of HiS on a variety of tasks\\nWe show improved performance of HiS on the original HySR\\nreal robot table tennis task. Additionally, we investigate to\\nwhat extent HiS can be applied to manipulation tasks and\\nreport more ef\\ufb01cient training in these regimes (see Fig. 1\\nfor an overview of all experiments). Another \\ufb01nding is that\\nHiS allows for more ef\\ufb01cient training in entirely simulated\\nenvironments w.r.t. wall-clock time.\\nII. R ELATED WORK\\nOur new method, HiS, shares some aspects with and takes\\ninspiration from other works. In the following section, we\\ndiscuss these similarities.\\nHindsight Experience Replay (HER) [1] improves sample\\nef\\ufb01ciency for sparse goal-conditioned RL tasks by relabeling\\nthe goal states of transitions in hindsight. The method replaces\\nthe original goal with a state reached by the agent. Then it\\ntrains an off-policy RL agent on a combination of original and\\nrelabeled data. This way, the agent receives additional positive\\nfeedback for reaching the relabeled goal, which can be moreinstructive than the negative feedback for missing the original\\ngoal. Rauber et al. [31] extend the idea to on-policy algorithms\\nthrough the use of importance sampling. Dynamic Hindsight\\nExperience Replay (DHER) [9] is a version of HER that\\nsupports dynamic goals, which change during the episode. The\\nmethod makes the idea of relabeled goals applicable to tasks\\nlike grasping moving objects. While HER samples hindsight\\ngoals uniformly, recent methods prioritize goals based on\\nhow instructive the resulting transitions are for the agent.\\nThese approaches sample hindsight goals that guide the agent\\ntoward the true goal [2, 32, 10] or result in a large temporal\\ndifference (TD) error [22]. Other methods sample hindsight\\ngoals uniformly and prioritize more informative transitions\\nwhen sampling from the replay buffer, akin to PER [33].\\nZhao and Tresp [40] prioritize trajectories that demonstrate\\ndif\\ufb01cult behavior. They quantify dif\\ufb01culty by the increase of\\nthe system\\u2019s energy over the course of the trajectory. Beyene\\nand Han [3] primarily sample hindsight transitions that incur a\\nlarge TD error. Similar to HER and its extensions, our method\\nalso relabels in hindsight. However, it focuses on exchanging\\nthe virtual part of the state rather than the goal.\\nHiS and sim-to-real techniques utilize simulations to devise\\na policy that transfers to reality. Different techniques, such as the virtual part of the state rather than the goal.\\nHiS and sim-to-real techniques utilize simulations to devise\\na policy that transfers to reality. Different techniques, such as\\ndomain randomization [26, 28, 23] and domain adaptation [15,\\n14, 39] were proposed to mitigate the impact of the reality\\ngap. HiS, on the other hand, assumes the reality gap to be\\nsuf\\ufb01ciently small for certain components of the environment\\nand devises a way to be more ef\\ufb01cient in the hybrid sim and\\nreal setting. Sim-to-real methods could be used on top of HiS\\nto close the remaining gap.\\nOther works have investigated leveraging real and simulated\\ndata of the same task to improve learning performance. For\\ninstance, Kang et al. [16] learn a perception system using\\nsimulated and a reward model based on real data. They use\\nmodel predictive control with the learned reward model to\\nsolve quadrotor navigation tasks. Di-Castro et al. [8] combine\\ncheap but imprecise simulated with expensive real transitions\\nto learn a policy for the real task. HiS, in contrast, investigates\\nmore ef\\ufb01cient training by mixing simulated and real compo-\\nnents in each transition. Algorithm 1 Hindsight States (HiS)\\n1:Initialize off-policy RL algorithm A, replay buffer R,\\nchoose criterion c\\n2:forepisode i=1;2;:::do\\n3: Sample sv\\n1:T\\u0018D recand initialize real system to sr\\n1\\n4: fort=1;:::; Tdo\\n5: Sample action atfromAgiven st= [sr\\nt;sv\\nt]\\n6: Apply atto the real system and observe sr\\nt+1\\n7: sv\\nt+1\\u0018sim([sr\\nt;sv\\nt];at)or take sv\\nt+1from sv\\n1:T\\n8: end for\\n9: fort=1;:::; Tdo .standard replay\\n10: Store ([sr\\nt;sv\\nt];at;r([sr\\nt;sv\\nt];at);[sr\\nt+1;sv\\nt+1]) inR\\n11: end for\\n12: Initialize temporary buffer Rtemp .HiS replay\\n13: forj=1;2;:::do\\n14: Sample svm\\n1:T\\u0018D rec\\n15: fort=1;:::; Tdo\\n16: svm\\nt+1\\u0018sim([sr\\nt;svmt];at)or take svm\\nt+1from svm\\n1:T\\n17: em\\nt= ([sr\\nt;svmt];at;r([sr\\nt;svmt];at);[sr\\nt+1;svm\\nt+1])\\n18: Store relabeled transition em\\ntinRtemp\\n19: end for\\n20: end for\\n21: Every few episodes:\\nAdd transitions from Rtemp toRusing criterion c\\n22: Every few steps:\\nPerform optimization on Awith replay buffer R\\n23:end for\\nUtilizing of\\ufb02ine RL [19, 20] can also alleviate some of the\\nproblems related to training on real systems by making use\\nof existing datasets. The central issue of of\\ufb02ine RL is the\\ndistribution shift between the behavior policy, which collected\\nthe dataset, and the trained policy [20]. Recent works constrain\\nthe trained policy to be similar to the behavior policy [36, 34,\\n17], modify the objective of the Q-function to be conservative\\nwith respect to samples not in the dataset [18, 38], or penalize\\nuncertainty in rollouts of learned models [37, 29].\\nIII. M ETHOD\\nThe core components of HiS comprise the generation of\\nparallel virtual trajectories, as well as the criteria to select\\namong the additional transitions generated with HiS. This\\nsection introduces each of these components, as well as the\\nHySR setup that HiS builds upon.\\nA. RL Preliminaries\\nWe consider tasks formulated as a discounted Markov deci-\\nsion process, which is de\\ufb01ned by the tuple M= (S;A;P;g;r),\\nwhereSdenotes the state space and Athe action space.\\nThe transition dynamics P(st+1jst;at)represent the probability\\nof reaching state st+12S after executing action at2A in\\nstate st2S. Furthermore, the agent receives a scalar reward\\nrt=r(st;at)at each time step. The goal of RL is to \\ufb01nd anoptimal policy p?, which maximizes the discounted total return\\nR=Ep\\\"\\nT\\n\\u00e5\\ntgtrt#\\n; (1)\\nwhere Tis the time horizon and g2[0;1)is the discount\\nfactor. The agent collects transitions et= (st;at;rt;st+1)by\\ninteracting with the environment and uses them to iteratively\\nupdate its policy p. Many popular off-policy RL algorithms,\\nsuch as SAC [12] or Deep Q-Networks (DQN) [24], store these\\ntransitions in a buffer D[21] and replay them when updating,\\nfor instance, the action value function Q(st;at). This replay\\nbuffer is \\ufb01xed-sized and contains the most recent transitions.\\nA ring buffer, which replaces the oldest with the most recent\\ntransition, is a popular implementation. The Q-function update\\ninvolves minimizing a variant of the TD error\\nd(s;a;r;s0) =r+gmax\\na02AQ(s0;a0)\\u0000Q(s;a); (2)\\nwhere variations might include an alternative way to select\\nthe next action a0or an n-step TD error formulation dn=\\n\\u00e5n\\u00001\\nk=0gkrt+k+gnmax a0Q(st+n;a0)\\u0000Q(st;at). The TD error\\nrepresents a measure of surprise: the higher the absolute value\\nofd, the more does this particular transition in\\ufb02uence the\\nupdate of the Q-function. For this reason, the TD error metric\\nis commonly used to establish a ranking among transitions,\\nfor example in prioritized sweeping [25, 27] and prioritized\\nexperience replay (PER) [33]. The Q-function loss employs\\nthe replay bufferDand the squared TD error\\nJ(q) =E(s;a;r;s0)\\u0018D\\u0014\\n(r+gmax\\na02AQqold(s0;a0)\\u0000Qq(s;a))2\\u0015\\n;\\n(3)\\nwhere the old parameters qoldare kept \\ufb01xed during optimiza-\\ntion of the current policy parameters qand transitions are\\nsampled fromDaccording to a distribution that traditionally\\nis uniform but can vary such as in PER.\\nB. Hybrid Sim and Real Training\\nHybrid Sim and Real (HySR) training [5] is the foundation\\nfor Hindsight States (HiS). Intuitively, the idea of HySR is is uniform but can vary such as in PER.\\nB. Hybrid Sim and Real Training\\nHybrid Sim and Real (HySR) training [5] is the foundation\\nfor Hindsight States (HiS). Intuitively, the idea of HySR is\\nto gain practical bene\\ufb01ts by of\\ufb02oading the part of the task\\nthat is easier to model to the simulation. At the same time,\\nthe dif\\ufb01cult part is kept real during training. More formally,\\nHySR assumes that the Markovian state ssplits into a real\\nstate srand a virtual state sv\\ns= [sr;sv] (4)\\nand that the dynamics governing the real part p(sr\\nt+1j[sr\\nt;sv\\nt];at)\\nare more complex than those governing the virtual part\\np(sv\\nt+1j[sr\\nt;sv\\nt];at). To better transfer the learned policy after\\nHySR training to the fully real setup, HySR keeps the com-\\nplicated dynamics real and expects the virtual dynamics to be\\nsuf\\ufb01ciently accurate. Another requirement is that the virtual\\nstate does not in\\ufb02uence the real state\\np(sr\\nt+1j[sr\\nt;sv\\nt];at) =p(sr\\nt+1jsr\\nt;at) (5) s s0\\na a0\\nr r0\\n(a) MDPs s0\\na a0\\nr r0 r r0g g0\\n(b) HERsrsr0\\na a0\\nr r0svsv0\\n(c) HySRsrsr0\\na a0\\nr r0svn svn0\\nr r0\\n(d) HiS\\nFig. 2: Graphical models of a general MDP, HER, HySR, and HiS. All blue and red elements indicate a difference to the\\ngeneral MDP. Red indicates components that are relabeled in hindsight. HER extends the MDP with a goal state and relabels\\ngoals. HySR separates the state into a virtual and a real part. The additional arrows indicate the dynamics. HiS combines key\\nideas from HySR and HER that are state separation and dynamics as well as relabeling, respectively.\\nsince mapping forces from the virtual to the real part can be\\nintricate, especially onto complex or unknown dynamics.\\nThe virtual dynamics do not necessarily simulate the part\\nfor the full duration of the training. Before the \\ufb01rst contact\\nbetween real and virtual part, HySR replays recorded data\\nof the virtual part instead of simulating it to minimize the\\naccumulation of model error. To that end, HySR \\ufb01rst uni-\\nformly samples a virtual trajectory tvrecfrom a databaseDrec\\ncontaining Nrecrecorded trajectories.\\nDrec=h\\nsvrecn\\n1;:::; svrecn\\nTiNrec\\nn=1(6)\\nAfter contact of the virtual and the real part, the simulated\\ndynamics determine the consecutive motion, as no information\\nabout the latter part can be deduced from the recordings. A\\ncomplete HySR training trajectory,\\ntHySR= [erec\\n1;:::; erec\\ntc;esim\\ntc+1;:::; esim\\nT]; (7)\\nhence, consists of both transitions replayed from the database\\nerec\\nt=\\u0010\\n[sr\\nt;svrec\\nt];at;r([sr\\nt;svrec\\nt];at);[sr\\nt+1;svrec\\nt+1]\\u0011\\nand simulated\\ntransitions esim\\nt=\\u0010\\n[sr\\nt;svsim\\nt];at;r([sr\\nt;svsim\\nt];at);[sr\\nt+1;svsim\\nt+1]\\u0011\\n,\\nassuming that the contact happened at time tc. During training,\\nthe policy samples actions conditioned on the complete state\\nfrom Eq. (4), in which the information of whether the virtual\\nparts are replayed or simulated is not included.\\nHySR is particularly suitable for applications where (i) the\\ncomplexity of the virtual and real dynamics is unbalanced, (ii)\\ntraining with the real instance of the virtual part is handy or\\nsafer, (iii) contact of the separated parts occurs relatively far\\ninto each roll-out to make better use of the recorded data, (iv)\\nand the in\\ufb02uence that the virtual part has on the real part is\\nnegligible and the effect in the opposite direction is known and\\ntransfers well to the real world. All these points apply to many\\nball games, such as football, cricket, baseball, or basketball,\\nwhere an object has to reach a goal state through an interaction\\nwith the player. Although, the ef\\ufb01cacy of HySR has only been\\nshown for robot table tennis so far, training in the HySR settingis not limited to ball games. For instance, for tasks like slide-\\npushing or throwing objects (in contrast to picking them up),\\nwe could combine objects of different weights or materials\\nwith a single arm motion. Points (i) to (iv) also apply to tasks\\nsuch as avoiding moving obstacles since contact should be\\navoided in the \\ufb01rst place. For example, a robot waking in\\na cluttered and dynamic environment or an autonomous car\\ndriving through a crowded city.\\nC. Hindsight States\\nHySR also enables the generation of additional data, and\\nHindsight States (HiS) is our proposed way to leverage them\\nfor ef\\ufb01cient training. HiS implements the idea that we can gen-\\nerate additional virtual data in the HySR setting. Speci\\ufb01cally,\\nit generates trajectories tHiS= [em\\n1;:::; em\\nT]with m=1;:::; M\\nfor each HySR trajectory tHySR= [e1;:::; eT]. For each of\\nthese Mtrajectories, we sample a different series of virtual\\nstates svm\\n1:T\\u0018D rec. We augment the HySR trajectory to re\\ufb02ect\\nwhat would have happened if the virtual states were svm\\n1:T. In\\nparticular, we generate the hindsight transitions\\nem\\nt=\\u0000\\n[sr\\nt;svmt];at;r([sr\\nt;svmt];at);[sr\\nt+1;svm\\nt+1]\\u0001\\n(8)\\nfor each transition et=\\u0000\\n[sr\\nt;sv\\nt];at;r([sr\\nt;sv\\nt];at);[sr\\nt+1;sv\\nt+1]\\u0001\\ngenerated by HySR. The real states sr\\nt;sr\\nt+1and the action at\\nremain the same as in the HySR transition, but the virtual for each transition et=\\u0000\\n[sr\\nt;sv\\nt];at;r([sr\\nt;sv\\nt];at);[sr\\nt+1;sv\\nt+1]\\u0001\\ngenerated by HySR. The real states sr\\nt;sr\\nt+1and the action at\\nremain the same as in the HySR transition, but the virtual\\nstates are relabeled in hindsight with the different instances\\nof the virtual states svmt;svm\\nt+1, which are either taken from the\\npre-recorded or simulated data. The reward of the hindsight\\ntransitions is then calculated according to the relabeled state.\\nNote that in this manner, HiS generates a wide bouquet of\\nhow the real part could have interacted with instances of the\\nvirtual object. For this reason, HiS provides valuable feedback\\non the quality of this particular action sequence in different\\nsituations. In the table tennis example depicted in Fig. 1c, the\\nracket hits a whole cloud of balls with different speeds and\\ntrajectories; hence, the resulting balls arrive at very distinct\\nlanding points. a) Picking Strategy: The naive approach in which all\\nHiS trajectories are added to a replay buffer and sampled\\nby an off-policy algorithm can be detrimental to the learning\\nprocess. A critical in\\ufb02uence on the sample complexity of RL\\nalgorithms is the degree of off-policyness of transitions in the\\nreplay buffer [11]. Off-policy data are challenging because\\nthey have been collected under multiple dissimilar policies, but\\nthe expectation in the return in Eq. (1) is computed w.r.t. the\\ndistribution induced by the current policy p. This distributional\\nshift adds bias and variance to the estimation of the return and\\nleads to incorrect estimations of the Qvalues. Typically, the\\nsize of the buffer determines the degree of off-policyness of\\nthe buffer since older transitions are more likely to stem from\\nolder and hence dissimilar policies [11]. The additional HiS\\ntrajectories add to the off-policyness of the buffer since the\\nactions in the HiS trajectories were generated conditioned on\\nthe original HySR states instead of the relabeled states.\\nFor this reason, HiS training includes a selection strategy\\nthat chooses the transitions that accelerate the learning process\\nmost. We propose to rank HiS transitions based on the reward\\nr(s;a)or the TD error d(s;a;r;s0)from Eq. (2). The reason-\\ning behind prioritizing higher rewards is to experience high\\nrewards early on, which is especially useful in a sparse reward\\nsetting. The TD error corresponds to the amount of in\\ufb02uence\\non the Q-function update. Hence, it is a sensible choice for\\nranking transitions. The selection process is based on the\\nabsolute value of the TD error jd(s;a;r;s0)jsince updating\\nwith both high and low-value transition ascribes to bringing\\nthe action value function closer to its optimal version Q\\u0003. It\\nis instrumental how these measures (TD error or reward) are\\napplied to the transitions of the HiS trajectories. We study\\ntwo ways: (i) applying the measure on each transition and (ii)\\non the whole set of transitions of a trajectory by taking the\\nsum of the reward or TD error over the whole trajectory. The\\n\\ufb01rst implementation represents the straightforward approach,\\nwhereas the second way tests the hypothesis that all transitions\\nof a trajectory with high value of the measure are signi\\ufb01cant.\\nThe idea of the latter point has been explored in the context\\nof prioritized replay, where it turned out that transitions\\ncorrelating in time have correlating TD errors [4].\\nHaving de\\ufb01ned the criteria to rank HiS transitions, we now\\npresent the methodology for determining the number of HiS\\ntransitions to be added to the replay buffer. For inclusion into\\nthe buffer, criterion chas to both (i) exceed a threshold ycand\\n(ii) be among the kcbest transitions according to the criterion.\\nCase (i) prevents transitions from being added that are among\\nthe best of their peers for this round but too low in value.\\nCondition (ii) controls the maximum amount of off-policy data\\nin the replay buffer. Algorithm 1 summarizes HiS in detail.\\nb) Relation to HER: HiS shares some aspects with\\nHER [1]. HER trains a goal-conditioned policy and relabels\\nthe desired goal with the actually achieved goal in hindsight.\\nBoth HER and HiS use hindsight data to learn more sample\\nef\\ufb01ciently, and are especially useful in a sparse reward setting.\\nIn contrast, HiS relabels virtual states instead of goals.\\nFig. 2 depicts the differences between a standard MDP,0 0:2 0:4 0:6 0:8 1 1:2 1:4\\n\\u000110400:20:40:60:8success ratevan. SAC HiS\\n(a) simulated robot\\n0 0:2 0:4 0:6 0:8 1 1:2 1:4\\n\\u000110400:20:4\\nepisodessuccess rate\\n(b) real robot\\nFig. 3: Results on the table tennis task. HiS increases sample-\\nef\\ufb01ciency and asymptotic performance with respect to the\\nnumber of robot steps.\\nHySR, HER, and HiS with respect to their corresponding\\ngraphical models. Note that the HER setting also allows for\\nmultiple goals in each transition and [9] introduced dynamics\\ninto the goal space of HER. As can be seen in Fig. 2, graphical models. Note that the HER setting also allows for\\nmultiple goals in each transition and [9] introduced dynamics\\ninto the goal space of HER. As can be seen in Fig. 2,\\ngoals and dynamic goals [9] can be seen as special cases\\nof virtual states. Therefore, our method can also be applied\\nto the goal-conditional setting. The difference to HER, when\\napplied to this setting, is that HER will sample goals such\\nthat they correspond to speci\\ufb01c achieved goals (e.g., the\\ngoal sampled in hindsight is the \\ufb01nal position of the robot\\ngripper). HiS, however, will sample from the initial goal\\ndistribution and then use a prioritization strategy to select from\\nthe set of sampled goals. With enough goals sampled and\\na corresponding prioritization strategy, HiS could converge\\ntoward doing the same as HER (e.g., HiS samples enough\\ngoals, such that the prioritization strategy can pick goals that\\nare similar to what the HER strategy would have chosen).\\nIn this setting, the additional complexity of HiS compared\\nto HER seems unnecessary. However, in the general HySR\\nsetting, HER cannot be applied to virtual states, while the\\napplication of HiS is possible.\\nOn a task, that has both a goal state and a virtual state, HER\\ncan be applied on top of HiS. c) Combining HiS and HER: HER and HiS are address-\\ning the problem of sparsity during learning from two different\\ndirections. HER generates additional experiences that can be\\nbene\\ufb01cial to the learning process because they solve the goal\\nand have high rewards, but because only goals are relabeled,\\nHER does not directly help exploring the state space. On\\nthe other hand, by creating additional virtual states of the\\nenvironment, HiS helps exploring the state space. Contrary\\nto HER, experiences created by HiS might not reach the goal\\nor be of high reward. To get some of the bene\\ufb01ts of HER\\nin a goal-directed HySR setting, HiS could sample numerous\\nhindsight trajectories until some of these trajectories also\\nreach the goal. For a challenging and sparse task, however,\\nthis strategy might be computationally expensive or even\\ninfeasible.\\nInstead, a combination of HER and HiS can be bene\\ufb01cial\\nfor such tasks, and utilize the strengths of both. Practically, the\\ncombination can be implemented by following Algorithm 1,\\nand adding complete HiS episodes into the replay buffer that\\nis used by HER. HER relabels the goal of the HiS trajectories,\\nand the resulting trajectories with relabeled virtual state and\\nrelabeled goal can then be sampled to optimize the policy.\\nIV. E XPERIMENTS\\nOur new method HiS promises improved sample ef\\ufb01ciency\\nin the HySR setting. The evaluation of HiS is carried out in two\\nregimes: We run extensive experiments on 1) several simulated\\ntasks, where the HySR assumptions apply, and 2) a challenging\\nreal robot table tennis task. Furthermore, we investigate to\\nwhat extent HiS is useful in manipulation tasks and illustrate\\nthe ef\\ufb01ciency of the combination of HER and HiS.\\nA. Applying HiS to the Original HySR Table Tennis Task\\nThe work that originally introduced HySR [5] learned to\\nplay table tennis with a muscular robot and naturally separated\\nbetween the hard-to-control soft muscular robot [7, 6] and the\\nrelatively simple ball model. Different from the original task,\\nwe refrain from using the hand-crafted dense reward function\\nand use a simple sparse reward that returns one in case the ball\\nlands within a circle of radius 40 cm and zero otherwise. To\\napply HiS, we replay 20 parallel instances of the virtual ball\\nsampled from a set of 100 recorded ball trajectories. We apply\\nHiS on top of SAC [12] and compare with vanilla SAC as a\\nbaseline. For trajectory selection, we use the reward criterion\\nwith threshold yc=0:5. The Appendix contains an in-depth\\ndescription of the experiment details and the parameters of\\neach experiment. It also contains an evaluation of different\\nselection criteria.\\nThe experiments depicted in Fig. 3 demonstrate that HiS\\nenables us to learn this task with fewer robot time steps in\\nboth, the fully simulated and the HySR setting. The simulated\\nexperiments from Fig. 3a show the mean and variance of ten\\nruns with different random seeds. HiS achieves a maximum\\naverage performance of appr. 70% success rate, whereas\\nvanilla SAC reaches appr. 40%. HiS matches the asymptotic\\nperformance of SAC with appr. 29% of the samples that0 1;000 2;000 3;000 4;00000:20:40:60:81success rateHER + HiS HiS HER van. SAC\\n(a)FetchPush\\n0 2;000 4;000 6;00000:20:40:6\\nepisodessuccess rate\\n(b)FetchSlide\\nFig. 4: Results on the simulated Gym robotics tasks\\nFetchPush andFetchSlide . HER and HiS both learn\\nfaster than SAC with respect to the number of robot steps.\\nCombining HER and HiS gives the best results.\\nSAC needs to reach its asymptotic performance (2k and 7k\\nepisodes). Another observation is that the variance between\\ndifferent runs is smaller compared to SAC.\\nHiS is also more ef\\ufb01cient when using the real robot. Fig. 3b\\nshows one training run for vanilla SAC and HiS. HiS achieves\\naround 45% success rate, while SAC reaches only appr. 35%.\\nSimilarly to the simulated experiment, HiS matches SAC\\u2019s\\nmaximum success rate with appr. 38% of the number of\\nsamples collected on the real robot (3k and 8k episodes). Similarly to the simulated experiment, HiS matches SAC\\u2019s\\nmaximum success rate with appr. 38% of the number of\\nsamples collected on the real robot (3k and 8k episodes).\\nB. HySR and HiS on Manipulation Tasks\\nFetchPush andFetchSlide [1] are two benchmark\\nmanipulation tasks for goal-conditioned RL. These two tasks\\nare based on existing robot hardware and simulated using\\nMuJoCo [35]. In the Pushing task, the goal is to move a\\nbox to a target location. The robot\\u2019s gripper is locked to\\nprevent grasping. The Sliding task is slightly different: Here\\nthe objective is to move a puck to a goal position that is outside\\nthe robot\\u2019s reach, thus, necessitating a sliding strategy. Both\\ntasks satisfy the HySR assumptions and, hence, our method\\ncould be applied to learn a policy on a real robot. In both tasks,\\nthe manipulated object adheres to simple dynamics. The arm is rigid and small disturbances of the contact with the object\\nare quickly compensated by a position controller. Thus, the\\nin\\ufb02uence of forces transferred from the object to the robot is\\nsmall.\\nHowever, both tasks use a robot with relatively simple\\ndynamics that can be simulated accurately. Sim-to-real training\\nhas been shown to be successful in solving these tasks [1] and\\nwould therefore be the better choice for this kind of system.\\nHowever, when executing similar tasks with a more complex\\nrobot, for instance one equipped with soft parts, or a less\\naccurate position controller, the larger discrepancies between\\nsimulation and reality would likely degrade the performance of\\nthe transferred policy. HySR would be well-suited for learning\\na policy on such a robot. Our experiments show that on top\\nof the practical bene\\ufb01ts coming with HySR, HiS could greatly\\nimprove sample ef\\ufb01ciency for these tasks, especially when\\ncombined with HER. We will analyze the results in detail\\nin the next subsection. Furthermore, we will show, that when\\nlearning a policy in a sim-to-real fashion, HiS can still help\\nto speed up learning during the simulation stage.\\nC. Combining HER and HiS\\nIn Section III-C, we have discussed the differences and sim-\\nilarities between HER and HiS. Due to the goal-conditioned\\nnature of the FetchPush andFetchSlide tasks, they\\nare suitable to experimentally investigate the relationship of\\nHER and HiS. Fig. 4 shows the results on the Pushing and\\nSliding tasks. To apply HiS, we simulate 100 parallel instances\\nof the virtual object. For virtual trajectory selection, we use\\ntrajectories where the robot moves the object. Similar to the\\ntable tennis task, we apply HiS and\\/or HER on top of SAC,\\nwhich also serves as a baseline.\\nOn both tasks, HER and HiS learn faster than vanilla SAC.\\nHiS learns signi\\ufb01cantly faster than HER on the Pushing task,\\nsolving it almost perfectly in only 4k episodes, while HER\\nsurpasses HiS on the Sliding task. The best performance,\\nhowever, is achieved by HER and HiS in combination on both\\ntasks. Similar to the table tennis task, we also \\ufb01nd that HiS\\nalone as well as in combination with HER signi\\ufb01cantly reduces\\nthe variance between different runs.\\nA key reason why hindsight methods seem to work well\\nis that they generate additional positively labeled experiences.\\nTo illustrate this phenomena, we look at a simple metric, the\\nnumber of successful trajectories put into the RL buffer during\\nthe \\ufb01rst 1000 episodes of training. These trajectories are either\\ncollected on-policy or generated in hindsight if HiS and\\/or\\nHER are involved. We \\ufb01lter out trivial episodes that are labeled\\nsuccessful but where the object does not change position.\\nSuch episodes contain little useful information. We note a\\nstrong correlation between the number of trajectories labeled\\nsuccessful in Fig. 5 and the results from Fig. 4. For both\\ntasks, they show the same arrangement between the algorithms\\nthat we compare. This \\ufb01nding illustrates well how HER and\\nHiS complement each other. For the two tasks studied, at the\\ninitial phases of the training, the combination of HER and HiS\\ngenerates an order of magnitude more successful trajectoriesHER + HiS HiS HER van. SAC\\n0 500 1;00010\\u00001100101102103104\\nepisodesno. successful episodes in buffer\\n(a)FetchPush0 500 1;00010\\u00001100101102103104\\nepisodes\\n(b)FetchSlide\\nFig. 5: Number of successful trajectories added to the RL\\nbuffer within the \\ufb01rst 1000 episodes of learning for the\\nFetchPush andFetchSlide tasks. HER and HiS gen-\\nerates an order of magnitude more successful trajectories than\\nHER or HiS on their own.\\nthan HER or HiS individually. The intuition behind this is that\\nHiS generates a lot of hindsight trajectories, and HER labels\\nthem successful.\\nD. Ef\\ufb01ciency of HiS for Entirely Simulated Tasks\\nHiS was originally designed to speed up learning in a\\nhybrid sim and real setting, where real data is expensive and\\nsimulated data is cheap. Therefore, the experiments so far, HiS was originally designed to speed up learning in a\\nhybrid sim and real setting, where real data is expensive and\\nsimulated data is cheap. Therefore, the experiments so far,\\neven those run only in simulation have been evaluated on\\nthe number of steps that would have been executed on a real\\nrobot. But does it make sense to apply HiS in purely simulated\\ntasks, e.g., in a sim-to-real setting for the simulation stage?\\nIn this setting, instead of minimizing the number of robot\\nsteps, the goal is usually to decrease total wall-clock time\\nand computation. Compared to the baseline RL algorithm,\\nHiS increases computation because it simulates extra virtual\\nstates, sorts this data according to the selection criterion and\\nthen feeds it into the replay buffer of the algorithm. However,\\nas shown earlier, HiS also increases sample ef\\ufb01ciency during\\nlearning.\\nTab. I shows the evaluation of HiS regarding its ef\\ufb01ciency\\nfor simulated tasks. We compare HiS to vanilla SAC and the\\ncombination of HER and HiS to only HER. To compare total\\ncomputational cost, we restrict the learning to only one CPU\\n(Intel Xeon W-2145 with 3.70GHz clock speed). The effect of\\nincreased computation depends on the task and the number of\\nadditional virtual objects. Wall-clock time for the same number\\nof steps when using HiS increases only by about 2% for the\\ntable tennis task but by about 60% for the manipulation tasks.\\nTo quantify improvements in sample ef\\ufb01ciency, we look at the\\nnumber of robot time steps HiS takes to reach the asymptotic\\nor \\ufb01nal performance of vanilla SAC and similarly the number\\nof robot time steps HER and HiS takes to reach the asymptotic\\nor \\ufb01nal performance of HER. As we have shown in Fig. 3 and Fig. 4, we achieve signi\\ufb01cant improvements by using HiS.\\nTherefore, combining these two effects, even when evaluated\\nin wall-clock time instead of in robot time steps, HiS compares\\nfavorably. In conclusion, we have shown that HiS can be\\nbene\\ufb01cial for settings where we want to optimize a policy\\nonly in simulation.\\nFor this reason, HiS\\u2019 range of applications extends beyond\\ntasks that ful\\ufb01ll all the HySR assumptions as discussed in\\nSection III-B. In simulated environments, differences in the\\ncomplexity of the dynamics of different parts of the task do\\nnot need to be considered. However, HiS also assumes that\\nthe virtual state should not affect the real state. For purely\\nsimulated tasks, these effects onto the robot can be modeled,\\nbut a remaining issue is that multiple objects, that do not\\nin\\ufb02uence each other, can exert forces onto the robot simultane-\\nously. To overcome these discrepancies, one potential solution\\nfor tasks where contacts are sparse, could be to split the\\ntraining into two stages: First, training with HiS and ignoring\\nthese discrepancies. This stage could help getting into contact\\nwith interesting objects. And later \\ufb01ne-tuning those contacts\\nwithout HiS.\\nV. C ONCLUSION AND FUTURE WORK\\nIn this work, we presented Hindsight States (HiS), a method\\nfor sample-ef\\ufb01cient training in a hybrid sim and real setup,\\nwhere multiple instances of the virtual part of a task are paired\\nwith the single real part. HiS leverages this additional data\\nby selectively choosing the data to train on. We evaluated\\nHiS on a variety of tasks in simulation and showed that it\\nimproves sample ef\\ufb01ciency on a complex real-world muscular\\nrobot task. We further showed that combining HER and HiS\\nleads to even better performance than applying each method\\nindividually.\\nA limitation of our work is the requirement that the virtual\\npart does not in\\ufb02uence the real part. In many tasks, this\\nin\\ufb02uence cannot be neglected, particularly when dealing with\\nheavier objects. One way around, could be to map the forces\\nfrom the simulation back onto the joint torques of a real robot.\\nAnother strategy would be to split learning into multiple\\nstages. Stages where the condition is ful\\ufb01lled could be learned\\nwith HiS and completely real training otherwise. As an exam-\\nple, for a grasping task, the stage where the robot learns to\\nget to the position just before the gripper touches the object\\ncould be learned using HiS.\\nFor complex tasks that require long interactions, sampling\\nvirtual objects throughout the episode (instead of only at the\\nbeginning) could produce better results. The advantage of\\nthis approach is that resampling objects close to the robot\\ngenerates more important experiences with a higher chance\\ndue to additional contacts.\\nThe focus of this work was to devise a more sample\\nef\\ufb01cient training scheme in the HySR setting. However, as\\nshown in Section IV-D, HiS can also be bene\\ufb01cial for purely\\nsimulated tasks. Finding the best strategies to apply HiS to\\nsimulated tasks, especially to those that do not satisfy the\\nHySR conditions holds a lot of potential.TABLE I: Evaluation of HiS regarding its ef\\ufb01ciency for purely\\nsimulated tasks. There are two effects. First, using HiS alone\\nor HiS in combination with HER increases total wall-clock\\ntime to perform the same number of time steps. However,\\nsecondly, on average HiS needs less of those time steps to\\nreach matching performance (here: the asymptotic or \\ufb01nal\\nperformance of SAC as evaluated during the experiments),\\nand HER and HiS also need less time steps to reach matching\\nperformance (the asymptotic or \\ufb01nal performance of HER as\\nevaluated during the experiments). For the tasks studied in\\nthis work, the second effect is larger, which results in less\\naverage wall-clock time to reach the same performance, even\\nfor learning purely in simulation.\\nTask\\nPushing Sliding Table Tennis\\nComparison of HiS to SAC\\ntotal wall-clock time 163.9 % 156.3 % 102.3 %\\nrobot time steps for\\nmatching performance16.7 % 50.8 % 30.7 % Task\\nPushing Sliding Table Tennis\\nComparison of HiS to SAC\\ntotal wall-clock time 163.9 % 156.3 % 102.3 %\\nrobot time steps for\\nmatching performance16.7 % 50.8 % 30.7 %\\nwall-clock time for\\nmatching performance27.4 % 79.4 % 31.4 %\\nComparison of HER+HiS to HER\\ntotal wall-clock time 165.6 % 162.3 %\\nrobot time steps for\\nmatching performance11.7 % 42.6 %\\nwall-clock time for\\nmatching performance19.4 % 69.1 %\\nFuture work can also focus on combining HiS with curricu-\\nlum learning by using data augmentations. Data augmentations\\nsuch as transformations as well as perturbations of the virtual\\nobjects, which itself can be physically plausible, facilitates\\ngeneralization. In addition, one can adapt the laws of physics\\nof the virtual part such as gravity or the speed of time to\\nchange the dif\\ufb01culty of the whole task. In this manner, a\\ncurriculum could be added that, e.g., slows down or accelerates\\ntime during the training, so the robot reaches high reward\\nregions faster. Subsequently, adjusting time gradually to its\\ntrue speed toward the end of the training improves the transfer\\nof the performance to the real world.\\nREFERENCES\\n[1] Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas\\nSchneider, Rachel Fong, Peter Welinder, Bob Mc-\\nGrew, Josh Tobin, OpenAI Pieter Abbeel, and Wojciech\\nZaremba. Hindsight experience replay. Advances in\\nNeural Information Processing Systems , 30, 2017.\\n[2] Chenjia Bai, Peng Liu, Wei Zhao, and Xianglong Tang.\\nGuided goal generation for hindsight multi-goal re-\\ninforcement learning. Neurocomputing , 359:353\\u2013367,\\n2019.\\n[3] Sofanit Wubeshet Beyene and Ji-Hyeong Han. Priori-\\ntized hindsight with dual buffer for meta-reinforcement\\nlearning. Electronics , 11(24):4192, 2022. [4] Marc Brittain, Josh Bertram, Xuxi Yang, and Peng Wei.\\nPrioritized sequence experience replay. arXiv preprint\\narXiv:1905.12726 , 2019.\\n[5] Dieter B \\u00a8uchler, Simon Guist, Roberto Calandra, Vincent\\nBerenz, Bernhard Sch \\u00a8olkopf, and Jan Peters. Learning\\nto play table tennis from scratch using muscular robots.\\nIEEE Transactions on Robotics , 38(6):3850\\u20133860, 2022.\\n[6] Dieter B \\u00a8uchler, Roberto Calandra, and Jan Peters. Learn-\\ning to control highly accelerated ballistic movements on\\nmuscular robots. Robotics and Autonomous Systems , 159:\\n104230, 2023.\\n[7] Dieter B \\u00a8uchler, Heiko Ott, and Jan Peters. A Lightweight\\nRobotic Arm with Pneumatic Muscles for Robot Learn-\\ning. In International Conference on Robotics and Au-\\ntomation (ICRA) , Stockholm, May 2016. doi: 10.1109\\/\\nicra.2016.7487599.\\n[8] Shirli Di-Castro, Dotan Di Castro, and Shie Mannor. Sim\\nand real: Better together. Advances in Neural Information\\nProcessing Systems , 34:6868\\u20136880, 2021.\\n[9] Meng Fang, Cheng Zhou, Bei Shi, Boqing Gong, Jia Xu,\\nand Tong Zhang. DHER: Hindsight experience replay for\\ndynamic goals. In International Conference on Learning\\nRepresentations , September 2018.\\n[10] Meng Fang, Tianyi Zhou, Yali Du, Lei Han, and\\nZhengyou Zhang. Curriculum-guided hindsight experi-\\nence replay. Advances in Neural Information Processing\\nSystems , 32, 2019.\\n[11] William Fedus, Prajit Ramachandran, Rishabh Agarwal,\\nYoshua Bengio, Hugo Larochelle, Mark Rowland, and\\nWill Dabney. Revisiting fundamentals of experience re-\\nplay. In International Conference on Machine Learning ,\\npages 3061\\u20133071. PMLR, 2020.\\n[12] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and\\nSergey Levine. Soft Actor-Critic: Off-policy maximum\\nentropy deep reinforcement learning with a stochastic\\nactor. In International Conference on Machine Learning ,\\npages 1861\\u20131870. PMLR, 2018.\\n[13] D. F. B. Haeu\\ufb02e, M. G \\u00a8unther, A. Bayer, and S. Schmitt.\\nHill-type muscle model with serial damping and eccen-\\ntric force\\u2013velocity relation. Journal of Biomechanics ,\\n47(6):1531\\u20131536, April 2014. ISSN 0021-9290. doi:\\n10.1016\\/j.jbiomech.2014.02.009.\\n[14] Irina Higgins, Arka Pal, Andrei Rusu, Loic Matthey,\\nChristopher Burgess, Alexander Pritzel, Matthew\\nBotvinick, Charles Blundell, and Alexander Lerchner.\\nDarla: Improving zero-shot transfer in reinforcement\\nlearning. In International Conference on Machine\\nLearning , pages 1480\\u20131490. PMLR, 2017.\\n[15] Stephen James, Paul Wohlhart, Mrinal Kalakrishnan,\\nDmitry Kalashnikov, Alex Irpan, Julian Ibarz, Sergey\\nLevine, Raia Hadsell, and Konstantinos Bousmalis. Sim-\\nto-real via sim-to-sim: Data-ef\\ufb01cient robotic grasping\\nvia randomized-to-canonical adaptation networks. In\\nProceedings of the IEEE\\/CVF Conference on Computer\\nVision and Pattern Recognition , pages 12627\\u201312637,\\n2019.[16] Katie Kang, Suneel Belkhale, Gregory Kahn, Pieter\\nAbbeel, and Sergey Levine. Generalization through sim-\\nulation: Integrating simulated and real data into deep re-\\ninforcement learning for vision-based autonomous \\ufb02ight.\\nInInternational Conference on Robotics and Automation\\n(ICRA) , pages 6008\\u20136014. IEEE, 2019.\\n[17] Aviral Kumar, Justin Fu, Matthew Soh, George Tucker,\\nand Sergey Levine. Stabilizing off-policy q-learning\\nvia bootstrapping error reduction. Advances in Neural\\nInformation Processing Systems , 32, 2019.\\n[18] Aviral Kumar, Aurick Zhou, George Tucker, and Sergey\\nLevine. Conservative Q-learning for of\\ufb02ine reinforce-\\nment learning. Advances in Neural Information Process-\\ning Systems , 33:1179\\u20131191, 2020.\\n[19] Sascha Lange, Thomas Gabel, and Martin Riedmiller.\\nBatch reinforcement learning. Reinforcement learning:\\nState-of-the-art , pages 45\\u201373, 2012.\\n[20] Sergey Levine, Aviral Kumar, George Tucker, and Justin\\nFu. Of\\ufb02ine reinforcement learning: Tutorial, review, and\\nperspectives on open problems. arXiv:2005.01643 [cs,\\nstat], November 2020. URL http:\\/\\/arxiv.org\\/abs\\/2005.\\n01643. arXiv: 2005.01643.\\n[21] Long-Ji Lin. Self-improving reactive agents based on\\nreinforcement learning, planning and teaching. Machine stat], November 2020. URL http:\\/\\/arxiv.org\\/abs\\/2005.\\n01643. arXiv: 2005.01643.\\n[21] Long-Ji Lin. Self-improving reactive agents based on\\nreinforcement learning, planning and teaching. Machine\\nlearning , 8(3):293\\u2013321, 1992. Publisher: Springer.\\n[22] Peng Liu, Chenjia Bai, Yingnan Zhao, Chenyao Bai, Wei\\nZhao, and Xianglong Tang. Generating attentive goals for\\nprioritized hindsight reinforcement learning. Knowledge-\\nBased Systems , 203:106140, 2020.\\n[23] Jeffrey Mahler, Matthew Matl, Vishal Satish, Michael\\nDanielczuk, Bill DeRose, Stephen McKinley, and Ken\\nGoldberg. Learning ambidextrous robot grasping poli-\\ncies. Science Robotics , 4(26), 2019.\\n[24] V olodymyr Mnih, Koray Kavukcuoglu, David Silver,\\nAlex Graves, Ioannis Antonoglou, Daan Wierstra, and\\nMartin Riedmiller. Playing Atari with deep reinforcement\\nlearning. arXiv preprint arXiv:1312.5602 , 2013. URL\\nhttp:\\/\\/arxiv.org\\/abs\\/1312.5602.\\n[25] Andrew W. Moore and Christopher G. Atkeson. Prior-\\nitized sweeping: Reinforcement learning with less data\\nand less time. Mach Learn , 13(1):103\\u2013130, October\\n1993. ISSN 1573-0565. doi: 10.1007\\/BF00993104. URL\\nhttps:\\/\\/doi.org\\/10.1007\\/BF00993104.\\n[26] Fabio Muratore, Fabio Ramos, Greg Turk, Wenhao Yu,\\nMichael Gienger, and Jan Peters. Robot learning from\\nrandomized simulations: A review. Frontiers in Robotics\\nand AI , page 31, 2022.\\n[27] Jing Peng and Ronald J. Williams. Ef\\ufb01cient learning\\nand planning within the Dyna framework. Adaptive\\nBehavior , 1(4):437\\u2013454, March 1993. ISSN 1059-7123.\\ndoi: 10.1177\\/105971239300100403. Publisher: SAGE\\nPublications Ltd STM.\\n[28] X. B. Peng, M. Andrychowicz, W. Zaremba, and\\nP. Abbeel. Sim-to-real transfer of robotic control with\\ndynamics randomization. In 2018 IEEE International\\nConference on Robotics and Automation (ICRA) , pages 1\\u20138, May 2018. doi: 10.1109\\/ICRA.2018.8460528.\\n[29] Rafael Rafailov, Tianhe Yu, Aravind Rajeswaran, and\\nChelsea Finn. Of\\ufb02ine reinforcement learning from im-\\nages with latent space models. In Learning for Dynamics\\nand Control , pages 1154\\u20131168. PMLR, 2021.\\n[30] Antonin Raf\\ufb01n. RL Baselines3 Zoo. github.com\\/\\nDLR-RM\\/rl-baselines3-zoo, 2020.\\n[31] Paulo Rauber, Avinash Ummadisingu, Filipe Mutz,\\nand J \\u00a8urgen Schmidhuber. Hindsight policy gradients.\\narXiv:1711.06006 [cs] , February 2019. URL http:\\/\\/arxiv.\\norg\\/abs\\/1711.06006. arXiv: 1711.06006.\\n[32] Zhizhou Ren, Kefan Dong, Yuan Zhou, Qiang Liu, and\\nJian Peng. Exploration via hindsight goal generation.\\nAdvances in Neural Information Processing Systems , 32,\\n2019.\\n[33] Tom Schaul, John Quan, Ioannis Antonoglou, and David\\nSilver. Prioritized experience replay. arXiv preprint\\narXiv:1511.05952 , 2015.\\n[34] Noah Y Siegel, Jost Tobias Springenberg, Felix\\nBerkenkamp, Abbas Abdolmaleki, Michael Neunert,\\nThomas Lampe, Roland Hafner, Nicolas Heess, and\\nMartin Riedmiller. Keep doing what worked: Behavioral\\nmodelling priors for of\\ufb02ine reinforcement learning. arXiv\\npreprint arXiv:2002.08396 , 2020.\\n[35] Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco:\\nA physics engine for model-based control. In Intelligent\\nRobots and Systems (IROS), 2012 IEEE\\/RSJ Interna-\\ntional Conference on , pages 5026\\u20135033. IEEE, 2012.\\n[36] Yifan Wu, George Tucker, and O\\ufb01r Nachum. Behavior\\nregularized of\\ufb02ine reinforcement learning. arXiv preprint\\narXiv:1911.11361 , 2019.\\n[37] Tianhe Yu, Garrett Thomas, Lantao Yu, Stefano Ermon,\\nJames Zou, Sergey Levine, Chelsea Finn, and Tengyu\\nMa. MOPO: Model-based of\\ufb02ine policy optimization.\\narXiv:2005.13239 [cs, stat] , May 2020. URL http:\\/\\/arxiv.\\norg\\/abs\\/2005.13239. arXiv: 2005.13239.\\n[38] Tianhe Yu, Aviral Kumar, Rafael Rafailov, Aravind Ra-\\njeswaran, Sergey Levine, and Chelsea Finn. COMBO:\\nConservative of\\ufb02ine model-based policy optimization.\\nAdvances in Neural Information Processing Systems , 34:\\n28954\\u201328967, 2021.\\n[39] Jingwei Zhang, Lei Tai, Peng Yun, Yufeng Xiong, Ming\\nLiu, Joschka Boedecker, and Wolfram Burgard. VR-\\ngoggles for robots: Real-to-sim domain adaptation for\\nvisual control. IEEE Robotics and Automation Letters , 4\\n(2):1148\\u20131155, 2019.\\n[40] Rui Zhao and V olker Tresp. Energy-based hindsight ex-\\nperience prioritization. In Conference on Robot Learning ,\\npages 113\\u2013122. PMLR, 2018.\\n[41] Wei Zhu, Xian Guo, Dai Owaki, Kyo Kutsuzawa, and\\nMitsuhiro Hayashibe. A survey of sim-to-real transfer\\ntechniques applied to reinforcement learning for bioin-\\nspired robots. IEEE Transactions on Neural Networks\\nand Learning Systems , pages 1\\u201316, 2021. ISSN 2162-\\n2388. doi: 10.1109\\/TNNLS.2021.3112718. VI. A PPENDIX\\nA. Experiment Details\\nIn this section, we elaborate on the details of the tasks that\\nwe used for evaluating our method.\\n1) Simulated Table Tennis Task: We run our experiments in\\na custom MuJoCo environment, simulating the muscles with a\\nHill-type muscle model [13]. Similar to B \\u00a8uchler et al. [5], we\\nreplay ball trajectories until contact and simulate afterwards.\\nThe average episode length is 37-39 time steps, depending on\\nthe policy. The episode ends when the ball touches the racket\\nor gets out of reach of the racket.\\nFor HiS, parallel episodes continue after the main episode\\nended. The problem is that we sample the policy distribution\\nconditioned on the state containing the main virtual object. In\\nthese cases, we sample the policy with one of the other virtual\\nobjects, until the episode is \\ufb01nished for all virtual objects.\\n2) Real Robot Table Tennis Task: We keep most of the task\\ndetails similar to the original experiment in B \\u00a8uchler et al. [5]\\nand list our adaptations.\\n\\u000fWe employ our policy on a new iteration of the robot\\nhardware.\\n\\u000fThe initialization procedure is different: Instead of using\\na position controller, we send \\ufb01xed target pressures to get\\nthe robot to an initial position.\\n\\u000fWe send actions to the robot at a frequency of 25 Hz\\ninstead of 100 Hz because we found this to lead to\\nsigni\\ufb01cantly better results with SAC.\\nCollecting 15k episodes takes about 24 hours on the real robot.\\n3) FetchPush and FetchSlide: Each episode has a \\ufb01xed\\nlength of 50 time steps. To apply HiS, we sample additional\\nvirtual objects according to the same distribution that is used\\nto sample the main object.\\nB. Hyperparameters\\nThe hyperparameters for the three tasks are shown in\\nTables II, III, and IV. The SAC hyperparameters for the table\\ntennis task were tuned on the simulated task without HiS. The\\ngym robotics experiments incorporate the hyperparameters\\nfound in [30].\\nC. Ablation Study: Selection Criterion\\nUsing the reward criterion, which selects mainly successful\\ntrajectories, works well for HiS on all our tasks. Because\\nHER relabels trajectories as successful, it is less relevant to\\nselect successful HiS trajectories when applying HER and\\nHiS together. Therefore, picking trajectories where the virtual\\nobject is moved, works even better in this case. In this work,\\nwe focused on these criteria to showcase our method. Our\\nexperiments indicate that the TD error criterion helps the\\nlearning progress, but less then the other two criteria. Table V\\nshows an evaluation of the different selection criteria. Note that\\nthe real table tennis task was evaluated for only one training\\nrun. The simulated table tennis task was averaged over ten\\ntraining runs with different random seeds, and the sliding task\\nover \\ufb01ve training runs. The table tennis task, both real andTABLE II: Hyperparameters table tennis\\nhyperparameter value\\nSACgamma 0.9999\\nentcoef 0\\nlearning rate 0.0003\\nbatch size 256\\npolicy network MLP\\nnum layers 1\\nnum hidden 200\\ngradient steps 500\\ntrain freq 1\\ntrain freq unit episode\\nbuffer size 5000000\\nlearning starts 10000\\nHiScriterion reward per episode\\nkc 3\\nyc 0.5\\nTABLE III: Hyperparameters FetchPush\\nhyperparameter value\\nSACgamma 0.95\\nentcoef auto\\nlearning rate 0.001\\nbatch size 256\\npolicy network MLP\\nnum layers 2\\nnum hidden 64\\ngradient steps 1\\ntrain freq 1\\ntrain freq unit step\\nbuffer size 5000000\\nlearning starts 1000\\nHERgoal selection strategy future\\nnsampled goal 4\\nHiScriterion Dxvirt. object\\nkc 3\\nyc 0.02\\nTABLE IV: Hyperparameters FetchSlide\\nhyperparameter value\\nSACgamma 0.95\\nentcoef auto\\nlearning rate 0.001\\nbatch size 2048\\npolicy network MLP\\nnum layers 3\\nnum hidden 512\\ngradient steps 1\\ntrain freq 1\\ntrain freq unit step\\nbuffer size 5000000\\nlearning starts 1000\\nHERgoal selection strategy future\\nnsampled goal 4\\nHiScriterion Dxvirt. object\\nkc 3\\nyc 0.02 simulated, was evaluated after 15000 and the sliding task after\\n5000 episodes.\\nTABLE V: Evaluation of different selection criteria for HiS\\nand HER+HiS on \\ufb01nal success rate.\\nTask\\nSim. Table Tennis Real Table Tennis Sliding\\nvan. SAC 41.2 % 34.5 % 9.9 %\\nHiS reward 70.6 % 44.7 % 33.7 %\\nHiSDxvirt. object 25.5 %\\nHiS TD error 50.8 % 34.2 % 13.2 %\\nHER+HIS reward 53.8 %\\nHER+HIS Dxvirt. object 64.5 %\",\"32\":\" ChatGPT and Other Large Language Models as Evolutionary\\nEngines for Online Interactive Collaborative Game Design\\nPier Luca Lanzi\\npierluca.lanzi@polimi.it\\nPolitecnico di Milano\\nMilano, ItalyDaniele Loiacono\\ndaniele.loiacono@polimi.it\\nPolitecnico di Milano\\nMilano, Italy\\nABSTRACT\\nLarge language models (LLMs) have taken the scientific world by\\nstorm, changing the landscape of natural language processing and\\nhuman-computer interaction. These powerful tools can answer\\ncomplex questions and, surprisingly, perform challenging creative\\ntasks (e.g., generate code and applications to solve problems, write\\nstories, pieces of music, etc.). In this paper, we present a collabo-\\nrative design framework that combines interactive evolution and\\nlarge language models to simulate the typical human design pro-\\ncess. We use the former to exploit users\\u2019 feedback for selecting\\nthe most promising ideas and large language models for a very\\ncomplex creative task\\u2014the recombination and variation of ideas. In\\nour framework, the process starts with a brief and a set of candidate\\ndesigns, either generated using a language model or proposed by\\nthe users. Next, users collaborate on the design process by pro-\\nviding feedback to an interactive genetic algorithm that selects,\\nrecombines, and mutates the most promising designs. We evaluated\\nour framework on three game design tasks with human designers\\nwho collaborated remotely.\\nCCS CONCEPTS\\n\\u2022Computing methodologies \\u2192Genetic algorithms ;\\u2022The-\\nory of computation \\u2192Interactive computation ;\\u2022Human-\\ncentered computing \\u2192Systems and tools for interaction de-\\nsign .\\nKEYWORDS\\nCollaborative Design, Large Language Models, Interactive Evolu-\\ntion\\n1 INTRODUCTION\\nDesign is a collaborative problem-solving activity that begins with\\na design brief providing an approximate description of a target\\nobjective. In the opening (divergent) phase, several initial ideas of\\npossible solutions are proposed. These are the seeds for the next\\nexploration (emergent) phase in which ideas are modified and com-\\nbined to create new solutions over a series of iterations. Finally,\\nin the closing (convergent) phase, developed ideas are critically\\nexamined, assessed, and the process result is synthesized [ 15,37].\\nInteractive evolution [ 47] can model collaborative design processes\\neffectively, and it has been widely applied to design and creative ac-\\ntivities like fashion [ 24,25,34,46], music composition [ 3,18,19,54],\\nvideo games [ 17,43] and art in general [ 26,27]. However, its applica-\\ntion has been limited to domains where admissible design solutions\\ncould be encoded, randomly generated, mutated, and recombined.\\nFor example, it has never been applied to evolve free-form texts thatwould need advanced operators to maintain semantic coherence in\\ngenerating the results.\\nLarge Language Models (LLMs) have changed the landscape\\nof natural language processing and human-computer interaction.\\nThese powerful tools can answer complex questions and perform\\nchallenging creative tasks such as writing screenplays and theater\\nscripts [ 32], solving complex math problems [ 9], theorem proofing\\n[41], generating code to solve specific tasks [ 8], generate descrip-\\ntions from images [ 7], etc. More interestingly, they can also generate\\nrandom texts from high-level instructions, combine two or more\\ntexts, and create variations of existing texts\\u2014all while maintaining\\nsemantic coherence in the generated texts. These functionalities can\\neasily map to the genetic operators at the core of most evolution-\\nary algorithms. Thus, we can use LLMs to implement evolutionary\\noperators that can manage free-form text individuals while also\\nmaintaining semantic coherence\\u2014something that would be difficult,\\nif not impossible, to achieve with traditional representations.\\nIn this paper, we present an online collaborative design frame-\\nwork that combines interactive evolution with large language mod-\\nels for evolving game design ideas ( game concepts ) represented In this paper, we present an online collaborative design frame-\\nwork that combines interactive evolution with large language mod-\\nels for evolving game design ideas ( game concepts ) represented\\nas free-form texts. The online interactive evolutionary algorithm\\nshows the ideas to the human designers and collects their feedback;\\nthe large language model implements all the operators for the ran-\\ndom generation, the recombination (crossover), and the variation\\n(mutation) of ideas. Our framework is very general in that it can\\nsupport any design process based on textual information (and could\\nbe easily extended to integrate visuals integrating tools like Dall-E\\n2 [39]); it can use any viable LLM and therefore, it is intrinsically\\nmultilingual; it can be accessed using any medium that can visualize\\ntexts like a webpage [ 5,6], a 2D-grid displayed on screen [ 1,53],\\nor any viable messaging platform that supports user polls or other\\nways to collect users\\u2019 feedback.\\nWe performed a preliminary evaluation of our framework on\\nthree creative tasks, the design of a board game, the design of a\\nvideo game, and the generation of game ideas during the 2023\\nGlobal Game Jam. The evaluation involved around 80 users (junior\\nand senior game designers). The goal of the two game design tasks\\nwas to simulate a typical idea-generation process; accordingly, we\\nset a fixed time limit of four working days for each task. The in-\\nteractive evolutionary algorithm communicated with the human\\ndesigners using the Telegram messaging platform [ 31] while the\\ngenetic operators were implemented using ChatGPT [ 38]. Each\\ntask was initiated with a design brief [ 15] that was sent to the de-\\nsigners as the first message on a Telegram group. The brief was\\nfollowed by a series of initial design ideas (some randomly gener-\\nated using the LLM, some collected from existing designs) \\u2014 the\\nopening (divergent) phase [ 15]. Then, over the next four days, usersarXiv:2303.02155v1  [cs.AI]  9 Feb 2023 Pier Luca Lanzi and Daniele Loiacono\\nevaluated the design ideas they received using the Telegram poll\\nsystem; the evolutionary algorithm employed the users\\u2019 feedback\\nto select, recombine and mutate the existing ideas into new ones\\npresented next. Instead, the evaluation performed during the 2023\\nGlobal Game Jam was conceived as an open-ended brainstorming\\nactivity that lasted less than 24 hours to fit the tight schedule of\\nthe event.\\nThere are few examples of applications of LLMs to idea gen-\\neration and creativity in games. For example, Zhu and Luo [ 56]\\nexplored a generative approach for design ideation combining a lan-\\nguage model and a knowledge based of existing patents to generate\\nnew ideas. van Stegeren and Myundefinedliwiec [48] investigated\\nthe application of GPT-2 for generating dialogues in role-playing\\ngames for non-player characters while V\\u00e4rtinen et al. [ 50] applied\\nGPT-2 [ 42] and GPT-3 [ 4] to generate description of quests in the\\nsame type of games. Frans [ 12] presented a framework for building\\nLanguage Model Games whose main mechanic involved players\\nmanipulating a language model into behaving in a desired manner\\nwith a demo game called AI Charades. However, to the best of our\\nknowledge, our framework is the first example of interactive evolu-\\ntionary algorithm that apply large language models to implement\\ngenetic operator and combines them with interactive evolution to\\nmimic a full-fledged design session from start to end.\\n2 RELATED WORK\\nInteractive evolution [ 47] dates back to Richard Dawkins\\u2019s The Blind\\nWatchmaker andBiomorphs [10]. Over the years, this approach has\\nbeen applied to several creative tasks, for which human evaluation\\nmight provide a better (if not the only) alternative to the definition\\nof an objective (or fitness) function to evaluate candidate solution,\\nsuch as fashion [ 24,25,34,46], ergonomics [ 2,51], yacht design [ 23],\\nvisual effects [ 11,21], synthetic images [ 45], music [ 3,18,19,54],\\nvideo games [17, 43] and art in general [26, 27].\\nInteractive evolutionary frameworks collect human evaluations\\neither implicitly or explicitly. In the former case, the systems track\\nhuman actions, sometimes building a model of users\\u2019 behavior and\\nderiving an assessment of their preferences [ 20,55]. Galactic Arms\\nRace (GAR) [ 17] is an example of implicit evaluation. It was the first\\nvideo game using interactive evolution for content generation; it\\nincluded a weapon system that automatically evolved new weapons\\nbased on players\\u2019 behavior. Petalz [ 43] was a Facebook game about\\ngrowing a garden of procedurally generated flowers; players would\\nselect what flowers they liked most and wanted to breed; human\\nevaluation used by the interactive evolutionary algorithm was im-\\nplicitly implemented within the farming mechanics.\\nMost often, interactive evolutionary frameworks involve an\\nexplicit human-based evaluation of candidates and, similarly to\\nDawkins\\u2019 Biomorphs [10], have a grid-like interface that allows\\nusers to evaluate candidate designs iteratively. Hoover et al. [ 18,19]\\nintegrated interactive evolution with functional scaffolding for gen-\\nerating drum tracks [ 18] and accompaniments for existing musical\\npieces [ 19] using a compositional pattern-producing network. At\\nfirst, users select a musical piece (in MIDI format) and which parts\\nshould be elaborated by the evolutionary algorithm (e.g., the pi-\\nano, guitar, or bass guitar). Next, users customize and refine thecomputer-generated accompaniment through an interactive selec-\\ntion and mutation of compositional networks. Noticeably, the user\\ninterface in [ 19] is similar to other music composition tools instead\\nof the grid-based structure used in most frameworks. Cardamone\\net al. [ 5,6] developed a tool that combined an interactive genetic\\nalgorithm with procedural content generation to evolve tracks for\\ntwo open-source racing games. The tool had a web frontend that\\nremained active for several years and currently appears to be dis- algorithm with procedural content generation to evolve tracks for\\ntwo open-source racing games. The tool had a web frontend that\\nremained active for several years and currently appears to be dis-\\ncontinued, although still online. \\u00d8lsted et al. [ 57] developed a tool\\nfor the interactive evolution of first-person-shooter maps inspired\\nby the popular multiplayer game Counter-Strike. While the tool of\\n[5,6] is external to the actual racing games, the tool of \\u00d8lsted et al.\\n[57] is integrated within the game (FPSEvolver) and operated live,\\ncollecting players\\u2019 feedback during the usual map voting and selec-\\ntion process\\u2014a typical phase in multiplayer first-person-shooter\\ngames. Pirovano et al. [ 40] developed a tool for creating 3D sword\\nmodels that combined procedural content generation, interactive\\nevolution, and a grid-based interface to let users explore such a pe-\\nculiar design space. Generative Adversarial Networks (GANs) [ 14]\\nrepresented a milestone in procedural content generation and have\\nbeen applied to various creative tasks. They learn the probability\\ndistribution that generates a collection of training examples and\\ncan generate more samples from randomly generated input vectors\\nof Gaussian noise, which form the latent space. Several authors\\ndeveloped methods to explore the latent space to guide the gener-\\nation of examples [ 13,49], including interactive evolution [ 1,44].\\nBontrager et al. [ 1] combined generative adversarial networks for\\nimage generation with interactive evolution to explore the latent\\nspace to improve the quality of sampled images for specific targets.\\nSchrum et al. [ 44] applied interactive evolution for exploring the\\nlatent space for generating levels for the video games Super Mario\\nBros. andThe Legend of Zelda . Both [ 1] and [ 44] require an explicit\\nhuman-based evaluation using a grid-like interface similar to the\\nones used in [5, 6, 10, 40].\\nMost approaches ask users to select the designs they like best\\n[1,44]; however, some authors proposed ways to improve the users\\u2019\\nagency over the process [ 30,53]. Wooley et al. [ 53] combine in-\\nteractive human evaluation with novelty search to facilitate the\\nserendipitous discovery of designs by allowing the users to seed the\\nforthcoming generation with novel descendants. Liapis et al. [ 30]\\nask the users to rank the individuals instead of evaluating the single\\nindividuals; this provides more information about the users\\u2019 pref-\\nerences to the underlying evolutionary engine and appears to be\\nfaster and more robust than preference-based evaluation. Mitchel et\\nal. [33] introduced an interactive evolutionary framework (PETRI)\\nwith an interface based on physical tangibles elements. Each tangi-\\nble represented a design in the population, and users could interact\\nwith them using specific gestures. The goal was to enhance the\\nuser experience while alleviating fatigue. Mitchel at al. [ 33] applied\\nPETRI to reproduce the evolution of biomorph images [10] and to\\ndevelop an application to explore quantum phenomena through\\nmovement and dance. Later, Ivanov et al. [ 22] developed EvoIsland,\\na framework that maintained localized populations of designs and\\nlet users manipulate the subpopulations using a touch interface.\\nEvoIsland structure is inspired by spatial isolation of land masses\\non Earth and resembles several similarities to the island models in\\nparallel genetic algorithms [16]. ChatGPT and Other Large Language Models as Evolutionary Engines for Online Interactive Collaborative Game Design\\nFigure 1: The framework architecture when collaboration is\\nbased on the Telegram messaging platform.\\n3 THE FRAMEWORK ARCHITECTURE\\nOur framework combines interactive evolution and large language\\nmodels to simulate the typical human design process. It comprises\\nthree components: (i) the online database of designs; (ii) the evolu-\\ntionary algorithm; and (iii) the agent responsible for publishing the\\ndesigns and collecting users\\u2019 feedback. Figure 1 shows the structure\\nof our framework; in this case, the interaction with the human\\ndesigners uses the Telegram messaging system, which is the one\\nwe employed in the experiments discussed in this study.\\nThe database contains (i) a list of design ideas that have been pub-\\nlished and are currently under evaluation; (ii) the list of submitted\\nevaluations that are used to compute the fitness; (iii) the current\\npopulation of active design ideas; this includes both the designs\\nthat been already published and under evaluation by the users and\\nthe ones that have been already generated but not published yet.\\nThe evolutionary engine is implemented as a steady-state ge-\\nnetic algorithm that, at each iteration, (i) selects promising design\\nideas from the current population; (ii) applies recombination and\\nmutation (implemented using a large language model like CharGPT\\nor another LLM like DaVinci GPT-3 [ 4]); (iii) inserts the newly gen-\\nerated individuals in the population; finally, (iv) deletes individuals\\nto keep the population size constant.\\nThe interaction agent manages the communication between the\\nframework and the publishing server; it publishes the design con-\\ncepts in the population and collects the users\\u2019 feedback. In Figure 1,\\nwe show the architecture used in the experiments discussed in this\\npaper with an interaction agent implemented as a Telegram bot\\nthat publishes the design concepts on a Telegram group through\\nthe Telegram servers. However, our framework also supports pub-\\nlishing using web servers (as done in [6, 45]).Note that, when using a publishing-subscribe platform like Tele-\\ngram, the publication of design concepts is timed to limit the num-\\nber of concepts (and message notifications) the users receive at\\nany given time (see Section 5). Accordingly, we maintain two sepa-\\nrate tables, one for the population of active designs (Figure 1) and\\none for the published designs currently being evaluated. In con-\\ntrast, when interacting through a web server, the agent will publish\\nthe population all at once; users will autonomously decide how\\nmany concepts they want to view and evaluate using the online\\nnavigation interface (similarly to [6, 45]).\\nUser Evaluation. Most of the interactive evolutionary algorithms\\nwe examined (Section 2) ask for qualitative evaluations based, for\\nexample, on basic voting systems [ 44,57], scales with a limited\\nnumber of values [ 5,6], or let users actively select what individual\\nshould go to the next generation [ 1,53]. In our framework, we\\nadopted a simple three-value scale (positive, neutral, and negative)\\nthat let users express how they felt about a given design idea. Note\\nthat, the evaluation is anonymous and users cannot see how other\\npeople evaluated an idea before they submitted their evaluation.\\nWhen using the Telegram poll-based interface, only the summary\\nstatistics about the submitted evaluations will appear on the group\\nchat; when using the web-based interface, users can see only their\\nevaluation.\\n3.1 The Workflow\\nThe process starts with a short design brief , describing the activity\\ngoals [ 52]. The evolutionary algorithm uses the brief to gener-\\nate the initial population either (i) by asking ChatGPT or another\\nlanguage model to generate random solutions (similarly to the tra-\\nditional population initialization in evolutionary algorithms); (ii) by\\nintroducing human-designed concepts originating from the brief ditional population initialization in evolutionary algorithms); (ii) by\\nintroducing human-designed concepts originating from the brief\\n(e.g., by asking participants to propose solutions or taking existing\\nwell-known solutions); or (iii) by combining the previous options\\n(similarly to the half and half initialization in Genetic Program-\\nming [ 28]). The individuals in the initial population are published\\neither at once (when collaboration is web-based) or timely (when\\nusing a publishing-subscribe platform like Telegram). Next, the\\nframework waits for feedback from the users, who can score each\\npublished concept. When sufficient evaluations have been submit-\\nted for all the published concepts, the evolutionary algorithm starts\\nits select-recombine-mutate cycle: it selects two individuals from\\nthe population using tournament selection of size two; with a given\\nprobability, it applies recombination to generate one offspring that\\nis then mutated; finally, the offspring is added to the population\\nwhile the worst individual is deleted to keep the population size\\nconstant. The new individual becomes eligible for publication; de-\\npending on the platform used, the individual is published immedi-\\nately or in a given time frame. The next iteration of the evolutionary\\nalgorithm will start when a sufficient number of evaluations have\\nbeen submitted for the newly published individuals.\\n4 IMPLEMENTING GENETIC OPERATORS\\nWITH LARGE LANGUAGE MODELS\\nLLMs are queried using questions (i.e., prompts ) expressed in nat-\\nural language. The quality of their answers heavily relies on how\\nwell the prompts can specify the objective for the target LLM. In Pier Luca Lanzi and Daniele Loiacono\\nour framework, individuals represent game design ideas expressed\\nas free-form text; all the operators manipulating them are imple-\\nmented by querying an LLM. Accordingly, we initially studied how\\nto design prompts so that the LLM answers could mimic the behav-\\nior of the three operators we needed to implement the evolutionary\\nalgorithm: random initialization of individuals, recombination or\\ncrossover, and variation or mutation.\\n4.1 Random Initialization of Individual\\nIn our framework, the population can be initialized (i) with random\\nindividuals, (ii) with existing designs, or (iii) with a combination of\\nthe previous approaches. Random individuals (game ideas) ignite\\nthe design process and need to provide an approximate indication\\nof the target objective. In our initial analysis of feasible prompts to\\nimplement evolutionary operators we noted that it is convenient to\\nask the system to structure the answer. Accordingly, the prompts\\nused to generate the initial random game ideas consist of (i) a design\\nbrief defining the objective and its constraints in an organized\\nstructure; and (ii) a request to act as a game designer and generate\\na novel design based on the brief. Table 1a shows a prompt to\\ngenerate a random idea for a generic board game. The bold text\\nshows the actual request while the remaining part specifies how the\\nanswer should be structured. Note that, the prompt also specifies the\\nlength of each section to avoid very long game ideas that could be\\nmore difficult to evaluate by the designers. When the request is less\\ngeneric, the prompt begins with a text that serves as inspiration\\nor as an example of the objective. Table 1b shows a prompt to\\ngenerate a video game that uses a minimal amount of resources.\\nThis example is inspired to the study of minimalism in video game\\ndesign [ 29,35,36]. The first sentence of the prompt is a statement\\nthat serves as an inspiration and sets the basis of what is going to\\nbe asked. The second sentence (in bold) shows the actual request\\n(as in the previous case). The remaining text specifies the answer\\nstructure. Table 1c shows an example of prompt that was used to\\ngenerate random individuals for an experiment run during the 2023\\nGlobal Game Jam1in which the goal was to create a game based on\\nthe theme roots . Also in this case, the first sentence was used as a sort\\nof inspiration. However, since there could be many interpretations\\nofroots (it could refer to cultural heritage, botanic, etc.), the sentence\\ncontained the marker \\u27e8\\ud835\\udc3c\\ud835\\udc41\\ud835\\udc47\\ud835\\udc38\\ud835\\udc45\\ud835\\udc43\\ud835\\udc45\\ud835\\udc38\\ud835\\udc47\\ud835\\udc34\\ud835\\udc47\\ud835\\udc3c\\ud835\\udc42\\ud835\\udc41 \\u27e9which was replaced\\nwith sentences suggesting different interpretations (Table 1d). The\\nprocess generated a great variety of prompts for the same theme.\\n4.2 Recombination (Crossover)\\nThe prompt to implement the recombination of two individuals\\nhas a general structure (Table 1e). It starts with the same brief used\\nto generate random individuals. For example, when recombining\\ntwo video games using minimal resources, the marker \\u27e8\\ud835\\udc35\\ud835\\udc45\\ud835\\udc3c\\ud835\\udc38\\ud835\\udc39\\u27e9\\nwould be replaced with the sentence \\u201cA white pixel is the minimum\\namount [...]\\u201d from Table 1b. The brief is followed by two examples\\nof games in which the markers \\u27e8\\ud835\\udc3c\\ud835\\udc41\\ud835\\udc37\\ud835\\udc3c\\ud835\\udc49\\ud835\\udc3c\\ud835\\udc37\\ud835\\udc48\\ud835\\udc34\\ud835\\udc3f\\u27e9are replaced with\\nthe text description of the two parents selected for recombination.\\nNext, the actual request (highlighted in bold) and two sentences\\nspecifying a series of constraints to avoid answers that are blunt\\nrephrasing of the parents. Note that, the recombination prompt\\n1https:\\/\\/globalgamejam.orggenerates one single offspring and to generate a second offspring\\n(as in traditional crossover) another prompt using two different\\nparents should be used. We tried several prompts to generate two\\noffspring at once but we noted that such approach would lead with\\nvery similar offspring so we opted for a recombination prompt that\\nwould generate only one offspring.\\n4.3 Variation (Mutation)\\nThe prompt for variation also starts with the brief used to generate\\nrandom individuals (marker \\u27e8\\ud835\\udc35\\ud835\\udc45\\ud835\\udc3c\\ud835\\udc38\\ud835\\udc39\\u27e9in Table 1f) followed by (i) would generate only one offspring.\\n4.3 Variation (Mutation)\\nThe prompt for variation also starts with the brief used to generate\\nrandom individuals (marker \\u27e8\\ud835\\udc35\\ud835\\udc45\\ud835\\udc3c\\ud835\\udc38\\ud835\\udc39\\u27e9in Table 1f) followed by (i)\\nan existing game idea to be mutated (marker \\u27e8\\ud835\\udc3c\\ud835\\udc41\\ud835\\udc37\\ud835\\udc3c\\ud835\\udc49\\ud835\\udc3c\\ud835\\udc37\\ud835\\udc48\\ud835\\udc34\\ud835\\udc3f\\u27e9), (ii)\\nthe request to the LLM (highlighted in bold) specifying the focus of\\nthe mutation (marker \\u27e8\\ud835\\udc40\\ud835\\udc48\\ud835\\udc47\\ud835\\udc34\\ud835\\udc47\\ud835\\udc3c\\ud835\\udc42\\ud835\\udc41\\u27e9), and (iv) two sentences spec-\\nifying a series of constraints to avoid answers that would be simple\\nrephrasing of the parent. Marker \\u27e8\\ud835\\udc40\\ud835\\udc48\\ud835\\udc47\\ud835\\udc34\\ud835\\udc47\\ud835\\udc3c\\ud835\\udc42\\ud835\\udc41\\u27e9specifies the focus\\nof the mutation and it is randomly replaced with a gameplay aspects\\nsuch as, the goal of the game ,the game\\u2019s resources ,the level design ,\\nthe player\\u2019s input , etc. We experimented with mutation operators\\nthat would work at a lower level of detail (e.g., by substituting some\\nkeywords of the game idea with synonyms or antonyms gener-\\nated using word embeddings); however, the approach proved less\\neffective as it did not introduce enough variation.\\n5 EXPERIMENTAL EVALUATION\\nWe performed a preliminary evaluation of our framework with\\nhuman subjects (junior and senior game designers) on three de-\\nsign tasks. One focused on designing a video game with the least\\namount of resources possible while having interesting mechanics\\u2014\\na minimalistic video game [ 29,35,36]. One concerned the creation\\nof a generic board game using only one board of any layout and\\na set of pieces. Finally, we evaluated our framework with people\\nparticipating in the 2023 Global Game Jam as a tool for the initial\\nbrainstorming phase of the jam. All the experiments were run using\\nthe Telegram-based version of our framework (Figure 1).\\n5.1 Board Game and Minimalistic Video Game\\nDesign\\nThese two experiments were aimed at simulating the typical idea-\\ngeneration process in a controlled environment (e.g., a design stu-\\ndio) that usually is well-structured [ 15], has clear goals, and has a\\nfixed duration. We set up two Telegram groups, one for each task.\\nWe sent a call for volunteers to game design Facebook groups and\\nmailing lists with the link to join each Telegram group; around\\n44 people participated in the two experiments; most people sub-\\nscribed to both groups. Participants could join using their identity\\nor a nickname. Several volunteers are employed in the game in-\\ndustry and told us they would not be able to provide continuous\\nfeedback over the day. Accordingly, we decided to schedule the\\npublication of design ideas during three periods, morning, noon,\\nand late afternoon. The experiments began with a brief that we\\nposted on each Telegram group with information about the timing\\nof the postings so that people could plan their participation. The\\nexperiments lasted four days, Tuesday to Friday; we allowed a grace\\nperiod of few hours, until Saturday morning, to let participants sub-\\nmit their final evaluations. We wanted to focus the design process\\nand limit the number of ideas that the participants had to evaluate ChatGPT and Other Large Language Models as Evolutionary Engines for Online Interactive Collaborative Game Design\\nAct as a board game designer and create a unique board game with the following restrictions: the game can involve only a board (of any layot) and a set\\nof pieces (of any number, and type). Organize your response as follows: 1) Name of the Game; 2) Number of players and cooperative\\/competitive game; 3) Game\\nBoard: describe the layout of the game board, the number and shape of its tiles (max 200 characters); 4) Game Pieces: describe the number of pieces, their type, and\\ntheir distribution among players (max 300 characters); 5) Rules: provide a detailed explanation of how the game is played, including how to move pieces and take\\nturns (max 800 characters); 6) Objective: provide a clear description of the win and lose conditions of the game (max 200 characters). Keep the game description as\\nsimple as possible, without including visual details, theme and story.\\n(a)\\nA white pixel is the minimum amount of information we can show on-screen, and pressing a key (or a button) is the least interaction we can ask players. Act as\\na game designer and design a minimalist game. Organize your response as follows: 1) the name of the game; 2) game concept: describe the idea, the core\\nmechanics, the goal of the game, and player controls (max 1000 chars); 3) game resources: describe the graphical elements involved by the game, which type of\\nplayer input is required, and any additional resources required by the game (e.g., sound); 4) level design: describe how to do the level design for this game (max 300\\nchars); 5) game instructions: provide the game instructions for the player (max 300 chars). Do not add additional information.\\n(b)\\nAct as a game designer and describe (max 1000 chars), a concept for a video game about the theme \\\"roots:\\u27e8\\ud835\\udc3c\\ud835\\udc41\\ud835\\udc47\\ud835\\udc38\\ud835\\udc45\\ud835\\udc43\\ud835\\udc45\\ud835\\udc38\\ud835\\udc47\\ud835\\udc34\\ud835\\udc47\\ud835\\udc3c\\ud835\\udc42\\ud835\\udc41 \\u27e9\\\". Both the game\\nmechanics and the interaction of the players with the game should involve elements directly connected to the theme of the game.\\nOrganize your response as follows: 1) name of the game; 2) game concept: describe the idea, the core mechanics, the goal of the game, and player controls (max\\n800 chars); 3) level design: describe how to do the level design for this game (max 400 chars); 4) game instructions: provide the game instructions for the player\\n(max 300 chars). Do not add additional information.\\n(c)\\n\\u2022\\\"the part of a plant that is below the ground and that absorbs water and minerals from the soil\\\"\\n\\u2022\\\"the part of a tooth within the socket\\\"\\n\\u2022\\\"one or more progenitors of a group of descendants\\\"\\n\\u2022\\\"the essential core\\\"\\n\\u2022...\\n(d)\\n\\u27e8\\ud835\\udc35\\ud835\\udc45\\ud835\\udc3c\\ud835\\udc38\\ud835\\udc39\\u27e9. Given these two examples of minimalist game.\\nEXAMPLE1:\\u27e8\\ud835\\udc3c\\ud835\\udc41\\ud835\\udc37\\ud835\\udc3c\\ud835\\udc49\\ud835\\udc3c\\ud835\\udc37\\ud835\\udc48\\ud835\\udc34\\ud835\\udc3f\\u27e9\\nEXAMPLE2:\\u27e8\\ud835\\udc3c\\ud835\\udc41\\ud835\\udc37\\ud835\\udc3c\\ud835\\udc49\\ud835\\udc3c\\ud835\\udc37\\ud835\\udc48\\ud835\\udc34\\ud835\\udc3f\\u27e9\\nAct as a game designer and recombine these two games to create a novel game. In your response do not include any introduction or final comment. In the\\nresponse, please avoid references to the two games recombined.\\n(e)\\n\\u27e8\\ud835\\udc35\\ud835\\udc45\\ud835\\udc3c\\ud835\\udc38\\ud835\\udc39\\u27e9. Given the following example of minimalist game.\\nEXAMPLE:\\u27e8\\ud835\\udc3c\\ud835\\udc41\\ud835\\udc37\\ud835\\udc3c\\ud835\\udc49\\ud835\\udc3c\\ud835\\udc37\\ud835\\udc48\\ud835\\udc34\\ud835\\udc3f\\u27e9\\nAct as a game designer and design a novel game as a variation of the given example, by changing only \\u27e8\\ud835\\udc40\\ud835\\udc48\\ud835\\udc47\\ud835\\udc34\\ud835\\udc47\\ud835\\udc3c\\ud835\\udc42\\ud835\\udc41\\u27e9. In your response do not\\ninclude any introduction or final comment. In the response, please avoid explicit reference to the original game.\\n(f)\\nTable 1: Example of prompts for generating (a) a generic board game; (b) a minimalistic video game; (c) a game based on the\\ntheme roots in which the marker \\u27e8\\ud835\\udc3c\\ud835\\udc41\\ud835\\udc47\\ud835\\udc38\\ud835\\udc45\\ud835\\udc43\\ud835\\udc45\\ud835\\udc38\\ud835\\udc47\\ud835\\udc34\\ud835\\udc47\\ud835\\udc3c\\ud835\\udc42\\ud835\\udc41 \\u27e9can be replaced with sentences suggesting different theme interpre-\\ntation, like the ones listed in (d). Prompts to implement (e) recombination (crossover) and (f) variation (mutation); \\u27e8\\ud835\\udc35\\ud835\\udc45\\ud835\\udc3c\\ud835\\udc38\\ud835\\udc39\\u27e9\\nidentify the brief used to generate the random individuals, \\u27e8\\ud835\\udc3c\\ud835\\udc41\\ud835\\udc37\\ud835\\udc3c\\ud835\\udc49\\ud835\\udc3c\\ud835\\udc37\\ud835\\udc48\\ud835\\udc34\\ud835\\udc3f\\u27e9an individual selected from the population, and\\n\\u27e8\\ud835\\udc40\\ud835\\udc48\\ud835\\udc47\\ud835\\udc34\\ud835\\udc47\\ud835\\udc3c\\ud835\\udc42\\ud835\\udc41\\u27e9a high-level characteristic of gameplay.\\ninitially (Section 3). Accordingly, we chose a small population size\\n(10 individuals); selection, recombination, and variation were im- \\u27e8\\ud835\\udc40\\ud835\\udc48\\ud835\\udc47\\ud835\\udc34\\ud835\\udc47\\ud835\\udc3c\\ud835\\udc42\\ud835\\udc41\\u27e9a high-level characteristic of gameplay.\\ninitially (Section 3). Accordingly, we chose a small population size\\n(10 individuals); selection, recombination, and variation were im-\\nplemented using ChatGPT [38]; they were triggered as soon as 25\\nnew evaluations were submitted; selection was implemented usinga tournament selection of size 2; recombination probability was set\\nto 0.7; variation was always applied.\\nOver the four days, the evolutionary algorithm performed 30\\niterations and a total of 40 game ideas were evaluated for each task; Pier Luca Lanzi and Daniele Loiacono\\nFigure 2: Number of evaluations submitted during the exper-\\niments.\\nFigure 3: Average length of the game concepts in the pop-\\nulation computed as the moving average over the last five\\nactivation of the evolutionary algorithm.\\nboard game ideas received a total of 799 evaluations; video game\\nideas received 1025 evaluations overall. Figure 2 shows the number\\nof evaluations submitted over the four and half days. The plots\\nfollow the schedule with peaks around the publication slots and no\\nevaluations during the night time. Figure 3 shows the average length\\nof the concepts in the population computed as the moving average\\nover the last 5 activation of the evolutionary algorithm. As can be\\nnoted, board game concepts tend to be longer on average (Figure 3)\\nand therefore require more effort to evaluate; as a consequence,\\nthe board game concepts received less evaluations than the ones of\\nminimalistic video games as Figure 2 shows.\\nFigure 4 compares the word cloud generated using all the board\\ngame ideas in the original population (4a) with the one generated\\nfrom the final population (4b). The clouds share some general terms\\n(e.g., moves, starts), but the game concepts in the final population\\n(a)\\n(b)\\nFigure 4: Word cloud for (a) the initial population and (b) the\\nfinal population of the board game design task.\\nfocus on completely different concepts like the terms related to\\necosystem maintenance (e.g., island, conservation, resources). Fig-\\nure 5 and Figure 6 show the word clouds generated during the\\nprocess for the board game and the video game respectively. At the\\nend, we asked participants to complete a form asking them which\\ngame idea they liked most and why they liked it. Many games in\\nthe final population shared similar elements and mechanics; thus,\\nwhile participant did not preferred a single game over the 40 ones\\nevaluated, there was a common preference over one specific me-\\nchanic for each task. In particular, in the board game design task,\\nmost participants liked a cooperative mechanic that asked players\\nto maintain the equilibrium of an ecosystem by placing tiles on a\\nboard. Interestingly, no component of this mechanic was present in\\nthe initial population but the mechanic emerged later in the experi-\\nment in games with different narratives (e.g., one set in an island\\nenvironment, one in an urban environment). In the video game task,\\nmost participants liked a game mechanic involving reflection in its\\ndifferent interpretations (e.g., reflection of light in some games,\\nreflection of bouncing bullets in other ones). In this case, the initial\\npopulation contained one game description using reflection of bul-\\nlets however the light reflection mechanics appeared much later\\nin the population. Table 2 shows examples of a board game and a\\nvideo game from the final population.\\n5.2 2023 Global Game Jam\\nWe performed a final experiment during the 2023 Global Game\\nJam that ran February 3-5, 2023, from Friday afternoon to Sunday ChatGPT and Other Large Language Models as Evolutionary Engines for Online Interactive Collaborative Game Design\\nevening. The event always starts at 5 pm on Friday with the publi-\\ncation of the theme (this year was roots ) and ends Sunday evening\\nwith the presentation of the games. The jam is a global event run\\nwith a local timing in that it starts at 5 pm and ends Sunday around\\n6 pm in every time zone ; jam sites in the Asia continent are the\\nfirst to start, those in the Hawaiian time zone are the last ones. We\\ncontacted organizers of the Global Game Jam sites in our country,\\nsent them a link to a dedicated Telegram group, and asked them to\\nadvertise the tool\\u2019s availability to the participants of their jam sites;\\naround 35 people joined the group, most of them anonymously. In\\nthe previous experiments, at the beginning, participants received\\na brief with a list of constraints and were instructed to cooperate\\ntoward a common objective. In contrast, in this case, participants\\nhad a theme ( roots ) that could lead to several different interpreta-\\ntions; they did not have constraints and did not have to collaborate\\nbut they brainstormed for ideas to develop their own game. In ad-\\ndition, the time window was more limited since the game must be\\nimplemented by Sunday afternoon, so teams need to come up with\\na game idea by Saturday around noon or early afternoon at the\\nlatest. When the theme was revealed on February 3 at 5 pm, we\\npublished the first ten game ideas randomly generated using the\\nprocedure discussed in Section 4; the framework remained online\\nuntil Saturday night. The evolutionary operators were implemented\\nusing DaVinci GPT-3 [ 4]; the selection was triggered every 5 new\\nevaluations; recombination and mutation probabilities were set as\\nin the previous experiments. The evolutionary algorithm ran for\\n15 iterations with a number of evaluations much lower than what\\nwas recorded in the previous experiments. This is probably due\\nto the short span that jammers can dedicate to the brainstorming\\nprocess and the in-person nature of the event. Global Game Jam is a\\nphysical jam that encourages meeting, working, and sharing ideas\\nwith new people. Thus, it is no surprise that an online framework\\nlike ours did not fit well in this in-person scenario.\\n6 CONCLUSIONS\\nWe presented an online collaborative design framework combining\\ninteractive evolution and large language models for helping users\\nevolve design ideas represented as free-form text. Our framework\\ncan support collaboration through traditional web-based grid in-\\nterfaces [ 6,45] as well as messaging systems, like Telegram, which\\nin our opinion might fit free-text content better. We evaluated our\\nframework in three scenarios involving around 80 participants.\\nTwo scenarios mimicked the typical collaborative design process\\nwith objectives specified at the start and a fixed duration. The third\\none was conceived as an open-ended brainstorming activity that\\ntook place during the 2023 Global Game Jam. The first two used an\\nevolutionary engine based on ChatGPT; the third one employed\\nDaVinci GPT-3 for the same purpose. All the experiments were run\\nusing the Telegram-based interface of our framework. Overall, we\\nreceived positive feedback from the participants. In our experience,\\nwe did not notice any difference between the two large language\\nmodels that are, in our opinion, both viable options. We focused\\nour evaluation on game design since we had access to several ju-\\nnior and senior game designers employed in the industry. However,\\nthe framework can be applied to any design task which can be de-\\nscribed in free-text form. And if needed, it could be easily extendedwith text-to-image platforms (like Dall \\u00b7E 2[39] or Midjourney2) to\\nenrich design ideas with captivating visuals.\\nREFERENCES\\n[1]Philip Bontrager, Wending Lin, Julian Togelius, and Sebastian Risi. 2018. Deep\\nInteractive Evolution. In Computational Intelligence in Music, Sound, Art and REFERENCES\\n[1]Philip Bontrager, Wending Lin, Julian Togelius, and Sebastian Risi. 2018. Deep\\nInteractive Evolution. In Computational Intelligence in Music, Sound, Art and\\nDesign - 7th International Conference, EvoMUSART 2018, Parma, Italy, April 4-\\n6, 2018, Proceedings (Lecture Notes in Computer Science, Vol. 10783) , Antonios\\nLiapis, Juan Jes\\u00fas Romero Cardalda, and Anik\\u00f3 Ek\\u00e1rt (Eds.). Springer, 267\\u2013282.\\nhttps:\\/\\/doi.org\\/10.1007\\/978-3-319-77583-8_18\\n[2]A.M. Brintrup, J. Ramsden, H. Takagi, and A. Tiwari. 2008. Ergonomic Chair\\nDesign by Fusing Qualitative and Quantitative Criteria Using Interactive Genetic\\nAlgorithms. Evolutionary Computation, IEEE Transactions on 12, 3 (June 2008),\\n343 \\u2013354. https:\\/\\/doi.org\\/10.1109\\/TEVC.2007.904343\\n[3]Andrew R. Brown, Matthew Horrigan, Arne Eigenfeldt, Toby Gifford, Daniel\\nField, and Jon McCormack. 2018. Interacting with Musebots. In 18th International\\nConference on New Interfaces for Musical Expression, NIME 2018, Blacksburg, VA,\\nUSA, June 3-6, 2018 . nime.org, 19\\u201324. http:\\/\\/www.nime.org\\/proceedings\\/2018\\/\\nnime2018_paper0004.pdf\\n[4]Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan,\\nPrafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda\\nAskell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan,\\nRewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter,\\nChristopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin\\nChess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya\\nSutskever, and Dario Amodei. 2020. Language Models are Few-Shot Learners.\\nhttps:\\/\\/doi.org\\/10.48550\\/ARXIV.2005.14165\\n[5]Luigi Cardamone, Pier Luca Lanzi, and Daniele Loiacono. 2015. TrackGen: An\\ninteractive track generator for TORCS and Speed-Dreams. Appl. Soft Comput. 28\\n(2015), 550\\u2013558. https:\\/\\/doi.org\\/10.1016\\/j.asoc.2014.11.010\\n[6]Luigi Cardamone, Daniele Loiacono, and Pier Luca Lanzi. 2011. Interactive\\nevolution for the procedural generation of tracks in a high-end racing game.\\nIn13th Annual Genetic and Evolutionary Computation Conference, GECCO 2011,\\nProceedings, Dublin, Ireland, July 12-16, 2011 , Natalio Krasnogor and Pier Luca\\nLanzi (Eds.). ACM, 395\\u2013402. https:\\/\\/doi.org\\/10.1145\\/2001576.2001631\\n[7]Mark Chen, Alec Radford, Rewon Child, Jeffrey Wu, Heewoo Jun, David Luan,\\nand Ilya Sutskever. 2020. Generative Pretraining From Pixels. In Proceedings of\\nthe 37th International Conference on Machine Learning, ICML 2020, 13-18 July\\n2020, Virtual Event (Proceedings of Machine Learning Research, Vol. 119) . PMLR,\\n1691\\u20131703. http:\\/\\/proceedings.mlr.press\\/v119\\/chen20s.html\\n[8]Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira\\nPinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman,\\nAlex Ray, Raul Puri, Gretchen Krueger, Michael Petrov, Heidy Khlaaf, Girish\\nSastry, Pamela Mishkin, Brooke Chan, Scott Gray, Nick Ryder, Mikhail Pavlov,\\nAlethea Power, Lukasz Kaiser, Mohammad Bavarian, Clemens Winter, Philippe\\nTillet, Felipe Petroski Such, Dave Cummings, Matthias Plappert, Fotios Chantzis,\\nElizabeth Barnes, Ariel Herbert-Voss, William Hebgen Guss, Alex Nichol, Alex\\nPaino, Nikolas Tezak, Jie Tang, Igor Babuschkin, Suchir Balaji, Shantanu Jain,\\nWilliam Saunders, Christopher Hesse, Andrew N. Carr, Jan Leike, Josh Achiam,\\nVedant Misra, Evan Morikawa, Alec Radford, Matthew Knight, Miles Brundage,\\nMira Murati, Katie Mayer, Peter Welinder, Bob McGrew, Dario Amodei, Sam\\nMcCandlish, Ilya Sutskever, and Wojciech Zaremba. 2021. Evaluating Large\\nLanguage Models Trained on Code. https:\\/\\/doi.org\\/10.48550\\/ARXIV.2107.03374\\n[9]Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun,\\nLukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano,\\nChristopher Hesse, and John Schulman. 2021. Training Verifiers to Solve Math\\nWord Problems. https:\\/\\/doi.org\\/10.48550\\/ARXIV.2110.14168\\n[10] Richard Dawkins. 1986. The Blind Watchmaker: Why the Evidence of Evolution Word Problems. https:\\/\\/doi.org\\/10.48550\\/ARXIV.2110.14168\\n[10] Richard Dawkins. 1986. The Blind Watchmaker: Why the Evidence of Evolution\\nReveals a Universe without Design . Norton & Company, Inc.\\n[11] Marc Ebner, Markus Reinhardt, and J\\u00fcrgen Albert. 2005. Evolution of Vertex and\\nPixel Shaders. In Genetic Programming, 8th European Conference, EuroGP2005,\\nLausanne, Switzerland, March 30 - April 1, 2005, Proceedings (Lecture Notes in\\nComputer Science, Vol. 3447) , Maarten Keijzer, Andrea Tettamanzi, Pierre Collet,\\nJano I. van Hemert, and Marco Tomassini (Eds.). Springer, 261\\u2013270. https:\\n\\/\\/doi.org\\/10.1007\\/978-3-540-31989-4_23\\n[12] Kevin Frans. 2021. AI Charades: Language Models as Interactive Game Environ-\\nments. In 2021 IEEE Conference on Games (CoG) . 1\\u20132. https:\\/\\/doi.org\\/10.1109\\/\\nCoG52621.2021.9619126\\n[13] Edoardo Giacomello, Pier Luca Lanzi, and Daniele Loiacono. 2019. Searching the\\nLatent Space of a Generative Adversarial Network to Generate DOOM Levels.\\nInIEEE Conference on Games, CoG 2019, London, United Kingdom, August 20-23,\\n2019. IEEE, 1\\u20138. https:\\/\\/doi.org\\/10.1109\\/CIG.2019.8848011\\n[14] Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-\\nFarley, Sherjil Ozair, Aaron C. Courville, and Yoshua Bengio. 2020. Generative\\n2https:\\/\\/www.midjourney.com Pier Luca Lanzi and Daniele Loiacono\\nadversarial networks. Commun. ACM 63, 11 (2020), 139\\u2013144. https:\\/\\/doi.org\\/10.\\n1145\\/3422622\\n[15] Dave Gray and Sunni Brown. 2010. Gamestorming: A Playbook for Innovators,\\nRulebreakers, and Changemakers . O\\u2019Reilly.\\n[16] Tomohiro Harada and Enrique Alba. 2020. Parallel Genetic Algorithms: A Useful\\nSurvey. ACM Comput. Surv. 53, 4, Article 86 (aug 2020), 39 pages. https:\\/\\/doi.\\norg\\/10.1145\\/3400031\\n[17] Erin J. Hastings, Ratan K. Guha, , and Kenneth O. Stanley. 2009. Automatic\\nContent Generation in the Galactic Arms Race Video Game. IEEE Transactions\\non Computational Intelligence and AI in Games 4, 1 (2009), 245\\u2013263.\\n[18] Amy K. Hoover and Kenneth O. Stanley. 2009. Exploiting functional relationships\\nin musical composition. Connect. Sci. 21, 2&3 (2009), 227\\u2013251. https:\\/\\/doi.org\\/10.\\n1080\\/09540090902733871\\n[19] Amy K. Hoover, Paul A. Szerlip, and Kenneth O. Stanley. 2011. Interactively\\nevolving harmonies through functional scaffolding. In 13th Annual Genetic and\\nEvolutionary Computation Conference, GECCO 2011, Proceedings, Dublin, Ireland,\\nJuly 12-16, 2011 , Natalio Krasnogor and Pier Luca Lanzi (Eds.). ACM, 387\\u2013394.\\nhttps:\\/\\/doi.org\\/10.1145\\/2001576.2001630\\n[20] Gregory S. Hornby and Josh C. Bongard. 2012. Accelerating human-computer\\ncollaborative search through learning comparative and predictive user models.\\nInGenetic and Evolutionary Computation Conference, GECCO \\u201912, Philadelphia,\\nPA, USA, July 7-11, 2012 , Terence Soule and Jason H. Moore (Eds.). ACM, 225\\u2013232.\\nhttps:\\/\\/doi.org\\/10.1145\\/2330163.2330196\\n[21] Andrew Howlett, Simon Colton, and Cameron Browne. 2010. Evolving pixel\\nshaders for the prototype video game Subversion. In Proceedings of the 3rd Inter-\\nnational Symposium on AI and Games - A Symposium at the AISB 2010 Convention\\n(Proceedings of the 3rd International Symposium on AI and Games - A Symposium\\nat the AISB 2010 Convention) . 41\\u201346. AISB Symposium on AI and Games 2010 ;\\nConference date: 29-03-2010 Through 01-04-2010.\\n[22] Alexander Ivanov, Wesley Willett, and Christian Jacob. 2022. EvoIsland: Inter-\\nactive Evolution via an Island-Inspired Spatial User Interface Framework. In\\nProceedings of the Genetic and Evolutionary Computation Conference (Boston,\\nMassachusetts) (GECCO \\u201922) . Association for Computing Machinery, New York,\\nNY, USA, 1200\\u20131208. https:\\/\\/doi.org\\/10.1145\\/3512290.3528722\\n[23] Shahroz Khan, Erkan Gunpinar, and Bekir Sener. 2019. GenYacht: An inter-\\nactive generative design system for computer-aided yacht hull design. Ocean\\nEngineering 191 (2019), 106462. https:\\/\\/doi.org\\/10.1016\\/j.oceaneng.2019.106462\\n[24] Hee-Su Kim and Sung-Bae Cho. 2000. Application of interactive genetic algorithm\\nto fashion design. Engineering Applications of Artificial Intelligence 13, 6 (2000),\\n635 \\u2013 644. https:\\/\\/doi.org\\/DOI:10.1016\\/S0952-1976(00)00045-2\\n[25] Hee-Su Kim and Sung-Bae Cho. 2005. Fashion Design Using Interactive Genetic\\nAlgorithm with Knowledge-based Encoding . Springer Berlin Heidelberg, Berlin,\\nHeidelberg, 411\\u2013434. https:\\/\\/doi.org\\/10.1007\\/978-3-540-44511-1_19\\n[26] Taras Kowaliw, Alan Dorin, and Jon McCormack. 2012. Promoting Creative\\nDesign in Interactive Evolutionary Computation. IEEE Trans. Evol. Comput. 16, 4\\n(2012), 523\\u2013536. https:\\/\\/doi.org\\/10.1109\\/TEVC.2011.2166764\\n[27] Taras Kowaliw, Jon McCormack, and Alan Dorin. 2011. An interactive electronic\\nart system based on artificial ecosystemics. In 2011 IEEE Symposium on Artificial\\nLife, ALIFE 2011, Paris, France, April 13-15, 2011 . IEEE, 162\\u2013169. https:\\/\\/doi.org\\/\\n10.1109\\/ALIFE.2011.5954645\\n[28] John R. Koza. 1992. Genetic Programming: On the Programming of Computers by\\nMeans of Natural Selection . The MIT Press.\\n[29] Pier Luca Lanzi, Daniele Loiacono, Alberto Arosio, Dorian Bucur, Davide Caio,\\nLuca Capecchi, Maria Giulietta Cappelletti, Lorenzo Carnaghi, Marco Giuseppe\\nCaruso, Valerio Ceraudo, Luca Contato, Luca Cornaggia, Christian Costanza,\\nTommaso Grilli, Sumero Lira, Luca Marchetti, Giulia Olivares, Barbara Pagano, Caruso, Valerio Ceraudo, Luca Contato, Luca Cornaggia, Christian Costanza,\\nTommaso Grilli, Sumero Lira, Luca Marchetti, Giulia Olivares, Barbara Pagano,\\nDavide Pons, Michele Pirovano, and Valentina Tosto. 2022. One Pixel, One\\nInteraction, One Game: An Experiment in Minimalist Game Design. https:\\n\\/\\/doi.org\\/10.48550\\/ARXIV.2207.03827\\n[30] Antonios Liapis, H\\u00e9ctor P. Mart\\u00ednez, Julian Togelius, and Georgios N. Yannakakis.\\n2013. Adaptive game level creation through rank-based interactive evolution. In\\n2013 IEEE Conference on Computational Inteligence in Games (CIG) . 1\\u20138. https:\\n\\/\\/doi.org\\/10.1109\\/CIG.2013.6633651\\n[31] Telegram FZ LLC and Telegram Messenger Inc. 2023. Telegram. https:\\/\\/telegram.\\norg\\n[32] Piotr Mirowski, Kory W. Mathewson, Jaylen Pittman, and Richard Evans. 2022. Co-\\nWriting Screenplays and Theatre Scripts with Language Models: An Evaluation\\nby Industry Professionals. https:\\/\\/doi.org\\/10.48550\\/ARXIV.2209.14958\\n[33] Thomas Mitchell, Peter Bennett, Sebastian Madgwick, Edward Davies, and Philip\\nTew. 2016. Tangible Interfaces for Interactive Evolutionary Computation. In\\nProceedings of the 2016 CHI Conference Extended Abstracts on Human Factors in\\nComputing Systems (San Jose, California, USA) (CHI EA \\u201916) . Association for\\nComputing Machinery, New York, NY, USA, 2609\\u20132616. https:\\/\\/doi.org\\/10.1145\\/\\n2851581.2892405\\n[34] P.Y. Mok, Jie Xu, X.X. Wang, J.T. Fan, Y.L. Kwok, and John H. Xin. 2013. An IGA-\\nbased design support system for realistic and practical fashion designs. Computer-\\nAided Design 45, 11 (2013), 1442\\u20131458. https:\\/\\/doi.org\\/10.1016\\/j.cad.2013.06.014[35] David Myers. 2009. In search of a minimalist game. In Proceedings of the 2009\\nDiGRA International Conference: Breaking New Ground: Innovation in Games,\\nPlay, Practice and Theory, DiGRA 2009, London, UK, September 1-4, 2009 , Tanya\\nKrzywinska, Helen W. Kennedy, and Barry Atkins (Eds.). Digital Games Research\\nAssociation. http:\\/\\/www.digra.org\\/digital-library\\/publications\\/in-search-of-a-\\nminimalist-game\\/\\n[36] Andrew Nealen, Adam Saltsman, and Eddy Boxerman. 2011. Towards minimalist\\ngame design. In Foundations of Digital Games, FDG\\u201911, Bordeaux, France, June 28\\n- July 1, 2011 , Marc Cavazza, Katherine Isbister, and Charles Rich (Eds.). ACM,\\n38\\u201345. https:\\/\\/doi.org\\/10.1145\\/2159365.2159371\\n[37] Don Norman. 2013. The Design Of Everyday Things . Basic Books. Revised edition.\\n[38] OpenAI. 2021. ChatGPT: a Generative Pre-training Transformer Model for\\nConversational AI. https:\\/\\/openai.com\\/models\\/gpt-3-citation\\/.\\n[39] OpenAI. 2022. DALL \\u00b7E 2. https:\\/\\/openai.com\\/dall-e-2\\/. Retrieved 2 February\\n2023.\\n[40] Michele Pirovano, Renato Mainetti, and Daniele Loiacono. 2015. Volcano: An\\ninteractive sword generator. In 2015 IEEE Games Entertainment Media Confer-\\nence, GEM 2015, Toronto, ON, Canada, October 14-16, 2015 , Elena G. Bertozzi,\\nBill Kapralos, Nahum D. Gershon, and Jim R. Parker (Eds.). IEEE, 1\\u20138. https:\\n\\/\\/doi.org\\/10.1109\\/GEM.2015.7377226\\n[41] Stanislas Polu and Ilya Sutskever. 2020. Generative Language Modeling for\\nAutomated Theorem Proving. https:\\/\\/doi.org\\/10.48550\\/ARXIV.2009.03393\\n[42] Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya\\nSutskever. 2019. Language Models are Unsupervised Multitask Learners. (2019).\\nhttps:\\/\\/openai.com\\/blog\\/better-language-models\\/\\n[43] Sebastian Risi, Joel Lehman, David B. D\\u2019Ambrosio, Ryan Hall, and Kenneth O.\\nStanley. 2012. Combining Search-Based Procedural Content Generation and So-\\ncial Gaming in the Petalz Video Game. In AIIDE , Mark Riedl and Gita Sukthankar\\n(Eds.). The AAAI Press.\\n[44] Jacob Schrum, Jake Gutierrez, Vanessa Volz, Jialin Liu, Simon Lucas, and Sebastian\\nRisi. 2020. Interactive Evolution and Exploration within Latent Level-Design\\nSpace of Generative Adversarial Networks. In Proceedings of the 2020 Genetic and\\nEvolutionary Computation Conference (Canc\\u00fan, Mexico) (GECCO \\u201920) . Association\\nfor Computing Machinery, New York, NY, USA, 148\\u2013156. https:\\/\\/doi.org\\/10.\\n1145\\/3377930.3389821 Evolutionary Computation Conference (Canc\\u00fan, Mexico) (GECCO \\u201920) . Association\\nfor Computing Machinery, New York, NY, USA, 148\\u2013156. https:\\/\\/doi.org\\/10.\\n1145\\/3377930.3389821\\n[45] Jimmy Secretan and Nicholas Beato. 2008. Picbreeder: evolving pictures collabo-\\nratively online. In CHI. 1759\\u20131768.\\n[46] Nazanin Alsadat Tabatabaei Anaraki. 2017. Fashion Design Aid System with\\nApplication of Interactive Genetic Algorithms. In Computational Intelligence in\\nMusic, Sound, Art and Design , Jo\\u00e3o Correia, Vic Ciesielski, and Antonios Liapis\\n(Eds.). Springer International Publishing, Cham, 289\\u2013303.\\n[47] H. Takagi. 2001. Interactive evolutionary computation: fusion of the capabilities\\nof EC optimization and human evaluation. Proc. IEEE 89, 9 (2001), 1275\\u20131296.\\nhttps:\\/\\/doi.org\\/10.1109\\/5.949485\\n[48] Judith van Stegeren and Jakub Myundefinedliwiec. 2021. Fine-Tuning GPT-2\\non Annotated RPG Quests for NPC Dialogue Generation. In Proceedings of the\\n16th International Conference on the Foundations of Digital Games (Montreal, QC,\\nCanada) (FDG \\u201921) . Association for Computing Machinery, New York, NY, USA,\\nArticle 2, 8 pages. https:\\/\\/doi.org\\/10.1145\\/3472538.3472595\\n[49] Vanessa Volz, Jacob Schrum, Jialin Liu, Simon M. Lucas, Adam M. Smith, and\\nSebastian Risi. 2018. Evolving mario levels in the latent space of a deep con-\\nvolutional generative adversarial network. In Proceedings of the Genetic and\\nEvolutionary Computation Conference, GECCO 2018, Kyoto, Japan, July 15-19,\\n2018, Hern\\u00e1n E. Aguirre and Keiki Takadama (Eds.). ACM, 221\\u2013228. https:\\n\\/\\/doi.org\\/10.1145\\/3205455.3205517\\n[50] Susanna V\\u00e4rtinen, Perttu H\\u00e4m\\u00e4l\\u00e4inen, and Christian Guckelsberger. 2022. Gener-\\nating Role-Playing Game Quests With GPT Language Models. IEEE Transactions\\non Games (2022), 1\\u201312. https:\\/\\/doi.org\\/10.1109\\/TG.2022.3228480\\n[51] Tianxiong Wang and Meiyu Zhou. 2020. A method for product form design of\\nintegrating interactive genetic algorithm with the interval hesitation time and\\nuser satisfaction. International Journal of Industrial Ergonomics 76 (2020), 102901.\\nhttps:\\/\\/doi.org\\/10.1016\\/j.ergon.2019.102901\\n[52] Wikipedia. 2022. Design Brief (Wikipedia, The Free Encyclopedia). https:\\n\\/\\/en.wikipedia.org\\/wiki\\/Design_brief\\n[53] Brian G. Woolley and Kenneth O. Stanley. 2014. A Novel Human-Computer\\nCollaboration: Combining Novelty Search with Interactive Evolution. In Pro-\\nceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation\\n(Vancouver, BC, Canada) (GECCO \\u201914) . Association for Computing Machinery,\\nNew York, NY, USA, 233\\u2013240. https:\\/\\/doi.org\\/10.1145\\/2576768.2598353\\n[54] Bin Xu, Shangfei Wang, and Xian Li. 2010. An emotional harmony generation\\nsystem. In Evolutionary Computation (CEC), 2010 IEEE Congress on . 1 \\u20137. https:\\n\\/\\/doi.org\\/10.1109\\/CEC.2010.5586210\\n[55] Georgios N. Yannakakis and Julian Togelius. 2011. Experience-Driven Procedural\\nContent Generation. IEEE Transactions on Affective Computing 2, 3 (2011), 147\\u2013\\n161. https:\\/\\/doi.org\\/10.1109\\/T-AFFC.2011.6\\n[56] Qihao Zhu and Jianxi Luo. 2022. Generative Design Ideation: A Natural Language\\nGeneration Approach. https:\\/\\/doi.org\\/10.48550\\/ARXIV.2204.09658 ChatGPT and Other Large Language Models as Evolutionary Engines for Online Interactive Collaborative Game Design\\n[57] Peter Thorup \\u00d8lsted, Benjamin Ma, and Sebastian Risi. 2015. Interactive evolution\\nof levels for a competitive multiplayer FPS. In 2015 IEEE Congress on Evolutionary\\nComputation (CEC) . 1527\\u20131534. https:\\/\\/doi.org\\/10.1109\\/CEC.2015.7257069 Pier Luca Lanzi and Daniele Loiacono\\nName of the game : Eco-Island\\nNumber of players : 2-4. Cooperative\\/competitive game: Cooperative.\\nGame Board : a square board with 64 tiles in an 8x8 grid, representing a tropical island. Each tile has a different terrain, such as a beach, a\\nrainforest, a volcano, a coral reef and a mangrove. The board also includes a starting point and a research station tile.\\nGame Pieces : Each player starts with a pawn-shaped game piece, representing their scientist and a set of worker pieces, which are small circles.\\nThe players also have a set of cards representing different research equipment, such as a microscope, a GPS, a sample container and a camera,\\nand different conservation measures, such as marine protected areas, reforestation and community education.\\nRules :\\nPlayers take turns moving their game pieces on the board, starting from the starting point and trying to reach the research station tile. Each tile\\nhas its own set of environmental issues, such as habitat destruction, pollution, and overfishing. Landing on a tile can either help or harm the\\nplayer\\u2019s research. Players can use their cards to study and mitigate the issues and increase their chances of reaching the research station. Players\\ncan also place their conservation measures on any tile, but they have to abide by certain rules such as: Marine protected areas need to be placed\\nnear coral reefs to protect them; Reforestation needs to be placed in areas with high soil erosion; Community education needs to be placed near\\nvulnerable communities; Players also have a limited number of workers, they can place on the board to help them implement conservation\\nmeasures faster. Players can also form alliances with other players to increase their chances of reaching the research station, but they can also\\nbetray their allies to gain an advantage. Players can also choose to remove their conservation measures to make room for new measures. The\\ngame ends when one player reaches the research station tile or all the tiles on the board are filled with conservation measures.\\nObjective : The game is won by the player who reaches the research station tile first and has implemented conservation measures that meet\\ncertain criteria for biodiversity and sustainability. The players lose if the island becomes unsustainable or all players run out of cards before\\nreaching the research station tile.\\n(a)\\nName of the game : Reflectron\\nGame Concept : The game is a minimalist puzzle platformer where the player controls a character that can reflect both light beams and\\nprojectiles to interact with the environment and solve puzzles. The goal of the game is to reach the end of each level by reflecting both light\\nbeams and projectiles to solve puzzles and overcome obstacles. The core mechanics involve using the arrow keys to move the character, the left\\nmouse button to reflect incoming projectiles, and the right mouse button to manipulate the angle of reflection of incoming light beams.\\nGame resources : The game only requires a simple background, a character that can reflect incoming projectiles and light beams, objects that\\ncan be interacted with by reflecting both projectiles and light beams, and obstacles that can be overcome by reflecting both projectiles and light\\nbeams. Player input is required through the use of the arrow keys, left mouse button, and right mouse button.\\nLevel design : The levels are designed by creating obstacles and challenges that require the player to reflect both projectiles and light beams in\\ndifferent ways to overcome them, adding diversity to the gameplay.\\nGame instructions : Use the arrow keys to move the character, left mouse button to reflect incoming projectiles, and the right mouse button to\\nmanipulate the angle of reflection of incoming light beams. Reach the end of each level by reflecting both projectiles and light beams to solve\\npuzzles and overcome obstacles.\\n(b) manipulate the angle of reflection of incoming light beams. Reach the end of each level by reflecting both projectiles and light beams to solve\\npuzzles and overcome obstacles.\\n(b)\\nTable 2: Example of evolved games for (a) the board game design task and (b) the video game design task. ChatGPT and Other Large Language Models as Evolutionary Engines for Online Interactive Collaborative Game Design\\n(a) (b)\\n(c) (d)\\n(e) (f)\\nFigure 5: Sequence of word clouds for the board game design task starting from (a) the initial population to the (f) the final\\npopulation. Pier Luca Lanzi and Daniele Loiacono\\n(a) (b)\\n(c) (d)\\n(e) (f)\\nFigure 6: Sequence of word clouds for the video game design task starting from (a) the initial population to the (f) the final\\npopulation.\",\"33\":\" PDSketch: Integrated Planning Domain\\nProgramming and Learning\\nJiayuan Mao1Tom\\u00e1s Lozano-P\\u00e9rez1Joshua B. Tenenbaum1;2;3Leslie Pack Kaelbling1\\n1MIT Computer Science & Arti\\ufb01cial Intelligence Laboratory\\n2MIT Department of Brain and Cognitive Sciences\\n3Center for Brains, Minds and Machines\\nAbstract\\nThis paper studies a model learning and online planning approach towards building\\n\\ufb02exible and general robots. Speci\\ufb01cally, we investigate how to exploit the locality\\nandsparsity structures in the underlying environmental transition model to improve\\nmodel generalization, data-ef\\ufb01ciency, and runtime-ef\\ufb01ciency. We present a new\\ndomain de\\ufb01nition language, named PDSketch. It allows users to \\ufb02exibly de\\ufb01ne\\nhigh-level structures in the transition models, such as object and feature dependen-\\ncies, in a way similar to how programmers use TensorFlow or PyTorch to specify\\nkernel sizes and hidden dimensions of a convolutional neural network. The details\\nof the transition model will be \\ufb01lled in by trainable neural networks. Based on\\nthe de\\ufb01ned structures and learned parameters, PDSketch automatically generates\\ndomain-independent planning heuristics without additional training. The derived\\nheuristics accelerate the performance-time planning for novel goals.\\n1 Introduction\\nA long-standing goal in AI is to build robotic agents that are \\ufb02exible and general, able to accomplish\\na diverse set of tasks in novel and complex environments. Such tasks generally require a robot to\\ngenerate long-horizon plans for novel goals in novel situations, by reasoning about many objects\\nand the ways in which their state-changes depend on one another. A promising solution strategy is\\nto combine model learning with online planning: the agent forms an internal representation of the\\nenvironment\\u2019s states and dynamics by learning from external or actively-collected data, and then\\napplies planning algorithms to generate actions, given a new situation and goal at performance time.\\nThere are two primary desiderata for a system based on model-learning and planning. First, the\\nlearning process should be data ef\\ufb01cient , especially because of the combinatorial complexity of\\npossible con\\ufb01guration of in the real world. Second, the learned model should be computationally\\nef\\ufb01cient , making online planning a feasible runtime execution strategy.\\nA critical strategy for learning models that generalize well from small amounts of data and that\\ncan be deployed ef\\ufb01ciently at runtime is to introduce inductive biases. In image processing, we\\nleverage translation invariance and equivariance by using convolutions. In graph learning, we leverage\\npermutation invariance by using graph neural networks. Two essential forms of structure we can\\nleverage in dynamic models of the physical world are locality andsparsity . Consider a robot picking\\nan object up off a table. At the object level, the operation only changes the states of the object being\\npicked up and the robot, and only depends on a few other nearby objects, such as the table (local). At\\nthe object feature level, only the con\\ufb01guration of the robot and pose of the object are changed, but\\ntheir colors and frictional properties are unaffected (sparse).\\nClassical hand-engineered approaches to robot task and motion planning have designed represen-\\ntations that expose and exploit locality through lifting (orobject-centrism ), which allows relational\\nCorrespondence to: jiayuanm@mit.edu. Project page: https:\\/\\/pdsketch.csail.mit.edu.\\n36th Conference on Neural Information Processing Systems (NeurIPS 2022).arXiv:2303.05501v1  [cs.AI]  9 Mar 2023 (:action move-into:parameters (?o1?o2):effects(and(pose::assign?o1(??f(pose ?o2)))......PDSketch(Structure)\\ud835\\udf03Neural Modules(Parameters)Data(Offline Demonstration or Online Interaction)ModelGoal(e.g.,paint all blocks yellow)Domain-IndependentPlanner (A*)PrimitivePolicies(pick, place, etc.)Action: move-into(Item#1, Item#5)ModelTraining\\nEnvironment(e.g.PyBullet)StateRobotCommandPolicy\\nTrainingTimePerformanceTimeFigure 1: The life cycle of a PDSketch model. A PDSketch model is composed of a model structure\\nde\\ufb01nition and a collection of trainable neural modules. The model parameters can be learned from\\ndata. During performance time, the model is used by a domain-independent planner to form a policy\\nthat directly interacts with the environment.\\ndefmove_into(o,c):o.pose=c.pose+[0,0,0.1]ifis_block(o)andis_painter(c):o.color=c.colordefmove_into(o,c):o.prop1=??(c.prop1)if??(o,c):o.prop2=??(c.prop2)(a)Afullspecificationofthetransitionmodel.(b)Astructure-onlyspecification.\\nFigure 2: De\\ufb01ning both the transition model structure and\\nimplementation in Python (a) vs. de\\ufb01ning only the structure\\nwhile leaving details (the ??functions) to be learned (b).descriptions of objects and abstrac-\\ntion over them, and through factoring ,\\nwhich represents different attributes of\\nan object in a disentangled way [Gar-\\nrett et al., 2021]. These representa-\\ntions are powerful and effective artic-\\nulations of locality and sparsity, but\\nthey are traditionally laboriously hand-\\ndesigned in a process that is very dif-\\n\\ufb01cult to get correct, similar to writing\\na full state-transition function as in Fig. 2a. This approach is not directly applicable to problems\\ninvolving perception or environmental dynamics that are unknown or dif\\ufb01cult to specify. In this\\npaper, we present PDSketch, a model-speci\\ufb01cation language that integrates human speci\\ufb01cation of\\nstructural sparsity priors and machine learning of continuous and symbolic aspects of the model. Just\\nas human users may de\\ufb01ne the structure of a convolutional neural network in TensorFlow [Abadi\\net al., 2016] or PyTorch [Paszke et al., 2019], PDSketch allows users to specify high-level structures\\nof the transition model as in Fig. 2b (analogous to setting the kernel sizes), and uses machine learning\\nto \\ufb01ll in the details (analogous to learning the convolution kernels).\\nFig. 1 depicts the life-cycle of a PDSketch model, PDSketch uses an object-centric, factored, symbolic\\nlanguage to \\ufb02exibly describe structural inductive biases in planning domains (i.e., the model structure).\\nA PDSketch model is associated with a collection of neural modules whose parameters can be learned\\nfrom robot trajectory data that are either collected of\\ufb02ine by experts or actively-collected by interacting\\nwith the environment. During performance time, a PDSketch is paired with a domain-independent\\nplanner, such as A\\u0003, and as a whole forms a goal-conditioned policy. The planner receives the\\nenvironmental state and the trained PDSketch model, makes plans in an abstract action space, and\\ninvokes primitive policies that actually generate robot joint commands.\\nCompared to unstructured models, such as a single multi-layer perceptron that models the complete\\nstate transition monolithically, the structures speci\\ufb01ed in PDSketch substantially improve model\\ngeneralization and data-ef\\ufb01ciency in training. In addition, they enable the computation of powerful\\ndomain-independent planning heuristics : these are estimates of the cost-to-go from each state to a\\nstate satisfying the goal speci\\ufb01cation, which can be obtained from the structured transition model\\nwithout any additional learning . They can be leveraged by A\\u0003to ef\\ufb01ciently plan for unseen goals,\\nspeci\\ufb01ed in a \\ufb01rst-order logic language.\\nWe experimentally verify the ef\\ufb01ciency and effectiveness of PDSketch in two domains: BabyAI, an\\n2D grid-world environment that focuses on navigation, and Painting Factory, a simulated table-top We experimentally verify the ef\\ufb01ciency and effectiveness of PDSketch in two domains: BabyAI, an\\n2D grid-world environment that focuses on navigation, and Painting Factory, a simulated table-top\\nrobotic environment that paints and moves blocks. Our results suggest that 1) locality and sparsity\\nstructures, speci\\ufb01ed economically in a few lines of code, can signi\\ufb01cantly improve the data ef\\ufb01ciency\\nof model learning; 2) the model learning and planning paradigm enables strong generalization to\\nunseen goal speci\\ufb01cations. Finally, the domain-independent heuristics automatically induced from the\\nstructures dramatically improve performance-time ef\\ufb01ciency, especially for novel goal speci\\ufb01cations.\\n2 Input ImageSegmentation(a) Raw Observations(b) Object-CentricState (Observation)\\nItem#1(423.0, 420.0)\\nItem#2(159.0, 256.5)\\nItem#3(543.0, 381.5)\\nItem#6(275.5, 273.5)pose       Item#1color      Item#1wetness    Item#1dirtiness  Item#1pose       Item#6color      Item#6wetness    Item#6dirtiness  Item#6......type       Item#1\\ntype       Item#6pose       Item#1color      Item#1wetness    Item#1dirtiness  Item#1pose       Item#6color      Item#6wetness    Item#6dirtiness  Item#6......type       Item#1\\ntype       Item#6(c) Factored State (Internal)(d) Factored State(Next Timestep)\\nItem#1(273.0, 273.0)\\nItem#2(159.0, 256.5)\\nItem#3(543.0, 381.5)\\nItem#6(275.5, 273.5)(e) Object-CentricState (Next Observation)Actionmove-into(Item#1, Item#6)\\nTransition Function \\ud835\\udcaf!SupervisionEncoder \\u2130!......(:action move-into:parameters (?o1?o2):effects(and(pose::assign?o1(??f(pose ?o2)))(when(??g(type ?o1)(type ?o2))(wetness::assign ?o1(??h(wetness ?o1)))............Figure 3: A factorized state representation and transition model for the robot painting domain.\\nThe raw observation (a) is \\ufb01rst processed by an external perception module into an object-centric\\nrepresentation (b). This representation is further transformed into a \\ufb01ne-grained factorization (c).\\nThe transition function T\\u0012can be de\\ufb01ned over this factored state representation: each action may\\nonly change a few factors of the state. Executing a speci\\ufb01c action move-into(item#1, item#6)\\nproduces the predicted factored state at the next timestep. During training time, we will be using the\\nobject-centric observation from the next timestep to supervise the learning of T\\u0012.\\n2 PDSketch\\nWe focus on the problem of learning models for a robot that operates in a space Sof world states that\\nplans to achieve goal conditions that are subsets of S. A planning problem is a tuple hS;s0;g;A;Ti,\\nwheres02S is the initial state, gis a goal speci\\ufb01cation in \\ufb01rst-order logic, Ais a set of actions\\nthat the agent can execute, and Tis a environmental transition model T:S\\u0002A!S . The task of\\nplanning is to output a sequence of actions \\u0016a=fai2Ag in which the terminal state sTinduced\\nby applying aisequentially following Tsatis\\ufb01es the goal speci\\ufb01cation g:eval(g;sT) = 1 . The\\nfunction eval(g;s)determines whether state ssatis\\ufb01es the goal condition gby recursively evaluating\\nthe logical expression and using learned neural groundings of the primitive terms in the expression.\\nAt execution time, the agent will observe s0and be given gfrom human input, such as a \\ufb01rst-order\\nlogic expression corresponding to \\u201call the apples are in a blue bowl.\\u201d However, we do not assume\\nthat the agent knows, in advance, the groundings of g(i.e. the underlying eval(g;s)function) or the\\ntransition modelT. Thus, we need to learn gandTfrom data, in the form of observed trajectories\\nthat achieve goal states of interest.\\nFormally, we assume the training data given to the agent is a collection of tuples hs;a;g; succi,\\nwheres=hsiiis a sequence of world states, a=haiiis the sequence of actions taken by the\\nrobot,gis a goal speci\\ufb01cation, and succ=hsucciiis the \\u201ctask-success\\u201d signal. Each succi2f0;1g\\nindicates whether the goal gis satis\\ufb01ed at state si:succi=eval(g;si). The data sequences should\\nbe representative of the dynamics of the domain but need not be optimal goal-reaching trajectories.\\nIt can be dif\\ufb01cult to learn a transition model that is accurate over the long term on some types of\\nstate representations. For this reason, we generally assume an arbitrary latent space, \\b, for planning.\\nThe learning problem, then, is to \\ufb01nd three parametric functions, collectively parameterized by \\u0012:\\nstate encoderE\\u0012:S! \\b, goal-evaluation function eval\\u0012:G\\u0002\\b!f0;1g, and transition model\\nT\\u0012: \\b\\u0002A! \\b. Although the domain might be mildly partially observable or stochastic, our goal\\nwill be to recover the most accurate possible deterministic model on the latent space.\\nLocal, sparse structure. We need models that will generalize very broadly to scenarios with different will be to recover the most accurate possible deterministic model on the latent space.\\nLocal, sparse structure. We need models that will generalize very broadly to scenarios with different\\nnumbers and types of objects in widely varying arrangements. To achieve this, we exploit structure to\\nenable compositional generalization: throughout this work we will be committing to an object-centric\\nrepresentation for s, a logical language for goals g, and a sparse, local model of action effects.\\nWe begin by factoring the environmental state sinto a set of object states. Each s2S is a tuple\\n(Us;fs), whereUsis the set of objects in state s, denoted by arbitrary unique names (e.g., item#1 ,\\nitem#2 ). The object setUsis assumed to be constant within a single episode, but may differ in\\ndifferent episodes. The second component, fs, is a dictionary mapping each object name to a \\ufb01xed-\\ndimensional object-state representation, such as a local image crop of the object and its position. We\\n3 can extend this representation to relations among objects, for example by adding gs(x;y);x;y2U\\nas a mapping from each object pair to a vector representation. We assume the detection and tracking\\nof objects through time is done by external perception modules (e.g., object detectors and trackers).\\nWe carry the object-centric representation through to actions, goals, and the transition model. Speci\\ufb01-\\ncally, we de\\ufb01ne a predicate as a tuplehname;args;groundingi, where args is a list ofkarguments and\\ngrounding is a function from the latent representations of the objects corresponding to its arguments\\n(in\\tk) into a scalar or vector value. (This is a generalization of the typical use of the term \\\"predicate,\\\"\\nwhich is better suited for use in robotics domains in which many quantities we must reason about are\\ncontinuous.) For example, as illustrated in Fig. 3c, the predicate wetness takes a single argument\\nas its input, and returns a (latent) vector representation of its wetness property; its grounding might\\nbe a neural network that maps from the visual appearance of the object to the latent wetness value.\\nGiven a set of predicates, we de\\ufb01ne the language of possible goal speci\\ufb01cations to be all \\ufb01rst-order\\nlogic formulas over the subset of the predicates whose output type is Boolean. To evaluate a goal\\nspeci\\ufb01cation in a state (U;f), quanti\\ufb01cation is interpreted in the \\ufb01nite domain Uandfsprovides an\\ninterpretation of object names into representations that can serve as input to the grounded predicates.\\nThe transition model T\\u0012is speci\\ufb01ed in terms of a set object-parameterized action schema\\nhname;args;precond;effect;\\u0019i, where name is a symbol, args is a list of symbols, precond and\\neffect are descriptions of the action\\u2019s effects, described in section 2.1, and \\u0019is a parameterized\\nprimitive policy for carrying out the action in terms of raw perception and motor commands. These\\nlocal policies can be learned via demonstration or reinforcement learning in a phase prior to the\\nmodel-learning phase, constructed using principles of control theory, or a combination of these\\nmethods. The set of concrete actions Aavailable in a state sis formed by instantiating the action\\nscheme with objects in universe Us. We assume the transition dynamics of the domain (i.e., the effect\\nof each action schema) are well characterized in terms of the changes of properties and relations of\\nobjects and that the transition model is lifted in the sense that it can be applied to domain instances\\nwith different numbers and types of objects. In addition, we assume the dynamics are local and sparse,\\nin the sense that effects of any individual action depend on and change only a small number attributes\\nand relations of a few objects, and that by default all other objects and attributes are unaffected.\\nTaking again action schema move-into as an example, shown in Fig. 3d, only the states of object #1\\nand #6 are relevant to this action (but not #2, #3, etc.), and furthermore, the action only changes the\\npose andwetness properties of item#1 (but not the color and the type).\\nThe factored representation also introduces a factored learning problem: instead of learning a\\nmonolithic neural network for T\\u0012andeval\\u0012, the problem is factored into learning the grounding of\\nindividual predicates that appear in goal formulas, as well as the transition function for individual\\nfactors that were changed by an action.\\n2.1 Representation Language\\nThe overall speci\\ufb01cation of eval\\u0012andT\\u0012can be decomposed into two parts: 1) the locality and\\nsparsity structures and 2) the actual model parameters, \\u0012, such as neural network weights. We provide\\na symbolic language for human programmers to specify the locality and sparsity structure of the\\ndomain and methods for representing and learning \\u0012. If the human provides no structure, the model\\nfalls back to a plain object-centric dynamics model Zhu et al. [2018]. However, we will show that domain and methods for representing and learning \\u0012. If the human provides no structure, the model\\nfalls back to a plain object-centric dynamics model Zhu et al. [2018]. However, we will show that\\nexplicit encoding of locality and sparsity structures can substantially improve the data ef\\ufb01ciency of\\nlearning and the computational ef\\ufb01ciency of planning with the resulting models.\\nPDSketch is an extension of the planning-domain de\\ufb01nition language [Fikes and Nilsson, 1971, Fox\\nand Long, 2003], a widely used formalism that focuses exposing locality and sparsity structure in\\nsymbolic planning domains. The key extensions are 1) allowing vector values in the computation\\ngraph and 2) enabling the programmer to use \\u201cblanks\\u201d, which are unspeci\\ufb01ed functions that will\\nbe \\ufb01lled in with neural networks learned from data. Thus, rather than specifying the model in full\\ndetail, the programmer provides only a \\u201csketch\\u201d [Solar-Lezama, 2008]. The two key representational\\ncomponents of PDDL are predicates and action schemas (operators).\\nFig. 4 shows a simple example of PDSketch de\\ufb01nition of predicates. All three predicates take a single\\nobject ?oof type item as their argument, and return either a \\ufb02oating-point vector or a scalar value\\nfrom 0 to 1, indicating the score of a binary classi\\ufb01er. The image predicate simply refers to the raw\\nimage crop feature of the object. The is-yellow predicate\\u2019s grounding takes a very simple form\\n\\u201c(??f (color ?o)) \\u201d. The term ??fde\\ufb01nes a slot whose name is f. It takes only one argument, the\\n4 (a)A PDSketchdefinition of an input feature of each objects: (image?o),and two derived feature\\/predicates:(color?o)and (is-yellow?o).(:predicates; inputfeatures(image [return_type=vector[float32],input] ?o-item))(:derived(color[return_type=vector[float32]]?o-item)(??f(image?o)))(:derived(is-yellow?o-item) ; parameter and type(??f(color?o)); function body)RawObservation(Image)image?o(per-obj.)color::f(ConvNet)color?o(per-obj.)is-yellow::f(ConvNet)is-yellow?o(per-obj.score)(b) Thecorresponding\\u201ccomputationgraph\\u201dinducedbythedefinitions.color::fandis-yellow::farecustomizableCNNsthatareappliedidenticallytoeachobjectintheinputstate.\\n0.0\\n0.9\\n0.1\\n0.0\\n0.9\\n0.0Figure 4: A minimal example of de\\ufb01ning derived features and predicates with blanks \\u201c ??\\u201d.\\ncolor of the object ?o, and outputs a classi\\ufb01cation score, which can be interpreted as the score of the\\nobject ?obeing yellow. The actual computation of the yellow predicate from the color value (as\\nwell as the computation of the color value from the image value) is instantiated in a neural network\\nwith trained parameters. The computation graph for the whole model can be built by recursively\\nchaining the function bodies of predicate de\\ufb01nitions.\\nNext, we illustrate how locality andsparsity structures can be speci\\ufb01ed for an action schema. Fig. 5\\nde\\ufb01nes an action schema name move-into with two effect components. First, highlighted in blue,\\nthe action changes the pose of object ?o1to a new pose that depends on the current pose of the\\nsecond object ?o2. Rather than hand-coding this detailed dependence, we leave the grounding blank.\\nIn addition, in our domain, the wetness of an object may be changed when the object is placed into\\na speci\\ufb01c type of container. This is encoded by specifying a conditional effect using the keyword\\nwhen , with two parts: 1) a Boolean-valued condition gof some other predicates on the state (in this\\ncase, the types of the two objects), and 2) the actual \\u201ceffect\\u201d, in this case, to change the wetness of\\n?o1based on a function that considers the current wetness of?o1. The update will be applied only\\nif the condition is true. To ensure the computation is differentiable, we make this condition \\u201csoft\\u201d:\\nletwbe the current wetness, w0be the new wetness computed by function ??h, andcbe the scalar\\ncondition value computed by function ??g. The updated value of the wetness will be cw0+ (1\\u0000c)w.\\nNote that all the pose, wetness, and type representations can be arbitrary latent vectors computed by\\nan encoder from the raw input. Thus, the \\u201cblanks\\u201d ??f,??g,??hare indeed general neural networks.\\nThe effect de\\ufb01nition here induces a corresponding computation graph of neural network weights and\\nstate representation tensors.\\nLike PDDL, PDSketch has full support of \\ufb01rst-order logic, including Boolean operations (and, or,\\nnot) and \\ufb01nite-domain quanti\\ufb01ers ( 8and9). They allow us to de\\ufb01ne more complex structures in the\\ndomain of interest. We present our full language and more examples in the supplementary material.\\n2.2 Model Learning and Planning with PDSketch\\nLet\\u0012denote the collection of all learnable parameters required to complete a PDSketch domain\\nde\\ufb01nition into a full model. This includes the parameters of the state encoder, all predicate groundings,\\nand the de\\ufb01nitions of slot-update functions that were left blank in the sketch. Recall that our training\\ndata are tuples of three sequences and a goal formula hs;a;g; succi. Fundamentally, our objective is\\nto minimize a sum of two losses, one related to predicting the truth values of the goal formulas and\\none related to predicting the next state, given the previous state and action:\\nL(\\u0012) =X\\nhs;a;g; succi2D(X\\niBCE(eval\\u0012(g;E\\u0012(si));succi) +X\\niL1(T\\u0012(E\\u0012(si);ai);E\\u0012(si+1)))\\n;\\nwhere BCE is the binary cross-entropy classi\\ufb01cation loss and L1is a regression loss. To avoid\\ndegenerate local optima, we add a \\u201clookahead\\u201d loss term that combines both aspects, as detailed in ;\\nwhere BCE is the binary cross-entropy classi\\ufb01cation loss and L1is a regression loss. To avoid\\ndegenerate local optima, we add a \\u201clookahead\\u201d loss term that combines both aspects, as detailed in\\nthe supplement. If the encoder E\\u0012is constrained to be the identity, and there is no predicate-level\\nstructure, then the transition model essentially learns to be a per-object next-image predictor. More\\ngenerally, including the encoder turns this into a bisimulation objective [Li et al., 2006]: we want to\\nuncover a latent transition model that accurately predicts the reward signal (in this case, whether the\\ngoal is satis\\ufb01ed or not) but does not necessarily reconstruct the input state representation.\\n5 pose       Item#1pose       Item#6color      Item#1color      Item#6wetness    Item#1wetness    Item#6dirtiness  Item#1dirtiness  Item#6type       Item#1type       Item#6Timestep \\ud835\\udc61Timestep \\ud835\\udc61+1move-into::f\\nmove-into::gActionmove-into(Item#1, Item#6)(:action move-into:parameters (?o1?o2):effects(and(pose::assign?o1(??f(pose ?o2)))(when (??g(type ?o1)(type ?o2))(wetness::assign ?o1(??h(wetness ?o1))))......pose       Item#1pose       Item#6color      Item#1color      Item#6wetness    Item#1wetness    Item#6dirtiness  Item#1dirtiness  Item#6type       Item#1type       Item#6move-into::hTheupdateonwetnessisconditionedon.theoutputofg.\\nItem#1(423.0, 420.0)\\nItem#6(275.5, 273.5)\\nItem#1(273.0, 273.0)\\nItem#6(275.5, 273.5)............Figure 5: A computation graph of a partial de\\ufb01nition of the action \\u201cmove-into.\\u201d\\nIn order to adjust \\u0012to minimize this loss, we must establish a differentiable computation graph. The\\nencoderE\\u0012will generally be a relatively standard combination of convolutional and fully-connected\\nfeed-forward neural network. Fig. 5 illustrates the computation graph associated with T\\u0012for a\\nparticular choice of action aiand objects that serve as its arguments. Importantly, note that there\\nis a substantial amount of parameter-tying: the same predicate-grounding network might appear in\\nmultiple times, even when characterizing T\\u0012for a single action, if that predicate appears multiple\\ntimes (applied to different objects) in the preconditions or effects of the action. The computation\\ngraph shown here, as well as those necessary to compute eval\\u0012(g;E\\u0012(si)), involve Boolean operators,\\nwhich do not have useful derivatives for optimization. We address this by representing truth values\\nas elements of the interval [0;1]and approximate logical operations with the differentiable G\\u00f6del\\nt-norms: not(p) = 1\\u0000p,and(p1;p2) = min(p1;p2),8x:p(x) =minxp(x).\\nOnce we have estimated \\u0012from data, we can solve any planning problem in the domain given any\\nstarting state s02S and goalgexpressed in terms of predicates for which we have groundings. The\\nresulting transition model Tcan be used by a variety of different planners. We will focus on forward\\nsearch, constructing a tree rooted at E(s0)with branches corresponding to the possible instantiations,\\na, of action templates Awith the objects in the universe Uassociated with s0, and next latent states\\ncomputed by applying T. The search terminates when it reaches a node nin which eval\\u0012(n;g)>:5.\\nUnguided forward search can be very slow when the planning horizon is long or the branching factor\\nis large (e.g., when there are many objects in an environment). To address this, we will use the A\\u0003\\nheuristic search algorithm using a domain-independent search heuristic directly derivable from the\\nlocality and sparsity structure de\\ufb01ned in PDSketch.\\n2.3 Inducing Domain-Independent Heuristics\\nBecause of the rich representational capacity of PDSketch models, in which values can be continuous\\nand multidimensional, we cannot take advantage of the planning algorithms that operate on PDDL\\ninput, such as Fast Downward [Helmert, 2006], which derive much of their ef\\ufb01ciency from domain-\\nindependent heuristics. We cannot use their strategies in detail, but we take inspiration from the idea\\nof deriving an optimistic estimate of the cost to reach the goal from a state by solving a \\u201crelaxed\\u201d\\nversion of the problem, which is computationally easier than the original Bonet and Geffner [2001].\\nOne way to construct a relaxed planning problem is to allow each predicate instance to take on\\nmultiple values at the same time. For example, the robot or an object can effectively be in multiple\\nplaces at the same time. In this relaxation, computing the number of steps needed to change the\\nvalue of a predicate instance can be done in polynomial time. We can use such a relaxation in the\\nhFF heuristic [Hoffmann and Nebel, 2001] to get an estimate of the cost to goal, by \\ufb01rst chaining value of a predicate instance can be done in polynomial time. We can use such a relaxation in the\\nhFF heuristic [Hoffmann and Nebel, 2001] to get an estimate of the cost to goal, by \\ufb01rst chaining\\nthe actions forward until all the components of the goal condition have been made true, and then\\nsearching to recover a small set of actions that can collectively achieve the goal under the relaxation.\\nTo use hFF, however, we must reduce our continuous-space problem to a discrete-space one. Specif-\\nically, we discretizes all continuous state variables (poses, etc.) into a designated number of bins\\n(e.g., 128). Next, for each externally-de\\ufb01ned functions, we learn a \\ufb01rst-order decision tree to approxi-\\nmate the computation (e.g., to approximate the neural network). Concretely, we use VQV AE [Van\\n6 Figure 6: A screenshot of\\nthe BabyAI environment.ModelInductive Biases Succ. Rate\\nObj-Centric. Facing Rob. Dyn. Obj. Prop. Basic #Obj.Gen.\\nBC Y N N N 0.93 0.79\\nDT (S) Y N N N 0.91 0.82\\nDT (S+F) Y N N N 0.32 0.19\\nDreamerV2 N N N N 0.96 0.79\\nPDS-Base Y N N N 0.82 0.62\\nPDS-Abs Y Y N N 0.99 0.98\\nPDS-Rob Y Y Y N 1.00 1.00\\nTable 1: The planning success rate of different models on BabyAI.\\nThe PDS-Base failed to understand the obstacles.5xMoreEfficient\\nFigure 7: (a) and (b) Data ef\\ufb01ciency comparison for model learning. We compare three structures\\nwith different levels of abstractness. (c) Planning ef\\ufb01ciency, measured as the number of expanded\\nnodes for different heuristic computation methods.\\nDen Oord et al., 2017] to discretize the feature vectors. For all predicates whose output is a latent\\nembedding, we add a vector quantization layer after the encoding layers. We initialize the quantized\\nembeddings by running a K-means clustering over the item feature embedding from a small dataset,\\nand \\ufb01netune the weights on the entire training dataset for one epoch. We describe this in more detail\\nin the supplementary material. Although our discretizations are inherently lossy on non-Boolean\\npredicate values, since they are only used for search guidance, and not in the forward simulation of\\nthe actual actions during planning, this does not affect the correctness of the overall algorithm.\\n3 Experiments\\nWe evaluate PDSketch in two domains: BabyAI, a 2D grid-world and Painting Factory, a simu-\\nlated tabletop manipulation task. We compare our model with two model-free methods: Behavior\\nCloning [BC; Bain and Sammut, 1995] and Decision Transformer [DT; Chen et al., 2021]. We\\nimplement two DT variants: DT-S with only successful demonstrations, and DT-S+F with successful\\nand failed demonstrations. We use graph neural networks [Gori et al., 2005, Battaglia et al., 2018] as\\ntheir state encoder.\\n3.1 BabyAI\\nBabyAI [Chevalier-Boisvert et al., 2019] is an image-based 2D grid-world environment where an\\nagent can navigate around obstacles, pick and place objects, and toggle doors. In this paper, we focus\\non a speci\\ufb01c level of BabyAI, namely ActionObjDoor . At this level, the agent navigates within a 7x7\\ngrid. The goals include go to an X ,pick up an X ,open an X , where Xis a noun phrase, such as \\u201cblue\\nkey\\u201d. We train all models on environments with 4 doors and 4 objects. The of\\ufb02ine dataset Dcontains\\nboth successful and failure demonstrations obtained by A\\u0003search. We extend PDSketch to interactive\\ndata gathering in the supplementary material. Additionally, we test generalization to environments\\nwith 6 doors and 8 objects. Since objects may block agents, the agent needs to successfully uncover\\nthe underlying dynamics and plan to navigate around them.\\nWe study three PDSketch models with various levels of locality andsparsity structures built in. The\\nPDS-Base model has no built-in structures: each object is represented as a holistic vector. This falls\\nback to an object-centric transition model [Zhu et al., 2018]. PDS-Abs model that disentangles poses\\nfrom object appearances. Importantly, it de\\ufb01nes a predicate facing and uses this concept to write\\naction de\\ufb01nitions (e.g., whether a robot move will be blocked by the object it is facing). However, the\\n7 (:actionforward-abs:parameters(?r -robot):precondition(and ):effect(robot-pose::cond-assign ?r(??f(robot-pose ?r) (robot-dir?r)(foreach (?o -item)(item-feature::cond-select ?o (facing?r ?o))))(??g(robot-pose ?r) (robot-dir?r))))robot-poseRrobot-poseRitem-feature#1item-feature#2facingR#1.......facingR#2forward::fforward::grobot-dirRFigure 8: The PDSketch de\\ufb01nition of the \\u201cforward\\u201d action and the corresponding computation graph\\nfor the PDS-Abs model. The mentioned predicate facing is an derived predicate represented as a\\nneural network that is jointly learned (omitted in the graph for brevity).\\ngrounding of facing is to be learned. Figure 8 shows the detail de\\ufb01nition of the forward action in the\\nPDS-Abs model. PDS-Rob contains prede\\ufb01ned rules for robot movements but the object recognition\\nmodules are learned. We provide their de\\ufb01nitions in the supplementary material.\\nResults. Table 1 shows the results. Overall, PDSketch with more structure (PDS-Abs and PDS-Rob)\\noutperforms baselines by a signi\\ufb01cant margin. From the performance of PDS-Abs, we see that even a\\ntiny amount of additional structure (e.g., an ungrounded predicate \\u201cfacing\\u201d) signi\\ufb01cantly improves\\nthe performance, especially when it comes to generalization to more complex environments. See\\nbelow for a zoomed-in analysis for model learning. Furthermore, we hypothesize that the inferior\\nperformance of decision transformer in the mixed training data (Succ+Fail) setting is due to the\\nreward sparsity: the agent only gets reward 1 when it reaches the goal. Thus, the failed demonstrations\\nare generally hard to model as they are irrelevant to the goal speci\\ufb01cation. In addition, we compare\\nour models with DreamerV2, a state-of-the-art model-based reinforce learning algorithm for image-\\nbased environments. Compared with BC and DT, we see DreamerV2 achieves slightly improved\\nperformance for the basic task, but does not show stronger generalization to environments with more\\nobjects. We hypothesize this is because Dreamer still learns a \\ufb01xed policy for execution.\\nData ef\\ufb01ciency. Fig. 7a-b shows model learning curves for the three PDSketch models, in the cases\\nwhere there is a single object in the environment and when there are 4 objects. . When the number\\nof object is small, the environmental dynamics is easier to learn: both Base and Abs have a similar\\nperformance. However, when the number of object increases, PDSketch can leverage the inductive\\nbias to learn the dynamics faster. Shown in the \\ufb01gure, even at the end of training, the model PDS-Base\\nhas not successfully learned the correct movement dynamics, leading to its inferior performance\\nduring generalization to more objects.\\nPlanning runtime ef\\ufb01ciency. Fig. 7c quanti\\ufb01es the number of nodes expanded by A\\u0003when using\\ndifferent heuristic computations. Speci\\ufb01cally, we compare our model with the blind heuristic (i.e., the\\nheuristic value of a state is 0 when it satis\\ufb01es the goal and otherwise 1.) and a GNN-parameterized\\nheuristic learned from successful demonstrations. All results are based on the PDS-Abs model. First,\\nlearning-based heuristics (GNN) shows improvement over the baseline \\u201cblind\\u201d heuristic, especially\\nwhen the task is easy (requiring a small number of expansions). Second, the heuristic derived from\\nPDSketch signi\\ufb01cantly improves the search ef\\ufb01ciency. At the success rate of 0.8, our method is 5\\ntimes more ef\\ufb01cient than learning-based heuristics.\\n3.2 Robot Painting\\nFinally, we extend the framework to a tabletop manipulation task, built based on the tabletop\\nenvironment of Zeng et al. [2020]. There are three zones, several bowls, and several blocks on the\\ntable. The robot can use its suction gripper to pick-and-place objects into a designated location.\\nBlocks have 8 possible colors. Placing objects in a bowl will paint the object to be the same color as\\nthe bowl. The task is to paint the blocks and organize them in the target brown box. Our training-time the bowl. The task is to paint the blocks and organize them in the target brown box. Our training-time\\ngoal requires the robot to paint-and-place two objects. The goal contains their colors and their\\nrelationship (e.g., a pink block is left of ayellow block. Demonstration are collected using hand-\\ncrafted oracle policies following Zeng et al. [2020]. The of\\ufb02ine dataset contains only successful\\ntrajectories. There are two built-in actions that the robot can execute. Fig. 9 shows the computation\\ngraph derived from PDSketch.\\n8 pose  Item#1stateItem#1pose  Item#2stateItem#2......FactoredStateActionspose  Item#1stateItem#1pose  Item#2stateItem#2PredictedStatemove-into(?o1?o2)move-to(?o?pose)is-yellow ?ois-brown ?oin ?o1 ?o2left ?o1 ?o2......PredicatesFigure 9: Our encoding of the painting factory domain. Each\\nobject has a pose and a state. Action move-to directly\\nchanges the object pose. move-into moves an object into\\na container, which possible changes the state (e.g., color) of\\nthe object. Goals are represented using relational predicates.Model Succ. Rate\\n@100Succ Rate\\n@1000\\nBC 0.70 0.95\\nDT 0.56 0.97\\nPDS 0.91 0.99\\nTable 2: Model performance on Paint-\\ning Factory, measured as plan success\\nrate. The number after @ is the num-\\nber of demonstration trajectories. PDS-\\nketch models show stronger data ef\\ufb01-\\nciency than baselines.\\n(exists ?x ?y (and(purple ?x) (yellow ?y)(left-of ?y ?x) (in-target ?x)))(exists ?x ?y ?z (and(yellow ?x) (cyan ?y) (red ?z)))(forall?x (and(yellow ?x) (in-target ?x)))(exists ?x ?y ?z (and(pink ?x) (purple ?y) (cyan ?z)(in-target ?x)(on ?y ?x) (on ?z ?y)))TrainingGeneralization (Unseen GoalsandMoreObjects)\\nPlace a yellow block left of a purple block in the target box.Paint a yellow, a cyan, and a red block (no need to put in the target box).Paint all blocks yellow and put them in the target box.Paint a pink, a purple, and a cyan block,and stack them in the target box.\\nFigure 10: From only one training task (left), a model speci\\ufb01ed in PDSketch learns primitive concepts\\nincluding object colors and spatial relations. They can be used in planning for unseen tasks given new\\n\\ufb01rst-order logic descriptions of the goal. The natural language descriptions are shown for readability.\\nState and action space. The state space in the painting factory is composed of a list of objects, and\\na list of containers. Both objects and containers are represented as a tuple of a 3D xyz location, and a\\nimage crop. The image crop is generated by \\ufb01rst computing the 2D bounding box of the object in the\\ncamera plain, cropping out the image patch, and resizing it to 32 by 32. The action space contains\\ntwo primitives. First, move-into is an action de\\ufb01ned for each pair of item and container. It changes\\nthe pose of the item (now the item will be inside the container). When the item is placed into a bowl,\\nit will be painted into the same color as the bowl. The second action takes an object and a 3D location\\nas input, and moves the object directly to a designated position (ignoring the rotation).\\nResults. Table 2 shows the planning success rate on the training task with different amounts of\\ntraining data. Overall, PDSketch is more data ef\\ufb01cient than both baselines and achieves strong overall\\nperformance in this task. Much more importantly, Fig. 10 shows our generalization to novel task\\nspeci\\ufb01cations , which involve more objects or new speci\\ufb01ers (e.g., forall ). Our model generalizes\\ndirectly to these novel scenarios without any additional training . The quantitative performance for\\nthese three generalization tasks are: 0.99, 0.98, and 0.87, respectively, measured as success rate after\\nexecuting the plan. The last task has lower success rate because stacking objects may fail due to\\ncontroller and physical noises. Future work may consider building closed-loop controller that can\\nrecover such failures. We specify implementation details in the supplementary material.\\n4 Related Work\\nIntegrated model learning and planning is a promising strategy for building robots that can gener-\\nalize to novel situations and novel goals. Speci\\ufb01cally, Chiappa et al. [2017], Zhang et al. [2021],\\nSchrittwieser et al. [2020] learn dynamics from raw pixels; Jetchev et al. [2013], Pasula et al. [2007],\\nKonidaris et al. [2018], Chitnis et al. [2021], Bonet and Geffner [2020], Asai and Muise [2020],\\nSilver et al. [2021] assume access to the underlying factored states of objects, such as object colors\\nand other physical properties. Our work bridges the gap between two groups: we do not assume Silver et al. [2021] assume access to the underlying factored states of objects, such as object colors\\nand other physical properties. Our work bridges the gap between two groups: we do not assume\\npre-factored state representations given as the input, but learn to ground different factors of object\\nstates. More importantly, instead of relying on off-the-shelf models for predicting pixels or for\\n9 learning \\ufb01rst-order rules in symbolic domains, our work considers how human-programmed locality\\nand sparsity structures can improve the model learning and planning ef\\ufb01ciency.\\nOur model learns an object-factored transition model, which generally falls into the category of\\nlearning object-oriented MDPs [OO-MDPs; Guestrin et al., 2003, Diuk et al., 2008]. OO-MDPs have\\nbeen applied to neural network-based representations for visual domains [Walsh, 2010, Kansky et al.,\\n2017, Zhu et al., 2018, Xia et al., 2019, Veerapaneni et al., 2020] and textual game domains [Liu\\net al., 2021]. Our model leverages object-centric representations to encode permutation-invariant\\nstructures. Furthremore, we exploit \\ufb01ne-grained local and sparse structures in model learning.\\nThe objective of model learning and planning is to obtain a goal-conditioned policy [Kaelbling,\\n1993b,a]. While many others have studied model-free [Dayan and Hinton, 1992, Schaul et al., 2015]\\nor hybrid model-free and model-based approaches [Pong et al., 2018, Nasiriany et al., 2019], in this\\npaper, we focus on learning structured models from data and leveraging domain-independent planners\\nfor planning in latent representations. Such structured models support novel goal speci\\ufb01cations\\nvia a \\ufb01rst-order logic language, and domain-independent heuristics to accelerate search. Another\\nalternative approach towards learning and planning is to learning to predict subgoals that needs to be\\nachieved before other goals [Xu et al., 2019]. However, their work assumes that all preconditions and\\neffects can be represented in a prede\\ufb01ned symbolic language, and there are controllers for achieving\\nindividual subgoals. By contrast, PDSketch supports generic neural-network-based representations\\nfor predicates and action effects.\\nOur framework PDSketch combines model de\\ufb01nition in a structured language (e.g., \\ufb01rst-order logic)\\nand neural network learning. It is closely related to the idea of neuro-symbolic programming and\\ndifferentiable programming for relational reasoning [Manhaeve et al., 2018, Riegel et al., 2020,\\nHuang et al., 2021] and policy learning [Sun et al., 2019]. Our language PDSketch borrows ideas\\nfrom earlier work on \\u201csoft\\u201d execution of logical formulas but works on planning domains.\\n5 Conclusions and Limitations\\nPDSketch supports \\ufb02exible and effective speci\\ufb01cation of locality andsparsity structures of en-\\nvironment transition models. Leveraging these structures enables more data-ef\\ufb01cient learning,\\ncompositional generalization to novel goal speci\\ufb01cations and environmental states, and also domain-\\nindependent heuristics that accelerates performance-time planning. Limitations we hope to address\\ninclude the lack of hierarchy and the inability of the method to discover novel factorizations.\\nIn terms of societal impact, PDSketch suggests a hybrid method for building intelligent robots, by\\ncombining human programs to specify high-level structures with learning, to reduce the amount of\\ndata and computation required for learning complex and long-horizon behaviors. It also enables more\\nmodularized systems: users can specify new tasks using the predicates and actions that have already\\nbeen de\\ufb01ned more easily than in unstructured approaches. However, PDSketch may require more\\nexpertise (or, at least, a different type of expertise) from programmers than other approaches.\\nAcknowledgement. We thank all group members of the MIT Learning & Intelligent Systems Group\\nfor helpful comments on an early version of the project. This work is in part supported by ONR\\nMURI N00014-16-1-2007, the Center for Brain, Minds, and Machines (CBMM, funded by NSF\\nSTC award CCF-1231216), NSF grant 2214177, AFOSR grant FA9550-22-1-0249, ONR grant\\nN00014-18-1-2847, the MIT Quest for Intelligence, MIT\\u2013IBM Watson Lab. Any opinions, \\ufb01ndings,\\nand conclusions or recommendations expressed in this material are those of the authors and do not\\nnecessarily re\\ufb02ect the views of our sponsors.\\nReferences and conclusions or recommendations expressed in this material are those of the authors and do not\\nnecessarily re\\ufb02ect the views of our sponsors.\\nReferences\\nMartin Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay\\nGhemawat, Geoffrey Irving, Michael Isard, Manjunath Kudlur, Josh Levenberg, Rajat Monga, Sherry Moore,\\nDerek G. Murray, Benoit Steiner, Paul Tucker, Vijay Vasudevan, Pete Warden, Martin Wicke, Yuan Yu, and\\nXiaoqiang Zheng. TensorFlow: A System for Large-Scale Machine Learning. In OSDI , pages 265\\u2013283, 2016.\\n2\\nMasataro Asai and Christian Muise. Learning Neural-Symbolic Descriptive Planning Models via Cube-Space\\nPriors: The V oyage Home (To Strips). arXiv:2004.12850 , 2020. 9\\n10 Michael Bain and Claude Sammut. A Framework for Behavioural Cloning. Machine Intelligence , pages\\n103\\u2013129, 1995. 7\\nPeter W Battaglia, Jessica B Hamrick, Victor Bapst, Alvaro Sanchez-Gonzalez, Vinicius Zambaldi, Mateusz\\nMalinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, et al. Relational Inductive\\nBiases, Deep Learning, and Graph Networks. arXiv:1806.01261 , 2018. 7\\nBlai Bonet and H\\u00e9ctor Geffner. Planning as Heuristic Search. Artif. Intell. , 129(1-2):5\\u201333, 2001. 6\\nBlai Bonet and Hector Geffner. Learning First-Order Symbolic Representations for Planning from the Structure\\nof the State Space. In ECAI , 2020. 9\\nLili Chen, Kevin Lu, Aravind Rajeswaran, Kimin Lee, Aditya Grover, Misha Laskin, Pieter Abbeel, Aravind\\nSrinivas, and Igor Mordatch. Decision Transformer: Reinforcement Learning via Sequence Modeling. In\\nNeurIPS , 2021. 7\\nMaxime Chevalier-Boisvert, Dzmitry Bahdanau, Salem Lahlou, Lucas Willems, Chitwan Saharia, Thien Huu\\nNguyen, and Yoshua Bengio. BabyAI: First Steps Towards Grounded Language Learning With a Human In\\nthe Loop. In ICLR , 2019. 7, 14, 28\\nSilvia Chiappa, S\\u00e9bastien Racaniere, Daan Wierstra, and Shakir Mohamed. Recurrent Environment Simulators.\\nInICLR , 2017. 9\\nRohan Chitnis, Tom Silver, Joshua B Tenenbaum, Tomas Lozano-Perez, and Leslie Pack Kaelbling. Learning\\nNeuro-Symbolic Relational Transition Models for Bilevel Planning. arXiv:2105.14074 , 2021. 9\\nPeter Dayan and Geoffrey E Hinton. Feudal Reinforcement Learning. In NeurIPS , 1992. 10\\nCarlos Diuk, Andre Cohen, and Michael L Littman. An Object-Oriented Representation for Ef\\ufb01cient Reinforce-\\nment Learning. In ICML , 2008. 10\\nRichard E Fikes and Nils J Nilsson. STRIPS: A New Approach to the Application of Theorem Proving to\\nProblem Solving. Artif. Intell. , 2(3-4):189\\u2013208, 1971. 4, 15\\nMaria Fox and Derek Long. PDDL2.1: An Extension to PDDL for Expressing Temporal Planning Domains.\\nJAIR , 20:61\\u2013124, 2003. 4, 15\\nCaelan Reed Garrett, Tom\\u00e1s Lozano-P\\u00e9rez, and Leslie Pack Kaelbling. Pddlstream: Integrating symbolic\\nplanners and blackbox samplers via optimistic adaptive planning. In ICAPS , 2020. 30\\nCaelan Reed Garrett, Rohan Chitnis, Rachel Holladay, Beomjoon Kim, Leslie Pack Kaelbling, and Tom\\u00e1s\\nLozano-P\\u00e9rez. Integrated task and motion planning. Annual Review of Control, Robotics, & Autonomous\\nSystems , 4:265\\u2013293, 2021. 2\\nMarco Gori, Gabriele Monfardini, and Franco Scarselli. A new model for learning in graph domains. In IJCNN ,\\n2005. 7\\nCarlos Guestrin, Daphne Koller, Chris Gearhart, and Neal Kanodia. Generalizing Plans to New Environments in\\nRelational MDPs. In IJCAI , 2003. 10\\nDanijar Hafner, Timothy Lillicrap, Mohammad Norouzi, and Jimmy Ba. Mastering Atari with Discrete World\\nModels. In ICLR , 2021. 14\\nMalte Helmert. The Fast Downward Planning System. JAIR , 26:191\\u2013246, 2006. 6\\nJ\\u00f6rg Hoffmann and Bernhard Nebel. The FF Planning System: Fast Plan Generation through Heuristic Search.\\nJAIR , 14:253\\u2013302, 2001. 6\\nJiani Huang, Ziyang Li, Binghong Chen, Karan Samel, Mayur Naik, Le Song, and Xujie Si. Scallop: From\\nProbabilistic Deductive Databases to Scalable Differentiable Reasoning. In NeurIPS , 2021. 10\\nNikolay Jetchev, Tobias Lang, and Marc Toussaint. Learning Grounded Relational Symbols from Continuous\\nData for Abstract Reasoning. In ICRA Workshop , 2013. 9\\nLeslie Pack Kaelbling. Hierarchical Learning in Stochastic Domains: Preliminary Results. In ICML , 1993a. 10\\nLeslie Pack Kaelbling. Learning to Achieve Goals. In IJCAI , 1993b. 10\\nKen Kansky, Tom Silver, David A M\\u00e9ly, Mohamed Eldawy, Miguel L\\u00e1zaro-Gredilla, Xinghua Lou, Nimrod\\nDorfman, Szymon Sidor, Scott Phoenix, and Dileep George. Schema Networks: Zero-Shot Transfer with a\\nGenerative Causal Model of Intuitive Physics. In ICML , 2017. 10\\nGeorge Konidaris, Leslie Pack Kaelbling, and Tomas Lozano-Perez. From skills to symbols: Learning symbolic\\nrepresentations for abstract high-level planning. JAIR , 61:215\\u2013289, 2018. 9 George Konidaris, Leslie Pack Kaelbling, and Tomas Lozano-Perez. From skills to symbols: Learning symbolic\\nrepresentations for abstract high-level planning. JAIR , 61:215\\u2013289, 2018. 9\\nLihong Li, Thomas J Walsh, and Michael L Littman. Towards a Uni\\ufb01ed Theory of State Abstraction for MDPs.\\nISAIM , 4(5):9, 2006. 5\\nGuiliang Liu, Ashutosh Adhikari, Amir-massoud Farahmand, and Pascal Poupart. Learning Object-Oriented\\nDynamics for Planning from Text. In ICLR , 2021. 10\\n11 Robin Manhaeve, Sebastijan Dumancic, Angelika Kimmig, Thomas Demeester, and Luc De Raedt. Deep-\\nProbLog: Neural Probabilistic Logic Programming. In NeurIPS , 2018. 10\\nSoroush Nasiriany, Vitchyr Pong, Steven Lin, and Sergey Levine. Planning with Goal-Conditioned Policies. In\\nNeurIPS , 2019. 10\\nHanna M Pasula, Luke S Zettlemoyer, and Leslie Pack Kaelbling. Learning Symbolic Models of Stochastic\\nDomains. JAIR , 29:309\\u2013352, 2007. 9\\nAdam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen,\\nZeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary\\nDeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and\\nSoumith Chintala. PyTorch: An Imperative Style, High-Performance Deep Learning Library. In NeurIPS ,\\n2019. 2\\nVitchyr Pong, Shixiang Gu, Murtaza Dalal, and Sergey Levine. Temporal Difference Models: Model-Free Deep\\nRL for Model-Based Control. In ICLR , 2018. 10\\nJ. Ross Quinlan. Learning Logical De\\ufb01nitions from Relations. MLJ, 5(3):239\\u2013266, 1990. 25\\nRyan Riegel, Alexander Gray, Francois Luus, Naweed Khan, Ndivhuwo Makondo, Ismail Yunus Akhalwaya,\\nHaifeng Qian, Ronald Fagin, Francisco Barahona, Udit Sharma, et al. Logical Neural Networks. In NeurIPS ,\\n2020. 10\\nTom Schaul, Daniel Horgan, Karol Gregor, and David Silver. Universal Value Function Approximators. In\\nICML , 2015. 10\\nJulian Schrittwieser, Ioannis Antonoglou, Thomas Hubert, Karen Simonyan, Laurent Sifre, Simon Schmitt,\\nArthur Guez, Edward Lockhart, Demis Hassabis, Thore Graepel, et al. Mastering Atari, Go, Chess and Shogi\\nby Planning with a Learned Model. Nat., 588(7839):604\\u2013609, 2020. 9, 14\\nJohn Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization\\nalgorithms. arXiv:1707.06347 , 2017. 14\\nTom Silver, Rohan Chitnis, Joshua Tenenbaum, Leslie Pack Kaelbling, and Tomas Lozano-Perez. Learning\\nSymbolic Operators for Task and Motion Planning. In IROS , 2021. 9\\nArmando Solar-Lezama. Program Synthesis by Sketching . PhD thesis, University of California, Berkeley, 2008.\\n4\\nShao-Hua Sun, Te-Lin Wu, and Joseph J Lim. Program Guided Agent. In ICLR , 2019. 10\\nAaron Van Den Oord, Oriol Vinyals, et al. Neural Discrete Representation Learning. In NeurIPS , 2017. 6, 25\\nRishi Veerapaneni, John D Co-Reyes, Michael Chang, Michael Janner, Chelsea Finn, Jiajun Wu, Joshua\\nTenenbaum, and Sergey Levine. Entity Abstraction in Visual Model-Based Reinforcement Learning. In\\nCoRL , 2020. 10\\nThomas J Walsh. Ef\\ufb01cient Learning of Relational Models for Sequential Decision Making . PhD thesis, Rutgers\\nThe State University of New Jersey-New Brunswick, 2010. 10\\nVictoria Xia, Zi Wang, and Leslie Pack Kaelbling. Learning Sparse Relational Transition Models. In ICLR ,\\n2019. 10\\nDanfei Xu, Roberto Mart\\u00edn-Mart\\u00edn, De-An Huang, Yuke Zhu, Silvio Savarese, and Li Fei-Fei. Regression\\nplanning networks. NeurIPS , 2019. 10\\nAndy Zeng, Pete Florence, Jonathan Tompson, Stefan Welker, Jonathan Chien, Maria Attarian, Travis Armstrong,\\nIvan Krasin, Dan Duong, Vikas Sindhwani, et al. Transporter Networks: Rearranging the Visual World for\\nRobotic Manipulation. In CoRL , 2020. 8\\nAmy Zhang, Rowan McAllister, Roberto Calandra, Yarin Gal, and Sergey Levine. Learning invariant representa-\\ntions for reinforcement learning without reconstruction. In ICLR , 2021. 9\\nGuangxiang Zhu, Zhiao Huang, and Chongjie Zhang. Object-Oriented Dynamics Predictor. In NeurIPS , 2018.\\n4, 7, 10\\n12 Supplementary Material for PDSketch:\\nIntegrated Planning Domain Programming and Learning\\nThe rest of the supplementary material is organized as the following. First, in Section A, we provide\\nadditional results on data ef\\ufb01ciency and on-policy learning for the BabyAI environment studied in\\nthe paper. Then, in Section B, we present an example-based formal introduction of the PDSketch\\nlanguage. Next, in Section C, we describe the implementation of our domain-independent heuristic.\\nWe also discuss why sparsity and locality structures are particularly important in inducing domain-\\nindependent heuristics. Finally, in Section D, we present our experiment setups. Both Section C and\\nSection D are presented based on the language de\\ufb01nition in Section B.\\nA Additional BabyAI Results\\nIn this section, we supplement two additional results on the BabyAI environment. First, in Section A.1,\\nwe present the data ef\\ufb01ciency study of different models in terms of their performance-time success\\nrate. We also make variance analysis of different models across different runs. Next, in Section A.2,\\nwe present how our model can be integrated with on-policy learning methods to make active data\\ngathering.\\nA.1 Data Ef\\ufb01ciency and Variance Analysis\\nWe further quantify the data ef\\ufb01ciency of different models in terms of their performance-time success\\nrate. Note that this is different from the Figure 7 in the main paper, where we have studied the\\ndata ef\\ufb01ciency in terms of model learning accuracy. In this section, we show that, 1) compared to\\nmodel-free baselines, integrate model learning and planning does improve the overall data ef\\ufb01ciency\\nin the domain we consider, and 2) the same is true for different PDSketch models with different levels\\nof abstractions and prior knowledge.\\nConcretely, we generate three datasets, with 100 demonstration episodes, 1000 episodes, and 10,000\\nepisodes, respectively. We train the model on different datasets until converge, and evaluate their\\nperformance. We have repeated all experiments three times. The reported value are the average score\\nand the standard deviation.\\nResults are summarized in Table 3. Here we summarize our \\ufb01ndings as the following.\\n1.First, in general, behavior cloning and decision transformer achieve similar performance on\\nthis task. When the amount of training data is small, the model perform noticeably worse\\nthan model-based methods.\\n2.The base model, even if trained with a large amount of data, struggle to capture the robotic\\nmovement transitions when facing obstacles. This results in that the performance gets stuck\\nat around 80%.\\n3.Compared to recognizing object properties, a signi\\ufb01cant amount of data is required to learn\\nthe transition model. When the transition model is given (in PDS-Rob), learning the visual\\nrecognition models in this simple grid-world domain is very data-ef\\ufb01cient: using only 100\\nepisodes, the model successfully solves 70% of the tasks.\\nModel 100 episodes 1000 episodes 10000 episodes\\nBC 0.24 \\u00060.01 0.24 \\u00060.01 0.39 \\u00060.03\\nDT 0.23 \\u00060.02 0.24 \\u00060.01 0.30 \\u00060.05\\nPDS-Base 0.04 \\u00060.00 0.11 \\u00060.01 0.80 \\u00060.07\\nPDS-Abs 0.04 \\u00060.02 0.98 \\u00060.03 0.99\\u00060.01\\nPDS-Rob 0.72\\u00060.09 0.99\\u00060.01 0.99\\u00060.01\\nTable 3: The planning success rate for different models using different amount of training data.\\n13 A.2 On-Policy Learning\\nOur system can also be integrated into an on-policy learning setting. This is compatible with the\\nstandard model-based reinforcement learning setting. Speci\\ufb01cally, recall that in an interactive learning\\nsetting, the system receives the current state and a goal expression. The agent itself should decide the\\nnext action to take. In the PDSketch case, we run the planner based on the current model parameters\\n\\u0012to generate a plan. We follow the plan in the simulator and collect the resulting trajectory as well\\nas the \\u201cdone\\u201d signals from the environment. Finally, we update the model parameter \\u0012using the\\nnewly collected data. Although our model may leverage of\\ufb02ine data, to keep the algorithm simple,\\nwe do not implement data reusing. Within 10,000 episodes, our model PDS-Rob is able to reach\\n0.99 performance-time success rate. Comparatively, the reported performance of Proximal Policy\\nOptimization (PPO) [Schulman et al., 2017, Chevalier-Boisvert et al., 2019] needs 200,000 episodes.\\nWe found that direct application of the on-policy algorithm on models with less inductive biases (i.e.,\\nPDS-Abs and PDS-Base) does not yield successful results. The success rate stuck around 6%. This is\\nprimarily because of their failure in exploration: direct application of the planner generates successful\\ntrajectories at a very low probability. This can be potentially alleviated by adding random exploration\\nfactors or replanning when the agent fails to reach the goal.\\nNote that, since our model solely focuses on representing and learning the model, it is possible\\nto integrate our framework with other model-based reinforcement learning algorithms, such as\\njoint model and policy learning [Schrittwieser et al., 2020], or world-model-based reinforcement\\nlearning [Hafner et al., 2021]. We leave these extensions as a future work.\\n14 B PDSketch Language\\nIn this section, we detail the design, the syntax, the semantics, and the implementation of the PDSketch\\nde\\ufb01nition language. We present it in two parts: the predicate and action de\\ufb01nition (Appendix B.1),\\nand PDSketch expressions (Appendix B.2).\\nB.1 Predicate and Action De\\ufb01nition\\nPDSketch is based on the Planning-Domain De\\ufb01nition Language [Fikes and Nilsson, 1971, Fox\\nand Long, 2003], which is a language specialized and derived from LISP. We choose a LISP-style\\nlanguage for two important reasons. First, compared to other procedure- or object-centric languages\\nsuch as Python, LISP features an easy de\\ufb01nition of \\ufb01rst-order logic, especially for recursively de\\ufb01ned\\nrules. Second, a signi\\ufb01cant amount of planning domains have been written in PDDL. Extending\\nthe language reduces the learning cost for programmers and also make old PDDL de\\ufb01nitions easily\\nreusable in PDSketch.\\nFor readers who are familiar with PDDL, a PDSketch \\ufb01le is a domain de\\ufb01nition \\ufb01le (in contrast to\\na problem de\\ufb01nition \\ufb01le). PDSketch only contains the de\\ufb01nition of predicates and actions in the\\ndomain. It does not contain problem-speci\\ufb01c information, such as the current state of the world, or\\nthe goal speci\\ufb01cation.\\nThus, a PDSketch \\ufb01le contains three parts. A PDSketch de\\ufb01nition, at the highest level, takes the\\nfollowing form:\\ndefinition: \\\"(\\\" \\\"define\\\" \\\"domain\\\" domain-name-def content-def* \\\")\\\"\\ndomain-name-def: \\\"(\\\" \\\"domain\\\" string \\\")\\\"\\ncontent-def: type-def | predicate-def | derived-def | action-def\\nAn empty domain \\ufb01le can be written as:\\n(define domain\\n(domain my-domain-name)\\n)\\nThis de\\ufb01nes a domain with name \\u201cmy-domain-name,\\u201d with no predicates and actions de\\ufb01ned.\\nAcontent-def can be either a type de\\ufb01nition, a predicate de\\ufb01nition, or an action schema de\\ufb01nition.\\nFirst, type de\\ufb01nition:\\ntype-def: \\\"(\\\" \\\":types\\\" single-type-def \\\")\\\"\\nsingle-type-def: type-name-list - base-type\\ntype-name-list: type-name+\\ntype-name: string\\nbase-type: type-name | prim-type | vector-type\\nprim-type: \\\"bool\\\" | \\\"int64\\\" | \\\"float32\\\"\\nvector-type: \\\"vector\\\" \\\"[\\\" prim-type (, int)? \\\"]\\\"\\nBelow, we will be using the BabyAI world as an example, the example type de\\ufb01nition of the BabyAI\\ndomain is:\\n(:types\\n15 robot item - object\\npose - vector[float32, 2]\\ndirection - vector[int64, 1]\\n)\\nSpeci\\ufb01cally, a type de\\ufb01nition contains multiple lines, each line being a pair of a type name list, and a\\nbase type. There are two types of types in PDSketch: object type and value type. Intuitively, object\\ntype refers to a concrete object in the world, while a value type usually denotes the return type of a\\nfeature function over the objects. Currently, we do not support the full hierarchical structure of type\\nde\\ufb01nitions: each type either inherits \\u201cobject\\u201d, which indicates this is an object type, or inherits a\\nprimitive value type (Boolean, integer, or \\ufb02oating-point numbers), or a vector type. In a vector-type\\nde\\ufb01nition, the second argument (e.g., the 2invector[float32, 2] ) denotes the dimension of the\\nvector. This can be omitted. Overall, the de\\ufb01nition above de\\ufb01nes four types: an object type called\\n\\u201crobot,\\u201d an object type called \\u201citem,\\u201d a value type that is a 2D vector, named \\u201cpose,\\u201d and a value type\\nthat is a 1D vector of integer (i.e., a single integer), named \\u201cdirection.\\u201d\\nThepredicate-def andderived-def are jointly used to de\\ufb01ne predicates. A predicate can be\\neither a predicate directly observable from the environment, or a \\u201cderived\\u201d predicate, which is a\\npredicate whose value is computed based on the value of other predicates. We \\ufb01rst show the grammar\\nde\\ufb01nition.\\npredicate-def: \\\"(\\\" \\\":predicates\\\" single-predicate-def \\\")\\\"\\nsingle-predicate-def: \\\"(\\\" predicate-name kwargs? variable-list \\\")\\\"\\npredicate-name: string\\nkwargs: \\\"[\\\" kwarg \\\"]\\\"\\nkwarg: kwarg-key \\\"=\\\" kwarg-value\\nkwarg-value: \\\"\\\\\\\"\\\" string \\\"\\\\\\\"\\\" | int | bool | float | base-type\\nvariable-list: typed-variable*\\ntyped-variable: variable-name \\\"-\\\" type-name\\nvariable-name: string\\n16 An example de\\ufb01nition of some input predicates is the following:\\n(:predicates\\n(robot-pose [return_type=pose] ?r - robot)\\n(robot-direction [return_type=direction] ?r - robot)\\n(item-pose [return_type=pose] ?o - item)\\n(item-image [return_type=vector[float32]] ?o - item)\\n)\\nIn this case, we have de\\ufb01ned four predicates: a 2D position of the robot, the direction that the robot is\\nfacing, the 2D position of the item, and \\ufb01nally, an image representing a local crop of the item.\\nBased on the input predicates, users can de\\ufb01ne many \\u201cderived\\u201d predicates, whose values are automat-\\nically computed based on other input and derived predicates. The formal syntax is:\\nderived-def: \\\"(\\\" \\\":derived\\\" single-predicate-def expr \\\")\\\"\\nin which expr is a syntax for expressions, which we will detail later. As an example, we look at the\\nde\\ufb01nition of several item property-related predicates.\\n(:derived (item-feature [return_type=vector[float32, 64]] ?o - item)\\n(??f (item-image ?o))\\n)\\nThis de\\ufb01nition de\\ufb01nes a predicate named item-feature . It takes an item as its argument, and returns\\na 64-dimensional vector embedding associated with the object. Its expression is (??f (item-image\\n?o)) . That is, an unknown mapping from the input image of the item to a vector embedding. This\\nde\\ufb01nition will implicitly de\\ufb01ne a function whose name is derived::item-feature::f . This\\nfunction has no default implementation. The programmer, after loading the PDSketch de\\ufb01nition,\\nshould register the corresponding implementation of this function. For example, one option is to\\nassociate this function with a convolutional neural network (CNN) that takes the image crop of each\\nobject ?oand computes the corresponding item feature. Note that, in this de\\ufb01nition, we are also\\nimplicitly de\\ufb01ning a parameter sharing strategy: for all objects in the environment (independent of\\nits absolute index), we will be applying the same CNN identically to them. We will detail possible\\nexpressions PDSketch supports later.\\nBased on this item-feature de\\ufb01nition, users can de\\ufb01ne several classi\\ufb01ers that will be useful in\\nde\\ufb01ning goal predicates.\\n(:derived (is-red ?o - item) (??f (item-feature ?o)))\\n(:derived (is-blue ?o - item) (??f (item-feature ?o)))\\n; ... more colors\\n(:derived (is-ball ?o - item) (??f (item-feature ?o)))\\n(:derived (is-door ?o - item) (??f (item-feature ?o)))\\n; ... more shapes\\n(:derived (is-open ?o - item) (??f (item-feature ?o))) ; for doors\\nBy default, when the return type is not speci\\ufb01ed, the function returns Boolean values, which is\\nconsistent with the original PDDL syntax.\\nIn BabyAI, items that the agent is holding has a special 2D position which is [-1, -1] . Thus,\\nwhether the agent is holding an object can be classi\\ufb01ed by\\n17 (:derived (robot-holding ?r - robot ?o - item)\\n(??f (item-pose ?o))\\n)\\nFinally, we want to de\\ufb01ne a function indicating whether the robot is facing the object,\\n(:derived (robot-facing ?r - robot ?o - item)\\n(??f (robot-pose ?r) (item-pose ?o))\\n)\\nIn its expression, we are de\\ufb01ning an unspeci\\ufb01ed function ??f, which takes two arguments, the robot\\nposition, and the item position. With all these predicates, we can now de\\ufb01ne the tasks. Recall that the\\nActionObjDoor environment has the following three tasks:\\ngo to a red box:\\n(exists (?o - item) (and\\n(robot-facing agent ?o) (is-red ?o) (is-box ?o)\\n))\\npick up a red box:\\n(exists (?o - item) (and\\n(robot-holding agent ?o) (is-red ?o) (is-box ?o)\\n))\\nopen a red door:\\n(exists (?o - item) (and\\n(is-red ?o) (is-door ?o) (is-open ?o)\\n))\\nIn the \\ufb01rst two goal speci\\ufb01cations, we are using a name constant \\u201cagent\\u201d to denote the robot that we\\nare considering.\\nNext, we are going to de\\ufb01ne actions. There are four actions the agent can perform. The syntax for\\naction de\\ufb01nition is:\\naction-def: \\\"(\\\" \\\":action\\\" (action-name kwargs)\\n\\\":parameter\\\" \\\"(\\\" variable-list \\\")\\\"\\n\\\":precondition\\\" expr\\n\\\":effect\\\" expr\\n\\\")\\\"\\nAn action de\\ufb01nition is composed of a name, a list of parameters, a precondition, and an effect. The\\nparameter are the arguments to the action. The precondition is a Boolean output value, indicating the\\n18 situation where this action can be executed. Note that, the precondition is not part of the transition\\nmodel of the environment. Instead, it\\u2019s a property that is associated with the underlying policy of\\nthis action. We will come back to this point when we discuss concrete examples. The effect is an\\nexpression that change the state variables according to certain rules.\\nConcretely, we start with a simple action: \\u201clturn,\\u201d which models the behavior that the robot turns left.\\n(:action lturn\\n:parameters (?r - robot)\\n:precondition (and )\\n:effect (assign (robot-direction ?r) (??f (robot-direction ?r)))\\n; could also be written as:\\n; :effect (robot-direction::assign ?r (??f (robot-direction ?r)))\\n)\\nThis de\\ufb01nition has de\\ufb01ned an action named \\u201clturn,\\u201d it takes a single parameter \\u201c?r\\u201d which indicates\\nthe robot that we want to control. It has an \\u201cempty\\u201d precondition, indicating that this primitive action\\ncan be executed anytime. Meanwhile, the effect is to change the \\u201crobot-direction\\u201d state variable\\nassociated with the robot ?r. The rule is a based on an (unspeci\\ufb01ed) function that takes the current\\nrobot facing direction and returns the new direction: (??f (robot-direction ?r)) .\\nWith \\ufb01rst-order logic formula, one can de\\ufb01ne more complex actions. One good example is the\\nde\\ufb01nition of the \\u201cforward\\u201d action. In BabyAI, the forward action takes the following rule. When\\nthere is no object facing the agent, the agent can move forward. When there is an object facing the\\nrobot, the robot may be blocked by the object, in which case the robot position will not change. Note\\nthat not all map items will block robots\\u2019 movement. Which type of objects will block the robot is to\\nbe learned by the learning algorithm. In this case, we can de\\ufb01ne the robot action in an intuitive way\\nas the following:\\n19 (:action forward\\n:parameters (?r - robot)\\n:precondition (and )\\n:effect (robot-pose::assign ?r (??f\\n(robot-pose ?r)\\n(robot-direction ?r)\\n(foreach (?o - item) ; enumerate all items in the map\\n(when (robot-facing ?r ?o) ; if the robot is facing this item\\n(item-feature ?o) ; consider the item-feature of ?o\\n)\\n)\\n))\\n)\\nWe now break down this de\\ufb01nition into pieces. First, this is an action that takes only one argument (the\\nrobot \\u201c?r\\u201d). Similar to the case of \\u201clturn,\\u201d this action does not have a precondition, indicating that the\\naction can be applied in any situations. The effect de\\ufb01nition for this action is slightly more complex.\\nFirst, the action changes the state variable (robot-pose ?r) . The new value is computed based on\\nthree things, the current robot-pose , the current robot-direction , and the feature of all objects\\n?othat the robot is facing. Here, we are using two special keywords, foreach andwhen . The \\ufb01rst\\none iterates over all items in the environment, and the second one selects the feature of object ?oif the\\ncondition is satis\\ufb01ed. Importantly, in this case, the corresponding function fshould be implemented\\nas a variable-length-input function. That is, depending on how many objects are selected, the number\\nof input features will change. In our implementation, this function is implemented as a graph neural\\nnetwork (GNN). Of course, if the programmer knows more about the underlying transition model,\\nthey can specify more detailed structure, such as the following:\\n; define a helper derived predicate.\\n(:derived (is-obstacle ?o - item) (??f (item-feature ?o)))\\n(:action forward-detail\\n:parameters (?r - robot)\\n:precondition (and )\\n:effect (when\\n; when there is no item ?o such that\\n; ?o is an obstacle and the robot is facing ?o\\n(not (exists (?o - item) (and (is-obstacle ?o) (robot-facing ?r ?o)) ))\\n; the robot pose will move forward and the new pose\\n; will be computed by the current pose and the facing direction.\\n(assign (robot-pose ?r)\\n(??f (robot-pose ?r) (robot-direction ?r))\\n)\\n)\\n)\\n20 Similarly, we can have the following de\\ufb01nition for \\u201cthe robot executes picking up.\\u201d\\n; define a helper predicate indicating whether the object\\n; can be picked up.\\n(:derived (can-pickup ?o) (??f (item-feature ?o)))\\n(:action pickup\\n:parameters (?r - robot)\\n:precondition (and )\\n:effect (foreach (?o - item)\\n(when (and (robot-facing ?r ?o) (can-pickup ?o) )\\n(assign (item-pose ?o)\\n; a dummy function, that should be implemented to return\\n; [-1, -1], indicating the item is in robot\\u2019s inventory.\\n(??f )\\n)\\n)\\n)\\n)\\nOverall, the de\\ufb01nition language allows users to encode various kinds of prior knowledge about the\\ntransition model into the representation. Meanwhile, it does not require human programmers to fully\\nspecify the details, especially the recognition and classi\\ufb01cation problems. The learning algorithm\\nwill follow the structure and learns the missing pieces.\\nImportantly, depend on the knowledge that the programmer has about the environment, such de\\ufb01nition\\ncan be very detailed, or very abstract. At an extreme case, we can have the following de\\ufb01nition:\\n(:action mysterious-action\\n:parameter (?r - robot)\\n:precondition (and )\\n:effect (and\\n(robot-pose::assign ?r (??f1\\n(robot-pose ?r) (robot-direction ?r) (item-pose ??) (item-feature ??)\\n))\\n(robot-direction::assign ?r (??f2\\n(robot-pose ?r) (robot-direction ?r) (item-pose ??) (item-feature ??)\\n))\\n(foreach (?o - item) (item-pose::assign ?o (??f3\\n(robot-pose ?r) (robot-direction ?r) (item-pose ??) (item-feature ??)\\n)))\\n(foreach (?o - item) (item-feature::assign ?o (??f4\\n(robot-pose ?r) (robot-direction ?r) (item-pose ??) (item-feature ??)\\n)))\\n)\\n)\\nHere we are using a syntax sugar: the notation (item-pose ??) is equivalent to (foreach (?x -\\nitem) (item-pose ?x)) . That is, this function takes the pose of all items into consideration. Note\\nthat, in this case, we have nostructure built in to the system: the action can change any state variable,\\nand the change depends on all other variables in the state representation. In this case, depending on\\nthe actual implementations of f1tof4, this transition model can be implemented in any form, such\\nas a graph neural network (thus an object-centric transition model).\\n21 Remark: Preconditions vs. conditional effects. There are two seemingly similar ways to de\\ufb01ne\\nthe condition under which an action takes effect: preconditions and conditional effects. To better\\nunderstand the different between these two options, consider the following two examples:\\n(:predicates\\n(p ?o - item)\\n(q ?o - item)\\n)\\n(:action op1\\n:parameters (?o - item)\\n:precondition (p ?o)\\n:effect (q:assign (??f (q ?o)))\\n)\\n(:action op2\\n:parameters (?o - item)\\n:precondition (and )\\n:effect (when\\n(p ?o)\\n(q:assign (??f (q ?o)))\\n)\\n)\\nThe key difference is that, a \\u201cprecondition\\u201d is the property associated with the corresponding low-level\\ncontroller for the action, that is \\u0019op1. Its semantics is that, the \\u0019op1can only be executed when (p\\n?o)returns true. It does not specify what will happen when we execute \\u0019op1in such a situation. In\\ncontrast, the semantics of the second de\\ufb01nition is that, the corresponding \\u0019op2can be executed under\\nany circumstances. However, only if the condition for the conditional effect is true (i.e., (p ?o) , the\\nstate value for qwill be changed.\\nThus, preconditions and conditional effects should be learned from different signals. Preconditions\\nshould be learned from the signal from the execution of the corresponding controller. More specif-\\nically, the controller \\u0019op1should return a Boolean value indicating whether the controller can be\\nexecuted. By contrast, the conditional effects should be learned from the outcome of the execution of\\nthe controller \\u0019op2.\\nNote that, the conditional effect formulation is the one that \\ufb01ts most of the concurrent environment\\nsetups for reinforcement learning. That is, the de\\ufb01ned actions correspond to the primitive actions\\nthat the robot can execute. Thus, usually all actions are always applicable at any state (i.e., the\\nprecondition of all actions are empty).\\n22 B.2 Expressions\\nPDSketch has the following built-in operations.\\nPropositional logic operations: and,or,not, and implies .The propositional logic operations\\ntake the following syntax:\\nand-expr-def: \\\"(\\\" \\\"and\\\" expr* \\\")\\\"\\nor-expr-def: \\\"(\\\" \\\"or\\\" expr* \\\")\\\"\\nnot-expr-def: \\\"(\\\" \\\"not\\\" expr \\\")\\\"\\nimplies-expr-def: \\\"(\\\" \\\"implies\\\" expr expr \\\")\\\"\\nThe semantics of these operators follows the convention of Boolean operations: conjunction,\\ndisjunction, negation, and implication. Their implementation are based on the differentiable\\nG\\u00f6del t-norms: not(p) = 1\\u0000p,and(p1;p2) = min(p1;p2),or(p1;p2) = max(p1;p2), and\\nimplies (p1;p2) = max(1\\u0000p1;p2).\\nThere are two conventional notations for Boolean operators used in the de\\ufb01nition of action effects.\\nFirst, when an action has multiple effects, they will be written within a big andblock. For example,\\nthe following de\\ufb01nition:\\n(:action set-pqr\\n:parameters (?r - robot)\\n:precondition (and )\\n:effect (and\\n(assign (p ?r) (??f))\\n(assign (q ?r) (??f))\\n(assign (r ?r) (??f))\\n)\\n)\\nMeanwhile, when a Boolean predicate is directly written in the effect, its semantics is that this\\nBoolean state variable will be set to true. Similarly, when we write (not (t ?r)) where tis a\\nBoolean predicate, it is equivalent to (assign (t ?r) true) .\\nBoolean quanti\\ufb01cation: forall andexists .Similarly, the quanti\\ufb01cation operations are de\\ufb01ne\\nas the following:\\nforall-expr-def: \\\"(\\\" \\\"forall\\\" (typed-variable) expr \\\")\\\"\\nexists-expr-def: \\\"(\\\" \\\"exists\\\" (typed-variable) expr \\\")\\\"\\nAs an example, the following expression:\\n(exists (?o - item) (is-red ?o))\\ncomputes whether there is an item in the environment that is red.\\nSimilarly, the implementation of Boolean quanti\\ufb01ers are: 8x:p(x) = minxp(x)and9x:p(x) =\\nmaxxp(x).\\n23 Assignment. There is only one assignment operation\\nassign-expr-def: \\\"(\\\" \\\"assign\\\" \\\"(\\\" predicate-name variable-list \\\")\\\" expr \\\")\\\"\\nThe \\ufb01rst term is the target state variable, while the second term is the expression of the value. For\\nexample, (assign (p ?r) (??f)) means that the value of (p ?r) will be assigned to the return\\nvalue of function f.\\nForeach and condition. There are two special operators: for-each and condition.\\nforeach-expr-def: \\\"(\\\" \\\"foreach\\\" (typed-variable) expr \\\")\\\"\\nwhen-expr-def: \\\"(\\\" \\\"when\\\" expr expr \\\")\\\"\\nWhen they are used in a precondition or an expression (in contrast to being under the effect de\\ufb01nition),\\nthey means:\\n\\u2022foreach : it selects all objects of the speci\\ufb01ed type. See the previous de\\ufb01nition of action\\nforward as an example.\\n\\u2022when : it speci\\ufb01es the scenario in which a state variable is relevant to the computation. See\\nthe previous de\\ufb01nition of action forward as an example.\\nWhen they are used in effect de\\ufb01nitions, they means:\\n\\u2022foreach : it applies the effect formula to objects of the speci\\ufb01ed type. See the previous\\nde\\ufb01nition of action mysterious-action as an example.\\n\\u2022when : it indicates a conditional effect. The effect will be applied if and only if the condition\\nis true. See the previous de\\ufb01nition of action mysterious-action as an example.\\nSyntax sugars. PDSketch includes the following syntax sugars:\\n(p::assign ?a1 ?a2 ... VALUE) is equivalent to\\n(assign (p ?a1 ?a2 ...) VALUE) .\\n(p::cond-assign ?a1 ?a2 ... COND VALUE) is equivalent to\\n(when COND (assign (p ?a1 ?a2 ...) VALUE)) .\\n(p::cond-select ?a1 ?a2 ... COND) is equivalent to\\n(when COND (p ?a1 ?a2 ...)) .\\nBlank notation ??. The??operation can be used to replace any expressions. It takes the following\\nsyntax:\\nslot-expr-def: \\\"(\\\" \\\"??\\\" predicate-name kwargs expr* \\\")\\\"\\nFor example, the de\\ufb01nition: (??f (item-feature ?o)) de\\ufb01nes an function that will be externally\\nimplemented. It has name f. It takes only one input, which is (item-feature ?o) .\\n24 inc-p(1)inc-q(1)inc-p(2)inc-q(2)(:actioninc-p:parameters(?o-item):effect(p::assign ?o(add (p ?o)1)))(:actioninc-q:parameters(?o-item):effect(q::assign ?o(add (q ?o) 1)))(:actionset-r:parameters(?o-item):effect(r::assign ?o(mul(p ?o) (q ?o))))\\ud835\\udc5d!=1\\ud835\\udc5e!=1\\nInitialState\\ud835\\udc5d!=2\\ud835\\udc5e!=2\\ud835\\udc5d\\\"=1\\ud835\\udc5e\\\"=1\\ud835\\udc5d\\\"=2\\ud835\\udc5e\\\"=2inc-p(1)inc-q(1)inc-p(2)inc-q(2)\\ud835\\udc5f!=1\\ud835\\udc5f\\\"=1\\ud835\\udc5d!=1\\ud835\\udc5e!=1\\ud835\\udc5d\\\"=1\\ud835\\udc5e\\\"=1\\ud835\\udc5f!=1\\ud835\\udc5f\\\"=1\\ud835\\udc5d!=2\\ud835\\udc5e!=2\\ud835\\udc5d\\\"=2\\ud835\\udc5e\\\"=2\\ud835\\udc5d!=1\\ud835\\udc5e!=1\\ud835\\udc5d\\\"=1\\ud835\\udc5e\\\"=1\\ud835\\udc5f!=1\\ud835\\udc5f\\\"=1\\ud835\\udc5d!=3\\ud835\\udc5e!=3\\ud835\\udc5d\\\"=3\\ud835\\udc5e\\\"=3\\ud835\\udc5f!=4set-r(1)\\ud835\\udc5f\\\"=4set-r(2)......Goal: (foreach?o(eq(r?o) 4)Figure 11: A graphical illustration of the hFF heuristics. The goal is to set the rvalue of all objects\\nto 4. The heuristic value hFF = 6, which is the number of selected operators (highlighted in red).\\nC Domain-Independent Heuristic\\nIn this paper, we focus on the hFF heuristic. Based on relaxed operators, it heuristically selects a\\nset of operators that should be applied to accomplish the goal. Shown in Fig. 11, hFF sequentially\\ntries to apply operators that yield novel values of the features. Once the goal condition is satis\\ufb01ed,\\nit back-traces the used operators. Note that such computation is general and applies to domains\\nwith arbitrarily complex feature dependencies and number of objects. We present details of our\\nimplementation in the supplementary. However, such heuristics does not naturally work with vector\\nembeddings and neural network predictors. Next, we talk about how to \\u201ccompile\\u201d trained neural\\nfeatures and modules into hFF-compatible representations.\\nIn this paper, we explore two strategies for performing this reduction. In the \\ufb01rst, we simply change\\nall the action models so that the result is to set all of the predicate instances that they effect to have a\\nspecial value called optimistic , which is assumed to satisfy any goal test on that predicate instance.\\n(So, for example, after we have placed an object somewhere, we immediately \\\"believe\\\" it could be\\nanywhere.) An alternative strategy, which is more computationally complex but yields \\\"tighter\\\" (and\\ntherefore more helpful) heuristic values, is to explicitly discretize the value space of the predicates\\nand directly reduce to the standard discrete case. Both of these strategies are described in more\\ndetail in the supplementary material. It is important to note that these methods are inherently lossy\\non non-Boolean predicate values, but since they are only used for search guidance, and not in the\\nforward simulation of the actual actions during planning, they do not affect the correctness of the\\noverall algorithm.\\nOptimistic compilation: leveraging locality. The \\ufb01rst approach, optimistic compilation (OPT),\\ncompiles each non-Boolean state variable, e.g., (color ?o) , into a boolean predicate color-opt .\\nAny operator that changes the value of color (e.g., press-button ) will set color-opt to true.\\nMeanwhile, any Boolean expressions, such as (?? (color item#1)) will return true as long as\\n(color-opt item#1) is true. Intuitively, any operator that changes feature values will make the\\nchange optimistically .\\nThe optimistic compilation ignores the \\ufb01ne-grained structure inside each state variables, but does\\nleverage the object-level locality. For example, in the case of (forall ?o (eq (r ?o) 4) , it will\\nselect two operators set-r(1) andset-r(2) , leading to hFF = 2.\\nAnd-Or compilation. The And-Or compilation (AO) works in two steps. First, it discretizes all\\ncontinuous state variables (poses, etc.) into a designated number of bins (e.g., 128). Next, for each\\nexternally-de\\ufb01ned functions, it learns a \\ufb01rst-order decision tree to approximate the computation\\n(e.g., to approximate a neural network). Concretely, we use VQV AE [Van Den Oord et al., 2017]\\nto discretize the feature vectors. For all predicates whose output is a latent embedding, e.g., the\\n(item-feature ?o) we introduced in our earlier example, we add a vector quantization layer after\\nthe encoding layers. We initialize the quantized embeddings by running a K-means clustering over (item-feature ?o) we introduced in our earlier example, we add a vector quantization layer after\\nthe encoding layers. We initialize the quantized embeddings by running a K-means clustering over\\nthe item feature embedding from a small dataset, and \\ufb01netune the weights on the entire training\\ndataset for one epoch.\\nFor all predicates whose inputs and outputs are both continuous parameters. We use the FOIL\\nalgorithm, the \\ufb01rst-order decision tree learning algorithm [Quinlan, 1990] to extract \\ufb01rst-order\\ndecision rules. We need to use FOIL instead of a plain (propositional) decision tree learning algorithm\\n25 because the input to decision functions may have a variable number of objects. Here we are showing\\nsome examples of the quantized predicates:\\n(is-red ?o - item) = (or\\n(item-feature@3 ?o)\\n(item-feature@14 ?o)\\n(item-feature@16 ?o)\\n(item-feature@10 ?o)\\n)\\nBelow we are showing a slightly more complex example, corresponding to the learned rule for action\\n\\u201cforward.\\u201d\\n26 (((robot-pose ?r - robot) <- (SAS\\n37 <- (or\\n(and\\n(not (robot-direction@2 ?r - robot))\\n(not (robot-direction@3 ?r - robot))\\n(not (robot-direction@1 ?r - robot))\\n(robot-pose@37 ?r - robot)\\n)\\n(and (robot-direction@3 ?r - robot) (robot-pose@45 ?r - robot))\\n(and (robot-pose@36 ?r - robot) (robot-direction@0 ?r - robot))\\n(and (robot-pose@29 ?r - robot) (robot-direction@1 ?r - robot))\\n)\\n4 <- (and (robot-pose@12 ?r - robot) (robot-direction@3 ?r - robot))\\n40 <- (and (robot-pose@41 ?r - robot) (robot-direction@2 ?r - robot))\\n21 <- (and (robot-pose@29 ?r - robot) (robot-direction@3 ?r - robot))\\n32 <- (and (robot-pose@33 ?r - robot) (robot-direction@2 ?r - robot))\\n3 <- (and\\n(robot-pose@11 ?r - robot) (robot-direction@3 ?r - robot))\\n(and\\n(not (robot-pose@27 ?r - robot))\\n(not (exists (_t0 - item) (and (robot-is-facing ?r - robot _t0 - item)\\n(item-feature@9 _t0 - item))))\\n(not (exists (_t0 - item) (and (robot-is-facing ?r - robot _t0 - item)\\n(item-feature@0 _t0 - item))))\\n(not (exists (_t0 - item) (and (robot-is-facing ?r - robot _t0 - item)\\n(item-feature@6 _t0 - item))))\\n(not (exists (_t0 - item) (and (robot-is-facing ?r - robot _t0 - item)\\n(item-feature@11 _t0 - item))))\\n; ... more rules omitted.\\nNote in the last section of the shown rules, the decision rule is written in \\ufb01rst-order logic, where we\\nneed to quantify over objects.\\nThe AO compilation further leverages the \\ufb01ne-grained structures of state variables. For example, the\\nreal value of features p,q, and rwill be discretized into bins, such as p=1,p=2,q=1,q=2. A decision\\ntree will be used to represent the transition model. Thus, the resulting model will support \\ufb01ne-grained\\nsimulation and back-tracing.\\nThe role of sparsity and locality structures in heuristics. Although the discretization and heuris-\\ntic computation itself does not assume any particular structure of the action de\\ufb01nitions, their perfor-\\nmance does rely on the programmed sparsity and locality structures, in particular, the \\u201cfactorization\\u201d\\nstructure of the feature representation.\\nSpeci\\ufb01cally, without factorization, the entire state is described with one single vector embedding.\\nThis introduces a exponential number of possible states of the state. For example, let\\u2019s assume an\\nobject state can be factorized into its x and y position (7 possible states per dimension), the color (6\\npossible states), and the shape (4 possible shapes). With factorization, we only need 24 possible state\\ncodes to represent each object. However, without factorization, we need 7\\u00027\\u00026\\u00024 = 1176 states.\\nRecall that, during the computation of heuristics, we will be solving a relaxed version of the planning\\nprogram. If there is no factorization, the search problem reduces to a plain path-\\ufb01nding problem in\\nthe graph induced by the connectivity between these states, which is very inef\\ufb01cient to solve.\\n27 D Experimental Setup.\\nIn this section, we detail the environment setup and domain de\\ufb01nition of two environments: BabyAI\\nand Painting Factory.\\nD.1 BabyAI\\nSetup. Following the original BabyAI setup, we use a 7 by 7 grid as the physical world. The agent\\nis initially positioned at the center, all other object locations are randomly selected.\\nOf\\ufb02ine data. The data contains both successful demonstrations and unsuccessful demonstrations.\\nFor successful ones, we use the grid-world A\\u0003search to generate the optimal trajectory from the\\noriginal agent position to the target. For unsuccessful demonstrations, we \\ufb01rst choose an incorrect\\nobject, approach the selected object, and then runs 5 steps of random walk.\\nBaselines. For all baselines and our models, we use the following ways to encode object states,\\nrobot positions, and facing directions.\\n1.object state: following the FiLM model presented in Chevalier-Boisvert et al. [2019], we\\nuse an integer embedding for object states.\\n2.object position and robot position: we directly use the raw value as the input to the neural\\nnetworks. We have tried to use embedding based methods to encode the values, but have\\nseen consistently worse performance.\\n3.robot facing direction: there are four directions. We use a 4 learnable embeddings for them.\\nThus, the input to the baselines are: 1) the robot state embedding, 2) the task goal speci\\ufb01cation,\\nencoded as the concatenation of three word embeddings: the verb, the adjective, and the noun, and 3)\\nthe object state embeddings.\\nFor both baselines BC and DT, we use a two-layer graph neural network to encode the world state\\ninto a 128-dimensional vector embedding. For BC, we use a single linear layer to predict the action to\\ntake. For DT, we use a transformer layer to aggregate the history (especially encoding the cost-to-go),\\nand use another linear layer to predict the action in the next take.\\nModels. For models written in PDSketch, we have three variants.\\n1.In PDS-Base, the robot state encodings are treated as a single vector, named\\nrobot-feature , and the item state encodings are also treated as a single vector (which is\\nthe concatenation of their positions and visual features). The action de\\ufb01nition encodes no\\nstructures. For example,\\n(:action forward\\n:parameters (?r - robot)\\n:precondition (and )\\n:effect (and\\n(robot-feature::assign ?r (??f (robot-feature ?r) (item-feature ??)))\\n)\\n)\\n28 2.In PDS-Abs, we have the abstraction of the idea \\u201crobot-facing.\\u201d This predicate is used to\\nde\\ufb01ne actions. For example,\\n(:action forward\\n:parameters (?r - robot)\\n:precondition (and )\\n:effect (robot-pose::cond-assign ?r\\n(??f\\n(robot-pose ?r) (robot-direction-feature ?r)\\n(foreach (?o - item)\\n(item-feature::cond-select ?o (robot-is-facing ?r ?o))\\n)\\n)\\n(??g (robot-pose ?r) (robot-direction-feature ?r))\\n)\\n)\\n3. In PDS-Rob, the robot\\u2019s movements have been encoded into the de\\ufb01nition. For example,\\n; The first function returns the 2D position that the robot is facing\\n; when the robot is at ?p and facing ?d.\\n; This function has been implemented externally.\\n; The second function returns whether an object is an obstacle (so\\n; it will block the robot\\u2019s movement.\\n; This information has also been given to the agent as a input.\\n(:predicates\\n(facing [return_type=pose] ?p - pose ?d - direction)\\n(is-obstacle ?o - item)\\n)\\n(:derived (_is-facing ?p - pose ?d - direction ?t - pose)\\n(equal (facing ?p ?d) ?t)\\n)\\n(:derived (_robot-facing [return_type=pose] ?r - robot)\\n(facing (robot-pose ?r) (robot-direction ?r))\\n)\\n(:derived (_robot-is-facing ?r - robot ?o - item)\\n(_is-facing (robot-pose ?r) (robot-direction ?r) (item-pose ?o))\\n)\\n(:derived (_robot-facing-clear ?r - robot)\\n(not (exists (?o - item) (and\\n(_robot-is-facing ?r ?o)\\n(is-obstacle ?o)\\n))\\n)\\n(:action forward\\n:parameters (?r - robot)\\n:precondition (_robot-facing-clear ?r)\\n:effect (and (robot-pose::assign ?r (_robot-facing ?r)) )\\n)\\nD.2 Painting Factory\\nState space. The state space in the painting factory is composed of a list of objects, and a list\\nof containers. Both objects and containers are represented as a tuple of a 3D xyz location, and a\\nimage crop. The image crop is generated by \\ufb01rst computing the 2D bounding box of the object in\\nthe camera plain, cropping out the image patch, and resizing it to 32 by 32. The image encoder is a\\n3-layer convolutional neural network followed by a linear transformation layer into a 128-dimensional\\nembedding. The xyz location are represented as three numbers, normalized into the range [0;1].\\n29 Action space. The action space in the painting factory contain two actions. We \\ufb01rst show the\\nPDSketch de\\ufb01nition of two actions:\\n(:action move-into\\n:parameters (?o - item ?c - container)\\n:precondition (and )\\n:effect (and\\n(item-pose::assign ?o (container-pose ?c))\\n(item-feature::cond-assign ?o\\n(??g (container-feature ?c)) ; when the container is a bowl,\\n(??h (container-feature ?c)) ; paint ?o to be the same color as ?c.\\n)\\n)\\n)\\n(:action move-to\\n:parameters (?o - item ?p - pose)\\n:precondition (and )\\n:effect (and (item-pose::assign ?o ?p))\\n)\\nThe \\ufb01rst action, move-into is an action de\\ufb01ned for each pair of item and container. It changes the\\npose of the item (now the item will be inside the container). When the item is placed into a bowl,\\nit will be painted into the same color as the bowl. The second action moves the object directly to a\\ndesignated position.\\nOf\\ufb02ine data. Recall that our task is to place two painted objects in the target location to form a\\ndesignated relationship. To generate demonstrations, we \\ufb01rst randomly select two blocks. Next, we\\npaint one of them and place it at the center of the target location. Then, we paint the second object to\\nthe designated color and place it with respect to the \\ufb01rst object.\\nBaselines. For all baselines, we use a two-layer graph neural network (GNN) to encode the objects.\\nWe generate the action in two steps. First, for each (block, container) pair, we generate a scalar logit\\nindicating the score of placing the block into the container. For each block, we also generate a scalar\\nlogit indicating the score of directly moving the block. We normalize these logits with a softmax to\\nformulate the action probability. If the chosen action is to directly move the block, we also predict\\nthe target location.\\nModels. The key feature in the painting factory domain is that now we need to plan for continuous\\nactions, instead of a discrete space. This is handled using a basic task and motion planning strategy\\nintroduced in PDDLStream [Garrett et al., 2020]. We do this in two steps.\\nFirst, during training, we build a structured model for classifying geometric relationships (on, left,\\nand right) in the following form:\\np(aonb) =\\u001b\\u0012kpose(a)\\u0000pose(b)\\u0000\\u0012k2\\n2\\u0000\\r\\n\\u001c\\u0013\\n;\\nwhere\\u0012,\\r,\\u001care learnable parameters. \\u001bis the Sigmoid function. That is, intuitively, we say the\\nblockais on blockbif the pose of ais close to pose(b) +\\u0012. This formulation allows us to \\u201csample\\u201d\\nposes of block agiven the pose of block b.\\nThus, during performance time, based on the initial pose of all objects, we sample a set of new poses\\nthat are on, left to, and right to the initial poses. We then use these poses to ground concrete operators.\\nThis strategy is also called \\u201cincremental\\u201d sampling in sampling-based task and motion planning\\nliterature.\\n30\",\"34\":\" Learning Rational Subgoals from Demonstrations and Instructions\\nZhezheng Luo*1, Jiayuan Mao*1, Jiajun Wu2,\\nTom \\u00b4as Lozano-P \\u00b4erez1, Joshua B. Tenenbaum1, Leslie Pack Kaelbling1\\n1Massachusetts Institute of Technology2Stanford University\\nAbstract\\nWe present a framework for learning useful subgoals that sup-\\nport ef\\ufb01cient long-term planning to achieve novel goals. At\\nthe core of our framework is a collection of rational subgoals\\n(RSGs), which are essentially binary classi\\ufb01ers over the envi-\\nronmental states. RSGs can be learned from weakly-annotated\\ndata, in the form of unsegmented demonstration trajectories,\\npaired with abstract task descriptions, which are composed of\\nterms initially unknown to the agent (e.g., collect-wood then\\ncraft-boat then go-across-river ). Our framework also discov-\\ners dependencies between RSGs, e.g., the task collect-wood\\nis a helpful subgoal for the task craft-boat . Given a goal de-\\nscription, the learned subgoals and the derived dependencies\\nfacilitate off-the-shelf planning algorithms, such as A\\u0003and\\nRRT, by setting helpful subgoals as waypoints to the plan-\\nner, which signi\\ufb01cantly improves performance-time ef\\ufb01ciency.\\nProject page: https:\\/\\/rsg.csail.mit.edu\\nIntroduction\\nBeing able to decompose complex tasks into subgoals is crit-\\nical for ef\\ufb01cient long-term planning. Consider the example\\nin Fig. 1: planning to craft a boat from scratch is hard, as it\\nrequires a long-term plan going from collecting materials to\\ncrafting boats, but it can be made easier if we know that hav-\\ning an axe andhaving wood are useful sub-goals. Planning\\nhierarchically with these subgoals can substantially reduce\\nthe search required. It is also helpful to understand the tem-\\nporal dependencies between these subgoals, such as having\\nwood being a useful subgoal to achieve prior to crafting boat\\nmakes long-term planning much more ef\\ufb01cient.\\nIn this work, we propose Rational Subgoals (RSGs), a\\nframework for learning useful subgoals and their temporal\\ndependencies from demonstrations. Our system learns with\\nvery weak supervision, in the form of a small number of un-\\nsegmented demonstrations of complex behaviors paired with\\nabstract task descriptions. The descriptions are composed of\\nterms that are initially unknown to the agent, much as an\\nadult might narrate the high-level steps when demonstrating\\na cooking recipe to a child. These action terms indicate im-\\nportant subgoals in the action sequence, and our agent learns\\n*These authors contributed equally.\\nCopyright \\u00a92023, Association for the Advancement of Arti\\ufb01cial\\nIntelligence (www.aaai.org). All rights reserved.to detect when these subgoals are true in the world, infer their\\ntemporal dependencies, and leverage them to plan ef\\ufb01ciently.\\nIllustrated in Fig. 1, our model learns from a dataset of\\npaired but unaligned low-level state-action sequences and\\nthe corresponding abstract task description ( collect-wood\\nthen craft-boat then go-across-river ). For each action term\\no(e.g., collect-wood ), our model learns a goal condition Go,\\nwhich maps any state to a binary random variable, indicat-\\ning whether the state satis\\ufb01es the goal condition. Given the\\ntraining data, we decompose the observed trajectory into frag-\\nments, each of which corresponds to a \\u201crational\\u201d sequence\\nof actions for achieving a subgoal in the description.\\nWhile this model-based approach enables great general-\\nity in generating behaviors, it suffers from the slow online\\ncomputation. To speed up online planning, we compute a de-\\npendency matrix whose entries encode which subgoals might\\nbe helpful to achieve before accomplishing another subgoal\\n(e.g., having wood is a helpful subgoal for the task crafting\\nboat, and thus the entry ( having wood ,crafting boat ) will\\nhave a higher weight). During test time, given a \\ufb01nal goal\\n(e.g., craft boat ) and the initial state, a hierarchical search\\nalgorithm is applied at both the subgoal level and the lower,\\nenvironmental-action level. (e.g., craft boat ) and the initial state, a hierarchical search\\nalgorithm is applied at both the subgoal level and the lower,\\nenvironmental-action level.\\nThe explicit learning of subgoals and their dependency\\nstructures brings two important advantages. First, the sub-\\ngoal dependency allows us to explicitly set helpful subgoals\\nas waypoints for planners. This signi\\ufb01cantly improves their\\nruntime ef\\ufb01ciency. Second, compared to alternative subgoal\\nparameterizations such as reward functions, subgoals in the\\nform of a state classi\\ufb01er allows us to use simple and ef\\ufb01cient\\nplanners. For example, in continuous spaces, we can use\\nRapidly-exploring Random Trees ( RRT) to search for plans\\nin the robot con\\ufb01guration space. These planers do not require\\ntraining and generalize immediately to novel environments.\\nWe evaluate RSGs in Crafting World (Chen, Gupta, and\\nMarino 2021), an image-based grid-world domain with a\\nrich set of object crafting tasks, and Playroom (Konidaris,\\nKaelbling, and Lozano-Perez 2018), a 2D continuous domain\\nwith geometric constraints. Our evaluation shows that our\\nmodel clearly outperforms baselines on planning tasks where\\nthe agent needs to generate trajectories to accomplish a given\\ntask. Another important application of RSGs is to create a\\nlanguage interface for human-robot communication, which\\nincludes robots interpreting human actions and humans in-arXiv:2303.05487v1  [cs.AI]  9 Mar 2023 MoveRightMoveDownCollectMoveLeftCraft\\nSubgoal1:collect-woodSubgoal2:craft-boat()*+!=,-.\\/'()*+!=,-.\\/'()*+!=,-.\\/'()*+!=#%&'()*+\\\"=,-.\\/'()*+\\\"=#%&'()*+\\\"=,-.\\/'Subgoal1:collect-wood=%&'(!Subgoal2:craft-boat=%&'(\\\"(a)(b)Figure 1: Interpreting a demonstration and its description in terms of RSGs: (a) Each RSG is represented as a subgoal Go. (b)\\nThe system infers a transition to the next subgoal if the Gcondition is satis\\ufb01ed. Such transition rules can be used to interpret\\ndemonstrations and to plan for tasks that require multiple steps to achieve.\\nstructing robots by specifying a sequence of subgoals. Our\\nmodel enables compositional generalization through \\ufb02exible\\nre-composition of learned subgoals, which allows the robot\\nto interpret and execute novel instructions.\\nRational Subgoal Learning and Planning\\nWe focus on learning rational subgoals from demonstration\\ndata and leveraging them for planning. Formally, our training\\ndata is a collection of paired unsegmented demonstrations\\n(i.e., state and action sequences) and abstract descriptions\\n(e.g., collect-wood then craft-boat ) composed of action terms\\n(collect-wood , etc.) and connectives ( then,or). Our ultimate\\ngoal is to recover the grounding (i.e., the corresponding sub-\\ngoal speci\\ufb01ed by the action term) for each individual action\\nterm. These subgoals will be leveraged by planning algo-\\nrithms to solve long-horizon planning problems.\\nWe begin this section with basic de\\ufb01nitions of the rational\\nsubgoal representations and the language TLfor abstract de-\\nscriptions. Second, we outline the planning algorithm we use\\nto re\\ufb01ne high-level instructions in TLinto environmental ac-\\ntions that agents can execute, given the RSGs. Although any\\nsearch algorithms or Markov Decision Process (MDP) solvers\\nare in principle applicable for our planning task, in this paper,\\nwe have focused on a simple extension to the A* algorithm.\\nNext, we present the algorithm we use to learn RSGs from\\ndata. Since we are working with unsegmented trajectories,\\nthe learning algorithm has two steps. It \\ufb01rst computes a ra-\\ntionality score for individual actions in the trajectory based\\non the optimal plan derived from the A* algorithm. Then,\\nit uses a dynamic programming algorithm to \\ufb01nd the best\\nsegmentation of the trajectory and updates the parameters.\\nFinally, we describe a dependency discovery algorithm for\\nRSGs and apply it to solve planning tasks given only a single\\ngoal action term (e.g., collect-gold ), in contrast to the earlier\\ncase where there are detailed step-by-step instructions.\\nWe call our representation rational subgoals because our\\nlearning algorithm is based on a rationality objective with\\n-(a))=+(b))!=,-.\\/01(d))=,2031(c))=,-.\\/01456-34561-1-127#7$7#7$7#7$-34561-17#13456-1-7$Figure 2: Illustrative example of how \\ufb01nite state machines\\n(FSM) are constructed from task descriptions. The super-\\nstarting node v0and the super-terminal node vTare high-\\nlighted.\\nrespect to demonstration trajectories, and our planning algo-\\nrithm chooses rational subgoals to accelerate the search.\\nFormally, a rational subgoal (RSG) is a classi\\ufb01er that\\nmaps an environmental state sto a Boolean value, indicat-\\ning whether the goal condition is satis\\ufb01ed at s. Each RSG\\nhas an atomic name o(e.g., collect-wood ), and the corre-\\nsponding goal classi\\ufb01er is denoted by Go. Depending on the\\nrepresentation of states, Gocan take various forms of neural\\nnetworks, such as convolutional neural networks (CNNs) for\\nimage-based state representations.\\nIn both learning and planning, we will be using an ab-\\nstract language to describe tasks, such as collect-wood then\\ncraft-boat . These descriptions are written in a formal task lan-\\nguageTL. Syntactically, all atomic subgoals are in TL; and\\nfor allt1;t22TL ,(t1thent2),(t1ort2), and (t1andt2)\\nare inTL. Semantically, a state sequence \\u0016ssatis\\ufb01es a task\\ndescriptiont, written \\u0016sj=twhen:\\n\\u2022Iftis aRSGo, then the \\ufb01rst state does not satisfy Go,\\nand the last state satis\\ufb01es Go. Note that this implies that descriptiont, written \\u0016sj=twhen:\\n\\u2022Iftis aRSGo, then the \\ufb01rst state does not satisfy Go,\\nand the last state satis\\ufb01es Go. Note that this implies that\\nthe sequence \\u0016smust have at least 2states.\\n\\u2022Ift= (t1thent2)then90< j < n such that\\n(s1;:::;sj)j=t1and(sj;:::;sn)j=t2: taskt1should\\nbe accomplished before t2.\\n\\u2022Ift= (t1ort2)then\\u0016sj=t1or\\u0016sj=t2: the agent should\\neither complete t1ort2. \\u2022Ift= (t1andt2)then \\u0016sj= (t1thent2)or\\u0016sj=\\n(t2thent1): the agent should complete both t1andt2,\\nbut in any order ( t1\\ufb01rst ort2\\ufb01rst) *.\\nNote that the relation \\u0016sj=tonly speci\\ufb01es whether \\u0016scom-\\npletestbut not how optimal \\u0016sis. Later on, when we de\\ufb01ne\\nthe planning problem, we will introduce the trajectory cost.\\nEach task description t2TL can be represented with a\\nnon-deterministic \\ufb01nite state machine ( FSM), representing\\nthe sequential and branching structures. Each FSMtis a\\ntuple (Vt;Et;VIt;VGt)which are subgoal nodes, edges, set\\nof possible starting nodes and set of terminal nodes. Each\\nnode corresponds to an action term in the description, and\\neach edge corresponds to a possible transition of changing\\nsubgoals. Fig. 2 illustrates the constructions for syntax in TL,\\nand we provide the follow algorithm for the construction.\\n\\u2022Single subgoal: A single subgoal sis corresponding FSM\\nwith a single node i.e. VIt=VGt=Vt=fsg, and\\nEt=;.\\n\\u2022t1thent2: We merge FSMt1and FSMt2by merging their\\nsubgoal nodes, edges and using VIt1as the new starting\\nnode set and VGt2as the new terminal node set. Then, we\\nadd all edges from VGt1toVIt2. Formally,\\nFSMt1thent2=\\n(Vt1[Vt2;Et1[Et2[(VGt1\\u0002VIt2);VIt1;VGt2);\\nwhere\\u0002indicates the Cartesian product, meaning that\\neach terminal node of FSMt1can transit to any starting\\nnode of FSMt2.\\n\\u2022t1or\\u0001\\u0001\\u0001ortn: Simply merge nFSMs without adding\\nany new edges. Formally,\\nFSMt1or\\u0001\\u0001\\u0001ortn= ([\\niVti;[\\niEti;[\\niVIti;[\\niVGti)\\n\\u2022t1and\\u0001\\u0001\\u0001andtn: Build 2n\\u00001nsub-FSMs over nlay-\\ners: thei-th layer contains n\\u0001\\u0000n\\u00001\\ni\\u00001\\u0001\\nsub-FSMs each la-\\nbeled by (s;D)wheresis the current subgoal to complete\\n(so this sub-FSM is a copy of FSMs), andDis the set\\nof subgoals that have been previously completed. Then\\nfor a sub-FSM (s1;D1)and a sub-FSM (s2;D2)in the\\nnext layer, if D2=D1[fs1g, we add all edges from ter-\\nminal nodes of the \\ufb01rst sub-FSM to starting nodes of the\\nsecond sub-FSM. After building layers of sub-FSMs and\\nconnecting them, we set the starting nodes to be the union\\nof starting nodes in the \\ufb01rst layer and terminal nodes to\\nbe the union of terminal nodes in the last layer.\\nNote that our framework requires the starting and terminal\\nnodes to be unique, but the construction above may output a\\nFSM with multiple starting\\/terminal nodes, so we introduce\\nthe virual super starting node v0and terminal node vTto\\nunify them.\\n*The operator and can be generalized be n-ary. In this case,\\naccomplishing them in any order is considered accomplishing the\\ncomposed task. For example, the task mine-wood and mine-gold\\nand mine-coal allows the agent to accomplish all three subgoals\\nin any order. Note that this is different from the speci\\ufb01cation with\\nparenthesis: (mine-wood and mine-gold) and mine-coal .\\nSkill1(!!):mine-goldSkill2(!\\\"):craft-boat\\n!7iscompleted.!8iscompleted.Theagentmaymakesomeprogress(minewoodfortheboat)towards!!evenif!\\\"hasnotbeencompletedyet.Task:!\\\"\\\"#$%!#Completing!!Completing!\\\"Completing!!Figure 3: An example of optimal interleaving subgoals: s1\\nis \\u201dmine gold\\u201d, and s2is \\u201dcraft boat\\u201d. It is valid that the\\nagent \\ufb01rst goes to collect wood (for accompolishing s2), and\\nthen mine gold (for accompolishing s1), and \\ufb01nally crafts\\nboat. In this case, the action sequences for completing s1and\\ns2are interleaved. However, they can are be recognized as\\ns2thens2becauses1is accomplished before s2.\\nRemark. In this paper, the language TLused for describ-\\ning tasks covers LTL f, a \\ufb01nite fragment of LTL that does not\\ncontain the always quanti\\ufb01er, so our fragment does not model\\ntask speci\\ufb01cations that contain in\\ufb01nite loops. Finite LTL for-\\nmulae can be converted to a \\ufb01nite automaton (De Giacomo\\nand Vardi 2013), represented using the FSM.\\nExecution steps for different subgoals can interleave.\\nRSGs does not simply run optimal policy for each individual\\nsubgoal sequentially. Rather, the semantic of s1thens2is:s1\\nshould be completed before s2. It does not restrict the agent\\nfrom making progress towards the subgoal before the subgoal subgoal sequentially. Rather, the semantic of s1thens2is:s1\\nshould be completed before s2. It does not restrict the agent\\nfrom making progress towards the subgoal before the subgoal\\nis completed. In some case, such interleaving is necessary to\\nobtain the globally optimal trajectory.\\nConsider the example shown in Figure 3, where s1is\\n\\u201dmine-gold\\u201d, and s2is \\u201dcraft-boat\\u201d. It is valid that the agent\\n\\ufb01rst goes to collect wood (for accompolishing s2), and then\\nmine gold (for accompolishing s1), and \\ufb01nally crafts boat.\\nIn this case, the action sequences for completing s1and\\ns2are interleaved. However, they can are be recognized as\\ns1thens2becauses1is accomplished before s2.\\nPlanning with RSGs\\nWe \\ufb01rst consider the problem of planning an action sequence\\nthat satis\\ufb01es a given task description twritten inTL. We\\nassume that the external world is well modeled as a deter-\\nministic, fully observable decision process with a known\\nstate space, an action space, a transition function, and a\\ncost functionhS;A;T;Ciand that we have a set of goal\\nclassi\\ufb01ersGoparameterized by \\u0012. Given a task t, we con-\\nstruct an FSM representation and then compose it with the\\nenvironment process to obtain an FSM-augmented process\\nhSt;At;Tt;Cti. Concretely,St=S\\u0002Vt, whereVtis the\\nset of nodes of FSM constructed from task t. We then denote\\neach task-augmented state as (s;v), wheresis the environ-\\nment state, and vindicates the current subgoal. The actions\\nAt=A[ FSMt, where each action either corresponds to a\\nprimitive action a2A or a transition in FSMt. An FSM tran- (,-!=0.3\\n(,-!=0\\n(,-!=0\\n(,-!=0.5(,-!=0.3\\n(,-!=0.6\\n(,-!=0\\n(,-!=0.4\\n(,-!=0.3\\n(,-!=0.6\\n(,-!=0.4\\n(,-!=0.6\\n(,-!=0.6\\n(,-!=0.7\\n(,-!=0.9\\n(,-!=0.9\\n(,-!=0.7\\n(,-!=0.8!!!\\\"\\n(,-!=0.8\\n!#:mine-gold&#!$:mine-coal&$!%:mine-wood&%\\nFigure 4: A running example of the FSM-A\\u0003algorithm for the task \\u201c (mine wood ormine coal) then mine gold .\\u201d For simplicity,\\nwe only show a subset of states visited on each FSM node. The blue arrows indicate transitions by primitive actions (in this\\nexample, each primitive action takes a cost of 0.1). The yellow arrows are transitions on the FSM, which can only be performed\\nwhenGv(\\u0001)andGv0(\\u0001)evaluates to False (in practice, the reward is computed as \\u0000(logGv(\\u0001) + log (1\\u0000Gv0(\\u0001)))). At the\\nsuper-terminal node vT, the state with minimum cost will be selected and we will back-trace the entire state-action sequence.\\nsition action indicates that the agent has achieved the current\\nsubgoal and will proceed to the next subgoal. We further de-\\n\\ufb01neTt((s;v);a) = (T(s;a);v)ifais a primitive action in\\nA, whileTt((s;v);a) = (s;v0)ifa= (v;v0)2FSMtis an\\nedge in the FSM. The former are environmental actions. They\\nonly change the environmental state sbut do not change the\\ncurrent subgoal v. The latter, namely FSM transitions, do not\\nchange the environmental state, but mark the current subgoal\\nas completed and switch to the next one. Similarly, for the\\ncost function,\\nC0((s;v);a) =8\\n<\\n:C(s;a) ifa2A;\\n\\u0000\\u0015(logGv(s;\\u0012)+ ifa= (v;v0)2FSM t\\nlog (1\\u0000Gv0(s;\\u0012)))\\nwhere\\u0015is a hyperparameter. The key intuition behind the\\nconstruction ofCtis that the cumulative cost from v0to\\nvTis the summation of all primitive action costs added to\\nthe log probability of the validity of subgoal transitions. At\\neach subgoal transition, the state sshould satisfy the goal\\ncondition of the current RSGs but should not satisfy the goal\\ncondition of the next RSGs\\u2014which enforces the sequential\\nconstraints speci\\ufb01ed in the task. In principle, when Gvare\\nBoolean-output classi\\ufb01ers, the cost is 0for a valid transition\\nand1for an invalid transition. In practice, we approximate\\nthe \\u201csoft\\u201d version of classi\\ufb01ers with neural networks: the\\noutputs are in [0;1], indicating how likely those conditions\\nare to be satis\\ufb01ed.\\nImportantly, our formulation of the RSG planning problem\\nis different from planning for each individual action term andstitching the sub-plans sequentially. Concretely, we are \\ufb01nd-\\ning a \\u201cglobally\\u201d optimal plan instead of achieving individual\\nsubgoals in a locally optimal way. Thus, we allow complex\\nbehaviors such as making progress for a later subgoal to\\nreduce the total cost. We include detailed examples in the\\nsupplementary material.\\nAt the input-output level, our planner receives the a task\\ndescriptiontrepresented as an FSM, an environmental tran-\\nsition modelT, and a cost function C, together with a set\\nof goal classi\\ufb01ersfGogparameterized by \\u0012. It generates a\\nsequence of actions \\u0016athat is a path from (s0;v0)to(sT;vT)\\nand minimizes the cumulative action costs de\\ufb01ned by Ct.\\nHere,s0is the initial environmental state, v0is the initial\\nstate of FSMt,sTis the last state of the trajectory, and vTis\\nthe terminal state of FSMt.\\nWe make plans using slightly modi\\ufb01ed versions of A\\u0003\\nsearch, with a learned domain-dependent heuristic for pre-\\nviously seen tasks and a uniform heuristic for unseen tasks.\\nThis algorithm can be viewed as doing a forward search to\\nconstruct a trajectory from a given state to a state that satis\\ufb01es\\nthe goal condition. Our extension to the algorithms handles\\nthe hierarchical task structure of the FSM.\\nOur modi\\ufb01ed A\\u0003search maintains a priority queue of nodes\\nto be expanded. At each step, instead of always popping the\\ntask-augmented state (s;v)with the optimal evaluation, we\\n\\ufb01rst sample a subgoal vuniformly in the FSM, and then\\nchoose the priority-queue node with the smallest evaluation\\nvalue among all states (\\u0001;v). This balances the time allocated\\nto \\ufb01nding a successful trajectory for each subgoals in the task\\ndescription. Our hierarchical search algorithm also extends to continu-\\nous domains by integrating Rapidly-Exploring Random Trees\\n(RRT) (LaValle et al. 1998). We include the implementation\\ndetails in the supplementary material. Any state-action se-\\nquence produced by planning in the augmented model is\\nlegal according to the environment transition model and is\\nguaranteed to satisfy the task speci\\ufb01cation t.\\nExample. Fig. 4 shows a running example of our FSM-A\\u0003\\nplanning given the task \\u201c mine wood ormine coal then mine\\ngold\\u201d from the state s0(shown as the left-most state in the\\n\\ufb01gure).\\n1.At the beginning, (s0;v0)is expanded to the node v1:mine\\nwood andv2:mine coal with FSM transition actions at no\\ncost.\\n2.We expand the search tree node on v1andv2and compute\\nthe cost for reaching each states on v1andv2.\\n3.For states that satisfy the goal conditions for v1andv2\\n(i.e.,G1andG2, respectively, and circled by green and\\nblue boxes) and the initial condition for v3(i.e.,1\\u0000G3),\\nwe make a transition to v3at no cost (the states that do\\nnot satisfy the conditions can also be expanded to v3but\\nwith a large cost.\\n4.Then search can be done in a similar way at v3and the\\nstates atv3that satisfyG3can reachvT.\\n5.For all states at vT, we back-trace the state sequence with\\nthe minimum cost.\\nLearning RSGs from Unsegmented Trajectories\\nand Descriptions\\nWe learn RSGs from weakly-annotated demonstrations, in the\\nform of unsegmented trajectories and paired task descriptions.\\nThe training dataset Dcontains tuples (\\u0016s;\\u0016a;t)where \\u0016sis a\\nsequence of environmental states, \\u0016ais a sequence of actions,\\nandt2TL is a task description.\\nOur goal is to recover the grounding of subgoal terms from\\nthese demonstrations. At a high level, our learning objective\\nis to \\ufb01nd a set of parameters for the goal classi\\ufb01ers Gothat\\nrationally explain the demonstration data: the actions taken\\nby the demonstrator should be \\u201cclose\\u201d in some sense to the\\noptimal actions that would be taken to achieve the goal. Let\\n\\u0012denote the collection of parameters in fGog. Thus, our\\ntraining objective takes the following form:\\n\\u0012\\u0003= arg max\\n\\u00121\\njDjX\\n(\\u0016s;\\u0016a;t)2Dscore (\\u0016s;\\u0016a;t;\\u0012): (1)\\nThe scoring function score combines the rationality of the\\nobserved trajectory with an additional term that emphasizes\\nthe appropriateness of FSM transitions given t:\\nscore (\\u0016s;\\u0016a;t;\\u0012):= max\\n\\u0016vflogY\\niRat(si;vi;ai;t;\\u0012) +\\nX\\n(vi;vi+1)2\\nFSM transitions\\b\\nlogGvi(si;\\u0012) + log\\u0000\\n1\\u0000Gvi+1(si;\\u0012)\\u0001\\tg\\n(2)The rationality score measures the likelihood that the ac-\\ntiona2Atin state (s;v)would have been chosen by a\\nnearly-optimal agent, who is executing a policy that assigns\\na probability to an action based on the optimal cost-to-go for\\ntasktin the FSM-augmented model after taking it:\\nRat(s;v;a;t ;\\u0012):=exp (\\u0000\\u000b\\u0001Jt(s;v;a ;\\u0012))R\\nx2A0exp (\\u0000\\u000b\\u0001Jt(s;v;x ;\\u0012));(3)\\nwhere\\u000bis a hyperparameter called inverse rationality. The\\nintegral is a \\ufb01nite sum for discrete actions and can be approx-\\nimated using Monte Carlo sampling for continuous actions.\\nIf\\u000bis small, the assumption is that the demonstrations may\\nbe highly noisy; if large, then they are near optimal.\\nThe cost-to-go (analogous to a value function) is de\\ufb01ned\\nrecursively as\\nJt(s;v;a ;\\u0012) =Ct((s;v);a) + max\\na02AtJt(T0((s;v);a);a;\\u0012):\\n(4)\\nIt need not be computed for the whole state space; rather, it\\ncan be computed using the planner on a tree of relevant states,\\nreachable from (s0;v0).\\nFigure 5 and Algorithm 1 summarize the learning process\\nof RSGs. First, we perform a A\\u0003search (or RRT for continu-\\nous domains) from the trajectory. Then, we backtrack in the\\nsearch tree\\/RRT to compute the shortest distance from each\\nnode to the terminal state, Jt, so that Rat(si;vi;ai;t;\\u0012)can\\nbe evaluated along the trajectory \\u0016s;\\u0016a.\\nAt learning time, we can observe the environmental state\\nand action sequence, but we cannot observe the FSM states or\\ntransitions. To ef\\ufb01ciently \\ufb01nd the optimal FSM states and tran-\\nsitions, given an environment state and action sequence as\\nwell as goal classi\\ufb01ers parameterized by the current \\u0012, we use transitions. To ef\\ufb01ciently \\ufb01nd the optimal FSM states and tran-\\nsitions, given an environment state and action sequence as\\nwell as goal classi\\ufb01ers parameterized by the current \\u0012, we use\\na dynamic programming method. Speci\\ufb01cally, we will \\ufb01rst la-\\nbel the FSM nodes from 0toTby sorting them topologically.\\nNext, we can use a two-dimensional dynamic programming\\nwith the transition equations based on RatandGvcan \\ufb01nd\\n\\u0016vthat maximizes score . Concretely, let f[i;j]denote the\\nmaximum score by aligning the trajectory si;ai;si+1;\\u0001\\u0001\\u0001\\nwith the last jnodes of the FSM. The dynamic programming\\nalgorithm iterates over iin the reversed order. At each step,\\nit tries to either assign the current (si;ai)pair to the cur-\\nrent FSM node j, or to create a new transition from another\\nFSM nodektoj. We present the detailed algorithm in the\\nsupplementary material. Although the transition model we\\nhave discussed so far is deterministic, the methods can all\\nbe extended straightforwardly to the stochastic case, as also\\ndescribed in the supplement.\\nTo improve the optimization, we add a contrastive loss\\nterm, encoding the idea that, for each demonstration (\\u0016s;\\u0016a),\\nthe corresponding task description tshould have a higher\\nrationality score compared to an unmatched task description\\nt0, yielding the \\ufb01nal objective to be maximized: \\ud835\\udc3a!RationalSubgoals\\ud835\\udc60\\u0305\\ud835\\udc4e$\\ud835\\udc61TrajectoryTaskSearch:A*\\/RRTCompute\\ud835\\udc3d!(\\ud835\\udc60,\\ud835\\udc63,\\ud835\\udc4e)DynamicProgrammingfor\\ud835\\udc60\\ud835\\udc50\\ud835\\udc5c\\ud835\\udc5f\\ud835\\udc52(\\ud835\\udc60\\u0305,\\ud835\\udc4e-,\\ud835\\udc61)\\u2112BackPropagationFigure 5: An overview of the training\\nparadigm for RSGs. See text for details.Algorithm 1: Overview of the training paradigm in pseudocode.\\nInitiate the goal condition Go(\\u0001;\\u0012)\\nfor(\\u0016s;\\u0016a;t)2Ddo\\nfort0in candidate task descriptions do\\nApply A* search from all states in \\u0016swith taskt0to compute a tree T.\\nforeach node (s;v;a;t0)2Tin reversed topological order do\\nComputeJt0(s;v;a ;\\u0012)on the node using Eq. 4.\\nend for\\nforeach node (s;v;a;t0)2Tin reversed topological order do\\nCompute Rat (s;v;a;t0;\\u0012)for each tree node using Eq. 3.\\nend for\\nCompute score (\\u0016s;\\u0016a;t0;\\u0012)using Eq. 2 based on Rat values of nodes in T.\\nend for\\nCompute the training objective J(\\u0012)using the score of all candidate task\\ndescriptions t0using Eq. 5.\\nUpdate\\u0012using gradient descent by maximizing J(\\u0012).\\nend for\\nmine-goldcollect-woodcraft-boat!!!\\\"\\nTask:(collect-woodormine-good)thencraft-boat\\nFigure 6: An example of the value function for task-\\naugmented states on a simple FSM.mina2AJt(s;v;a )are\\nplotted at each location at each FSM node. Deeper color in-\\ndicates larger cost. Red boxes and dotted lines illustrate the\\ngoal and a rational trajectory for each subgoal.\\nJ(\\u0012) =X\\n(\\u0016s;\\u0016a;t)2D(score (\\u0016s;\\u0016a;t;\\u0012)\\n+\\r\\u0001logexp (\\f\\u0001score (\\u0016s;\\u0016a;t;\\u0012))P\\nt0exp (\\f\\u0001score (\\u0016s;\\u0016a;t0;\\u0012))\\u0013\\n;(5)\\nwheret0s are uniformly sampled negative tasks in TL. This\\nloss function is fully differentiable w.r.t. \\u0012, which enables\\nus to apply gradient descent for optimization. Essentially,\\nwe are back-propagating through two dynamic programming\\ncomputation graphs: one that computes Jtbased on planning\\noptimal trajectories given goal classi\\ufb01ers parameterized by\\n\\u0012, and one that \\ufb01nds the optimal task-state transitions for the\\nobserved trajectory.\\nRSG Dependency Discovery and Planning\\nNext, we describe our algorithm for planning with a sin-\\ngle, \\ufb01nal goal term (e.g., craft-boat ) instead of step-by-stepinstructions. Since directly planning for the goal based on\\nthe corresponding goal classi\\ufb01er can be very slow due to\\nthe long horizon, our key idea here is to leverage the RSGs\\nlearned from data to perform a bilevel search. Our algorithm\\nbegins with discovering a dependency matrix between RSGs\\nduring training time. At performance time, we \\ufb01rst use the\\ndiscovered dependency model to suggest high-level plans, in\\nthe form of step-by-step instructions in TL. Next, we use\\nthese instructions to plan for environmental actions using our\\nplanning algorithm.\\nFor each possible subgoal o, we evaluate the associated\\nlearned goal classi\\ufb01er Goover all states along training tra-\\njectories that contain o. Next, we compute \\ufb01rst(\\u0016s;o)as the\\nsmallest index isuch thatGo(si)is true. If such idoes\\nnot exist (i.e., Gois never satis\\ufb01ed in \\u0016s) orois not men-\\ntioned in the task speci\\ufb01cation tassociated with \\u0016s, we de-\\n\\ufb01ne\\ufb01rst(\\u0016s;o) =1. For all tuples (\\u0016s;o1;o2), we sayo2is\\nachieved beforeo1if neither \\ufb01rst(\\u0016s;o1)nor\\ufb01rst(\\u0016s;o1)is\\nin\\ufb01nity, and \\ufb01rst(\\u0016s;o2)<\\ufb01rst(\\u0016s;o1).\\nLetbcount (o1;o2)be the number of \\u0016s2D such thato2is\\nachieved before o1in\\u0016s. We construct a dependency matrix d\\nby normalizing the bcount as:\\nd(o1;o2),bcount (o1;o2)P\\no0bcount (o1;o0); (6)\\nwhereo0sums over all RSGs.\\nThe derived dependency matrix can be interpreted as the\\nprobability that o2is a precondition for o1. Now, recall that\\nour task is to \\ufb01nd an action sequence \\u0016athat, starting from\\nthe initial state s0, yields a new state sTthat satis\\ufb01es the\\ngiven goal action term g, such as craft-boat . Our high-level\\nidea is to leverage the dependency matrix to suggest possible\\nstep-by-step instructions t, whose last action term is g. The\\nplanning algorithm will follow the suggested instructions to\\ngenerate low-level plans \\u0016a.\\nFormally, we only consider instructions that are action\\nterms connected by the then connective. Denote a candidate instructiont=o1theno2then\\u0001\\u0001\\u0001thenok. We de\\ufb01ne its\\npriority as:\\npriority (t) =\\u0015kk\\u00001Y\\ni=10\\n@1\\u0000kY\\nj=i+1(1\\u0000d(oj;oi))1\\nA;(7)\\nwhere\\u0015is a length bias constant which is set to 0:9because\\nwe prefer shorter instructions.\\nGiven the candidate instructions, we run the planning al-\\ngorithm for these instructions. We prioritize instructions t\\nwith high priorities priority (t), and these instructions are gen-\\nerated by a search approach (Algorithm 2) from the given\\n\\ufb01nal goal. The limit of instruction length, length limit, is set\\nto6for our experiment.. For more complicated domains, a\\npromising future direction is to learn a full abstract planning\\nmodel (symbolic or continuous) based on the subgoal terms\\nlearned from demonstrations.\\nAlgorithm 2: Overview of the search algorithm given only\\nthe \\ufb01nal goal.\\nBuild a priority queue of instructions H.\\nH f\\ufb01nal goalg\\nwhileHis not empty do\\nt H:pop ()\\nRun A* search on task t.\\nifthe A* search \\ufb01nds a solution then\\nReturn the solution.\\nend if\\niflength (t)\\u0014length limit then\\nforo2Odo\\nifo =2tand9o02t:d(o0;o)>0then\\nH:push (othent)# See Eq. 7.\\nend if\\nend for\\nend if\\nend while\\nExperiments\\nWe compare our model with other subgoal-learning ap-\\nproaches in Crafting World (Chen, Gupta, and Marino\\n2021), a Minecraft-inspired crafting environment, and Play-\\nroom (Konidaris, Kaelbling, and Lozano-Perez 2018), a 2D\\ncontinuous domain with geometric constraints.\\nCrafting World. In Crafting World, the agent can move in\\na 2D grid world and interact with objects next to it, including\\npicking up tools, mining resources, and crafting items. Min-\\ning in the environment typically requires tools, while crafting\\ntools and other objects have their own preconditions, such\\nas being close to a workstation or holding another speci\\ufb01c\\ntool. Thus, crafting a single item often takes multiple subgoal\\nsteps. There are also obstacles such as rivers (which require\\nboats to go across) and doors (which require speci\\ufb01c keys to\\nopen).\\nWe de\\ufb01ne 26 primitive tasks, instantiated from templates\\nofgrab-X ,toggle-switch ,mine-X , and craft-X . While gener-\\nating trajectories, all required items have been placed in the\\nFigure 7: An illustration of\\nthe Playroom environment\\nand a trajectory for the\\ntask: turn-on-music then\\nplay-with-ball then turn-\\noff-music .\\nagent\\u2019s inventory. For example, before mining wood, an axe\\nmust be already in the inventory. In this case, the agent is\\nexpected to move to a tree and execute the mining action. We\\nalso de\\ufb01ne 26 compositional tasks composed of the afore-\\nmentioned primitive tasks. For each task, we have 400 expert\\ndemonstrations.\\nAll models are trained using tuples of task description tand\\nexpert state-action sequences (\\u0016s;\\u0016a). In particular, we train all\\nmodels on primitive and compositional tasks and test them\\non two splits: compositional andnovel . The compositional\\nsplit contains novel state-action sequences of previously-seen\\ntasks. The novel split contains 12 novel tasks, where primitive\\ntasks are composed in ways never seen during training (i.e.,\\nnot in the 26 tasks from the compositional split).\\nPlayroom. Our second environment is Play-\\nroom (Konidaris, Kaelbling, and Lozano-Perez 2018),\\na 2D maze with continuous coordinates and geometric\\nconstraints. Fig. 7 shows an illustrative example of the\\nenvironment. Speci\\ufb01cally, a 2D robot can make moves in\\na small room with obstacles. The agent has three degrees\\nof freedom (DoFs): xandydirection movement, and a 1D\\nrotation. The environment invalidates movements that cause\\ncollisions between the agent and the obstacles. Additionally,\\nthere are six objects randomly placed in the room, which the\\nrobot can interact with. For simplicity, when the agent is\\nclose to an object, the corresponding robot-object interaction\\nwill be automatically triggered.\\nSimilar to the Crafting World, we have de\\ufb01ned six primi-\\ntive tasks (corresponding to the interaction with six objects in\\nthe environment) and eight compositional tasks (e.g., turn-on- Similar to the Crafting World, we have de\\ufb01ned six primi-\\ntive tasks (corresponding to the interaction with six objects in\\nthe environment) and eight compositional tasks (e.g., turn-on-\\nmusic then play-with-ball ). We have designed another eight\\nnovel tasks, and for each task, we have 400 expert demonstra-\\ntions. We train different models on rational demonstrations\\nfor both the primitive and compositional tasks, and evaluate\\nthem on the compositional and novel splits.\\nBaselines\\nWe compare our RSGs, which learns goal-based represen-\\ntations, with two baselines using different underlying repre-\\nsentations: IRL methods learn reward-based representations,\\nand behavior cloning methods directly learn policies. The\\nimplementation details are in the supplementary material.\\nOur max-entropy inverse reinforcement learning (IRL;\\nZiebart et al. 2008) baseline learns a task-conditioned re-\\nward function by trying to explain the demonstration. For\\nplanning, we use the built-in deep-Q-learning algorithm. The\\nbehavior cloning (BC; Torabi, Warnell, and Stone 2018) base-\\nline directly learns a task-conditioned policy that maps the ModelTask\\nInputEnv.\\nTran.Crafting World Playroom\\nCom. Novel Com. Novel\\nIRL Lang. Y 36.5 1.8 28.3 9.6\\nBC Lang. N 11.2 0.8 15.8 4.8\\nBC- FSM FSM N 5.2 0.3 38.2 31.5\\nRSGs FSM Y 99.6 97.8 82.0 78.2\\nTable 1: Results of the planning task, evaluated as the success\\nrate of task completion. IRL and BC take raw task speci\\ufb01ca-\\ntion and process them with LSTM, while BC- FSM and RSGs\\nuses the FSM directly. RSGs and IRL use the environmental\\ntransition model during training while BC and BC- FSM dot\\nnot. The maximum number of expanded nodes for all plan-\\nners is 5,000. All models are trained on the compositional\\nsplit, and tested on the compositional and the novel split.\\ncurrent state and the given task to an environment primitive\\naction. BC- FSM is the BC algorithm augmented with our\\nFSM description of tasks. Compared with RSGs, instead of\\nsegmenting the demonstration sequence based on rational-\\nity, BC- FSM segments them based on how consistent each\\nfragment is with the policy for the corresponding action term.\\nResults\\nTo evaluate planning, each algorithm is given a new task t,\\neither speci\\ufb01ed inTL, or as a black-box goal state classi\\ufb01er,\\nand generates a trajectory of actions to complete the task.\\nPlanning with instructions. Table 1 summarizes the results.\\nOverall, RSGs outperforms all baselines. On the composi-\\ntional split, our model achieves a nearly perfect success rate\\nin the Crafting World ( 99:6%). Comparatively, although the\\ntasks have been presented during training of all baselines,\\ntheir scores remain below 40%.\\nOn the novel split, RSGs outperforms all baselines by a\\nlarger margin than on the compositional split. We observe\\nthat since novel tasks contain longer descriptions than those\\nin the compositional set, all baselines have a success rate of\\nalmost zero. Compared with IRL methods, the more com-\\npositional structure in our goal-centric representation allows\\nit to perform better. Meanwhile, a key difference between\\nbehavior cloning methods (BC and BC- FSM) and ours is that\\nBC directly applies a learned policy, while our model runs an\\nA* search based on the learned goal classi\\ufb01er and leverages\\nthe access to the transition model. This suggests that learning\\ngoals is more sample-ef\\ufb01cient than learning policies in such\\ndomains and generalizes better to new maps.\\nOur model can be easily applied to environments with\\nimage-based states, simply by changing the inputs of Ioand\\nGomodels to images. We evaluate our model in an image-\\nbased Crafting World environment. It achieves 82.0% and\\n78.2% success rates on the compositional and novel splits,\\nrespectively. Comparatively, the best baseline BC- FSM gets\\n38.2% and 31.5%. Details are in the supplementary material.\\nPlanning with goals. We also evaluate RSGs on planning\\nwith a single goal action term. These problems require a long\\nsolution sequence, making them too dif\\ufb01cult to solve witha blind search from an initial state. Since there is no task\\nspeci\\ufb01cation given, in order to solve the problems ef\\ufb01ciently,\\nit is critical to use other dependent RSGs for search guidance.\\nWe use 8 manually designed goal tests, each of which can\\nbe decomposed into 2\\u20135 subgoals. We run our hierarchical\\nsearch based on RSGs and the discovered dependencies.\\nWe compare this method with two baselines: a blind\\nforward-search algorithm, and a hierarchical search based\\non RSGs without discovered dependencies (i.e., by setting\\nthe dependency matrix as a uniform distribution). We test\\nall three methods on 100 random initial states for each task.\\nFig. 8 summarizes the result. Overall, RSGs with discovered\\ndependencies enables ef\\ufb01cient searches for plans. On easier\\ntasks (2 or 3 subgoals), search with RSGs and dependencies\\nhas a similar runtime as the baseline that searches without\\ndependencies. Both of them outperform the blind-search base-\\nline (about 2.4\\u0002more ef\\ufb01cient when reaching a 70% success\\nrate). However, when the task becomes complex (4 or 5 sub-\\ngoals), searching with RSGs and the discovered dependencies line (about 2.4\\u0002more ef\\ufb01cient when reaching a 70% success\\nrate). However, when the task becomes complex (4 or 5 sub-\\ngoals), searching with RSGs and the discovered dependencies\\nsigni\\ufb01cantly outperforms other alternatives. For example, to\\nreach a 70% success rate, searching with RSGs needs only\\n4,311 expanded nodes. By contrast, searching without RSGs\\nneeds 19,220 (4.5 \\u0002) nodes. Interestingly, searching with\\nRSGs but without discovered dependencies performs worse\\nthan the blind-search baseline. We hypothesize that this is\\nbecause it wastes time on planning for unreasonable instruc-\\ntions. Overall, the effectiveness of RSGs with discovered\\ndependencies grows as the complexity of tasks grows.\\nRelated Work\\nModular policy learning and planning. Researchers have\\nbeen learning modular \\u201cpolicies\\u201d by simultaneously looking\\nat trajectories and reading task speci\\ufb01cations in the form of\\naction term sequences (Corona et al. 2021; Andreas, Klein,\\nand Levine 2017; Andreas and Klein 2015), programs (Sun,\\nWu, and Lim 2020), and linear temporal logic (LTL) formu-\\nlas (Bradley et al. 2021; Toro Icarte et al. 2018; Tellex et al.\\n2011). However, they either require additional annotation\\nfor segmenting the sequence and associating fragments with\\nlabels in the task description (Corona et al. 2021; Sun, Wu,\\nand Lim 2020), or cannot learn models for planning (Tellex\\net al. 2011). By contrast, RSGs learns useful subgoals from\\ndemonstrations. We use a small but expressive subset of LTL\\nfor task description, and jointly learn useful subgoals and\\nsegment the demonstration sequence.\\nOur subgoal representation is also related to other mod-\\nels in domain control knowledge (de la Rosa and McIlraith\\n2011), goal-centric policy primitives (Park et al. 2020), macro\\nlearning (Newton et al. 2007), options and hierarchical rein-\\nforcement learning (HRL; Sutton, Precup, and Singh 1999;\\nDietterich 2000; Barto and Mahadevan 2003; Mehta 2011),\\nand methods that combine reinforcement learning and plan-\\nning (Segovia-Aguas, Ferrer-Mestres, and Jonsson 2016;\\nWinder et al. 2020). However, the execution of subgoals\\nin RSGs is fundamentally different from options: each option\\nhas a policy that we can follow to achieve the short-term goal,\\nwhile subgoals in RSGs should be re\\ufb01ned with segments of\\nprimitives by planning algorithms. Our planning algorithm is\\nsimilar to other approaches: (de la Rosa and McIlraith 2011; 4.5\\u00d7Efficiency4311192203095123055111822.5\\u00d7Efficiency2.1\\u00d7EfficiencyFigure 8: RSGs applied to planning with a \\ufb01nal goal. We do evaluation on 3 groups of planning tasks in the Crafting World\\nenvironment. We use 100 random initial states for each task. Each search method can expand up to 25,000 nodes.\\nBotvinick and Weinstein 2014; Winder et al. 2020), but they\\ndo not leverage discovered dependencies between subgoals.\\nLearning from demonstration. Learning from demonstra-\\ntion generally refers to building agents that can interact with\\nthe environment by observing expert demonstrations (e.g.,\\nstate-action sequences). Techniques for learning from demon-\\nstration can be roughly categorized into four groups: pol-\\nicy function learning (Chernova and Veloso 2007; Torabi,\\nWarnell, and Stone 2018), cost and reward function learn-\\ning (Markus Wulfmeier and Posner 2015; Ziebart et al. 2008),\\ngenerative adversarial learning (Ho and Ermon 2016; Liu et al.\\n2022), and learning high-level plans (Ekvall and Kragic 2008;\\nKonidaris et al. 2012). We refer to Argall et al. (2009) and\\nRavichandar et al. (2020) as comprehensive surveys. In this\\npaper, we learn useful subgoals that support planning, and\\ncompare our model with methods that directly learn policies\\nand cost functions. Moreover, unlike those who use similari-\\nties between different actions (Niekum et al. 2012) to segment\\ndemonstrations, in RSGs, we segment the demonstration with\\nassociate action terms by rationality assumptions of the agent.\\nInverse planning. Our model is also related to inverse plan-\\nning algorithms that infer agent intentions from behavior by\\n\\ufb01nding a task description tthat maximizes the consistency\\nbetween the agent\\u2019s behavior and the synthesized plan (Baker,\\nSaxe, and Tenenbaum 2009). While existing work has largely\\nfocused on modeling the rationality of agents (Baker, Saxe,\\nand Tenenbaum 2009; Zhi-Xuan et al. 2020) and more ex-\\npressive task descriptions (Shah et al. 2018), our focus is on\\nleveraging the learned subgoals and their dependencies to\\nfacilitate agent planning for novel tasks.\\nUnsupervised subgoal discovery. Our method is also re-\\nlated to approaches for discovering subgoals from unlabelled\\ntrajectories (Paul, Vanbaar, and Roy-Chowdhury 2019; Tang\\net al. 2018; Kipf et al. 2019; Lu et al. 2021; Gopalakrish-\\nnan et al. 2021), mostly based on the assumption that the\\ntrajectory can be decomposed into segments, and each seg-\\nment corresponds to a subgoal. Some other approaches for\\ndiscovering subgoals are to detect \\u201cbottleneck\\u201d states (Men-\\nache, Mannor, and Shimkin 2002; S \\u00b8ims \\u00b8ek, Wolfe, and Barto\\n2005) based on the state transition graphs. RSG differs from\\nthese works in that we focus on learning the grounding of\\naction terms de\\ufb01ned in task descriptions. Thus, RSGs areassociated with action terms and thus can be recomposed by\\nhuman users to describe novel tasks. It is a meaningful future\\ndirection to combine learning from trajectory-only data and\\ntrajectories with descriptions to improve the data ef\\ufb01ciency.\\nConclusion\\nWe have presented a subgoal learning framework for long-\\nhorizon planning tasks. The rational subgoals (RSGs) can be\\nlearned by observing expert demonstrations and reading task\\nspeci\\ufb01cations described in a simple task language TL. Our\\nlearning algorithm simultaneously segments the trajectory\\ninto fragments corresponding to individual subgoals, and\\nlearns planning-compatible models for each subgoal. Our\\nexperiments suggest that our framework has strong composi-\\ntional generalization to novel tasks.\\nLimitation. The assumption of a deterministic environ-\\nment has allowed us to focus on the novel RSG formulation\\nof subgoal models. For domains with substantial stochastic-\\nity, the high-level concepts of RSGs could be retained (e.g.,\\nrationality), and algorithmic changes may be required such\\nas replacing maximum entropy IRL with maximum causal\\nentropy (Ziebart, Bagnell, and Dey 2010). Another limitation rationality), and algorithmic changes may be required such\\nas replacing maximum entropy IRL with maximum causal\\nentropy (Ziebart, Bagnell, and Dey 2010). Another limitation\\nof RSGs is that it can not leverage trajectories without la-\\nbeled task descriptions. Future work may consider the jointly\\nlearning of subgoals and subgoal structures of tasks (Vazquez-\\nChanlatte et al. 2018; Chou, Ozay, and Berenson 2022).\\nAcknowledgement. We thank Yunyun Wang for giving ad-\\nvice on making \\ufb01gures. We thank all group members of\\nthe MIT Learning & Intelligent Systems Group for help-\\nful comments on an early version of the project. This work\\nis in part supported by NSF grant 2214177, AFOSR grant\\nFA9550-22-1-0249, ONR MURI grant N00014-22-1-2740,\\nthe MIT-IBM Watson Lab, the MIT Quest for Intelligence,\\nthe Center for Brain, Minds, and Machines (CBMM, funded\\nby NSF STC award CCF-1231216), the Stanford Institute\\nfor Human-Centered Arti\\ufb01cial Intelligence (HAI), and Ana-\\nlog, Amazon, JPMC, Meta, Salesforce, and Samsung. Any\\nopinions, \\ufb01ndings, and conclusions or recommendations ex-\\npressed in this material are those of the authors and do not\\nnecessarily re\\ufb02ect the views of our sponsors. References\\nAndreas, J.; and Klein, D. 2015. Alignment-Based Composi-\\ntional Semantics for Instruction Following. In EMNLP .\\nAndreas, J.; Klein, D.; and Levine, S. 2017. Modular multi-\\ntask reinforcement learning with policy sketches. In ICML .\\nArgall, B. D.; Chernova, S.; Veloso, M.; and Browning, B.\\n2009. A survey of robot learning from demonstration. Rob\\nAuton Syst. , 57(5): 469\\u2013483.\\nBaker, C. L.; Saxe, R.; and Tenenbaum, J. B. 2009. Action\\nunderstanding as inverse planning. Cognition , 113(3): 329\\u2013\\n349.\\nBarto, A. G.; and Mahadevan, S. 2003. Recent advances in\\nhierarchical reinforcement learning. Discrete event dynamic\\nsystems , 13(1): 41\\u201377.\\nBotvinick, M.; and Weinstein, A. 2014. Model-based hierar-\\nchical reinforcement learning and human action control. Phi-\\nlos. Trans. R. Soc. Lond., B, Biol. Sci. , 369(1655): 20130480.\\nBradley, C.; Pacheck, A.; Stein, G. J.; Castro, S.; Kress-\\nGazit, H.; and Roy, N. 2021. Learning and Planning\\nfor Temporally Extended Tasks in Unknown Environments.\\narXiv:2104.10636.\\nChen, V .; Gupta, A.; and Marino, K. 2021. Ask Your Hu-\\nmans: Using Human Instructions to Improve Generalization\\nin Reinforcement Learning. In ICLR .\\nChernova, S.; and Veloso, M. 2007. Con\\ufb01dence-based policy\\nlearning from demonstration using gaussian mixture models.\\nInAAMAS .\\nChou, G.; Ozay, N.; and Berenson, D. 2022. Learning tem-\\nporal logic formulas from suboptimal demonstrations: theory\\nand experiments. Autonomous Robots , 46(1): 149\\u2013174.\\nCorona, R.; Fried, D.; Devin, C.; Klein, D.; and Darrell,\\nT. 2021. Modular Networks for Compositional Instruction\\nFollowing. In NAACL-HLT , 1033\\u20131040.\\nDe Giacomo, G.; and Vardi, M. Y . 2013. Linear temporal\\nlogic and linear dynamic logic on \\ufb01nite traces. In IJCAI .\\nde la Rosa, T.; and McIlraith, S. 2011. Learning domain\\ncontrol knowledge for TLPlan and beyond. In ICAPS 2011\\nWorkshop on Planning and Learning .\\nDietterich, T. G. 2000. Hierarchical reinforcement learning\\nwith the MAXQ value function decomposition. JAIR , 13:\\n227\\u2013303.\\nDong, H.; Mao, J.; Lin, T.; Wang, C.; Li, L.; and Zhou, D.\\n2019. Neural Logic Machines. In ICLR .\\nEkvall, S.; and Kragic, D. 2008. Robot learning from demon-\\nstration: a task-level planning approach. IJARS , 5(3): 33.\\nGopalakrishnan, A.; Irie, K.; Schmidhuber, J.; and van\\nSteenkiste, S. 2021. Unsupervised Learning of Temporal\\nAbstractions using Slot-based Transformers. In Deep RL\\nWorkshop at NeurIPS .\\nHo, J.; and Ermon, S. 2016. Generative adversarial imitation\\nlearning. In NeurIPS .\\nHochreiter, S.; and Schmidhuber, J. 1997. Long short-term\\nmemory. Neural Comput. , 9(8): 1735\\u20131780.Kipf, T.; Li, Y .; Dai, H.; Zambaldi, V .; Sanchez-Gonzalez, A.;\\nGrefenstette, E.; Kohli, P.; and Battaglia, P. 2019. Compile:\\nCompositional imitation learning and execution. In ICML .\\nKonidaris, G.; Kaelbling, L. P.; and Lozano-Perez, T. 2018.\\nFrom skills to symbols: Learning symbolic representations\\nfor abstract high-level planning. JAIR , 61: 215\\u2013289.\\nKonidaris, G.; Kuindersma, S.; Grupen, R.; and Barto, A.\\n2012. Robot learning from demonstration by constructing\\nskill trees. IJRR , 31(3): 360\\u2013375.\\nLaValle, S. M.; et al. 1998. Rapidly-exploring random trees:\\nA new tool for path planning. Technical report, Computer\\nScience Department, Iowa State University.\\nLiu, M.; Zhu, Z.; Zhuang, Y .; Zhang, W.; Hao, J.; Yu, Y .;\\nand Wang, J. 2022. Plan Your Target and Learn Your Skills:\\nTransferable State-Only Imitation Learning via Decoupled\\nPolicy Optimization. In ICML .\\nLu, Y .; Shen, Y .; Zhou, S.; Courville, A.; Tenenbaum, J. B.;\\nand Gan, C. 2021. Learning task decomposition with ordered\\nmemory policy network. In ICLR .\\nMarkus Wulfmeier, P. O.; and Posner, I. 2015. Maximum\\nEntropy Deep Inverse Reinforcement Learning. In NeurIPS\\nWorkshop .\\nMehta, N. 2011. Hierarchical structure discovery and trans-\\nfer in sequential decision problems . Oregon State University.\\nMenache, I.; Mannor, S.; and Shimkin, N. 2002. Q-\\ncut\\u2014dynamic discovery of sub-goals in reinforcement learn- fer in sequential decision problems . Oregon State University.\\nMenache, I.; Mannor, S.; and Shimkin, N. 2002. Q-\\ncut\\u2014dynamic discovery of sub-goals in reinforcement learn-\\ning. In European conference on machine learning , 295\\u2013306.\\nSpringer.\\nNewton, M. A. H.; Levine, J.; Fox, M.; and Long, D. 2007.\\nLearning Macro-Actions for Arbitrary Planners and Domains.\\nInICAPS .\\nNiekum, S.; Osentoski, S.; Konidaris, G.; and Barto, A. G.\\n2012. Learning and generalization of complex tasks from\\nunstructured demonstrations. In IROS . IEEE.\\nPark, D.; Noseworthy, M.; Paul, R.; Roy, S.; and Roy, N.\\n2020. Inferring task goals and constraints using bayesian\\nnonparametric inverse reinforcement learning. In JMLR .\\nPaul, S.; Vanbaar, J.; and Roy-Chowdhury, A. 2019. Learning\\nfrom trajectories via subgoal discovery. Advances in Neural\\nInformation Processing Systems , 32.\\nRavichandar, H.; Polydoros, A. S.; Chernova, S.; and Billard,\\nA. 2020. Recent advances in robot learning from demonstra-\\ntion. Annu Rev Control. , 3: 297\\u2013330.\\nSegovia-Aguas, J.; Ferrer-Mestres, J.; and Jonsson, A. 2016.\\nPlanning with partially speci\\ufb01ed behaviors. In Arti\\ufb01cial\\nIntelligence Research and Development , 263\\u2013272. IOS Press.\\nShah, A.; Kamath, P.; Shah, J. A.; and Li, S. 2018. Bayesian\\nInference of Temporal Task Speci\\ufb01cations from Demonstra-\\ntions. In NeurIPS .\\nS \\u00b8ims \\u00b8ek,\\u00a8O.; Wolfe, A. P.; and Barto, A. G. 2005. Identifying\\nuseful subgoals in reinforcement learning by local graph parti-\\ntioning. In Proceedings of the 22nd international conference\\non Machine learning , 816\\u2013823.\\nSun, S.-H.; Wu, T.-L.; and Lim, J. J. 2020. Program guided\\nagent. In ICLR . Sutton, R. S.; Precup, D.; and Singh, S. 1999. Between MDPs\\nand semi-MDPs: A framework for temporal abstraction in\\nreinforcement learning. Arti\\ufb01cial intelligence , 112(1-2): 181\\u2013\\n211.\\nTang, D.; Li, X.; Gao, J.; Wang, C.; Li, L.; and Jebara, T.\\n2018. Subgoal discovery for hierarchical dialogue policy\\nlearning. arXiv preprint arXiv:1804.07855 .\\nTellex, S.; Kollar, T.; Dickerson, S.; Walter, M.; Banerjee, A.;\\nTeller, S.; and Roy, N. 2011. Understanding natural language\\ncommands for robotic navigation and mobile manipulation.\\nInAAAI .\\nTorabi, F.; Warnell, G.; and Stone, P. 2018. Behavioral\\nCloning from Observation. In IJCAI .\\nToro Icarte, R.; Klassen, T. Q.; Valenzano, R.; and McIlraith,\\nS. A. 2018. Teaching Multiple Tasks to an RL Agent Using\\nLTL. In AAMAS .\\nVazquez-Chanlatte, M.; Jha, S.; Tiwari, A.; Ho, M. K.; and\\nSeshia, S. 2018. Learning task speci\\ufb01cations from demon-\\nstrations. In NeurIPS .\\nWinder, J.; Milani, S.; Landen, M.; Oh, E.; Parr, S.; Squire,\\nS.; Matuszek, C.; et al. 2020. Planning with abstract learned\\nmodels while learning transferable subtasks. In AAAI .\\nZhi-Xuan, T.; Mann, J. L.; Silver, T.; Tenenbaum, J. B.; and\\nMansinghka, V . K. 2020. Online bayesian goal inference for\\nboundedly-rational planning agents. In NeurIPS .\\nZiebart, B. D.; Bagnell, J. A.; and Dey, A. K. 2010. Modeling\\ninteraction via the principle of maximum causal entropy. In\\nICML .\\nZiebart, B. D.; Maas, A. L.; Bagnell, J. A.; and Dey, A. K.\\n2008. Maximum entropy inverse reinforcement learning. In\\nAAAI . Supplementary Material for\\nLearning Rational Subgoals from Demonstrations and Instructions\\nFirst, we elaborate how A\\u0003search is performed on the FSM-augmented transition models. We also discuss representation\\nchoices of RSGs, as well as the optimality, complexity and scalability of the search algorithms. Recall that we are using dynamic\\nprogramming obtain deterministic transitions, and there are other formulations such as using stochastic transitions, we talk\\nabout the comparison between our formulation and others in this section. In addition, we provide details for the dependency\\ndiscovering algorithm and hierarchical search algorithm used in planning for \\ufb01nal goals.\\nSecond, we discuss details about datasets and how we process the data, including how the features are extracted from the state\\nrepresentations. We also provide the list of task descriptions covered in each data split.\\nNext, we provide implementation details for baselines, and then discusses the limitation and future work of RSGs.\\nImplementation Details of RSGs\\nRe-parameterize 1\\u0000Go(\\u0001)using a separate neural network. In practice, instead of directly using 1\\u0000Go(\\u0001)to evaluate\\nthe probability that a subgoal has not been met, we parameterize 1\\u0000Go(\\u0001)using a separate neural network Io(\\u0001)that has the\\nsame architecture as Go(\\u0001). We observe that this re-parameterization stabilizes the training. In performance time, we will only be\\nusing the goal classi\\ufb01er Goand ignores the Io.\\nEmpirically, we \\ufb01nd that when using a single subgoal classi\\ufb01er instead of separate classi\\ufb01ers for IandG, some classi\\ufb01ers\\nusually get stuck at local optima. As a result, the overall planning performance, on average, drops from 99.6% to 75%. We\\nhypothesize that this is because using separate parameterization allows a broader set of possible solutions to the original problem,\\nwhich are practically equivalently helpful for planning. As a concrete example, consider the subgoal \\u201dmining-wood.\\u201d\\nIf we use only the Gand1\\u0000Gparameterization, the only feasible solution is:\\nG1= [wood in inventory ]\\nHowever, if we use separate parameterizations, the following solution will also be accepted:\\nI2= [tree on map and axe in inventory and wood not in inventory ]\\nG2= [tree on map and axe in inventory and wood in inventory ]\\nNote that, during planning, both classi\\ufb01ers G1andG2have the same effects, but the relaxed parameterization allows a broader\\nset of solutions.\\nFSM-A\\u0003\\nWe have implemented a extended version of the A\\u0003algorithm to handle FSM states in Crafting World.\\nA\\u0003at each FSM node. We start with the A\\u0003search process happening at each FSM node. For a given FSM state, the A\\u0003search\\nextends the tree search in two stages. The \\ufb01rst stage lasts for b= 3layers during training and b= 4layers during testing. In the\\n\\ufb01rstblayers of the search tree, we run a Breadth-First-Search so that every possible path with length bis explored. Then on\\nthe second stage lasts for c= 15 layers during training and 25layers in testing. In layer d2[b+ 1;b+c], we run A* from the\\nleaves in the \\ufb01rst stage based on the heuristic for each node. By enforcing the exploration at the early stage, we avoid imperfect\\nheuristic from misguiding the A* search at the beginning. For each FSM nodevand each layer d, we only keep the top k= 10 .\\nFinally, we run the value iteration on the entire search tree.\\nTo accelerate this search process, for all tasks tin the training set, we have initialized a dedicated value approximator Vt(\\u0016s),\\nconditioned on the historical state sequence. During training, we use the value iteration result on the generated search tree to\\nsupervise the learning of this approximator Vt. Meanwhile, we use the value prediction of Vtas the heuristic function for node\\npruning. During test, since we may encounter unseen tasks, the A\\u0003-FSM search uses a uniform heuristic function h\\u00110.\\nSearch on an FSM.For a given initial state s0and task description t, we \\ufb01rst build FSMtand add the search tree node (s0;v0) Search on an FSM.For a given initial state s0and task description t, we \\ufb01rst build FSMtand add the search tree node (s0;v0)\\nto the search tree, where v0is the initial node of the FSM. Then we expand the search tree nodes (s;v)by a topological order of v.\\nIt has two stages. First, for each FSM nodev, we run up to 5000 steps of A\\u0003search. Next, for all search tree nodes (s;v)atFSM\\nnodev, we try to make a transition from (s;v)to(s;v0)where (v;v0)is a valid edge in FSMt. Finally, we output a trajectory\\nending at the FSM nodevTwith minimum cost. Optimality of the A* algorithm on FSM.In the current implementation, RSGs might return sub-optimal solutions even with\\na perfect heuristic, because RSGs balance the expanded nodes across all FSM nodes: it \\ufb01rst samples an FSM node and then\\nexpand a search tree node with the best heuristic value on that node.\\nThe optimality can be guaranteed by either of the following simple modi\\ufb01cations, although at the cost of possibly increasing\\nthe running time:\\n\\u2022Always expand the search node with the globally best admissible heuristic value. (Because our heuristic is learned, this may\\nnot be practical.)\\n\\u2022Keep expanding nodes, even after \\ufb01nding a plan, until none of the unexpanded search tree nodes across all subgoal nodes in\\nthe FSM have better heuristic values than the current best solution.\\nTransitions on FSM\\nAlgorithm 3: The dynamic programming for computing score (\\u0016s;\\u0016a;t;\\u0012)givenGo(\\u0001;\\u0012)and Rat (si;vi;ai;t;\\u0012).\\nInitializef[1::n;0::T]to\\u00001\\nTopological sort all FSM nodes v0::Tso that for all 0\\u0014i<j\\u0014T, there is no path from vjtovion FSM. Clearly v0is still\\nthe start node and vTis the terminate node.\\nf[n;T] 0\\nfori=n::1do\\nforj=T::0do\\nifi<n then\\nf[i;j] Rat(si;vj;ai;t;\\u0012) +f[i+ 1;j]\\nend if\\nfor dok= 0::T:\\nif(vj;vk)2FSM transitions then :\\nf[i;j] maxf[i;j];logGvj(si;\\u0012) + log(1\\u0000Gvk(si;\\u0012) +f[i+ 1;k]\\nend if\\nend for\\nend for\\nend for\\nReturnscore (\\u0016s;\\u0016a;t;\\u0012) =f[1;0]\\nWhen encoding transitions on FSM, we use dynamic programming\\u2020to select a transition that maximize our score . Algorithm\\n3 shows the pseudo-code of the dynamic programming. If we consider IvandGvas \\u201csoft\\u201d probabilities, the computation of\\nscore \\ufb01nds \\u0016s0and\\u0016a0that maximize the rationality of primitive actions and the likelihood that FSM transitions are successful. We\\nusescore (\\u0016s;\\u0016a;t)to rank all candidate tasks.\\nThere is another formulation which is to consider the stochastic transitions using Gvas probabilities. These two formulations\\ncan be uni\\ufb01ed using a framework of latent transition models, though they are computed using different DP algorithms and may\\nlead to different results.\\nFirst of all, these two formulations are equivalent when the goal classi\\ufb01ers are binary (0\\/1). When the classi\\ufb01ers are\\napproximated by \\u201dsoft\\u201d functions that indicate the probability they are satis\\ufb01ed, the two formulations correspond to two\\napproaches of integrating reward (i.e. rationality in our model). The stochastic transition formulation computes the expected\\nrationality, and our formulation can be viewed as an approximation of maximum-likelihood estimated rationality \\u2013 we take\\nmax\\u001c\\u0015log Pr( transitions in \\u001care successful )+Rationality (\\u0016s;\\u0016a;\\u001c). It would be an interesting extension to adopt the stochastic\\ntransition formulation (i.e., E\\u001c\\u0015log Pr( transitions in \\u001care successful ) +Rationality (\\u0016s;\\u0016a;\\u001c)) and use a stochastic planner or\\nMDP solver, although the planning time might be substantially increased.\\nSecond, even if these two approaches behave differently in some cases, but it is unclear which one is better: this is a fundamental\\nchallenge in planning: how should the robot decide whether it has \\ufb01nished a task if there is no indication (such as rewards) from\\nthe environment?\\nDiscover and search with dependencies\\nEvaluate goal classi\\ufb01ers when computing \\ufb01rst(\\u0016s;o)for discovering dependencies. Recall that when discovering the de-\\npendencies, we need to compute \\ufb01rst(\\u0016s;o), the smallest index isuch thatGo(si)is true. We need to round Go(si)to Boolean\\nvalue because we internally use the soft version of Go(\\u0001).\\nFor each subgoal o, we evaluate Goon all states in the training set to obtain the minimum and maximum output MinGoand\\nMaxGo, and we consider Go(si)to evaluate to true iff Go(si)\\u0015pMinGo\\u0001MaxGo.\\n\\u2020The dynamic programming is similar to Dynamic time warping(DTW) warping trajectories into sequential subgoals, but the \\u201ccost\\u201d is\\ncomputed at each segment instead of at matched positions. Tuning hyperparameters. There are several hyper-parameters that need to be tuned: \\u0015;\\u000b;\\r;\\f , and we discuss them separately\\nbelow:\\n\\u2022\\u0015is determines the importance of satisfying the transition condition when transiting compared to regular action cost. In the\\nenvironment, the cost of a single move C(s;a)is set to 0:1, and\\u0015is set to 1.\\n\\u2022\\u000bis called the inverse rationality, the higher \\u000b, the more rationality is assumed of the agent i.e. the agent is assume to take the\\noptimal action with higher strategy. We set \\u000b= 1in our experiments.\\n\\u2022\\u0015;\\u000b and the environmental action costs jointly determine the weight of FSM transitions, as well as the tolerance of\\nsuboptimality in the demonstrations (the higher \\u000bthe lower tolerance). Our model is not sensitive to \\u000b(setting 0:1<\\u000b< 10\\nhave similar performance) because the trajectories in our dataset are close to optimal, and our model is a bit sensitive to the\\nratio of\\u0015to the action cost. We found that expected-number-of-steps-per-subgoal \\u0002action-cost is a good value for \\u0015\\u2014it\\ngives about the same weight to the transition appropriateness and the rationality along the path to achieve it.\\n\\u2022\\ris the weight of the classi\\ufb01cation loss in the loss function J(\\u0012). We set\\r= 0:1and we found that 0:01<\\r < 0:1all work\\nwell.\\n\\fis to adjust the base for the softmax function for the classi\\ufb01cation, which is set to 1. Note that\\fand\\rjointly determine the\\nweight of classi\\ufb01cation loss. Similarly, we have found that 0:1<\\f < 1all work well.\\nScalability and complexity of instruction generating. Meanwhile, the ef\\ufb01ciency of our hierarchical search can be justi\\ufb01ed\\ntheoretically, even at the worst case when we are doing enumerating search approach without well-discovered dependencies. Say\\nwe have a subgoal set O, a primitive action set A, and each subgoal can be completed in lactions, and the task can be achieved by\\nsequencingmsubgoals. Our two-level search generates up to O(jOjm)candidate subgoal sequences and searching each sequence\\ntakesO(mjAjl)time. Thus, the worst-case complexity is O(mjOjmjAjl)which is still better than a pure primitive-level search,\\nwhich is at worst O(jAjml), because the number of subgoals jOjis usually much smaller than jAjl(the number of all possible\\nlengthlsequences).\\nDataset\\nCrafting World\\nOur Crafting World environment is based on the Crafting environment introduced by (Chen, Gupta, and Marino 2021). The\\nenvironment has a crafting agent that can move in a grid world, collect resources, and craft items. In the original environment,\\nevery crafting rule is associated with a unique crafting station (e.g., paper must be crafted on the paper station). We modi\\ufb01ed the\\nrules such that some crafting rules can share a common crafting station (e.g., both arrows and swords can be crafted on a weapon\\nstation). We add additional tiles: doors and rivers into the environment. Toggling a speci\\ufb01c switch will open all doors. Otherwise,\\nthe agent can move across doors when they are holding a key. Meanwhile, the agent can move across rivers when they have a\\nboat in their inventory.\\nWe have used 47object types in Crafting World including obstacles (e.g., river tiles, doors), items (e.g., axe), resources (e.g.,\\ntrees), and crafting stations. We use 27rules for mining resources and crafting items. When the agent is at the same tile as\\nanother object, the toggle action will trigger the object-speci\\ufb01c interaction. For item, the toggle action will pick up the object. For\\nresource, the toggle action will mine the resource if the agent has space in their inventory and has the required tool for mining\\nthis type of resource (e.g., pickaxe is needed for mining iron ore).\\nState representation. The state representation of Crafting World consists of three parts.\\n1. The global feature contains the size of grid world, the location of the agent, and the inventory size of the agent.\\n2.Theinventory feature contains an unordered list of objects in the agent\\u2019s inventory. Each of them is represented as a one-hot 2.Theinventory feature contains an unordered list of objects in the agent\\u2019s inventory. Each of them is represented as a one-hot\\nvector indicating its object type.\\n3.Themap feature contains all objects on the map, including obstacles, items, resources, and crafting stations. Each of them is\\nrepresented by a one-hot type encoding, the location (as integer values), and state (e.g., open orclosed for doors).\\nAction. In Crafting World, there are 5primitive level actions: up,down ,left,right , and toggle . The \\ufb01rst four actions will move\\nthe agent in the environment, while the toggle action will try to interact with the object in the same cell as the agent.\\nState feature extractor. Since our state representation contain a varying number of objects, we extract a vector representation\\nof the environment with a relational neural network: Neural Logic Machines (Dong et al. 2019).\\nConcretely, we extract the inventory feature and the map feature separately. For each item in the inventory, we concatenate its\\ninput representation (i.e., the object type) with the global input feature. We process each item with the same fully-connected\\nlayer with ReLU activation. Following NLM (Dong et al. 2019), we use a max pooling operation to aggregate the feature for all\\ninventory objects, resulting in a 128-dim vector. We use a similar architecture (but different neural network weights) to process\\nall objects on the map. Finally, we concatenate the extracted inventory feature (128-dim), the map feature (128-dim), and the\\nglobal feature (4-dim) as the holistic state representation. Thus, the output feature dimension for each state is 260. Primitive\\ngrab-pickaxe grab-axe grab-key toggle-switch\\ncraft-wood-plank craft-stick craft-shears craft-bed\\ncraft-boat craft-sword craft-arrow craft-cooked-potato\\ncraft-iron-ingot craft-gold-ingot craft-bowl craft-beetroot-soup\\ncraft-paper mine-gold-ore mine-iron-ore mine-sugar-cane\\nmine-coal mine-wood mine-feather mine-wool\\nmine-potato mine-beetroot\\nCompositional\\ngrab-pickaxe grab-axe\\ngrab-key toggle-switch\\nmine-wood then craft-wood-plank craft-wood-plank then craft-stick\\ncraft-iron-ingot orcraft-gold-ingot then craft-shears mine-wool andcraft-wood-plank then craft-bed\\ncraft-wood-plank then craft-boat craft-iron-ingot andcraft-stick then craft-sword\\nmine-feather andcraft-stick then craft-arrow mine-potato andmine-coal then craft-cooked-potato\\nmine-iron-ore andmine-coal then craft-iron-ingot mine-gold-ore andmine-coal then craft-gold-ingot\\ncraft-wood-plank orcraft-iron-ingot then craft-bowl craft-bowl andmine-beetroot then craft-beetroot-soup\\nmine-sugar-cane then craft-paper grab-pickaxe then mine-gold-ore\\ngrab-pickaxe then mine-iron-ore grab-pickaxe orgrab-axe then mine-sugar-cane\\ngrab-pickaxe then mine-coal grab-axe then mine-wood\\ncraft-sword then mine-feather craft-shears orcraft-sword then mine-wool\\ngrab-axe ormine-coal then mine-potato grab-axe orgrab-pickaxe then mine-beetroot\\nNovel\\n1. mine-sugar-cane then craft-paper\\n2. mine-potato and(gran pickaxe then mine-coal) andcraft-cooked-potato\\n3. mine-beetroot and(grab-axe then mine-wood then craft-wood-plank then craft-bowl) then craft-\\nbeetroot-soup\\n4. grab-axe then mine-wood then craft-wood-plank then grab-pickaxe then mine-iron-ore andmine-\\ncoal then craft-iron-ingot then craft-shears then mine-wool then craft-bed\\n5. grab-axe then mine-wood then craft-wood-plank then craft-stick then grab-pickaxe then mine-\\niron-ore andmine-coal then craft-iron-ingot then craft-sword then mine-feather then mine-wood\\nthen craft-wood-plank then craft-stick then craft-arrow\\n6. grab-key then grab-axe\\n7. toggle-switch then mine-beetroot\\n8. grab-axe then mine-wood then craft-wood-plank then craft-boat then mine-sugar-cane\\n9. grab-axe then mine-wood then craft-wood-plank then craft-boat then grab-pickaxe\\n10. grab-key then grab-axe then mine-wood then craft-wood-plank then craft-boat then mine-potato\\n11. grab-key or(grab-axe then mine-wood then craft-wood-plank then craft-boat) then grab-pickaxe\\nthen mine-gold-ore\\n12. grab-axe then mine-wood then craft-wood-plank then craft-boat then grab-key ortoggle-switch\\nthen grab-pickaxe then mine-iron-ore andmine-coal then craft-iron-ingot\\nTable 2: Task descriptions in the primitive ,compositional andnovel sets for the Crafting World.\\nTask de\\ufb01nitions. We list the task descriptions in the primitive , the compositional , and the novel splits Table 2. Table 3 lists 8\\n\\ufb01nal goals and corresponding full instructions that are used in search for a single \\ufb01nal goal.\\nPlayroom\\nWe build our Playroom environment following Konidaris et al. (Konidaris, Kaelbling, and Lozano-Perez 2018). Speci\\ufb01cally, we\\nhave added obstacles into the environment. The environment contains an agent, 6 effectors (a ball, a bell, a light switch, a button\\nto turn on the music, a button to turn off the music and a monkey), and a \\ufb01x number of obstacles. The agent and the effectors\\nhave \\ufb01xed shapes. Thus, their geometry can be fully speci\\ufb01ed by their location and orientation. For simplicity, we have also \\ufb01xed\\nthe shape and the location of the obstacles.\\nState representation. We represent the pose of the agent by a 3D vector including the x, y coordinates (real-valued) and\\nits rotation (real-valued, in [\\u0000\\u0019;\\u0019). The state representation consist of the pose of the agent (as a 3-dimensional vector) and\\nthe locations of six effectors (as 6 2-dimensional vectors). Note that the state representation does not contain the shapes nor\\nthe locations of obstacles as they remain unchanged throughout the experiment. We concatenate these 7 vectors as the state\\nrepresentation. Final Goal Steps Example full instruction\\nmine-wood 2 grab-axe then mine-wood\\ncraft-paper 2 mine-sugar-cane then craft-paper\\ncraft-beetroot-soup 3 (mine-beetroot andcraft-bowl) then craft-beetroot-soup\\ncraft-bed 3 (craft-wood-plank andmine-wool) then craft-bed\\ncraft-gold-ingot 4grab-pickaxe then (mine-gold-ore andmine-coal)\\nthen craft-gold-ingot\\ncraft-boat 4grab-axe then mine-wood then craft-wood-plank\\nthen craft-boat\\ncraft-cooked-potato 4((grab-pickaxe then mine-coal) andmine-potato)\\nthen craft-cooked-potato\\ncraft-shears 5grab-pickaxe then (mine-coal andmine-iron-ore)\\nthen craft-iron-ingot then craft-shears\\nTable 3: Task descriptions used in search for a single \\ufb01nal goal in Crafting World.\\nAction. The agent has a 3-dimensional action space: [\\u00001;1]3. That is, for example, at each time step, the agent can at most\\nmove 1 meter along the x axis. We perform collision checking when the agent is trying to make a movement. If an action will\\nresult in a collision with objects or obstacles in the environment, the action will be treated as invalid and the state of the agent\\nwill not change.\\nTask de\\ufb01nitions. We list the task descriptions in each of the primitive ,compositional andnovel set of the Playroom in Table 4\\nPrimitive\\nplay-with-ball ring-bell turn-on-light\\ntouch the mounkey turn-off-music turn-on-music\\nCompositional (designed meaningful tasks)\\nplay-with-ball\\nturn-on-light then ring-bell\\nturn-on-music andplay-with-ball then touch the monkey\\nplay-with-ball then turn-on-light\\nturn-on-music andplay-with-ball then turn-off-music\\nturn-on-music orplay-with-ball\\nturn-off-music then play-with-ball then turn-on-music\\nturn-on-music andplay-with-ball andturn-on-light then ring-bell\\nNovel (randomly sampled)\\nplay-with-ball then turn-on-light orring-bell\\nturn-on-music then turn-on-light\\nturn-on-music then turn-on-light\\nplay-with-ball then touch the monkey\\nturn-on-music then turn-off-music\\nturn-on-music andring-bell then touch the monkey\\nring-bell then touch the monkey then turn-on-light\\nturn-on-light and(ring-bell orturn-on-music) then play-with-ball\\nTable 4: Task descriptions in the primitive ,compositional andnovel sets for the Playroom. Baseline Implementation Details\\nIn this section, we present the implementation details of RSGs and other baselines. Without further notes, through out this\\nsection, we will be using the same LSTM encoder for task descriptions in TL, and the same LSTM encoder for state sequences.\\nThe architecture of both encoders will be presented in Appendix .\\nLSTM\\nTask description encoder. We use a bi-directional LSTM (Hochreiter and Schmidhuber 1997) with a hidden dimension of\\n128 to encode the task description. The vocabulary contains all primitive subgoals, parentheses, and three connectives ( and,or,\\nandthen). We perform an average pooling on the encoded feature for both directions, and concatenate them as the encoding for\\nthe task description. Thus, the output dimension is 256.\\nState sequence encoder. For a given state sequence \\u0016s=fsig, we \\ufb01rst use a fully-connected layer to map each state siinto a\\n128-dimensional vector. Next, we feed the sequence into a bi-directional LSTM module. The hidden dimension of the LSTM is\\n128. We perform an average pooling on the encoded feature for both directions, and concatenate them as the encoding for the\\nstate sequence.\\nTraining. In our LSTM baseline for task recognition, we concatenate the state sequence feature and the task description feature,\\nand use a 2-layer multi-layer perceptron (MLP) to compute the score of the input tuple: (trajectory, task description). The LSTM\\nmodel is trained for 100 epochs on both environments. Each epoch contains 30 training batches that are randomly sampled from\\ntraining data. The batch size is 32. We use the RMSProp optimizer with a learning rate decay from 10\\u00003to10\\u00005.\\nInverse Reinforcement Learning (IRL)\\nThe IRL baseline uses an LSTM model to encode task descriptions. We use different parameterizations for the reward function\\nand the Q function in two datasets.\\nCrafting World Since the task description may have complex temporal structures, the reward value does not only condition on\\nthe current state and but all historical states. Therefore, instead of Q(s;ajt)andR(s;a;s0jt), we useQ(\\u0016s;ajt)andR(\\u0016s;a;s0jt)\\nto parameterize the Q function and reward function, where sis the current state, athe action,tthe task description, s0the next\\nstate, and \\u0016sthe historical state sequence from the initial state to the current state.\\nWe use neural networks to approximate the Q function and reward function. For both of them, \\u0016sis \\ufb01rst encoded by an\\nLSTM model into a \\ufb01xed-length vector embedding. We simply concatenate the historical state encoding and the task description\\nencoding, and then use a fully-connected layer to map the feature into a 5-dimensional vector. Each entry corresponds to the Q\\nvalue or the reward value for a primitive action.\\nPlayroom The Q function and reward function in Playroom also condition on all historical states. In Playroom, we parameterize\\nthe value of each state: V(\\u0016s), instead ofQ(\\u0016s;a). We parameterize R(\\u0016s;a;s0)asR(\\u0016s;s0).\\nThe input to our reward function network is composed of three parts: the vector encoding of the historical state sequence, the\\nvector encoding for the next state s0, and the task description encoding. We concatenate all three vectors and run a fully-connected\\nlayer with a logSigmoid activation function.\\u2021\\nIn Playroom, since we do not directly parameterize the Q value for all actions in the continuous action space, in order to\\nsample the best action at each state sfor plan execution, we \\ufb01rst randomly sample 20 valid actions from the action space (i.e.,\\nactions that do not lead to collision), and choose the action that maximizes the Q function: Q(\\u0016s[s0;a), where \\u0016sis the historical\\nstate seuqnce and s0is the next state after taking a.\\nValue iteration. Both environments have a very large (Crafting World) or even in\\ufb01nite (Playroom) state space. Thus it is\\nimpossible to run value iteration on the entire state space. Thus, at each iteration, for a given demonstration trajectory (\\u0016se;\\u0016ae), impossible to run value iteration on the entire state space. Thus, at each iteration, for a given demonstration trajectory (\\u0016se;\\u0016ae),\\nwe construct a self exploration trajectory (\\u0016sp;\\u0016ap)that share the same start state as \\u0016se\\u00a7. We run value iteration on f\\u0016seg[f \\u0016spg.\\nFor states not in this set, we use the Q function network to approximate their values.\\nTraining. For both Crafting World and Playroom, we train the IRL model for 60 epochs. We set the batch size to be 32 and\\neach epoch has 30 training batches. We use a replay buffer that can store 100,000 trajectories. For both environments, we use the\\nAdam optimizer with a learning rate decay from 10\\u00003to10\\u00005. We have found the IRL method unstable to train in the Playroom\\nenvironment. Thus, in Playroom, we use a warm-up training procedure. In the \\ufb01rst 18(30%) epochs, we set \\r= 0for a \\u201cwarm\\nstart\\u201d, and for rest of the epochs we use \\r= 0:5, where\\ris the discount factor in the Q function.\\n\\u2021We have experimented with no activation function, Sigmoid, and logSigmoid activations, and found that the logSigmoid activation works\\nthe best.\\n\\u00a7Since running self-exploration in Playroom is too slow, in practice, we only generate self-exploration trajectories for 4 trajectories in the\\ninput batch. Model #EpochsCrafting World Playroom\\nTraining Time\\n(minute\\/epoch)Planning Time\\n(second\\/sample)Training Time\\n(minute\\/epoch)Planning Time\\n(second\\/sample)\\nRSG 60 17.0 4.6962 32.4 0.4313\\nLSTM 200 0.4 N\\/A 0.2 N\\/A\\nIRL 60 34.8 0.7269 7.1 23.0187\\nBC 150 0.3 0.1615 0.4 0.1688\\nBC(FSM) 150 0.5 0.5423 0.5 0.1875\\nTable 5: Training time per epoch and planning time per sample for all models.\\nBehavior Cloning (BC)\\nBC learns a policy \\u0019(\\u0016s;ajt)from data, where tis the task description, aa primitive action, and \\u0016sthe historical state sequence.\\nThe state sequence \\u0016sis \\ufb01rst encoded by an LSTM model into a \\ufb01xed-length vector embedding.\\nIn Crafting World, we use a fully-connected layer with softmax activation to parameterize \\u0019(aj\\u0016s;t). Speci\\ufb01cally, the input to\\nthe fully-connected layer is the concatenation of the vector encoding of \\u0016sand the vector encoding of the task description t.\\nIn Playroom, we use two fully-connected (FC) layers to parameterize \\u0019(aj\\u0016s;t). Speci\\ufb01cally, we parameterize \\u0019(aj\\u0016s;t)as a\\nGaussian distribution. The \\ufb01rst FC layer has a Tanh activation and parameterizes the mean \\u0016of the Gaussian. The second FC\\nlayer has a Sigmoid activation and paramerizes the standard variance \\u001b2of the Gaussian.\\nTo make this model more consistent with our BC- FSM model, in both environments, we also train a module to compute the\\ntermination condition of the trajectory. That is, a neural network that maps \\u0016sto a real value in [0;1], indicating the probability of\\nterminating the execution. Denote the output of this network as stop(\\u0016s). At each time step, the agent will terminate its policy\\nwith probability stop(\\u0016s). We modulate the probability for other actions aas\\u0019(aj\\u0016s;t)\\u0001(1\\u0000stop(\\u0016s))\\nFor planning in Crafting World, at each step, we choose the action with the maximum probability (including the option\\nto \\u201cterminate\\u201d the execution). In Playroom, we always take the \\u201cmean\\u201d action parameterized by \\u0019(aj\\u0016s;t)until we reach the\\nmaximum allowed steps.\\nWe then de\\ufb01ne the score of a task given a trajectory, score (\\u0016s;\\u0016a;t), as the sum of log-probabilities of the actions taken at each\\nstep. We train this model with the same loss and training procedure as RSGs. We train the model for 100 epochs using the Adam\\noptimizer with a learning rate decay from 10\\u00003to10\\u00005.\\nBehavior Cloning with FSM (BC- FSM)\\nBC- FSM represents task description as an FSM, in the same way as our model RSGs. It represents each subgoal oas a tuple:\\nh\\u0019o(sa);stopo(s)i, corresponding to the subgoal-conditioned policy and the termination condition.\\nTask recognition. The task recognition procedure for BC- FSM jointly segments the trajectory and computes the consistency\\nscore between the task description and the input trajectory. In particular, our algorithm will assign an FSM statevito each state\\nsi, and insert several action. We use a dynamic programming algorithm (similar to the one used by our algorithm for RSGs) to\\n\\ufb01nd the assignment that maximize the overall score:\\nscore (\\u0016s;\\u0016v;\\u0016a):=Y\\nip(aijsi;vi;t)\\np(aijsi;vi;t) =\\u001a\\n\\u0019(aijsi;vi)\\u0001(1\\u0000stop(si;vi))ifa2A is a primitive action\\nstop(si;vi) ifa2Etis an FSM transition\\nPlanning. We use the same strategy as the basic Behavior Cloning model to choose actions at each step, conditioned on the\\ncurrent FSM state. BC- FSM handles branches in the FSM in the same way as our algorithm for RSGs.\\nComputational Source Used for Training and Testing\\nIn Table 5, we list the training time per epoch for our model and all baselines, and averaging time cost for each planning task\\nwhen testing. All models are trained until convergence, and we list the number of epochs after which the loss stops going down.\\nGenerative Adversarial Imitation Learning\\nWe have also tested the Generative Adversarial Imitation Learning (GAIL)(Ho and Ermon 2016) as a baseline on the planning\\ntask on seen instructions. Since GAIL does not natually generalize to structured subgoals (FSMs), we used an LSTM to encode task on seen instructions. Since GAIL does not natually generalize to structured subgoals (FSMs), we used an LSTM to encode\\nthe task description as in the BC and IRL baselines. On CraftingWorld, GAIL achieves 19.2% planning success rate on the Comp.\\nsplit, and 1.0% success rate on the Novel split. GAIL has a better performance compared to BC and BC-FSM, which we think is because it explores the environment during training. There is still a large performance gap between GAIL and RSG, mainly\\nbecause GAIL does not have goal representations that helps compositional generalization.\",\"35\":\" Knowledge-augmented Risk Assessment (KaRA): a hybrid-intelligence framework\\nfor supporting knowledge-intensive risk assessment of prospect candidates\\nCarlos Raoni Mendes, Emilio Vital Brazil, Vinicius Segura, Renato Cerqueira\\nIBM Research\\nRio de Janeiro, RJ, Brazil\\ncraoni@br.ibm.com, evital@br.ibm.com, vboas@br.ibm.com, rcerq@br.ibm.com\\nAbstract\\nEvaluating the potential of a prospective candidate is a com-\\nmon task in multiple decision-making processes in different\\nindustries. We refer to a prospect as something or someone\\nthat could potentially produce positive results in a given con-\\ntext, e.g., an area where an oil company could \\ufb01nd oil, a com-\\npound that, when synthesized, results in a material with re-\\nquired properties, and so on. In many contexts, assessing the\\nProbability of Success (PoS) of prospects heavily depends\\non experts\\u2019 knowledge, often leading to biased and inconsis-\\ntent assessments. We have developed the framework named\\nKARA (Knowledge-augmented Risk Assessment) to address\\nthese issues. It combines multiple AI techniques that con-\\nsider SMEs (Subject Matter Experts) feedback on top of a\\nstructured domain knowledge-base to support risk assessment\\nprocesses of prospect candidates in knowledge-intensive con-\\ntexts.\\nIntroduction\\nThe assessment of the risk of failure or its opposite, the Prob-\\nability of Success (POS), plays a crucial role in deciding\\nwhich prospects are worth investing in. In many critical con-\\ntexts, the prospect risk assessment process carries the char-\\nacteristics of the so-called Knowledge-intensive process (Ci-\\nccio, Marrella, and Russo 2015). When this is the case, there\\nis a strong dependency on the tacit knowledge of multiple\\nexperts in different areas, and a relevant part of the available\\ndata is heavily uncertain. Methods that don\\u2019t correctly han-\\ndle this inherent complexity often lead to biased and incon-\\nsistent assessments. The challenge is to develop a methodol-\\nogy and supporting technology for prospect risk assessment\\nin knowledge-intensive contexts that is consistent, reduces\\nbiases, controls uncertainty, and often leads to the selection\\nof successful prospects (or discoveries). This work describes\\na framework that aims to address these issues. We called it\\nthe Knowledge-augmented Risk Assessment (KaRA) frame-\\nwork.\\nThe recent advances in AI and Knowledge Engineering\\nprovided the basis for the development of KaRA. Knowl-\\nedge engineering practices and technologies are applied to\\nrepresent and integrate the domain knowledge from multi-\\nple data sources and stakeholders and provide easy access\\nCopyright \\u00a9 2023, Association for the Advancement of Arti\\ufb01cial\\nIntelligence (www.aaai.org). All rights reserved.to this knowledge. At the same time, AI provides the appro-\\npriate tools for inference, prediction, uncertainty reduction,\\nconsensus reaching, etc. Then, KaRA combines multiple AI\\ntechniques that consider SME\\u2019s (Subject Matter Experts)\\nfeedback on top of a structured domain KB (Knowledge\\nBase) to support the risk assessment processes of prospect\\ncandidates in knowledge-intensive contexts.\\nThe KaRA framework is a generalization and extension\\nof the work we developed for supporting the assessment of\\nthe geological success of prospects (see Silva et al. (2019)\\nand Vital Brazil et al. (2021)). Next, we detail the general\\nproblem that KaRA approaches.\\nKnowledge-intensive Prospect Risk Assessment\\nIn this work, we refer to a successful prospect as a dis-\\ncovery. Finding discoveries generally starts with many can-\\ndidates from which very little is known. These candidates\\npass through a triage process that selects those worth fur-\\nther investigation. Given the number of candidates, the triage\\noften involves some form of automatic \\ufb01ltering combined\\nwith experts\\u2019 simple evaluations. The candidates that pass\\ntriage became what we refer to as prospects. A prospect can\\npass through a detailed assessment that involves analyzing\\nthe many risk factors that could prevent it from succeeding. triage became what we refer to as prospects. A prospect can\\npass through a detailed assessment that involves analyzing\\nthe many risk factors that could prevent it from succeeding.\\nThis process requires experts\\u2019 analysis from different areas\\nand probably new data acquisition. These resource require-\\nments imply the existence of a limited number of prospects.\\nThen, a \\ufb01nal investment is made to an even smaller number\\nto become discoveries or failures. For instance, in oil and\\ngas exploration, this last investment would mean drilling a\\nwell, while in materials discovery, that would mean synthe-\\nsizing a new compound. Both examples are very resource-\\ndemanding, but they could generate high rewards on the\\nother side. Therefore, the prospect selection process is es-\\nsential to creating a successful business. At each step of\\nthe process the number of candidates decreases, while at\\nthe same time, more is known about them. Another essen-\\ntial aspect is that each step requires increasing resource in-\\nvestments. In Figure 1, we detail the work\\ufb02ow for assessing\\nprospects\\u2019 POS. It starts with acquiring new data related to\\nthe prospect to be analyzed. The available data is then pro-\\ncessed and interpreted by experts from different areas. This\\nprocess is carried out to detail relevant information and char-\\nacteristics resulting in a prospect characterization suited forarXiv:2303.05288v1  [cs.AI]  9 Mar 2023 Figure 1: Prospect Risk Assessment Work\\ufb02ow\\nthe risk assessment methodology. It\\u2019s worth noting that the\\navailable data is often limited and uncertain. The experts\\u2019 in-\\nterpretation compensates for these limitations with their ex-\\nperience and tacit knowledge. But the \\ufb01nal characterization\\nis still subject to uncertainty and may be biased by the ex-\\nperts\\u2019 opinions. The risk assessment methodology then takes\\nthe characterization as input and produces the prospect\\u2019s\\nPOS itself. It is also a process that depends on experts\\u2019 opin-\\nions and knowledge. It becomes only a guessing exercise for\\nexperts if done in an unstructured form, often leading to in-\\nconsistent and biased assessments (Milkov 2015).\\nThe prospect\\u2019s POS helps decision-makers decide if they\\nshould invest, discard, or need more information\\/data and\\nthen another round of assessment. This last decision is rep-\\nresented in Figure 1 by the backward arrow. Essentially an\\nenduring decision (invest or discard) is made when there is\\nenough con\\ufb01dence in the current knowledge of the prospect\\nand, consequently, in the assessment. Some risk assessment\\nmethodologies use the metric Level-of-Knowledge (LOK)\\nto represent how much is known about the prospect explic-\\nitly (Lowry, Suttill, and Taylor 2005).\\nStructure\\nIn the next section, we detail the related literature with a\\nfocus on the works in the Oil & Gas exploration space.\\nWe assume that it is the most advanced industry regarding\\nprospect risk assessment. In the section KaRA Framework ,\\nwe present an overview of the proposed methodology de-\\nveloped in KaRA. Subsequent sections are dedicated to de-\\ntail the methodology phases, that are, prospect characteri-\\nzation, Level-of-Knowledge assessment, and Probability-of-\\nSuccess assessment.\\nRelated Literature\\nThe generalization that we proposed for knowledge-\\nintensive prospect risk assessment was derived from the Oil\\n& Gas exploration domain. In this context, a prospect is a re-\\ngion containing a potentially recoverable oil accumulation.\\nTo con\\ufb01rm or discard the hypothesis of oil accumulation in\\nthe prospect region, the company must drill a well, which is\\nvery resource demanding. It is a high-risk, high-reward sit-uation, so this industry treats it very carefully. We believe\\nthat prospect risk assessment in the Oil & Gas exploration\\nspace is mature compared to other prospect exploration do-\\nmains. Therefore, we will use its literature as a reference in\\nthis section.\\nGiven the complexity involved in assessing a prospect, a\\ncommon approach adopted by the industry is to break the as-\\nsessment into different geological (or risk) factors, evaluat-\\ning the POS of each one and then combining the correspond-\\ning probabilities to obtain the POS for the prospect itself.\\nThe factors correspond to geological features that should ex-\\nist to form an oil accumulation. One common assumption to\\nfacilitate the analysis is to adopt independence between the\\nrisk factors. In this case, the prospect\\u2019s POS is the multi-\\nplication of all individual factor\\u2019s POS values. So, choosing\\nhow to break the assessment into different factors plays a\\ncrucial role in the methodology. An excessive number may\\nresult in very small and hard to differentiate prospects\\u2019 POS\\nvalues, while a small number may neglect essential dimen-\\nsions of the risk evaluation. So, the methods adopted across\\ndifferent organizations vary in the number of geological risk\\nfactors analyzed. Milkov (2015) presents a table comparing\\nvarious methodologies found in the literature. To illustrate\\nthis variation, the method proposed by Lowry, Suttill, and\\nTaylor (2005) considers the assessment of 19 geological risk\\nfactors, while Rose (2001) analyzes only \\ufb01ve.\\nThe methodologies also vary in the strategy that guides\\nPOS assessments for each geological factor. They must bal-\\nance the quality and quantity of the available data with the\\ngeological model characteristics to \\ufb01ne-tune the range of\\npossible values experts could adopt for the POS. In (Milkov ance the quality and quantity of the available data with the\\ngeological model characteristics to \\ufb01ne-tune the range of\\npossible values experts could adopt for the POS. In (Milkov\\n2015) and (Jan-Erik, Bundiit, and Simplicil 2000), the au-\\nthors propose using reference tables. To use a geological fac-\\ntor reference table, one must identify which cell the prospect\\nmaps based on the available data and geological model char-\\nacteristics. While in (Milkov 2015), the cell speci\\ufb01es the\\nPOS value itself, in (Jan-Erik, Bundiit, and Simplicil 2000),\\nthey provide a range of possible values that POS could as-\\nsume. Although adopting these tables may improve the con-\\nsistency and repeatability of the assessment process, they are\\nnot suitable to accommodate new disruptive knowledge. One\\nclear example of this problem is that the tables become out- Knowledge Base (KB)Structured Domain Knowledge \\nStructured Domain DataUnstructured Domain DataAI-discovery AgentsKnowledge-augmented Risk Assessment (KaRA)\\nDiscovery WorkbenchFigure 2: Overview of KaRA \\u2013 methodology work\\ufb02ow and features.\\ndated when new technologies are developed.\\nAnother popular method is the use of the metric Level-of-\\nKnowledge (LOK) to assess data availability, quantity, and\\nquality (see (Paltrinieri, Comfort, and Reniers 2019)). Some\\napproaches use the LOK metric to constrain the possible\\nrange of values assigned to the POS. In (Rose 2001), we \\ufb01nd\\na matrix specifying a relation where the POS is restricted to\\nthe middle of the scale (around \\ufb01fty percent) when the LOK\\nis low. At the same time, at high LOK levels, the POS value\\nshould be either very low (around zero percent) or very high\\n(around a hundred percent). The rationale is that in a low\\ncon\\ufb01dence situation, one should not be conclusive about the\\nPOS value, while in high con\\ufb01dence cases, one must be very\\nassertive. In (Lowry, Suttill, and Taylor 2005), they apply the\\nsame rationale and create a graphic that speci\\ufb01es a LOK and\\nPOS relationship by delimiting a region of allowed LOK and\\nPOS assessment pairs. The same work also presents alterna-\\ntive LOK\\/POS relationships that implement different ratio-\\nnales. The authors also recognize that making quantitative\\nassessments of LOK and using it to manipulate POS is chal-\\nlenging both in theory and practice.\\nThe KaRA framework aims at addressing the limitations\\nof the described methods. The main objective is to provide a\\nmethodology that guarantees consistency, decreases biases,\\nand continuously improves given new relevant knowledge\\nobtained from data sources and or SMEs, all supported by\\npractical and meaningful LOK quantitative scales.\\nKaRA Framework\\nThe KaRA framework is a generalization and extension of\\nthe work we developed for supporting the assessment of the\\ngeological success of prospects (see Silva et al. (2019), and\\nVital Brazil et al. (2021)). It combines multiple AI tech-\\nniques that consider SMEs\\u2019 feedback on top of a structured\\ndomain knowledge base to support the risk assessment pro-\\ncesses of candidates in knowledge-intensive contexts. The\\nrecent advances in AI and Knowledge Engineering providedthe basis for the development of KaRA. Knowledge en-\\ngineering practices and technologies are applied to repre-\\nsent and integrate the domain knowledge from multiple data\\nsources and stakeholders and provide easy access to this\\nknowledge. At the same time, AI provides the appropriate\\ntools for inference, prediction, uncertainty reduction, con-\\nsensus reaching, etc.\\nThe methodology implemented by KaRA divides the as-\\nsessment into multiple risk factors. Individual risk factor\\nevaluation encompasses three phases: risk factor characteri-\\nzation, LOK assessment, and POS assessment. Different AI\\nagents support expert decisions at each one of them. As men-\\ntioned, the domain knowledge is structured and integrated\\nby the KB. The KB also handles all the data used by KaRA\\ncomponents. KaRA aims to enable a co-creation environ-\\nment for SMEs and AI agents to support the risk assessment\\nprocess. Figure 2 summarizes the architecture, the method-\\nology work\\ufb02ow, and some of the main features of KaRA.\\nWe discuss those features in the sequence according to each\\nphase.\\nRisk Factor Characterization\\nWe adopted a similar approach to (Milkov 2015) and (Jan-\\nErik, Bundiit, and Simplicil 2000) in the risk factor char-\\nacterization phase. For each risk factor, experts should an-\\nswer a standard questionnaire showcasing the data analy-\\nsis work\\ufb02ow results that could in\\ufb02uence LOK or POS as-\\nsessments. The characterization questionnaires structure the\\ncandidates in the KB, providing an easy way to represent\\nthem for supervised and unsupervised machine learning al-\\ngorithms. One representative example used in the following\\nphases is retrieving similar candidates in terms of characteri- them for supervised and unsupervised machine learning al-\\ngorithms. One representative example used in the following\\nphases is retrieving similar candidates in terms of characteri-\\nzation for comparison purposes. The differentiator of KaRA\\nwill be the ability to query and reason through the KB to re-\\ntrieve and rank relevant evidence that may help the experts\\nanswer the characterization questions. The expert will curate\\nthe evidence recovered from the KB, keeping only the ones GlobalLOKExpert LOKCandidatesExpert LOK ComparisonsRule-based InferenceConsistent Expert LOK Comparisons \\nPeer-reviewed CandidatesUnder-review CandidatesReference LOK ML TrainingReference LOK ML PredictionReference LOK Trained ModelReference LOK ScoresLOK Calibration LP ModelExpert LOK Scores\\nAll Experts LOK ComparisonsConflicts Minimization IP ModelConsistent Consensus LOK Comparisons LOK Calibration LP ModelGlobal LOK ScoresFigure 3: Work\\ufb02ows for expert and global LOK scores.\\nrelevant to the answer. This feedback feeds the ranking algo-\\nrithms used in the evidence retrieval, which are continuously\\nimproved. This mechanism establishes a co-creation process\\nwhere AI and experts collaborate to accurately characterize\\nthe candidate risk factors taking advantage of available evi-\\ndence stored in the KB and the expert\\u2019s tacit knowledge.\\nLOK Assessment\\nThe role of the LOK assessment phase is to establish fair\\nand consistent LOK score scales dependent on the risk fac-\\ntor characterizations produced in the previous step to support\\nthe POS assessment process, such as done in (Lowry, Suttill,\\nand Taylor 2005) and (Rose 2001). The LOK scales are cre-\\nated based on pairwise comparisons made by the experts and\\npreviously assessed, and peer-reviewed candidates present\\nin KB. These candidates serve as training examples for a\\nsupervised machine learning algorithm that gives an initial\\nLOK estimate, the so-called reference LOK. Experts\\u2019 pair-\\nwise LOK comparisons calibrate this estimate by solving a\\nLinear Programming (LP) model that determines the small-\\nest overall adjustments in the LOK values needed to respect\\nthe comparisons. A rule-based inference strategy maintains\\nthe consistency of comparisons for a particular individual.\\nThe comparisons at the individual level capture the ranking\\nopinions of each expert. The reference LOK estimates cal-\\nibrated by these comparisons create the expert LOK scale.\\nThe system also determines the subset of consistent LOK\\ncomparisons that best represent experts\\u2019 consensus by solv-\\ning an Integer Programming (IP) model that minimizes the\\ncon\\ufb02icts between the output comparisons and the overall\\nexpert opinions. We obtain the so-called global LOK scale\\nwhen these comparisons calibrate the reference LOK esti-\\nmates with the same LP model previously described. Ad-\\nditional candidates and comparisons continuously adjust ex-\\npert and global LOK scales. Figure 3 summarizes expert and\\nglobal LOK work\\ufb02ows. This approach is another example of\\na human-AI co-creation strategy applied to support an essen-\\ntial step in the discovery process.LOK comparisons consistency trough rule-based infer-\\nence As previously mentioned, when evaluating a risk fac-\\ntor of a target prospect, experts compare pairwisely the risk\\nfactor characterization of the target prospect with other sim-\\nilar risk characterizations present in the KB. In the pairwise\\ncomparison, the expert answers which risk characterization\\nshould receive a higher LOK score or if their LOK scores\\nshould be the same. A rule-based approach keeps the con-\\nsistency of LOK comparisons for each expert. Given risk\\nfactor characterizations a,b, and c, and Lirepresenting the\\nLOK score of characterization i, the consistency rules for\\npairwise LOK comparisons are:\\n\\u2022(La< Lb)^(Lb< Lc) =)La< Lc\\n\\u2022(La=Lb)^(Lb=Lc) =)La=Lc\\n\\u2022(La< Lb)^(Lb=Lc) =)La< Lc\\nIn \\ufb01gure 4, we show a graph representation of canonical ex-\\npert\\u2019s LOK comparisons. The nodes represent the risk factor\\ncharacterizations, directed edges from a node ito a node\\njrepresent a LOK comparison where Li> L j, while the\\nbi-directional ones represent a LOK comparison Li=Lj.\\nFigure 5 shows the graph representation after applying LOK\\nA\\nB\\nC D E\\nFigure 4: Canonical LOK comparisons graph\\nconsistency rules. The inference of those rules could be ob-\\ntained by polynomial-time graph-based algorithms such as\\nthe ones found in (Mohr and Henderson 1986).\\nReference LOK and LOK Calibration LP Reference tained by polynomial-time graph-based algorithms such as\\nthe ones found in (Mohr and Henderson 1986).\\nReference LOK and LOK Calibration LP Reference\\nLOK scores provide a \\ufb01rst quantitative estimation of LOK A\\nB\\nC\\nD\\nEFigure 5: Consistent LOK comparisons graph after inference\\nbased on peer-reviewed examples, which means that a group\\nof experts validated their \\ufb01nal LOK assessment scores. Us-\\ning a one-hot encoding representation of risk-factor charac-\\nterizations, we apply supervised machine learning (ML) al-\\ngorithms to estimate the reference LOK scores. The model\\nis re-trained whenever a new peer-reviewed assessment is\\nstored in the KB. We use the average loss of a ten-fold\\ncross-validation test to choose the best ML model (i.e., al-\\ngorithm and parametrization). Despite the usefulness of ref-\\nerence LOK, we adopt in KaRA an approach that prioritizes\\nexpert feedback. We create individual LOK scales by cali-\\nbrating the reference LOK estimates with the expert\\u2019s con-\\nsistent LOK comparisons. Given a set of risk factor charac-\\nterizations P, reference LOK estimates LRi(8i2P), a set\\nof consistent greater than GTand equals EQpairwise (i; j)\\nLOK comparisons, the following LP is used to estimate the\\nexpert\\u2019s LOK scores Li(8i2P):\\nMinX\\ni2PjLi\\u0000LRij (1)\\ns:t:\\nLi\\u0000Lj\\u0015t8(i; j)2GT (2)\\nLi\\u0000Lj= 08(i; j)2EQ (3)\\n0\\u0014Li\\u001418i2P (4)\\nThe objective function (1) minimizes the absolute difference\\nbetween the expert\\u2019s LOK scores Liand the reference LOK\\nLRi. Constraints (2) guarantee that the LOK score of iis\\ngreater than the LOK score of jby at least a threshold t\\nfor each greater than comparison (i; j)2GT. Similarly,\\nconstraints (3) guarantee that equality pairwise comparisons\\n(i; j)2EQare respected. Constraints (4) de\\ufb01ne the domain\\nof LOK scores to be between 0and 1. Despite the absolute\\nvalue function (1), we can transform this formulation into a\\nstandard form LP following strategies such as described in\\n(Bertsimas and Tsitsiklis 1997).\\nGlobal LOK Con\\ufb02icts Minimization (or Consensus) IP\\nAs previously explained, the framework maintains a set of\\nconsistent LOK comparisons for each expert. So, the opin-\\nions of different experts may con\\ufb02ict regarding the LOK\\npairwise relation for pair of characterizations aandb, which\\ncould be higher ( La> L b), lower ( La< L b) or equal\\n(La=Lb). In this context, achieving overall consistencywould mean discovering the set of pairwise LOK relations\\nthat minimizes the con\\ufb02icts with all expert opinions. In\\n(Brancotte et al. 2015), authors proposed an Integer Pro-\\ngramming (IP) model to solve a ranking aggregation prob-\\nlem with ties and partial orderings. We used this formulation\\nas a reference to create a new one with fewer constraints\\nand suitable for our context. Next, we describe each element\\nused in the formulation and present it in the sequence:\\n\\u2022xa\\u0014b\\u2013 binary decision variable that indicates if La\\u0014Lb.\\n\\u2022xa=b\\u2013 binary decision variable that indicates if La=Lb.\\n\\u2022wa\\u0014b\\u2013 constant representing the number of experts that\\nchose lower than or equal as pairwise LOK relation be-\\ntween aandb.\\n\\u2022wa=b\\u2013 constant representing the number of experts that\\nchose equal as pairwise LOK relation between aandb.\\nMinX\\naX\\nb(wa\\u0014b\\u0003xb\\u0014a+\\nwb\\u0014a\\u0003xa\\u0014b\\u00002(wa=b)\\u0003xa=b)(5)\\ns:t:\\n\\u0000xa\\u0014b\\u0000xb\\u0014a\\u0014\\u0000 18a2P; b2P (6)\\n2xa=b\\u0000xa\\u0014b\\u0000xb\\u0014a\\u001408a2P; b2P (7)\\nxa\\u0014b+xb\\u0014a\\u0000xa=b\\u001418a2P; b2P (8)\\nxa\\u0014c+xc\\u0014b\\u0000xa\\u0014b\\u001418a2P; b2P; c2P (9)\\nxa\\u0014b2f0;1g 8a2P; b2P (10)\\nxa=b2f0;1g 8a2P; b2P (11)\\nThe rationale of this formulation is to guarantee that xa\\u0014b=\\n1andxb\\u0014a= 1 implies xa=b= 1. This way, a strict lower\\nrelation happens when xa\\u0014b= 1andxa=b= 0. The objec-\\ntive function (5) considers this assumption to represent the\\nminimization of con\\ufb02icts. The constraints (6) ensures that at\\nleast one relation is adopted for each pair of elements aand\\nb. The constraints (7) and (8) guarantees that, if both xa\\u0014b\\nandxb\\u0014aare one, than xa=bwill be as well. Finally, con-\\nstraint (9) ensures that transitivity is maintained throughout\\nthe solution.\\nPOS Assessment\\nIn the last phase, experts\\u2019 should collaborate to give a \\ufb01-\\nnal POS value to the risk factor of the target candidate.\\nThe process collects individual POS assessments of multi-\\nple experts, and then a peer-review evaluation is conducted nal POS value to the risk factor of the target candidate.\\nThe process collects individual POS assessments of multi-\\nple experts, and then a peer-review evaluation is conducted\\nto reach a consensus value. Both steps constrain the experts\\u2019\\npossible POS assessment values given a certain LOK level.\\nThe speci\\ufb01cation of how the LOK constrains the POS values\\nis part of the specialization of the framework. KaRA\\u2019s de-\\nfault method uses the so-called con\\ufb01dence-likelihood plot,\\na strategy devised in the oil and gas industry (see Rose\\n(2001) and Lowry, Suttill, and Taylor (2005)). In this ap-\\nproach, POS values are constrained to the middle of the scale\\n(around \\ufb01fty percent) when the LOK is low, while at high\\nLOK levels, the POS value should be either very low (around\\nzero percent) or very high (around a hundred percent). The\\nrationale is that in a low con\\ufb01dence situation, one should not\\nbe conclusive about the POS value, while in high con\\ufb01dence (a) Expert POS assessment\\n(b) Peer-review POS assessment\\nFigure 6: Con\\ufb01dence-likelihood plots.\\ncases, one must be very assertive. The plots shown in Figure\\n6 specify this relationship by delimiting a region (in white)\\nof allowed LOK (y-axis) and POS (x-axis) assessment pairs.\\nFigure 6a shows the supporting plot for the individual ex-\\npert\\u2019s POS assessment step, while Figure 6b presents the\\none for the peer-review POS assessment. In Figure 6a, we\\nuse the expert\\u2019s LOK score to position the line on the y-\\naxis. The expert is then constrained to give a POS value in\\nthe intersection between this line and the white region. The\\nblue squares represent the assessments of similar candidates,\\nwhere the blueness is proportional to the similarity with the\\ntarget candidate. They serve as a reference to possibly main-\\ntain consistency with prior assessments. In Figure 6b, the\\nglobal LOK score determines the position of the LOK line,\\nand the circles represent the multiple assessments of the dif-\\nferent experts. With both information, the peer-review step\\nmay reach a \\ufb01nal consensus driven by the multiple experts\\u2019\\nopinions, potentially decreasing the biases in the process.\\nConclusions and Future Work\\nWe described a knowledge-intensive prospect risk assess-\\nment that can be applied to critical business decisions in\\nmultiple domains. To the best of our knowledge, KaRA is\\nthe \\ufb01rst framework to address this problem in a general\\nsense. The methodology proposed in the KaRA framework\\ncombines AI and knowledge engineering to address the limi-\\ntations of the methods found in the literature. Guarantee con-\\nsistency, decreasing biases, capturing experts\\u2019 knowledge,\\nbeing \\ufb02exible to adapt to technology disruptions, and pro-\\nviding a practical and meaningful LOK quantitative scale are\\nsome of the most relevant objectives of the research around\\nKaRA. It may become essential and provide a competitive\\nedge in multiple accelerated discovery pipelines. We have\\ntested KaRA in two domains, petroleum exploration, and\\nmaterial discovery, demonstrating the \\ufb02exibility of our ap-\\nproach. Moreover, We have feedback from many experts in\\nboth areas about the positive impact of our system. How-\\never, unfortunately, we do not have quantity results to be\\ndiscussed at this point. Even though we have experts using\\nKaRA nowadays, capturing data to analyze the system\\u2019s ef-\\nfectiveness is problematic because it involves knowledge-\\nintensive tasks. To approach this issue, we plan to organize\\nstudies to measure the effects of the KaRA using controlled\\ndata and focus groups. We also will develop a link betweenthe process of creating the methodology (the questionary)\\nand the knowledge base. There is a synergy between creat-\\ning the questions and feeding an ontology that can enrich\\nthe system, enabling reasoning about data already captured\\nto support answering some questions.\\nReferences\\nBertsimas, D.; and Tsitsiklis, J. N. 1997. Introduction to\\nlinear optimization , volume 6. Athena Scienti\\ufb01c Belmont,\\nMA.\\nBrancotte, B.; Yang, B.; Blin, G.; Cohen-Boulakia, S.;\\nDenise, A.; and Hamel, S. 2015. Rank aggregation with\\nties: Experiments and analysis. Proceedings of the VLDB\\nEndowment (PVLDB) , 8(11): 1202\\u20131213.\\nCiccio, C. D.; Marrella, A.; and Russo, A. 2015.\\nKnowledge-Intensive Processes: Characteristics, Require-\\nments and Analysis of Contemporary Approaches. Journal\\non Data Semantics , 4: 29\\u201357.\\nJan-Erik, K.; Bundiit, C.; and Simplicil, P. 2000. The CCOP\\nguidelines for risk assessment of petroleum prospects. Pro-\\nfessional Paper of Coordinating Committee for Offshore\\nProspecting in Asia , 4\\u201321.\\nLowry, D.; Suttill, R.; and Taylor, R. 2005. Advances in\\nrisking exploration prospects. The APPEA Journal , 45(1):\\n143\\u2013158.\\nMilkov, A. V . 2015. Risk tables for less biased and more\\nconsistent estimation of probability of geological success\\n(PoS) for segments with conventional oil and gas prospec-\\ntive resources. Earth-Science Reviews , 150: 453\\u2013476. consistent estimation of probability of geological success\\n(PoS) for segments with conventional oil and gas prospec-\\ntive resources. Earth-Science Reviews , 150: 453\\u2013476.\\nMohr, R.; and Henderson, T. C. 1986. Arc and path consis-\\ntency revisited. Arti\\ufb01cial intelligence , 28(2): 225\\u2013233.\\nPaltrinieri, N.; Comfort, L.; and Reniers, G. 2019. Learn-\\ning about risk: Machine learning for risk assessment. Safety\\nscience , 118: 475\\u2013486.\\nRose, P. R. 2001. Risk analysis and management of\\npetroleum exploration ventures , volume 12. American As-\\nsociation of Petroleum Geologists Tulsa, OK.\\nSilva, F.; Fernandes, S.; Casac \\u02dcao, J.; Lib \\u00b4orio, C.; Almeida,\\nJ.; Cers \\u00b4osimo, S.; Mendes, C.; Brand \\u02dcao, R.; and Cerqueira,\\nR. 2019. Machine-Learning in Oil and Gas Exploration: A\\nNew Approach to Geological Risk Assessment. 2019(1):\\n1\\u20135. Vital Brazil, E.; Ferreira, R.; Torres, V .; Martins, L.; Raoni,\\nC.; Cerqueira, R.; Fernandez, A.; Almeida, J.; Fernandes,\\nS.; Carvalho, B.; Casac \\u02dcao, J.; Lib \\u00b4orio, C.; Quintela, M.;\\nRamos, M.; Patrocinio, D.; Ferraz, M.; and Cersosimo, S.\\n2021. GSA \\u2013 GeoScience Advisor. 2021(1): 1\\u20135.\",\"36\":\" Published as a conference paper at ICLR 2023\\nPLANNING WITH LARGE LANGUAGE MODELS\\nFOR CODE GENERATION\\nShun Zhang, Zhenfang Chen, Yikang Shen\\nMIT-IBM Watson AI LabMingyu Ding\\nThe University of Hong Kong\\nJoshua B. Tenenbaum\\nMIT BCS, CBMM, CSAILChuang Gan\\nUMass Amherst, MIT-IBM Watson AI Lab\\nABSTRACT\\nExisting large language model-based code generation pipelines typically use beam\\nsearch or sampling algorithms during the decoding process. Although the pro-\\ngrams they generate achieve high token-matching-based scores, they often fail to\\ncompile or generate incorrect outputs. The main reason is that conventional Trans-\\nformer decoding algorithms may not be the best choice for code generation. In\\nthis work, we propose a novel Transformer decoding algorithm, Planning-Guided\\nTransformer Decoding (PG-TD), that uses a planning algorithm to do lookahead\\nsearch and guide the Transformer to generate better programs. Speci\\ufb01cally, in-\\nstead of simply optimizing the likelihood of the generated sequences, the Trans-\\nformer makes use of a planner to generate candidate programs and test them on\\npublic test cases. The Transformer can therefore make more informed decisions\\nand generate tokens that will eventually lead to higher-quality programs. We also\\ndesign a mechanism that shares information between the Transformer and the\\nplanner to make our algorithm computationally ef\\ufb01cient. We empirically evaluate\\nour framework with several large language models as backbones on public coding\\nchallenge benchmarks, showing that 1) it can generate programs that consistently\\nachieve higher performance compared with competing baseline methods; 2) it en-\\nables controllable code generation, such as concise codes and highly-commented\\ncodes by optimizing modi\\ufb01ed objective1.\\n1 I NTRODUCTION\\nLarge language models like Transformer (Vaswani et al., 2017a) have shown successes in natural\\nlanguage processing, computer vision, and various other domains. Thanks to Transformer\\u2019s power\\non sequence modeling, it has been adopted for code generation (Wang et al., 2021; Ahmad et al.,\\n2021) by treating programs as text sequences. Transformer has achieved signi\\ufb01cant improvements\\non the benchmarking tasks of code translation (Roziere et al., 2022), code completion (Chen et al.,\\n2021a), and solving coding challenge problems (Hendrycks et al., 2021). Recently, AlphaCode (Li\\net al., 2022) even achieved a competitive-level performance in programming competitions with the\\nhelp of large Transformer models pre-trained on a large programming corpus.\\nTransformer-based pipelines like AlphaCode follow the tradition of natural language processing\\nand use sampling methods (Fan et al., 2018; Dabre & Fujita, 2020) during the generation process.\\nSpeci\\ufb01cally, they sample a large number of complete programs using a pre-trained code generation\\nTransformer, evaluate these programs using the public test cases provided in the dataset, and output\\nthe program that passes the most number of test cases. Compared with beam search-based methods,\\nthese sampling followed by \\ufb01ltering algorithms (which we will refer to as sampling + \\ufb01ltering ) can\\ntake advantage of test cases and indeed improve the quality of the generated programs. However,\\nduring the Transformer generation process, they do not consider the test cases at all. Instead, they\\nonly use the test cases to evaluate the programs after all the candidate programs are generated. This\\ncan make their algorithms sample inef\\ufb01cient. Different from natural languages, programs may fail\\n1Project page: https:\\/\\/codeaimcts.github.io\\n1arXiv:2303.05510v1  [cs.LG]  9 Mar 2023 Published as a conference paper at ICLR 2023\\ncompletely with even a single incorrect generated token. So these algorithms need to exhaustively\\nsample a large number of programs to \\ufb01nd a correct solution.\\nThe main reason behind the sample ef\\ufb01ciency issue of these algorithms is that the Transformer beam\\nsearch algorithm and the sampling algorithm (Vaswani et al., 2017b) may not be the best choices\\nfor code generation. An ideal code generation algorithm should stop early in the generation process\\nwhen it knows the program it currently generates would certainly fail, and bias the generation pro-\\ncess towards generating successful programs that pass more test cases. To achieve such a goal, we\\ncontribute to applying a planning algorithm in the Transformer generation process. Since a planning\\nalgorithm can use the pass rates of the generated programs as its objective, we use it to determine\\nthe quality of the generated codes and make the Transformer model make more informed decisions.\\nIn this paper, we investigate the following research question: Can we integrate a planning algo-\\nrithm with a pre-trained code generation Transformer, achieving an algorithm that generates better\\nprograms than the conventional Transformer generation algorithms and the well-accepted sam-\\npling + \\ufb01ltering scheme in the literature? To answer this question, we propose a novel algorithm,\\nPlanning-Guided Transformer Decoding (PG-TD). During the code generation process, a planner\\ndoes lookahead search and \\ufb01nds tokens that will lead to high-quality codes. The planner alone may\\nnot ef\\ufb01ciently \\ufb01nd high-quality codes due to the large search space of codes, and that is where a pre-\\ntrained code generation Transformer comes into play. Speci\\ufb01cally, the Transformer beam search\\nalgorithm and the next-token probabilities are used inside the planner to provide useful heuristics.\\nWe \\ufb01nd that a straightforward integration between the planner and the Transformer can be compu-\\ntationally inef\\ufb01cient. So we design mechanisms that allow the Transformer and the planner to share\\ntheir information to make the overall algorithm more ef\\ufb01cient.\\nWe emphasize that our algorithm is model-agnostic , that is, any standard code generation Trans-\\nformer model can be used as the backbone Transformer. Importantly, our algorithm does not require\\nacquiring more sample solutions or \\ufb01netuning the Transformer model to improve its performance.\\nWe empirically \\ufb01nd that our proposed algorithm generates higher-quality programs under multiple\\naccepted metrics compared with competing baseline methods. Additionally, we also empirically\\nshow that our algorithm has the following advantages. 1) By changing the reward function of the\\nplanner, our algorithm becomes versatile and can optimize different objective functions without the\\nnecessity of \\ufb01netuning the Transformer model. 2) Our algorithm can generate solutions that are used\\nto \\ufb01netune a code-generation Transformer model to improve the Transformer\\u2019s performance. More\\nprecisely, we have the following contributions in this paper.\\n\\u2022 First, we propose a novel algorithm, Planning-Guided Transformer Decoding (PG-TD), that\\nuses a planning algorithm for lookahead search and guide the Transformer to generate better\\ncodes. Our algorithm is model-agnostic, which can work with any standard Transformer model,\\nand does not require knowledge of the grammar of the generated programs.\\n\\u2022 Second, a direct integration of the planning algorithm with the Transformer decoding process\\ncan cause redundant uses of the Transformer beam search algorithm. We contribute to design-\\ning mechanisms that signi\\ufb01cantly improve the computational ef\\ufb01ciency of the algorithm.\\n\\u2022 Third, we evaluate our algorithm on competitive programming benchmarks and empirically\\nshow that our algorithm can consistently generate better programs in terms of the pass rate and\\nother metrics compared with the baseline methods. We also show that our algorithm is versatile show that our algorithm can consistently generate better programs in terms of the pass rate and\\nother metrics compared with the baseline methods. We also show that our algorithm is versatile\\nand can optimize objectives other than the pass rate for controllable code generation, such as\\ngenerating concise codes and codes with more comments.\\n2 R ELATED WORK\\nTransformers for program synthesis. Our work is based on Transformer for program synthe-\\nsis (Roziere et al., 2020; Austin et al., 2021). Inspired by their capacities on a range of natural\\nlanguage tasks, modern transformer-based language models (Devlin et al., 2019; Radford et al.,\\n2019; Raffel et al., 2020) have been adopted for program synthesis by treating programming lan-\\nguages in the same way as natural languages. A family of BERT-based Transformers are developed\\nfor code syntax (Kanade et al., 2020; Feng et al., 2020; Devlin et al., 2019; Guo et al., 2020). Later,\\nCodeX (Chen et al., 2021a) and CodeT5 (Wang et al., 2021) adopted GPT2 (Radford et al., 2019)\\nand T5 (Raffel et al., 2020), respectively, as backbones for both code understanding and generation.\\nDifferent learning methods including learning from examples (Ellis et al., 2021) and neural-symbolic\\nmethods (Nye et al., 2020) were explored. Recently, AlphaCode (Li et al., 2022) combined large\\n2 Published as a conference paper at ICLR 2023\\ntransformer models pre-trained on massive program data with large-scale sampling, showing com-\\npetitive performance in programming competitions. All these works mainly focused on training\\nmore powerful code-generation models and still used beam search (Graves, 2012) or sampling (Fan\\net al., 2018) during the Transformers\\u2019 generation process.\\nTest cases for program synthesis. Our work is also related to using unit tests (Myers, 1979) for\\nprogram synthesis. Tufano et al. (2020a;b) propose to generate test cases and corresponding accurate\\nassert statements with Transformer models (Vaswani et al., 2017a). Recently, Roziere et al. (2022)\\nleverage automatically generated unit tests to construct parallel training data for unsupervised code\\ntranslation. Chen et al. (2018); Gupta et al. (2020); Chen et al. (2021b) directly synthesize domain-\\nspeci\\ufb01c programs from input-output pairs without problem description. Ellis et al. (2019) use test\\ncases to train a reinforcement learning agent and use a sequential Monte-Carlo sampling method\\nfor code generation. Unlike the prior work, we use unit-testing results as reward signals for a\\ntree-search-based planning algorithm, which is further integrated with a Transformer-based large\\nlanguage model to generate better codes.\\nPlanning and reinforcement learning for code generation. The code generation problem has\\nbeen formulated as a sequential decision-making problem (Bunel et al., 2018; Chen et al., 2018),\\nwhich enables designing and applying reinforcement learning (RL) and planning algorithms for code\\ngeneration. RL has been used for both the training phase and the decoding phase of the Transformer\\nfor code generation. In the training phase, Le et al. (2022); Xu et al. (2019b) use an RL objective\\nthat optimizes the correctness of the generated programs instead of optimizing their similarity to the\\nreference solutions. In the decoding phase, the Monte-Carlo tree search (MCTS) algorithm has been\\napplied to search for high-quality codes. However, MCTS itself is unable to scale to larger domains.\\nIt is only used to generate domain-speci\\ufb01c languages or for restricted programming synthesis tasks\\nlike assembly code generation (Xu et al., 2019a), Java bytecode translation (Lim & Yoo, 2016), and\\nrobot planning (Matulewicz, 2022). These tasks have a much smaller scale than generating Python\\ncodes in our work. Therefore, their frameworks do not require a large language model and do not\\nneed to address the challenge of integrating a planning algorithm with a large language model.\\nMCTS is more ef\\ufb01cient and applicable to large domains when combined with deep learning or with\\na default policy as prior knowledge (Gelly & Silver, 2011; Silver et al., 2016; Simmons-Edler et al.,\\n2018). We consider the same overall recipe in our work. Speci\\ufb01cally, we contributed to integrating\\nthe large language model with a tree search algorithm to design a novel algorithm that is capable of\\nsolving competitive programming problems, and designed mechanisms to improve its ef\\ufb01ciency.\\nPlanning in natural language generation. Planning algorithms like MCTS have also been used\\nto \\ufb01nd the optimal text outputs for different natural language processing (NLP) tasks. For example,\\nScialom et al. (2021); Leblond et al. (2021); Chaf\\ufb01n et al. (2022) use pre-trained discriminators or\\npre-de\\ufb01ned metrics as reward functions. We want to emphasize that we are the \\ufb01rst to combine a\\ntree search algorithm with large language models for general-purpose programming language gener-\\nation. We design the interfaces between these two components and deal with the unique challenges\\nof making the framework computationally ef\\ufb01cient. Concurrently, other search algorithms like A*\\nalgorithm (Hart et al., 1968) are also applied in the Transformer decoding process. Lu et al. (2021)\\nconsider the constrained text generation problem and integrate A* with the beam search algorithm. algorithm (Hart et al., 1968) are also applied in the Transformer decoding process. Lu et al. (2021)\\nconsider the constrained text generation problem and integrate A* with the beam search algorithm.\\nHowever, their constraints are expressed as a formal logic expression, which is different from maxi-\\nmizing the pass rate in our problem.\\n3 M ETHOD\\n3.1 O VERVIEW\\nWe consider the code generation problem for competitive programming, illustrated in Fig. 1. An\\nagent is given the natural language description of a coding problem. It requires the agent to under-\\nstand the problem description and generate a program that solves the problem. Similar to Li et al.\\n(2022); Chen et al. (2018); Ellis et al. (2019), we assume that the agent has access to a set of test\\ncases, where a test case is a pair of input, output strings. Given the input string, the agent is expected\\nto generate a program that produces an output that exactly matches the test case\\u2019s output string. The\\nobjective is to generate a program that passes the most number of test cases. To determine if the\\nagent is able to generate programs that generalize to unseen test cases, we divide the test cases into\\npublic test cases and private test cases, following the terms in Li et al. (2022). The agent can only\\n3 Published as a conference paper at ICLR 2023\\nProblem Statement\\nGiven is a string S. Replace every character in S with xand print the result.\\nConstraints\\n(1).Sis a string consisting of lowercase English letters.\\n(2). The length of Sis between 1and100(inclusive).\\nInput\\nInput is given from Standard Input in the following format: S\\nOutput\\nReplace every character in Swithxand print the result.\\nSample Test Input\\nsardine\\nSample Test Output\\nxxxxxxx\\nBeam Search (Pass Rate: 0.00).\\n Sampling + Filtering (Pass Rate:\\n0.22).\\nPG-TD (Pass Rate: 1.00).\\nFigure 1: A code generation example for competitive programming, with the problem description\\n(top) and the programs generated by baseline algorithms and our PG-TD algorithm (bottom).\\naccess the public test cases during the program generation process, while we use the private test\\ncases to evaluate the programs it generates.\\nTransformer models (Li et al., 2022; Hendrycks et al., 2021) have been widely applied for code\\ngeneration thanks to their capacity in sequence-to-sequence modeling. In the Transformer\\u2019s genera-\\ntion process, beam search (Graves, 2012) and sampling (Fan et al., 2018; Dabre & Fujita, 2020) are\\nadopted to generate code sequences. However, these algorithms cannot easily optimize an objective\\ndifferent from what it is trained on (usually the similarity to the reference solutions). So we cannot\\ndirectly use these generation algorithms to generate programs aiming to pass more test cases.\\nOn the other hand, a planning algorithm can directly optimize the pass rate or any desirable pro-\\ngramming objective. To use a planning algorithm, we follow Bunel et al. (2018); Ellis et al. (2019)\\nto formulate the code generation problem as a Markov decision process (MDP) (Sutton & Barto,\\n2018). In this formulation, a statesis the concatenation of the problem description and a partial or\\ncomplete program, where a complete program ends with a special terminal token. An actionais a\\ntoken in the vocabulary set of the Transformer. There is a special termination action (the terminal\\ntoken) that indicates that the agent believes the program is complete. The transition function de-\\nterministically concatenates a state swith a token a, and an episode ends when the agent takes the\\ntermination action. The reward of statesis the pass rate of the program on the public test cases\\nwhensis a complete program ( i.e.,when the last token of sis the terminal token). The reward of a\\npartial program is always 0.\\nThere is rich literature on \\ufb01nding the optimal policy in an MDP. In this paper, we consider a tree\\nsearch-based planning algorithm inspired by Monte-Carlo tree search (MCTS), illustrated in Fig. 2.\\nIntuitively, the tree search algorithm maintains a tree structure where nodes correspond to states and\\nedges correspond to actions. The algorithm starts from the root node (the initial state) and searches\\nthe state space to \\ufb01nd terminal states with high rewards. It maintains 1) the number of times each\\nnode is visited and 2) a value function that maintains the maximum reward obtained by starting\\nin node (or state) sand taking action a. The algorithm would visit and expand nodes with either\\nhigher values (as they lead to higher-quality programs) or with smaller visit numbers (as they are\\nunder-explored). In the following part of this section, we describe how we integrate the tree search\\nalgorithm in the generation process of a Transformer.\\n4 Published as a conference paper at ICLR 2023\\na,,<PD>axa=x,ax=,<PD>axa=x,ax=,generates complete programs using beam searcha,ba,\\\\n\\na,,<PD>axa=x,ax=,a,,<PD>axa=x,ax=,suggests next tokens\\u2026SelectionExpansionEvaluationBackpropagation\\nTransformer \\nEvaluate onpublic test cases\\u2026\\nFigure 2: Illustration of using the Monte Carlo tree search algorithm in the Transformer generation\\nprocess for code generation. <PD>stands for problem description.\\n3.2 P LANNING -GUIDED TRANSFORMER DECODING\\nAlgorithm 1 The PG-TD algorithm.\\nRequire: root: the current state; c: P-UCB exploration pa-\\nrameter; k: the maximum number of children of any node;\\nb: the number of beams for Transformer beam search.\\n1:program dict=DICTIONARY ()\\n2:fori 1;2; : : : ; max rollouts do\\n3: node root\\n4: # Selection\\n5: whilejnode:childrenj>0do\\n6: node PUCB SELECT (node:children; c )\\n7: end while\\n8: # Expansion\\n9: next tokens TOP K(node; k )\\n10: fornext token2next tokens do\\n11: next state CONCAT (node; next token )\\n12: Create a node newnode fornext state\\n13: Add newnode to the children of node\\n14: end for\\n15: # Evaluation\\n16: p BEAM SEARCH (node; b )\\n17: r GET REWARD (p)\\n18: program dict[p] =r\\n19: # Backpropagation\\n20: Update and the values of node and its ancestors in the\\ntree with r\\n21:end for\\n22:return program in program dict with the highest rewardNow we are ready to answer the question\\nwe asked in the introduction: Can we in-\\ntegrate a planning algorithm with a pre-\\ntrained code generation Transformer to\\ngenerate better programs? We design a\\nTransformer generation algorithm where\\na tree search algorithm is used to perform\\nlookahead planning. The tree search al-\\ngorithm alone may not be able to \\ufb01nd\\nhigh-quality codes due to the large search\\nspace. So the conventional Transformer\\nbeam search algorithm and the next-token\\nprobabilities provided by the pre-trained\\nTransformer are used by the tree search\\nalgorithm to guide the search process.\\nWe provide the pseudocode of our\\nPlanning-Guided Transformer Decoding\\nalgorithm (PG-TD) in Algorithm 1 and\\nillustrate the whole process in Figure 2.\\nThe PG-TD algorithm follows the same\\nframework as the standard MCTS algo-\\nrithm, based on the implementation used\\nin Silver et al. (2017). Here, we focus on\\nhow the Transformer is used in the tree\\nsearch steps. We provide more details of\\nour algorithm in Sec. D.1.\\nIn the selection step, we follow Silver et al. (2017) and use the P-UCB algorithm to select which\\nbranch of the tree we want to explore. In P-UCB, we weigh the exploration term by the probability of\\nthe next tokens determined by the Transformer. So the tree search selects higher-probability tokens\\nmore often. The selection algorithm is parameterized by an exploration parameter, c, where a higher\\ncvalue leads to more exploration. We describe the details of P-UCB in Sec. D.1.\\nIn the expansion step, after a node in the tree is selected, we select the possible next tokens and add\\nthe corresponding next states as new nodes to its children list (for succinctness, a node also refers\\nto the state that it represents). Sampling a random token as in the standard MCTS may very likely\\ncause a syntax error. So we call TOP Kto get the most likely next tokens, where TOP K(s;k)returns\\nthekmost likely next tokens starting from s;kis the maximum number of children that any node\\nmay have. The corresponding knext states are the concatenations of the current state with each of\\n5 Published as a conference paper at ICLR 2023\\nthe next tokens suggested by the Transformer. These next states are added to the children list of the\\ncurrent node. (Line 9-14)\\nIn the evaluation step, we need to evaluate the selected node . Note thatnode may still be a partial\\nprogram. We cannot directly evaluate the quality of a partial program as we do not know how it\\nwill be completed and how many test cases it will pass. Here, we use the Transformer again by\\ncalling the BEAM SEARCH function to generate a complete program from the current node, where\\nBEAM SEARCH (s;b)generates a sequence using the Transformer beam search algorithm with the\\npre\\ufb01xsand beam size b. We run the generated program on the public test cases to get its reward,\\nand set it to be the value of node (Line 16-17). This value is backpropagated up in the tree so that\\nthe values of its ancestors are updated (Line 20).\\na,ba,ca,,<PD>axa=x,ax=,a,b,a,b=a,c=a,c,\\u2026\\u2026\\u2026\\u2026\\u2026a,,<PD>axa=x,ax=,a,b,a,b=a,bIteration tIteration t+1\\nFigure 3: Illustration for caching in\\nthe PG-TD algorithm. The tree search\\npart is visualized in black color and the\\nTransformer beam search part is in red\\ncolor.Information sharing between Transformer and tree\\nsearch. A keen reader may notice that if we follow the\\nalgorithm described above, there may be a lot of repeated\\ncomputations. The key observation is that the Trans-\\nformer beam search algorithm also implicitly builds a tree\\nstructure , which can be used by future iterations of tree\\nsearch. In the rest of this section, we describe how we im-\\nprove the algorithm\\u2019s ef\\ufb01ciency by sharing information in\\nthe Transformer beam search algorithm with tree search.\\nConsider the example in Fig. 3, which shows two iter-\\nations of PG-TD. In the evaluation step of the t-th iter-\\nation, the Transformer beam search algorithm implicitly\\nbuilds a tree to \\ufb01nd the most likely sequences within a\\nbeam (Fig. 3 (left)). Because we only keep bpartial pro-\\ngrams in the beam, it is a tree where only bnodes with the\\nhighest likelihood are expanded at each level (in the illus-\\ntration,b= 2). Other nodes are dropped and no longer\\nconsidered by the beam search algorithm. In the (t+1)-st\\niteration, if the tree search algorithm selects \\u201c a,\\u201d, such a\\nstate is already visited in the Transformer beam search in\\nthet-th iteration. When the algorithm needs to \\ufb01nd the\\ntop-k most likely next tokens, such information is already obtained in the t-th iteration and can be\\nreused without recomputation. In our implementation, we cache the tree structure generated by the\\nTransformer beam search algorithm. We call this implementation tree structure caching .\\nWe can also cache the complete programs generated during the PG-TD evaluation step. In the\\nevaluation step of the t-th iteration (Fig. 3), suppose the greedily-optimal sequence is \\u201c a,b=... \\u201d.\\nIn the (t+ 1)-st iteration, to generate a sequence starting with \\u201c a,b\\u201d, the Transformer beam search\\nalgorithm will generate the same sequence \\u201c a,b=... \\u201d as before. (This is not necessarily true\\nwhen the beam size is larger than 1. We will clarify this in Sec. D.2.) To improve the ef\\ufb01ciency of\\nBEAM SEARCH , we cache the sequences that have been generated during the evaluation step. In the\\nevaluation step of future iterations, PG-TD will check if the current state matches the pre\\ufb01x of any\\nsequence that has been generated before, and use the generated sequence directly without calling\\nthe Transformer beam search function. We call this implementation sequence caching . We will\\nempirically con\\ufb01rm the effectiveness of using these caching methods in the next section.\\n4 E MPIRICAL EVALUATION\\nIn this section, we empirically examine the effectiveness and ef\\ufb01ciency of our PG-TD algorithm\\nby answering the following questions. Q1: Does PG-TD generate better programs than using\\nthe Transformer beam search algorithm and other competing baseline methods? Is our algorithm\\nmodel-agnostic, showing improvement when applied to different Transformer models? Q2: Is the the Transformer beam search algorithm and other competing baseline methods? Is our algorithm\\nmodel-agnostic, showing improvement when applied to different Transformer models? Q2: Is the\\ntree search algorithm an effective planning algorithm in PG-TD? Is it better than other planning\\nor sampling-based algorithms? Q3: Is PG-TD ef\\ufb01cient in terms of the number of times it runs\\nthe Transformer beam search algorithm and also the computation time it consumes? Q4: Are the\\ncaching methods effective in saving computation time? Q5: Can we use the samples generated by\\nPG-TD in its search process and \\ufb01netune the Transformer to generate even better solutions?\\n6 Published as a conference paper at ICLR 2023\\nPass Rate (%) Strict Accuracy (%)\\nAPPS Intro. APPS Inter. APPS comp. CodeContests APPS Intro. APPS Inter. APPS comp. CodeContests\\nAPPS GPT-2 Beam Search 11.95 9.55 5.04 5.10 5.50 2.10 1.00 0.00\\nSampling+Filtering 25.19 24.13 11.92 20.40 13.80 5.70 2.30 3.64\\nSMCG-TD 24.10 21.98 10.37 17.47 11.70 5.50 2.10 4.24\\nPG-TD (c= 4) 26.70 24.92 12.89 24.05 13.10 6.10 3.10 4.85\\nAPPS GPT-Neo Beam Search 14.32 9.80 6.39 5.73 6.70 2.00 2.10 0.00\\nSampling+Filtering 27.71 24.85 12.55 25.26 15.50 5.80 3.00 4.24\\nSMCG-TD 25.09 20.34 9.16 15.44 13.80 5.10 1.80 3.03\\nPG-TD (c= 4) 29.27 25.69 13.55 26.07 15.50 6.43 3.50 4.85\\nTable 1: Results of PG-TD and other algorithms. The maximum number of Transformer generations\\nfor all algorithms is 256.\\n200 400 600\\n# of Transformer Generations22242628Pass Rate (%)\\n1000 2000 3000\\nComputation Time (sec.)22242628Pass Rate (%)\\nSMCG-TD\\nSampling + Filtering\\nPG-TD (c=2)\\nPG-TD (c=4)\\nPG-TD (c=6)\\nFigure 4: Pass rates of PG-TD and baseline algorithms vs. the number of Transformer generations\\n(left) and the computation time (middle) on the introductory problems in the APPS test dataset (1000\\nproblems), using the APPS GPT-2 Transformer model.\\nDatasets and models. Recently, several competitive programming datasets have been made avail-\\nable for benchmarking code generation algorithms. For each programming problem, they include\\nthe natural language program descriptions, sample solutions, and test cases. We evaluated PG-TD\\nand the baseline algorithms on some popular benchmark datasets: APPS (Hendrycks et al., 2021)\\nand CodeContests in AlphaCode (Li et al., 2022). The APPS dataset does not specify public vs. pri-\\nvate test cases. So we split all the test cases of a program evenly into two sets, where the \\ufb01rst set is\\nused as the public test cases for the algorithms to optimize the pass rate, and the second set is used as\\nthe private test cases for evaluating the generated programs. For CodeContests, we use their public\\nand generated test cases as our public test cases, and their private test cases as our private test cases.\\nTo show that PG-TD is model-agnostic and can be applied to different pre-trained Transformers and\\nachieve a better performance, we use two popular pre-trained code-generation Transformers in the\\nliterature: GPT-2 and GPT-Neo \\ufb01netuned on the APPS training dataset (Hendrycks et al., 2021).\\nAlgorithms. We compare PG-TD with the following algorithms. Beam search only uses the Trans-\\nformer beam search algorithm to generate the whole program, without using the test cases. This is\\nthe method used in Hendrycks et al. (2021); Chen et al. (2021a). We use the beam size of 5 in the\\nTransformer generation function, which is the same choice as in Hendrycks et al. (2021).\\nWe also implemented two baselines that use Transformer to sample programs and \\ufb01lter them using\\nthe test cases. Sampling + Filtering (S+F) generates a set of programs using the Transformer\\nsampling algorithm. Once the programs are all generated, it computes the pass rates of all the\\nprograms on the public test cases and returns the program with the highest pass rate. To avoid\\ngenerating low-probability tokens that may fail the program completely, we use top-3 sampling, that\\nis, the Transformer only samples the token that is in the top-3 most-likely tokens in each step. The\\nsampling temperature is 1. This method is similar to the algorithm in AlphaCode (Li et al., 2022),\\nexcept that we did not perform clustering as we generate a much smaller number of programs.\\nTo determine the effectiveness of the tree search algorithm, we considered another baseline, Se-\\nquential Monte-Carlo-Guided Transformer Decoding (SMCG-TD) , that replaces the tree search\\nalgorithm component with a sequential Monte-Carlo algorithm (Ellis et al., 2019). It is an iterative\\nalgorithm that maintains a population of partial programs. It determines the \\ufb01tness of a partial pro- algorithm that maintains a population of partial programs. It determines the \\ufb01tness of a partial pro-\\ngram using the Transformer in the same way as the evaluation step in PG-TD. Partial programs with\\nhigher \\ufb01tness scores are more likely to be selected in the next iteration. We leave more details about\\nthe baseline algorithms in Sec D.3.\\nFor PG-TD, we set the maximum number of children of any node ( k) to be 3, and the beam size\\n(b) to be 1 by default. For the baseline methods, we sample at most 512 programs in Sampling\\n7 Published as a conference paper at ICLR 2023\\nMethod Time (sec.)\\nWith both caching methods 1312.27\\nW\\/ only sequence caching 1430.63\\nW\\/ only tree structure caching 1899.17\\nW\\/o caching 2206.38\\nTable 2: Affects of using caching for PG-TD on\\nthe \\ufb01rst 100 problems on the APPS introductory\\ndataset, using the APPS GPT-2 (1.5B).Method Pass Rate (%) Strict Acc. (%)\\nOriginal 14.32 6.70\\nFT w\\/ S+F 15.24 7.90\\nFT w\\/ PG-TD 16.54 7.90\\nTable 3: Performance on APPS introduc-\\ntory dataset with \\ufb01netuned (FT) APPS GPT-2\\n(2.7B), using Beam Search for code genera-\\ntion.\\n+ Filtering, and maintain a set of 200 partial programs in each iteration for SMCG-TD. We will\\nuse these parameters for the following experiments unless noted otherwise. The experiments are\\nconducted on the test set of the APPS dataset (5000 problems) (Hendrycks et al., 2021) and the test\\nset of CodeContests (165 problems) (Li et al., 2022). We use the same metrics as in Hendrycks et al.\\n(2021), which are pass rates andstrict accuracies on the private test cases. Speci\\ufb01cally, the pass\\nrate is the average percentage of the private test cases that the generated programs pass over all the\\nproblems. The strict accuracy is the percentage of the problems where the generated programs pass\\nall the private test cases.\\nEffectiveness of PG-TD. To make a fair comparison, we evaluate the best programs found by the\\nalgorithms when they use the same number of Transformer generations . Speci\\ufb01cally, the num-\\nber of Transformer generations is the number of function calls of BEAM SEARCH in PG-TD and\\nSMCG-TD (when no cached sequences are found and used in sequence caching), and the number\\nof sampling function calls in S+F.\\nAs shown in Table 1, PG-TD consistently outperforms all the other baselines on all the datasets\\nunder the pass rate metric. As we optimize the pass rate, we expect our algorithm to outperform other\\nbaselines under strict accuracy as well. This is almost true except that our algorithm is matched or\\noutperformed by S+F on the APPS introductory set, mostly due to the fact that this set of programs\\nis less challenging. The gaps between different algorithms are also small on this set. Overall, these\\nresults con\\ufb01rm that our algorithm indeed generates programs that pass more test cases than the\\nbaseline methods (answering Q1). Speci\\ufb01cally, S+F and SMCG-TD both use test cases to \\ufb01lter\\nprograms generated by the Transformer, while their performance is overall outperformed by PG-\\nTD. So the tree search-based planning algorithm is indeed powerful enough and can be effectively\\nintegrated with Transformer (answering Q2). We also report the performance of PG-TD and S+F\\nunder the n@kandpass @kmetrics (Li et al., 2022) in Sec. A (Table 6).\\nEf\\ufb01ciency of PG-TD. To show PG-TD\\u2019s ef\\ufb01ciency, we report the pass rates of the best programs\\nfound by the algorithms using the same computation time (Fig. 4 (right)). The experiments are run\\non the introductory problems in the APPS test set. Since one may implement these algorithms dif-\\nferently (possibly using parallelization), we also report the results using the number of Transformer\\ngenerations as a budget (Fig. 4 (left)).\\nFor PG-TD, we set k= 3;b= 1and vary the P-UCB exploration parameter cto be 2, 4, and 6. We\\nsee that PG-TD with c= 4 has the best performance, while setting c= 2 tends to under-explore\\nand settingc= 6over-explores. With the same computational budget, S+F generates programs with\\nlower pass rates (answering Q3). This con\\ufb01rms our observation that S+F, like the algorithm used\\nin AlphaCode (Li et al., 2022), generates solutions only according to the Transformer\\u2019s generation\\nprobabilities, without taking their pass rates into consideration until all the programs are generated.\\nPG-TD actively considers the pass rates of the generated programs during the generation process,\\nwhich achieves better ef\\ufb01ciency. On the other hand, SMCG-TD also uses the Transformer and PG-TD actively considers the pass rates of the generated programs during the generation process,\\nwhich achieves better ef\\ufb01ciency. On the other hand, SMCG-TD also uses the Transformer and\\npublic test cases to guide the generation process. However, due to its sampling nature, it cannot do\\nmulti-step lookahead search as in PG-TD, which results in worse performance. We leave additional\\nresults of varying k;bfor PG-TD and varying temperatures for S+F in Sec. A.\\nEffectiveness of caching. In terms of the design choices of tree structure caching and sequence\\ncaching, we performed ablative studies to verify their ef\\ufb01ciencies. In Table 2, we compare versions\\nof PG-TD with and without tree structure caching and sequence caching. As we expect, without\\nsequence caching, the algorithm needs to regenerate whole sequences, ending up consuming much\\nmore time. Without tree structure caching, the algorithm is slightly slower as it needs to call Trans-\\nformer to get the most likely next tokens (answering Q4).\\n8 Published as a conference paper at ICLR 2023\\nFinetuning transformer with PG-TD-generated samples. Since we are able to generate solutions\\nwith high pass rates using our PG-TD algorithm, can we use these generated solutions to \\ufb01netune\\nthe code generation Transformers to further improve their performance? This may effectively solve\\nthe problem that high-quality human-written programs that can be used to train the code generation\\nTransformers are scarcely available. Concretely, we \\ufb01rst run PG-TD on the training set of APPS.\\nWe then add the generated solutions with pass rates larger than 80% to the APPS sample solution\\nset, and use the augmented solution set to \\ufb01netune the GPT-2 (2.7B) model. With the \\ufb01netuned\\nTransformer model, we run beam search to generate solutions on the test set to see if it has a better\\nperformance. We use the \\ufb01rst 500 problems in the interview-level problems in APPS test set for\\nvalidation and the introductory-level problems in APPS test set for testing. The results are reported\\nin Table 3. After \\ufb01netuning with PG-TD-generated solutions, we see improvement in both pass rate\\nand strict accuracy. More details are in Sec. D.4.\\nMethodsCode#Comment\\\"Pass\\\"length number rate\\nDefault 248.42 0.68 23.14\\nLength Penalty 190.73 - 22.82\\nComment Encouragement - 3.11 21.65\\nTable 4: Performance of controllable code generation.\\nCode length denotes the length of the generated code\\nstring and comment number denotes the number of code\\nlines that contains comments.Optimizing other code generation\\nobjectives. Beyond the pass rate,\\nwe can make the algorithm versatile\\nby using different reward functions.\\nWe consider two new objectives, code\\nlength penalty and comment encour-\\nagement . As shown in Table 4, the\\ncode length penalty reward function\\nmakes the model generate more con-\\ncise codes; the comment encourage-\\nment reward function encourages mod-\\nels to generate codes with more comments. Both objectives still achieve reasonable pass rates on\\nthe public test cases. We provide qualitative examples and more details of the reward functions in\\nSec. C of the appendix.\\nUsing automatically-generated test cases. The datasets we use in this paper both contain test\\ncases that can be used to compute rewards for PG-TD. However, in reality, human-speci\\ufb01ed test\\ncases are not always available. Recently, Chen et al. (2022) observe that Transformer pre-trained\\non code generation can also generate useful test cases by adding an assert keyword at the end of\\nthe prompt. We follow the prompt design in Chen et al. (2022) to automatically generate test cases\\nand run our PG-TD algorithm using the automatically-generated test cases. Empirically, we con\\ufb01rm\\nthat compared with beam search, PG-TD still has a higher strict accuracy by using automatically-\\ngenerated test cases to verify the generated programs. We provide the details of the method and the\\nexperiments are in Sec. B.\\n5 D ISCUSSION AND CONCLUSION\\nIn summary, we proposed a novel algorithm that uses the power of a pre-trained Transformer and\\na tree search algorithm inspired by Monte-Carlo tree search. We evaluate our algorithm and em-\\npirically show that our algorithm generates programs of higher quality compared with competing\\nbaseline algorithms in different settings. We also design model structure-speci\\ufb01c caching mecha-\\nnisms which contribute to saving computational expenses. We show that our algorithm is versatile\\nand can generate codes under objectives other than the pass rate without \\ufb01netuning the Transformer.\\nWe hope our work can inspire more ways of incorporating planning algorithms into the Transformer\\ngeneration process for code generation or other problems.\\nOne limitation of our algorithm is its reliance on test cases, although we \\ufb01nd that even a small\\nnumber of test cases can help \\ufb01nd better solutions (Sec. A, Table 7). To address this limitation, we\\nprovided results to show that our algorithm can take advantage of automatically-generated test cases provided results to show that our algorithm can take advantage of automatically-generated test cases\\n(Sec. B). Our algorithm is also more computationally expensive than the pure Transformer beam\\nsearch algorithm since ours needs to run the beam search algorithm multiple times within the tree\\nsearch algorithm. In the future, we consider improving the framework\\u2019s performance by using a\\nvalue function to estimate the programs\\u2019 pass rates, similar to the evaluation function in Silver et al.\\n(2016) for mastering the game of Go. We can learn a neural state representation for partial programs\\nas in Chen et al. (2018). We also consider using parallel tree search algorithms (Chaslot et al., 2008)\\nfor the evaluation step to parallelize the computation for different states.\\n9 Published as a conference paper at ICLR 2023\\nAcknowledgements. This work was supported by the MIT-IBM Watson AI Lab, DARPA MCS,\\nand gift funding from MERL, Cisco, and Amazon. We would also like to thank the computation\\nsupport from AiMOS, a server cluster for the IBM Research AI Hardware Center.\\nREFERENCES\\nWasi Ahmad, Saikat Chakraborty, Baishakhi Ray, and Kai-Wei Chang. Uni\\ufb01ed pre-training for pro-\\ngram understanding and generation. In Proceedings of the 2021 Conference of the North American\\nChapter of the Association for Computational Linguistics: Human Language Technologies , pp.\\n2655\\u20132668, Online, June 2021. Association for Computational Linguistics. 1\\nJacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan,\\nEllen Jiang, Carrie Cai, Michael Terry, Quoc Le, and Charles Sutton. Program Synthesis with\\nLarge Language Models. arXiv:2108.07732 [cs] , August 2021. URL http:\\/\\/arxiv.org\\/\\nabs\\/2108.07732 . arXiv: 2108.07732. 2\\nRudy Bunel, Matthew Hausknecht, Jacob Devlin, Rishabh Singh, and Pushmeet Kohli. Leveraging\\nGrammar and Reinforcement Learning for Neural Program Synthesis. ICLR , May 2018. URL\\nhttp:\\/\\/arxiv.org\\/abs\\/1805.04276 . arXiv: 1805.04276. 3, 4\\nAntoine Chaf\\ufb01n, Vincent Claveau, and Ewa Kijak. Ppl-mcts: Constrained textual generation through\\ndiscriminator-guided mcts decoding. In Proceedings of the 2022 Conference of the North Amer-\\nican Chapter of the Association for Computational Linguistics: Human Language Technologies ,\\npp. 2953\\u20132967, 2022. 3\\nGuillaume MJ-B Chaslot, Mark HM Winands, and HJVD Herik. Parallel monte-carlo tree search.\\nInInternational Conference on Computers and Games , pp. 60\\u201371. Springer, 2008. 9\\nBei Chen, Fengji Zhang, Anh Nguyen, Daoguang Zan, Zeqi Lin, Jian-Guang Lou, and Weizhu\\nChen. CodeT: Code Generation with Generated Tests, July 2022. URL http:\\/\\/arxiv.org\\/\\nabs\\/2207.10397 . arXiv:2207.10397 [cs]. 9, 17\\nMark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira Pinto, Jared\\nKaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, Alex Ray, Raul Puri,\\nGretchen Krueger, Michael Petrov, Heidy Khlaaf, Girish Sastry, Pamela Mishkin, Brooke Chan,\\nScott Gray, Nick Ryder, Mikhail Pavlov, Alethea Power, Lukasz Kaiser, Mohammad Bavarian,\\nClemens Winter, Philippe Tillet, Felipe Petroski Such, Dave Cummings, Matthias Plappert, Fo-\\ntios Chantzis, Elizabeth Barnes, Ariel Herbert-V oss, William Hebgen Guss, Alex Nichol, Alex\\nPaino, Nikolas Tezak, Jie Tang, Igor Babuschkin, Suchir Balaji, Shantanu Jain, William Saunders,\\nChristopher Hesse, Andrew N. Carr, Jan Leike, Josh Achiam, Vedant Misra, Evan Morikawa,\\nAlec Radford, Matthew Knight, Miles Brundage, Mira Murati, Katie Mayer, Peter Welinder,\\nBob McGrew, Dario Amodei, Sam McCandlish, Ilya Sutskever, and Wojciech Zaremba. Eval-\\nuating Large Language Models Trained on Code. arXiv:2107.03374 [cs] , July 2021a. URL\\nhttp:\\/\\/arxiv.org\\/abs\\/2107.03374 . arXiv: 2107.03374. 1, 2, 7, 15, 17\\nXinyun Chen, Chang Liu, and Dawn Song. Execution-guided neural program synthesis. In Interna-\\ntional Conference on Learning Representations , 2018. 3, 9\\nXinyun Chen, Dawn Song, and Yuandong Tian. Latent execution for neural program synthesis\\nbeyond domain-speci\\ufb01c languages. Advances in Neural Information Processing Systems , 2021b.\\n3\\nRaj Dabre and Atsushi Fujita. Softmax tempering for training neural machine translation models.\\narXiv , 2020. 1, 4\\nJacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep\\nbidirectional transformers for language understanding. In Proceedings of the 2019 Conference of\\nthe North American Chapter of the Association for Computational Linguistics: Human Language\\nTechnologies, Volume 1 (Long and Short Papers) , pp. 4171\\u20134186, Minneapolis, Minnesota, June\\n2019. Association for Computational Linguistics. doi: 10.18653\\/v1\\/N19-1423. URL https:\\n\\/\\/aclanthology.org\\/N19-1423 . 2\\n10 Published as a conference paper at ICLR 2023\\nKevin Ellis, Catherine Wong, Maxwell Nye, Mathias Sabl \\u00b4e-Meyer, Lucas Morales, Luke Hewitt,\\nLuc Cary, Armando Solar-Lezama, and Joshua B Tenenbaum. Dreamcoder: Bootstrapping in-\\nductive program synthesis with wake-sleep library learning. In Proceedings of the 42nd ACM\\nSIGPLAN International Conference on Programming Language Design and Implementation , pp.\\n835\\u2013850, 2021. 2\\nKevin M. Ellis, Maxwell Nye, Yewen Pu, Felix Sosa, Joshua Tenenbaum, and Armando Solar-\\nLezama. Write, execute, assess: Program synthesis with a repl. Neural Information Processing\\nSystems , 2019. 3, 4, 7\\nAngela Fan, Mike Lewis, and Yann Dauphin. Hierarchical neural story generation. In Proceedings\\nof the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long\\nPapers) , pp. 889\\u2013898, 2018. 1, 3, 4\\nZhangyin Feng, Daya Guo, Duyu Tang, Nan Duan, Xiaocheng Feng, Ming Gong, Linjun Shou,\\nBing Qin, Ting Liu, Daxin Jiang, et al. Codebert: A pre-trained model for programming and\\nnatural languages. In Findings of the Association for Computational Linguistics: EMNLP 2020 ,\\npp. 1536\\u20131547, 2020. 2\\nSylvain Gelly and David Silver. Monte-carlo tree search and rapid action value estimation in com-\\nputer go. Arti\\ufb01cial Intelligence , 175(11):1856\\u20131875, 2011. 3\\nAlex Graves. Sequence transduction with recurrent neural networks. In International Conference\\non Machine Learning: Representation Learning Workshop , 2012. 3, 4\\nDaya Guo, Shuo Ren, Shuai Lu, Zhangyin Feng, Duyu Tang, LIU Shujie, Long Zhou, Nan Duan,\\nAlexey Svyatkovskiy, Shengyu Fu, et al. Graphcodebert: Pre-training code representations with\\ndata \\ufb02ow. In International Conference on Learning Representations , 2020. 2\\nKavi Gupta, Peter Ebert Christensen, Xinyun Chen, and Dawn Song. Synthesize, execute and debug:\\nLearning to repair for neural program synthesis. Advances in Neural Information Processing\\nSystems , 2020. 3\\nPeter E Hart, Nils J Nilsson, and Bertram Raphael. A formal basis for the heuristic determination\\nof minimum cost paths. IEEE transactions on Systems Science and Cybernetics , 4(2):100\\u2013107,\\n1968. 3\\nDan Hendrycks, Steven Basart, Saurav Kadavath, Mantas Mazeika, Akul Arora, Ethan Guo, Collin\\nBurns, Samir Puranik, Horace He, Dawn Song, and Jacob Steinhardt. Measuring coding challenge\\ncompetence with APPS. NeurIPS , 2021. 1, 4, 7, 8, 14, 17\\nAditya Kanade, Petros Maniatis, Gogul Balakrishnan, and Kensen Shi. Learning and evaluating\\ncontextual embedding of source code. In International Conference on Machine Learning , pp.\\n5110\\u20135121. PMLR, 2020. 2\\nLevente Kocsis and Csaba Szepesv \\u00b4ari. Bandit based monte-carlo planning. In European conference\\non machine learning , pp. 282\\u2013293. Springer, 2006. 21\\nHung Le, Yue Wang, Akhilesh Deepak Gotmare, Silvio Savarese, and Steven C. H. Hoi. CodeRL:\\nMastering Code Generation through Pretrained Models and Deep Reinforcement Learning.\\narXiv, July 2022. doi: 10.48550\\/arXiv.2207.01780. URL http:\\/\\/arxiv.org\\/abs\\/2207.\\n01780 . arXiv:2207.01780 [cs]. 3, 24\\nR\\u00b4emi Leblond, Jean-Baptiste Alayrac, Laurent Sifre, Miruna Pislar, Jean-Baptiste Lespiau, Ioannis\\nAntonoglou, Karen Simonyan, and Oriol Vinyals. Machine translation decoding beyond beam\\nsearch. arXiv preprint arXiv:2104.05336 , 2021. 3\\nYujia Li, David Choi, Junyoung Chung, Nate Kushman, Julian Schrittwieser, R \\u00b4emi Leblond, Tom\\nEccles, James Keeling, Felix Gimeno, and Agustin Dal Lago. Competition-Level Code Genera-\\ntion with AlphaCode. arXiv preprint arXiv:2203.07814 , 2022. 1, 2, 3, 4, 7, 8, 14, 17\\nJinsuk Lim and Shin Yoo. Field report: Applying monte carlo tree search for program synthesis. In\\nInternational Symposium on Search Based Software Engineering , pp. 304\\u2013310. Springer, 2016. 3\\n11 Published as a conference paper at ICLR 2023\\nXiming Lu, Sean Welleck, Peter West, Liwei Jiang, Jungo Kasai, Daniel Khashabi, Ronan Le\\nBras, Lianhui Qin, Youngjae Yu, Rowan Zellers, Noah A. Smith, and Yejin Choi. NeuroLogic\\nA*esque Decoding: Constrained Text Generation with Lookahead Heuristics, December 2021.\\nURLhttp:\\/\\/arxiv.org\\/abs\\/2112.08726 . arXiv:2112.08726 [cs]. 3\\nNadia Matulewicz. Inductive program synthesis through using monte carlo tree search guided by a\\nheuristic-based loss function. 2022. 3\\nGlenford J Myers. The art of software testing. 1979. 3\\nMaxwell Nye, Armando Solar-Lezama, Josh Tenenbaum, and Brenden M Lake. Learning compo-\\nsitional rules via neural program synthesis. Advances in Neural Information Processing Systems ,\\n33:10832\\u201310842, 2020. 2\\nAlec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language\\nmodels are unsupervised multitask learners. 2019. 2, 28\\nColin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi\\nZhou, Wei Li, and Peter J. Liu. Exploring the limits of transfer learning with a uni\\ufb01ed text-to-\\ntext transformer. Journal of Machine Learning Research , 2020. URL http:\\/\\/jmlr.org\\/\\npapers\\/v21\\/20-074.html . 2\\nBaptiste Roziere, Marie-Anne Lachaux, Lowik Chanussot, and Guillaume Lample. Unsupervised\\ntranslation of programming languages. Advances in Neural Information Processing Systems , 33,\\n2020. 2\\nBaptiste Roziere, Jie M Zhang, Francois Charton, Mark Harman, Gabriel Synnaeve, and Guillaume\\nLample. Leveraging automated unit tests for unsupervised code translation. In International\\nConference on Learning Representations , 2022. 1, 3\\nThomas Scialom, Paul-Alexis Dray, Jacopo Staiano, Sylvain Lamprier, and Benjamin Piwowarski.\\nTo beam or not to beam: That is a question of cooperation for language gans. Advances in neural\\ninformation processing systems , 34:26585\\u201326597, 2021. 3\\nDavid Silver, Aja Huang, Chris J. Maddison, Arthur Guez, Laurent Sifre, George Van Den Driess-\\nche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, and Marc Lanctot. Mas-\\ntering the game of Go with deep neural networks and tree search. Nature , 529(7587):484, 2016.\\nPublisher: Nature Publishing Group. 3, 9\\nDavid Silver, Thomas Hubert, Julian Schrittwieser, Ioannis Antonoglou, Matthew Lai, Arthur Guez,\\nMarc Lanctot, Laurent Sifre, Dharshan Kumaran, Thore Graepel, Timothy Lillicrap, Karen Si-\\nmonyan, and Demis Hassabis. Mastering Chess and Shogi by Self-Play with a General Reinforce-\\nment Learning Algorithm, December 2017. URL http:\\/\\/arxiv.org\\/abs\\/1712.01815 .\\narXiv:1712.01815 [cs]. 5, 21\\nRiley Simmons-Edler, Anders Miltner, and Sebastian Seung. Program synthesis through reinforce-\\nment learning guided tree search. arXiv preprint arXiv:1806.02932 , 2018. 3\\nRichard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction . MIT Press,\\nOctober 2018. ISBN 978-0-262-35270-3. 4\\nMichele Tufano, Dawn Drain, Alexey Svyatkovskiy, Shao Kun Deng, and Neel Sundaresan. Unit\\ntest case generation with transformers and focal context. arXiv , 2020a. 3\\nMichele Tufano, Dawn Drain, Alexey Svyatkovskiy, and Neel Sundaresan. Generating accurate\\nassert statements for unit test cases using pretrained transformers. arXiv , 2020b. 3\\nAshish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez,\\n\\u0141ukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural informa-\\ntion processing systems , 2017a. 1, 3\\nAshish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez,\\nLukasz Kaiser, and Illia Polosukhin. Attention Is All You Need. arXiv:1706.03762 [cs] , De-\\ncember 2017b. URL http:\\/\\/arxiv.org\\/abs\\/1706.03762 . arXiv: 1706.03762. 2\\n12 Published as a conference paper at ICLR 2023\\nYue Wang, Weishi Wang, Sha\\ufb01q Joty, and Steven C.H. Hoi. CodeT5: Identi\\ufb01er-aware Uni\\ufb01ed\\nPre-trained Encoder-Decoder Models for Code Understanding and Generation. In Proceedings\\nof the 2021 Conference on Empirical Methods in Natural Language Processing , pp. 8696\\u20138708,\\nOnline and Punta Cana, Dominican Republic, November 2021. Association for Computational\\nLinguistics. doi: 10.18653\\/v1\\/2021.emnlp-main.685. URL https:\\/\\/aclanthology.org\\/\\n2021.emnlp-main.685 . 1, 2\\nThomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi,\\nPierric Cistac, Tim Rault, R \\u00b4emi Louf, and Morgan Funtowicz. Huggingface\\u2019s transformers: State-\\nof-the-art natural language processing. arXiv preprint arXiv:1910.03771 , 2019. 17\\nYifan Xu, Lu Dai, Udaikaran Singh, Kening Zhang, and Zhuowen Tu. Neural program synthesis by\\nself-learning. arXiv preprint arXiv:1910.05865 , 2019a. 3\\nYifan Xu, Lu Dai, Udaikaran Singh, Kening Zhang, and Zhuowen Tu. Neural Program Synthesis\\nBy Self-Learning. October 2019b. doi: 10.48550\\/arXiv.1910.05865. URL https:\\/\\/arxiv.\\norg\\/abs\\/1910.05865v1 . 3\\n13 Published as a conference paper at ICLR 2023\\nAPPENDIX\\nIn this appendix, we supplement the main paper by providing more thorough empirical evaluations to\\nback up our claims and more detailed descriptions of the algorithms to help readers better understand\\nour paper.\\nThis appendix is organized as follows.\\n\\u2022 In Sec. A, we provide more comprehensive results of our algorithm and the baseline algorithms.\\nWe also include the license information of the datasets we use.\\n\\u2022 In Sec. B, we consider the scenario where test cases are not provided. We evaluate our PG-TD\\nalgorithm using automatically-generated test cases.\\n\\u2022 In Sec. C, we provide empirical evidence for our claims in the discussion section that our\\nalgorithm is versatile and can be used to optimize different code generation objectives other\\nthan the pass rate. We consider the objectives of generating concise codes and generating\\ncodes with more comments.\\n\\u2022 In Sec. D, we provide more details on the components in PG-TD as well as the baseline algo-\\nrithms.\\n\\u2022 In Sec. E, we illustrate more examples of the codes generated by our PG-TD algorithm and the\\nbaseline algorithms.\\n\\u2022 In Sec. F, we discuss more on the advantages and the potential negative social impacts of our\\nalgorithm.\\nA E MPIRICAL EVALUATION\\nWe reported the performance of our algorithm and other baselines on the whole APPS dataset\\n(Hendrycks et al., 2021) and CodeContests (Li et al., 2022). The APPS test dataset contains coding\\nproblems at three levels: introductory (1000 problems), interview (3000 problems), and competition\\n(1000 problems).\\nIn addition to our results in Table 1 in the main paper, Table 5 shows the results where the budget\\nof the number of Transformer generations is 512. As we expect, with a larger number of generated\\nprograms, the gap between PG-TD and the rest is smaller. Sampling + Filtering has a better strict\\naccuracy than our algorithm on some subsets of datasets. We also experimented with other parameter\\nsettings of PG-TD. In Fig. 5 and 6, we vary the beam size ( b= 1;3;5) underc= 2;4. In Fig. 7, we\\nvary the maximum number of children of any node ( k= 2;3;4).\\nAlthough one may expect expanding the width of the tree (increasing k) can help \\ufb01nd better pro-\\ngrams, considering generating less-likely tokens suggested by the Transformer may not contribute\\nto \\ufb01nding better programs. In fact, it wastes the budget of the number of Transformer generations\\nand the computation time by considering less likely tokens. On the other hand, increasing the beam\\nsize (b) does help improve the pass rate when the number of rollouts (corresponding to the number\\nof Transformer generations) is small. However, increasing the beam size under a larger number of\\nrollouts does not always increase the pass rate (Fig. 7), which may be caused by an over\\ufb01tting issue.\\nIncreasingbalso costs more computation time even though sequence caching is used.\\nIn Fig. 8, we use different temperatures in the Sampling + Filtering algorithm ( t= 0:6;0:8;1),\\ncon\\ufb01rming that our choice of t= 1in the experiments in the main paper does have the best perfor-\\nmance.\\nn@k and pass@k metrics. We also report the performance of PG-TD and S+F under the n@kand\\npass @kmetrics (Li et al., 2022). Given a problem, each algorithm is asked to generate kprograms\\nand submit nprograms for evaluation (they submit the nprograms with the highest reward on the\\npublic test cases). n@kis the proportion of the problems where any of the nsubmitted programs\\npasses all the private test cases. pass @kis the proportion of the problems where any of the k\\ngenerated programs passes all the private test cases (effectively k@k). For PG-TD, the ksamples are\\nthe complete programs found in the \\ufb01rst krollouts. The results are reported in Table 6, evaluated on\\nthe APPS introductory problems. We observe that for smaller norkvalues, PG-TD \\ufb01nds programs\\n14 Published as a conference paper at ICLR 2023\\nPass Rate (%) Strict Accuracy (%)\\nAPPS Intro. APPS Inter. APPS comp. CodeContests APPS Intro. APPS Inter. APPS comp. CodeContests\\nAPPS GPT-2 Beam Search 11.95 9.55 5.04 5.10 5.50 2.10 1.00 0.00\\nSampling+Filtering 27.33 25.75 12.09 21.57 14.80 6.97 2.40 4.24\\nSMCG-TD 24.40 22.18 10.60 17.72 11.80 5.87 2.50 4.24\\nPG-TD (c= 4) 28.27 26.13 13.74 25.91 14.40 6.63 4.00 4.85\\nAPPS GPT-Neo Beam Search 14.32 9.80 6.39 5.73 6.70 2.00 2.10 0.00\\nSampling+Filtering 30.23 26.58 13.12 26.57 16.70 7.40 3.10 4.24\\nSMCG-TD 24.77 20.52 9.19 15.42 14.00 5.20 1.70 2.42\\nPG-TD (c= 4) 30.38 26.89 13.28 27.10 16.70 6.90 3.20 5.45\\nTable 5: Results of PG-TD and other algorithms. The maximum number of Transformer generations\\nfor all algorithms is 512.\\n1@10 1@50 1@100\\nPG-TD (c= 4) 8.1 10.2 11.6\\nS+F 6.5 9.7 11.4\\n5@10 5@50 5@100\\nPG-TD (c= 4) 12.5 14.9 16.2\\nS+F 11.9 14.6 18.2\\npass@10 pass@50 pass@100\\nPG-TD (c= 4) 14.7 23.7 27.3\\nS+F 14.7 23.2 28.3\\nTable 6: n@kandpass @kresults of PG-TD and Sampling, evaluated on the APPS introductory\\nproblems.\\nPass Rate Strict Accuracy\\n0 public test cases (the beam search baseline) 13.61 4.2\\n1 public test case 23.27 7.4\\n3 public test cases 33.02 13.4\\n5 public test cases 32.99 14.2\\nHalf of all test cases (10.44 public test cases on average) 37.71 14.8\\nTable 7: The performance of PG-TD using different numbers of public test cases, evaluated on the\\n\\ufb01rst 500 APPS introductory problems.\\nthat are more likely to pass all the private test cases. When more programs are generated ( k) or more\\nprograms are submitted for evaluation ( n), the performance of PG-TD is matched or outperformed\\nby S+F.\\nUsing different numbers of public test cases. On the APPS dataset, we split all the test cases\\nevenly into public test cases and private test cases. To investigate how many test cases are enough\\nfor PG-TD to achieve a satisfactory performance, we consider using 1, 3, and 5 public test cases and\\nusing the rest of the test cases as private test cases. The results are reported in Table 7, evaluated on\\nthe \\ufb01rst 500 APPS introductory problems. We observe that even with a small number of public test\\ncases, PG-TD is able to \\ufb01nd programs with higher a pass rate and strict accuracy than beam search.\\nThis also suggests that the Transformer can serve as a regularizer in code generation. With a small\\nnumber of public test cases, PG-TD still generates programs similar to human-written programs\\nwithout over\\ufb01tting the pubic test cases.\\nResults on Codex. To evaluate the effectiveness of our algorithms on more powerful code-\\ngeneration Transformers, we run both beam search and PG-TD algorithms using Codex (Chen et al.,\\n2021a) as the backbone. We follow the \\u201cbest practices\\u201d provided in the Codex guide to put the\\nproblem description into a block comment in the beginning, and ask Codex to complete the rest.\\nThe prompt looks like the following.\\n\\\"\\\"\\\"\\nPython 3\\n{problem description}\\n\\\"\\\"\\\"\\n15 Published as a conference paper at ICLR 2023\\n100 200 300 400 500\\n# of Transformer Generations22242628Pass Rate (%)\\n1000 2000 3000\\nComputation Time (sec.)22242628Pass Rate (%)\\nSMCG-TD\\nSampling + Filtering\\nPG-TD (b=1)\\nPG-TD (b=3)\\nPG-TD (b=5)\\nFigure 5: Results of PG-TD ( c= 2) on the APPS introductory dataset, using the APPS GPT-2\\nTransformer model.\\n200 400\\n# of Transformer Generations22242628Pass Rate (%)\\n1000 2000 3000\\nComputation Time (sec.)22242628Pass Rate (%)\\nSMCG-TD\\nSampling + Filtering\\nPG-TD (b=1)\\nPG-TD (b=3)\\nPG-TD (b=5)\\nFigure 6: Results of PG-TD ( c= 4) on the APPS introductory dataset, using the APPS GPT-2\\nTransformer model.\\n200 400\\n# of Transformer Generations22242628Pass Rate (%)\\n500 1000 1500 2000\\nComputation Time (sec.)22242628Pass Rate (%)\\nPG-TD (k=2)\\nPG-TD (k=3)\\nPG-TD (k=4)\\nFigure 7: Results of PG-TD ( c= 4) with different kon the APPS introductory dataset, using the\\nAPPS GPT-2 Transformer model..\\n100 200 300 400 500\\n# of Transformer Generations222426Pass Rate (%)\\n500 1000 1500 2000\\nComputation Time (sec.)222426Pass Rate (%)\\nS + F (t=0.6)\\nS + F (t=0.8)\\nS + F (t=1)\\nFigure 8: Results of Sampling + Filtering with different temperatures on the APPS introductory\\ndataset, using the APPS GPT-2 Transformer model.\\nAs we expect, PG-TD helps Codex generate better codes (Table 8). This further validates our claim\\nthat our method is model-agnostic and can help a pre-trained code generation Transformer generate\\nbetter codes. Due to the limits of the OpenAI APIs, we are only able to test the algorithms on a\\nsubset of data. We also show a concrete example where PG-TD outperforms beam search in the\\nfollowing sections in Fig. 16.\\nUsing sampling-based tree search. We could use a sampling approach in the evaluation step of\\nPG-TD instead of using beam search, which would be a more faithful implementation of MCTS.\\nWe experimented with the two versions of PG-TD and \\ufb01nd that overall using beam search indeed\\nhas a better performance (Table 9). The gap could be caused by that we only do sampling once\\nto evaluate any node. We could estimate the value of nodes more accurately by sampling multiple\\ntimes. However, it would make our algorithm more computationally expensive.\\n16 Published as a conference paper at ICLR 2023\\nModel Decoding Algorithm Pass Rate (%) Strict Accuracy (%)\\nAPPS GPT-2 (1.5B) Beam Search 11.65 2\\nAPPS GPT-2 (1.5B) PG-TD 36.42 6\\nCodex (code-davinci-002) Beam Search 33.26 14\\nCodex (code-davinci-002) PG-TD 59.14 31\\nTable 8: Comparison between using different Transformers (APPS and Codex) as the backbone for\\ncode generation, evaluated on the \\ufb01rst 100 APPS introductory problems.\\nDecoding Algorithm Pass Rate (%) Strict Accuracy (%)\\n# of Transformer Generations 128 256 512 128 256 512\\nPG-TD using Beam Search 36.19 37.71 38.57 14.4 14.8 15.6\\nPG-TD using Sampling 33.95 35.67 37.87 13.8 15.4 17.2\\nTable 9: PG-TD using beam search vs. sampling in the evaluation step, evaluated on the \\ufb01rst 500\\nAPPS introductory problems.\\nCompilation Error (%) Runtime Error (%)\\nAPPS GPT-2 (1.5B), Beam Search 5.58 32.95\\nAPPS GPT-2 (1.5B), Sampling + Filtering 4.68 27.38\\nAPPS GPT-2 (1.5B), PG-TD 1.93 19.5\\nTable 10: Failure case analysis. The percentages are averaged over the APPS introductory dataset.\\nStrict Accuracy (%)\\nBeam Search 26.82\\nPG-TD (using auto-generated test cases) 74.53\\nTable 11: Comparison between beam search and TG-PD using automatically-generated test cases,\\nusing the Codex model and evaluated on the HumanEval dataset.\\nFailure mode analysis. In Table 10, we report the average percentages of test cases where com-\\npilation errors and runtime errors occur. We use the error analysis utility provided in the original\\nAPPS codebase. As we expect, PG-TD executes the generated programs and \\ufb01nds the ones with\\nhigher pass rates, so it dramatically reduces the percentage of both types of errors.\\nAssets and licenses. Our experiments are run on machines with two Intel(R) Xeon(R) Gold 6258R\\nCPUs (@ 2.70GHz), and one V100-SXM2 GPU. The APPS dataset (Hendrycks et al., 2021) is\\nreleased under MIT License2. The CodeContests dataset (Li et al., 2022) is released under Apache\\nLicense 2.03. The Huggingface Transformer (Wolf et al., 2019) is released under Apache License\\n2.04.\\nB U SING AUTOMATICALLY -GENERATED TESTCASES\\nWhen human-speci\\ufb01ed test cases are not available, our algorithm can still rely on automatically-\\ngenerated test cases. We \\ufb01rst follow Chen et al. (2022) to use the Codex model and evaluate the\\ncode generation algorithms on the HumanEval dataset (Chen et al., 2021a). We adopt the same\\nprocedure as Chen et al. (2022) to automatically generate test cases. Concretely, we use a prompt\\nthat includes the natural language description of the problem and an assert statement at the end\\nof the prompt. We also removed the example input, output pairs in the problem description to avoid\\n2https:\\/\\/github.com\\/hendrycks\\/apps\\/blob\\/main\\/LICENSE\\n3https:\\/\\/github.com\\/deepmind\\/code_contests\\/blob\\/main\\/LICENSE\\n4https:\\/\\/github.com\\/huggingface\\/transformers\\/blob\\/main\\/LICENSE\\n17 Published as a conference paper at ICLR 2023\\nthe Transformer from directly copying the test cases in the problem description. The following is an\\nexample of the prompt for test case generation.\\nfrom typing import List\\ndef has_close_elements(numbers: List[float], threshold: float) -> bool:\\n\\\"\\\"\\\" Check if in given list of numbers, are any two numbers closer to\\neach other than given threshold.\\n\\\"\\\"\\\"\\npass\\n# check the correctness of has_close_elements\\ndef check(candidate):\\nassert candidate\\nDoes Codex generate correct test cases? In the HumanEval dataset, the generated solutions are\\nevaluated by the test script by calling the check function. So we use the strict accuracy (pass@1)\\nmetric that counts the percentage of the problems where the generated solutions pass the check\\nfunction. To evaluate the quality of the automatically-generated test cases, we compute the strict\\naccuracies of the sample solutions (correct solutions written by human programmers) on these test\\ncases. Clearly, the strict accuracies of the sample solutions on the ground-truth test cases should\\nbe 100%. We evaluate the sample solutions on the automatically-generated test cases and \\ufb01nd the\\ncorresponding strict accuracy is 72.56%, which con\\ufb01rms that the automatically-generated test cases\\nare mostly correct.\\nCan PG-TD take advantage of the automatically-generated test cases? We report the average\\nstrict accuracy of the generated programs on a subset of HumanEval problems in Table 11. We\\ncon\\ufb01rm that the strict accuracy of PG-TD is higher as it uses high-quality automatically-generated\\ntest cases to verify the generated programs.\\nC P LANNING FOR OTHER CODE GENERATION OBJECTIVES\\nBesides the default reward function that optimizes the pass rate, we show the versatility of the pro-\\nposed algorithm by setting two new objectives, code length penalty and comment encouragement.\\nCode length penalty . We make the planner generate more concise codes by using the following\\nreward function\\nRlength =p+\\u00151\\u0002e\\u0000lc=t; (1)\\nwherepis the average pass rate on the public test case set and lcis the length of the code string. \\u0015\\nandtare hyperparameters that control the weight of the code length penalty and are set to 0:1and\\n20, respectively. As shown in Figure 9, the generated solution becomes more concise and the code\\nstring length decreases from 187 to 78 while still passing all the test cases.\\nComment encouragement . We can also generate solutions with more comments. We achieve this\\ngoal by using the following reward function\\nRcomment =p+\\u00151\\u0002e\\u0000lc=t+\\u00152\\u0002min(1;Ncm\\nNmax); (2)\\nwhereNcmis the number of \\u201c #\\u201d in the code string and \\u00152is set to 0:2, controlling the weights of\\nthe comment lines. If we simply add \\u00152\\u0002Ncmas the comment encouragement term, the planner\\nwould generate code with repeated meaningless comment lines to get more rewards. To handle this\\nproblem, we add the code length penalty term, \\u00151\\u0002e\\u0000lc=t. We also upper-bound the rewards for the\\nnumber of comment lines by setting the comment encouragement term to be \\u00152\\u0002min(1;Ncm\\nNmax),\\nwhereNmax = 5 in the example shown in Figure 10. As shown in Figure 10, we can generate\\nsolutions with more comment lines with this designed reward function.\\n18 Published as a conference paper at ICLR 2023\\nProblem Statement\\nTakahashi loves palindromes. Non-palindromic strings are unacceptable to him. Each time he hugs\\na string, he can change one of its characters to any character of his choice. Given is a string S.\\nFind the minimum number of hugs needed to make Spalindromic.\\nConstraints\\n(1).Sis a string consisting of lowercase English letters.\\n(2). The length of Sis between 1and100(inclusive).\\nInput\\nInput is given from Standard Input in the following format: S.\\nOutput\\nPrint the minimum number of hugs needed to make Spalindromic.\\nSample Test Input 1\\nredcoder\\nSample Test Output 1\\n1\\nFor example, we can change the fourth character to oand get a palindrome redooder.\\n1# cook up a palindrome\\n2s=input()\\n3n=len(s)\\n4c=0\\n5ifn%\\n6 for iinrange(n\\/\\/2):\\n7 ifs[i]!=s[n-i-1]:\\n8 c+=1\\n9 print (c)\\n10else :\\n11 for iinrange(n\\/\\/2):\\n12 ifs[i]!=s[n-i-1]:\\n13 c+=1\\n14 print (c)\\nDefault reward function (Pass Rate: 1.00, code string\\nlength: 187).1s=input()\\n2ans=0\\n3for iinrange(len(s)\\/\\/2):\\n4 ifs[i]!=s[\\u02dci]: ans+=1\\n5print (ans)\\n6\\nReward function with code length penalty (Pass\\nRate: 1.00, code string length: 78).\\nFigure 9: A code generation example with code length penalty. The problem description is shown\\non the top. The programs generated by default PG-TD and PG-TD with code length penalty are\\nshown at the bottom. We can generate more concise code by code length penalty reward.\\n19 Published as a conference paper at ICLR 2023\\nProblem Statement\\nA+Bis often used as an example of the easiest problem possible to show some contest platform.\\nHowever, some scientists have observed that sometimes this problem is not so easy to get accepted.\\nWant to try?\\nInput\\nThe input contains two integers aandb(0\\u0014a;b\\u0014103), separated by a single space.\\nOutput\\nOutput the sum of the given integers.\\nSample Test Input 1\\n5 14\\nSample Test Output 1\\n19\\nSample Test Input 2\\n381 492\\nSample Test Output 2\\n873\\n1a, b = map(int, input().split())\\n2print (a + b)\\n3\\nDefault reward function.1#!\\/usrbin\\/env python3\\n2# coding =utf-8 #py3 runs on py2\\n3a,b=map(int,input().split())\\n4print (a+b)\\nReward function with comment encouragement\\n(code with comments).\\nFigure 10: A code generation example with code comment encouragement. We can generate codes\\nwith more comments by using the code comment encouragement reward.\\n20 Published as a conference paper at ICLR 2023\\nD A LGORITHM DETAILS\\nD.1 PG-TD\\nIn this section, we describe our PG-TD algorithm in more details. We show our framework in Fig. 12\\nagain for convenience. The algorithm maintains a search tree structure where nodes correspond to\\nstates and edges correspond to actions. For each node s, it maintains the number of times that it\\nhas been visited, denoted by s:visits . For a state, action pair, s;a(a stat, action pair corresponds\\nto an edge in the tree), the algorithm maintains a function Q(s;a), which is the maximum reward\\nobtained by starting in sand taking action a. Conventionally, Q(s;a)is the average return. Since\\nour code generation problem is deterministic (that is, there are no stochastic transitions), we \\ufb01nd the\\nperformance is better if we keep track of the pass rate of the best program generated from a node.\\nThe algorithm \\ufb01rst selects a node for expansion. It starts from the root node and selects subtrees\\nrecursively until it reaches a node that has not been expanded before (with no children). A com-\\nmon node selection criterion is upper con\\ufb01dence bound (UCB) (Kocsis & Szepesv \\u00b4ari, 2006), which\\nleverages between exploiting known-to-be-good states and exploring less-visited states, We adapt\\nthe P-UCB heuristic used in Silver et al. (2017), which is a variation of UCB, as follows,\\nP-UCB (s;a) =Q(s;a) +\\f(s)\\u0001PTransformer (ajs)\\u0001p\\nlog(s:visits )\\n1 +s0:visits; (3)\\nwheres0is the state (deterministically) reached by taking action ains;PTransformer (ajs)is the\\nprobability that token ais the next token given the partial program s, which is provided by a Trans-\\nformer model. \\fis the weight for exploration, which depends on the number of visits of s, de\\ufb01ned\\nas\\n\\f(s) = log(s:visits +cbase+ 1\\ncbase) +c: (4)\\nThe PUCB SELECT function in Alg. 1 returns the action that has the highest P-UCB value,\\nPUCB SELECT (s;c) = arg max\\naP-UCB (s;a): (5)\\nNote thatcparameterizes the exploration weight in Eq. 4. We omit it in Eq. 5 for notational con-\\nciseness.\\nIntuitively, PUCB SELECT (s;c)would return an action amore often if 1) Q(s;a)is large, which\\nmeans that it has found high-quality programs starting from s;a; or 2)PTransformer (ajs)is large,\\nwhich means that the Transformer believes that ais a highly likely next token; or 3) s:visits is large\\nbuts0:visits is small, which means s0is under-explored. We did a hyperparameter search and set\\ncbase= 10 , and usedc= 2;4;6in our experiments.\\nIn the following steps, the algorithm expands the node by adding its children to the tree and\\nevaluates the node using BEAM SEARCH , as we described in the main paper. Finally, the re-\\nward of the completed program risbackpropagated to its parents recursively until it reaches\\nthe root to update their Qvalues. For all state, action pairs, s00;a00, along the trajectory in the\\ntree from the root to the selected node, Q(s00;a00)is updated accordingly using the new value:\\nQ(s00;a00) max(Q(s00;a00);r).\\nComparison with the standard MCTS algorithm. It is worth noting that our implementation of\\nMCTS in PG-TD is different from the standard MCTS algorithm. If we use the standard MCTS\\nalgorithm, it is computationally intractable to search the large space of possible programs to \\ufb01nd\\nhigh-quality ones.\\nFor comparison, we visualize the standard MCTS algorithm in Fig. 11. In the expansion step, MCTS\\nselects the next available action in the action set, and adds the state that can be reached by following\\nthe action. In the example, action \\u201c b\\u201d is taken, and the new state that is added to the tree is \\u201c ab\\u201d. In\\nthe evaluation step, MCTS evaluates the new state by executing a random policy from the new state\\nand computes the value of the policy.\\nIt is impractical to apply the standard MCTS algorithm to domains with large state spaces or large\\naction spaces, as we cannot afford to try all possible actions in the expansion step. A random policy\\nin the evaluation step also has a high variance in estimating the value of the new state. In PG-TD, in the evaluation step also has a high variance in estimating the value of the new state. In PG-TD,\\nwe use a pre-trained Transformer to resolve these limitations of the standard MCTS algorithm.\\n21 Published as a conference paper at ICLR 2023\\nabb<PD>abaabaabaa<PD>abaabaabaaSelectionExpansionEvaluationBackpropagation\\nabb<PD>abaabaabaaabb<PD>abaabaabaaSelect the next action in the action setGenerate a random policyand evaluateit\\nFigure 11: The standard Monte-Carlo tree search algorithm, without using a pre-trained Trans-\\nformer.<PD>stands for problem description.\\na,,<PD>axa=x,ax=,<PD>axa=x,ax=,generates complete programs using beam searcha,ba,\\\\n\\na,,<PD>axa=x,ax=,a,,<PD>axa=x,ax=,suggests next tokens\\u2026SelectionExpansionEvaluationBackpropagation\\nTransformer \\nEvaluate onpublic test cases\\u2026\\nFigure 12: Our proposed framework, PG-TD, repeated here for convenience.\\na,ba,ca,a,b,a,b=a,ca,c=a,c,\\u2026a,ba,a,b,a,b=\\u2026\\nBeam 1Beam 2\\u2026\\u2026Beam 1Beam 2\\u2026\\u2026\\nFigure 13: Illustration of an example where sequence caching may not return the correct sequence\\nwhenb>1.\\nD.2 E FFICIENT IMPLEMENTATION BY INFORMATION SHARING\\nAs discussed in the main paper, there may be calls to the Transformer generation functions the\\nperform redundant computations. We have described tree structure caching in the main paper. We\\nprovide more details on sequence caching in this section.\\n22 Published as a conference paper at ICLR 2023\\nAlgorithm 2 Sampling + Filtering.\\nRequire:samplenum : the number of samples to generate using Transformer.\\n1:fori 1;2;:::;sample nums do\\n2:pi TRANSFORMER SAMPLE (hPDi;k= 5)\\n3:rewards [i] GET REWARD (pi)\\n4:end for\\n5:returnpiwith the largest rewards [i]\\nSequence caching. The goal of sequence caching is to reduce the computational cost of the eval-\\nuation step of PG-TD. In the evaluation step, the Transformer beam search algorithm is called to\\ngenerate a complete program from the current node. These complete programs and their rewards are\\nboth cached and used in a future iteration.\\nHowever, when b>1, using cached sequences in the previous iterations may be suboptimal. Let\\u2019s\\nconsider the example in Figure 13, where b= 2. Starting from \\u201c a,\\u201d (the left \\ufb01gure), suppose the\\nbeam search algorithm returns a sequence by searching two beams with the pre\\ufb01xes of \\u201c a,b\\u201d and\\n\\u201ca,c\\u201d, respectively (the bold lines). In the next iteration (the right \\ufb01gure), to evaluate \\u201c a,b\\u201d using\\nthe beam search algorithm, we may not use the sequence found in the previous iteration, which only\\nsearches the subtree of \\u201c a,b\\u201d with one beam . In other words, we may \\ufb01nd a sequence with a higher\\nlikelihood by re-running the beam search algorithm with b= 2starting from \\u201c a,b\\u201d.\\nAlthough using sequence caching with b > 1may underestimate the values of programs in the\\nevaluation step in PG-TD, we still use sequence caching for all choices of bfor the merits of com-\\nputational saving.\\nD.3 B ASELINE ALGORITHMS\\nSampling + Filtering. We have described the sampling algorithm in the main paper. The pseu-\\ndocode is shown in Alg. 2.\\nSequential Monte-Carlo-Guided Transformer Decoding. Sequential Monte-Carlo-Guided\\nTransformer Decoding (SMCG-TD) is another algorithm that we compare PG-TD with, which also\\nuses Transformer to evaluate partial programs. The pseudocode is shown in Alg. 3. One can think\\nof it from a view of genetic algorithms. SMCG-TD maintains a group of partial programs, called a\\ngeneration , and updates them iteratively. For each partial program in the generation, we sample the\\nnext token using the Transformer and append it to the partial program, and put it in a new generation.\\nWe then evaluate each partial program in the new generation in the same way as PG-TD: We use the\\nTransformer to generate a complete program from the partial program and compute its reward (pass\\nrate). We then use these rewards as the partial programs\\u2019 \\ufb01tness scores. At the end of an iteration,\\nwe re-sample from the partial programs according to their \\ufb01tness. So partial programs with higher\\n\\ufb01tness scores are more likely to remain in the generation. In the end, the algorithm outputs the\\ncomplete program it \\ufb01nds that has the largest reward.\\nOne may notice that SMCG-TD resembles PG-TD on a higher level. It also uses the Transformer\\nand the public test cases to decide which next token to generate in the code generation process. The\\nkey difference is that SMCG-TD uses a sampling-based approach and PG-TD uses a tree search\\nalgorithm. So in our experiments, the performance of SMCG-TD is worse than PG-TD.\\nIt is worth noting that sequence caching is also implemented for SMCG-TD when it calls\\nBEAM SEARCH . So even if multiple partial programs in a generation can be the same, the algo-\\nrithm would not call the Transformer to generate a complete program from the same pre\\ufb01x more\\nthan once, which effectively reduces the number of times that it needs to call the Transformer and\\nsaves the computation time.\\nD.4 F INETUNING\\nIn our experiments, we use the samples generated by the PG-TD to further \\ufb01netune the Transformer\\nmodel. We then generate solutions using the \\ufb01netuned Transformer model by running beam search,\\n23 Published as a conference paper at ICLR 2023\\nAlgorithm 3 Sequential Monte Carlo-Guided Transformer Decoding (SMCG-TD).\\nRequire:popsize: the population size\\n1:generation [hPDi]\\u0003popsize # initialize the population with the problem description\\n2:fitness [1:0=popsize]\\u0003popsize # uniform distribution\\n3:t 0\\n4:completeprograms NEW LIST()\\n5:whilejpopulationj>0andt<maxsteps do\\n6:newgeneration NEW LIST()\\n7:newfitness NEW LIST()\\n8: fori 1;:::; LEN(generation )do\\n9:s sampled from generation , wheregeneration [j]is selected with the probability of\\nfitness [j]\\n10:a sampled from PTransformer (\\u0001js)\\n11:s0 CONCAT (s;a)\\n12: ifs0[\\u00001] =<|endoftext|> then\\n13: # this sample \\ufb01nishes generation, save to output\\n14: adds0tocompleteprograms\\n15: else\\n16: adds0tonewgeneration\\n17: #BEAM SEARCH uses sequence cache as in PG-TD\\n18: p BEAM SEARCH (s0;b= 1)\\n19: r GET REWARD (p)\\n20: addrtonewfitness\\n21: end if\\n22: end for\\n23: normalizenewfitness\\n24:generation newgeneration\\n25:fitness newfitness\\n26:t t+ 1\\n27:end while\\n28:return the program in completeprograms that has the largest reward\\nand observe improvement in both pass rate and strict accuracy compared with running beam search\\nusing the original model.\\nOne challenge in our design is that solutions that PG-TD generates have different pass rates. We\\ncould use only solutions with a pass rate of 100%. However, that largely limits the number of\\nsamples we can use for \\ufb01netuning. We instead collect all solutions that have pass rates larger than\\n80%. To make the Transformer aware of the different pass rates of training samples, we use the loss\\nfunction similar to Le et al. (2022),\\nL(\\u0012) =\\u0000EW\\u0018p\\u0012[r(W)r\\u0012logp\\u0012(WjPD)]; (6)\\nwhereWdenotes a complete program; r(W)is the pass rate of the program; PDis the program de-\\nscription. In future work, we will investigate different loss functions and compare their effectiveness\\nin improving the Transformer model\\u2019s performance.\\nE I LLUSTRATIVE EXAMPLES\\nIn this section, we provide more qualitative examples generated by our PG-TD and the baseline\\nmethods, shown in Figure 14 and Figure 15. Our PG-TD algorithm generates solutions that have\\nhigher pass rates than the baseline methods. We also show a concrete example where PG-TD out-\\nperforms beam search in Fig. 16 using Codex as the backbone Transformer.\\nFor a step-by-step illustration of PG-TD, please visit our project website at https:\\/\\/\\ncodeaimcts.github.io .\\n24 Published as a conference paper at ICLR 2023\\nProblem Statement\\nTakahashi is meeting up with Aoki. They have planned to meet at a place that is D meters away\\nfrom Takahashi\\u2019s house in T minutes from now. Takahashi will leave his house now and go straight\\nto the place at a speed of S meters per minute. Will he arrive in time?\\nConstraints\\n(1).1\\u0014D\\u001410000\\n(2).1\\u0014T\\u001410000\\n(3).1\\u0014S\\u001410000\\n(4). All values in input are integers.\\nInput\\nInput is given from Standard Input in the following format: DT S\\nOutput\\nIf Takahashi will reach the place in time, print Yes; otherwise, print No.\\nSample Test Input\\n1000 15 80\\nSample Test Output\\nYes\\nIt takes 12.5 minutes to go 1000 meters to the place at a speed of 80 meters per minute. They have\\nplanned to meet in 15 minutes so he will arrive in time.\\n1# cook your dish here\\n2try:\\n3 a=list(map(int,\\ninput().split(\\n))),!\\n,!\\n4 if(a[2]==1):\\n5 print (\\\"Yes\\\")\\n6 else :\\n7 print (\\\"No\\\")\\n8except :\\n9 pass\\n10\\nBeam Search (Pass Rate: 0.125).1# cook your dish here\\n2d,t,s=list(map(int,\\ninput().split\\n())),!\\n,!\\n3if(d==t):\\n4 if(s*(s+1)\\/s==t):\\n5 print (\\\"Yes\\\")\\n6 else :\\n7 print (\\\"No\\\")\\n8else :\\n9 if(s*(s+1)\\/s>t):\\n10 print (\\\"Yes\\\")\\n11 else :\\n12 print (\\\"No\\\")\\nSampling + Filtering (Pass Rate:\\n0.625).1n, t, s=map(int,\\ninput().split()) ,!\\n2t1=n \\/ s\\n3ift1 <= t:\\n4 print (\\\"Yes\\\")\\n5else :\\n6 print (\\\"No\\\")\\n7\\nPG-TD (Pass Rate: 1.00).\\nFigure 14: A code generation example for competitive programming.The problem description is\\nshown on the top. The programs generated by baseline algorithms and our PG-TD algorithm are\\nshown at the bottom.\\n25 Published as a conference paper at ICLR 2023\\nProblem Statement\\nGiven is an integer r. How many times is the area of a circle of radius rlarger than the area\\nof a circle of radius 1? It can be proved that the answer is always an integer under the constraints\\ngiven.\\nConstraints\\n(1).1\\u0014r\\u0014100.\\n(2). All values in input are integers.\\nInput\\nInput is given from Standard Input in the following format: r.\\nOutput\\nPrint the area of a circle of radius r, divided by the area of a circle of radius 1, as an integer..\\nSample Test Input\\n2\\nSample Test Output\\n4\\nThe area of a circle of radius 2is4times larger than the area of a circle of radius 1. Note that\\noutput must be an integer - for example, 4:0will not be accepted.\\n1# cook your dish here\\n2r=int(input())\\n3if(r>=2 *r):\\n4 print (r*r*2)\\n5else :\\n6 print (0)\\nBeam Search (Pass Rate: 0.00).1# cook your dish here\\n2r=int(input())\\n3if(r>=2 *r):\\n4 print (r*r*2)\\n5else :\\n6 print (0)\\nSampling + Filtering (Pass Rate:\\n0.00).1# cook your dish here\\n2r=int( input() )\\n3print (2*(r**2)-r **2)\\n4\\nPG-TD (Pass Rate: 1.00).\\nFigure 15: A code generation example for competitive programming. The problem description is\\nshown on the top. The programs generated by baseline algorithms and our PG-TD algorithm are\\nshown at the bottom.\\n26 Published as a conference paper at ICLR 2023\\nProblem Statement\\nYou are given an array aconsisting of ninteger numbers.\\nLet instability of the array be the following value:nmax\\ni=1ai\\u0000n\\nmin\\ni=1ai.\\nYou have to remove exactly one element from this array to minimize instability of the resulting\\n(n\\u00001)-elements array. Your task is to calculate the minimum possible instability.\\nInput\\nThe \\ufb01rst line of the input contains one integer n(2\\u0014n\\u0014105) \\u2014 the number of elements in the\\narraya.\\nThe second line of the input contains nintegersa1;a2;:::;an(1\\u0014ai\\u0014105) \\u2014 elements of the\\narraya.\\nOutput\\nPrint one integer \\u2014 the minimum possible instability of the array if you have to remove exactly one\\nelement from the array a.\\nExamples\\nInput\\n4\\n1 3 3 7\\nOutput\\n2\\nInput\\n2\\n1 100000\\nOutput\\n0\\nNote\\nIn the \\ufb01rst example you can remove 7then instability of the remaining array will be 3\\u00001 = 2 .\\nIn the second example you can remove either 1or100000 then instability of the remaining array\\nwill be 100000\\u0000100000 = 0 and1\\u00001 = 0 correspondingly.\\n1n = int(input())\\n2a = list(map(int, input().split()))\\n3\\n4print (max(a) - min(a) - 1)\\nBeam search (Pass Rate 0.00).\\n1def main():\\n2 n = int(input())\\n3 a = list(map(int, input().split()))\\n4 a.sort()\\n5 print (min(a[n-1] - a[1], a[n-2] - a[0]))\\n6\\n7if__name__ == \\\"__main__\\\":\\n8 main()\\nPG-TD (Pass Rate 1.00).\\nFigure 16: A code generation example using Codex (code-davinci-002). Beam search \\ufb01nds an\\nincorrect solution while PG-TD \\ufb01nds a correct solution. This result further con\\ufb01rms that PG-TD is\\nmodel-agnostic and could be effective with different backbone Transformers (GPT-2 (1.5B), GPT-\\nNeo (2.7B) and Codex).\\n27 Published as a conference paper at ICLR 2023\\nF M ORE DISCUSSIONS\\nPotential bene\\ufb01ts. We have shown that our framework can not only help achieve higher pass rates\\ncompared with using only the Transformer generation process, but also generate codes that opti-\\nmize different objectives. These are done without \\ufb01ne-tuning the pre-trained Transformer models.\\nOur framework makes a pre-trained Transformer more versatile and adapts it to different tasks by\\ndesigning different reward functions used by the planning algorithm. If we re-train or \\ufb01ne-tune\\nTransformers like GPT-2 used in this paper, we will need to train billions of parameters (Radford\\net al., 2019). Our approach potentially saves the computational cost and energy consumption that is\\nneeded to train or \\ufb01ne-tune Transformers for different tasks.\\nPotential negative social impacts. Automatic code generation makes it easier for anyone to gen-\\nerate programs that meet a speci\\ufb01cation. Our hope is that this development will relieve the burden\\nof software engineers and enable AI-assisted programming or even end-to-end automatic program-\\nming. However, it may also make it easier to develop malware. Anyone that can specify a malicious\\ngoal in a natural language can use an automatic code generation tool to generate codes that meet the\\ngoal. When automatic code generation techniques become more mature, we may need a separate\\nmodule that screens the natural language description and rejects the code generation requests that\\ncan lead to harmful codes.\\n28\",\"37\":\" Published as a conference paper at ICLR 2023\\nTANGOS: R EGULARIZING TABULAR NEURAL NET-\\nWORKS THROUGH GRADIENT ORTHOGONALIZATION\\nAND SPECIALIZATION\\nAlan Jeffares\\u0003\\nUniversity of Cambridge\\naj659@cam.ac.ukTennison Liu\\u0003\\nUniversity of Cambridge\\ntl522@cam.ac.ukJonathan Crabb\\u00e9\\nUniversity of Cambridge\\njc2133@cam.ac.uk\\nFergus Imrie\\nUniversity of California, Los Angeles\\nimrie@ucla.eduMihaela van der Schaar\\nUniversity of Cambridge\\nAlan Turing Institute\\nmv472@cam.ac.uk\\nABSTRACT\\nDespite their success with unstructured data, deep neural networks are not yet a\\npanacea for structured tabular data. In the tabular domain, their ef\\ufb01ciency crucially\\nrelies on various forms of regularization to prevent over\\ufb01tting and provide strong\\ngeneralization performance. Existing regularization techniques include broad\\nmodelling decisions such as choice of architecture, loss functions, and optimization\\nmethods. In this work, we introduce Tabular Neural Gradient Orthogonalization\\nand Specialization ( TANGOS ), a novel framework for regularization in the tabular\\nsetting built on latent unit attributions. The gradient attribution of an activation\\nwith respect to a given input feature suggests how the neuron attends to that feature,\\nand is often employed to interpret the predictions of deep networks. In TANGOS ,\\nwe take a different approach and incorporate neuron attributions directly into\\ntraining to encourage orthogonalization and specialization of latent attributions\\nin a fully-connected network. Our regularizer encourages neurons to focus on\\nsparse, non-overlapping input features and results in a set of diverse and specialized\\nlatent units. In the tabular domain, we demonstrate that our approach can lead to\\nimproved out-of-sample generalization performance, outperforming other popular\\nregularization methods. We provide insight into whyour regularizer is effective and\\ndemonstrate that TANGOS can be applied jointly with existing methods to achieve\\neven greater generalization performance.\\n1 I NTRODUCTION\\nDespite its relative under-representation in deep learning research, tabular data is ubiquitous in\\nmany salient application areas including medicine, \\ufb01nance, climate science, and economics. Beyond\\nraw performance gains, deep learning provides a number of promising advantages over non-neural\\nmethods including multi-modal learning, meta-learning, and certain interpretability methods, which\\nwe expand upon in depth in Appendix C. Additionally, it is a domain in which general-purpose\\nregularizers are of particular importance. Unlike areas such as computer vision or natural language\\nprocessing, architectures for tabular data generally do not exploit the inherent structure in the input\\nfeatures (i.e. locality in images and sequential text, respectively) and lack the resulting inductive\\nbiases in their design. Consequentially, improvement over non-neural ensemble methods has been\\nless pervasive. Regularization methods that implicitly or explicitly encode inductive biases thus play\\na more signi\\ufb01cant role. Furthermore, adapting successful strategies from the ensemble literature\\nto neural networks may provide a path to success in the tabular domain (e.g. Wen et al., 2020).\\nRecent work in Kadra et al. (2021) has demonstrated that suitable regularization is essential to\\n\\u0003Equal contribution\\n1arXiv:2303.05506v1  [cs.LG]  9 Mar 2023 Published as a conference paper at ICLR 2023\\nFigure 1: TANGOS encourages specialization and orthogonalization. TANGOS penalizes neuron attribu-\\ntions during training. Here, indicates strong positive attribution and indicates strong negative attribution,\\nwhile interpolating colors re\\ufb02ect weaker attributions. Neurons are regularized to be specialized (attend to sparser\\nfeatures) and orthogonal (attend to non-overlapping features).\\noutperforming such methods and, furthermore, a balanced cocktail of regularizers results in neural\\nnetwork superiority.\\nRegularization methods employed in practice can be categorized into those that prevent over\\ufb01tting\\nthrough data augmentation (Krizhevsky et al., 2012; Zhang et al., 2018), network architecture choices\\n(Hinton et al., 2012; Ioffe & Szegedy, 2015), and penalty terms that explicitly in\\ufb02uence parameter\\nlearning (Hoerl & Kennard, 1970; Tibshirani, 1996; Jin et al., 2020), to name just a few. While all\\nsuch methods are uni\\ufb01ed in attempting to improve out-of-sample generalization, this is often achieved\\nin vastly different ways. For example, L1andL2penalties favor sparsity and shrinkage, respectively,\\non model weights, thus choosing more parsimonious solutions. Data perturbation techniques, on the\\nother hand, encourage smoothness in the system assuming that small perturbations in the input should\\nnot result in large changes in the output. Which method works best for a given task is generally\\nnot known a priori and considering different classes of regularizer is recommended in practice.\\nFurthermore, combining multiple forms of regularization simultaneously is often effective, especially\\nin lower data regimes (see e.g. Brigato & Iocchi, 2021 and Hu et al., 2017).\\nNeuroscience research has suggested that neurons are both selective (Johnston & Dark, 1986) and\\nhave limited capacity (Cowan et al., 2005) in reacting to speci\\ufb01c physiological stimuli. Speci\\ufb01cally,\\nneurons selectively choose to focus on a few chunks of information in the input stimulus. In deep\\nlearning, a similar concept, commonly described as a receptive \\ufb01eld , is employed in convolutional\\nlayers (Luo et al., 2016). Here, each convolutional unit has multiple \\ufb01lters, and each \\ufb01lter is only\\nsensitive to specialized features in a local region. The output of the \\ufb01lter will activate more strongly\\nif the feature is present. This stands in contrast to fully-connected networks, where the all-to-all\\nrelationships between neurons mean each unit depends on the entire input to the network. We\\nleverage this insight to propose a regularization method that can encourage arti\\ufb01cial neurons to be\\nmore specialized and orthogonal to each other.\\nContributions. (1) Novel regularization method for deep tabular models. In this work, we\\npropose TANGOS , a novel method based on regularizing neuron attributions. A visual depiction is\\ngiven in Figure 1. Speci\\ufb01cally, each neuron is more specialized , attending to sparse input features\\nwhile its attributions are more orthogonal to those of other neurons. In effect, different neurons pay\\nattention to non-overlapping subsets of input features resulting in better generalization performance.\\nWe demonstrate that this novel regularization method results in excellent generalization performance\\non tabular data when compared to other popular regularizers. (2) Distinct regularization objective.\\nWe explore how TANGOS results in distinct emergent characteristics in the model weights. We\\nfurther show that its improved performance is linked to increased diversity among weak learners in\\nan ensemble of latent units, which is generally in contrast to existing regularizers. (3) Combination\\nwith other regularizers. Based upon these insights, we demonstrate that deploying TANGOS in\\ntandem with other regularizers can further improve generalization of neural networks in the tabular\\nsetting beyond that of any individual regularizer.\\n2 Published as a conference paper at ICLR 2023\\n2 R ELATED WORK\\nGradient Attribution Regularization. A number of methods exist which incorporate a regulari-\\nsation term to penalize the network gradients in some way. Penalizing gradient attributions is a\\nnatural approach for achieving various desirable properties in a neural network. Such methods\\nhave been in use at least since Drucker & Le Cun (1992), where the authors improve robustness by\\nencouraging invariance to small perturbations in the input space. More recently, gradient attribution\\nregularization has been successfully applied across a broad range of application areas. Some notable\\nexamples include encouraging the learning of robust features in auto-encoders (Rifai et al., 2011),\\nimproving stability in the training of generative adversarial networks (Gulrajani et al., 2017), and\\nproviding robustness to adversarial perturbations (Moosavi-Dezfooli et al., 2019). While many works\\nhave applied a shrinkage penalty (L2) to input gradients, Ross et al. (2017a) explore the effects of\\nencouraging sparsity by considering an L1 penalty term. Gradient penalties may also be leveraged\\nto compel a network to attend to particular human-annotated input features (Ross et al., 2017b). A\\nrelated line of work considers the use of gradient aggregation methods such as Integrated Gradients\\n(Sundararajan et al., 2017) and, typically, penalizes their deviation from a given target value (see e.g.\\nLiu & Avci (2019) and Chen et al. (2019)). In contrast to these works, we do not require manually\\nannotated regions upon which we constrain the network to attend. Similarly, Erion et al. (2021)\\nprovide methods for encoding domain knowledge such as smoothness between adjacent pixels in\\nan image. We note that while these works have investigated penalizing a predictive model\\u2019s output\\nattributions, we are the \\ufb01rst to regularize attributions on latent neuron activations. We provide an\\nextended discussion of related works on neural network regularization more generally in Appendix A.\\n3 TANGOS\\n3.1 P ROBLEM FORMULATION\\nWe operate in the standard supervised learning setting, with dX-dimensional input variables X2\\nX\\u0012RdXand target output variable Y2Y\\u0012 R. LetPXYdenote the joint distribution between\\ninput and target variables. The goal of the supervised learning algorithm is to \\ufb01nd a predictive model,\\nf\\u0012:X!Y with learnable parameters \\u00122\\u0002. The predictive model belongs to a hypothesis space\\nf\\u00122H that can map from the input space to the output space.\\nThe predictive function is usually learned by optimizing a loss function L: \\u0002!Rusing empirical\\nrisk minimization (ERM). The empirical risk cannot be directly minimized since the data distribution\\nPXYis not known. Instead, we use a \\ufb01nite number of iid samples (x;y)\\u0018PXY, which we refer to\\nas the training data D=f(xi;yi)gN\\ni=1.\\nOnce the predictive model is trained on D, it should ideally predict well on out-of-sample data\\ngenerated from the same distribution. However, over\\ufb01tting can occur if the hypothesis space His\\ntoo complex and the sampling of training data does not fully represent the underlying distribution\\nPXY. Regularization is an approach that reduces the complexity of the hypothesis space so that more\\ngeneralized functions are learned to explain the data. This leads to the following ERM:\\n\\u0012\\u0003= arg min\\n\\u00122\\u00021\\njDjX\\n(x;y)2DL(f\\u0012(x);y) +R(\\u0012;x;y ); (1)\\nthat includes an additional regularization term Rwhich, generally, is a function of input x, the\\nlabely, the model parameters \\u0012, and re\\ufb02ects prior assumptions about the model. For example, L1\\nregularization re\\ufb02ects the belief that sparse solutions in parameter space are more desirable.\\n3.2 N EURON ATTRIBUTIONS\\nFormally, attribution methods aim to uncover the importance of each input feature of a given sample to\\nthe prediction of the neural network. Recent works have demonstrated that feature attribution methods\\ncan be incorporated into the training process (Lundberg & Lee, 2017; Erion et al., 2021). These the prediction of the neural network. Recent works have demonstrated that feature attribution methods\\ncan be incorporated into the training process (Lundberg & Lee, 2017; Erion et al., 2021). These\\nattribution priors optimize attributions to have desirable characteristics, including interpretability as\\nwell as smoothness and sparsity in predictions. However, these methods have exclusively investigated\\noutput attributions, i.e., contributions of input features to the output of a model. To the best of our\\nknowledge, we are the \\ufb01rst work to investigate regularization of latent attributions .\\n3 Published as a conference paper at ICLR 2023\\n=++\\/uni007C\\/uni007C\\/uni007C\\/uni007C1\\/uni2112spec\\/uni007C\\/uni007C\\/uni007C\\/uni007C1\\/uni007C\\/uni007C\\/uni007C\\/uni007C1=cos(,)+\\/uni2112orthcos(,)+cos(,)\\nInputFeaturesh1h2h3\\n\\/uni2202hi\\/uni2202xjInputGradients\\nFigure 2: Method illustration. TANGOS\\nregularizes the gradients with respect to\\neach of the latent units.We rewrite our predictive function fusing function com-\\npositionf=l\\u000eg. Hereg:X !H maps the input to a\\nrepresentation h=g(x)2H , whereH\\u0012RdHis adH-\\ndimensional latent space. Additionally, l:H!Y maps\\nthe latent representation to a label space y=l(h)2Y. We\\nlethi=gi(x), for,i2[dH]denote theithneuron in the\\nhidden layer of interest. Additionally, we use ai\\nj(x)2R\\nto denote the attribution of the ithneuron w.r.t. the feature\\nxj. With this notation, upper indices correspond to latent\\nunits and lower indices to features. In some cases, it will be\\nconvenient to stack all the feature attributions together in the\\nattribution vector ai(x) = [ai\\nj(x)]dX\\nj=12RdX.\\nAttribution methods work by using gradient signals to eval-\\nuate the contributions of the input features. In the most\\nsimplistic setting:\\nai\\nj(x)\\u0011@hi(x)\\n@xj: (2)\\nThis admits a simple interpretation through a \\ufb01rst-order\\nTaylor expansion: if the input feature xjwere to increase\\nby some small number \\u000f2R+, the neuron activation would\\nchange by\\u000f\\u0001ai\\nj(x)+O(\\u000f2). The larger the absolute value of\\nthe gradient, the stronger the effect of a change in the input\\nfeature. We emphasize that our method is agnostic to the gradient attribution method, as different\\nmethods may be more appropriate for different tasks. For a comprehensive review of different\\nmethods, assumptions, and trade-offs, see Ancona et al. (2017). For completeness, we also note\\nanother category of attribution methods is built around perturbations : this class of methods evaluates\\ncontributions of individual features through repeated perturbations. Generally speaking, they are\\nmore computationally inef\\ufb01cient due to the multiple forward passes through the neural network and\\nare dif\\ufb01cult to include directly in the training objective.\\n3.3 R EWARDING ORTHOGONALIZATION AND SPECIALIZATION\\nThe main contribution of this work is proposing regularization on neuron attributions. In the most\\ngeneral sense, any function of any neuron attribution method could be used as a regularization term,\\nthus encoding prior knowledge about the properties a model should have.\\nSpeci\\ufb01cally, the regularization term is a function of the network parameters \\u0012andx, i.e.,R(\\u0012;x), and\\nencourages prior assumptions on desired behavior of the learned function. Biological sensory neurons\\nare highly specialized. For example, certain visual neurons respond to a speci\\ufb01c set of visual features\\nincluding edges and orientations within a single receptive \\ufb01eld. They are thus highly selective with\\nlimited capacity to react to speci\\ufb01c physiological stimuli (Johnston & Dark, 1986; Cowan et al.,\\n2005). Similarly, we hypothesize that neurons that are more specialized and pay attention to sparser\\nsignals should exhibit better generalization performance. We propose the following desiderata and\\ncorresponding regularization terms:\\n\\u2022Specialization. The contribution of input features to the activation of a particular neuron should\\nbe sparse, i.e.,jjai(x)jjis small for all i2[dH]andx2X. Intuitively, in higher-dimensional\\nsettings, a few features should account for a large percentage of total attributions while others\\nare near zero, resulting in more specialized neurons. We write this as a regularization term for\\nmini-batch training:\\nLspec(x) =1\\nBBX\\nb=11\\ndHdHX\\ni=1kai(xb)k1;\\nwhereb2[B]is the batch index of xb2X andk\\u0001k1denotes thel1norm.\\n\\u2022Orthogonalization. Different neurons should attend to non-overlapping subsets of input features\\ngiven a particular input sample. To encourage this, we penalize the correlation between neuron\\nattributions\\u001a[ai(x);aj(x)]for alli6=jandx2X. In other words, for each particular input, we\\n4 Published as a conference paper at ICLR 2023\\nwant to discipline the latent units to attend to different aspects of the input. Then, expressing this\\nas a regularization term for mini-batch training, we obtain:\\nLorth(x) =1\\nBBX\\nb=11\\nCdHX\\ni=2i\\u00001X\\nj=1\\u001a\\u0002\\nai(xb);aj(xb)\\u0003\\n:\\nHere,Cis the number of pairwise correlations, C=dH\\u0001(dH\\u00001)\\n2, and\\u001a[ai(xb);aj(xb)]2[0;1]is\\ncalculated using the cosine similarityjai|(xb)aj(xb)j\\njjai(xb)jj2jjaj(xb)jj2wherek\\u0001k2denotes thel2norm.\\nThese terms can be combined into a single regularization term and incorporated into the training\\nobjective. The resulting TANGOS regularizer can be expressed as:\\nRTANGOS (x) =\\u00151Lspec(x) +\\u00152Lorth(x);\\nwhere\\u00151,\\u001522Ract as weighting terms. As this expression is computed using gradient signals, it\\ncan be ef\\ufb01ciently implemented and minimized in any auto-grad framework.\\n4How AND Why DOES TANGOS W ORK?\\nTo the best of our knowledge, TANGOS is the only work to explicitly regularize latent neuron\\nattributions. A natural question to ask is (1) How is TANGOS different from other regularization?\\nWhile intuitively it makes sense to enforce specialization of each unit and orthogonalization between\\nunits, we empirically investigate if other regularizers can achieve similar effects, revealing that our\\nmethod regularizes a unique objective. Having established that the TANGOS objective is unique, the\\nnext question is (2) Why does it work? To investigate this question, we frame the set of neurons as an\\nensemble, and demonstrate that our regularization improves diversity among weak learners , resulting\\nin improved out-of-sample generalization.\\n4.1 TANGOS R EGULARIZES A UNIQUE OBJECTIVE\\nTANGOS encourages generalization by explicitly decorrelating and sparsifying the attributions of\\nlatent units. A reasonable question to ask is if this is unique, or if other regularizers might achieve\\nthe same objective implicitly. Two alternative regularizers that one might consider are L2weight\\nregularization and Dropout . Like TANGOS , weight regularization methods implicitly and partially\\npenalize the gradients by shrinking the weights in the neural network. Additionally, Dropout trains\\nan ensemble of learners by forcing each neuron to be more independent. In Figure 3, we provide these\\nresults on the UCI temperature forecast dataset (Cho et al., 2020), in which data from 25 weather\\nstations in South Korea is used to predict next-day peak temperature. We train a fully connected\\nneural network for each regularization method. Speci\\ufb01cally, we plot LspecandLorthfor neurons in\\nthe penultimate layers and the corresponding generalization performance. We supply an extended\\nselection of these results on additional datasets and regularizers in Appendix J.\\nFirst, we observe that TANGOS signi\\ufb01cantly decreases correlation between different neuron attri-\\nbutions while other regularization terms, in fact, increase them. For L2weight regularization, this\\nsuggests that as the neural network weights are made smaller, the neurons increasingly attend to the\\nsame input features. A similar effect is observed for Dropout - which has a logical explanation. In-\\ndeed,Dropout creates redundancy by forcing each latent unit to be independent of others. Naturally,\\nthis encourages individual neurons to attend to overlapping features. In contrast, TANGOS aims to\\nachieve specialization, such that neurons pay attention to sparse, non-overlapping features.\\nAdditionally, we note that no alternative regularizers achieve greater attribution sparsity. This does\\nnot come as a surprise for Dropout , where the aim to induce redundancy in each neuron will\\nnaturally encourage individual neurons to attend to more features. While L2does achieve a similar\\nlevel of sparsity, this is paired with a high Lorthterm indicating that, although the latent units do\\nattend to sparse features, they appear to collapse to a solution in which they all attend to the same\\nweighted subset of the input features. This, as we will discover in \\u00a74.2, is unlikely to be optimal for weighted subset of the input features. This, as we will discover in \\u00a74.2, is unlikely to be optimal for\\nout-of-sample generalization.\\nTherefore, we conclude that the pairing of the specialization and orthogonality objectives in TANGOS\\nregularizes a unique objective.\\n5 Published as a conference paper at ICLR 2023\\n0150030000.050.150.250.35BaselineLtestLtrain015003000L2015003000Dropout015003000L-GOS\\n0.00.20.40.60.81.0Epoch0.00.20.40.60.81.0LossTANGOS\\n0150030000.00.20.40.60.81.0BaselineLorthLspec015003000L2015003000Dropout015003000L-GOS\\n0.00.20.40.60.81.0Epoch0.00.20.40.60.81.0Regularisation LossTANGOS\\nFigure 3: Comparison of regularization objectives. (Top) Generalization performance of key regularization\\ntechniques, (Bottom) corresponding neuron attributions evaluated on the test set. L2andDO can reduce over-\\n\\ufb01tting, but neuron attributions are in fact becoming more correlated. TANGOS achieves the best generalization\\nperformance by penalizing a different objective.\\n4.2 TANGOS G ENERALIZES BY INCREASING DIVERSITY AMONG LATENT UNITS\\nHaving established how TANGOS differs from existing regularization objectives, we now turn to\\nanswer whyit works. In this section, we provide an alternative perspective on the effect of TANGOS\\nregularization in the context of ensemble learning. A predictive model f(x)may be considered as\\nan ensemble model if it can be written in the form f(x) =P\\nTk2T\\u000bkTk(x), whereTrepresents a\\nset of basis functions sometimes referred to as weak learners and the\\u000bk\\u2019s represent their respective\\nscalar weights. It is therefore clear that each output of a typical neural network may be considered an\\nensemble predictor with every latent unit in its penultimate layer acting as a weak learner in their\\ncontribution to the model\\u2019s output. More formally, in this setting Tk(x)is the activation of latent unit\\nkwith respect to an input xand\\u000bkis the subsequent connection to the output activation. With this in\\nmind, we present the following de\\ufb01nition.\\nDe\\ufb01nition 4.1. Consider an ensemble regressor f(x) =P\\nTk2T\\u000bkTk(x)trained onD=\\nf(xi;yi)gN\\ni=1where each (x;y)is drawn randomly from PXY. Additionally, the weights are con-\\nstrained such thatP\\nk\\u000bk= 1. Then, for a given input-label pair (x;y), we de\\ufb01ne:\\n(a) The overall ensemble error as: Err = (f(x)\\u0000y)2.\\n(b) The weighted errors of the weak learners as: Err =P\\nk\\u000bk(Tk(x)\\u0000y)2.\\n(c) The ensemble diversity as: Div =P\\nk\\u000bk(Tk(x)\\u0000f(x))2.\\nIntuitively, Errprovides a measure of the strength of the ensemble members while Divmeasures the\\ndiversity of their outputs. To understand the relationship between these two terms and the overall\\nensemble performance, we consider Proposition 1.\\nProposition 1 (Krogh & Vedelsby 1994) .The overall ensemble error for an input-label pair (x;y)\\ncan be decomposed into the weighted errors of the weak learners and the ensemble diversity such\\nthat:\\nErr = Err\\u0000Div: (3)\\nThis decomposition provides a fundamental insight into the success of ensemble methods: an\\nensemble\\u2019s overall error is reduced by decreasing the average error of the individual weak learners and\\nincreasing the diversity of their outputs. Successful ensemble methods explicitly increase ensemble\\ndiversity when training weak learners by, for example, sub-sampling input features (random forest,\\nBreiman, 2001), sub-sampling from the training data (bagging, Breiman, 1996) or error-weighted\\ninput importance (boosting, B\\u00fchlmann, 2012).\\nReturning to the speci\\ufb01c case of neural networks, it is clear that TANGOS provides a similar mech-\\nanism of increasing diversity among the latent units that act as weak learners in the penultimate\\n6 Published as a conference paper at ICLR 2023\\n04080012BaselineDiversity (Div)Weighted Errors (Err)Ensemble Error (Err)04080L204080Dropout04080L-GOS\\n0.00.20.40.60.81.0Epoch0.00.20.40.60.81.0Mean scoreTANGOS\\nFigure 4: Neuron Diversity. Overall ensemble error and decomposition in terms of diversity and average error\\nof the weak learners. Note while all methods achieve low overall error, TANGOS is the only method that does so\\nby increasing the diversity among the latent units.\\nlayer. By forcing the latent units to attend to sparse, uncorrelated selections of features, the learned\\nensemble is encouraged to produce diverse learners whilst maintaining coverage of the entire input\\nspace in aggregate. In Figure 4, we demonstrate this phenomenon in practice by returning to the UCI\\ntemperature forecast regression task. We provide extended results in Appendix J. We train a fully\\nconnected neural network with two hidden layers with the output layer weights constrained such that\\nthey sum to 1. We observe that regularizing with TANGOS increases diversity of the latent activations\\nresulting in improved out-of-sample generalization. This is in contrast to other typical regularization\\napproaches which also improve model performance, but exclusively by attempting to reduce the error\\nof the individual ensemble members. This provides additional motivation for applying TANGOS in\\nthe tabular domain, an area where traditional ensemble methods have performed particularly well.\\nTable 1: Stand-Alone Regularization. Comparison of regularizers on regression and classi\\ufb01cation in terms\\nof test MSE and NLL. All models are trained on real-world datasets using 5-fold cross-validation and \\ufb01nal\\nevaluation reported on a held-out test set. Bold indicates the best performance. The average rank of each method\\nacross both regression and classi\\ufb01cation is included in the \\ufb01nal row of the respective tables.\\nDataset Baseline L1 L2 DO BN IN MU TANGOS\\nRegression (Mean Squared Error)\\nFB 0.037 0.081 0.029 0.060 0.699 0.043 0.147 0.032\\nBH 0.192 0.197 0.183 0.209 0.190 0.215 0.286 0.166\\nWE 0.118 0.096 0.099 0.097 0.090 0.101 0.146 0.093\\nBC 0.323 0.263 0.277 0.282 0.294 0.308 0.323 0.244\\nWQ 0.673 0.641 0.644 0.658 0.639 0.669 0.713 0.637\\nSC 0.422 0.408 0.411 0.423 0.410 0.434 0.547 0.387\\nFF 1.274 1.280 1.274 1.266 1.330 1.201 1.289 1.276\\nPR 0.624 0.611 0.580 0.592 0.647 0.591 0.745 0.573\\nST 0.419 0.416 0.418 0.387 0.461 0.539 0.380 0.382\\nAB 0.345 0.319 0.332 0.312 0.348 0.355 0.366 0.325\\nAvg Rank 5.4 3.8 3.4 4.0 5.0 5.5 7.1 1.9\\nClassi\\ufb01cation (Mean Negative Log-likelihood)\\nHE 0.490 0.472 0.431 0.428 0.459 0.435 0.416 0.426\\nBR 0.074 0.070 0.070 0.078 0.080 0.071 0.095 0.069\\nCE 0.519 0.395 0.407 0.436 0.604 0.457 0.472 0.408\\nCR 0.464 0.405 0.402 0.456 0.460 0.481 0.448 0.369\\nHC 0.320 0.222 0.226 0.237 0.257 0.312 0.248 0.215\\nAU 0.448 0.442 0.385 0.405 0.549 0.479 0.478 0.379\\nTU 1.649 1.633 1.613 1.621 1.484 1.646 1.657 1.495\\nEN 1.040 1.040 1.042 1.058 1.098 1.072 1.065 0.974\\nTH 0.700 0.506 0.500 0.714 0.785 0.638 0.618 0.513\\nSO 0.606 0.238 0.382 0.567 0.484 0.540 0.412 0.371\\nAvg Rank 6.4 3.0 2.7 4.8 6.3 6.0 5.2 1.7\\n5 E XPERIMENTS\\nIn this section, we empirically evaluate TANGOS as a regularization method for improving gen-\\neralization performance. We present our benchmark methods and training architecture, followed\\nby extensive results on real-world datasets. There are a few main aspects that deserve empirical\\n7 Published as a conference paper at ICLR 2023\\ninvestigation, which we investigate in turn: IStand-alone performance. \\u00a75.1 Comparing the\\nperformance of TANGOS , where the focus is on applying it as a stand-alone regularizer, to a variety\\nof benchmarks on a suite of real-world datasets. IIn tandem performance. \\u00a75.2 Motivated by our\\nunique regularization objective and our analysis in \\u00a74, we demonstrate that applying TANGOS in\\nconjunction with other regularizers can lead to even greater gains in generalization performance. I\\nModern architectures. \\u00a75.3 We evaluate performance on a state-of-the-art tabular architecture and\\ncompare to boosting. All experiments were run on NVIDIA RTX A4000 GPUs. Code is provided on\\nGithub12.\\nTANGOS. We train TANGOS regularized models as described in Algorithm 1 in Appendix F. For the\\nspecialization parameter we search for \\u001512f1;10;100gand for the orthogonalization parameter we\\nsearch for\\u001522f0:1;1g. For computational ef\\ufb01ciency, we apply a sub-sampling scheme where 50\\nneuron pairs are randomly sampled for each input (for further details see Appendix F).\\nBenchmarks. We evaluate TANGOS against a selection of popular regularizer benchmarks. First, we\\nconsider weight decay methods L1andL2regularization, which sparsify and shrink the learnable pa-\\nrameters. For the regularizers coef\\ufb01cients, we search for \\u00152f0:1;0:01;0:001gwhere regularization\\nis applied to all layers. Next, we consider Dropout ( DO), with drop rate p2f10%;25%;50%g, and\\napply DO after every dense layer during training. We also consider implicit regularization in batch\\nnormalization ( BN). Lastly, we evaluate data augmentation techniques Input Noise ( IN), where we\\nuse additive Gaussian noise with mean 0 and standard deviation \\u001b2f0:1;0:05;0:01gand MixUp\\n(MU). Furthermore, each training run applies early stopping with patience of 30 epochs. In all\\nexperiments, we use 5-fold cross-validation to train and validate each benchmark. We select the\\nmodel which achieves the lowest validation error and provide a \\ufb01nal evaluation on a held-out test set.\\n5.1 G ENERALIZATION : STAND -ALONE REGULARIZATION\\nFor the \\ufb01rst set of experiments, we are interested in investigating the individual regularization effect\\nofTANGOS . To ensure a fair comparison, we evaluate the generalization performance on held-out\\ntest sets across a variety of datasets.\\nDatasets. We employ 20real-world tabular datasets from the UCI machine learning repository.\\nEach dataset is split into 80% for cross-validation and the remaining 20% for testing. The splits are\\nstandardized on just the training data, such that features have mean 0and standard deviation 1and\\ncategorical variables are one-hot encoded. See Appendix L for further details on the 20datasets used.\\nTraining and Evaluation. To ensure a fair comparison, all regularizers are applied to an MLP with\\ntwo ReLU-activated hidden layers, where each hidden layer has dH+ 1neurons. The models are\\ntrained using Adam optimizer with a dataset-dependent learning rate from f0:01;0:001;0:0001gand\\nare trained for up to a maximum of 200epochs. For regression tasks, we report the average Mean\\nSquare Error (MSE) and, on classi\\ufb01cation tasks, we report the average negative log-likelihood (NLL).\\nResults. Table 1 provides the benchmarking results for individual regularizers. We observe that\\nTANGOS achieves the best performance on 10=20of the datasets. We also observe that on 6of the\\nremaining datasets, TANGOS ranks second. This is also illustrated by the ranking plot in Appendix H.\\nThere we also provide a table displaying standard errors. As several results have overlapping error\\nintervals, we also assess the magnitude of improvement by performing a non-parametric Wilcoxon\\nsigned-rank sum test (Wilcoxon, 1992) paired at the dataset level. We compare TANGOS to the\\nbest-performing baseline method (L2) as a one-tailed test for both the regression and classi\\ufb01cation signed-rank sum test (Wilcoxon, 1992) paired at the dataset level. We compare TANGOS to the\\nbest-performing baseline method (L2) as a one-tailed test for both the regression and classi\\ufb01cation\\nresults obtaining p-values of 0.006 and 0.026 respectively. This can be interpreted as strong evidence\\nto suggest the difference is statistically signi\\ufb01cant in both cases. Note that a single regularizer is\\nseldom used by itself. In addition to a stand-alone method, it remains to be shown that TANGOS\\nbrings value when used with other regularization methods. This is explored in the next section.\\n5.2 G ENERALISATION : INTANDEM REGULARIZATION\\nMotivated by the insights described in \\u00a74, a natural next question is whether TANGOS can be applied\\nin conjunction with existing regularization to unlock even greater generalization performance. In this\\nset of experiments, we investigate this question.\\n1https:\\/\\/github.com\\/alanjeffares\\/TANGOS\\n2https:\\/\\/github.com\\/vanderschaarlab\\/TANGOS\\n8 Published as a conference paper at ICLR 2023\\nL1 L2 DO BN IN MU0.380.450.52MSERegression\\nStandalone\\n+ TANGOS\\nL1 L2 DO BN IN MU0.450.550.65NLLClassi\\ufb01cation\\nFigure 5: In Tandem Regularization. Aggregated errors across the 10 regression datasets (left) and the 10\\nclassi\\ufb01cation datasets (right). In all cases, the addition of TANGOS provides superior performance over the\\nstandalone regularizer.\\nSetup. The setting for this experiment is identical to \\u00a75.1 except now we consider the six baseline\\nregularizers in tandem withTANGOS . We examine if pairing our proposed regularizer with existing\\nmethods results in even greater generalization performance. We again run 5-fold cross-validation,\\nsearching over the same hyperparameters, with the \\ufb01nal models evaluated on a held-out test set.\\nResults. We summarize the aggregated results over the datasets for each of the six baseline regu-\\nlarizers in combination with TANGOS in Figure 5. Consistently across all regularizers in both the\\nregression and the classi\\ufb01cation settings, we observe that adding TANGOS regularization improves\\ntest performance. We provide the full table of results in the supplementary material. We also note an\\napparent interaction effect between certain regularizers (i.e. input noise for regression and dropout for\\nclassi\\ufb01cation), where methods that seemed to not be particularly effective as stand-alone regularizers\\nbecome the best-performing method when evaluated in tandem. The relationship between such\\nregularizers provides an interesting direction for future work.\\n5.3 C LOSING THE GAP ON BOOSTING\\nIn this experiment, we apply TANGOS regularization to a state-of-the-art deep learning architecture\\nfor tabular data (Gorishniy et al., 2021) and evaluate its contribution towards producing competitive\\nperformance against leading boosting methods. We provide an extended description of this experiment\\nin Appendix B and results in Table 2. We \\ufb01nd that TANGOS provides moderate gains in this setting,\\nimproving performance relative to state-of-the-art boosting methods. Although boosting approaches\\nstill match or outperform deep learning in this setting, in Appendix C we argue that deep learning\\nmay also be worth pursuing in the tabular modality for its other distinct advantages.\\nTable 2: FT-Transformer Architecture and Boosting. Adding TANGOS regularization can contribute to\\nclosing the gap between state-of-the-art tabular architectures and leading boosting methods. We report mean\\naccuracy \\u0006standard deviation.\\nSetting DatasetFT-Transformer Boosting\\nBaseline + TANGOS XGBoost CatBoost\\nDefaultJannis 0:714\\u00060:002 0:720\\u00060:000 0:711\\u00060:000 0:724\\u00060:001\\nHiggs 0:721\\u00060:002 0:723\\u00060:000 0:717\\u00060:000 0:728\\u00060:001\\nTunedJannis 0:720\\u00060:001 0:727\\u00060:001 0:724\\u00060:000 0:727\\u00060:001\\nHiggs 0:727\\u00060:002 0:729\\u00060:002 0:728\\u00060:001 0:729\\u00060:002\\n6 D ISCUSSION\\nIn this work, we have introduced TANGOS , a novel regularization method that promotes specialization\\nand orthogonalization among the gradient attributions of the latent units of a neural network. We\\nshowed how this regularization objective is distinct from other popular methods and motivated whyit\\nprovides out-of-sample generalization. We empirically demonstrated TANGOS utility with extensive\\nexperiments. This work raises several exciting avenues for future work including (1) developing\\nTANGOS beyond the tabular setting (e.g. images), (2) investigating alternative ef\\ufb01cient methods\\nfor achieving specialization and orthogonalization, (3) proposing other latent gradient attribution\\nregularizers, (4) augmenting TANGOS for speci\\ufb01c applications such as multi-modal learning or\\nincreased interpretability (see Appendix C).\\n9 Published as a conference paper at ICLR 2023\\nACKNOWLEDGMENTS\\nWe thank the anonymous ICLR reviewers as well as members of the van der Schaar lab for many\\ninsightful comments and suggestions. Alan Jeffares is funded by the Cystic Fibrosis Trust. Tennison\\nLiu would like to thank AstraZeneca for their sponsorship and support. Fergus Imrie and Mihaela van\\nder Schaar are supported by the National Science Foundation (NSF, grant number 1722516). Mihaela\\nvan der Schaar is additionally supported by the Of\\ufb01ce of Naval Research (ONR).\\nREPRODUCIBILITY STATEMENT\\nWe have attempted to make our experimental results easily reproducible by both a detailed description\\nof our experimental procedure and providing the code used to produce our results ( https:\\/\\/\\ngithub.com\\/alanjeffares\\/TANGOS ). Experiments are described in Section 5 with further\\ndetails in Appendices F and L. All datasets used in this work can be freely downloaded from the UCI\\nrepository (Dua et al., 2017) with speci\\ufb01c details provided in Appendix L.\\nREFERENCES\\nJuli\\u00e1n N Acosta, Guido J Falcone, Pranav Rajpurkar, and Eric J Topol. Multimodal biomedical AI.\\nNature Medicine , 28(9):1773\\u20131784, 2022.\\nMarco Ancona, Enea Ceolini, Cengiz \\u00d6ztireli, and Markus Gross. Towards better understanding of\\ngradient-based attribution methods for deep neural networks. arXiv preprint arXiv:1711.06104 ,\\n2017.\\nDara Bahri, Heinrich Jiang, Yi Tay, and Donald Metzler. SCARF: Self-supervised contrastive learning\\nusing random feature corruption. arXiv preprint arXiv:2106.15147 , 2021.\\nPierre Baldi, Peter Sadowski, and Daniel Whiteson. Searching for exotic particles in high-energy\\nphysics with deep learning. Nature Communications , 5(1):1\\u20139, 2014.\\nNitin Bansal, Xiaohan Chen, and Zhangyang Wang. Can we gain more from orthogonality regu-\\nlarizations in training deep networks? Advances in Neural Information Processing Systems , 31,\\n2018.\\nLeo Breiman. Bagging predictors. Machine learning , 24(2):123\\u2013140, 1996.\\nLeo Breiman. Random forests. Machine learning , 45(1):5\\u201332, 2001.\\nLorenzo Brigato and Luca Iocchi. A close look at deep learning with small data. In 2020 25th\\nInternational Conference on Pattern Recognition (ICPR) , pp. 2490\\u20132497. IEEE, 2021.\\nPeter B\\u00fchlmann. Bagging, boosting and ensemble methods. In Handbook of computational statistics ,\\npp. 985\\u20131022. Springer, 2012.\\nJiefeng Chen, Xi Wu, Vaibhav Rastogi, Yingyu Liang, and Somesh Jha. Robust attribution regular-\\nization. Advances in Neural Information Processing Systems , 32, 2019.\\nTianqi Chen and Carlos Guestrin. Xgboost: A scalable tree boosting system. In Proceedings of the\\n22nd acm sigkdd international conference on knowledge discovery and data mining , pp. 785\\u2013794,\\n2016.\\nDongjin Cho, Cheolhee Yoo, Jungho Im, and Dong-Hyun Cha. Comparative assessment of various\\nmachine learning-based bias correction methods for numerical weather prediction model forecasts\\nof extreme air temperatures in urban areas. Earth and Space Science , 7(4):e2019EA000740, 2020.\\nPaulo Cortez and An\\u00edbal de Jesus Raimundo Morais. A data mining approach to predict forest \\ufb01res\\nusing meteorological data. 2007.\\nPaulo Cortez and Alice Maria Gon\\u00e7alves Silva. Using data mining to predict secondary school\\nstudent performance. 2008.\\n10 Published as a conference paper at ICLR 2023\\nPaulo Cortez, Ant\\u00f3nio Cerdeira, Fernando Almeida, Telmo Matos, and Jos\\u00e9 Reis. Modeling wine\\npreferences by data mining from physicochemical properties. Decision support systems , 47(4):\\n547\\u2013553, 2009.\\nNelson Cowan, Emily M Elliott, J Scott Saults, Candice C Morey, Sam Mattox, Anna Hismjatullina,\\nand Andrew RA Conway. On the capacity of attention: Its estimation and its role in working\\nmemory and cognitive aptitudes. Cognitive psychology , 51(1):42\\u2013100, 2005.\\nJonathan Crabb\\u00e9 and Mihaela van der Schaar. Label-free explainability for unsupervised models.\\narXiv preprint arXiv:2203.01928 , 2022.\\nJonathan Crabb\\u00e9, Zhaozhi Qian, Fergus Imrie, and Mihaela van der Schaar. Explaining latent\\nrepresentations with a corpus of examples. Advances in Neural Information Processing Systems ,\\n34:12154\\u201312166, 2021.\\nHarris Drucker and Yann Le Cun. Improving generalization performance using double backpropaga-\\ntion. IEEE Transactions on Neural Networks , 3(6):991\\u2013997, 1992.\\nDheeru Dua, Casey Graff, et al. UCI machine learning repository. 2017.\\nGabriel Erion, Joseph D Janizek, Pascal Sturmfels, Scott M Lundberg, and Su-In Lee. Improving\\nperformance of deep learning models with axiomatic attribution priors and expected gradients.\\nNature machine intelligence , 3(7):620\\u2013631, 2021.\\nKelwin Fernandes, Jaime S Cardoso, and Jessica Fernandes. Transfer learning with partial observabil-\\nity applied to cervical cancer screening. In Iberian conference on pattern recognition and image\\nanalysis , pp. 243\\u2013250. Springer, 2017.\\nDouglas H Fisher and Jeffrey C Schlimmer. Concept simpli\\ufb01cation and prediction accuracy. In\\nMachine Learning Proceedings 1988 , pp. 22\\u201328. Elsevier, 1988.\\nYura Gorishniy, Ivan Rubachev, and Artem Babenko. On embeddings for numerical features in\\ntabular deep learning. arXiv preprint arXiv:2203.05556 , 2022.\\nYury Gorishniy, Ivan Rubachev, Valentin Khrulkov, and Artem Babenko. Revisiting deep learning\\nmodels for tabular data. Advances in Neural Information Processing Systems , 34:18932\\u201318943,\\n2021.\\nL\\u00e9o Grinsztajn, Edouard Oyallon, and Ga\\u00ebl Varoquaux. Why do tree-based models still outperform\\ndeep learning on tabular data? arXiv preprint arXiv:2207.08815 , 2022.\\nFrancesca Grisoni, Viviana Consonni, Sara Villa, Marco Vighi, and Roberto Todeschini. Qsar models\\nfor bioconcentration: is the increase in the complexity justi\\ufb01ed by more accurate predictions?\\nChemosphere , 127:171\\u2013179, 2015.\\nIshaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville.\\nImproved training of Wasserstein GANs. Advances in Neural Information Processing Systems , 30,\\n2017.\\nWenzhong Guo, Jianwen Wang, and Shiping Wang. Deep multimodal representation learning: A\\nsurvey. IEEE Access , 7:63373\\u201363394, 2019.\\nIsabelle Guyon, Lisheng Sun-Hosoya, Marc Boull\\u00e9, Hugo Jair Escalante, Sergio Escalera, Zhengying\\nLiu, Damir Jajetic, Bisakha Ray, Mehreen Saeed, Mich\\u00e8le Sebag, et al. Analysis of the automl\\nchallenge series. Automated Machine Learning , pp. 177, 2019.\\nDavid Harrison Jr and Daniel L Rubinfeld. Hedonic housing prices and the demand for clean air.\\nJournal of environmental economics and management , 5(1):81\\u2013102, 1978.\\nKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into recti\\ufb01ers: Surpassing\\nhuman-level performance on imagenet classi\\ufb01cation. In Proceedings of the IEEE international\\nconference on computer vision , pp. 1026\\u20131034, 2015.\\nYuanqin He, Yan Kang, Jiahuan Luo, Lixin Fan, and Qiang Yang. A hybrid self-supervised learning\\nframework for vertical federated learning. arXiv preprint arXiv:2208.08934 , 2022.\\n11 Published as a conference paper at ICLR 2023\\nGeoffrey E Hinton, Nitish Srivastava, Alex Krizhevsky, Ilya Sutskever, and Ruslan R Salakhutdinov.\\nImproving neural networks by preventing co-adaptation of feature detectors. arXiv preprint\\narXiv:1207.0580 , 2012.\\nArthur E Hoerl and Robert W Kennard. Ridge regression: Biased estimation for nonorthogonal\\nproblems. Technometrics , 12(1):55\\u201367, 1970.\\nGeorg Hoffmann, Andreas Bietenbeck, Ralf Lichtinghagen, and Frank Klawonn. Using machine\\nlearning techniques to generate laboratory diagnostic pathways\\u2014a case study. J Lab Precis Med ,\\n3:58, 2018.\\nTimothy Hospedales, Antreas Antoniou, Paul Micaelli, and Amos Storkey. Meta-learning in neural\\nnetworks: A survey. IEEE transactions on pattern analysis and machine intelligence , 44(9):\\n5149\\u20135169, 2021.\\nGuosheng Hu, Xiaojiang Peng, Yongxin Yang, Timothy M Hospedales, and Jakob Verbeek. Franken-\\nstein: Learning deep face representations using small data. IEEE Transactions on Image Processing ,\\n27(1):293\\u2013303, 2017.\\nSadiq Hussain, Rasha Atallah, Amirrudin Kamsin, and Jiten Hazarika. Classi\\ufb01cation, clustering and\\nassociation rule mining in educational datasets using data mining tools: A case study. In Computer\\nScience On-line Conference , pp. 196\\u2013211. Springer, 2018.\\nSergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by\\nreducing internal covariate shift. In International conference on machine learning , pp. 448\\u2013456.\\nPMLR, 2015.\\nGaojie Jin, Xinping Yi, Liang Zhang, Lijun Zhang, Sven Schewe, and Xiaowei Huang. How does\\nweight correlation affect generalisation ability of deep neural networks? Advances in Neural\\nInformation Processing Systems , 33:21346\\u201321356, 2020.\\nWilliam A Johnston and Veronica J Dark. Selective attention. Annual review of psychology , 37(1):\\n43\\u201375, 1986.\\nArlind Kadra, Marius Lindauer, Frank Hutter, and Josif Grabocka. Well-tuned simple nets excel on\\ntabular datasets. Advances in Neural Information Processing Systems , 34, 2021.\\nBeen Kim, Martin Wattenberg, Justin Gilmer, Carrie Cai, James Wexler, Fernanda Viegas, et al.\\nInterpretability beyond feature attribution: Quantitative testing with concept activation vectors\\n(tcav). In International conference on machine learning , pp. 2668\\u20132677. PMLR, 2018.\\nAlex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classi\\ufb01cation with deep convolu-\\ntional neural networks. Advances in Neural Information Processing Systems , 25, 2012.\\nAnders Krogh and Jesper Vedelsby. Neural network ensembles, cross validation, and active learning.\\nAdvances in Neural Information Processing Systems , 7, 1994.\\nJan Kuka \\u02c7cka, Vladimir Golkov, and Daniel Cremers. Regularization for deep learning: A taxonomy.\\narXiv preprint arXiv:1710.10686 , 2017.\\nYann LeCun, L\\u00e9on Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to\\ndocument recognition. Proceedings of the IEEE , 86(11):2278\\u20132324, 1998.\\nChanghee Lee, Fergus Imrie, and Mihaela van der Schaar. Self-supervision enhanced feature selection\\nwith correlated gates. In International Conference on Learning Representations , 2022.\\nRoman Levin, Valeriia Cherepanova, Avi Schwarzschild, Arpit Bansal, C Bayan Bruss, Tom Gold-\\nstein, Andrew Gordon Wilson, and Micah Goldblum. Transfer learning with deep tabular models.\\narXiv preprint arXiv:2206.15306 , 2022.\\nDong Liang, Jun Wang, Xiaoyu Gao, Jiahui Wang, Xiaoyong Zhao, and Lei Wang. Self-supervised\\npretraining isolated forest for outlier detection. In 2022 International Conference on Big Data,\\nInformation and Computer Network (BDICN) , pp. 306\\u2013310. IEEE, 2022.\\n12 Published as a conference paper at ICLR 2023\\nFrederick Liu and Besim Avci. Incorporating priors with feature attribution on text classi\\ufb01cation.\\narXiv preprint arXiv:1906.08286 , 2019.\\nWeiyang Liu, Rongmei Lin, Zhen Liu, James M Rehg, Liam Paull, Li Xiong, Le Song, and Adrian\\nWeller. Orthogonal over-parameterized training. In Proceedings of the IEEE\\/CVF Conference on\\nComputer Vision and Pattern Recognition , pp. 7251\\u20137260, 2021.\\nIlya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint\\narXiv:1711.05101 , 2017.\\nScott M Lundberg and Su-In Lee. A uni\\ufb01ed approach to interpreting model predictions. Advances in\\nNeural Information Processing Systems , 30, 2017.\\nWenjie Luo, Yujia Li, Raquel Urtasun, and Richard Zemel. Understanding the effective receptive\\n\\ufb01eld in deep convolutional neural networks. Advances in Neural Information Processing Systems ,\\n29, 2016.\\nRyszard S Michalski, Igor Mozetic, Jiarong Hong, and Nada Lavrac. The multi-purpose incremental\\nlearning system aq15 and its testing application to three medical domains. In Proc. AAAI , volume\\n1986, pp. 1\\u2013041, 1986.\\nSeyed-Mohsen Moosavi-Dezfooli, Alhussein Fawzi, Jonathan Uesato, and Pascal Frossard. Robust-\\nness via curvature regularization, and vice versa. In Proceedings of the IEEE\\/CVF Conference on\\nComputer Vision and Pattern Recognition , pp. 9078\\u20139086, 2019.\\nS\\u00e9rgio Moro, Paulo Rita, and Bernardo Vala. Predicting social media performance metrics and\\nevaluation of the impact on brand building: A data mining approach. Journal of Business Research ,\\n69(9):3341\\u20133351, 2016.\\nAdam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor\\nKilleen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. PyTorch: An imperative style,\\nhigh-performance deep learning library. Advances in Neural Information Processing Systems , 32,\\n2019.\\nGabriel Pereyra, George Tucker, Jan Chorowski, \\u0141ukasz Kaiser, and Geoffrey Hinton. Regularizing\\nneural networks by penalizing con\\ufb01dent output distributions. arXiv preprint arXiv:1701.06548 ,\\n2017.\\nLiudmila Prokhorenkova, Gleb Gusev, Aleksandr V orobev, Anna Veronika Dorogush, and Andrey\\nGulin. Catboost: unbiased boosting with categorical features. Advances in Neural Information\\nProcessing Systems , 31, 2018.\\nJ. Ross Quinlan. Simplifying decision trees. International journal of man-machine studies , 27(3):\\n221\\u2013234, 1987.\\nDhanesh Ramachandram and Graham W Taylor. Deep multimodal learning: A survey on recent\\nadvances and trends. IEEE Signal Processing Magazine , 34(6):96\\u2013108, 2017.\\nSalah Rifai, Pascal Vincent, Xavier Muller, Xavier Glorot, and Yoshua Bengio. Contractive auto-\\nencoders: Explicit invariance during feature extraction. In Icml, 2011.\\nAndrew Ross, Isaac Lage, and Finale Doshi-Velez. The neural LASSO: Local linear sparsity for\\ninterpretable explanations. In Workshop on Transparent and Interpretable Machine Learning\\nin Safety Critical Environments, 31st Conference on Neural Information Processing Systems ,\\nvolume 4, 2017a.\\nAndrew Slavin Ross, Michael C Hughes, and Finale Doshi-Velez. Right for the right reasons: Training\\ndifferentiable models by constraining their explanations. arXiv preprint arXiv:1703.03717 , 2017b.\\nIvan Rubachev, Artem Alekberov, Yury Gorishniy, and Artem Babenko. Revisiting pretraining\\nobjectives for tabular deep learning. arXiv preprint arXiv:2207.03208 , 2022.\\nNabeel Seedat, Alan Jeffares, Fergus Imrie, and Mihaela van der Schaar. Improving adaptive\\nconformal prediction using self-supervised learning. arXiv preprint arXiv:2302.12238 , 2023.\\n13 Published as a conference paper at ICLR 2023\\nRavid Shwartz-Ziv and Amitai Armon. Tabular data: Deep learning is not all you need. Information\\nFusion , 81:84\\u201390, 2022.\\nW Nick Street, William H Wolberg, and Olvi L Mangasarian. Nuclear feature extraction for breast\\ntumor diagnosis. In Biomedical Image Processing and Biomedical Visualization , volume 1905, pp.\\n861\\u2013870. SPIE, 1993.\\nBaohua Sun, Lin Yang, Wenhan Zhang, Michael Lin, Patrick Dong, Charles Young, and Jason Dong.\\nSuperTML: Two-dimensional word embedding for the precognition on structured tabular data. In\\nProceedings of the IEEE\\/CVF Conference on Computer Vision and Pattern Recognition Workshops ,\\npp. 0\\u20130, 2019.\\nMukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. In\\nInternational conference on machine learning , pp. 3319\\u20133328. PMLR, 2017.\\nMichelle Tang, Pulkit Kumar, Hao Chen, and Abhinav Shrivastava. Deep multimodal learning for the\\ndiagnosis of autism spectrum disorder. Journal of Imaging , 6(6):47, 2020.\\nJoseph J Thompson, Mark R Blair, Lihan Chen, and Andrew J Henrey. Video game telemetry as a\\ncritical tool in the study of complex skill learning. PloS one , 8(9):e75129, 2013.\\nRobert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical\\nSociety: Series B (Methodological) , 58(1):267\\u2013288, 1996.\\nTalip Ucar, Ehsan Hajiramezanali, and Lindsay Edwards. SubTab: Subsetting features of tabular data\\nfor self-supervised representation learning. Advances in Neural Information Processing Systems ,\\n34:18853\\u201318865, 2021.\\nAshish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, \\u0141ukasz\\nKaiser, and Illia Polosukhin. Attention is all you need. Advances in Neural Information Processing\\nSystems , 30, 2017.\\nZifeng Wang and Jimeng Sun. TransTab: Learning transferable tabular transformers across tables.\\nAdvances in Neural Information Processing Systems , 2022. URL https:\\/\\/openreview.\\nnet\\/forum?id=A1yGs_SWiIi .\\nSamuel George Waugh. Extending and benchmarking Cascade-Correlation: extensions to the\\nCascade-Correlation architecture and benchmarking of feed-forward supervised arti\\ufb01cial neural\\nnetworks . PhD thesis, University of Tasmania, 1995.\\nYeming Wen, Dustin Tran, and Jimmy Ba. BatchEnsemble: an alternative approach to ef\\ufb01cient\\nensemble and lifelong learning. arXiv preprint arXiv:2002.06715 , 2020.\\nFrank Wilcoxon. Individual comparisons by ranking methods. In Breakthroughs in Statistics , pp.\\n196\\u2013202. Springer, 1992.\\nXinglong Wu, Mengying Li, Xin-wu Cui, and Guoping Xu. Deep multimodal learning for lymph\\nnode metastasis prediction of primary thyroid cancer. Physics in Medicine & Biology , 67(3):\\n035008, 2022.\\nJinsung Yoon, Yao Zhang, James Jordon, and Mihaela van der Schaar. VIME: Extending the success\\nof self-and semi-supervised learning to tabular domain. Advances in Neural Information Processing\\nSystems , 33:11033\\u201311043, 2020.\\nHongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical\\nrisk minimization. In International Conference on Learning Representations , 2018.\\nYu Zhang, Peter Ti \\u02c7no, Ale\\u0161 Leonardis, and Ke Tang. A survey on neural network interpretability.\\nIEEE Transactions on Emerging Topics in Computational Intelligence , 2021.\\nYitan Zhu, Thomas Brettin, Fangfang Xia, Alexander Partin, Maulik Shukla, Hyunseung Yoo,\\nYvonne A Evrard, James H Doroshow, and Rick L Stevens. Converting tabular data into images\\nfor deep learning with convolutional neural networks. Scienti\\ufb01c reports , 11(1):11325, 2021.\\n14 Published as a conference paper at ICLR 2023\\nMaciej Zikeba, Jakub M Tomczak, Marek Lubicz, and Jerzy \\u2019Swikatek. Boosted SVM for extracting\\nrules from imbalanced data in application to prediction of the post-operative life expectancy in the\\nlung cancer patients. Applied Soft Computing , 2013.\\nHui Zou and Trevor Hastie. Regularization and variable selection via the elastic net. Journal of the\\nroyal statistical society: series B (statistical methodology) , 67(2):301\\u2013320, 2005.\\n15 Published as a conference paper at ICLR 2023\\nA E XTENDED RELATED WORKS\\nNeural Network Regularization. Regularization methods seek to penalize complexity and impose a\\nform of smoothness on a model. This may be cast as expressing a prior belief over the hypothesis space\\nof a neural network which attempts to aid generalization. ICategories. A vast array of regularization\\nmethods have been proposed throughout the literature (for a comprehensive taxonomy see e.g.\\nKuka \\u02c7cka et al., 2017). Modern nomenclature typically includes broad modeling decisions such as\\nchoice of architecture, loss function, and optimization method under the umbrella of regularization.\\nAdditionally, many regularization techniques augment the training data using methods such as input\\nnoise (Krizhevsky et al., 2012) or MixUp (Zhang et al., 2018). Dropout (Hinton et al., 2012) and\\nrelated approaches that augment a hidden representation of the input may also be included in this\\ncategory. Possibly a more conventional category of regularization is that which adds explicit penalty\\nterm(s) to the loss function. These terms might penalize the network weights directly to shrink\\nor sparsify their values as in L2 (Hoerl & Kennard, 1970) and L1 (Tibshirani, 1996), respectively.\\nAlternatively, network outputs may be penalized to, for example, reduce overcon\\ufb01dence (Pereyra\\net al., 2017). IWeight Orthogonalization. A number of works have studied the orthogonalization\\nof network weights via various weight penalization methods (Bansal et al., 2018). More recent work in\\nLiu et al. (2021) proposed to learn an orthogonal transformation of the randomly initialized incoming\\nweights to a given neuron. In contrast, this work seeks to ensure that the gradients of different\\nlatent neurons with respect to a given input vector are orthogonal. ICombination. Compositions\\nof multiple regularization methods are extensively applied in practice. An early example in the\\nregression setting is the elastic net penalty (Zou & Hastie, 2005) which attempts to combine sparsity\\nwith shrinkage in the coef\\ufb01cients. More recent work has demonstrated the effectiveness of combining\\nseveral regularization terms on tabular data (Kadra et al., 2021), a domain in which neural networks\\nsuperiority had previously been less convincing.\\nB T ABULAR ARCHITECTURES AND BOOSTING\\nWhile non-neural methods such as XGBoost (Chen & Guestrin, 2016) and CatBoost (Prokhorenkova\\net al., 2018) are still considered state of the art for tabular data (Grinsztajn et al., 2022), much progress\\nhas been made in recent years to close the gap. Furthermore, differing learning paradigms have\\nvarious strengths and weaknesses outside of maximum generalization performance, which is often\\na consideration in practical applications. While boosting methods boast excellent computational\\nef\\ufb01ciency and strong out-of-the-box performance, neural networks have unique utility in, for example,\\nmulti-modal learning (Ramachandram & Taylor, 2017), meta-learning (Hospedales et al., 2021)\\nand certain interpretability methods (Zhang et al., 2021). In this section, we provide additional\\nexperiments applying TANGOS to a state-of-the-art transformer architecture for tabular data proposed\\nin Gorishniy et al. (2021). Speci\\ufb01cally, this architecture combines a Feature Tokenizer which\\ntransforms features into embeddings with a multi-layer Transformer (Vaswani et al., 2017). We\\ncompare this FT-Transformer architecture to boosting methods in the default setting where we\\nevaluate out-of-the-box performance and the tuned setting where we jointly optimize the Transformer\\nalong with its baseline regularizers. We describe these two settings in more detail next.\\nDefault Setting. In this setting, we use a 3-layer Transformer with a 32-dimensional feature embed-\\nding size and 4 attention heads. Following the original paper we use Reglu activations, a hidden\\nlayer size of 43 corresponding to a ratio of4\\n3with the embedding size, Kaiming initialization (He ding size and 4 attention heads. Following the original paper we use Reglu activations, a hidden\\nlayer size of 43 corresponding to a ratio of4\\n3with the embedding size, Kaiming initialization (He\\net al., 2015), and AdamW optimizer (Loshchilov & Hutter, 2017). Finally, we apply a learning rate of\\n0.001. We compare this architecture with and without TANGOS regularization applied which we refer\\nto as \\u201cBaseline\\u201d and \\u201c+ TANGOS \\u201d respectively. We set \\u00151= 1and\\u00152= 0:01which were found to\\nbe reasonable default values for specialization and orthogonalization in our experiments in Section 5.\\nTuned Setting. Here we apply ten iterations of random search tuning over the same hyperparameters\\nas in the original work with those achieving the best validation performance selected. We then evaluate\\nthis combination by training over three seeds and perform their \\ufb01nal evaluations on a held-out test set.\\nWe search using the same distributions as in the original work and consider the following ranges. L2\\nregularization2[1e\\u000006;1e\\u000003], residual dropout2[0:0;0:2], hidden layer dropout 2[0:0;0:5],\\nattention dropout 2[0:0;0:5], hidden layer to feature embedding dimension ratio 2[1:0;3:0],\\nembedding dimension 2[16;48], number of layers 2[1;3], learning rate2[1e\\u000004;1e\\u000003]. In\\nthe \\u201c+ TANGOS \\u201d setting we also include \\u001512[0:001;10]and\\u001522[0:0001;1]with a log uniform\\n16 Published as a conference paper at ICLR 2023\\ndistribution. All remaining architecture choices are consistent with the default setting and the original\\nwork.\\nWe ran our experiments on the Jannis (Guyon et al., 2019) and Higgs (Baldi et al., 2014) datasets.\\nThese are both classi\\ufb01cation datasets consisting of 83733 and98050 examples respectively. These\\ndatasets were selected as they represent a signi\\ufb01cant number of input examples along with a middling\\nnumber of input features relative to the other tabular datasets explored in this work ( 54and28\\nrespectively). We follow the experimental protocol of the boosting comparison in Grinsztajn et al.\\n(2022) using the same training, validation, and test splits and reporting mean test accuracy over three\\nruns. Therefore we obtain the same results for boosting as reported in that work.\\nThe results of this experiment are reported in Table 2 where we \\ufb01nd that TANGOS does indeed have a\\npositive effect on the FT-Transformer performance although, consistent with the original work, we\\nfound that regularization only provides modest gains at best with this architecture. While we do not\\nclaim that TANGOS regularization results in neural networks that outperform Boosting methods, these\\nresults indicate that TANGOS regularization can contribute to closing the gap and may play a key role\\nwhen combined with other methods as highlighted in Kadra et al. (2021). We believe this to be an\\nimportant area for future research and, in particular, expect that architecture-speci\\ufb01c developments of\\nthe ideas presented in this work may provide further improvements on the results obtained in this\\nsection.\\nC M OTIVATION FOR DEEPLEARNING ON TABULAR DATA\\nSeveral works have argued that boosting methods generally achieve superior performance to even\\nstate-of-the-art deep learning architectures for tabular data (Grinsztajn et al., 2022; Shwartz-Ziv &\\nArmon, 2022). However, this is in contrast to recent \\ufb01ndings for transformer style architectures in\\nGorishniy et al. (2021), especially with appropriate feature embeddings (Gorishniy et al., 2022) and\\nsuf\\ufb01cient pretraining (Rubachev et al., 2022). We defer from this discussion to highlight a selection\\nof reasons to consider deep learning methods for tabular data beyond straightforward improvements\\nin predictive performance. In particular, we include a number of deep learning paradigms that are\\ndif\\ufb01cult to analogize for non-neural models and have been successfully applied to tabular data.\\nMulti-modal learning refers to the task of modeling data inputs that consist of multiple data\\nmodalities (e.g. image, text, tabular). As one might intuit, jointly modeling these multiple modalities\\ncan result in better performance than independently predicting from each of them (Ramachandram\\n& Taylor, 2017; Guo et al., 2019). Deep learning provides a uniquely natural method of combining\\nmodalities with the advantages of (1) modality-speci\\ufb01c encoders, (2) that are fused into a joint\\ndownstream representation and trained end-to-end with backpropagation, and (3) superior modeling\\nperformance in many modalities such as images and natural language. Healthcare is a domain\\nin which multi-modal learning is particularly salient (Acosta et al., 2022). Recent work in Wu\\net al. (2022) showed that jointly modeling tabular clinical records using an MLP together with\\nmedical images using a CNN outperforms the non-multi-modal baselines. Elsewhere in Tang et al.\\n(2020), a multi-modal approach is taken in combining input modalities based on the preprocessing\\nof functional magnetic resonance imaging and region of interest time series data for the diagnosis\\nof autism spectrum disorder. A resnet-18 encodes one modality while an MLP encodes the other,\\nresulting in superior performance when analyzed in an ablation study. In this setting, progress in\\nmodeling each of the individual modalities is likely to result in better performance of the system resulting in superior performance when analyzed in an ablation study. In this setting, progress in\\nmodeling each of the individual modalities is likely to result in better performance of the system\\nas a whole. Interestingly, Ramachandram & Taylor (2017) identi\\ufb01ed regularization techniques\\nfor improved cross-modality learning as an important research direction. We believe that further\\ndevelopment of the ideas presented in this work could provide a powerful tool for balancing how\\nmodels attend to multiple input modalities.\\nMeta-learning aims to distill the experience of multiple learning episodes across a distribution\\nof related tasks to improve learning performance on future tasks (Hospedales et al., 2021). Deep\\nlearning-based approaches have seen great success as a solution to this problem in a variety of \\ufb01elds.\\nIn the tabular domain, with careful consideration of the shared information between tasks, recent\\nworks have also shown promising results in this direction by developing methods for transferring\\ndeep tabular models across tables (Wang & Sun, 2022; Levin et al., 2022). In particular, in Levin\\net al. (2022) it was noted that \\u201crepresentation learning with deep tabular models provides signi\\ufb01cant\\n17 Published as a conference paper at ICLR 2023\\ngains over strong GBDT baselines\\u201d, also \\ufb01nding that \\u201cthe gains are especially pronounced in low\\ndata regimes\\u201d.\\nInterpretability is an important area of deep learning research aiming to provide users with the\\nability to understand and reason about model outputs. Certain classes of interpretability methods\\nhave recently been developed that provide distinct forms of interpretability relying on the hidden\\nrepresentations of neural networks. In such models, probing the representation space of a deep model\\npermits a new type of interpretation. For instance, Kim et al. (2018) studies how human concepts\\nare represented by deep classi\\ufb01ers. This makes it possible to analyze how the classes predicted\\nby the model relate to human understandable concepts. For example, one can verify if the stripe\\nconcept is relevant for a CNN classi\\ufb01er to identify a zebra, as demonstrated in the paper. Another\\nexample is Crabb\\u00e9 et al. (2021), which proposes to explain a given example with reference to a freely\\nselected set of other examples (potentially from the same dataset). A user study was carried out in\\nthis work which concluded that, among non-technical users, this method of explanation does affect\\ntheir con\\ufb01dence in the model\\u2019s prediction. These powerful methods crucially rely on the model\\u2019s\\nrepresentation space, which effectively assumes that the model is a deep neural network.\\nRepresentation learning more generally provides access to several other methods from deep learning\\nto the tabular domain. A number of works have used deep learning approaches to map inputs to\\nembeddings which can be useful for downstream applications. SuperTML (Sun et al., 2019) and\\nZhu et al. (2021) map tabular inputs to image-like embeddings that can therefore be passed to image\\narchitectures such as CNNs. Other self-supervised methods include VIME (Yoon et al., 2020) which\\napplies input reconstruction, SubTab (Ucar et al., 2021) which suggests a multi-view reconstruction\\ntask and SCARF (Bahri et al., 2021) which takes a contrastive approach. Representation learning\\napproaches such as these have proven successful on downstream tabular data tasks such as uncertainty\\nquanti\\ufb01cation (Seedat et al., 2023), federated learning (He et al., 2022), anomaly detection (Liang\\net al., 2022), and feature selection (Lee et al., 2022).\\nD TANGOS B EHAVIOR ANALYSIS\\nIn this section, we apply TANGOS to a simple image classi\\ufb01cation task using a convolutional neural\\nnetwork (CNN) and provide a qualitative analysis of the behavior of the learned network. This\\nanalysis is conducted on the MNIST dataset (LeCun et al., 1998) using the recommended split\\nresulting in 60,000 training and 10,000 validation examples.\\nIn this experiment, we train a standard CNN architecture (as described in Table 3) with a penultimate\\nhidden layer of 10 neurons for 10 epochs with Adam optimizer and a learning rate of 0.001. We also\\napply L2 regularization with weight 0.001. After each epoch, the model is evaluated on the validation\\nset where the epoch achieving the best validation performance is stored for further analysis. Two\\nmodels are trained under this protocol. One model which applied TANGOS to the penultimate hidden\\nlayer with\\u00151= 100 ,\\u00152= 0:1andM= 25 and a baseline model which does not apply TANGOS .\\nTable 3: MNIST Convolutional Neural Network Architecture.\\nLayer Type Hyperparameters Activation Function\\nConv2d Input Channels:1 ; Output Channels:16 ; Kernel Size:5 ; Stride:2 ; Padding:1 ReLU\\nConv2d Input Channels:16 ; Output Channels:32 ; Kernel Size:5 ; Stride:2 ; Padding:1 ReLU\\nFlatten Start Dimension:1\\nLinear Input Dimension: 512 ; Output Dimension: 256 ReLU\\nLinear Input Dimension: 256 ; Output Dimension: 10\\nLinear Input Dimension: 10 ; Output Dimension: 10\\nIn this section, we examine the gradients of each of the 10 neurons in the penultimate hidden\\nlayer with respect to each of the input dimensions of a given image. TANGOS is designed to reward In this section, we examine the gradients of each of the 10 neurons in the penultimate hidden\\nlayer with respect to each of the input dimensions of a given image. TANGOS is designed to reward\\northogonalization and specialization of these gradient attributions which can be evaluated qualitatively\\nby inspection. In all plots that follow we apply a min-max scaling across all hidden units for a fair\\ncomparison. Both strong positive and strong negative values for attributions may be interpreted as\\n18 Published as a conference paper at ICLR 2023\\na latent unit attending to a given input dimension. In Figure 6 we provide results for the baseline\\nmodel applied to a test image where, in line with similar analyses in previous works such as Crabb\\u00e9\\n& van der Schaar (2022), we note that the way in which hidden units attend to the input is highly\\nentangled. In contrast to this, in Figure 7, we include the same plot for the TANGOS trained model on\\nthe same image. In this case, each hidden unit does indeed produce relatively sparse and orthogonal\\nattributions as desired. These results were consistent across the test images.\\nH1\\n H2\\n H3\\n H4\\n H5\\nH6\\n H7\\n H8\\n H9\\n H10\\nStrong\\nNegative0Strong\\nPositive\\nFigure 6: Without TANGOS Training. Gradient attributions with respect to each of the 10 hidden\\nneurons. These results suggest signi\\ufb01cant overlap among the gradient attributions.\\nH1\\n H2\\n H3\\n H4\\n H5\\nH6\\n H7\\n H8\\n H9\\n H10\\nStrong\\nNegative0Strong\\nPositive\\nFigure 7: With TANGOS Training. TANGOS encourages gradient attributions to be sparse with\\nminimal overlap.\\nWe can glean further insight into the TANGOS trained model by examining the role of individual\\nneurons across multiple test images. In Figure 8, we provide the gradient attributions for hidden\\nneuron 5 (H5) from our previous discussion across twelve test images. This neuron appears to\\ndiscriminate between an open or a closed loop at the lower left of the digit. Indeed this is a key\\naspect of distinction between the set of digits f2;6;8;0g(\\ufb01rst row) andf9;5;3g(second row). We\\nalso include digits where this visual feature is less useful as they contain no lower-left loop either\\nopen or closed (third row). This hypothesis can be further examined by analyzing the values of these\\nactivations. We note that the \\ufb01rst two rows typically have higher magnitude with opposite signs while\\nthe third row has lower magnitude activations. In Table 4 we summarize the effect of these activation\\nscores on class probabilities by accounting for the weights connecting to each of the ten classes. As\\none might expect, the weight connections between the hidden neuron and classes on the \\ufb01rst row and\\nthe second row have opposite signs indicating that neuron 5 does indeed discriminate between these\\nclasses.\\nE P ERFORMANCE WITH INCREASING DATA SIZE\\nIn this section, we evaluate TANGOS performance with an increasing number of input examples. To\\ndo this we use the Dionis dataset, which was the largest benchmark dataset proposed in Kadra et al.\\n(2021) with 416,188 examples. As in that work, we set aside 20% for testing with the remaining\\ndata further split into 80% training and 20% validation. The data was standardized to have zero\\nmean and unit variance with statistics calculated on the training data. We then consider using\\nvarious proportions (10%, 50%, 100%) of the training data to train an MLP with and without\\nTANGOS regularization. We also evaluate the best-performing regularization method, L2, from our\\nexperiments in Section 5. For both regularization methods, we train three hyperperameter settings\\nat each proportion and evaluate the best performing of the three on the test set. For TANGOS we\\nconsiderf(\\u00151= 1;\\u00152= 0:01);(\\u00151= 1;\\u00152= 0:1);(\\u00151= 10;\\u00152= 0:1)gand for L2 we consider\\n19 Published as a conference paper at ICLR 2023\\nActivation = 1.40 Activation = 0.81 Activation = 0.50 Activation = 1.05\\nActivation = -1.01 Activation = -0.96 Activation = -0.54 Activation = -1.12\\nActivation = -0.34 Activation = -0.26 Activation = 0.26 Activation = 0.27\\nFigure 8: Hidden Neuron 5. This neuron attempts to discriminate whether inputs contain a closed\\nloop on the lower left of their digit. Inputs with a closed lower loop incur highly positive activations\\n(\\ufb01rst row). Inputs with open lower loops incur highly negative activations (second row). While\\nambiguous inputs with no lower loop at all tend to produce low-magnitude activations (third row).\\nTable 4: Neuron 5 Class Weights. Weights connecting neuron 5 to each of the ten classes and a\\nsummary of their combined effect with the neuron activation. The classi\\ufb01cation in\\ufb02uence column\\nprovides a categorical indication of the magnitude of the contribution to each class output for a \\ufb01xed\\nactivation magnitude. This is determined by the magnitude of the connecting weight where: Low\\n2[0;0:59], Medium2[0:59;1:17], and High2[1:17;1:76].\\nClass label Weight connectionIncreases class probability\\nif activation isClassi\\ufb01cation in\\ufb02uence\\n0 1.0883 Positive Medium\\n1 -0.6826 Negative Medium\\n2 1.5298 Positive High\\n3 -1.5862 Negative High\\n4 -0.2516 Negative Low\\n5 -0.6008 Negative Medium\\n6 0.9065 Positive Medium\\n7 0.3524 Positive Low\\n8 0.7362 Positive Medium\\n9 -1.7608 Negative High\\n\\u00152f0:01;0:001;0:0001g. We repeat this procedure for 6 runs and report the mean test accuracy.\\nThe MLP contained three ReLU-activated hidden layers of 400, 100, and 10 hidden units, respectively.\\nWe include the results of this experiment in Figure 9. Consistent with our experiments in Section 5,\\nwe \\ufb01nd that TANGOS outperforms both the baseline model and the strongest baseline regularization\\nmethod across all proportions of the data. These results are indicative that TANGOS remains similarly\\neffective across both small and large datasets in the tabular domain.\\nF A PPROXIMATION AND ALGORITHM\\nCalculating the attribution of the latent units with respect to the input involves computing the\\nJacobian matrix, which can be computed in O(1)time and has memory complexity O(dHdX). The\\ncomputational complexity of calculating Lorth isO(d2\\nH)(i.e. all pairwise computation between\\nlatent units). While the calculation can be ef\\ufb01ciently parallelized, this still becomes impractically\\nexpensive with higher dimensional layers. To address this, we introduce a relaxation by randomly\\nsubsampling pairs of neurons to calculate attribution similarity. We denote by Idenote the set of\\nall possible pairs of neuron indices, I=f(i;j)8i;j2[dH]andi6=jg. Further, we let Mdenote\\na randomly sampled subset of I,M\\u0012I. We devise an approximation to the regularization term,\\ndenoted byL0\\north, by estimating the penalty on the subset M, where the size of Mcan be chosen to\\n20 Published as a conference paper at ICLR 2023\\n10 50 100\\n% Training Data Used76808488% Test Accuracy\\nTANGOS\\nL2\\nBaseline\\nFigure 9: Performance Gains With Increasing Data Size. Training with various proportions\\nof training data from the 416,188 examples of the Dionis dataset, we \\ufb01nd the relative boost in\\nperformance from TANGOS to be consistent.\\nbalance computational burden with more faithful estimation:\\nL0\\north(x) =1\\nBBX\\nb=11\\njMjX\\n(i;j)2M\\u001a[ai(xb);aj(xb)]\\nThis reduces the complexity of calculating LorthfromO(d2\\nH)toO(M). For our experimental results\\ndescribed in Tables 1, 6 and 7, we use jMj= 50 . We empirically demonstrate that this approximation\\nstill leads to strong results in real-world experiments. The overall training procedure is described in\\nAlgorithm 1.\\nAlgorithm 1 TANGOS regularization\\nResult: Learned parameters \\u0012\\nInput:\\u00151;\\u00152;training dataD;learning rate \\u0011;\\nInitialise\\u0012;\\nwhile not converged do\\nSampleDmini fromD;\\n^L(f\\u0012(x);y) =E(x;y)\\u0018Dmini[L(f\\u0012(x);y)];\\n^R(x) =\\u00151Ex\\u0018Dmini[Lspec(x)] +\\u00152Ex\\u0018Dmini[L0\\north(x)];\\n\\u0012 \\u0012+\\u0011r\\u0012h\\n^L(f\\u0012(x);y) +^R(x)i\\n;\\nend while\\nAdditionally, we provide an empirical analysis of TANGOS designed to evaluate the effectiveness of\\nour proposed subsampling approximation with respect to generalization performance and compu-\\ntational ef\\ufb01ciency as the number of sampled neuron pairs Mgrows. Furthermore, we analyze the\\ncomputational ef\\ufb01ciency of our method as the number of latent units grows, evaluating the method\\u2019s\\ncapacity to scale to large models.\\nAll experiments are run on the BC dataset which we split into 80% training and 20% validation. We\\n\\ufb01x\\u00151= 100 and\\u00152= 0:1throughout our experiments. We run each experiment over 10 random\\nseeds and report the mean and standard deviation. All remaining experimental details are consistent\\nwith our experiments in Section 5. We note that our implementation of TANGOS is not optimized to\\nthe same extent as the Pytorch (Paszke et al., 2019) implementation of L2 to which we compare, and\\ntherefore we may consider our relative computational performance to be a loose upper bound on a\\ntruly optimized version.\\nIn Figure 10 (left), we report the relative increase in compute time per epoch as we increase the\\nnumber of sampled pairs. As theory would suggest, this growth is linear. A natural follow-up question\\n21 Published as a conference paper at ICLR 2023\\n25 50 75 100 125 150\\nNumber of sampled pairs\\u00d71\\u00d72\\u00d73\\u00d74\\u00d75Relative Runtime IncreaseTANGOS\\nL2\\n25 50 75 100 125 150\\nNumber of sampled pairs0.2250.2400.255Loss\\nFigure 10: Sampling ef\\ufb01ciency. Runtime increases linearly with the number of sampled pairs (left)\\nwhile better generalization performance is maintained even for low sampling rates (right). The\\nbene\\ufb01ts of TANGOS can be realized using our proposed sampling approximation with comparable\\nruntime to even the most ef\\ufb01cient existing regularization approaches.\\nis the extent to which model performance is affected by decreasing the number of sampled pairs. In\\nFigure 10 (right), we observe that even very low sampling rates still result in excellent performance.\\nBased on these results, our recommendation for practitioners is that while increasing the sampling\\nrate can lead to marginal improvements in performance, relatively low sampling rates appear to be\\ngenerally suf\\ufb01cient and do not require prohibitive computational overhead.\\nGiven the results in Figure 10, we next wish to evaluate if the proposed sampling scheme enables\\nTANGOS to scale to much bigger models. In order to evaluate this we vary the number of neurons\\nin the relevant hidden layer while maintaining a \\ufb01xed sampling rate of 50 pairs (consistent with our\\nexperiments in Section 5). Other experimental parameters are consistent with the previous experiment.\\nThe results are provided in Figure 11 where we observe a relatively slow increase in runtime as\\nthe model grows. These results demonstrate that TANGOS can ef\\ufb01ciently be applied to much larger\\nmodels by using our proposed sampling scheme.\\n200 400 600 800 1000\\nNumber of Latent Units-2%0%2%4%6%8%% Runtime Increase\\nFigure 11: Scaling to large models. With a subsampling rate \\ufb01xed at M= 50 ,TANGOS incurs only\\na small percentage increase in runtime as the number of neurons in the penultimate hidden layer\\nincreases dramatically.\\nG A BLATION STUDY\\nTANGOS is designed with joint application of both regularization on specialization and orthogonal-\\nization in mind. Having empirically demonstrated strong overall results, an immediate question\\nis the dynamics of the two regularizers, and how they interact to affect performance. Speci\\ufb01cally,\\nwe consider the performance gain due to joint regularization effects over applying each regularizer\\nseparately.\\nThis includes three separate settings: 1)when the specialization regularizer is applied independently\\n(SpecOnly ), here we set \\u00152= 0and search over \\u001512f1;10;100g;2)when the orthogonalization\\nis applied separately ( OrthOnly ), we set\\u00151= 0and search over \\u001522f0:1;1g; and lastly 3)when\\nboth are applied jointly ( TANGOS ), i.e. searching over \\u001512f1;10;100gand\\u001522f0:1;1g. We\\n22 Published as a conference paper at ICLR 2023\\nreport the result of the ablation study in Table 5. We empirically observe that the joint effects of both\\nregularizers (i.e. TANGOS ) are crucial to achieve consistently good performance.\\nCombining these results with what we observed in Figure 3, we hypothesize that applying just\\nspecialization regularization, with no regard for diversity, can inadvertently force the neurons to\\nattend to overlapping regions in the input space. Correspondingly, simply enforcing orthogonalization,\\nwith no regard for sparsity, will likely result in neurons attending to non-overlapping yet spurious\\nregions in the input. Thus, we conclude that the two regularizers have distinct, but complementary,\\neffects that work together to achieve the desired regularization effect.\\nTable 5: Ablation study. Generalization performance on different ablation settings.\\nClassi\\ufb01cation (Mean NLL) Regression (MSE)\\nDataset BR CR HC BC BH WQ\\nNoReg 0:0726 0:4633 0:3321 0:3343 0:1977 0:6732\\nSpecOnly 0:0742 0:4466 0:3837 0:3099 0:1842 0:6714\\nOrthOnly 0:0716 0:3696 0:2073 0:2692 0:1916 0:6529\\nTANGOS 0:070 0:3633 0:2191 0:2472 0:1826 0:6379\\n23 Published as a conference paper at ICLR 2023\\nH R ANKING PLOT\\nIn Table 1, we reported the generalization performance of TANGOS compared to other regularizers\\nin a stand-alone setting. To gain a better understanding of relative performance, we visually depict\\nthe relative ranking of regularizers across all 20datasets. Figure 12 demonstrates that TANGOS\\nconsistently ranks as one of the better-performing regularizers, while performance of benchmark\\nmethods tend to \\ufb02uctuate depending on the dataset.\\nFB BH WE BC WQ SC AB PR ST FF1\\n2\\n3\\n4\\n5\\n6\\n7\\n8Regression\\nHE BR CE CR HC AU TU EN TH SO1\\n2\\n3\\n4\\n5\\n6\\n7\\n8Classi\\ufb01cation\\nTANGOS\\nBaseline\\nL1\\nL2\\nDropout\\nBatch Norm\\nInput Noise\\nMixUp\\n0.0 0.2 0.4 0.6 0.8 1.0\\nDataset0.00.20.40.60.81.0Rank\\nFigure 12: Ranking of stand-alone regularizers. Relative ranking of regularizer performance\\nacross the 20datasets, as reported in Table 1. TANGOS consistently ranks among the best-performing\\nregularizers.\\nI S TAND -ALONE UNCERTAINTY\\nIn Table 6, we report the standard deviation on generalization performance reported in Table 1. The\\nstandard errors are computed using 10seeded runs.\\nTable 6: Standard error on generalization performance. Standard errors with respect to the\\nrandom seed after retraining models from Table 1 experiments 10times.\\nDataset Baseline L1 L2 DO BN IN MU TANGOS\\nRegression (Mean Squared Error)\\nFB 0.051 0.016 0.009 0.051 0.627 0.076 0.041 0.042\\nBH 0.023 0.021 0.029 0.022 0.025 0.023 0.011 0.028\\nWE 0.006 0.010 0.008 0.008 0.008 0.013 0.010 0.009\\nBC 0.013 0.007 0.005 0.009 0.020 0.024 0.012 0.009\\nWQ 0.016 0.019 0.005 0.014 0.021 0.015 0.019 0.008\\nSC 0.026 0.014 0.019 0.025 0.023 0.168 0.067 0.017\\nFF 0.034 0.029 0.035 0.033 0.041 0.036 0.031 0.035\\nPR 0.042 0.029 0.020 0.031 0.032 0.072 0.016 0.026\\nST 0.090 0.084 0.085 0.064 0.076 0.080 0.082 0.029\\nAB 0.016 0.016 0.008 0.011 0.026 0.014 0.012 0.006\\nClassi\\ufb01cation (Mean Negative Log-likelihood)\\nHE 0.057 0.049 0.009 0.033 0.163 0.038 0.067 0.032\\nBR 0.086 0.005 0.002 0.133 0.034 0.060 0.010 0.031\\nCE 0.060 0.007 0.008 0.043 0.051 0.044 0.056 0.023\\nCR 0.029 0.094 0.004 0.034 0.041 0.027 0.027 0.019\\nHC 0.091 0.014 0.019 0.026 0.054 0.111 0.024 0.014\\nAU 0.038 0.030 0.002 0.030 0.081 0.031 0.037 0.019\\nTU 0.075 0.075 0.087 0.077 0.087 0.096 0.078 0.064\\nEN 0.036 0.038 0.033 0.045 0.082 0.049 0.050 0.046\\nTH 0.048 0.021 0.002 0.065 0.096 0.047 0.039 0.001\\nSO 0.038 0.028 0.016 0.046 0.029 0.025 0.023 0.008\\n24 Published as a conference paper at ICLR 2023\\nJ I NSIGHTS - EXTENDED RESULTS\\nIn this section, we present extended results of the decomposition of overall model error into diversity\\nand weighted error among an ensemble of latent units from Section 4.2. We include all eight\\nregularizers as described in Table 9 and three datasets (WE, ST, and BC) as described in Table 8. The\\nresults are included in Figure 13.\\n012BaselineDiversity (Div)Weighted Errors (Err)Ensemble Error (Err)L1L2Dropout\\n04080012Batch Norm04080Input Noise04080Mixup04080L-GOS\\n0.00.20.40.60.81.0Epoch0.00.20.40.60.81.0Mean scoreTANGOS\\n(a) Weather (WE) dataset.\\n01.53BaselineDiversity (Div)Weighted Errors (Err)Ensemble Error (Err)L1L2Dropout\\n010020001.53Batch Norm0100200Input Noise0100200Mixup0100200L-GOS\\n0.00.20.40.60.81.0Epoch0.00.20.40.60.81.0Mean scoreTANGOS\\n(b) Student (ST) dataset.\\n012BaselineDiversity (Div)Weighted Errors (Err)Ensemble Error (Err)L1L2Dropout\\n04080012Batch Norm04080Input Noise04080Mixup04080L-GOS\\n0.00.20.40.60.81.0Epoch0.00.20.40.60.81.0Mean scoreTANGOS\\n(c) Bioconcentration (BC) dataset.\\nFigure 13: Neuron diversity. Further examples of ensemble decomposition Err = Err\\u0000Divas\\ndiscussed in Section 4.2.\\n25 Published as a conference paper at ICLR 2023\\nK I NTANDEM RESULTS\\nIn Table 7, we provide a detailed breakdown of Figure 5, speci\\ufb01cally by reporting in tandem\\nperformance when benchmarks are paired with TANGOS across all datasets.\\nTable 7: In tandem performance. Mean\\u0006standard deviation of generalization performance when\\neach regularizer is employed in tandem with TANGOS .\\nDataset L1 L2 DO BN IN MU\\nRegression (Mean Squared Error)\\nFB 0.033\\u00060:018 0.018\\u00060:009 0.023\\u00060:215 0.042\\u00060:268 0.028\\u00060:047 0.046\\u00060:073\\nBH 0.192\\u00060:022 0.176\\u00060:024 0.196\\u00060:024 0.220\\u00060:021 0.191\\u00060:023 0.178\\u00060:019\\nWE 0.093\\u00060:011 0.091\\u00060:009 0.092\\u00060:009 0.076\\u00060:009 0.081\\u00060:012 0.077\\u00060:013\\nWQ 0.637\\u00060:014 0.639\\u00060:006 0.628\\u00060:008 0.630\\u00060:015 0.644\\u00060:011 0.649\\u00060:018\\nBC 0.227\\u00060:011 0.236\\u00060:011 0.243\\u00060:013 0.275\\u00060:027 0.235\\u00060:009 0.260\\u00060:014\\nSC 0.407\\u00060:044 0.399\\u00060:072 0.370\\u00060:031 0.408\\u00060:049 0.391\\u00060:317 0.425\\u00060:061\\nAB 0.309\\u00060:028 0.312\\u00060:028 0.312\\u00060:026 0.311\\u00060:027 0.319\\u00060:037 0.308\\u00060:03\\nFF 1.281\\u00060:029 1.276\\u00060:043 1.268\\u00060:028 1.297\\u00060:027 1.203\\u00060:046 1.207\\u00060:011\\nPR 0.553\\u00060:047 0.572\\u00060:031 0.642\\u00060:093 0.561\\u00060:069 0.565\\u00060:025 0.568\\u00060:081\\nST 0.392\\u00060:006 0.382\\u00060:006 0.388\\u00060:016 0.446\\u00060:018 0.417\\u00060:012 0.447\\u00060:011\\nClassi\\ufb01cation (Mean Negative Log-likelihood)\\nHE 0.441\\u00060:046 0.427\\u00060:049 0.454\\u00060:095 0.407\\u00060:047 0.377\\u00060:067 0.397\\u00060:075\\nBR 0.074\\u00060:005 0.070\\u00060:002 0.068\\u00060:006 0.062\\u00060:003 0.065\\u00060:01 0.078\\u00060:011\\nCE 0.389\\u00060:007 0.396\\u00060:007 0.394\\u00060:033 0.446\\u00060:024 0.422\\u00060:075 0.422\\u00060:062\\nCR 0.362\\u00060:021 0.366\\u00060:015 0.364\\u00060:003 0.406\\u00060:023 0.367\\u00060:01 0.384\\u00060:036\\nHC 0.200\\u00060:056 0.179\\u00060:012 0.185\\u00060:009 0.186\\u00060:021 0.211\\u00060:008 0.181\\u00060:037\\nAU 0.368\\u00060:028 0.360\\u00060:138 0.352\\u00060:011 0.344\\u00060:013 0.379\\u00060:016 0.368\\u00060:041\\nTU 1.519\\u00060:082 1.506\\u00060:045 1.481\\u00060:048 1.506\\u00060:067 1.522\\u00060:049 1.503\\u00060:087\\nSO 0.227\\u00060:036 0.268\\u00060:071 0.233\\u00060:021 0.353\\u00060:052 0.257\\u00060:032 0.304\\u00060:054\\nEN 0.990\\u00060:024 0.971\\u00060:002 0.945\\u00060:001 1.004\\u00060:026 0.995\\u00060:001 1.007\\u00060:04\\nTH 0.506\\u00060:031 0.503\\u00060:008 0.513\\u00060:01 0.524\\u00060:008 0.512\\u00060:045 0.514\\u00060:03\\nL D ATASET AND REGULARIZER DETAILS\\nWe perform our experiments on 20 real-world publicly available datasets obtained from (Dua et al.,\\n2017). They are summarized in Table 8. Further information and the source \\ufb01les used for each of\\nthe respective datasets can be found at: https:\\/\\/archive.ics.uci.edu\\/ml\\/machine-\\nlearning-databases\\/<UCISource>\\/ where <UCI Source> denotes the datasets unique\\nidenti\\ufb01er as listed in Table 8. Standard preprocessing was applied including standardization of\\nfeatures, one hot encoding of categorical variables, median imputation of missing values and log\\ntransformations of highly skewed feature distributions. Furthermore, for computational feasibility,\\ndatasets with over 1000 samples were reduced in size. In these cases the \\ufb01rst 1000 samples from\\nthe original UCI Source \\ufb01le were used. In Table 9 we summarize the regularizers considered in this\\nwork.\\n26 Published as a conference paper at ICLR 2023\\nTable 8: Dataset descriptions. Summary of the datasets considered in this work.\\nDataset UCI Source Type Feature size Sample Size Reference\\nFacebook (FB) \\u201c00368\\u201d Regression 21 495 Moro et al. (2016)\\nBoston (BH) [1] Regression 13 506 Harrison Jr & Rubinfeld (1978)\\nWeather (WE) \\u201c00514\\u201d Regression 45 1000 Cho et al. (2020)\\nWine Quality (WQ) \\u201cwine-quality\\u201d Regression 11 1000 Cortez et al. (2009)\\nBioconcentration (BC) \\u201c00510\\u201d Regression 45 779 Grisoni et al. (2015)\\nSkillcraft (SC) \\u201c00272\\u201d Regression 18 1000 Thompson et al. (2013)\\nForest Fire (FF) \\u201cforest-\\ufb01res\\u201d Regression 39 517 Cortez & Morais (2007)\\nProtein (PR) \\u201c00265\\u201d Regression 9 1000 Dua et al. (2017)\\nStudent (ST) \\u201c00320\\u201d Regression 56 649 Cortez & Silva (2008)\\nAbalone (AB) \\u201cabalone\\u201d Regression 9 1000 Waugh (1995)\\nHeart (HE) \\u201cstatlog\\u201d Classi\\ufb01cation 20 270 Dua et al. (2017)\\nBreast (BR) \\u201cbreast-cancer-wisconsin\\u201d Classi\\ufb01cation 9 699 Street et al. (1993)\\nCervical (CE) \\u201c00383\\u201d Classi\\ufb01cation 136 858 Fernandes et al. (2017)\\nCredit (CR) \\u201ccredit-screening\\u201d Classi\\ufb01cation 40 677 Dua et al. (2017)\\nHCV (HC) \\u201c00571\\u201d Classi\\ufb01cation 12 615 Hoffmann et al. (2018)\\nAustralian (AU) \\u201cstatlog\\u201d Classi\\ufb01cation 55 690 Quinlan (1987)\\nTumor (TU) \\u201cprimary-tumor\\u201d Classi\\ufb01cation 25 339 Michalski et al. (1986)\\nEntrance (EN) \\u201c00582\\u201d Classi\\ufb01cation 38 666 Hussain et al. (2018)\\nThoracic (TH) \\u201c00277\\u201d Classi\\ufb01cation 24 470 Zikeba et al. (2013)\\nSoybean (SO) \\u201csoybean\\u201d Classi\\ufb01cation 484 683 Fisher & Schlimmer (1988)\\n[1] This dataset has now been removed due to ethical issues. For more information see the following url\\nhttps:\\/\\/medium.com\\/@docintangible\\/racist-data-destruction-113e3eff54a8\\nTable 9: Overview of regularizers. Description of benchmarks considered in this work and their\\nimplementations.\\nRegularizer Reference Implementation\\nBaseline NA Paszke et al. (2019)\\nL1 Tibshirani (1996) Paszke et al. (2019)\\nL2 Hoerl & Kennard (1970) Paszke et al. (2019)\\nDropout Hinton et al. (2012) Paszke et al. (2019)\\nBatch Norm Ioffe & Szegedy (2015) Paszke et al. (2019)\\nInput Noise Krizhevsky et al. (2012) Paszke et al. (2019)\\nMixup Zhang et al. (2018) Zhang et al. (2018)\\nTANGOS This work Paszke et al. (2019)\\n27\",\"38\":\" Accepted to the ICLR 2023 TrustML-(un)Limited workshop\\nMARK MYWORDS : DANGERS OF WATERMARKED\\nIMAGES IN IMAGE NET\\nKirill Bykov1, 2, *& Klaus-Robert M \\u00a8uller1, 3, 4, 5& Marina M.-C. H \\u00a8ohne1, 2, 3, 6, 7\\n1Technische Universit \\u00a8at Berlin, Machine Learning Group, 10587 Berlin, Germany\\n2Understandable Machine Intelligence Lab, ATB, 14469 Potsdam, Germany\\n3BIFOLD \\u2013 Berlin Institute for the Foundations of Learning and Data, 10587 Berlin, Germany\\n4Korea University, Department of Arti\\ufb01cial Intelligence, Seoul 136-713, Korea\\n5Max Planck Institute for Informatics, 66123 Saarbr \\u00a8ucken, Germany\\n6Machine Learning Group, UiT the Arctic University of Norway, 9037 Troms\\u00f8, Norway\\n7Department of Computer Science, University of Potsdam, 14476 Potsdam, Germany\\n*Corresponding Author: KBykov@atb-potsdam.de\\nABSTRACT\\nThe utilization of pre-trained networks, especially those trained on ImageNet, has\\nbecome a common practice in Computer Vision. However, prior research has\\nindicated that a signi\\ufb01cant number of images in the ImageNet dataset contain wa-\\ntermarks, making pre-trained networks susceptible to learning artifacts such as\\nwatermark patterns within their latent spaces. In this paper, we aim to assess the\\nextent to which popular pre-trained architectures display such behavior and to de-\\ntermine which classes are most affected. Additionally, we examine the impact\\nof watermarks on the extracted features. Contrary to the popular belief that the\\nChinese logographic watermarks impact the \\u201ccarton\\u201d class only, our analysis re-\\nveals that a variety of ImageNet classes, such as \\u201cmonitor\\u201d, \\u201cbroom\\u201d, \\u201capron\\u201d\\nand \\u201csafe\\u201d rely on spurious correlations. Finally, we propose a simple approach\\nto mitigate this issue in \\ufb01ne-tuned networks by ignoring the encodings from the\\nfeature-extractor layer of ImageNet pre-trained networks that are most susceptible\\nto watermark imprints.\\n1 I NTRODUCTION\\nIn recent years, the utilization of ImageNet Deng et al. (2009) pre-trained models has become a\\nstandard practice in Computer Vision applications Kornblith et al. (2019). Trained on the large and\\ndiverse collection of images, these models obtain the ability to extract high-level visual features that\\nlater could be transferred to a different task. This technique, referred to as transfer learning (see\\ne.g. Weiss et al. (2016) for a review), has proven to be highly effective, leading to signi\\ufb01cant ad-\\nvancements in various computer vision applications, such as object detection Talukdar et al. (2018),\\nsemantic segmentation Van Opbroek et al. (2018) and classi\\ufb01cation Yuan et al. (2021).\\nDeep Neural Networks (DNNs), despite being highly effective across a variety of applications, are\\nprone to learning spurious correlations, i.e., erroneous relationships between variables that seem to\\nbe associated based on a given dataset but in reality lack a causal relationship Izmailov et al. (2022).\\nThis phenomenon, referred to as the \\u201cClever-Hans effect\\u201d Lapuschkin et al. (2019) or \\u201cshortcut-\\nlearning\\u201d Geirhos et al. (2020), impairs the model\\u2019s ability to generalize. In Computer Vision (CV),\\nsuch correlations may manifest as DNNs\\u2019 dependence on background information for image classi-\\n\\ufb01cation Xiao et al. (2020), textural information Geirhos et al. (2018), secondary objects Rosenfeld\\net al. (2018), or unintended artifacts, such as human pen markings in skin cancer detection Anders\\net al. (2022) and patient information in X-ray images for pneumonia detection Zech et al. (2018).\\nRecent studies have uncovered the presence of spurious correlations in the ImageNet dataset, specif-\\nically, the connection of the Chinese logographic watermarks to the \\u201ccarton\\u201d class Anders et al.\\n(2022); Li et al. (2022). These correlations make ImageNet-trained networks vulnerable to learn\\nwatermark detectors in their latent space, leading to incorrect predictions when encountering similar\\npatterns in the data. Furthermore, it has been shown that this behavior persists even after \\ufb01ne-tuning patterns in the data. Furthermore, it has been shown that this behavior persists even after \\ufb01ne-tuning\\n1arXiv:2303.05498v1  [cs.LG]  9 Mar 2023 Accepted to the ICLR 2023 TrustML-(un)Limited workshop\\non different datasets Bykov et al. (2022), indicating that the vulnerability to watermarks is not ex-\\nclusive to ImageNet networks but possibly extends to all \\ufb01ne-tuned models.\\nWith this study, we aim to examine which speci\\ufb01c ImageNet classes are in\\ufb02uenced by the artifactual\\nbehavior of watermarks. We analyze the extent to which commonly used pre-trained architectures\\nexhibit this phenomenon and propose a straightforward solution for reducing such behavior in trans-\\nfer learning by eliminating the most artifact-sensitive representations, with negligible effect on the\\nmodel\\u2019s performance.\\n2 M ETHOD\\nFigure 1: The illustration shows the images\\nin the baseline dataset and their correspond-\\ning watermarked versions.In this work, we de\\ufb01ne neural representations as sub-\\nfunctions of a model that map the input domain to\\na scalar value indicating the activation of a speci\\ufb01c\\nneuron. Our analysis focuses on two primary sce-\\nnarios: scalar representations of output classes and\\nfeature-extractor representations, which correspond\\nto the layer preceding the output logit layer1.\\nTo evaluate the susceptibility of individual represen-\\ntations to watermarks, we created binary classi\\ufb01ca-\\ntion datasets between normal and watermarked im-\\nages and assessed their ability to distinguish between\\nthe two classes. We followed the approach outlined\\nin Bykov et al. (2022) and used a baseline dataset\\nof 998 ImageNet images2. We created four probing\\ndatasets by inserting random textual watermarks in\\nthe three most popular languages (Chinese, Latin, Hindi) Sanches (2014) and Arabic numerals, as il-\\nlustrated in Figure 1. We evaluated the representations\\u2019 ability to differentiate between watermarked\\nand normal classes using AUC ROC, a widely used performance metric for binary classi\\ufb01ers. To\\ndo so, we utilized the true labels provided by the two datasets, where class 1 represents images\\nwith a watermark and class 0 represents those without. We \\ufb01rst calculated the scalar activations\\nfrom a speci\\ufb01c neural representation for all images from both classes. Then, utilizing the binary\\nlabels, we calculated the AUC ROC classi\\ufb01cation score based on the differences in activations.\\nAUC ROC score of 1 indicates a perfect classi\\ufb01er, ranking the watermarked images consistently\\nhigher than normal ones, and 0.5 a random classi\\ufb01er. However, we can also observe scores less\\nthan 0.5, such as the score of 0 illustrating the perfect classi\\ufb01er, that is de-activated by the water-\\nmarked images. To measure the general ability of representations to differentiate between the two\\nclasses and provide evidence that the concept has been learned, we de\\ufb01ned a differentiability mea-\\nsured= max (A;1\\u0000A);whereAis the AUC ROC score of the representation in the particular\\nbinary classi\\ufb01cation problem.\\n3 R ESULTS\\nTo analyze the effects of watermarked images on learned representations, we employed 20 popular\\nImageNet-pre-trained Computer Vision architectures, namely AlexNet Krizhevsky (2014), ResNet\\n18, 50, 101, and 152 He et al. (2016), ResNext 101 Xie et al. (2017), WideResNet 101 Zagoruyko\\n& Komodakis (2016), ViT Dosovitskiy et al. (2020), BEiT Bao et al. (2021), Inception V3 Szegedy\\net al. (2016), DenseNet 121, 161, and 201 Huang et al. (2017), GoogLeNet Szegedy et al. (2015),\\nMobileNet V2 Sandler et al. (2018), Shuf\\ufb02eNet V2 Ma et al. (2018), VGG 11, 13, 16, and 19\\nSimonyan & Zisserman (2014).\\n1In the case of neurons that produce multi-dimensional activations in feature-extractor representations, such\\nas convolutional neurons, the channel neurons were analyzed by taking the average of the activation maps per\\neach channel.\\n2Images were obtained from https:\\/\\/github.com\\/EliSchwartz\\/\\nimagenet-sample-images , excluding 2 images that already contained Chinese logographic water-\\nmarks.\\n2 Accepted to the ICLR 2023 TrustML-(un)Limited workshop\\nFigure 2: ImageNet classes with the highest mean AUC ROC scores across the models analyzed in 4\\ndifferent scenarios (Chinese, Latin, Hindi, and Numeric watermarks). Each dot represents the AUC\\nROC performance of the class representation for a single model.\\nFor the 4 different scenarios, we collected the AUC ROC scores for every class logit representation\\nacross all 20 ImageNet pre-trained networks. Figure 2 illustrates the top-5 ImageNet classes by the\\nhighest average AUC ROC across the 20 models. We can observe the clear distinction between the\\ndifferent scenarios \\u2013 Chinese watermarks show signi\\ufb01cantly higher average classi\\ufb01cation scores,\\ncompared to the other three watermarks, namely Latin, Hindi, and Arabic numerals. Furthermore,\\nit can be observed that classes with a high capability for detecting Chinese watermarks are not\\ninherently linked to textual objects, whereas classes for other watermarks have a natural association\\nwith text, such as \\u201cweb site\\u201d or \\u201cbook jacket\\u201d. This observation supports the conclusion that the\\nability of DNNs to detect Chinese logograms results from the Clever-Hans effect and is not desirable,\\nwhereas this cannot be said for other text detectors. Interestingly, by analyzing the classes with the\\nlowest average AUC ROC we could even reveal \\u2014 for the \\ufb01rst time \\u2014 the ability of ImageNet\\nclasses to detect the absence of the Chinese watermarks in images, which was not given for the\\nother types of watermarks (illustrated in the Appendix 6).\\nFigure 3 illustrates the number of representations that are sensitive to the Chinese symbols, across\\nthe logit and feature-extractor layers (layers of representations, preceding the last prediction layer)\\nof different networks. From the left \\ufb01gure, which represents the sensitivity of output logits, we can\\nobserve that nearly all of the networks exhibit sensitive logit representations. This could be the rea-\\nson for the average drop of 10.6% in model performance when transparent Chinese watermarks are\\nadded to the ImageNet validation dataset, as reported in Li et al. (2022). Some networks, such as\\nGoogleNet, have up to 285 output classes (out of 1000) that are susceptible to Chinese watermarks.\\nThe right \\ufb01gure, which represents the ratio of sensitive representations to the total number of rep-\\nresentations in the feature-extractor layers, reveals a signi\\ufb01cant proportion of representations that\\nhave a high degree of differentiability toward the Chinese watermarks. Furthermore, we can observe\\nthat several networks, including DenseNet-161, ResNet-18, and GoogLeNet exhibit at least several\\nrepresentations with very high watermark differentiability scores ( d >0:95), which is in line with\\nthe reported high number of Chinese-sensitive class representations across output logit layer.\\n4 I GNORING SENSITIVE EMBEDDINGS DURING FINE -TUNING\\nPre-trained ImageNet models are frequently utilized as feature extractors, where the pre-trained\\nweights are kept \\ufb01xed and only the \\ufb01nal layer of the network is trained on a new task-speci\\ufb01c\\ndataset. To disable the undesired, but inherent correlations of the classes in \\ufb01ne-tuned networks,\\n3 Accepted to the ICLR 2023 TrustML-(un)Limited workshop\\nFigure 3: Left: Number of output class representations that exhibit a high degree of differentiability\\ntowards Chinese watermarks across various ImageNet models. Right : Percentage of representations\\nin the feature-extractor layers of various networks that demonstrate a high degree of differentiability\\ntowards Chinese watermarks.\\nwe propose the method that simply ignores the most sensitive representations from the feature-\\nextractor model. To demonstrate this, we conduct an experiment, where we employed a pre-trained\\nDenseNet-161 model as a \\ufb01xed feature-extractor and \\ufb01ne-tuned the last linear layer on the CalTech-\\n256 image classi\\ufb01cation dataset Grif\\ufb01n et al. (2007) while varying the amount of the most sensitive\\nrepresentations omitted from the embeddings. Speci\\ufb01cally, we ranked the representations from the\\nDenseNet-161 feature-extractor layer based on the differentiability towards Chinese watermarks and\\nretrained the last linear layer while ignoring a varying amount of the most sensitive representations.\\nTo determine the effect of this procedure, we evaluated both the accuracy of each \\ufb01ne-tuned model,\\nas well as the distribution of AUC ROC and differentiability scores across 256 output representa-\\ntions. The results of the experiment, displayed in Figure 4, demonstrate that by excluding 0.5%\\nof the most sensitive representations from the DenseNet-161 feature extractor, the dependence of\\nthe newly learned logit representations on Chinese watermarks can be signi\\ufb01cantly reduced. Fur-\\nthermore, omitting up to 10% of the most sensitive embeddings has no signi\\ufb01cant impact on the\\nperformance of the \\ufb01ne-tuned model while signi\\ufb01cantly suppressing the Clever-Hans effect of the\\nnew model. Additionally, it can be observed that excluding the most sensitive representations from\\nthe feature-extractor layer narrows the distribution of AUC ROC scores, making the output classes\\nless likely to be highly differentiable towards spurious concepts.\\nFigure 4: Left: The accuracy of the \\ufb01ne-tuned model and the maximum differential ability towards\\nChinese symbols across output representations, with respect to the number of representations ig-\\nnored in the DenseNet-161 feature-extractor layer. Right : The distribution of AUC ROC scores\\nacross output representations, with respect to the number of representations omitted from the fea-\\nture extractor.\\n5 D ISCUSSION AND CONCLUSION\\nWith this paper, we aim to bring awareness to the potential risks of watermarked images present in\\nImageNet and their impact on popular DNNs trained on this dataset. It is known that the \\u201ccarton\\u201d\\n4 Accepted to the ICLR 2023 TrustML-(un)Limited workshop\\nclass is impacted by the Chinese watermarks - however, we were able for the \\ufb01rst time to demonstrate\\nand identify the signi\\ufb01cant amount of other ImageNet classes, which are affected by the Chinese\\nwatermarks across popular ImageNet pre-trained models. Our results indicate that the sensitivity to\\nwatermarks is a common trait among all studied networks and this poses signi\\ufb01cant risks for transfer\\nlearning, as new models could be also vulnerable to unintended concepts. We demonstrate that by\\nsimply omitting the most watermark-sensitive representations, \\ufb01ne-tuned networks can suppress the\\nreliance on the watermarks without incurring a signi\\ufb01cant decline in model performance. Overall,\\nthis study highlights the importance of paying attention to the presence of watermarks in image\\ndatasets and their impact on the performance of machine learning models.\\nACKNOWLEDGEMENTS\\nThis work was partly funded by the German Ministry for Education and Research through the project\\nExplaining 4.0 (ref. 01IS200551), the German Research Foundation (ref. DFG KI-FOR 5363), the\\nInvestitionsbank Berlin through BerDiBa (grant no. 10174498), and the European Union\\u2019s Horizon\\n2020 programme through iToBoS (grant no. 965221). KRM was partly funded by the German Min-\\nistry for Education and Research (under refs 01IS14013A-E, 01GQ1115, 01GQ0850, 01IS18056A,\\n01IS18025A and 01IS18037A) and BBDC\\/BZML and BIFOLD and by the Institute of Information\\n& Communications Technology Planning & Evaluation (IITP) grants funded by the Korea Govern-\\nment (MSIT) (No. 2019-0-00079, Arti\\ufb01cial Intelligence Graduate School Program, Korea Univer-\\nsity and No. 2022-0-00984, Development of Arti\\ufb01cial Intelligence Technology for Personalized\\nPlug-and-Play Explanation and Veri\\ufb01cation of Explanation).\\nREFERENCES\\nChristopher J Anders, Leander Weber, David Neumann, Wojciech Samek, Klaus-Robert M \\u00a8uller, and\\nSebastian Lapuschkin. Finding and removing clever hans: Using explanation methods to debug\\nand improve deep models. Information Fusion , 77:261\\u2013295, 2022.\\nHangbo Bao, Li Dong, and Furu Wei. BEIT: BERT pre-training of image transformers. arXiv\\npreprint arXiv:2106.08254 , 2021.\\nKirill Bykov, Mayukh Deb, Dennis Grinwald, Klaus-Robert M \\u00a8uller, and Marina M-C H \\u00a8ohne. Dora:\\nExploring outlier representations in deep neural networks. arXiv preprint arXiv:2206.04530 ,\\n2022.\\nJun Da. A corpus-based study of character and bigram frequencies in chinese e-texts and its impli-\\ncations for chinese language instruction. In Proceedings of the fourth International Conference\\non new technologies in teaching and learning Chinese , pp. 501\\u2013511. Citeseer, 2004.\\nJia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hier-\\narchical image database. In 2009 IEEE Conference on Computer Vision and Pattern Recognition ,\\npp. 248\\u2013255. IEEE, 2009.\\nAlexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas\\nUnterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An\\nimage is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint\\narXiv:2010.11929 , 2020.\\nRobert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A Wichmann, and\\nWieland Brendel. ImageNet-trained CNNs are biased towards texture; increasing shape bias\\nimproves accuracy and robustness. arXiv preprint arXiv:1811.12231 , 2018.\\nRobert Geirhos, J \\u00a8orn-Henrik Jacobsen, Claudio Michaelis, Richard Zemel, Wieland Brendel,\\nMatthias Bethge, and Felix A Wichmann. Shortcut learning in deep neural networks. Nature\\nMachine Intelligence , 2(11):665\\u2013673, 2020.\\nGregory Grif\\ufb01n, Alex Holub, and Pietro Perona. Caltech-256 object category dataset. 2007.\\n5 Accepted to the ICLR 2023 TrustML-(un)Limited workshop\\nKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog-\\nnition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition , pp.\\n770\\u2013778, 2016.\\nGao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely Connected\\nConvolutional Networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern\\nRecognition , pp. 4700\\u20134708, 2017.\\nPavel Izmailov, Polina Kirichenko, Nate Gruver, and Andrew Gordon Wilson. On feature learning\\nin the presence of spurious correlations. arXiv preprint arXiv:2210.11369 , 2022.\\nSimon Kornblith, Jonathon Shlens, and Quoc V Le. Do better imagenet models transfer better? In\\nProceedings of the IEEE\\/CVF conference on computer vision and pattern recognition , pp. 2661\\u2013\\n2671, 2019.\\nAlex Krizhevsky. One weird trick for parallelizing convolutional neural networks. arXiv preprint\\narXiv:1404.5997 , 2014.\\nSebastian Lapuschkin, Stephan W \\u00a8aldchen, Alexander Binder, Gr \\u00b4egoire Montavon, Wojciech Samek,\\nand Klaus-Robert M \\u00a8uller. Unmasking clever hans predictors and assessing what machines really\\nlearn. Nature Communications , 10, 03 2019. doi: 10.1038\\/s41467-019-08987-4.\\nZhiheng Li, Ivan Evtimov, Albert Gordo, Caner Hazirbas, Tal Hassner, Cristian Canton Ferrer,\\nChenliang Xu, and Mark Ibrahim. A whac-a-mole dilemma: Shortcuts come in multiples where\\nmitigating one ampli\\ufb01es others, 2022.\\nNingning Ma, Xiangyu Zhang, Hai-Tao Zheng, and Jian Sun. Shuf\\ufb02enet v2: Practical guidelines\\nfor ef\\ufb01cient cnn architecture design. In Proceedings of the European Conference on Computer\\nVision (ECCV) , pp. 116\\u2013131, 2018.\\nAmir Rosenfeld, Richard Zemel, and John K Tsotsos. The elephant in the room. arXiv preprint\\narXiv:1808.03305 , 2018.\\nEdalina Rodrigues Sanches. The community of portuguese language speaking countries: The role\\nof language in a globalizing world. In Workshop, University of Pretoria (South Africa) , 2014.\\nMark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mo-\\nbilenetv2: Inverted residuals and linear bottlenecks. In Proceedings of the IEEE Conference on\\nComputer Vision and Pattern Recognition , pp. 4510\\u20134520, 2018.\\nKaren Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image\\nrecognition. arXiv preprint arXiv:1409.1556 , 2014.\\nChristian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Du-\\nmitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In\\nProceedings of the IEEE Conference on Computer Vision and Pattern Recognition , pp. 1\\u20139, 2015.\\nChristian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethink-\\ning the inception architecture for computer vision. In Proceedings of the IEEE Conference on\\nComputer Vision and Pattern Recognition , pp. 2818\\u20132826, 2016.\\nJonti Talukdar, Sanchit Gupta, PS Rajpura, and Ravi S Hegde. Transfer learning for object detec-\\ntion using state-of-the-art deep neural networks. In 2018 5th international conference on signal\\nprocessing and integrated networks (SPIN) , pp. 78\\u201383. IEEE, 2018.\\nStefan Trost. Wordcreator, 2023. URL https:\\/\\/www.sttmedia.com\\/\\ncharacterfrequency-latin .\\nAnnegreet Van Opbroek, Hakim C Achterberg, Meike W Vernooij, and Marleen De Bruijne. Trans-\\nfer learning for image segmentation by combining image weighting and kernel learning. IEEE\\ntransactions on medical imaging , 38(1):213\\u2013224, 2018.\\nKarl Weiss, Taghi M Khoshgoftaar, and DingDing Wang. A survey of transfer learning. Journal of\\nBig data , 3(1):1\\u201340, 2016.\\n6 Accepted to the ICLR 2023 TrustML-(un)Limited workshop\\nKai Xiao, Logan Engstrom, Andrew Ilyas, and Aleksander Madry. Noise or signal: The role of\\nimage backgrounds in object recognition. arXiv preprint arXiv:2006.09994 , 2020.\\nSaining Xie, Ross Girshick, Piotr Doll \\u00b4ar, Zhuowen Tu, and Kaiming He. Aggregated residual trans-\\nformations for deep neural networks. In Proceedings of the IEEE conference on computer vision\\nand pattern recognition , pp. 1492\\u20131500, 2017.\\nZhuoning Yuan, Yan Yan, Milan Sonka, and Tianbao Yang. Large-scale robust deep auc maximiza-\\ntion: A new surrogate loss and empirical studies on medical image classi\\ufb01cation. In Proceedings\\nof the IEEE\\/CVF International Conference on Computer Vision , pp. 3040\\u20133049, 2021.\\nSergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint\\narXiv:1605.07146 , 2016.\\nJohn R Zech, Marcus A Badgeley, Manway Liu, Anthony B Costa, Joseph J Titano, and Eric Karl\\nOermann. Variable generalization performance of a deep learning model to detect pneumonia in\\nchest radiographs: a cross-sectional study. PLoS medicine , 15(11):e1002683, 2018.\\n7 Accepted to the ICLR 2023 TrustML-(un)Limited workshop\\nA A PPENDIX\\nA.1 D ATASET GENERATION\\nIn generating the dataset, our approach is similar to that outlined in Bykov et al. (2022). We imple-\\nment 4 distinct scenarios, namely Chinese characters, Latin characters, Hindi characters, and Nu-\\nmeric watermarks. For each image in the baseline dataset, we insert a random string of 7 symbols,\\nselected from the set of the 20 most frequently occurring characters in each language Da (2004);\\nTrost (2023) (for Arabic numerals we sample digits out of 10 available numbers). The watermark is\\nplaced randomly within the image, subject to the requirement of full visibility. The font size for all\\nwatermarks has been set to 30, while the image dimensions remain standard at 224 \\u0002224 pixels.\\nFigure 5: Multiple images with watermarks observed in the ImageNet training dataset.\\nA.2 R ESULTS\\nFigure 6 depicts the top-5 ImageNet classes ranked by the lowest average AUC ROC. It can be\\nseen that, similarly to the classes with the highest AUC ROC, the ImageNet classes demonstrate a\\nsigni\\ufb01cantly better ability to differentiate between watermarked and normal images in the case of\\nChinese watermarks, compared to other scenarios.\\nFigure 6: Top-5 ImageNet ranked by the lowest average AUC ROC across 20 analyzed models for\\n4 different scenarios.\\n8 Accepted to the ICLR 2023 TrustML-(un)Limited workshop\\nFigure 7 displays the top-30 ImageNet classes with the lowest (left) and highest (right) AUC ROC\\nscores for the task of detection of Chinese characters.\\nFigure 7: Left: The top-30 ImageNet classes ranked by the lowest average AUC ROC for the de-\\ntection of Chinese symbols. Right :the top-30 ImageNet classes ranked by the highest average AUC\\nROC for the detection of Chinese symbols.\\nA.3 I GNORING SENSITIVE EMBEDDINGS DURING FINE -TUNING\\nFor this experiment, we utilized DenseNet-161 Huang et al. (2017), a well-known pre-trained model\\non ImageNet, to extract features from the images. The features were then subjected to an average\\npooling layer to yield a 2204-value embedding for each image. The embeddings were ranked based\\n9 Accepted to the ICLR 2023 TrustML-(un)Limited workshop\\non their differentiability, i.e., their ability to distinguish between normal images and those with\\nChinese symbols.\\nTo classify images on the CalTech-256 Grif\\ufb01n et al. (2007) dataset with 256 classes, we added a\\nlinear layer to the extracted features and trained the network with 10 different scenarios. In each\\nscenario, we excluded a fraction \\u000bof the most differentiable embeddings from training, where \\u000b\\nwas varied across the values of 0(baseline);0:005;0:01;0:02;0:03;0:05;0:1;0:15;0:25;0:5, and\\ntrained with the same set of hyperparameters for all scenarios.\\n10\",\"39\":\" Multiplexed gradient descent: Fast online training of\\nmodern datasets on hardware neural networks without\\nbackpropagation\\nA. N. McCaughan1, B. G. Oripov2, N. Ganesh1, S. W. Nam1,\\nA. Dienstfrey1, S. M. Buckley1\\n1National Institute of Standards and Technology, Boulder, CO 80305\\n2University Colorado Boulder, Boulder, CO 80309\\nAbstract\\nWe present multiplexed gradient descent (MGD), a gradient descent framework\\ndesigned to easily train analog or digital neural networks in hardware. MGD utilizes\\nzero-order optimization techniques for online training of hardware neural networks. We\\ndemonstrate its ability to train neural networks on modern machine learning datasets,\\nincluding CIFAR-10 and Fashion-MNIST, and compare its performance to backprop-\\nagation. Assuming realistic timescales and hardware parameters, our results indicate\\nthat these optimization techniques can train a network on emerging hardware platforms\\norders of magnitude faster than the wall-clock time of training via backpropagation on\\na standard GPU, even in the presence of imperfect weight updates or device-to-device\\nvariations in the hardware. We additionally describe how it can be applied to existing\\nhardware as part of chip-in-the-loop training, or integrated directly at the hardware\\nlevel. Crucially, the MGD framework is highly \\rexible, and its gradient descent pro-\\ncess can be optimized to compensate for speci\\fc hardware limitations such as slow\\nparameter-update speeds or limited input bandwidth.\\n1arXiv:2303.03986v1  [cs.LG]  5 Mar 2023 1 Introduction\\nMachine learning has proven an invaluable tool for a variety of applications [1]. However,\\nmachine learning on traditional digital hardware is ine\\u000ecient, leading to a signi\\fcant e\\u000bort\\ntowards building custom hardware that can perform machine learning tasks at high speeds\\nwith lower energy costs [2]. A number of hardware platforms have emerged using analog [3],\\ndigital [4, 5], or mixed-signal processing [6] that will potentially o\\u000ber increased operational\\nspeeds and\\/or reduced energy costs [7]. However, many of the most promising hardware\\ninstantiations only perform the inference part of the machine learning algorithm. Meanwhile\\nthe larger portion of the energy cost is spent training on datasets [8], usually via gradient\\ndescent. Backpropagation is by far the most commonly used method of computing the gra-\\ndient for gradient descent, but has proved to be challenging to implement in novel hardware\\nplatforms [9].\\nThough often con\\rated, training via gradient descent does not require backpropagation\\n{ backpropagation is only used to calculate the gradient. Other methods for computing the\\ngradient in neural networks exist, but are much less e\\u000ecient in software than backpropagation\\nand so are rarely used in today's machine learning applications. This is not generally true in\\nhardware, where backpropagation may not only be challenging to implement, but also may\\nnot be the most e\\u000ecient way to compute the gradient.\\nOf particular interest in hardware are model-free methods, in which we require no knowl-\\nedge of the internal structure of the network (e.g topology, activation function, derivatives,\\netc), only the ability to perturb the network's parameters and measure the network's re-\\nsponse. The simplest example of such a method is \\fnite-di\\u000berence [10], which has been\\nemployed for chip-in-the-loop training [11]. However, \\fnite-di\\u000berence has several other dis-\\nadvantages that prevent its widespread implementation in hardware, including the require-\\nments for extra memory at every synapse and global synchronization. Fortunately, there\\nare a variety of other model-free methods that overcome some of the issues associated with\\n\\fnite-di\\u000berence [12, 13].\\n2 In this paper, we show that model-free perturbative methods can be used to e\\u000eciently\\ntrain modern neural network architectures in a way that can be implemented natively within\\nemerging hardware. These methods were investigated for training VLSI neural networks\\nbeginning in the 1990s [14, 15, 16, 17, 18, 19, 20, 21, 22, 23], and more recently on memristive\\ncrossbars [24] and photonic hardware [25], but all these demonstrations have been very\\nlimited in scale, comprising small datasets with only a few neurons. Below we describe a\\nframework for applying these techniques to existing hardware at much larger scales, with\\nan emphasis on creating simple, highly-localized circuits that could be implemented on-chip\\nif desired. The framework is also extensible to training existing hardware systems via a\\nchip-in-the-loop technique. We note that these methods have also been adapted in forward\\ngradient approaches using auto-di\\u000berentiation, which have attracted recent interest in the\\nmachine learning literature [26, 27, 28].\\nWe show that under realistic assumptions of the operating timescales of analog and digital\\nhardware neural networks, one can train hardware to solve modern datasets such as CIFAR-\\n10 faster than training a software network on a GPU, even in the presence of signal noise\\nand device-to-device variations in the hardware. A major advantage of this framework is\\nthat it can be used to perform online training of hardware platforms originally designed only\\nfor inference while making minimal hardware modi\\fcations.\\n2 Multiplexed gradient descent\\n2.1 Computing the gradient with perturbations\\nWe begin with the basic assumption that we have some hardware with programmable pa-\\nrameters (e.g. weights and biases) that can perform inference. Our goal is to augment the\\nhardware minimally such that it can also be trained via gradient descent. We will show\\nhow to con\\fgure the hardware such that the network as a whole automatically performs\\ngradient descent, without backpropagation. As an example, assume we have a hardware\\n3 instantiation of a feedforward multi-layer neural network as shown in Fig. 1. The hardware\\ntakes time-varying inputs x(t), training target ^ y(t), has variable parameters \\u0012, outputs the\\ninferencey(t), and computes a cost C(y(t);^y(t)). To allow us to compute the gradient of\\nsuch a system, we \\frst add a small time-varying perturbation ~\\u0012i(t) to each parameter base\\nvalue\\u0012i(Fig. 1a, inset). This perturbation will slightly modulate the cost C, and that mod-\\nulation will be fed back to the parameters. This process will ultimately allow us to extract\\nthe gradient of the system.\\nC\\nneuron\\nsynapse\\n...\\n\\u0283\\n update \\n(a)\\n(c)(b)\\n...... x1\\nx2\\nx3sinusoidalt\\nsequential\\ncode\\nFigure 1: (a) Schematic diagram showing the operation of the MGD framework in a feed-\\nforward neural network using example sinusoidal perturbations. (a, inset) Each parameter\\niis modulated slightly from its base value \\u0012iby the perturbation ~\\u0012i. The result of these\\nperturbations causes a modulation in the cost ~C, which is globally broadcast back to all\\nthe parameters. (b) A homodyne detection process is used to compute the partial gradient\\napproximations Gifrom the integrated product of \\u0012iand ~C. This partial gradient is then\\nused to update \\u0012iin the approximate direction of the gradient. (c) Example perturbation\\ntypes that can be used with this process.\\nAlthough the perturbations can take a variety of di\\u000berent forms, we will \\frst describe\\nthis process by using sinusoidal perturbations as they are conceptually straightforward to\\nunderstand. In this scenario, each parameter \\u0012iis slightly modulated at a unique frequency\\n!iand amplitude \\u0001 \\u0012, giving the perturbation ~\\u0012i(t) = \\u0001\\u0012sin(!it). As each parameter is\\n4 modulated, it slightly changes y(t) which in turn changes the cost. Thus, if the parameters\\nare modulated by frequencies !1,!2,!3, etc, those same frequencies will necessarily appear\\nas small modulations in the cost ~C(t) on top of the baseline (unperturbed) cost value C0,\\nsuch that\\nC(t) =C0+~C(t) =C0+X\\ni\\u0001Cisin(!it) (1)\\nIf we remove C0, we are left with a time varying signal ~C(t) =P\\ni\\u0001Cisin(!it) corre-\\nsponding only to the e\\u000bects of our parameter perturbations. The amplitude \\u0001 Ciis simply\\nthe amplitude of change in the cost due to ~\\u0012i(t), the perturbation of parameter i.\\nSince the gradient with respect to the cost dC=d\\u0012 is composed solely from the partial\\ngradientsdC=d\\u0012 = (@C=@\\u0012 1; @C=@\\u0012 2; :::), if we can extract \\u0001 Cifor each parameter we\\ncan produce an estimate of the complete gradient G= (\\u0001C1=\\u0001\\u00121;\\u0001C2=\\u0001\\u00122;:::). Now the\\ntask becomes to extract individual \\u0001 Ciout of the summed signal ~C(t). Fortunately, to\\nextract a given \\u0001 Ci, all we need to do is integrate the product of the input perturbation\\n~\\u0012i(t) with ~C(t). The integration takes the form of a homodyne detection, where unwanted\\nperturbations (frequencies) from other parameters are eliminated via integration:\\nGi=1\\n\\u0001\\u00122\\ni1\\nTZT\\nt=0X\\nk\\u0001Cksin(!kt)\\u0001\\u0012isin(!it)dt\\n=\\u0001Ci\\n\\u0001\\u0012iasT!1(2)\\nwhere 1=\\u0001\\u00122\\niis a normalization constant.\\nThe valueGiis the approximation for the partial gradient for parameter i.Gapproaches\\nthe exact gradient when both T!1 and the amplitude of the perturbation \\u0001 \\u0012iapproaches\\nzero, and is only an approximation otherwise. Fortunately, even at realistic timescales and\\namplitudes, Gcontains meaningful information and can be used to perform gradient de-\\nscent [12].\\n5 For illustrative purposes we have described the algorithm using sinusoidal parameter\\nperturbations. However, any collection of orthogonal, mean zero perturbations can be used\\n[13], including a variety of analog and discrete perturbations as shown in Fig. 1c. In general,\\nwe will be integrating the product ei(t) = ~C(t)~\\u0012i(t)=\\u0001\\u00122\\ni, which we refer to as the error\\nsignal, and Giwill be given by1\\nGi=ZT\\nt=0~C(t)~\\u0012i(t)\\n\\u0001\\u00122\\nidt (3)\\nWe discuss the e\\u000bects of changing the perturbation type in Section 3.4. We also note that\\nalthough many of the quantities described here are time-varying, in the following sections\\nwe will drop the explicit time dependence notation for the sake of brevity.\\n2.2 Gradient descent in the MGD framework\\nHere we describe the practical implementation of a model-free gradient descent framework\\nin hardware, which we term multiplexed gradient descent (MGD). To better understand\\nthe algorithm from a hardware perspective, we will run through the same computation\\npreviously described, but from the viewpoint of a single parameter (e.g. a synapse weight in\\na hardware neural network). The process begins with the application of a local perturbation\\n~\\u0012ithat slightly modi\\fes the base value of the parameter \\u0012i(Fig. 1a, inset). As previously\\ndescribed, this perturbation { and any other perturbations from other parameters { induce\\na change in the cost ~Con top of the baseline cost C0such that the cost at the output is\\nC=C0+~C.~Cmay be extracted from Ceither by direct subtraction of C0or, in some\\nanalog scenarios, by a simple highpass \\flter. The resulting ~Csignal is broadcast globally\\nto all parameters, so our parameter \\u0012ihas access to it. (Note that although Fig. 1 shows a\\nwireless broadcast tower for purposes of clarity, in most hardware platforms this will be a\\n1Note that here and in the simulation results, Giis being accumulated with time and is not normalized by\\n1=T, unlike Eq. 2. As described later, this allows us to vary the integration time without greatly impacting\\nthe rate of training{equivalently, one can integrate for a long time resulting in a large step down the gradient,\\nor one can take a series of shorter steps instead and travel approximately the same distance along the gradient.\\n6 wired connection). However, we must assume that parameters other than the ith are also\\ncausing modulations in the cost as well. To our parameter \\u0012i, these other modulations are\\nunwanted and must be \\fltered out. As shown in Fig. 1b, for the parameter ito extract only\\nits own e\\u000bect on the cost, it can just integrate the product of its local perturbation ~\\u0012iand\\nthe global cost signal ~Cit receives. This has the e\\u000bect of isolating the contribution from \\u0012i\\ndue to the pairwise orthogonality of the perturbation signals. From Eq. 3, this integration\\nproduces the partial gradient approximation Gi\\/\\u0001Ci=\\u0001\\u0012i. The parameter can then use\\ntheGivalue to directly to reduce the cost by updating itself according to a gradient descent\\nstep\\n\\u0012i!\\u0012i\\u0000\\u0011Gi (4)\\nwhere\\u0011is the learning rate. When all the parameters of a system perform this operation\\nin parallel, the resulting weight update corresponds to gradient descent training of the entire\\nnetwork. Because all the parameters are being perturbed and updated simultaneously, we\\ncall the framework multiplexed.\\nLooking at this process from the hardware perspective, one must also examine several\\npractical considerations such as when to perform the weight update, how long to integrate\\nthe gradient, and so forth. We introduce the following time constants to provide a framework\\nfor managing these considerations in the MGD context.\\n\\u2022\\u001cpis the timescale over which perturbations occur. In a digital system, the pertur-\\nbations of each parameter would be updated to new values every \\u001cp. In an analog\\n(continuous) system, \\u001cpcorresponds to the characteristic timescale of the perturba-\\ntions. For instance, if using sinusoidal perturbations, it corresponds approximately to\\nthe total bandwidth the frequencies occupy.\\n\\u2022\\u001c\\u0012is the gradient-integration time (i.e. Tin Eq. 3). It sets how often parameter-\\nupdates occur and also determines how accurate the gradient approximation will be.\\n7 For each time period \\u001c\\u0012, the gradient approximation Gis integrated, and at the end of\\nthat period the parameters are updated according to Eq. 4.\\n\\u2022\\u001cxcontrols how often new training samples x, ^yare shown to the hardware. After\\neach\\u001cxperiod, the old sample is discarded and a new one is applied, generating new\\noutputsyand costC.\\n0 4 8 0 4 8 0 4 8 0 4 8Finite difference Coordinate descent SPSA Analog(a) (b) (c) (d)\\n0\\n0\\n0\\n0\\n0\\n0 0\\n0 0\\n0\\n0\\n0\\n0\\n0\\n0\\n0\\n0\\n0\\n0\\n00\\n0\\n0\\nTime Time Time Time320 2020 20\\n33 3\\nFigure 2: Di\\u000berent optimization algorithms can be implemented in the MGD framework\\nby changing the values the time constants, \\u001cp,\\u001c\\u0012and\\u001cx, and varying the perturbation\\ntype. Shown here are the parameter values \\u0012that are updated every \\u001c\\u0012, perturbation ~\\u0012\\nwith characteristic time constant \\u001cp, computed cost C, and cost-modulation ~C. Varying\\u001c\\u0012\\nand\\u001cxand perturbation type allows MGD to implement a variety of optimization algorithms\\nincluding (a) \\fnite-di\\u000berence (b) coordinate descent (c) simultaneous perturbation stochastic\\napproximation (SPSA) and (d) an analog implementation. Each line color in the upper two\\nplots corresponds to one of the three parameters in this simulation.\\nThe values of the time constants \\u001c\\u0012and\\u001cphave a large impact on the particulars of the\\ntraining, and particular choices of \\u001c\\u0012and\\u001cpallow the implementation of certain conventional\\nalgorithms in numerical analysis. For instance, consider the implementation of the forward\\n\\fnite-di\\u000berence algorithm within this framework. To do this we start with discrete pertur-\\nbations, such that every \\u001cponly a single parameter is perturbed by \\u0001 \\u0012, and the parameters\\nare perturbed sequentially as shown in Fig. 1c. When parameter iis perturbed by \\u0001 \\u0012i, the\\ncost changes by \\u0001 Cand the resulting partial gradient \\u0001 C=\\u0001\\u0012i\\u0019@C=@\\u0012iis stored in Gi.\\n8 Now if we set \\u001c\\u0012=P\\u001cp, wherePis the number of parameters in the network, this is ex-\\nactly equivalent to computing a \\fnite-di\\u000berence approximation of the gradient: each \\u001cp, one\\nelement of the gradient is approximated, and after P\\u001cp, every partial gradient Gihas been\\nmeasured and stored. Since \\u001c\\u0012=P\\u001cp, the weight update only comes after all the partials\\nofGhas been collected, just as in \\fnite-di\\u000berence. This process is shown schematically in\\nFig. 2a.\\nSimilarly, other optimization algorithms can be implemented simply by modifying the\\nvalues of the time constants. For example, using same procedure as above but with the\\nintegration time reduced to a single timestep, i.e. \\u001c\\u0012=\\u001cp. This corresponds exactly to\\ncoordinate descent. In this case, rather than storing each Giuntil all the partials of the\\ngradient are fully assembled, the weight update is applied immediately after each Giis\\ncomputed. This may be more appealing from a hardware perspective than a \\fnite-di\\u000berence\\napproach, as Gican be used for the weight update and immediately discarded { unlike \\fnite-\\ndi\\u000berence, it does not require a per-parameter memory to store Giuntil the weight update\\nis executed. This coordinate-descent process is shown schematically in Fig. 2b.\\nAs a third example, it is possible to implement simultaneous-perturbation stochastic\\napproximation (SPSA) [12] by only changing the values of the time constants and the form\\nof the perturbation. In this case, \\u001c\\u0012=\\u001cpand a random, discrete f+\\u0001\\u0012;\\u0000\\u0001\\u0012gperturbation\\nis applied to every parameter every \\u001cp, as shown in Fig. 2c. Similar to coordinate descent,\\nthis con\\fguration avoids the need for additional per-parameter memories, as Givalues do not\\nneed to be stored. This method avoids the need for global synchronization of the parameters\\n{ the perturbations do not need to be sequential, and instead can be generated locally and\\nrandomly at the parameter.\\nIn addition to being able to choose these speci\\fc optimization algorithms, by varying the\\ntime constants \\u001cpand\\u001c\\u0012one can also implement entirely new optimization algorithms. For\\ninstance, in Fig. 2d we show an MGD implementation on an analog system using sinusoidal\\nperturbations that does not correspond to any of the aforementioned methods. In this\\n9 case,\\u001cpcorresponds to the timescale 1 =\\u0001f, where \\u0001fis the perturbation bandwidth, the\\ndi\\u000berence between the maximum and minimum perturbation frequency. Additionally, in the\\nanalog case there is no discrete update of the parameters and instead \\u001c\\u0012is an integration\\ntime constant. Unlike the discrete case which accumulates Gfor\\u001c\\u0012time then resets it to\\nzero,\\u0012is continuously updated with the output of an lowpass \\flter with time constant \\u001c\\u0012\\n(see Algorithm 2).\\nBecause modern machine learning datasets are composed of many training examples\\n(often tens of thousands), \\u001cx, the time constant that controls how often training samples\\nare changed, is critical. In fact, by setting \\u001cxappropriately, mini-batching can even be\\nimplemented in hardware that only allows 1 sample input at a time. The batch size is\\ndetermined by \\u001c\\u0012=\\u001cx, the ratio of the gradient integration time \\u001c\\u0012and the sample update\\ntime\\u001cx. When\\u001cxis shorter than \\u001c\\u0012, multiple samples are shown to the network during a\\nsingle gradient integration period. As the sample changes, the gradient approximation G\\nwill then include gradient information from each of those samples. This integration-in-time\\nprocess is arithmetically identical to summing multiple examples in parallel (as is often done\\non a GPU). As a concrete example, for hardware which accepts one input example at a time,\\nif\\u001c\\u0012=\\u001cxthe batch size is 1, but if \\u001c\\u0012= 4\\u001cx, the batch size is 4. This is shown in Fig. 3 for\\nthe same network as in Fig. 2 and using simultaneous discrete perturbations. In Section 3\\nwe demonstrate that batch sizes as large as 1000 function correctly in MGD.\\nIn summary, setting just three time constants and the perturbation type allows the MGD\\nframework to implement a wide range of variations of gradient descent and, depending on\\nthe desired approach, one can tailor their training to the needs of the problem and the\\nhardware. For example, MGD can match backpropagation by making \\u001c\\u0012arbitrarily long2,\\nas we show in Section 3.2. At the same time, we show that short \\u001c\\u0012values can work equally\\nwell. In practice, hardware considerations such as write-speed limitations may require an\\n2The MGD approximation to a partial derivative converges to the true partial derivative as \\u001c\\u0012becomes\\narbitrarily long. Thus the various parameter update steps presented above can be made arbitrarily close to\\ntheir analytical de\\fnitions as implemented using digital arithmetic and backpropagation.\\n10 Time 0\\nx\\n40 80 120Figure 3: Illustration of batching in a small network with three parameters and two x\\ninputs, training on a dataset with four samples. The parameters \\u0012are updated every \\u001c\\u0012, and\\nduring that time all four training samples are shown to the network and integrated into the\\ngradient approximation (batch size \\u001c\\u0012=\\u001cx= 4). The gradient approximation Gaccumulates\\neach timestep, and is reset during the weight-update process after each \\u001c\\u0012period. Note that\\nthe updates to \\u0012occurs in the opposite direction of G, per Eq. 4.\\nintermediate \\u001c\\u0012value, as we discuss in Section 4. These features allow the MGD framework\\nto leverage the same training techniques that have been so successful for training machine\\nlearning models, while also being compatible with new and emergent hardware.\\n3 Simulation of MGD\\n3.1 Intro\\nTo characterize the utility of the MGD framework, we \\frst simulated its performance on\\nmodern machine learning datasets. The goal of the simulator was not to perform gradient\\ndescent as fast as possible on a CPU or GPU, but rather to emulate hardware implement-\\ning MGD and evaluate its potential performance in a hardware context. In particular, we\\nused the simulator to estimate the speed, accuracy, and resilience to noise and fabrication\\nimperfections. The simulator was written in the Julia language and can be run on a CPU\\n11 or GPU and is available online3. The algorithms used in the simulation are shown in the\\nAlgorithm 1 and Algorithm 2 boxes, with the parameters and their types in software shown\\nin Table 1.\\nAlgorithm 1 Discrete algorithm\\n1:Initialize parameters \\u0012\\n2:forninnum iterations do\\n3: if(nmod\\u001cx= 0)then\\n4: Input new training sample x, ^y\\n5: if(nmod\\u001cx= 0) or (nmod\\u001c\\u0012= 0)then\\n6: Set perturbations to zero ~\\u0012 0\\n7: Update baseline cost C0 C(f(x;\\u0012);^y)\\n8: if(nmod\\u001cp= 0)then\\n9: Update perturbations ~\\u0012\\n10: Compute output y f(x;\\u0012+~\\u0012)\\n11: Compute cost C C(y;^y)\\n12: Compute change in cost ~C C\\u0000C0\\n13: Compute instantaneous error signal e ~C~\\u0012=\\u0001\\u00122\\n14: Accumulate gradient approximation G G+e\\n15: if(nmod\\u001c\\u0012= 0)then\\n16: Update parameters \\u0012 \\u0012\\u0000\\u0011G\\n17: Reset gradient approximation G 0\\nAlgorithm 2 Analog algorithm\\n1:Initialize parameters \\u0012\\n2:fort= 0toTstepdtdo\\n3: if(tmod\\u001cx= 0)then\\n4: Input new training sample x, ^y\\n5: Update perturbations ~\\u0012\\n6: Compute output y f(x;\\u0012+~\\u0012)\\n7: Compute cost C(t) C(y;^y)\\n8: Compute discretized highpass ~C(t) \\u001chp\\n\\u001chp+dt\\u0010\\n~C(t\\u0000dt) +C(t)\\u0000C(t\\u0000dt)\\u0011\\n9: Compute instantaneous error signal e(t) ~C~\\u0012dt=\\u0001\\u00122\\n10: Update gradient approximation G(t) dt\\n\\u001c\\u0012+dt\\u0000\\ne(t) +\\u001c\\u0012\\ndtG(t\\u0000dt)\\u0001\\n11: Update parameters \\u0012 \\u0012\\u0000\\u0011G\\n3Code available at https:\\/\\/github.com\\/amccaugh\\/multiplexed-gradient-descent-paper\\n12 Description Symbol analog or digital\\nChange in the cost due to\\nperturbation~C both\\nPerturbation to parameters ~\\u0012 both\\nParameters \\u0012 both\\nInput sample x both\\nTarget output ^y both\\nNetwork output y both\\nCost C both\\nUnperturbed baseline cost C0 digital\\nGradient approximation G both\\nInstantaneous error signal e both\\nLearning rate \\u0011 both\\nPerturbation amplitude \\u0001\\u0012 both\\nInput-sample change time\\nconstant\\u001cx both\\nParameter update time\\nconstant\\u001c\\u0012 both\\nPerturbation time constant \\u001cp digital\\nHighpass \\flter time\\nconstant\\u001chp analog\\nTable 1: Algorithm parameters and variables used in the simulations\\n3.2 Equivalence to backpropagation\\nWe \\frst veri\\fed that the simulation was able to minimize the cost for a sample problem, and\\nthat it was equivalent to gradient descent via backpropagation with appropriate parameter\\nchoices. For this initial comparison, we chose to solve the 2-bit parity problem by training a\\n2-2-1 feedforward network with 9 parameters (6 weights, 3 biases). We simulated the problem\\nwith a very large value for \\u001c\\u0012and\\u001c\\u0012=\\u001cxsuch that we achieved a very good approximation\\nof the gradient in Gfor each training sample. We then ran the same problem using a \\u001c\\u0012\\nvalue of 1 so that the gradient approximation Gfor each sample was relatively poor. We\\nmeasured both the number of epochs and the amount of time (number of iterations of the\\nsimulation) for the two experiments, and the results are shown in Fig. 4. Here, an epoch was\\nde\\fned in the typical way { equivalent to the network being shown all training examples of\\na dataset.\\nComparing the two scenarios in terms of epochs in Fig. 4a, one can observe that a value\\n13 0.3\\n0\\n10\\n0\\n0.3\\n0\\n0\\n2\\n1\\n(a)\\n10\\n1\\n10\\n2\\n10\\n3\\n10\\n4\\n10\\n6\\n10\\n5\\nx10\\n6\\nbackprop\\n= 1\\n= 1000\\n \\nx\\n=\\n \\nx\\n=\\n= 1\\n= 1000\\n \\nx\\n=\\n \\nx\\n=\\n(b)\\nMean cost\\nMean cost\\nEpochs\\nTime (iterations)Figure 4: Solving the 2-bit parity (XOR) problem using a 2-2-1 network with 9 parameters,\\n\\u001cp= 1, and batchsize \\u001c\\u0012\\/\\u001cx= 1. (a) Mean cost over dataset versus epochs the on a 2-2-1\\nfeedforward network averaged over 1000 random initializations and trained with MGD with\\ndi\\u000berent gradient integration times \\u001c\\u0012(solid lines). The dashed line shows the same network\\nand dataset trained using backpropagation. (b) The same MGD training data as in part (a)\\nplotted versus time (number of perturbations \\u001cp) instead of epochs.\\nof\\u001c\\u0012=\\u001cx= 1000 resulted in the system following a nearly identical training trajectory as\\nbackpropagation. This is as expected { for each sample shown to the network, the gradient\\napproximation Ghas 1000 timesteps to integrate a very accurate estimate that should be\\nvery close to the true gradient computed by backpropagation. When \\u001c\\u0012=\\u001cx= 1, however,\\neach sample only has a single timestep to estimate the gradient before moving on to the\\nnext sample. As a result, the samples have to be shown to the network many more times to\\nminimize the cost, resulting in a much larger number of epochs.\\nHowever, while the \\u001c\\u0012=\\u001cx= 1 case uses the sample data less e\\u000eciently (requiring more\\nepochs), there is actually a tradeo\\u000b for data e\\u000eciency and run time. If we plot the cost\\nversus iterations rather than versus epoch, we get a more accurate estimate of how long it\\nwill take hardware to train in terms of real time. As shown in Fig. 4b, one can see that the\\nshorter\\u001c\\u0012and\\u001cxvalues take approximately half the time to minimize the cost as the longer\\nvalues. These examples serve to highlight that while longer integration times produce a more\\naccurate gradient approximation, integration times as short as \\u001cpmay also be used to train\\na network, consistent with the \\fndings of Ref. [12]. In fact, shorter integration times may\\n14 even be faster to train in terms of operational time.\\n100101102103104100\\n02040608090\\u00ba\\n2-bit parity\\n4-bit parity\\nAngle (deg)\\nTimeNIST7x7 dataset\\nFigure 5: Angle between the gradient approximation Gand the true gradient versus time.\\nThen-bit parity networks used n-n-1 networks, while the NIST7x7 networks used a 49-4-4\\nnetwork . The networks have 9, 25 and 220 parameters respectively, and the datasets have\\n4, 16 and 44136 training examples respectively. The solid line shows the median angle value\\nfor 100 random initializations for the n-bit parity and and 15 random initializations for the\\nNIST7x7 dataset. The shaded regions show the 1st to 3rd quartile values.\\nTo quantify the e\\u000bect of longer integration times on the accuracy of the gradient ap-\\nproximation, we measured how the gradient approximation Gconverged to the true gradient\\n@C=@\\u0012 (as computed by backpropagation) as a function of time. For this experiment, we ran\\nthe simulation with \\u001c\\u0012=1and\\u001cx=\\u001cp= 1, so that it continuously integrated Gwithout\\never resetting or updating the parameters. As the simulation ran, we repeatedly computed\\nthe angle between the true gradient @C=@\\u0012 and the approximation G. The results are shown\\nin Fig. 5, showing the solution to the 2-bit parity, 4-bit parity, and NIST7x7 problems. The\\nNIST7x7 dataset is a small image recognition problem based on identifying the letters N,\\nI, S, and T on a 7 \\u00027 pixel plane. The dataset has the property that it cannot be solved\\nto greater than 93% with a linear solve for a 49-4-4 feedforward network with sigmoidal\\nactivation functions. We also chose this network and dataset because it was small enough\\nto perform many di\\u000berent simulations to acquire statistics, and the problem space is large\\nenough that random solutions are unlikely.\\n15 As expected, the angle decreased with time as Galigned with the true gradient. The\\ntime axis is in units of \\u001cp, which is the minimum discrete timestep in this system. For a\\nreal hardware platform, this timestep is approximately the inference time of the system. In\\ngeneral, the more parameters the network has, the longer it takes to converge to the true\\ngradient.\\nBatch size              =  1\\nBatch size              =  4\\n0 200 400 600 800150550x103Training time\\n0 200 400 600Max \\u03b7100101\\nMinimum training time104105\\n(a) (b)\\nBatch size              =  1\\nBatch size              =  4\\nFigure 6: E\\u000bect of \\u001c\\u0012on the training time of the 2-bit parity (XOR) problem. (a) Training\\ntime as a function of \\u001c\\u0012and batch size. (b) E\\u000bect of \\u001c\\u0012on the maximum achievable learning\\nrate\\u0011and corresponding minimum training time.\\n3.3 Details of mini-batching\\nWe next investigated in more detail how \\u001c\\u0012and\\u001cxa\\u000bect the training time. Longer \\u001c\\u0012values\\nresult in a more accurate gradient approximation, but reduce the frequency of parameter\\nupdates. Using a \\fxed, low \\u0011value, we trained a 2-2-1 network to solve 2-bit parity (XOR)\\nfor 100 di\\u000berent random parameter initializations, varying \\u001c\\u0012but keeping the batch size \\u001c\\u0012=\\u001cx\\nconstant at either 4 or 1. Since the 2-bit parity dataset is composed of four ( x, ^y) pairs,\\n\\u001cx= 4\\u001c\\u0012is analogous to gradient descent { all four samples are integrated into the gradient\\napproximation Gbefore performing a weight update. When \\u001cx=\\u001c\\u0012, the network performs\\nstochastic gradient descent (SGD) with a batch size of 1. Fig. 6a shows the training time as\\na function of \\u001c\\u0012for these two cases. Here, training time corresponds to the time at which\\n16 the total cost Cdropped below 0.04, indicating the problem was solved successfully. In the\\ncase where the batch size was 1, we observed that increasing \\u001c\\u0012increased the training time.\\nHowever, when the batch size was 4, increasing \\u001c\\u0012had little e\\u000bect on the training time.\\nAs with any training process, the training can become unstable at higher \\u0011values and\\nfail to solve the task. Since the results in Fig. 6a were only for a \\fxed learning rate, we\\nalso wanted to examine the e\\u000bect of \\u001c\\u0012on the maximum achievable \\u0011. Here, we de\\fned\\nthe \\\\max\\u0011\\\" as the maximum learning rate where the network successfully solved the 2-bit\\nparity problem for at least 50 out of 100 random initializations. As seen in Fig. 6b, as \\u001c\\u0012\\nis increased the max \\u0011decreases, resulting in longer minimum training times. This was\\ntrue whether the batch size was large or small, although the larger batch sizes had higher\\nachievable\\u0011values overall.\\nFrom these results, we infer that a poor gradient approximation taken with respect to all\\ntraining examples is more useful than collecting an accurate gradient with respect to a single\\nexample. Stated another way, waiting a long time for an extremely accurate gradient then\\ntaking a large step is less productive than taking a series of short (but less accurate) steps.\\nThis is an important conclusion for hardware, as it shows that implementing an e\\u000bective\\ngradient descent process in MGD does not necessarily require additional memory to store\\naccurate, high-bit-depth gradient values. Note that in our implementation Gaccumulates\\nwith time and so the size of the parameter update \\u0011Gfrom Eq. 4 grows proportionally to\\nthe integration time. This meant that when \\u001c\\u0012was larger the e\\u000bective step in the direction\\nof the gradient was also larger, and so for \\fxed \\u0011the rate of training therefore remains\\napproximately constant. If this was not the case, whenever \\u001c\\u0012was doubled we would also\\nneed to reduce \\u0011by half to maintain the same approximate rate of training.\\n3.4 Analog and digital perturbations\\nThe parameter perturbations can take many di\\u000berent forms, as long as they are low-amplitude\\nand their time averages are pairwise orthogonal or, in a statistical setting, are uncorre-\\n17 lated [13]. We have implemented four types of perturbations: sinusoidal perturbations,\\nsequential discrete perturbations, discrete code perturbations, and random code perturba-\\ntions. Sinusoidal perturbations are like those shown in Fig. 1a, where each parameter is\\nassigned a unique frequency. Sequential discrete perturbations refer to the case where where\\nparameters are sequentially perturbed, one at a time, by +\\u0001 \\u0012, as described in the \\fnite-\\ndi\\u000berence implementation of Section 2.2. \\\\Code\\\" perturbations are simultaneous discrete\\nperturbations off\\u0000\\u0001\\u0012;+\\u0001\\u0012gfor every parameter every \\u001cptimesteps. We call them code\\nperturbations due to their similarity to code-multiplexing (spread-spectrum) techniques used\\nin wireless communication technologies. There are two \\ravors of code-perturbations: the \\frst\\nconsists of a prede\\fned set of pairwise-orthogonal square wave functions that take the values\\noff\\u0000\\u0001\\u0012;+\\u0001\\u0012g. Each of these perturbation patterns is a deterministic sequence, and no\\ntwo parameters have the same sequence. One example of these are the Walsh codes used in\\nmodern cell-phone communications. The second consists of randomly-generated sequences\\noff\\u0000\\u0001\\u0012;+\\u0001\\u0012gthat are pairwise uncorrelated. We call these \\\\statistically orthogonal.\\\"\\nThe statistically orthogonal case is slightly less e\\u000ecient than the deterministic orthogonal\\ncodes since perturbations from multiple parameters interfere with each other more in ~C{\\nany \\fnite sample of the perturbations will have a non-zero correlation that decreases to zero\\nwith sample size. However, the use of the statistically orthogonal version allows the per-\\nturbations to be generated locally and randomly. These perturbations may be very useful\\nin hardware implementations, as they are spread-spectrum and single-frequency noise from\\nexternal sources is unlikely to corrupt the feedback signals.\\nTo compare the training performance between di\\u000berent perturbation types, we applied\\nfour di\\u000berent perturbation types to the same 2-2-1 network described in the last section in\\norder to solve the 2-bit parity problem. In particular, we aimed to show that training can\\nhappen in both a purely analog or purely digital way. In practice, many systems have hybrid\\nanalog-digital features, and a combination approach may be used.\\nFig. 7 shows the training time distributions for the 2-bit parity problem using the dif-\\n18 x 105\\nTraining time\\nsinusoidal code sequential8\\n6\\n4\\n2Figure 7: Demonstrating equivalence between the various perturbation types by measuring\\ntheir time to train a 2-2-1 network on the 2-bit parity (XOR) problem. Each box plot\\nrepresents the results of 100 random initializations. The hyperparameters were \\u001cx= 250,\\n\\u001c\\u0012= 1,\\u0011= 0:05,\\u001cp= 1 (discrete), \\u0001 f= 0:3 (analog)\\n.\\nferent perturbation types. The bandwidth for sinusoidal perturbations was set to be 1 =2\\u001cp.\\nAs expected, we found that the di\\u000berent perturbation types are approximately equivalent\\nin terms of speed of training. This equivalence makes sense when one considers that the\\nfeedback from ~Chas a \\fnite bandwidth that must be shared between all the parameters {\\nno matter the encoding (perturbation) scheme, the information carried in that feedback to\\nthe parameters will be limited by that \\fnite bandwidth.\\n3.5 Operation on noisy or imperfect hardware\\nThe fabrication defects and signal noise present in emerging hardware platforms pose signi\\f-\\ncant challenges for current training techniques in hardware. For example, weight parameters\\nmay di\\u000ber from design or update non-deterministically, causing a discrepancy between the\\nmodel and actual hardware that can negatively a\\u000bect training [29, 30, 9]. For example,\\nin Ref. [30], a simulated network achieves 97.6% accuracy for the MNIST dataset, but the\\nhardware di\\u000bractive optical network implementation only obtains 63.9% accuracy after di-\\n19 rect transfer of the pre-trained model to their hardware without some in-situ training, while\\nRef. [31] shows that a 3% component variation can lead to a 7% drop in classi\\fcation accu-\\nracy without error correction. Ref. [9] shows how small discrepancies in modeled parameters\\ncan blow-up in general hierarchical problems. In their toy problem, a 0.5% error in a model\\nparameter leads to a 30% error in output after 20 layers. The solutions to these non-idealities\\ncan be cumbersome for o\\u000fine training. For example, the actual value of the weights may\\nhave to be regularly measured during training [29], imperfections may be carefully measured\\nand accounted for in the models [32, 31], or weights may be quantized with low bit-depth,\\nwhere the bit depth is chosen such that the system can be modeled and controlled accu-\\nrately despite device-to-device variations. This is common in hardware platforms such as\\nmemristive crossbar arrays [33] or phase change materials [34], where bit depths of 6-8 can\\nbe achieved.\\nHere we investigate the e\\u000bects of three di\\u000berent types of imperfections and noise that\\ncould a\\u000bect hardware systems: (1) stochastic noise on the output cost Cnoise, (2) stochastic\\nnoise on the parameter update \\u0012noise, (3) per-neuron defects in the activation function, where\\neach neuron has a randomly scaled and o\\u000bset sigmoidal activation function that is static in\\ntime. These tests were performed on the NIST7x7 dataset using the previously described\\n49-4-4 network with 220 parameters and with \\u001cx=\\u001c\\u0012= 1 unless otherwise stated.\\nIn the \\frst test, we added Gaussian noise with mean zero and standard-deviation \\u001bCto\\nthe cost, applied every timestep such that noise C(t) =Cideal(t) +Cnoise(t;\\u001bC). For example,\\nin optical hardware, this could be noise due to laser \\ructuations. Noise in the cost is also\\nequivalent to any other gaussian mean-zero noise a\\u000becting the accumulated gradient G.\\nFig. 8a shows the e\\u000bect on the training time for di\\u000berent learning rates as \\u001bCwas increased.\\nFor a given learning rate, there is a threshold amount of noise below which the training\\ntime is minimally changed. However, as cost noise increases, the training time eventually\\nincreases and ultimately stops converging.\\nTo see how this noise would a\\u000bect the minimum training time for optimized learning\\n20 0 0.08\\n\\u03b7 = 0.1\\n\\u03b7 = 1\\n107\\n107\\n106\\n105100\\n10-1\\n10-2101\\n0.20\\n106\\nTraining time\\nMax \\u03b7 \\nMinimum training time\\n(a) (b)Figure 8: E\\u000bect of noise applied to the cost signal. (a) Training time versus \\u001bC, the amplitude\\nof the Gaussian noise added to the cost signal C(normalized to the perturbation magnitude\\nj~\\u0012j). The training time is the median time to >80% accuracy for 10 random parameter\\ninitializations. (b) E\\u000bect of noise on maximum \\u0011and training time. The maximum \\u0011is\\nthe largest learning rate for which >80% of the 10 random initializations converged; the\\ncorresponding training time at that maximum learning rate is shown on the right axis.\\nrates, we also measured the maximum achievable \\u0011value for a range of \\u001bC. Fig. 8b shows\\nthis maximum \\u0011value versus cost noise, and corresponding minimum training time. The\\ntrend indicates that the lower the cost noise \\u001bC, the higher we can make \\u0011and the faster we\\ncan train. Stated in reverse, one can compensate for large amounts of cost noise in a system\\nsimply by reducing the learning rate.\\nIn the next test, we analyzed the e\\u000bect of noisy parameter updates on the training\\nprocess. For this experiment, whenever any parameter was updated, the update included a\\nrandomly-applied deviation. Thus, the update rule became\\n\\u0012 \\u0012\\u0000\\u0011G+\\u0012noise (5)\\nwhere\\u0012noisewas Gaussian with mean zero and standard deviation \\u001b\\u0012, normalized by \\u0001 \\u0012,\\nsuch that\\u0012noise\\u0018N(0;\\u001b\\u0012=\\u0001\\u0012). This kind of noise is often seen in analog memories without\\nclosed-loop feedback [35, 36].\\n21 1.0\\n00 14 2 4 6 8 10 12\\n38\\n4567x105\\n38\\n4567x105\\n0 2 4 6 0 2 4 6\\n= 1\\n= 1\\n= 100Converged fraction(a)\\n(d)Training time\\nTraining time\\nConverged fraction1.0\\n00 14 2 4 6 8 10 12\\n= 0\\n= 0.1\\n= 0.3\\n= 100(b)\\n(c)Figure 9: E\\u000bect of noisy (stochastic) parameter updates on solving XOR in a 2-2-1 feedfor-\\nward network, measured for various noise amplitudes \\u001b\\u0012. (a,b) E\\u000bect of parameter update\\nnoise on the probability of the training converging, as a function of learning rate \\u0011. Noise\\nwas much more likely to prevent convergence for (a) \\u001c\\u0012= 1 than (b) \\u001c\\u0012= 100. Convergence\\nwas de\\fned as reaching 93% accuracy within 5 \\u0002107iterations. Statistics are from 25 ran-\\ndom parameter initializations. (c,d) E\\u000bect of parameter update noise on training time as a\\nfunction of the learning rate \\u0011for integration times (c) \\u001c\\u0012= 1 and (d) \\u001c\\u0012= 100. Only \\u0011\\nvalues sets with >50% convergence are shown { in (c) \\u001b\\u0012= 0.3 did not have \\u0011values that\\nmet this criteria.\\nWe found that larger values of \\u001b\\u0012can prevent convergence entirely (Fig. 9a). Curiously, in\\nthe presence of this noise, increasing \\u0011can actually improve the convergence of the problem,\\nas highlighted by the \\u001b\\u0012= 0:1 and\\u001b\\u0012= 0:3 lines in Fig. 9a. We believe this is likely because\\nat very small \\u0011values\\u0012noisewill overwhelm the the very small \\u0011Gin Eq. 5. By making the\\n\\u0011term larger, one can prevent \\u0011Gfrom being drowned out by \\u0012noise. Obviously, at very\\nlarge\\u0011values, the usual gradient-descent instability starts to dominate and the convergence\\napproaches zero. For a given \\u0011, we found that small values of \\u001b\\u0012marginally increase the\\ntraining time, but the e\\u000bect is less signi\\fcant than simply changing the learning rate \\u0011\\n(Fig. 9c).\\n22 Another way to reduce the impact of \\u0012noiseis to increase the integration time of the\\ngradient. When \\u001c\\u0012is increased, the value of Gis accumulated for a longer time and becomes\\nproportionally larger. For instance, when \\u001c\\u0012is 100 times larger, the value of \\u0011Gbecomes\\n100 times larger, meaning the e\\u000bect of \\u0012noisein Eq. 5 is relatively 100 times smaller. The\\nresult can be seen in Fig. 9b,d, where even the largest \\u001b\\u0012value has little e\\u000bect on the result.\\n0 0.25\\u03b7 = 0.1\\n\\u03b7 = 1 106Training time\\n(a) (b)\\n0.2\\n1 Converged fraction\\n0 0.5\\n\\u03b7 = 0.1\\n\\u03b7 = 1\\nFigure 10: E\\u000bect of adding random o\\u000bsets and scaling to each neuron's sigmoid activation\\nfunction. (a) Training time versus \\u001ba, the standard deviation of activation-function o\\u000bsets\\nand scaling. (b) Converged fraction versus the standard deviation of the logistic function\\nparameters. The results are statistics of 25 di\\u000berent random network initializations and\\nactivation-function randomizations. Convergence is de\\fned as reaching 80% accuracy within\\nthe time of the simulation.\\nIn the \\fnal test, we analyzed the e\\u000bect of including \\\\defects\\\" in the neuronal activation\\nfunctions. Here, neuronal activation functions were no longer identical sigmoid functions,\\nbut had \\fxed random o\\u000bsets and scaling that were static in time. These variations were\\nmeant to emulate device-to-device variations that may be found in hardware, for instance in\\nanalog VLSI neurons [37]. The sigmoid activation function for each neuron kwas modi\\fed\\nto a general logistic function fk(a) =\\u000bk(1\\u0000e\\u0000\\fk(a\\u0000ak))\\u00001+bk. The variations were all\\nGaussian, and the scaling factors \\u000bk,\\fkhad a standard deviation \\u001baand a mean of 1, while\\nthe o\\u000bset factors ak,bkalso had a standard deviation of \\u001babut were mean-zero.\\nAs seen in Fig. 10a, adding defects to the network's activation functions had a relatively\\nsmall e\\u000bect on the training time. Even with relatively large variations in the activation\\n23 Setup Parameters Accuracy\\nTask Networkj\\u0012j\\u001c\\u0012\\u001cp\\u0011 batch size 104steps 105steps 106steps 107steps backprop\\n2-bit parity 2-2-1 9 1 15 1 100% 100% 100% 100% 100%\\nN-I-S-T 49-4-4 220 1 13 1 38% 81% 94% 97.7 % 99.8%\\nN-I-S-T 49-4-4 220 1 10.5 1 22% 45% 93% 98.7 % 99.8%\\nFashion-MNIST 2-layer CNN 14378 1 19 1000 34.2% 66.3% 79.3 % 83.5% 88.6%\\nFashion-MNIST 2-layer CNN 14378 10 19 1000 34.3% 66.3% 79.2 % 83.4% 88.6%\\nFashion-MNIST 2-layer CNN 14378 100 19 1000 35.3% 66.3% 77.7 % 84.7% 88.6%\\nFashion-MNIST 2-layer CNN 14378 1000 19 1000 35.3% 59.6% 79.1 % 86.1% 88.6%\\nCIFAR-10 3-layer CNN 26154 1 19 1000 12% 23% 43.8% 60.7% 68%\\nTable 2: Performance of MGD training on four di\\u000berent datasets. The setup, MGD param-\\neters and achieved test accuracies are shown. The best accuracy achieved with training via\\nbackpropagation for the same network is shown in the \\fnal column for comparison.\\nfunctions (\\u001ba= 0:25), the network only took about twice as much time to fully train the\\nNIST7x7 dataset. However, deviations that were too large ( \\u001ba>0:25) caused issues with\\nthe training being able to converge. This is likely due to the output neurons being so scaled\\nand o\\u000bset that it was no longer possible for them to produce the correct output.\\nOverall, we found that MGD is robust against di\\u000berent types of noise and fabrication\\nimperfections of a reasonable scale as those that would be seen in a real hardware system. In\\ngeneral, these non-idealities a\\u000bected the training speed of the network, but did not prevent\\nthe desired problem from being solved.\\n3.6 Modern dataset results\\nTo assess the scalability and relevance of MGD for larger machine learning problems, we com-\\npared MGD and backpropagation on a variety of tasks for di\\u000berent network architectures\\nand hyperparameters. Table 2 compares the accuracies obtained with MGD and backpropa-\\ngation for di\\u000berent datasets and various hyperparameter choices ( \\u001c\\u0012,\\u001cp,\\u0011, batch size), with\\n\\u001cx\\fxed at 1. To make an e\\u000bective comparison, for the backpropagation results we used a\\nbasic stochastic gradient descent (SGD) optimizer without momentum. Both strategies used\\na mean squared error (MSE) cost function. These choices allowed us to avoid confounding\\ne\\u000bects from more complex strategies, although we note that MGD is capable of implement-\\ning several of these more complicated feature (e.g., momentum, dropout, etc.). In all cases\\nwe used the same network architectures for both sets of results. The networks architectures\\n24 used were not selected to reach state-of-the-art accuracy values, but instead smaller networks\\nwere chosen for purposes of rapid testing and statistical analysis. To improve the accuracy\\nfurther, more advanced network architectures (e.g. more layers) or optimizers (e.g. adding\\nmomentum) would be needed. For the purposes of meaningful comparison, we measured the\\naccuracy obtained in the MGD process after \\fxed numbers of timesteps. This allowed us to\\nestimate actual training time for various hardware con\\fgurations in Section 4.1.\\nThe CIFAR-10 network was composed of 3 convolution and max-pool layers followed\\nby a fully-connected layer. The convolutional \\flters were 3 \\u00023 with 16, 32 and 64 output\\nchannels respectively, stride 1, and relu activation function outputs. Each convolution was\\nfollowed by a 2\\u00022 max-pool layer. The \\fnal fully-connected layer converted the 256 outputs\\nof the convolutional layers to the 10 classes, and no softmax was used. The Fashion-MNIST\\nnetwork was similarly composed, comprising two convolution and max-pool layers followed\\nby a (32\\u000210) fully-connected layer and no softmax. The networks used to solve NIST7x7\\nand XOR were fully connected feedforward networks with sigmoidal activation functions. To\\nmake the comparisons in Table 2 as fair as possible, the backpropagation accuracies of each\\nrow were maximized by training the networks with the listed batch size and a variety of \\u0011\\nvalues. The backpropagation accuracy reported is the highest median accuracy obtained for\\nall the\\u0011values tested, with the median taken over \\fve random initializations that each ran\\nfor 2500 epochs (long after the accuracy had converged).\\nBy training networks on the 2-bit parity, NIST7x7, Fashion-MNIST, and CIFAR-10\\ndatasets, we found that MGD approached the solution-accuracy of backpropagation. How-\\never, even after 107steps we observed the MGD-trained networks still had slightly lower\\naccuracy than could be achieved by backpropagation. This is likely related to convergence\\ncriteria of the gradient approximation not being met, due to \\fxed \\u0011values { for instance,\\nthe theory underlying the SPSA process only guarantees convergence with a learning rate\\nthat asymptotically approaches zero [12]. In general, we observed that while larger learning\\nrates achieved higher accuracies earlier on, lower learning rates achieved higher accuracies at\\n25 longer times (see e.g. NIST7x7 result). Therefore custom learning rates are likely to achieve\\nmore optimal training time and accuracy.\\nThe simulations also indicate that changing \\u001c\\u0012had a marginal e\\u000bect on the maximum\\naccuracy of MGD for a \\fxed \\u0011value for the larger datasets shown here, with increased values\\nof\\u001c\\u0012leading to slightly higher \\fnal accuracies. This may be because longer \\u001c\\u0012values produce\\nmore-accurate gradient estimates and allow the network to better optimize the weights as the\\nsystem approaches a local minima in the cost, although more experiments and statistics are\\nneeded to verify if this e\\u000bect is occurs in general. Fortunately, our results show this e\\u000bect is\\nrelatively small, allowing the hardware-designer \\rexibility in how often weight updates need\\nto be performed.\\nThese results show explicitly that MGD is a viable technique for training emerging hard-\\nware platforms on real machine learning datasets. Previous theoretical studies have focused\\non convergence proofs and computation of gradient approximations to varying degrees of\\naccuracy [12, 13] or on simpli\\fed problems, such as linear activation functions [38]. While\\nthese theoretical results are clearly important and necessary, they are not su\\u000ecient to give\\ninsight into the applicability of these optimization techniques to real machine learning prob-\\nlems on experimental hardware platforms. Meanwhile, previous experimental and simulation\\nstudies have for the most part focused on small scale problems, for which 100% accuracy is\\ntrivial, such as XOR [14, 15], few-bit parity [16, 20, 21], the iris dataset [24] or other similar\\ntoy problems [17, 18, 25]. These are not representative of larger machine learning datasets,\\nwhich typically have on the order of tens of thousands of training examples and are trained\\non networks with even more parameters.\\nThese results go further than any of these previous studies, by elucidating the util-\\nity of these techniques for larger, modern, machine learning datasets, and implementing a\\nframework that can connect di\\u000berent optimization techniques (SPSA, \\fnite-di\\u000berence, ap-\\nproximate gradient descent) via speci\\fc hardware parameters.\\nIt is also worth noting that the random weight change (RWC) algorithm has been imple-\\n26 mented in memristor networks [23, 39] previously, and while super\\fcially similar to MGD is\\nfunctionally very di\\u000berent. RWC is not an approximate gradient descent technique, since the\\nweight update is not scaled by the magnitude of the change in the cost, but rather random\\nperturbations are either kept or discarded based on whether or not they improve the cost.\\nBecause of this, it scales more poorly with number of parameters than the optimization\\ntechniques we have implemented in the MGD framework.\\n4 Hardware considerations\\n4.1 Time constants\\nHW1 HW2 HW3 backprop\\n\\u001cx(input-sample update time) 100 ns 1 ns 10 ps n\\/a\\n\\u001cp(perturbation time) 1 ms 10 ns 200 ps n\\/a\\n\\u001c\\u0012(parameter-update time) 1 ms 1\\u00b5s 200 ps n\\/a\\n2-bit parity training time ( 104steps) 20 s 200\\u00b5s 4\\u00b5s 70 msy\\nFashion-MNIST training time ( 106steps) 33 min 20 ms 400\\u00b5s 54 s\\u0003\\nCIFAR-10 training time ( 107steps) 5.6 hours 200 ms 4 ms 480 s\\u0003\\nExamples of hardwarechip-in-the-loop,\\nintegrated photonics\\nwith thermo-optic tuning[40, 11]Mem-compute devices [41],\\nanalog VLSI [42]superconducting devices [43],\\nathermal resonant\\nsilicon photonic modulator [44]yCPU\\/\\u0003GPU\\nTable 3: Approximate training time using MGD for estimated hardware time constants. Data\\nshown is estimated using the same dataset and network architectures as used in Table 2. The\\n\\fnal column shows the time it takes to get to that same accuracy using backpropagation on\\na standard GPU or CPU.\\nWhile the MGD framework is general, individual hardware platforms may have di\\u000berent\\npractical considerations that dictate the optimal choices for the di\\u000berent time constants.\\nHere we discuss potential issues for implementation on hardware, and point out tradeo\\u000bs\\nthat may exist.\\nPerturbation speed ( \\u001cp):In many hardware platforms, perturbing parameters directly\\nmay be di\\u000ecult or undesired either due to memory-endurance or update-speed concerns.\\nHowever, it is not necessary to to perturb parameters directly{instead, one may perturb\\na separate element that is in series with the parameter. For instance, in a photonic MZI\\narray, the weights may be implemented with slow ( \\u0018millisecond) thermal or mechanical phase\\nshifters, but the small perturbation could be implemented with a fast ( \\u0018nanosecond) electro-\\n27 optic modulator such as a depletion [44] or opto-mechanical [45] modulator. Alternatively,\\nin a memristor system, the perturbation could be implemented with a single sub-threshold\\ntransistor in series with the memristor.\\nFor perturbative-style techniques like MGD to function properly, \\u001cpshould be slower than\\nthe system's inference time [13] which includes the cost calculation and broadcast processes.\\nIf\\u001cpis too short, it can introduce misalignment between the input perturbation and resulting\\ncost feedback.\\nParameter update ( \\u001c\\u0012);When performing gradient descent on hardware, parameters\\nmust be updated multiple times. For a training period of length T, the weights will be\\nupdatedT=\\u001c \\u0012times. For some hardware platforms it may be advantageous to reduce the\\nnumber of parameter updates, for instance if parameters have a limited lifetime, take a\\nlong time to update, or the update process requires a large amount of energy. Fortunately,\\nTable 2 shows that for networks with large numbers of parameters, \\u001c\\u0012can be increased\\nwithout greatly impacting the overall training speed or \\fnal accuracy.\\nChanging training examples ( \\u001cx):The value of \\u001cxis limited by the speed with which\\ntraining examples can be input to the network. For most technologies, \\u001c\\u0012or\\u001cxwill not be\\nlimiting factors, as xand ^ysamples can be provided by a conventional computer or FPGA\\nat nanosecond timescales. However, it should be noted that MGD will not function correctly\\nif\\u001cxis shorter than the inference time of the hardware, and this will likely be the practical\\nlimit for\\u001cx.\\u001cxis also used as a way of controlling the batch size (given by \\u001c\\u0012=\\u001cx). The\\nparticulars of the task and dataset may determine the desired batch size, and this may have\\nsome e\\u000bect on \\u001cx, for example if the minimum allowable value of \\u001c\\u0012is very long.\\nCalculation of cost: The calculation of the cost is performed only at the output of the\\nsystem, and therefore it is acceptable that the hardware used in its computation be more\\nexpensive and complex. In integrated electronic systems, the cost can be straightforward\\nto implement, although it may occupy more relatively more chip real-estate than other\\nelements. However, for very exotic hardware platforms, this cost computation may still be\\n28 di\\u000ecult to perform on-chip. This can be addressed by using a non-standard cost function\\nor by using a computer to calculate the cost o\\u000b-chip. However, the input and output to an\\no\\u000b-chip computation may limit the speed of the cost computation and also limit the speed of\\nthe global broadcast of ~C, and may be another speed bottleneck in some hardware systems.\\nBased on the literature, some estimates of plausible time constants that could be imple-\\nmented in hardware are shown in Table 3 with the corresponding times to solve benchmark\\ntasks, based on the results from Table 2.Examples from the literature of hardware platforms\\nthat could implement these sets of parameters are shown in the \\fnal row. By comparison\\nof these results with the \\fnal column in Table 3, we see that using realistic estimates for\\nemerging hardware, MGD could be signi\\fcantly faster than current implementations using\\nbackpropagation.\\n4.2 Analog and digital implementations\\nThere are a few notable di\\u000berences for an analog versus a digital hardware implementation of\\nthe MGD framework. The discrete case requires one memory element at the network output\\nto storeC0(sample-and-hold), and a simple subtraction operation to compute ~C=C\\u0000C0.\\nThe network also requires some timesteps to be devoted to the measurement of C0. This\\nmeans that for the simple case of \\u001c\\u0012=\\u001cp, only a single additional memory element is required\\nfor training the entire hardware system, located at the network cost output.\\nHowever, in the case where \\u001c\\u0012>\\u001cp, an additional memory element is required for every\\nparameter (for instance, an analog capacitor or discrete memory) to perform the gradient\\nintegration. The analog case requires a lowpass \\flter at every parameter element, and a single\\nhighpass \\flter on the network output to convert Cto~C. These could be created with simple\\nanalog circuits like RC or LR \\flters. However, we note the use of a continuous highpass \\flter\\ncan cause implementation issues is some cases. For instance, if \\u0011is too large, rapid changes\\nin\\u0012can generate unwanted frequency components that mix with the perturbation input and\\nmay negatively a\\u000bect the gradient approximation. Moreover, if the dataset is discrete, jumps\\n29 inxcan propagate high frequency noise through Cand ~C, again negatively a\\u000becting the\\ngradient approximation. The simulations demonstrate that in principle MGD can be used\\non di\\u000berent hardware platforms of either an analog or digital nature for network training.\\n5 Discussion\\nAn interesting analogy can be drawn between the MGD training framework and wireless\\ncommunication systems. In wireless communications, cell phone users must be able to extract\\nthe signal that is relevant to their own communication from a received broadcast that contains\\nmultiple users' signals. MGD operates similarly{each parameter is analogous to an individual\\ncell phone user and the global cost broadcast ~Cis analogous to the cell tower transmitter.\\nThe broadcasted signal ~Cis available to all parameters, and each parameter extracts its\\nrelevant (gradient) information { analogous to voice data { from that broadcasted signal.\\nThe encoding and multiplexing techniques used in wireless communications are broadly\\ntermed \\\"multiple access\\\" techniques in communications, and there are several varieties of\\nthem including frequency, time, and code multiplexing. Similarly, in MGD, these multi-\\nplexing techniques are directly analogous to the di\\u000berent perturbation types discussed in\\nSection 3.4. For instance, sinusoidal perturbations can be considered a type of frequency-\\nmultiplexing: each \\\\user\\\" (parameter) has a unique frequency containing information rel-\\nevant to it, and must perform a time-integration to extract that information from the ~C\\nsignal which contains many other, irrelevant signals. Likewise, as mentioned in Section 3.4,\\nthe discretef\\u0000\\u0001\\u0012;+\\u0001\\u0012gperturbation type is analogous to the code-multiplexing techniques\\nused in modern cell phones.\\nIt is important to note that for a \\fxed bandwidth broadcast, all multiplexing techniques\\nhave the same nominal spectral e\\u000eciency. However, the particulars of real systems can\\ngreatly a\\u000bect which multiplexing techniques are most e\\u000bective. For example, because code-\\nmultiplexing encodes a user's signal out over a wide time window and a spread spectrum, it\\n30 has been shown to have an improved robustness to multipath distortion and timing errors\\ncompared to time-multiplexing.\\nSimilarly, the various perturbation types described earlier may operate better or worse\\ndepending on the particulars of the hardware network and environment. Newly developed\\ncommunications techniques may also be directly applicable to MGD training in real hardware\\nsystems. Key to its usage in hardware, each parameter can operate like a cell phone |\\nas an independent device with a receiver, processor and memory using only information\\navailable locally and from the global broadcast. This contrasts with most other training\\nalgorithms, where error signals must propagate through the entire network [46]. While trivial\\nin software, this type of upstream information-\\row means that hardware neurons must (1)\\nhave additional memory to process the error signal, (2) perform di\\u000berent operations for\\nforward-inference and reverse error propagation, and (3) operate in a synchronized fashion\\nto propagate the error correctly.\\nIn systolic arrays \\\\broadcast\\\" is often discouraged because the time and energy of com-\\nmunication is proportional to the number of inputs by CV2(with C growing with the number\\nof inputs). Furthermore, as the capacitance is larger, then a larger ampli\\fer is needed for\\nthe communication which also requires energy and area. However in other types of analog\\nor emergent hardware, broadcast may be a more energy and space e\\u000ecient process. For\\nexample, in optical hardware, a global optical signal could be accessible in a simple slab\\nwaveguide mode. Alternatively, in analog electrical hardware, a single electrical plane could\\nbe used to carry the ~Csignal back to the individual parameters.\\nThe MGD framework may also be much more biologically-plausible than other algo-\\nrithms. There are a host of algorithms being developed to address the mystery of how the\\nbrain learns, and to use biology as inspiration to \\fnd more hardware-friendly approaches\\n[46, 47, 48]. One of the major issues with backpropagation from both a practical and bio-\\nplausibility standpoint is the need for a separate type of operation in the forward (inference)\\nand backward (learning) phases. This is not the case for MGD: in MGD, the training is\\n31 done in the same con\\fguration used for inference. In a biological analogy, the a global cost\\nsignal ~Ccould be modeled as some type of quasi-global chemical signal in the brain.\\nIn future, the MGD algorithm could be extended such that the cost is not global to every\\nsynapse, but rather to speci\\fc clusters of neurons\\/synapses. There are chemical signals in\\nthe brain with these types of properties. The time scales needed for perturbation may line\\nup with biologically observed e\\u000bects such as short-term plasticity (minutes) [49, 50] and\\nsynaptic noise [51]. MGD is also consistent with \\\\three-factor\\\" learning rules, for which\\nthere is mounting experimental evidence [50]. Additionally, although not explored in this\\nwork, MGD should work in a spiking model. Similar algorithms that use perturbations in\\nneuron conductance [52], probabilistic synapse transmission [53] or the stochastic nature\\nof spike-timing arrival [54] for reward-based learning have been explored in small spiking\\nnetworks.\\n6 Conclusions\\nIn this paper we show that with realistic timescales for emerging hardware, training via\\nMGD could be orders of magnitude faster than backpropagation in terms of wall-clock time-\\nto-solution on a standard GPU\\/CPU. The MGD framework allows the implementation of\\nmultiple optimization algorithms using a single, global, cost signal and local parameter\\nupdates. The style of algorithm used (e.g. \\fnite-di\\u000berence, coordinate-descent, SPSA, etc.)\\ncan be adjusted via the tuning of the MGD time-constants, and can even be adjusted during\\ntraining if desired. Because it is a model-free perturbative technique (sometimes called\\nzeroth order optimization), it is applicable to a wide range of systems { it can be applied to\\nboth analog and digital hardware platforms, and it can be used in the presence of noise and\\ndevice imperfections. This overcomes a major barrier to using hardware platforms based on\\nemerging technologies, which are often di\\u000ecult to train.\\nGoing forward, there are many opportunities to explore the application of MGD on both\\n32 large and small hardware systems. The most direct next step will be to test the training per-\\nformance in existing hardware, for example on photonic [40] or memristive crossbar hardware\\nusing a chip-in-the-loop process. This can occur without even redesigning the hardware, as\\nMGD only requires the ability to input samples, capture inference output, and modify pa-\\nrameters when in a chip-in-the-loop con\\fguration. When used in this fashion, the speed will\\nmost likely be limited by system I\\/O. For example, perturbations can be injected directly\\nto the hardware from an external computer, and that same computer could capture the\\nchanges in cost and perform the gradient approximation and calculate the weight updates.\\nThis would allow testing of the algorithm without any changes to the hardware. Ultimately,\\nto overcome I\\/O limitations local circuits can be designed be implemented for autonomous\\nonline training. Although not examined in detail here, in this case many of the hardware\\nplatforms examined would likely have several orders of magnitude improvement in terms of\\nenergy usage as well.\\nThere is also lots of room for improvement on tuning the technique and its hyper-\\nparameters. In this paper we performed very little hyperparameter optimization or reg-\\nularization, and so there are likely opportunities for further optimization by examining tech-\\nniques such as dropout, momentum, and regularization. Also of interest is examining the\\nnode-perturbation version of the algorithm [55] on large datasets, which could signi\\fcantly\\nreduce the number of perturbative elements and speed up the training process.\\nWhile in this paper we only examined feed forward networks, it has already been demon-\\nstrated that is possible to use perturbative techniques to train recurrent neural networks [56],\\nspiking networks [53] and other non-standard networks at small scale; future work remains\\nto demonstrate their utility on problems of modern interest. This will have applicability to\\na wide range of neuromorphic hardware and other physical neural networks.\\nUltimately, MGD is well-suited to implementation directly on-chip with local, autonomous\\ncircuits. Although signi\\fcant work remains before autonomous learning can be truly imple-\\nmented without the participation of an external computer, such a device would be truly\\n33 enabling for remote and edge-computing applications, allowing training in-the-\\feld on real-\\ntime data.\\n7 Acknowledgements\\nThe authors acknowledge helpful discussions with the NIST NCSG community. This re-\\nsearch was funded by NIST (https:\\/\\/ror.org\\/05xpvk416) and University Colorado Boulder\\n(https:\\/\\/ror.org\\/02ttsq026).\\nReferences\\n[1] G Hinton Y LeCun, Y Bengio. Deep learning. Nature , 521(7553):436{444, may 2015.\\n[2] Catherine D. Schuman, Thomas E. Potok, Robert M. Patton, J. Douglas Birdwell,\\nMark E. Dean, Garrett S. Rose, and James S. Plank. A Survey of Neuromorphic\\nComputing and Neural Networks in Hardware. may 2017.\\n[3] Wilfried Haensch, Tayfun Gokmen, and Ruchir Puri. The Next Generation of Deep\\nLearning Hardware: Analog Computing. Proceedings of the IEEE , 107(1):108{122, jan\\n2019.\\n[4] Paul A Merolla, John V Arthur, Rodrigo Alvarez-icaza, Andrew S Cassidy, Jun Sawada,\\nFilipp Akopyan, Bryan L Jackson, Nabil Imam, Chen Guo, Yutaka Nakamura, Bernard\\nBrezzo, Ivan Vo, Steven K Esser, Rathinakumar Appuswamy, Brian Taba, Arnon Amir,\\nMyron D Flickner, William P Risk, Rajit Manohar, and Dharmendra S Modha. A\\nmillion spiking-neuron integrated circuit with a scalable communication network and\\ninterface. Science , 345:668, 2014.\\n[5] Mike Davies, Narayan Srinivasa, Tsung-Han Lin, Gautham Chinya, Yongqiang Cao,\\nHarsha Choday, Georgios Dimou, Prasad Joshi, Nabil Imam, Shweta Jain, Yuyun Liao,\\n34 Chit-Kwan Lin, Andrew Lines, Ruokun Liu, Deepak Mathaikutty, Steve Mccoy, Arnab\\nPaul, Jonathan Tse, Guruguhanathan Venkataramanan, Yi-Hsin Weng, Andreas Wild,\\nYoonseok Yang, and Hong Wang. Loihi: A neuromorphic manycore processor with on-\\nchip learning; loihi: A neuromorphic manycore processor with on-chip learning. IEEE\\nMicro , 38, 2018.\\n[6] Karlheinz Meier. A mixed-signal universal neuromorphic computing system. In 2015\\nIEEE International Electron Devices Meeting (IEDM) , 2015.\\n[7] A. Mehonic and A. J. Kenyon. Brain-inspired computing needs a master plan. Nature ,\\n604:255{260, 4 2022.\\n[8] Neil C Thompson, Kristjan Greenewald, Keeheon Lee, and Gabriel F Manso. The\\ncomputational limits of deep learning. arXiv:2007.05558, 2020.\\n[9] Logan G. Wright, Tatsuhiro Onodera, Martin M. Stein, Tianyu Wang, Darren T.\\nSchachter, Zoey Hu, and Peter L. McMahon. Deep physical neural networks trained\\nwith backpropagation. Nature , 601:549{555, 2022.\\n[10] J. Kiefer and J. Wolfowitz. Stochastic Estimation of the Maximum of a Regression\\nFunction. The Annals of Mathematical Statistics , 23(3):462{466, 1952.\\n[11] Yichen Shen, Nicholas C. Harris, Scott Skirlo, Mihika Prabhu, Tom Baehr-Jones,\\nMichael Hochberg, Xin Sun, Shijie Zhao, Hugo Larochelle, Dirk Englund, and Marin\\nSoljacic. Deep learning with coherent nanophotonic circuits. Nature Photonics , 11:441{\\n446, 2017.\\n[12] James C. Spall. Multivariate stochastic approximation using a simultaneous perturba-\\ntion gradient approximation. IEEE Transactions on Automatic Control , 37:332{341,\\n1992.\\n35 [13] Amir Dembo and Thomas Kailath. Model-Free Distributed Learning. IEEE Transac-\\ntions on Neural Networks , 1(1):58{70, 1990.\\n[14] T. Matsumoto and M. Koga. Novel learning method for analogue neural networks. ElL,\\n26(15):1136, sep 1990.\\n[15] Marwan Jabri and Barry Flower. Weight Perturbation: An Optimal Architecture and\\nLearning Technique for Analog VLSI Feedforward and Recurrent Multilayer Networks.\\nIEEE Transactions on Neural Networks , 3(1):154{157, 1992.\\n[16] J. Alspector, R. Meir, B. Yuhas, A. Jayakumar, and D. Lippe. A parallel gradient\\ndescent method for learning in analog VLSI neural networks | Proceedings of the 5th\\nInternational Conference on Neural Information Processing Systems. In NIPS'92: Pro-\\nceedings of the 5th International Conference on Neural Information Processing Systems ,\\npages 836{844, 1992.\\n[17] David B. Kirk and Douglas Kerns. Analog VLSI Implementation of Multi-dimensional\\nGradient Descent. In Advances in Neural Information Processing Systems 5 (NIPS\\n1992) , pages 789{796, 1992.\\n[18] Yutaka Maeda, Hiroaki Hirano, and Yakichi Kanata. A learning rule of neural net-\\nworks via simultaneous perturbation and its hardware implementation. Neural Net-\\nworks , 8(2):251{259, jan 1995.\\n[19] Gert Cauwenberghs. An analog VLSI recurrent neural network learning a continuous-\\ntime trajectory. IEEE Transactions on Neural Networks , 7(2):346{361, 1996.\\n[20] Hiroshi Miyao, Kazuhiro Noguchi, Masafumi Koga, and Takao Matsumoto. Multifre-\\nquency oscillation learning method for analog neural network: Its implementation in a\\nlearning LSI. Electronics and Communications in Japan (Part III: Fundamental Elec-\\ntronic Science) , 80(5), 1997.\\n36 [21] Antonio J. Montalvo, Ronald S. Gyurcsik, and John J. Paulos. Toward a general-\\npurpose analog VLSI neural network with on-chip learning. IEEE Transactions on\\nNeural Networks , 8(2):413{423, 1997.\\n[22] Yutaka Maeda and Toshiki Tada. FPGA implementation of a pulse density neural\\nnetwork with learning ability using simultaneous perturbation. IEEE Transactions on\\nNeural Networks , 14(3):688{695, may 2003.\\n[23] Shyam Prasad Adhikari, Hyongsuk Kim, Ram Kaji Budhathoki, Changju Yang, and\\nLeon O. Chua. A circuit-based learning architecture for multilayer neural networks\\nwith memristor bridge synapses. IEEE Transactions on Circuits and Systems I: Regular\\nPapers , 62(1):215{223, jan 2015.\\n[24] Chunhua Wang, Lin Xiong, Jingru Sun, and Wei Yao. Memristor-based neural networks\\nwith weight simultaneous perturbation training. Nonlinear Dynamics , 95(4):2893{2906,\\n2019.\\n[25] Saumil Bandyopadhyay, Alexander Sludds, Stefan Krastanov, Ryan Hamerly, Nicholas\\nHarris, Darius Bunandar, Matthew Streshinsky, Michael Hochberg, and Dirk Englund.\\nSingle chip photonic deep neural network with accelerated training. pages 1{21, aug\\n2022.\\n[26] At\\u0010l\\u0010m G\u007f unes Baydin, Barak A. Pearlmutter, Don Syme, Frank Wood, and Philip Torr.\\nGradients without backpropagation, 2022.\\n[27] David Silver, Anirudh Goyal, Ivo Danihelka, Matteo Hessel, and Hado van Hasselt.\\nLearning by directional gradient descent. In Proceedings of the International Conference\\non Learning Representations (ICLR) , 2022.\\n[28] Mengye Ren, Simon Kornblith, Renjie Liao, and Geo\\u000brey Hinton. Scaling forward\\ngradient with local losses, 2022.\\n37 [29] Brian Hoskins, Mitchell Fream, Matthew Daniels, Jonathan Goodwill, Advait Mad-\\nhavan, Jabez Mcclelland, Osama Yousuf, Gina Adam, Wen Ma, Tung Hoang, Mark\\nBranstad, Muqing Liu, Rasmus Madsen, Martin Lueker-Boden, and Martin Lueker. A\\nsystem for validating resistive neural network prototypes. International Conference on\\nNeuromorphic Systems 2021 , 5.\\n[30] Tiankuang Zhou, Xing Lin, Jiamin Wu, Yitong Chen, Hao Xie, Yipeng Li, Jingtao Fan,\\nHuaqiang Wu, Lu Fang, and Qionghai Dai. Large-scale neuromorphic optoelectronic\\ncomputing with a recon\\fgurable di\\u000bractive processing unit. Nature Photonics , 15:367{\\n373, 2021.\\n[31] Saumil Bandyopadhyay, Ryan Hamerly, Ryan Hamerly, and Dirk Englund. Hardware\\nerror correction for programmable photonics. Optica, Vol. 8, Issue 10, pp. 1247-1255 ,\\n8:1247{1255, 2021.\\n[32] Amir Khosrowshahi, Casimir Wierzynski, Michael R. DeWeese, Michael Y.-S. Fang, and\\nSasikanth Manipatruni. Design of optical neural networks with component imprecisions.\\nOptics Express, Vol. 27, Issue 10, pp. 14009-14029 , 27:14009{14029, 2019.\\n[33] Miao Hu, Catherine E. Graves, Can Li, Yunning Li, Ning Ge, Eric Montgomery, Noraica\\nDavila, Hao Jiang, R. Stanley Williams, J. Joshua Yang, Qiangfei Xia, and John Paul\\nStrachan. Memristor-based analog computation and neural network classi\\fcation with\\na dot product engine. Advanced Materials , 30:1705914, 2018.\\n[34] Stefano Ambrogio, Pritish Narayanan, Hsinyu Tsai, Robert M. Shelby, Irem Boybat,\\nCarmelo Di Nolfo, Severin Sidler, Massimo Giordano, Martina Bodini, Nathan C.P.\\nFarinha, Benjamin Killeen, Christina Cheng, Yassine Jaoudi, and Geo\\u000brey W. Burr.\\nEquivalent-accuracy accelerated neural-network training using analogue memory. Na-\\nture, 558:60{67, 2018.\\n38 [35] Stefano Ambrogio, Simone Balatti, Antonio Cubeta, Alessandro Calderoni, Nirmal Ra-\\nmaswamy, and Daniele Ielmini. Statistical \\ructuations in hfox resistive-switching mem-\\nory: Part i-set\\/reset variability. IEEE Transactions on Electron Devices , 61:2912{2919,\\n2014.\\n[36] Jubin Hazra, Maximilian Liehr, Karsten Beckmann, Sarah Ra\\fq, and Nathaniel Cady.\\nImproving the memory window\\/resistance variability trade-o\\u000b for 65nm cmos integrated\\nhfo2 based nanoscale rram devices. volume 2019-October. Institute of Electrical and\\nElectronics Engineers Inc., 10 2019.\\n[37] Cory Merkel and Dhireesha Kudithipudi. Comparison of o\\u000b-chip training methods\\nfor neuromemristive systems. volume 2015-February, pages 99{104. IEEE Computer\\nSociety, 2 2015.\\n[38] Justin Werfel, Xiaohui Xie, and H. Sebastian Seung. Learning Curves for Stochastic\\nGradient Descent in Linear Feedforward Networks. Neural Computation , 17(12):2699{\\n2718, dec 2005.\\n[39] Changju Yang, Hyongsuk Kim, Shyam Prasad Adhikari, and Leon O. Chua. A Circuit-\\nBased Neural Network with Hybrid Learning of Backpropagation and Random Weight\\nChange Algorithms. Sensors 2017, Vol. 17, Page 16 , 17(1):16, dec 2016.\\n[40] Alexander N. Tait, Thomas Ferreira De Lima, Ellen Zhou, Allie X. Wu, Mitchell A.\\nNahmias, Bhavin J. Shastri, and Paul R. Prucnal. Neuromorphic photonic networks\\nusing silicon photonic weight banks. Scienti\\fc Reports , 7, 2017.\\n[41] Geo\\u000brey W Burr, Robert M Shelby, Abu Sebastian, Sangbum Kim, Seyoung Kim, Sev-\\nerin Sidler, Kumar Virwani, Masatoshi Ishii, Pritish Narayanan, Alessandro Fumarola,\\nLucas L Sanches, Irem Boybat, Manuel Le Gallo, Kibong Moon, Jiyoo Woo, Hyunsang\\nHwang, and Yusuf Leblebici. Neuromorphic computing using non-volatile memory.\\n2:89{124, 2017.\\n39 [42] Y. Kohda, Y. Li, K. Hosokawa, S. Kim, R. Khaddam-Aljameh, Z. Ren, P. Solomon,\\nT. Gokmen, S. Rajalingam, C. Baks, W. Haensch, and E. Leobandung. Unassisted true\\nanalog neural network training chip. Technical Digest - International Electron Devices\\nMeeting, IEDM , 2020-December:36.2.1{36.2.4, 12 2020.\\n[43] Michael Schneider, Emily Toomey, Graham Rowlands, Je\\u000b Shainline, Paul Tschirhart,\\nand Ken Segall. Supermind: a survey of the potential of superconducting electronics\\nfor neuromorphic computing. Superconductor Science and Technology , 35:053001, 2022.\\n[44] Erman Timurdogan, Cheryl M. Sorace-Agaskar, Jie Sun, Ehsan Shah Hosseini, Alek-\\nsandr Biberman, and Michael R. Watts. An ultralow power athermal silicon modulator.\\nNature Communications , 5:1{11, 2014.\\n[45] Mark Dong, Genevieve Clark, Andrew J Leenheer, Matthew Zimmermann, Daniel\\nDominguez, Adrian J Menssen, David Heim, Gerald Gilbert, Dirk Englund, and Matt\\nEichen\\feld. High-speed programmable photonic circuits in a cryogenically compatible,\\nvisible-near-infrared 200 mm cmos architecture. Nature Photonics , 16:59{65, 2022.\\n[46] Timothy P. Lillicrap, Adam Santoro, Luke Marris, Colin J. Akerman, and Geo\\u000brey\\nHinton. Backpropagation and the brain. Nature Reviews Neuroscience 2020 21:6 ,\\n21(6):335{346, apr 2020.\\n[47] Friedemann Zenke and Emre O. Neftci. Brain-inspired learning on neuromorphic sub-\\nstrates. Proceedings of the IEEE , 109:935{950, 5 2021.\\n[48] Benjamin Scellier and Yoshua Bengio. Equilibrium propagation: Bridging the gap\\nbetween energy-based models and backpropagation. Frontiers in Computational Neu-\\nroscience , 11:24, 5 2017.\\n[49] Andrea Soltoggio. Short-term plasticity as cause{e\\u000bect hypothesis testing in distal\\nreward learning. Biological Cybernetics , 109:75{94, 2 2015.\\n40 [50] Eligibility Traces and Plasticity on Behavioral Time Scales: Experimental Support of\\nNeoHebbian Three-Factor Learning Rules. Frontiers in Neural Circuits , 0:53, jul 2018.\\n[51] Benjamin A. Rowland, Anthony S. Maida, and Istvan S. N. Berkeley. Synaptic noise as\\na means of implementing weight-perturbation learning. 18(1):69{79, mar 2007.\\n[52] Ila R. Fiete and H. Sebastian Seung. Gradient learning in spiking neural networks by\\ndynamic perturbation of conductances. Phys. Rev. Lett. , 97:048104, Jul 2006.\\n[53] H. Sebastian Seung. Learning in Spiking Neural Networks by Reinforcement of Stochas-\\ntic Synaptic Transmission. Neuron , 40(6):1063{1073, dec 2003.\\n[54] Xiaohui Xie and H. Sebastian Seung. Learning in neural networks by reinforcement of\\nirregular spiking. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related\\nInterdisciplinary Topics , 69(4):10, 2004.\\n[55] Barry Flower and Marwan Jabri. Summed weight neuron perturbation. In NIPS'92:\\nProceedings of the 5th International Conference on Neural Information Processing Sys-\\ntems, pages 212{219, 1992.\\n[56] Gert Cauwenberghs. A Fast Stochastic Error-Descent Algorithm for Supervised Learn-\\ning and Optimization. In Advances in Neural Information Processing Systems 5 (NIPS\\n1992) , pages 244{251, 1992.\\n41\",\"40\":\" SYMBOLIC SYNTHESIS OF NEURAL NETWORKS\\nEli Whitehouse\\nNew York, NY 10024\\neliw55@gmail.com\\nABSTRACT\\nNeural networks adapt very well to distributed and continuous representations, but struggle to learn\\nand generalize from small amounts of data. Symbolic systems commonly achieve data ef\\ufb01cient\\ngeneralization by exploiting modularity to bene\\ufb01t from local and discrete features of a representation.\\nThese features allow symbolic programs to be improved one module at a time and to experience\\ncombinatorial growth in the values they can successfully process. However, it is dif\\ufb01cult to design\\ncomponents that can be used to form symbolic abstractions and which are highly-overparametrized\\nlike neural networks, as the adjustment of parameters makes the semantics of modules unstable.\\nI present Graph-based Symbolically Synthesized Neural Networks (G-SSNNs), a form of neural\\nnetwork whose topology and parameters are informed by the output of a symbolic program. I\\ndemonstrate that by developing symbolic abstractions at a population level, and applying gradient-\\nbased optimization to such neural models at an individual level, I can elicit reliable patterns of\\nimproved generalization with small quantities of data known to contain local and discrete features.\\nThe paradigm embodied by G-SSNNs offers a route towards the communal development of compact\\nand composable abstractions which can be \\ufb02exibly repurposed for a variety of tasks and high-\\ndimensional media. In future work, I hope to pursue these bene\\ufb01ts by exploring more ambitious\\nG-SSNN designs based on more complex classes of symbolic programs. The code and data associated\\nwith the reported results are available here.\\nKeywords neural networks\\u0001symbolic programs \\u0001graph neural networks \\u0001library learning\\u0001distributional program\\nsearch\\n1 Introduction\\nMost conventional modes of human communication naturally occur in a high-dimensional medium such as text, audio,\\nor images. When processing these media, slight errors can easily accrue across many dimensions of the input where\\nfeatures are distributed. With adequate data, neural networks adapt effectively to these patterns by adjusting many\\ninterdependent real-valued parameters. In their basic form, however, neural models are data inef\\ufb01cient. In settings\\nwhere more data cannot be sourced, pretraining has arisen as the most general and data-driven answer to this challenge.\\nPretraining allows practitioners to repurpose data that is not speci\\ufb01cally suited to a task to enrich a representation or\\ninitialize parameters of a neural model. While pretraining grants a model exposure to out-of-distribution data which\\nmay be irrelevant or inappropriate to the task at hand, the bene\\ufb01ts of this exposure can outweigh the costs.\\nIn symbolic systems, modularity is often used to stimulate greater data ef\\ufb01ciency and generalization. By examining the\\nparticular dimensions and values at which systems fail, developers can trace issues back to speci\\ufb01c modules. When a\\nfailure in one module is isolated and corrected, the system bene\\ufb01ts from this across all combinations of values and\\ndimensions, witnessing potentially exponential growth in the number of inputs on which it succeeds. This approach\\nrelies on the locality and discreteness of the representations associated with the task so that modular functionalities\\ncan be developed and debugged. Unfortunately, it is dif\\ufb01cult to design a fundamental unit of a system that gains the\\nbene\\ufb01ts of modularity and of neural networks, as it is dif\\ufb01cult to compose stable abstractions from components that\\ncontain many adjustable parameters. Nonetheless, this is a more direct solution to the problem of data inef\\ufb01ciency\\nand inadequate generalization in circumstances where high-dimensional data are known to exhibit both locality and\\ndiscreteness.arXiv:2303.03340v1  [cs.NE]  6 Mar 2023 Symbolic Synthesis of Neural Networks\\nIn this work, I propose a technique for structuring modular neural networks that utilizes symbolic programs to\\ninform the network\\u2019s topology and to determine the values of some of its parameters. This allows the symbolic\\nprograms themselves to be developed so as to exhibit increasing degrees of abstraction at a population level, while\\nallowing resulting individual networks to be trained and evaluated with standard techniques. I present Graph-based\\nSymbolically-Synthesized Neural Networks (G-SSNNs) and apply populations of G-SSNNs to a binary prediction task\\nover high-dimensional data containing many local and discrete features. I show that these populations exhibit reliable\\npatterns of improved generalization from small quantities of data compared to a population of baseline models which\\nare not structured according to the output of symbolic programs.\\n2 Related Work\\n2.1 Neural Architecture Search\\nNeural Architecture Search (NAS) is the process of using automated methods to explore the discrete space of neural\\nnetwork topologies. Though varying in the design of their search space [Jin et al., 2022], search strategies [Real\\net al., 2020], and model evaluation strategies [Abdelfattah et al., 2021], all approaches I reviewed assume that model\\nparameters learnable through gradient-based optimization will be learned in this fashion once search is complete. That\\nis, they serve to explore a (potentially hierarchical) space of hyperparameters in which topology often plays a central\\nrole. This differs from G-SSNNs, in which symbolic programs play a role in determining both the topology and the\\nparameters of a model.\\n2.2 Embedding Spaces and Preprocessing\\nThe act of adding \\ufb01xed transformations into a neural network that could otherwise be performed with the help of\\nlearnable parameters seems more analogous to an act of preprocessing, as might be performed with word embeddings\\n[Mikolov et al., 2013] or positional encoding [Vaswani et al., 2017]. However, I found no works in which similar acts\\nof preprocessing were designed with the help of automated program synthesis.\\n2.3 Program Synthesis\\nThe program synthesis apparatus used in this work is inspired by the DreamCoder system [Ellis et al., 2021] and uses\\nthe same cycle of library learning and distributional program search between iterations of G-SSNN training, though I\\nutilize the more recent STITCH tool [Bowers et al., 2023] and heap search algorithm [Matricon et al., 2022] for these\\nrespective functionalities. G-SSNNs have a deeper analogy with the DreamCoder system in that the process of training\\na G-SSNN on a dataset generalizes the evaluation mechanisms for tasks and symbolic programs used in DreamCoder\\u2019s\\ndomains. The greater generality of G-SSNNs introduces concerns about performance on unseen data that are not\\nrelevant when measuring the performance of purely symbolic programs, and must be addressed by separate mechanisms.\\nHowever, in principle, nothing prevents the multitask Bayesian program learning framework of DreamCoder from being\\napplied to the development of multiple, parallel populations of G-SSNNs.\\nIn the broader neurosymbolic literature, I did not \\ufb01nd other examples of systems in which adjacent parametric and\\nnonparametric modules are separately optimized through gradient-based learning and the development of increasingly\\nhierarchical symbolic abstractions.\\n3 Methods\\n3.1 Evolutionary Framework\\nWhile the parameters of a G-SSNN may be optimized with ordinary training methods, we must utilize a separate\\nmechanism to optimize their symbolic structure. Since a G-SSNN is characterized by a single symbolic program, we\\nmust retain a population of G-SSNNs in order to conduct program search. This search is guided in two respects: by the\\nstructure of the DSL, and by the parameters of a distribution over programs. The structure of the DSL is determined by structure of the DSL, and by the parameters of a distribution over programs. The structure of the DSL is determined by\\nthe results of library learning performed with STITCH [Bowers et al., 2023]. STITCH\\u2019s library learning functionality\\noperates by extracting novel primitives from a corpus according to which provides the greatest degree of compression\\n(i.e. reduction in aggregate program size). A distribution over programs may also be inferred from such a corpus by\\ncounting how many times n-grams of primitives from the DSL are used.1Thus, if we are able to select an approximately\\noptimal corpus of symbolic programs based on the training performance of a population of G-SSNNs, we may both\\n1In experiments, I utilize a unigram distribution.\\n2 Symbolic Synthesis of Neural Networks\\nbias compression towards the discovery of modules common to successful programs and bias program search towards\\nthe discovery of increasingly complex programs which utilize those modules.\\nBecause there is always the possibility that a G-SSNN performs differently on unseen data, there is no heuristic that is\\nunquestionably best for encouraging the greatest generalization. Perhaps the simplest and most generally agreeable,\\nhowever, are those based on discarding G-SSNNs with the Lowest Training Performance (LTP). I refer to the heuristic I\\nuse in the experiments presented here as rank-LTP-50, which involves discarding the bottom half of G-SSNNs in terms\\nof the rank of their training performance among the population.\\n3.2 G-SSNNs\\nThere are many ways that the output of a symbolic program could be incorporated into the topology and parameter\\nvalues of a neural network. However, the use of learned abstractions forces a consideration: how do we make symbolic\\nprograms increasingly sophisticated without simply making their corresponding neural networks larger? In order that\\nour models don\\u2019t become increasingly costly to train as the symbolic programs that structure them utilize increasingly\\ncomplex modules, the number of parameters optimized with gradient-based learning should remain \\ufb01xed regardless of\\nhow the symbolic program \\ufb01xes their topology or values. This can be achieved by repurposing generic graph neural\\nnetworks (GNNs).\\nLets2Skbe relational structures whose elements are accompanied by symbolic feature vectors v2Nk. let\\ne:Nk\\u0002Rm!Rnbe an embedding function that maps each symbolic feature vector vand a vector x2Rmto\\nanother real-valued feature vector. Lastly, let graph_map :Sk\\u0002Rm!Gnbe the function that maps from relational\\nstructures and input vectors to graphs with the corresponding real-valued features given by e(v; x)for each v2s. We\\nmay then de\\ufb01ne a G-SSNN as a model m:Rm!Rqconsisting of a GNN mg:Gn!Rq, a relational structure s,\\nand an embedding function e, for which\\nm(x) =mg(graph_map (e; x; s )):\\nProvided that eis differentiable with respect to x, a G-SSNN may function as a generic neural module.\\nThere are many ways one could design the class of relational structures Skand the embedding function edepending on\\nthe degree and forms of control we want to grant symbolic programs over the parameter space and topology of the\\nnetwork. For the experiments conducted in this paper, I adopt a simple design: I utilize a subset of the class S2of\\nsingle-component undirected graphs with node and edge features. The natural numbers associated with each graph\\nelement correspond to a unique ID and a neighborhood rank. IDs are 0-indexed and re\\ufb02ect the order in which the\\nelements were created. I denote the number of unique IDs in a graph with d\\u0003= 1+max v2sv2. The neighborhood ranks\\nare natural numbers in a limited range which distinguish the edges of a particular neighborhood. The neighborhood rank\\nof a node is 0, while the neighborhood rank of an edge ranges from 1 to a \\ufb01xed maximum degree. Since neighborhood\\nranks function to distinguish edges, even if their endpoints are the same, this class of relational structures could be\\nunderstood as permitting multiple edges. I denote the number of distinct neighborhood ranks with r\\u0003= 1 + max v2sv2.\\nTo process these symbolic features, I utilize the embedding function\\ne(v; x) =tile_expand (xidx(v1):idx(v1+1)) +2d_pos (v1; v2)\\nwhere idx(w) = ( mmodl) +w\\u0001lforl=bm=d\\u0003c. The function tile_expand :Rt!Rnimplements the\\ndifferentiable operation of concatenating dn=tecopies of its input and truncating the result to a dimensionality of n,\\nand the function 2d_pos :N2!Rnproduces computes a vector b=2d_pos (d; r)such that2\\nbi=8\\n>><\\n>>:sin(reorder_1 (d; r)=10000imod ( n=4)) 1 < i\\u0014n=4\\ncos(reorder_1 (d; r)=10000imod ( n=4))n=4< i\\u00142n=4\\nsin(reorder_2 (d; r)=10000imod ( n=4)) 2n=4< i\\u00143n=4\\ncos(reorder_2 (d; r)=10000imod ( n=4)) 3n=4< i\\u0014n: cos(reorder_1 (d; r)=10000imod ( n=4))n=4< i\\u00142n=4\\nsin(reorder_2 (d; r)=10000imod ( n=4)) 2n=4< i\\u00143n=4\\ncos(reorder_2 (d; r)=10000imod ( n=4)) 3n=4< i\\u0014n:\\nThis embedding function constructs node and edge features by selecting values from a subset of the dimensions of x\\nand adding them to a bias vector, where both the subset of dimensions and the value of the vector are chosen based\\non the unique values of each graph element\\u2019s symbolic feature vector. This incentivizes greater modularity in mby\\nensuring that the parameters of mgfunction differently when applied to the same values depending on the dimensions\\n2The functions reorder_1 andreorder_2 are an implementation detail due to mistaken indexing which did not affect the validity\\nof the experiment. These functions are de\\ufb01ned as reorder_1 (d; r) = (d\\u0001r\\u0003+r) mod d\\u0003andreorder_2 (d; r) =b(d\\u0001r\\u0003+r)=d\\u0003c\\nwhere d\\u0003andr\\u0003are the maximal values of dandr. These functions may be replaced by f1(d; r) =dandf2(d; r) =rrespectively\\nto simplify the computation of 2d_pos .\\n3 Symbolic Synthesis of Neural Networks\\nofxin which they appear. Similarly, this behavior incentivizes antecedent neural modules to exhibit greater modularity\\nby ensuring that their parameters function differently based on the dimensions of xthat they in\\ufb02uence. These incentives\\nrely crucially on the bias term of e, which allows the symbolic program generating the relational structure sto in\\ufb02uence\\nthe parameter space of m. Without this term, the symbolic program would still in\\ufb02uence the topology of mthrough the\\nstructure of the graph convolutions applied by mgand the distribution of values from speci\\ufb01c dimensions of xinto\\nspeci\\ufb01c node and edge features under e.3However, this alone would create no obligation for different parameters of\\nmgor antecedent neural modules to behave differently with regard to the various dimensions of x. In general, while\\ntopology may allow or disallow particular parameters from in\\ufb02uencing each other, it cannot guarantee that they behave\\ndifferently with respect to the output for the same input. This can only be achieved by systematically varying the values\\ndifferent parameters may take.\\n3.3 Symbolic Programs\\nThe symbolic programs used to generate the relational structures described above are of type S2!S2and are all\\nevaluated by application to the same initial relational structure. The primitive operations of the initial DSL are such that\\nsymbolic programs may only make decisions based on the information contained in the relational structure. Since the\\ninitial relational structure is uninformative, symbolic programs therefore only have access to information about their\\nown semantics as it manifests in the relational structure they construct. More detailed information on the operations of\\nthe initial DSL is available in the documentation of the antireduce-graphs library.\\nDuring program search, symbolic programs are considered novel if they generate a relational structure that is unique\\nunder isomorphism when symbolic features are not considered. Isomorphism testing is performed with the VF2++\\nalgorithm [J\\u00fcttner and Madarasi, 2018].\\n4 Experiments\\n4.1 Setup & Hyperparameters\\nTo facilitate comparison, I utilize a base model architecture consisting of a transformer with a convolutional stem [Xiao\\net al., 2021]. I use subpixel convolution to perform downsampling from 128\\u0002128to8\\u00028[Shi et al., 2016], followed\\nby a pointwise convolution and three residual convolutional units. The patches are then \\ufb02attened and fed to a simpli\\ufb01ed\\nVision Transformer (ViT) [Beyer et al., 2022] with 2 transformer blocks. Each block has an MLP dimension of 512 and\\n4 128-dimensional attention heads. Global average pooling is then applied to produce an output of the target dimension\\nfor baseline models, or with the dimensionality of a graph element for experimental models in which the output is fed\\nto a G-SSNN unit.\\nTo construct G-SSNNs, I use the Graph Isomorphism Network with Edge features (GINE) of Hu et al. [2020] as\\nimplemented in the DGL library [Wang et al., 2019]. I use 512-dimensional node and edge representations, which\\nare passed through three GINE units parametrized by an MLP layer with identical input and output dimensionalities.\\nBetween GINE units, I apply 30% droput [Srivastava et al., 2014]. To produce the \\ufb01nal output, I apply average pooling\\nfollowed by another MLP layer which projects its input to the target output.\\nFor both baseline and experimental models I use a batch size of 8, train each model for 16 epochs with the Adam\\noptimizer [Kingma and Ba, 2015], and reduce the learning rate by a factor of 1\\/2 when a new low in the loss has not\\nbeen experienced in the \\ufb01rst 50 batches, or in the most recent 65 batches since the learning rate was last reduced.\\nAcross iterations of evolutionary selection, I retain a population of at most 50 and apply the rank-LTP-50 heuristic if\\nthe size of the population is greater than or equal to 25. I run distributional program search for a 15 second period. In the size of the population is greater than or equal to 25. I run distributional program search for a 15 second period. In\\nthe course of this run, I retain only the most likely 50\\u0000population_size novel programs; however, if programs are\\ndiscovered that are shorter than, but semantically equivalent to, programs currently in the population, these shorter\\nreformulations are kept and their equivalent\\u2019s are set aside. I do not consider programs that consist of more than 150\\nprimitives from the most updated DSL. Prior to program search, I reweight the parameters of the program distribution\\nsuch that the log likelihoods of the most and least likely primitives differ by no more than 0.5. I implemented this\\ntransformation such that it preserves the relative distance of each primitive\\u2019s log likelihood from the mean and therefore\\n3The latter of these two behaviors is included both because it makes greater use of the relational structure s, and also because it\\naverts a potential perverse incentive that could be created by distributing the entirety of xwithin each node and edge representation.\\nNamely, this has the consequence of making more copies of the information in xavailable if a graph has more elements, making\\nit less important to depend on the contribution of e\\u2019s bias term. This is especially undesirable since library learning acts as an\\nindiscriminate force for increased complexity in relational structure; thus, to an extent, this effect would be dif\\ufb01cult to avoid whether\\nor not this perverse incentive was directly appealing.\\n4 Symbolic Synthesis of Neural Networks\\n0 10 20 30 40 50\\nRank (Descending)0.650.700.750.800.850.900.951.00AccuracyBaseline\\nDataset\\ntraining\\nvalidation\\nvalidation\\n0.0 0.2 0.4 0.6 0.8 1.0\\nRank (Descending)0.600.650.700.750.800.850.900.95AccuracyExperimental (Iteration 1)\\nDataset\\ntraining\\nvalidation\\n0 1 2 3 4 5 6\\nRank (Descending)0.60.70.80.9AccuracyExperimental (Iteration 2)\\nDataset\\ntraining\\nvalidation\\n0 10 20 30 40\\nRank (Descending)0.40.50.60.70.80.91.0AccuracyExperimental (Iteration 3)\\nDataset\\ntraining\\nvalidation\\nvalidation\\n0 10 20 30 40 50\\nRank (Descending)0.50.60.70.80.9AccuracyExperimental (Iteration 4)\\nDataset\\ntraining\\nvalidation\\nvalidation\\n0 10 20 30 40 50\\nRank (Descending)0.40.50.60.70.80.9AccuracyExperimental (Iteration 5)\\nDataset\\ntraining\\nvalidation\\nvalidation\\nFigure 1: Training and validation set performances of populations of neural networks. Each model of a population is\\nrepresented in both trend lines of a plot at parallel points with respect to the x-axis. The baseline plot depicts the spread\\nof training set and validation set accuracies for 50 instantiations of a baseline model. The experimental plots depict\\nthe spread of training set and validation set accuracies for populations of G-SSNNs at various stages of evolution. At\\nstage n, the population has undergone nrounds of heuristic selection\\/culling and program synthesis. At iterations 1-5,\\nthe populations were of sizes 2, 7, 44, 50 and 50, respectively. Dotted lines represent unsmoothed data and solid lines\\nrepresent smoothed data with a window size of 6. I present validation set accuracies in both smooth and unsmoothed\\nform except where the population of models seem too small to meaningfully bene\\ufb01t from smoothing ( n <15).\\ndoes not alter the DSL\\u2019s total probability mass.4During library learning, I perform up to three rounds of compression,\\neach producing a primitive of arity 2 or less. All other parameters of the STITCH compression routine were allowed to\\ntake their default values.5\\n4.2 Dataset\\nThe RA VEN dataset [Zhang et al., 2019] is an arti\\ufb01cial dataset consisting of visual puzzles known as Raven\\u2019s Progressive\\nMatrices (RPMs). Originated as a psychometric test, RPMs have since been extensively studied as tests of Abstract\\nVisual Reasoning (A VR) in both symbolic and neural AI systems. An RPM is a 9\\u00029matrix of panels depicting simple\\narrangements of polygons. Within each row, panels exhibit one of a handful of consistent patterns with respect to each\\nattribute\\u2013size, shape, color, angle, position, and multiplicity\\u2013of their polygons. Due to the independence between these\\ncharacteristics, their limited sets of values, and the segregated placement of the polygons, RPMs are naturally local and\\ndiscrete high-dimensional representations.\\nClassically, the \\ufb01nal panel of the last sequence (bottom right) is omitted and some number of candidate panels are\\nprovided, only one of which contains polygons which obey the same patterns exhibited in the \\ufb01rst two rows for all\\nattributes. The goal of the solver is then to distinguish the correct panel from the pool of candidates. Odouard and\\nMitchell [2022] have demonstrated that several recent high-performing neural models from the RA VEN literature\\nperform poorly when more exhaustively evaluated on their understanding of the various instantiations of an individual\\npattern. This inspired the Prediction of Constant Color Pattern (PCCP) task, a binary prediction task in which a model\\nexamines a complete and correct RPM and predicts whether it exhibits constancy of polygon color across rows. The\\ntwo classes in this task\\u2013constant and non-constant\\u2013were relatively evenly-distributed among generated data without\\n4This and other domain-general aspects of program search are implemented in the antireduce library.\\n5Explanation of these settings may be found in the documentation of the stitch_core Python package.\\n5 Symbolic Synthesis of Neural Networks\\nadditional constraints on pattern instantiation or attribute values.6As with any binary RPM prediction task based on the\\npresence of a single pattern-attribute pairing, indiscernible spurious correlation with the presence or absence of other\\npattern-attribute pairings is unlikely, even with small data.\\nThe RA VEN-like data used in my experiments were generated with a reimplementation of the original RA VEN data\\ngeneration code extended to support creation of RPMs at lower resolutions than were originally accommodated. I use a\\ntraining dataset size of 128 RPMs and a validation set of 1200 RPMs. These sizes were well suited to demonstrating the\\ngeneralization performance of the baseline and experimental models and were also helpful for practical performance\\nwhen serially training populations of G-SSNNs. In general, the G-SSNNs of a population are not interdependent and\\nmay be trained in parallel. Serial training was instead a logistical limitation of the current study.\\n4.3 Results\\nFigure 1 presents comparative results for populations of baseline and experimental models. Across these plots, we see\\nthat validation set accuracy tends to be especially high for models whose training set accuracy is in the range of 80-90%.\\nAmongst baseline models less than 20% fall in this range, whereas after iteration 1, 30% or more of the population\\nof experimental models fall in this range. The populations of experimental models are centered on this \\u201csweet spot\\\"\\nof training performances, and this portion of the distribution gets \\ufb02atter and broader across iterations 3-5. These data\\ndemonstrate that the use of G-SSNNs reduces the performance gap between seen and unseen data for at least one form\\nof local and discrete high-dimensional representation. Moreover, the models for which this results in peak validation set\\naccuracies are signi\\ufb01cantly more numerous and fall in a reliable part of the distribution of validation set accuracies.\\nThis is a markedly different pattern of improved generalization which is easier to anticipate and exploit.\\nThe distributions of validation set accuracies illustrated in these data suggest that it is actually models with a training\\nset performance of a more intermediate rank that are most likely to exhibit peak generalization. In future work and\\napplications of G-SSNNs, it may therefore be bene\\ufb01cial to explore the space of Intermediate Training Performance\\n(ITP) heuristics for selecting from the population. This pattern also suggests that in successful G-SSNNs, symbolic\\nprograms offer a form of regularization. It is possible for a relational structure to permit a G-SSNN to over\\ufb01t, and\\nalso for it to prevent a G-SSNN from \\ufb01tting very well at all; the optimal relational structures would then be those that\\ndo not inhabit either extreme, making performance on seen and unseen data more similar without lowering training\\nperformance more than necessary.\\n5 Discussion\\nAcross high-dimensional media, there are tasks that involve extracting information that manifests in local and discrete\\nfeatures. In adapting to such tasks, modular models may exhibit an advantage as they are more likely to specialize in\\ndetecting a small number of patterns. In the case of G-SSNNs, we encourage modularity by requiring that models\\ncontextualize data-independent symbols in a task-speci\\ufb01c manner. This has the auxiliary bene\\ufb01t of contributing\\ntransferable knowledge about how symbols should be composed. In a world where symbolically-informed, modular\\nneural models are widely used, the community of model developers would bene\\ufb01t from the sharing of novel abstractions\\nin a way that mirrors present-day communal processes of software development. However, unlike current practice,\\nthese abstractions could be \\ufb02exibly adapted to the circumstances in which they are applied in ways that cannot always\\nbe concisely expressed with symbols. In advancing this possibility, I believe that G-SSNNs capture much of what be concisely expressed with symbols. In advancing this possibility, I believe that G-SSNNs capture much of what\\nis intuitively desirable about the promise of common sense: the ability to quickly adapt and elaborate on collective\\nknowledge in a wide range of circumstances.\\nIn future work, I intend to investigate other G-SSNN designs incorporating more complex embedding functions and\\nclasses of relational structures, as well as more complex classes of symbolic programs. Alternative embedding functions\\ncould draw parameter values from more interesting sources, such as pretrained models, and could embed these values\\ninto more interesting topological structures such as convolutions or attention mechanisms. These choices could be\\nsupported by more complex logic on the part of symbolic programs, which could be enhanced with access to richer\\nexternal databases of symbolic information.\\n6Some constraints on pattern instantiation and attribute values were necessary to ensure legibility at the desired resolution of\\n128\\u0002128. Speci\\ufb01cally, polygon size was kept large and not allowed to vary and oblique angles of polygon rotation were disallowed.\\nNeither of these constraints contribute to determining the color constancy of an RPM.\\n6 Symbolic Synthesis of Neural Networks\\n6 Conclusion\\nI have presented G-SSNNs, a class of neural modules whose topology and parameter space are in\\ufb02uenced by the output\\nof a modular symbolic program. I presented the results of applying neural models containing G-SSNN modules to a\\nbinary prediction task over Raven\\u2019s Progressive Matrices (RPMs), psychometric tests of Abstract Visual Reasoning\\n(A VR) which contain local and discrete features. These results demonstrate that G-SSNNs produce reliable patterns of\\nimproved generalization that are not exhibited by baseline models. In addition to their potential to improve performance\\non tasks involving local and discrete features across high-dimensional media, G-SSNNs offer a route towards communal\\ndevelopment of compact and novel abstractions that can be \\ufb02exibly applied to numerous tasks. In future work, I intend\\nto investigate more complex G-SSNN designs and more complex classes of associated symbolic programs.\\nReferences\\nCharles Jin, Phitchaya Mangpo Phothilimthana, and Sudip Roy. Neural architecture search using property guided\\nsynthesis. Proc. ACM Program. Lang. , 6(OOPSLA2):1150\\u20131179, 2022. doi:10.1145\\/3563329. URL https:\\n\\/\\/doi.org\\/10.1145\\/3563329 .\\nEsteban Real, Chen Liang, David R. So, and Quoc V . Le. Automl-zero: Evolving machine learning algorithms from\\nscratch. CoRR , abs\\/2003.03384, 2020. URL https:\\/\\/arxiv.org\\/abs\\/2003.03384 .\\nMohamed S. Abdelfattah, Abhinav Mehrotra, Lukasz Dudziak, and Nicholas D. Lane. Zero-cost proxies for lightweight\\nNAS. CoRR , abs\\/2101.08134, 2021. URL https:\\/\\/arxiv.org\\/abs\\/2101.08134 .\\nTom\\u00e1s Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Ef\\ufb01cient estimation of word representations in vector\\nspace. In Yoshua Bengio and Yann LeCun, editors, 1st International Conference on Learning Representations, ICLR\\n2013, Scottsdale, Arizona, USA, May 2-4, 2013, Workshop Track Proceedings , 2013. URL http:\\/\\/arxiv.org\\/\\nabs\\/1301.3781 .\\nAshish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia\\nPolosukhin. Attention is all you need. CoRR , abs\\/1706.03762, 2017. URL http:\\/\\/arxiv.org\\/abs\\/1706.03762 .\\nKevin Ellis, Catherine Wong, Maxwell I. Nye, Mathias Sabl\\u00e9-Meyer, Lucas Morales, Luke B. Hewitt, Luc Cary,\\nArmando Solar-Lezama, and Joshua B. Tenenbaum. Dreamcoder: bootstrapping inductive program synthesis with\\nwake-sleep library learning. In Stephen N. Freund and Eran Yahav, editors, PLDI \\u201921: 42nd ACM SIGPLAN\\nInternational Conference on Programming Language Design and Implementation, Virtual Event, Canada, June\\n20-25, 2021 , pages 835\\u2013850. ACM, 2021. doi:10.1145\\/3453483.3454080. URL https:\\/\\/doi.org\\/10.1145\\/\\n3453483.3454080 .\\nMatthew Bowers, Theo X. Olausson, Lionel Wong, Gabriel Grand, Joshua B. Tenenbaum, Kevin Ellis, and Armando\\nSolar-Lezama. Top-down synthesis for library learning. Proc. ACM Program. Lang. , 7(POPL):1182\\u20131213, 2023.\\ndoi:10.1145\\/3571234. URL https:\\/\\/doi.org\\/10.1145\\/3571234 .\\nTh\\u00e9o Matricon, Nathana\\u00ebl Fijalkow, Guillaume Lagarde, and Kevin Ellis. Deepsynth: Scaling neural program\\nsynthesis with distribution-based search. J. Open Source Softw. , 7(78):4151, 2022. doi:10.21105\\/joss.04151. URL\\nhttps:\\/\\/doi.org\\/10.21105\\/joss.04151 .\\nAlp\\u00e1r J\\u00fcttner and P\\u00e9ter Madarasi. VF2++ - an improved subgraph isomorphism algorithm. Discret. Appl. Math. , 242:\\n69\\u201381, 2018. doi:10.1016\\/j.dam.2018.02.018. URL https:\\/\\/doi.org\\/10.1016\\/j.dam.2018.02.018 .\\nTete Xiao, Mannat Singh, Eric Mintun, Trevor Darrell, Piotr Doll\\u00e1r, and Ross B. Girshick. Early con-\\nvolutions help transformers see better. In Marc\\u2019Aurelio Ranzato, Alina Beygelzimer, Yann N. Dauphin,\\nPercy Liang, and Jennifer Wortman Vaughan, editors, Advances in Neural Information Processing Systems\\n34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14,\\n2021, virtual , pages 30392\\u201330400, 2021. URL https:\\/\\/proceedings.neurips.cc\\/paper\\/2021\\/hash\\/\\nff1418e8cc993fe8abcfe3ce2003e5c5-Abstract.html . 2021, virtual , pages 30392\\u201330400, 2021. URL https:\\/\\/proceedings.neurips.cc\\/paper\\/2021\\/hash\\/\\nff1418e8cc993fe8abcfe3ce2003e5c5-Abstract.html .\\nWenzhe Shi, Jose Caballero, Ferenc Huszar, Johannes Totz, Andrew P. Aitken, Rob Bishop, Daniel Rueckert, and\\nZehan Wang. Real-time single image and video super-resolution using an ef\\ufb01cient sub-pixel convolutional neural\\nnetwork. In 2016 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2016, Las Vegas, NV ,\\nUSA, June 27-30, 2016 , pages 1874\\u20131883. IEEE Computer Society, 2016. doi:10.1109\\/CVPR.2016.207. URL\\nhttps:\\/\\/doi.org\\/10.1109\\/CVPR.2016.207 .\\nLucas Beyer, Xiaohua Zhai, and Alexander Kolesnikov. Better plain vit baselines for imagenet-1k. CoRR ,\\nabs\\/2205.01580, 2022. doi:10.48550\\/arXiv.2205.01580. URL https:\\/\\/doi.org\\/10.48550\\/arXiv.2205.\\n01580 .\\n7 Symbolic Synthesis of Neural Networks\\nWeihua Hu, Bowen Liu, Joseph Gomes, Marinka Zitnik, Percy Liang, Vijay S. Pande, and Jure Leskovec. Strategies\\nfor pre-training graph neural networks. In 8th International Conference on Learning Representations, ICLR 2020,\\nAddis Ababa, Ethiopia, April 26-30, 2020 . OpenReview.net, 2020. URL https:\\/\\/openreview.net\\/forum?id=\\nHJlWWJSFDH .\\nMinjie Wang, Lingfan Yu, Da Zheng, Quan Gan, Yu Gai, Zihao Ye, Mufei Li, Jinjing Zhou, Qi Huang, Chao Ma,\\nZiyue Huang, Qipeng Guo, Hao Zhang, Haibin Lin, Junbo Zhao, Jinyang Li, Alexander J. Smola, and Zheng Zhang.\\nDeep graph library: Towards ef\\ufb01cient and scalable deep learning on graphs. CoRR , abs\\/1909.01315, 2019. URL\\nhttp:\\/\\/arxiv.org\\/abs\\/1909.01315 .\\nNitish Srivastava, Geoffrey E. Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout:\\na simple way to prevent neural networks from over\\ufb01tting. J. Mach. Learn. Res. , 15(1):1929\\u20131958, 2014.\\ndoi:10.5555\\/2627435.2670313. URL https:\\/\\/dl.acm.org\\/doi\\/10.5555\\/2627435.2670313 .\\nDiederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun,\\neditors, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015,\\nConference Track Proceedings , 2015. URL http:\\/\\/arxiv.org\\/abs\\/1412.6980 .\\nChi Zhang, Feng Gao, Baoxiong Jia, Yixin Zhu, and Song-Chun Zhu. RA VEN: A dataset for relational and\\nanalogical visual reasoning. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2019,\\nLong Beach, CA, USA, June 16-20, 2019 , pages 5317\\u20135327. Computer Vision Foundation \\/ IEEE, 2019.\\ndoi:10.1109\\/CVPR.2019.00546. URL http:\\/\\/openaccess.thecvf.com\\/content_CVPR_2019\\/html\\/Zhang_\\nRAVEN_A_Dataset_for_Relational_and_Analogical_Visual_REasoNing_CVPR_2019_paper.html .\\nVictor Vikram Odouard and Melanie Mitchell. Evaluating understanding on conceptual abstraction benchmarks. In\\nJos\\u00e9 Hern\\u00e1ndez-Orallo, Lucy Cheke, Joshua Tenebaum, Tomer D. Ullman, Fernando Mart\\u00ednez-Plumed, Danaja\\nRutar, John Burden, Ryan Burnell, and Wout Schellaert, editors, Proceedings of the Workshop on AI Evaluation\\nBeyond Metrics co-located with the 31st International Joint Conference on Arti\\ufb01cial Intelligence (IJCAI-ECAI\\n2022), Vienna, Austria, July 25th, 2022 , volume 3169 of CEUR Workshop Proceedings . CEUR-WS.org, 2022. URL\\nhttp:\\/\\/ceur-ws.org\\/Vol-3169\\/paper8.pdf .\\n8\",\"41\":\" MAP-Elites with Descriptor-Conditioned Gradients and\\nArchive Distillation into a Single Policy\\nMaxence Faldor\\nm.faldor22@imperial.ac.uk\\nImperial College London\\nLondon, United KingdomF\\u00e9lix Chalumeau\\nf.chalumeau@instadeep.com\\nInstaDeep\\nCape Town, South Africa\\nManon Flageat\\nmanon.flageat18@imperial.ac.uk\\nImperial College London\\nLondon, United KingdomAntoine Cully\\na.cully@imperial.ac.uk\\nImperial College London\\nLondon, United Kingdom\\nABSTRACT\\nQuality-Diversity algorithms, such as MAP-Elites, are a branch of\\nEvolutionary Computation generating collections of diverse and\\nhigh-performing solutions, that have been successfully applied\\nto a variety of domains and particularly in evolutionary robotics.\\nHowever, MAP-Elites performs a divergent search based on random\\nmutations originating from Genetic Algorithms, and thus, is limited\\nto evolving populations of low-dimensional solutions. PGA-MAP-\\nElites overcomes this limitation by integrating a gradient-based\\nvariation operator inspired by Deep Reinforcement Learning which\\nenables the evolution of large neural networks. Although high-\\nperforming in many environments, PGA-MAP-Elites fails on several\\ntasks where the convergent search of the gradient-based operator\\ndoes not direct mutations towards archive-improving solutions. In\\nthis work, we present two contributions: (1) we enhance the Policy\\nGradient variation operator with a descriptor-conditioned critic\\nthat improves the archive across the entire descriptor space, (2) we\\nexploit the actor-critic training to learn a descriptor-conditioned\\npolicy at no additional cost, distilling the knowledge of the archive\\ninto one single versatile policy that can execute the entire range\\nof behaviors contained in the archive. Our algorithm, DCG-MAP-\\nElites improves the QD score over PGA-MAP-Elites by 82% on\\naverage, on a set of challenging locomotion tasks.\\n1 INTRODUCTION\\nA fascinating aspect of evolution is its ability to generate a large\\nvariety of different species, each being adapted to their ecological\\nniche. Inspired by this idea, Quality-Diversity (QD) optimization is\\na family of evolutionary algorithms that aims to generate a set of\\nboth diverse and high-performing solutions to a problem [ 3,7,32].\\nContrary to traditional optimization methods that return a single\\nhigh-performing solution, the goal of QD algorithms is to illuminate\\na search space of interest called descriptor space [27]. Producing\\na large collection of diverse and effective solutions enables to get\\nmultiple alternatives to solve a single problem which is useful\\nin robotics to improve robustness, recover from damage [ 6] or\\nreduce the reality gap [ 4]. Furthermore, conventional optimization\\nmethods are prone to get stuck in local optima whereas keeping\\ndiverse solutions to a problem can help to find stepping stones that\\nlead to globally better solutions [ 27,28]. Another benefit of diversity\\nFigure 1: DCG-MAP-Elites performs a standard MAP-Elites\\nloop of selection, variation, evaluation and addition. Two\\ncomplementary variation operators are applied: (1) a stan-\\ndard Genetic Algorithm (GA) variation operator for explo-\\nration, (2) a Descriptor-Conditioned Policy Gradient (PG)\\nvariation operator for fitness improvement. Concurrently to\\nthe critic\\u2019s training, the knowledge of the archive is distilled\\nin the descriptor-conditioned actor as by-product.\\nsearch is efficient exploration in problems where the reward signal\\nis sparse or deceptive [2, 8, 31].\\nMAP-Elites [ 27] is a conceptually simple but effective QD opti-\\nmization algorithm that has shown competitive results in a variety\\nof applications, to generate large collections of diverse skills. How-\\never, MAP-Elites relies on random variations that can cause slow\\nconvergence in large search space [ 5,28,31], making it inadequate\\nto evolve neural networks with a large number of parameters.\\nIn contrast, Deep Reinforcement Learning (DRL) [ 25,26] algo-\\nrithms combine reinforcement learning with the directed search to evolve neural networks with a large number of parameters.\\nIn contrast, Deep Reinforcement Learning (DRL) [ 25,26] algo-\\nrithms combine reinforcement learning with the directed search\\npower of gradient-based methods in order to learn a single solu-\\ntion. DRL can surpass human performance at video games [ 40],\\nbeat world champions in board games [ 35] and control complex\\nrobots in continuous action spaces [ 17], which is a long-standing\\nchallenge in artificial intelligence. Policy Gradient methods have\\nshown state-of-the-art results to learn large neural network policies\\nwith thousands of parameters in high-dimensional state space and\\ncontinuous action space [18, 24, 36].\\nPGA-MAP-Elites [ 28] is an extension of MAP-Elites that inte-\\ngrates the sample efficiency of DRL using the TD3 algorithm [ 15].arXiv:2303.03832v1  [cs.NE]  7 Mar 2023 This algorithm uses a Policy Gradient (PG) variation operator for\\nefficient fitness improvement, coupled with the usual Genetic Algo-\\nrithm (GA) variation operator. The PG variation operator leverages\\ngradients derived from DRL to improve fitness and drive mutations\\ntowards the global optimum and is supported by the divergent\\nsearch of the GA variation operator for both exploration and opti-\\nmization [ 10]. Other recent works have also introduced methods to\\ncombine the strength of QD algorithms with reinforcement learn-\\ning [31, 38] on complex robotics tasks.\\nPGA-MAP-Elites achieves state-of-the-art performances in most\\nof the environments considered so far in the literature [ 28,31,38].\\nHowever, the PG variation operator becomes ineffective in tasks\\nwhere the global optimum is in an area of the search space that is\\nnot likely to produce offspring that are added to the archive. For\\nexample, consider a locomotion task where the fitness is the oppo-\\nsite of the energy consumption and the descriptor is defined as the\\nfinal position of the robot. The global optimum for the fitness is the\\nsolution that does not move in order to minimize energy consump-\\ntion. Thus, the PG variation operator will encourage solutions to\\nstay motionless, collapsing their descriptors to a single point, the\\ndescriptor of the global optimum. Consequently, the PG variation\\noperator generates offspring that are discarded and no interesting\\nstepping stone is found, thereby hindering diversity.\\nIn this work, we introduce Descriptor-Conditioned Gradients\\nMAP-Elites (DCG-MAP-Elites) that builds upon PGA-MAP-Elites\\nalgorithm by enhancing the PG variation operator with a descriptor-\\nconditioned critic that provides gradients depending on a target de-\\nscriptor. The descriptor-conditioned critic takes as input a state and\\na descriptor to evaluate actions. With such a descriptor-conditioned\\ncritic, the PG variation operator can mutate solutions to produce\\noffspring with higher fitness while targeting a desired descriptor,\\nthereby avoiding to collapse the descriptor to a single point.\\nFurthermore, TD3 which is the DRL algorithm used by the PG\\nvariation operator, requires to train an actor and a critic in parallel.\\nWe take advantage of this intertwined actor-critic training to make\\nthe actor descriptor-conditioned as well, allowing it to take actions\\nbased on the current state and on an input descriptor we want to\\nachieve. Thus, instead of taking actions that maximize the fitness\\nglobally, the actor now takes actions that maximize the fitness\\nwhile achieving a desired descriptor. At the end of training, we\\ncan condition the actor on a desired descriptor to execute a policy\\nthat takes actions that achieve the desired descriptor. On half of\\nthe tasks, we observe that the descriptor-conditioned actor can\\nachieve the entire range of descriptors contained in the archive\\nwith a similar QD-score, negating the burden of dealing with a\\ncollection of thousands of solutions.\\nIn summary, we present two contributions: (1) we enhance the\\nPG variation operator with a descriptor-conditioned critic, (2) we\\ndistill the knowledge of the archive into one single versatile policy\\nat no additional cost. We compare our algorithm to four state-of-the-\\nart QD algorithms on four high-dimensional robotics locomotion\\ntasks. The results demonstrate that DCG-MAP-Elites has a QD-\\nscore 82% higher than PGA-MAP-Elites on average.2 BACKGROUND\\n2.1 Problem Statement\\nWe consider an agent sequentially interacting with an environment\\nat discrete time steps \\ud835\\udc61for an episode of length \\ud835\\udc47. At each time step\\n\\ud835\\udc61, the agent observes a state \\ud835\\udc60\\ud835\\udc61, takes an action \\ud835\\udc4e\\ud835\\udc61and receives a\\nscalar reward \\ud835\\udc5f\\ud835\\udc61. We model it as a Markov Decision Process (MDP)\\nwhich comprises a state spaceS, a continuous action spaceA, a\\nstationary transition dynamics distribution \\ud835\\udc5d(\\ud835\\udc60\\ud835\\udc61+1|\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61)and a\\nreward function \\ud835\\udc5f:S\\u00d7A\\u2192 R. In this work, a policy (also called\\nsolution ) is a deterministic neural network parameterized by \\ud835\\udf19\\u2208\\u03a6, stationary transition dynamics distribution \\ud835\\udc5d(\\ud835\\udc60\\ud835\\udc61+1|\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61)and a\\nreward function \\ud835\\udc5f:S\\u00d7A\\u2192 R. In this work, a policy (also called\\nsolution ) is a deterministic neural network parameterized by \\ud835\\udf19\\u2208\\u03a6,\\nand denoted \\ud835\\udf0b\\ud835\\udf19:S\\u2192A . The agent uses its policy to select actions\\nand interact with the environment to give a trajectory of states,\\nactions and rewards. The fitness of a solution is given by \\ud835\\udc39:\\u03a6\\u2192R,\\ndefined as the expected discounted return E\\ud835\\udf0b\\ud835\\udf19\\u0002\\u00cd\\ud835\\udc47\\u22121\\n\\ud835\\udc61=0\\ud835\\udefe\\ud835\\udc61\\ud835\\udc5f\\ud835\\udc61\\u0003\\n.\\nThe objective of QD algorithms in this MDP setting is to find\\nthe highest-fitness solutions in each point of the descriptor space\\nD. The descriptor function \\ud835\\udc37:\\u03a6\\u2192D is generally defined by the\\nuser and characterize solutions in a meaningful way for the type of\\ndiversity desired. With this notation, our objective is to evolve a\\npopulation of solutions that are both high-performing with respect\\nto\\ud835\\udc39and diverse with respect to \\ud835\\udc37.\\n2.2 MAP-Elites\\nMulti-dimensional Archive of Phenotypic Elites (MAP-Elites) [ 27]\\nis a simple yet effective QD algorithm that discretizes the descriptor\\nspaceDinto a multi-dimensional grid of cells called archive Xand\\nsearches for the best solution in each cell, see Alg. 4. The goal of the\\nalgorithm is to return an archive that is filled as much as possible\\nwith high-fitness solutions. MAP-Elites starts by generating ran-\\ndom solutions and adding them to the archive. The algorithm then\\nrepeats the following steps until a budget of \\ud835\\udc3csolutions have been\\nevaluated: (1) a batch of solutions from the archive are uniformly\\nselected and modified through mutations and\\/or crossovers to pro-\\nduce offspring, (2) the fitnesses and descriptors of the offspring are\\nevaluated, and each offspring is placed in its corresponding cell if\\nand only if the cell is empty or if the offspring has a better fitness\\nthan the current solution in that cell, in which case the current\\nsolution is replaced. As most evolutionary methods, MAP-Elites\\nrelies on undirected updates that are agnostic to the fitness objec-\\ntive. With a Genetic Algorithm (GA) variation operator, MAP-Elites\\nperforms a divergent search that may cause slow convergence in\\nhigh-dimensional problems due to a lack of directed search power,\\nand thus, is performing best on low-dimensional search space [ 28].\\n2.3 Deep Reinforcement Learning\\nDeep Reinforcement Learning (DRL) [ 25,26] combines the rein-\\nforcement learning framework with the function approximation\\ncapabilities of deep neural networks to represent policies and value\\nfunctions in high-dimensional state and action spaces. In opposition\\nto black-box optimization methods like evolutionary algorithms,\\nDRL leverages the structure of the MDP in the form of the Bellman\\nequation to achieve better sample efficiency. The objective is to\\nfind an optimal policy \\ud835\\udf0b\\ud835\\udf19, which maximizes the expected return or\\nfitness\\ud835\\udc39(\\ud835\\udf0b\\ud835\\udf19). In reinforcement learning, many approaches try to\\nestimate the action-value function \\ud835\\udc44\\ud835\\udf0b(\\ud835\\udc60,\\ud835\\udc4e)=E\\ud835\\udf0b\\u0002\\u00cd\\ud835\\udc47\\u22121\\n\\ud835\\udc61=0\\ud835\\udefe\\ud835\\udc61\\ud835\\udc5f\\ud835\\udc61|\\ud835\\udc60,\\ud835\\udc4e\\u0003\\n2 defined as the expected discounted return starting from state \\ud835\\udc60,\\ntaking action \\ud835\\udc4eand thereafter following policy \\ud835\\udf0b.\\nThe Twin Delayed Deep Deterministic Policy Gradient (TD3)\\nalgorithm [ 15] is an actor-critic, off-policy reinforcement learn-\\ning method that achieves state-of-the-art results in environments\\nwith large and continuous action space. TD3 indirectly learns a\\npolicy\\ud835\\udf0b\\ud835\\udf19via maximization of the action-value function \\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e).\\nThe approach is closely connected to Q-learning [ 15] and tries\\nto approximate the optimal action-value function \\ud835\\udc44\\u2217(\\ud835\\udc60,\\ud835\\udc4e)in or-\\nder to find the optimal action \\ud835\\udc4e\\u2217(\\ud835\\udc60)=arg max\\ud835\\udc4e\\ud835\\udc44\\u2217(\\ud835\\udc60,\\ud835\\udc4e). How-\\never, computing the maximum over action in max\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e)is in-\\ntractable in continuous action space, so it is approximated using\\nmax\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e)=\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60)). In TD3, the policy \\ud835\\udf0b\\ud835\\udf19takes actions\\nin the environment and the transitions are stored in a replay buffer.\\nThe collected experience is then used to train a pair of critics \\ud835\\udc44\\ud835\\udf031,\\n\\ud835\\udc44\\ud835\\udf032using temporal difference and target networks \\ud835\\udc44\\ud835\\udf031\\u2032,\\ud835\\udc44\\ud835\\udf032\\u2032are\\nupdated to slowly track the main networks. Both critics use a single\\nregression target \\ud835\\udc66, calculated using whichever of the two critics\\ngives a smaller target value and using target policy smoothing by\\nsampling a noise \\ud835\\udf16\\u223cclip(N(0,\\ud835\\udf0e),\\u2212\\ud835\\udc50,\\ud835\\udc50):\\n\\ud835\\udc66=\\ud835\\udc5f(\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61)+\\ud835\\udefemin\\n\\ud835\\udc56=1,2\\ud835\\udc44\\ud835\\udf03\\ud835\\udc56\\u2032\\u0010\\n\\ud835\\udc60\\ud835\\udc61+1,\\ud835\\udf0b\\ud835\\udf19\\u2032(\\ud835\\udc60\\ud835\\udc61+1)+\\ud835\\udf16\\u0011\\n(1)\\nBoth critics are learned by regression to this target and the policy\\nis learned with a delay, only updated every \\u0394iterations simply by\\nmaximizing \\ud835\\udc44\\ud835\\udf031withmax\\ud835\\udf19E\\u0002\\n\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60))\\u0003\\n. The actor is updated\\nusing the deterministic policy gradient:\\n\\u2207\\ud835\\udf19\\ud835\\udc3d(\\ud835\\udf19)=Eh\\n\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60)\\u2207\\ud835\\udf19\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60)i\\n(2)\\n2.4 PGA-MAP-Elites\\nPolicy Gradient Assisted MAP-Elites (PGA-MAP-Elites) [ 28] is an\\nextension of MAP-Elites that is designed to evolve deep neural\\nnetworks by combining the directed search power and sample effi-\\nciency of DRL methods with the exploration capabilities of genetic\\nalgorithms, see Alg. 5. The algorithm follows the usual MAP-Elites\\nloop of selection, variation, evaluation and addition for a budget of\\n\\ud835\\udc3citerations, but uses two parallel variation operators: half of the\\noffspring are generated using a standard Genetic Algorithm (GA)\\nvariation operator and half of the offspring are generated using\\na Policy Gradient (PG) variation operator. During each iteration\\nof the loop, PGA-MAP-Elites stores the transitions from offspring\\nevaluation in a replay buffer Band uses it to train a pair of critics\\nbased on the TD3 algorithm, described in Alg. 6. The trained critic\\nis then used in the PG variation operator to update the selected\\nsolutions from the archive for \\ud835\\udc5agradient steps to select actions\\nthat maximize the approximated action-value function, as described\\nin Alg. 7. At each iteration, the critics are trained for \\ud835\\udc5bsteps of\\ngradients descents towards the target described in Eq. 1 averaged\\nover\\ud835\\udc41transitions of experience sampled uniformly from the replay\\nbufferB. The actor (also named greedy actor [ 28]) learns with a\\ndelay \\u0394via maximization of the critic according to Eq. 2.\\n3 RELATED WORK\\n3.1 Scaling QD to Neuroevolution\\nThe challenge of evolving diverse solutions in a high-dimensional\\nsearch space has been an active research subject over the recentyears. ME-ES [ 5] scales MAP-Elites to high-dimensional solutions\\nparameterized by large neural networks. This algorithm leverages\\nEvolution Strategies to perform a directed search that is more effi-\\ncient than random mutations used in Genetic Algorithms. Fitness\\ngradients are estimated locally from many perturbed versions of\\nthe parent solution to generate a new one. The population tends\\ntowards regions of the parameter space with higher fitness but it\\nrequires to sample and evaluate a large number of solutions, mak-\\ning it particularly data inefficient. In order to use the time step\\nlevel information and hence improve data efficiency, methods that\\ncombine MAP-Elites with Reinforcement Learning [ 28,30,31,38]\\nhave emerged and proved to efficiently evolve populations of high- level information and hence improve data efficiency, methods that\\ncombine MAP-Elites with Reinforcement Learning [ 28,30,31,38]\\nhave emerged and proved to efficiently evolve populations of high-\\nperforming and diverse neural network for complex tasks. PGA-\\nMAP-Elites [ 28] uses policy gradients for part of its mutations, see\\nsection 2.4 for details. CMA-MEGA [ 38] estimates descriptor gradi-\\nents with Evolution Strategies and combines the fitness gradient\\nand the descriptor gradients with a CMA-ES mechanism [ 12,19].\\nQD-PG [ 31] introduces a diversity reward based on the novelty of\\nthe states visited and derives a policy gradient for the maximiza-\\ntion of those diversity rewards which helps exploration in settings\\nwhere the reward is sparse or deceptive. PBT-MAP-Elites [ 30] mixes\\nMAP-Elites with a population based training process [ 21] to opti-\\nmize hyper-parameters of diverse RL agents. Interestingly, recent\\nwork [ 37] scales the algorithm CMA-MAE [ 13] to high-dimensional\\npolicies on robotics tasks with pure Evolution Strategies while\\nshowing comparable data efficiency to QD-RL approaches. It shows\\ncompetitiveness but is still outperformed by PGA-MAP-Elites.\\n3.2 Conditioning the critic\\nNone of the above methods takes a descriptor into account when\\nderiving policy gradients used to mutate solutions. In other words,\\nthey do not use descriptor-conditioned policies nor descriptor-\\nconditioned critics as our method DCG-MAP-Elites does. The con-\\ncept of descriptor-conditioned critic is related to the concept of\\nUniversal Value Function Approximators [ 33] and the most related\\nfield to Quality-Diversity that uses it is Skill Discovery Reinforce-\\nment Learning [ 1]. In VIC, DIAYN, DADS, SMERL [ 9,16,23,34],\\nconditioned actor-critic are used but the condition is a sampled prior\\nand does not correspond to a real posterior like in DCG-MAP-Elites.\\nFurthermore, those methods use diversity at the step level and not\\nexplicitly at the trajectory level like ours. Finally, they do not use an\\narchive to store their population, resulting in much smaller sets of\\nfinal policies. Ultimately, it has been shown that behaviors evolved\\nby QD methods are competitive with skills learned by this family of\\nmethods [ 1], in regards to their use for adaptation and hierarchical\\nlearning.\\n3.3 Archive distillation\\nDistilling the knowledge of an archive into a single neural model\\nis an alluring process that reduces the number of parameters out-\\nputted by the algorithm and enables generalization and interpo-\\nlation\\/extrapolation. Although distillation is usually referring to\\npolicy distillation \\u2014 learning the observation\\/action mapping from\\na teacher policy \\u2014 we present archive distillation as a general term\\nreferring to any kind of knowledge transfer from an archive to\\n3 another model, should it be the policies, transitions experienced in\\nthe environment, full trajectories or discovered descriptors.\\nTo the best of our knowledge, two QD-related works use the\\nconcept of archive distillation. Go-Explore [ 8] stores an archive of\\nreached states and trains a goal-conditioned policy to reproduce the\\ntrajectory of the policy that reached that state. Another interesting\\napproach to archive distillation is to learn a generative policy net-\\nwork [ 22] over the policy contained in the archive. Our approach\\nDCG-MAP-Elites distills the experience of the archive into a single\\nversatile policy.\\n4 DCG-MAP-ELITES\\nOur new method Descriptor-Conditioned Gradients MAP-Elites\\n(DCG-MAP-Elites) overcomes limitations of PGA-MAP-Elites by\\nleveraging a descriptor-conditioned critic to improve the PG varia-\\ntion operator and concurrently distills the knowledge of the archive\\nin a single versatile policy as a by-product of the actor-critic train-\\ning. The pseudocode is provided in Alg. 1. The algorithm follows\\nthe usual MAP-Elites loop of selection, variation, evaluation and\\naddition for a budget of \\ud835\\udc3citerations. Two complementary and inde-\\npendent variation operators are used in parallel: 1) a standard GA\\noperator 2) a descriptor-conditioned PG operator. At each iteration,\\nthe transitions from the evaluation step are stored in a replay buffer\\nand used to train an actor-critic pair based on TD3.\\nContrary to PGA-MAP-Elites, the actor-critic pair is descriptor-\\nconditioned. In addition to the state \\ud835\\udc60and action\\ud835\\udc4e, the critic\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\n\\ud835\\udc51)also depends on the descriptor \\ud835\\udc51and estimates the expected dis-\\ncounted return starting from state \\ud835\\udc60, taking action \\ud835\\udc4eand thereafter\\nfollowing policy \\ud835\\udf0bandachieving descriptor \\ud835\\udc51. Achieving descrip-\\ntor\\ud835\\udc51means that the descriptor of the trajectory generated by the\\npolicy\\ud835\\udf0bshould have descriptor \\ud835\\udc51. In addition to the state \\ud835\\udc60, the\\nactor\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51)also depends on the descriptor \\ud835\\udc51and maximizes the\\nexpected discounted return conditioned on achieving descriptor \\ud835\\udc51.\\nThus, the goal of the descriptor-conditioned actor is to achieve the\\ninput descriptor \\ud835\\udc51while maximizing fitness.\\n4.1 Descriptor-Conditioned Critic\\nInstead of estimating the action-value function with \\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e), we\\nwant to estimate the descriptor-conditioned action-value function\\nwith\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51). When a policy \\ud835\\udf0binteracts with the environment\\nfor an episode of length T, it generates a trajectory \\ud835\\udf0f, which is a\\nsequence of transitions:\\n(\\ud835\\udc600,\\ud835\\udc4e0,\\ud835\\udc5f0,\\ud835\\udc601),...,(\\ud835\\udc60\\ud835\\udc47\\u22121,\\ud835\\udc4e\\ud835\\udc47\\u22121,\\ud835\\udc5f\\ud835\\udc47\\u22121,\\ud835\\udc60\\ud835\\udc47)\\nwith descriptor \\ud835\\udc37(\\ud835\\udf0b)=\\ud835\\udc51. We extend the definition of a transition\\n(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f,\\ud835\\udc60\\u2032)to include the descriptor \\ud835\\udc51of the policy(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f,\\ud835\\udc60\\u2032,\\ud835\\udc51).\\nThus, a trajectory \\ud835\\udf0fwith descriptor \\ud835\\udc51gives a sequence of transitions:\\n(\\ud835\\udc600,\\ud835\\udc4e0,\\ud835\\udc5f0,\\ud835\\udc601,\\ud835\\udc51),...,(\\ud835\\udc60\\ud835\\udc47\\u22121,\\ud835\\udc4e\\ud835\\udc47\\u22121,\\ud835\\udc5f\\ud835\\udc47\\u22121,\\ud835\\udc60\\ud835\\udc47,\\ud835\\udc51)\\nHowever, the descriptor is only available at the end of the episode,\\ntherefore the transitions can only be augmented with the descriptor\\nafter the episode is done. In all the tasks we consider, the reward\\nfunction is positive \\ud835\\udc5f:S\\u00d7A\\u2192 R+and hence, the fitness function\\n\\ud835\\udc39and action-value function are positive as well. Thus, for any\\nsampled descriptor \\ud835\\udc51\\u2032\\u2208D, we define the descriptor-conditioned\\ncritic as equal to the normal action-value function when the policy\\nachieves the sampled descriptor \\ud835\\udc51\\u2032and as equal to zero when theAlgorithm 1 DCG-MAP-Elites\\nInput: batch size\\ud835\\udc4f, number of GA variations \\ud835\\udc54\\u2264\\ud835\\udc4f\\nInitialize archiveXwith\\ud835\\udc4frandom solutions and replay buffer B\\nInitialize critic networks \\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032and actor network \\ud835\\udf0b\\ud835\\udf19\\n\\ud835\\udc56\\u21900\\nwhile\\ud835\\udc56<\\ud835\\udc3cdo\\ntrain_actor_critic (\\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032,\\ud835\\udf0b\\ud835\\udf19,B)\\n\\ud835\\udf0b\\ud835\\udf131,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc4f\\u2190selection(X)\\n\\ud835\\udf0bb\\ud835\\udf131,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc54\\u2190variation_ga(\\ud835\\udf0b\\ud835\\udf131,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc54)\\n\\ud835\\udf0bb\\ud835\\udf13\\ud835\\udc54+1,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc4f\\u2190variation_pg(\\ud835\\udf0b\\ud835\\udf13\\ud835\\udc54+1,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc4f,\\ud835\\udc44\\ud835\\udf031,B)\\naddition(\\ud835\\udf0bb\\ud835\\udf131,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc4f,X,B)\\n\\ud835\\udc56\\u2190\\ud835\\udc56+\\ud835\\udc4f\\nfunction addition (X,B,\\ud835\\udf0b\\ud835\\udf19,\\ud835\\udf0bb\\ud835\\udf13...)\\nfor\\ud835\\udc51\\u2032\\u2208D sampled from \\ud835\\udc4fsolutions inXdo\\n(\\ud835\\udc53,transitions)\\u2190\\ud835\\udc39(\\ud835\\udf0b\\ud835\\udf19(.|\\ud835\\udc51\\u2032))\\ninsert(B,transitions)\\nfor\\ud835\\udf0bb\\ud835\\udf13...do\\n(\\ud835\\udc53,transitions)\\u2190\\ud835\\udc39(\\ud835\\udf0bb\\ud835\\udf13),\\ud835\\udc51\\u2190\\ud835\\udc37(\\ud835\\udf0bb\\ud835\\udf13)\\ninsert(B,transitions)\\nifX(\\ud835\\udc51)=\\u2205or\\ud835\\udc39(X(\\ud835\\udc51))<\\ud835\\udc53then\\nX(\\ud835\\udc51)\\u2190\\ud835\\udf0bb\\ud835\\udf13\\npolicy does not achieve the sampled descriptor \\ud835\\udc51\\u2032. Given a transition\\n(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f,\\ud835\\udc60\\u2032,\\ud835\\udc51), and\\ud835\\udc51\\u2032\\u2208D,\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\u2032)=(\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e),if\\ud835\\udc51=\\ud835\\udc51\\u2032\\n0, if\\ud835\\udc51\\u2260\\ud835\\udc51\\u2032(3) insert(B,transitions)\\nifX(\\ud835\\udc51)=\\u2205or\\ud835\\udc39(X(\\ud835\\udc51))<\\ud835\\udc53then\\nX(\\ud835\\udc51)\\u2190\\ud835\\udf0bb\\ud835\\udf13\\npolicy does not achieve the sampled descriptor \\ud835\\udc51\\u2032. Given a transition\\n(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f,\\ud835\\udc60\\u2032,\\ud835\\udc51), and\\ud835\\udc51\\u2032\\u2208D,\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\u2032)=(\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e),if\\ud835\\udc51=\\ud835\\udc51\\u2032\\n0, if\\ud835\\udc51\\u2260\\ud835\\udc51\\u2032(3)\\nHowever, with this piecewise definition, the descriptor-conditioned\\naction-value function is not continuous and violates the universal\\napproximation theorem continuity hypothesis [ 20]. To address this\\nissue, we introduce a similarity function \\ud835\\udc46:D2\\u2192]0,1]defined\\nas\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)=\\ud835\\udc52\\u2212||\\ud835\\udc51\\u2212\\ud835\\udc51\\u2032||D\\n\\ud835\\udc59 to smooth the descriptor-conditioned critic\\nand relax Eq. 3 into:\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\u2032)=\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e)=\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)E\\ud835\\udf0b\\\"\\ud835\\udc47\\u22121\\u2211\\ufe01\\n\\ud835\\udc61=0\\ud835\\udefe\\ud835\\udc61\\ud835\\udc5f\\ud835\\udc61\\f\\f\\f\\f\\f\\ud835\\udc60,\\ud835\\udc4e#\\n=E\\ud835\\udf0b\\\"\\ud835\\udc47\\u22121\\u2211\\ufe01\\n\\ud835\\udc61=0\\ud835\\udefe\\ud835\\udc61\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\ud835\\udc5f\\ud835\\udc61\\f\\f\\f\\f\\f\\ud835\\udc60,\\ud835\\udc4e#\\n(4)\\nWith Eq. 4, we demonstrate that learning the descriptor-conditioned\\ncritic is equivalent to scaling the reward by the similarity \\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\nbetween the descriptor of the trajectory \\ud835\\udc51and the sampled descrip-\\ntor\\ud835\\udc51\\u2032. Therefore, the critic target in Eq. 1 is modified to include the\\nsimilarity scaling and the descriptor-conditioned actor:\\n\\ud835\\udc66=\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\ud835\\udc5f(\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61)+\\ud835\\udefemin\\n\\ud835\\udc56=1,2\\ud835\\udc44\\ud835\\udf03\\ud835\\udc56\\u2032\\u0010\\n\\ud835\\udc60\\ud835\\udc61+1,\\ud835\\udf0b\\ud835\\udf19\\u2032(\\ud835\\udc60\\ud835\\udc61+1|\\ud835\\udc51\\u2032)+\\ud835\\udf16|\\ud835\\udc51\\u2032\\u0011\\n(5)\\nIf the sampled descriptor \\ud835\\udc51\\u2032is approximately equal to the observed\\ndescriptor\\ud835\\udc51of the trajectory \\ud835\\udc51\\u2248\\ud835\\udc51\\u2032then we have \\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\u22481so the\\nreward is unchanged. However, if the descriptor \\ud835\\udc51\\u2032is very different\\nfrom the observed descriptor \\ud835\\udc51then, the reward is scaled down to\\n\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\ud835\\udc5f(\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61)\\u22480. The scaling ensures that the magnitude of the\\nreward depends not only on the quality of the action \\ud835\\udc4ewith regards\\nto the fitness function \\ud835\\udc39, but also on achieving the descriptor \\ud835\\udc51\\u2032.\\nGiven one transition (\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f,\\ud835\\udc60\\u2032,\\ud835\\udc51), we can generate infinitely many\\ncritic updates by sampling \\ud835\\udc51\\u2032\\u2208D. This is leveraged in the new\\n4 actor-critic training introduced with DCG-MAP-Elites, which is\\ndetailed in Alg. 2 and section 4.3.\\nAlgorithm 2 Descriptor-conditioned Actor-Critic Training\\nfunction train_actor_critic (\\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032,\\ud835\\udf0b\\ud835\\udf19,B)\\nfor\\ud835\\udc61=1\\u2192\\ud835\\udc5bdo\\nSample\\ud835\\udc41transitions(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e),\\ud835\\udc60\\u2032,\\ud835\\udc51,\\ud835\\udc51\\u2032)fromB\\nSample smoothing noise \\ud835\\udf16\\n\\ud835\\udc66\\u2190\\ud835\\udc46(\\ud835\\udc51,\\ud835\\udc51\\u2032)\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e)+\\ud835\\udefemin\\n\\ud835\\udc56=1,2\\ud835\\udc44\\ud835\\udf03\\u2032\\n\\ud835\\udc56\\u0000\\ud835\\udc60\\u2032,\\ud835\\udf0b\\ud835\\udf19\\u2032(\\ud835\\udc60\\u2032|\\ud835\\udc51\\u2032)+\\ud835\\udf16|\\ud835\\udc51\\u2032\\u0001\\nUpdate both critics by regression to \\ud835\\udc66\\nif\\ud835\\udc61mod\\u0394then\\nUpdate actor using the deterministic policy gradient:\\n1\\n\\ud835\\udc41\\u00cd\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\u2032)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51\\u2032)\\u2207\\ud835\\udf19\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51\\u2032)\\nSoft-update target networks \\ud835\\udc44\\ud835\\udf03\\ud835\\udc56\\u2032and\\ud835\\udf0b\\ud835\\udf19\\u2032\\n4.2 Descriptor-Conditioned Actor and\\nArchive Distillation\\nThe training of the critic requires to train an actor \\ud835\\udf0b\\ud835\\udf19to approx-\\nimate the optimal action \\ud835\\udc4e\\u2217as explained in section 2.3. However,\\nin this work, the action-value function estimated by the critic\\nis conditioned on a descriptor \\ud835\\udc51. Hence, we don\\u2019t want to esti-\\nmate the best action globally, but the best action given that the\\npolicy should achieve descriptor \\ud835\\udc51by the end of the trajectory.\\nTherefore, the actor is extended to a descriptor-conditioned policy\\n\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51), that maximizes the descriptor-conditioned critic\\u2019s value\\nwithmax\\ud835\\udf19E\\u0002\\n\\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51)|\\ud835\\udc51)\\u0003\\n. The actor is updated using the\\ndeterministic policy gradient, see Alg. 2:\\n\\u2207\\ud835\\udf19\\ud835\\udc3d(\\ud835\\udf19)=1\\n\\ud835\\udc41\\u2211\\ufe01\\n\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\u2032)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51\\u2032)\\u2207\\ud835\\udf19\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51\\u2032)(6)\\nThe policy \\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60|\\ud835\\udc51)learns to suggest actions \\ud835\\udc4ethat optimize\\nreward\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e)while generating a trajectory achieving descriptor \\ud835\\udc51.\\nConsequently, the descriptor-conditioned actor can exhibit a wide\\nrange of descriptors, effectively distilling some of the capabilities\\nof the archive into a single versatile policy.\\n4.3 Actor-Critic Training\\nIn section 4.1, we show that the new descriptor-conditioned critic\\ntarget in Eq. 5 requires a sampled descriptor \\ud835\\udc51\\u2032. At each iteration,\\nthe transitions(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e),\\ud835\\udc60\\u2032,\\ud835\\udc51)generated from the evaluation of\\noffspring are stored in a replay buffer B. To learn the relation\\nbetween being in state \\ud835\\udc60, taking action \\ud835\\udc4eand achieving descriptor\\n\\ud835\\udc51\\u2032, the critic needs to be trained on \\u201cpositive\\u201d samples i.e. samples\\nwith\\ud835\\udc51\\u2032=\\ud835\\udc51and \\u201cnegative\\u201d samples i.e. samples with \\ud835\\udc51\\u2032\\u2260\\ud835\\udc51. Positive\\nsamples are easy to generate because we can sample \\ud835\\udc51\\u2032as equal\\nto the observed descriptor \\ud835\\udc51of the trajectory. However, negative\\nsamples require to define an arbitrary sampling strategy that can\\nimpact the performance of the algorithm.\\nMoreover, learning to act from data without active interactions in\\nthe environment is difficult [ 29]. If the actor only learns passively on\\nrollouts performed by offspring from policies stored in the archive,\\nstrong deviation from the state-action distribution can happen,\\npreventing convergence. To solve this passive learning problem\\nand generate negative samples at the same time, we evaluate the\\nactor\\ud835\\udf0b\\ud835\\udf19in the environment. At each iteration, we sample a batch of\\ud835\\udc4fdescriptors\\ud835\\udc51\\u2032inDand evaluate \\ud835\\udf0b\\ud835\\udf19(.|\\ud835\\udc51\\u2032). The actor evaluation\\nstep gives\\ud835\\udc4ftrajectories with transitions (\\ud835\\udc60\\ud835\\udc61,\\ud835\\udc4e\\ud835\\udc61,\\ud835\\udc5f\\ud835\\udc61,\\ud835\\udc60\\ud835\\udc61+1,\\ud835\\udc51,\\ud835\\udc51\\u2032)with\\n\\ud835\\udc51the observed descriptor achieved by the trajectory and \\ud835\\udc51\\u2032the\\ninput descriptor that is sampled for evaluation. In general, the\\nobserved descriptor and the sampled descriptor are different, \\ud835\\udc51\\u2032\\u2260\\n\\ud835\\udc51, providing negative samples to the critic training. To get the\\ndescriptors\\ud835\\udc51\\u2032, we uniformly sample with replacement \\ud835\\udc4fsolutions\\n\\ud835\\udf0b\\ud835\\udf13from the archive and take their descriptors with an additional\\nGaussian noise. The solutions have descriptors \\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)=\\ud835\\udc51\\ud835\\udf13and we\\ncompute a batch of descriptors \\ud835\\udc51\\u2032=\\ud835\\udc51\\ud835\\udf13+N(0D,\\ud835\\udf0e\\ud835\\udc51\\ud835\\udc3c). The negative\\nsamples from the actor evaluation are stored in the replay buffer\\nalongside the positive samples from offspring evaluation for which\\nwe take\\ud835\\udc51\\u2032=\\ud835\\udc51, as detailed in the addition function of Alg. 1.\\n4.4 Descriptor-Conditioned PG variation\\nOnce the critic \\ud835\\udc44\\ud835\\udf03(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51)is learned, it can be used to improve the\\nfitness of solutions in the archive, as described in Alg. 3. A parent\\nsolution\\ud835\\udf0b\\ud835\\udf13with descriptor \\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)=\\ud835\\udc51\\ud835\\udf13is selected from the archive.\\nThen, we apply the PG variation operator, using the descriptor \\ud835\\udc51\\ud835\\udf13\\nto condition the critic, and thus, to apply \\ud835\\udc5adescriptor-conditioned solution\\ud835\\udf0b\\ud835\\udf13with descriptor \\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)=\\ud835\\udc51\\ud835\\udf13is selected from the archive.\\nThen, we apply the PG variation operator, using the descriptor \\ud835\\udc51\\ud835\\udf13\\nto condition the critic, and thus, to apply \\ud835\\udc5adescriptor-conditioned\\ngradient steps using the deterministic policy gradient:\\n\\u2207\\ud835\\udf13\\ud835\\udc3d(\\ud835\\udf13)=1\\n\\ud835\\udc41\\u2211\\ufe01\\n\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\ud835\\udf13)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60)\\u2207\\ud835\\udf13\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60) (7)\\nThe goal is to improve the fitness of the solution while remaining\\nin the same cell as the parent.\\nAlgorithm 3 Descriptor-conditioned PG Variation\\nfunction variation_pg (\\ud835\\udf0b\\ud835\\udf13...,\\ud835\\udc44 \\ud835\\udf031,B)\\nfor\\ud835\\udf0b\\ud835\\udf13...do\\n\\ud835\\udc51\\ud835\\udf13\\u2190\\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)\\nfor\\ud835\\udc56=1\\u2192\\ud835\\udc5ado\\nSample\\ud835\\udc41transitions(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e),\\ud835\\udc60\\u2032,\\ud835\\udc51,\\ud835\\udc51\\u2032)fromB\\nUpdate actor using the deterministic policy gradient:\\n1\\n\\ud835\\udc41\\u00cd\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e|\\ud835\\udc51\\ud835\\udf13)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60)\\u2207\\ud835\\udf13\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60)\\nreturn\\ud835\\udf0bb\\ud835\\udf19...\\n5 EXPERIMENTS\\n5.1 Evaluation Tasks\\nWe evaluate DCG-MAP-Elites on four QDGym tasks [ 28] imple-\\nmented in Brax [ 14] and derived from standard DRL benchmarks\\nfor robot locomotion. We call these tasks \\u201cAnt-Omni\\u201d, \\u201cAntTrap-\\nOmni\\u201d, \\u201cHexapod-Omni\\u201d, and \\u201cWalker-Uni\\u201d. Three omnidirectional\\ntasks and one unidirectional task are considered for evaluation.\\nAnt-Omni, AntTrap-Omni and Hexapod-Omni are omnidirectional\\ntasks in which the objective is for the robot to reach all possible\\npoints in the plane while minimizing energy consumption. Walker-\\nUni is a unidirectional task in which the objective is for the robot\\nto discover all possible ways to walk forward while maximizing a\\ntrade-off between speed and energy consumption. Walker-Uni is\\nexactly the same tasks used in PGA-MAP-Elites paper [ 28], Ant-\\nOmni and Hexapod-Omni have been used by Flageat and Lim et\\nal. [11] and AntTrap-Omni is adapted from QD-PG paper [ 31]. The\\ndifference is the elimination of the forward reward term in the\\nreward function. The goal of AntTrap-Omni is to reach every point\\n5 in the plane while minimizing energy consumption in a deceptive\\nenvironment. This new task is designed to show that our algorithm\\nperforms well in environment where there is a discontinuity on the\\nfitness landscape in descriptor space caused by the trap. The trap\\nis causing a discontinuity that the descriptor-conditioned critic has\\nto learn. Indeed, points on both sides of the trap will be close in the\\ndescriptor space, but distant in terms of trajectory to achieve these\\ndescriptors. We detail these tasks in Tab. 1.\\nPGA-MAP-Elites has previously shown state-of-the-art results\\non some of these tasks, in particular Walker-Uni but tends to strug-\\ngle in omnidirectional tasks. In those tasks, the global optimal of\\nthe fitness function is a solution that prevents the robot to move,\\nwhich is directly opposed to the objective of discovering how to\\nreach different locations. Hence, most of the offspring generated by\\nthe PG variation will tend to move less and travel a shorter distance.\\nInstead, DCG-MAP-Elites aims to improve the energy consumption\\nwhile maintaining the ability to reach distant locations.\\nTable 1: Evaluation Tasks\\nAnt AntTrap Hexapod Walker\\nState Position and velocity of body and joints\\nAction Torques for each joint\\nType Omni Omni Omni Uni\\nDescriptor Final(\\ud835\\udc65,\\ud835\\udc66)position Feet contact\\nFitness Surviving reward + Energy consumption penalty\\nState dim 29 29 48 19\\nAction dim 8 8 18 6\\nEpisode len 250 250 250 1000\\nParameters 21,384 21,384 25,106 19,846\\n5.2 Baselines\\nWe compare DCG-MAP-Elites with four algorithms: CVT-MAP-\\nElites [ 39], MAP-Elites-ES [ 5], PGA-MAP-Elites [ 28] and QD-PG [ 31].\\nWe could not compare to PBT-MAP-Elites [ 30] as its implementa-\\ntion is not yet available. We use QDax implementations and each\\nexperiment is replicated 20 times with random seeds, over one\\nmillion evaluations. The source code of DCG-MAP-Elites is avail-\\nable at github.com\\/adaptive-intelligent-robotics\\/DCG-MAP-Elites\\nincluding a container in which all the experiments can be replicated.\\n5.3 Evaluation Metrics\\nWe consider three main metrics to evaluate the archives and two\\nadditional metrics to evaluate the descriptor-conditioned actor. We\\ncompute p-values based on the Wilcoxon rank-sum test with a\\nBonferroni correction for all metrics.\\n\\u2022QD-score: The sum of fitness of all solutions stored in the archive.\\nIn the task considered, fitnesses are always positive, which avoid\\npenalizing algorithms for finding additional solutions. This score\\nencompasses both the quality and diversity of the population.\\n\\u2022Coverage: The proportion of filled cells in the archive, indicating\\nthe success of descriptor space illumination.\\u2022Maximum fitness: The fitness of the best solution in the archive.\\nWe also consider two additional metrics to evaluate the descriptor-\\nconditioned policy. The purpose of these metrics is to quantify how\\nsimilar the archive and the descriptor-conditioned policy are in\\nterms of QD-score and coverage of the descriptor space.\\n\\u2022Descriptor-conditioned QD-score: The sum of fitness achieved\\nby the descriptor-conditioned policy evaluated on each descriptor\\nof the solutions in the archive.\\u2211\\ufe01\\n\\ud835\\udf0b\\ud835\\udf13\\u2208X\\ud835\\udc39(\\ud835\\udf0b\\ud835\\udf19(.|\\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)))\\n\\u2022Descriptor error mean: (i) for the archive, the average differ-\\nence over all solutions, between the stored descriptor and the\\nreevaluated descriptor, (ii) for the descriptor-conditioned policy,\\nthe average difference over all solutions, between the stored de-\\nscriptor and the reevaluated descriptor achieved by the policy\\nconditioned on the stored descriptor. ( |X|number of solutions in\\nthe archive).\\n1\\n|X|\\u2211\\ufe01\\n\\ud835\\udf0b\\ud835\\udf13\\u2208X||\\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf19(.|\\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)))\\u2212\\ud835\\udc37(\\ud835\\udf0b\\ud835\\udf13)||D\\n5.4 Results\\n5.4.1 Archive. The experimental results on Fig. 2 demonstrate that\\nDCG-MAP-Elites achieves the best archive in the two Ant tasks in\\nterms of QD-score ( \\ud835\\udc5d<10\\u22126) and coverage ( \\ud835\\udc5d<10\\u22124) compared\\nto all other baselines. On the two other tasks, the QD-score of\\nour algorithm is similar to PGA-MAP-Elites, the previous state-\\nof-the-art, while achieving a significantly better coverage than to all other baselines. On the two other tasks, the QD-score of\\nour algorithm is similar to PGA-MAP-Elites, the previous state-\\nof-the-art, while achieving a significantly better coverage than\\nall baselines ( \\ud835\\udc5d<10\\u22123). The coverage metric shows that DCG-\\nMAP-Elites surpasses the exploration capabilities of QD-PG on\\nall tasks (\\ud835\\udc5d<10\\u22124). We also show that our method still benefits\\nfrom the exploration power of the GA operator even in deceptive\\nenvironment like AntTrap-Omni.\\n30\\n 20\\n 10\\n 0 10 20 30\\nx position30\\n20\\n10\\n0102030y positionPGA-MAP-Elites\\n30\\n 20\\n 10\\n 0 10 20 30\\nx positionDCG-MAP-Elites\\n6006507007508008509009501000\\nFigure 3: Final archives for PGA-MAP-Elites and DCG-MAP-\\nElites on Ant-Omni after one million evaluations.\\nThe experimental results confirm that DCG-MAP-Elites is able\\nto overcome the limits of PGA-MAP-Elites on omnidirectional tasks\\nwhile still performing on par when evaluated on the unidirectional\\ntask (Walker-Uni) where no significant improvement was expected.\\nThus confirming the interest of having a descriptor-conditioned\\ngradient to make the PG variation operator fruitful in a wider range\\n6 0.00.20.40.60.8QD-Score (1e6)Ant-Omni\\n0.00.20.40.60.8AntTrap-Omni\\n0.00.20.40.6Hexapod-Omni\\n0.00.51.01.5Walker-Uni\\n020406080Coverage (%)\\n020406080\\n020406080\\n020406080\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e67008009001000Maximum fitness\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e67008009001000\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e6950100010501100\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e60100020003000\\nDCG-MAP-Elites PGA-MAP-Elites QD-PG MAP-Elites MAP-Elites ESFigure 2: QD-score, coverage and maximum fitness for DCG-MAP-Elites and the baselines on all tasks. Each experiment is\\nreplicated 20 times with different random seeds. The solid line is the median and the shaded area is the first and third quartiles.\\nof tasks. Overall, DCG-MAP-Elites shows best (or competitive) per-\\nformance on all metrics and tasks, hence proving to be the first\\nsuccessful effort in the QD-RL literature to achieve well on both\\nthe unidirectional and omnidirectional tasks. Previous efforts were\\nusually adapted to either one or the other [ 28,31,38]. Nevertheless,\\nour results show that there is still room for improvement. In partic-\\nular, we can see that DCG-MAP-Elites was not able to outperform\\nthe baselines with the same margin on Hexapod-Omni that it has\\nbeen able to do on the other omnidirectional tasks. We hypothesize\\nthat the environment dynamics and the higher-dimensional search\\nspace of Hexapod-Omni makes it much more challenging.\\nTable 2: Median of the QD-score and Descriptor Error Mean\\n(DEM) after one million evaluations for the archive and for\\nthe descriptor-conditioned policy over 20 replications.\\nMetric Type Ant AntTrap Hexapod Walker\\nQD-score\\n(105)Archive 8.7 7.2 6.5 9.8\\nPolicy 8.3 7.0 0.96 5.2\\nDEMArchive 4.04 3.44 0.078 0.12\\nPolicy 6.25 5.8 1.39 0.27\\n5.4.2 Descriptor-Conditioned Policy. In Tab. 2 we provide the final\\nQD-score and Descriptor Error Mean (DEM) of the archive and\\nthe descriptor-conditioned policy, see section 5.3 for definitions.\\nFirst, on Ant-Omni and AntTrap-Omni, the descriptor-conditioned\\npolicy achieves the same QD-score as the archive, measured by\\nthe descriptor-conditioned QD-score (decreases by less than 3%).\\nThis shows that the single-conditioned policy is able to restore\\nthe quality of the archive although having compressed the infor-\\nmation in a single network. Second, it is interesting to cross this\\nobservation with the fact that, at the same time, the Descriptor\\nError Mean obtained by the conditioned policy is reasonable. We\\nobserve a DEM of 6.25, which is less than 8% of the typical size ofthe environment (i.e. the diagonal of the square considered). The\\nreader can refer to Fig. 3 to get an idea of what that represents. In\\naddition, the DEM of the archive is of 4.04 (4.7% of size of the envi-\\nronment), showing an inevitable stochasticity in the evaluation that\\nis related to the simulator used [ 14]. Those two combined observa-\\ntions (QD-score and DEM) tend to show that our archive and our\\ndescriptor-conditioned policy have similar properties on Ant-Omni\\nand AntTrap-Omni and hence that the versatile policy could be a\\npractical summary of our archive. Third, we can observe that on the\\nHexapod-Omni and Walker-Uni tasks, the descriptor-conditioned\\npolicy struggles to recover the QD-score of the archive (reduced\\nby more than 40% in both cases). On the Hexapod-Omni, the DEM\\nof the descriptor-conditioned policy is significantly larger than the\\ntypical error obtained by the archive (1.39 vs 0.078). This task is\\nmore challenging than Ant-Omni because of a higher number of\\ndimensions (see Tab. 1) and we hypothesize that the optimization\\nalgorithm used (TD3) could be the limitation. For Walker-Uni, the\\nDEM of the archive (0.12) shows that there is also an inherent vari-\\nability of the descriptor, we are hence less surprised by the DEM\\nobtained by the single-conditioned policy (0.27) although it could\\nmake it less usable in practice. ability of the descriptor, we are hence less surprised by the DEM\\nobtained by the single-conditioned policy (0.27) although it could\\nmake it less usable in practice.\\nOverall, those results shows that our single descriptor-conditioned\\npolicy can already be seen as a promising summary of our archive,\\nshowing very similar properties on half our tasks. The other tasks\\nprove that there is still room for improvement of our method.\\n5.4.3 Ablations. We perform additional ablation experiments to\\nshow the importance of three components of DCG-MAP-Elites,\\nnamely: using negative samples for the critic\\u2019s training, actively\\ncollecting experiences with the actor, and conditioning the actor\\non the descriptor. In Ablation (1) , we remove the actor evaluation\\nin the environment. This allows to study the impact of both pas-\\nsive learning and of lack of negative samples, see section 4.3 for a\\n7 0.00.20.40.60.8QD-Score (1e6)Ant-Omni\\n0.00.20.40.60.8AntTrap-Omni\\n0.00.20.40.6Hexapod-Omni\\n0.00.51.01.5Walker-Uni\\n020406080Coverage (%)\\n020406080\\n020406080\\n020406080\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e699099510001005Maximum fitness\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e699099510001005\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e6111011201130\\n0.0 0.2 0.4 0.6 0.8 1.0\\nNumber of evaluations 1e6100020003000\\nDCG-MAP-Elites Ablation 1 Ablation 2 Ablation 3Figure 4: QD-score, coverage and maximum fitness for DCG-MAP-Elites and the ablations on all tasks. Each experiment is\\nreplicated 20 times with different random seeds. The solid line is the median and the shaded area is the first and third quar-\\ntiles. Ablation (1) is DCG-MAP-Elites without actor evaluation and without negative samples, Ablation (2) is DCG-MAP-Elites\\nwithout actor evaluation but with negative samples, Ablation (3) is DCG-MAP-Elites without a descriptor-conditioned actor.\\ndetailed explanation. The lack of active learning deprives the learn-\\ning of negative samples that are necessary to learn the descriptor-\\nconditioned action-value function. We observe that ablation (1) is\\nperforming worse than DCG-MAP-Elites on all omnidirectional\\ntasks (\\ud835\\udc5d<0.01). However it performs similarly on Walker-Uni,\\nwhich can be explained by the fact that non-conditioned PG emitter\\nalready prove efficient in solving this task. This first ablation result\\nhighlight the importance of active learning and negative examples\\nfor the critic\\u2019s training. To complement this result, in Ablation (2) ,\\nwe remove the actor evaluation but we generate negative samples\\nartificially by sampling in the descriptor space. We show that ab-\\nlation (2) is performing significantly worse than DCG-MAP-Elites\\non Ant tasks ( \\ud835\\udc5d<10\\u22123) but performs similarly on Hexapod-Omni\\nand Walker-Uni. However, ablation (2) is performing significantly\\nbetter than ablation (1) on Ant tasks ( \\ud835\\udc5d<10\\u22125), demonstrating the\\nimportance of both active learning andnegative samples. Finally, in\\nAblation (3) we replace the descriptor-conditioned actor \\ud835\\udf0b\\ud835\\udf19(.|\\ud835\\udc51)\\nwith a normal actor \\ud835\\udf0b\\ud835\\udf19(.). Ablation (3) is performing significantly\\nworse on all omnidirectional tasks ( \\ud835\\udc5d<0.05), demonstrating the\\nimportance of the descriptor-conditioned actor for the critic\\u2019s train-\\ning. Ablation (3) performs equivalently to DCG-MAP-Elites on\\nWalker-Uni.\\n6 CONCLUSION\\nIn this work, we introduce DCG-MAP-Elites and demonstrate the\\nbenefits of having descriptor-conditioned gradients to evolve popu-\\nlations of large neural networks. We concurrently train a descriptor-\\nconditioned actor, as a by-product of the critic\\u2019s training, that can\\nachieve a wide range of descriptors. Our method performs better\\nthan PGA-MAP-Elites by a significant margin in omnidirectional\\ntasks while maintaining similar performance to PGA-MAP-Elitesin unidirectional tasks. We also show that the synergy between the\\nfitness improvement capabilities of the PG variations and the explo-\\nration capabilities of the GA variations is preserved, even in decep-\\ntive environments. On some tasks, the descriptor-conditioned pol-\\nicy demonstrates properties that are similar to the discrete archive,\\nsummarizing its capabilities into one single neural network and\\nacting as a continuous archive. We think that distilling the archive\\ninto a single policy is a promising method as it enables to have\\nless redundancy compared to a discrete archive in which most of\\nthe solutions can be similar, especially between close cells. The\\ndescriptor-conditioned policy could also negate the burden of deal-\\ning with archive of thousands of solutions in practical applications.\\nThe benefits of combining DRL methods with MAP-Elites come\\nwith the limitations of MDP settings. Specifically, we are limited\\nto evolving differentiable solutions and the foundations of DRL\\nalgorithms rely on the Markov property and full observability. In\\nthis work in particular, we face challenges with the Markov property to evolving differentiable solutions and the foundations of DRL\\nalgorithms rely on the Markov property and full observability. In\\nthis work in particular, we face challenges with the Markov property\\nbecause the descriptors depend on full trajectories. Thus, the scaled\\nreward introduced in our method depends on the full trajectory\\nand not only on the current state and action. The performance of\\nthe descriptor-conditioned policy also shows that there is room for\\nimprovement to better distill the knowledge of the archive.\\nFor future work, we would like to investigate the generaliza-\\ntion capabilities of the descriptor-conditioned policy trained with\\nDCG-MAP-Elites and try to produce solutions with descriptors\\nthat are not in the archive, effectively performing descriptor space\\ninterpolation. In our method, the critic attempts to mutate solu-\\ntions to produce offspring with higher fitness while keeping their\\ndescriptors constant. We think that we could use the descriptor-\\nconditioned critic to mutate solutions to produce offspring towards\\ndifferent descriptors, thereby explicitly promoting diversity.\\n8 REFERENCES\\n[1]Felix Chalumeau, Raphael Boige, Bryan Lim, Valentin Mac\\u00e9, Maxime Allard,\\nArthur Flajolet, Antoine Cully, and Thomas Pierrot. 2022. Neuroevolution is a\\nCompetitive Alternative to Reinforcement Learning for Skill Discovery. https:\\n\\/\\/doi.org\\/10.48550\\/arXiv.2210.03516 arXiv:2210.03516 [cs].\\n[2]Felix Chalumeau, Thomas Pierrot, Valentin Mac\\u00e9, Arthur Flajolet, Karim Beguir,\\nAntoine Cully, and Nicolas Perrin-Gilbert. 2022. Assessing Quality-Diversity\\nNeuro-Evolution Algorithms Performance in Hard Exploration Problems. http:\\n\\/\\/arxiv.org\\/abs\\/2211.13742 arXiv:2211.13742 [cs].\\n[3]Konstantinos Chatzilygeroudis, Antoine Cully, Vassilis Vassiliades, and Jean-\\nBaptiste Mouret. 2020. Quality-Diversity Optimization: a novel branch of sto-\\nchastic optimization. http:\\/\\/arxiv.org\\/abs\\/2012.04322 arXiv:2012.04322 [cs, math,\\nstat].\\n[4]Konstantinos Chatzilygeroudis, Vassilis Vassiliades, and Jean-Baptiste Mouret.\\n2018. Reset-free Trial-and-Error Learning for Robot Damage Recovery. Robotics\\nand Autonomous Systems 100 (Feb. 2018), 236\\u2013250. https:\\/\\/doi.org\\/10.1016\\/j.\\nrobot.2017.11.010 arXiv:1610.04213 [cs].\\n[5]C\\u00e9dric Colas, Joost Huizinga, Vashisht Madhavan, and Jeff Clune. 2020. Scal-\\ning MAP-Elites to Deep Neuroevolution. CoRR abs\\/2003.01825 (2020), 67\\u201375.\\narXiv:2003.01825 https:\\/\\/arxiv.org\\/abs\\/2003.01825\\n[6]Antoine Cully, Jeff Clune, Danesh Tarapore, and Jean-Baptiste Mouret. 2015.\\nRobots that can adapt like animals. Nature 521, 7553 (May 2015), 503\\u2013507. https:\\n\\/\\/doi.org\\/10.1038\\/nature14422 arXiv:1407.3501 [cs, q-bio].\\n[7]Antoine Cully and Yiannis Demiris. 2017. Quality and Diversity Optimization: A\\nUnifying Modular Framework. http:\\/\\/arxiv.org\\/abs\\/1708.09251 arXiv:1708.09251\\n[cs].\\n[8]Adrien Ecoffet, Joost Huizinga, Joel Lehman, Kenneth O. Stanley, and Jeff Clune.\\n2021. Go-Explore: a New Approach for Hard-Exploration Problems. http:\\n\\/\\/arxiv.org\\/abs\\/1901.10995 arXiv:1901.10995 [cs, stat].\\n[9]Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine. 2018.\\nDiversity is All You Need: Learning Skills without a Reward Function. http:\\n\\/\\/arxiv.org\\/abs\\/1802.06070 arXiv:1802.06070 [cs].\\n[10] Manon Flageat, Felix Chalumeau, and Antoine Cully. 2022. Empirical analysis of\\nPGA-MAP-Elites for Neuroevolution in Uncertain Domains. http:\\/\\/arxiv.org\\/\\nabs\\/2210.13156 arXiv:2210.13156 [cs].\\n[11] Manon Flageat, Bryan Lim, Luca Grillotti, Maxime Allard, Sim\\u00f3n C. Smith,\\nand Antoine Cully. 2022. Benchmarking Quality-Diversity Algorithms on\\nNeuroevolution for Reinforcement Learning. http:\\/\\/arxiv.org\\/abs\\/2211.02193\\narXiv:2211.02193 [cs].\\n[12] Matthew C. Fontaine and Stefanos Nikolaidis. 2021. Differentiable Quality Diver-\\nsity. https:\\/\\/doi.org\\/10.48550\\/arXiv.2106.03894 arXiv:2106.03894 [cs].\\n[13] Matthew C. Fontaine and Stefanos Nikolaidis. 2023. Covariance Matrix\\nAdaptation MAP-Annealing. https:\\/\\/doi.org\\/10.48550\\/arXiv.2205.10752\\narXiv:2205.10752 [cs].\\n[14] C. Daniel Freeman, Erik Frey, Anton Raichuk, Sertan Girgin, Igor Mordatch, and\\nOlivier Bachem. 2021. Brax \\u2013 A Differentiable Physics Engine for Large Scale\\nRigid Body Simulation. http:\\/\\/arxiv.org\\/abs\\/2106.13281 arXiv:2106.13281 [cs].\\n[15] Scott Fujimoto, Herke van Hoof, and David Meger. 2018. Addressing Function\\nApproximation Error in Actor-Critic Methods. http:\\/\\/arxiv.org\\/abs\\/1802.09477\\narXiv:1802.09477 [cs, stat].\\n[16] Karol Gregor, Danilo Jimenez Rezende, and Daan Wierstra. 2016. Variational\\nIntrinsic Control. http:\\/\\/arxiv.org\\/abs\\/1611.07507 arXiv:1611.07507 [cs].\\n[17] Shixiang Gu, Ethan Holly, Timothy Lillicrap, and Sergey Levine. 2016. Deep\\nReinforcement Learning for Robotic Manipulation with Asynchronous Off-Policy\\nUpdates. http:\\/\\/arxiv.org\\/abs\\/1610.00633 arXiv:1610.00633 [cs].\\n[18] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. 2018. Soft\\nActor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a\\nStochastic Actor. http:\\/\\/arxiv.org\\/abs\\/1801.01290 arXiv:1801.01290 [cs, stat].\\n[19] Nikolaus Hansen. 2016. The CMA Evolution Strategy: A Tutorial. http:\\/\\/arxiv. Stochastic Actor. http:\\/\\/arxiv.org\\/abs\\/1801.01290 arXiv:1801.01290 [cs, stat].\\n[19] Nikolaus Hansen. 2016. The CMA Evolution Strategy: A Tutorial. http:\\/\\/arxiv.\\norg\\/abs\\/1604.00772 arXiv:1604.00772 [cs, stat].\\n[20] Kurt Hornik, Maxwell Stinchcombe, and Halbert White. 1989. Multilayer feedfor-\\nward networks are universal approximators. Neural Networks 2, 5 (1989), 359\\u2013366.\\nhttps:\\/\\/doi.org\\/10.1016\\/0893-6080(89)90020-8\\n[21] Max Jaderberg, Valentin Dalibard, Simon Osindero, Wojciech M. Czarnecki, Jeff\\nDonahue, Ali Razavi, Oriol Vinyals, Tim Green, Iain Dunning, Karen Simonyan,\\nChrisantha Fernando, and Koray Kavukcuoglu. 2017. Population Based Training\\nof Neural Networks. https:\\/\\/doi.org\\/10.48550\\/arXiv.1711.09846 arXiv:1711.09846\\n[cs].\\n[22] Marija Jegorova, St\\u00e9phane Doncieux, and Timothy Hospedales. 2019. Behavioural\\nRepertoire via Generative Adversarial Policy Networks. In 2019 Joint IEEE 9th\\nInternational Conference on Development and Learning and Epigenetic Robotics\\n(ICDL-EpiRob) . https:\\/\\/doi.org\\/10.1109\\/ICDL-EpiRob44920.2019 arXiv:1811.02945\\n[cs, stat].\\n[23] Saurabh Kumar, Aviral Kumar, Sergey Levine, and Chelsea Finn. 2020. One\\nSolution is Not All You Need: Few-Shot Extrapolation via Structured MaxEnt RL.\\nhttp:\\/\\/arxiv.org\\/abs\\/2010.14484 arXiv:2010.14484 [cs].[24] Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom\\nErez, Yuval Tassa, David Silver, and Daan Wierstra. 2019. Continuous control with\\ndeep reinforcement learning. http:\\/\\/arxiv.org\\/abs\\/1509.02971 arXiv:1509.02971\\n[cs, stat].\\n[25] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis\\nAntonoglou, Daan Wierstra, and Martin Riedmiller. 2013. Playing Atari with\\nDeep Reinforcement Learning. http:\\/\\/arxiv.org\\/abs\\/1312.5602 arXiv:1312.5602\\n[cs].\\n[26] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness,\\nMarc G. Bellemare, Alex Graves, Martin Riedmiller, Andreas K. Fidjeland, Georg\\nOstrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen\\nKing, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. 2015.\\nHuman-level control through deep reinforcement learning. Nature 518, 7540 (Feb.\\n2015), 529\\u2013533. https:\\/\\/doi.org\\/10.1038\\/nature14236 Number: 7540 Publisher:\\nNature Publishing Group.\\n[27] Jean-Baptiste Mouret and Jeff Clune. 2015. Illuminating search spaces by mapping\\nelites. http:\\/\\/arxiv.org\\/abs\\/1504.04909 arXiv:1504.04909 [cs, q-bio].\\n[28] Olle Nilsson and Antoine Cully. 2021. Policy gradient assisted MAP-Elites. In\\nProceedings of the Genetic and Evolutionary Computation Conference . ACM, Lille\\nFrance, 866\\u2013875. https:\\/\\/doi.org\\/10.1145\\/3449639.3459304\\n[29] Georg Ostrovski, Pablo Samuel Castro, and Will Dabney. 2021. The Difficulty of\\nPassive Learning in Deep Reinforcement Learning. http:\\/\\/arxiv.org\\/abs\\/2110.\\n14020 arXiv:2110.14020 [cs].\\n[30] Thomas Pierrot and Arthur Flajolet. 2023. Evolving Populations of Diverse RL\\nAgents with MAP-Elites. https:\\/\\/openreview.net\\/forum?id=CBfYffLqWqb\\n[31] Thomas Pierrot, Valentin Mac\\u00e9, F\\u00e9lix Chalumeau, Arthur Flajolet, Geoffrey\\nCideron, Karim Beguir, Antoine Cully, Olivier Sigaud, and Nicolas Perrin-Gilbert.\\n2022. Diversity Policy Gradient for Sample Efficient Quality-Diversity Opti-\\nmization. In Proceedings of the Genetic and Evolutionary Computation Conference .\\n1075\\u20131083. https:\\/\\/doi.org\\/10.1145\\/3512290.3528845 arXiv:2006.08505 [cs].\\n[32] Justin K. Pugh, Lisa B. Soros, and Kenneth O. Stanley. 2016. Quality Diversity:\\nA New Frontier for Evolutionary Computation. Frontiers in Robotics and AI 3\\n(2016). https:\\/\\/www.frontiersin.org\\/articles\\/10.3389\\/frobt.2016.00040\\n[33] Tom Schaul, Daniel Horgan, Karol Gregor, and David Silver. 2015. Universal Value\\nFunction Approximators. In Proceedings of the 32nd International Conference on\\nMachine Learning (Proceedings of Machine Learning Research, Vol. 37) , Francis\\nBach and David Blei (Eds.). PMLR, Lille, France, 1312\\u20131320. https:\\/\\/proceedings.\\nmlr.press\\/v37\\/schaul15.html Machine Learning (Proceedings of Machine Learning Research, Vol. 37) , Francis\\nBach and David Blei (Eds.). PMLR, Lille, France, 1312\\u20131320. https:\\/\\/proceedings.\\nmlr.press\\/v37\\/schaul15.html\\n[34] Archit Sharma, Shixiang Gu, Sergey Levine, Vikash Kumar, and Karol Hausman.\\n2020. Dynamics-Aware Unsupervised Discovery of Skills. http:\\/\\/arxiv.org\\/abs\\/\\n1907.01657 arXiv:1907.01657 [cs, stat].\\n[35] David Silver, Aja Huang, Chris J. Maddison, Arthur Guez, Laurent Sifre, George\\nvan den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershel-\\nvam, Marc Lanctot, Sander Dieleman, Dominik Grewe, John Nham, Nal Kalch-\\nbrenner, Ilya Sutskever, Timothy Lillicrap, Madeleine Leach, Koray Kavukcuoglu,\\nThore Graepel, and Demis Hassabis. 2016. Mastering the game of Go with\\ndeep neural networks and tree search. Nature 529, 7587 (Jan. 2016), 484\\u2013489.\\nhttps:\\/\\/doi.org\\/10.1038\\/nature16961 Number: 7587 Publisher: Nature Publishing\\nGroup.\\n[36] David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and\\nMartin Riedmiller. 2014. Deterministic Policy Gradient Algorithms. In Proceedings\\nof the 31st International Conference on Machine Learning (Proceedings of Machine\\nLearning Research, Vol. 32) , Eric P. Xing and Tony Jebara (Eds.). PMLR, Bejing,\\nChina, 387\\u2013395. https:\\/\\/proceedings.mlr.press\\/v32\\/silver14.html\\n[37] Bryon Tjanaka, Matthew C. Fontaine, Aniruddha Kalkar, and Stefanos Nikolaidis.\\n2022. Training Diverse High-Dimensional Controllers by Scaling Covariance\\nMatrix Adaptation MAP-Annealing. https:\\/\\/doi.org\\/10.48550\\/arXiv.2210.02622\\narXiv:2210.02622 [cs].\\n[38] Bryon Tjanaka, Matthew C. Fontaine, Julian Togelius, and Stefanos Nikolaidis.\\n2022. Approximating Gradients for Differentiable Quality Diversity in Reinforce-\\nment Learning. https:\\/\\/doi.org\\/10.48550\\/arXiv.2202.03666 arXiv:2202.03666\\n[cs].\\n[39] Vassilis Vassiliades, Konstantinos Chatzilygeroudis, and Jean-Baptiste Mouret.\\n2017. Using Centroidal Voronoi Tessellations to Scale Up the Multi-dimensional\\nArchive of Phenotypic Elites Algorithm. http:\\/\\/arxiv.org\\/abs\\/1610.05729\\narXiv:1610.05729 [cs].\\n[40] Oriol Vinyals, Igor Babuschkin, Wojciech M. Czarnecki, Micha\\u00ebl Mathieu, An-\\ndrew Dudzik, Junyoung Chung, David H. Choi, Richard Powell, Timo Ewalds,\\nPetko Georgiev, Junhyuk Oh, Dan Horgan, Manuel Kroiss, Ivo Danihelka, Aja\\nHuang, Laurent Sifre, Trevor Cai, John P. Agapiou, Max Jaderberg, Alexander S.\\nVezhnevets, R\\u00e9mi Leblond, Tobias Pohlen, Valentin Dalibard, David Budden, Yury\\nSulsky, James Molloy, Tom L. Paine, Caglar Gulcehre, Ziyu Wang, Tobias Pfaff,\\nYuhuai Wu, Roman Ring, Dani Yogatama, Dario W\\u00fcnsch, Katrina McKinney,\\nOliver Smith, Tom Schaul, Timothy Lillicrap, Koray Kavukcuoglu, Demis Has-\\nsabis, Chris Apps, and David Silver. 2019. Grandmaster level in StarCraft II\\nusing multi-agent reinforcement learning. Nature 575, 7782 (Nov. 2019), 350\\u2013\\n354. https:\\/\\/doi.org\\/10.1038\\/s41586-019-1724-z Number: 7782 Publisher: Nature\\nPublishing Group.\\n9 SUPPLEMENTARY MATERIAL \\u2014 MAP-ELITES\\nWITH DESCRIPTOR-CONDITIONED\\nGRADIENTS AND ARCHIVE DISTILLATION\\nINTO A SINGLE POLICY\\nA ALGORITHMS\\nA.1 MAP-Elites\\nAlgorithm 4 MAP-Elites\\nInput: batch size\\ud835\\udc4f\\nInitialize archiveXwith\\ud835\\udc4frandom solutions\\n\\ud835\\udc56\\u21900\\nwhile\\ud835\\udc56<\\ud835\\udc3cdo\\n\\ud835\\udc651,...,\\ud835\\udc65 \\ud835\\udc4f\\u2190selection(X)\\nb\\ud835\\udc651,...,b\\ud835\\udc65\\ud835\\udc4f\\u2190variation(\\ud835\\udc651,...,\\ud835\\udc65 \\ud835\\udc4f)\\naddition(X,b\\ud835\\udc651,...,b\\ud835\\udc65\\ud835\\udc4f)\\n\\ud835\\udc56\\u2190\\ud835\\udc56+\\ud835\\udc4f\\nfunction addition (X,b\\ud835\\udc65... ) :\\nforb\\ud835\\udc65... do\\n\\ud835\\udc53\\u2190\\ud835\\udc39(b\\ud835\\udc65),\\ud835\\udc51\\u2190\\ud835\\udc37(b\\ud835\\udc65)\\nifX(\\ud835\\udc51)=\\u2205or\\ud835\\udc39(X(\\ud835\\udc51))<\\ud835\\udc53then\\nX(\\ud835\\udc51)\\u2190b\\ud835\\udc65\\nA.2 PGA-MAP-Elites\\nAlgorithm 5 PGA-MAP-Elites\\nInput: batch size\\ud835\\udc4f, number of GA variations \\ud835\\udc54\\u2264\\ud835\\udc4f\\nInitialize archiveXwith\\ud835\\udc4frandom solutions and replay buffer B\\nInitialize critic networks \\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032and actor network \\ud835\\udf0b\\ud835\\udf19\\n\\ud835\\udc56\\u21900\\nwhile\\ud835\\udc56<\\ud835\\udc3cdo\\ntrain_actor_critic (\\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032,\\ud835\\udf0b\\ud835\\udf19,B)\\n\\ud835\\udf0b\\ud835\\udf131,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc4f\\u22121\\u2190selection(X)\\n\\ud835\\udf0bb\\ud835\\udf131,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc54\\u2190variation_ga(\\ud835\\udf0b\\ud835\\udf131,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc54)\\n\\ud835\\udf0bb\\ud835\\udf13\\ud835\\udc54+1,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc4f\\u22121\\u2190variation_pg(\\ud835\\udf0b\\ud835\\udf13\\ud835\\udc54+1,...,\\ud835\\udf0b \\ud835\\udf13\\ud835\\udc4f\\u22121,\\ud835\\udc44\\ud835\\udf031,B)\\naddition(\\ud835\\udf0bb\\ud835\\udf131,...,\\ud835\\udf0b b\\ud835\\udf13\\ud835\\udc4f\\u22121,\\ud835\\udf0b\\ud835\\udf19,X,B)\\n\\ud835\\udc56\\u2190\\ud835\\udc56+\\ud835\\udc4f\\nfunction addition (X,B,\\ud835\\udf0bb\\ud835\\udf13...)\\nfor\\ud835\\udf0bb\\ud835\\udf13...do\\n(\\ud835\\udc53,transitions)\\u2190\\ud835\\udc39(\\ud835\\udf0bb\\ud835\\udf13),\\ud835\\udc51\\u2190\\ud835\\udc37(\\ud835\\udf0bb\\ud835\\udf13)\\ninsert(B,transitions)\\nifX(\\ud835\\udc51)=\\u2205or\\ud835\\udc39(X(\\ud835\\udc51))<\\ud835\\udc53then\\nX(\\ud835\\udc51)\\u2190\\ud835\\udf0bb\\ud835\\udf13\\nAlgorithm 6 Actor-Critic Training\\nfunction train_actor_critic (\\ud835\\udc44\\ud835\\udf031,\\ud835\\udc44\\ud835\\udf032,\\ud835\\udf0b\\ud835\\udf19,B)\\nfor\\ud835\\udc61=1\\u2192\\ud835\\udc5bdo\\nSample\\ud835\\udc41transitions(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e),\\ud835\\udc60\\u2032)fromB\\nSample smoothing noise \\ud835\\udf16\\n\\ud835\\udc66\\u2190\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e)+\\ud835\\udefemin\\n\\ud835\\udc56=1,2\\ud835\\udc44\\ud835\\udf03\\u2032\\n\\ud835\\udc56\\u0000\\ud835\\udc60\\u2032,\\ud835\\udf0b\\ud835\\udf19\\u2032(\\ud835\\udc60\\u2032)+\\ud835\\udf16\\u0001\\nUpdate both critics by regression to \\ud835\\udc66\\nif\\ud835\\udc61mod\\u0394then\\nUpdate actor using the deterministic policy gradient:\\n1\\n\\ud835\\udc41\\u00cd\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60)\\u2207\\ud835\\udf19\\ud835\\udf0b\\ud835\\udf19(\\ud835\\udc60)\\nSoft-update target networks \\ud835\\udc44\\ud835\\udf03\\ud835\\udc56\\u2032and\\ud835\\udf0b\\ud835\\udf19\\u2032Algorithm 7 PG Variation\\nfunction variation_pg (\\ud835\\udf0b\\ud835\\udf13...,\\ud835\\udc44 \\ud835\\udf031,B)\\nfor\\ud835\\udf0b\\ud835\\udf13...do\\nfor\\ud835\\udc56=1\\u2192\\ud835\\udc5ado\\nSample\\ud835\\udc41transitions(\\ud835\\udc60,\\ud835\\udc4e,\\ud835\\udc5f(\\ud835\\udc60,\\ud835\\udc4e),\\ud835\\udc60\\u2032)fromB\\nUpdate actor using the deterministic policy gradient:\\n1\\n\\ud835\\udc41\\u00cd\\u2207\\ud835\\udc4e\\ud835\\udc44\\ud835\\udf031(\\ud835\\udc60,\\ud835\\udc4e)|\\ud835\\udc4e=\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60)\\u2207\\ud835\\udf13\\ud835\\udf0b\\ud835\\udf13(\\ud835\\udc60)\\nreturn\\ud835\\udf0bb\\ud835\\udf13...\\nB HYPERPARAMETERS\\nIn Tab. 3 we provide the hyperparameters used for DCG-MAP-Elites\\nand the baselines on all tasks.\\n10 Table 3: Hyperparameters for DCG-MAP-Elites and all baselines\\nParameter MAP-Elites MAP-Elites ES PGA-MAP-Elites QD-PG DCG-MAP-Elites\\nNumber of centroids 1024 1024 1024 1024 1024\\nEvaluation batch size \\ud835\\udc4f 256 1050 256 256 256\\nPolicy networks [128, 128,|A|] [128, 128,|A|] [128, 128,|A|] [128, 128,|A|] [128, 128,|A|]\\nNumber of GA variations \\ud835\\udc54 256 0 128 86 128\\nGA variation param. 1 \\ud835\\udf0e1 0.005 0 .005 0 .005 0 .005\\nGA variation param. 2 \\ud835\\udf0e2 0.05 0 .05 0 .05 0 .05\\nActor network [256, 256,|A|] [256, 256,|A|] [256, 256,|A|]\\nCritic network [256, 256, 1] [256, 256, 1] [256, 256, 1]\\nTD3 batch size \\ud835\\udc41 100 100 100\\nQuality critic training steps \\ud835\\udc5b 3000 3000 3000\\nDiversity critic training steps \\ud835\\udc5b 300\\nPG training steps \\ud835\\udc5a 150 150 150\\nPolicy learning rate 5\\u00d710\\u221235\\u00d710\\u221235\\u00d710\\u22123\\nActor learning rate 3\\u00d710\\u221243\\u00d710\\u221243\\u00d710\\u22124\\nCritic learning rate 3\\u00d710\\u221243\\u00d710\\u221243\\u00d710\\u22124\\nReplay buffer size 106106106\\nDiscount factor \\ud835\\udefe 0.99 0 .99 0 .99\\nActor delay \\u0394 2 2 2\\nTarget update rate 0.005 0 .005 0 .005\\nSmoothing noise var. \\ud835\\udf0e 0.2 0 .2 0 .2\\nSmoothing noise clip 0.5 0 .5 0 .5\\nSample number 1000\\nSample sigma 0.02\\nlengthscale \\ud835\\udc59 0.008\\nDescriptor sigma \\ud835\\udf0e\\ud835\\udc51 0.0004\\n11\",\"42\":\" A Computer Vision Enabled damage detection model with\\nimproved YOLOv5 based on Transformer Prediction Head\\nArunabha M. Roy1and Jayabrata Bhaduri2\\n1Aerospace Engineering Department, University of Michigan, Ann Arbor, MI\\n48109, USA\\n2Capacloud AI, Deep Learning &Data Science Division, Kolkata, WB 711103,\\nIndia.\\nAbstract\\nObjective. Computer vision-based up-to-date accurate damage classification and localization\\nare of decisive importance for infrastructure monitoring, safety, and the serviceability of civil\\ninfrastructure. Current state-of-the-art deep learning (DL)-based damage detection models,\\nhowever, often lack superior feature extraction capability in complex and noisy environments,\\nlimiting the development of accurate and reliable object distinction. Method. To this end,\\nwe present DenseSPH-YOLOv5, a real-time DL-based high-performance damage detection\\nmodel where DenseNet blocks have been integrated with the backbone to improve in preserving\\nand reusing critical feature information. Additionally, convolutional block attention modules\\n(CBAM) have been implemented to improve attention performance mechanisms for strong and\\ndiscriminating deep spatial feature extraction that results in superior detection under various\\nchallenging environments. Moreover, an additional feature fusion layers and a Swin-Transformer\\nPrediction Head (SPH) have been added leveraging advanced self-attention mechanism for more\\nefficient detection of multiscale object sizes and simultaneously reducing the computational\\ncomplexity. Results. Evaluating the model performance in large-scale Road Damage Dataset\\n1arXiv:2303.04275v1  [cs.CV]  7 Mar 2023 2\\n(RDD-2018), at a detection rate of 62.4 FPS, DenseSPH-YOLOv5 obtains a mean average\\nprecision (mAP) value of 85 :25%, F1-score of 81 :18%, and precision (P) value of 89 :51%\\noutperforming current state-of-the-art models. Significance. The present research provides\\nan effective and efficient damage localization model addressing the shortcoming of existing\\nDL-based damage detection models by providing highly accurate localized bounding box\\nprediction. Current work constitutes a step towards an accurate and robust automated damage\\ndetection system in real-time in-field applications.\\nKeywords: Automated damage detection; You Only Look Once (YOLOv5) algorithm; Swin\\nTransformer Object Detection (OD); Computer vision; Deep Learning (DL)\\n1. Introduction :\\nIn recent years, automated damage detection plays an important role in various industrial\\napplications including product quality assessment (Agarwal and Singh, 2015; Hanzaei et al.,\\n2017), infrastructure monitoring (Eisenbach et al., 2017; Gopalakrishnan, 2018), safety and\\nthe serviceability of civil infrastructure (Koch et al., 2015; Hartmann and Trappey, 2020).\\nEarly-stage accurate crack detection is critical for pavement damage rating and subsequent\\nsealing or rehabilitation activities (Chen et al., 2021; Chen and Cho, 2022). Therefore, it\\nis important for roadway infrastructure engineers to detect pavement cracks accurately so\\nthat the best cost-effective plans of maintenance and rehabilitation could be employed (Ni\\net al., 2022; Dong et al., 2021). While traditional damage detection techniques mainly include\\nvisual inspection, however, such labor-intensive methods have disadvantages due to their low\\nefficiency, high cost, and individual biases (Xu et al., 2022; Shang et al., 2023). Additionally, it\\nis also limited in reproducibility, reliability, and objectivity due to the requirement of qualified\\npersonnel for domain-specific experience, knowledge, and skill sets (Fang et al., 2020). To\\ncircumvent such issues, more recently, various automatic and semi-automatic crack detection\\nalgorithms have gained significant attraction (Gopalakrishnan, 2018).\\nFor the last three decades, image-based crack detection (Mohan and Poobal, 2018; Koch\\net al., 2015) that include various image processing approaches, such as edge detection (Zhao 3\\net al., 2010; Hanzaei et al., 2017; Li et al., 2022), dynamic thresholding (Oliveira and Correia,\\n2009; Wang et al., 2021b), and different morphological operations (Anitha et al., 2021; Li\\nand Zhao, 2021) have been the central focus for detecting damage in challenging real-world\\nscenarios. However, the aforementioned methods are quite sensitive to noise and varying\\nillumination intensities, and hence, not suitable in real-world complex conditions (Koch et al.,\\n2015). To circumvent such issues, later, conventional machine learning (ML) approaches\\nhave been introduced for damage detection (Mohan and Poobal, 2018; Koch et al., 2015). In\\ngeneral, such methods utilize a trained classifier such as a Support Vector Machine (SVM)\\non local feature descriptors that can be either Local Binary Patterns (LBP) (Varadharajan\\net al., 2014; Quintana et al., 2015) or Histogram of Oriented Gradient (HOG) (Kapela et al.,\\n2015). Although, compared to conventional image processing approaches, ML-based models\\nsignificantly improve the accuracy and efficiency of the damage detection, however, due to large\\nerrors in classification performances remains a serious bottleneck for deploying such models in\\nreal-world applications (Fang et al., 2020).\\nMore recently, deep learning (DL) characterized by multilayer neural networks (NN) (LeCun\\net al., 2015) has shown remarkable breakthroughs in pattern recognition for various fields\\nincluding image classification (Rawat and Wang, 2017; Jamil et al., 2022; Khan et al., 2022b,a),\\ncomputer vision (Voulodimos et al., 2018; Roy and Bhaduri, 2021; Roy et al., 2022c; Roy and\\nBhaduri, 2022; Roy et al., 2022a), object detection (Zhao et al., 2019a; Chandio et al., 2022;\\nRoy et al., 2022b; Singh et al., 2023a), brain-computer interfaces (Roy, 2022b,a,c; Singh et al.,\\n2023b), signal classification (Jamil and Roy, 2023, 2022) and across diverse scientific disciplines\\n(Bose and Roy, 2022; Roy and Bose, 2023b; Roy and Guha, 2022; Roy and Bose, 2023a; Roy\\nand Guha, 2023). Following the success, there is an increasing thrust of research works geared\\ntowards damage classification tasks employing DL techniques, mostly convolutional neural\\nnetworks (CNN), such as ResNet (Bang et al., 2018), AlexNet (Dorafshan et al., 2018; Li\\net al., 2018), VGG-net (Gopalakrishnan et al., 2017; Silva and Lucena, 2018) and various\\nothers (Chow et al., 2020; Nath et al., 2022; Li et al., 2021). Particularly in object localization,\\nDL methods have demonstrated superior accuracy (Han et al., 2018) that can be categorized\\ninto two classes: two-stage and one-stage detector (Lin et al., 2017a). Two-stage detectors\\nincluding Region Convolution Neural Network (R-CNN) (Girshick, 2015), faster R-CNN (Ren 4\\net al., 2016), mask R-CNN (He et al., 2017) etc that have shown a significant improvement\\nin accuracy in object localization. In recent times, You Only Look Once (YOLO) variants\\n(Redmon et al., 2016; Redmon and Farhadi, 2017, 2018; Bochkovskiy et al., 2020) have been\\nproposed that unify target classification and localization. In (Roy et al., 2022c; Roy and\\nBhaduri, 2022, 2021), DenseNet (Huang et al., 2017) blocks attaching Spatial Pyramid Pooling\\n(SPP) (He et al., 2015) with an improved modified Path Aggregation Network (PANet) (Liu\\net al., 2018) has been integrated to enhance the representation of receptive fields and extraction\\nof important contextual features in the original YOLOv4 leading to significant improvement in\\nthe detection speed and accuracy. In order to enhance gradient performance and reduce the\\ncomputational cost, YOLOv4 (Bochkovskiy et al., 2020) designs a cross-stage partial (CSP)\\nnetwork. To further improve detection accuracy, YOLOv4 implements Mish activation (Misra,\\n2020) and CIoU loss (Zheng et al., 2020). Recently, Scaled-YOLOv4 (Wang et al., 2021a)\\ndemonstrated its superior performance in detecting the vast range of linearly scaled objects for\\nvarious applications. As the latest generation of the YOLO series, the YOLOv5 (Jocher et al.,\\n2021) has been rated top among state-of-the-art object detection models which inherit all the\\nabove advantages. Thus, in the present work, YOLOv5 has been considered a benchmark model\\nfor multiclass damage detection. More recently, improved YOLOv5 Based on Transformer\\nPrediction Head (TPH-YOLOv5) (Zhu et al., 2021) has been proposed integrating convolutional\\nblock attention module (CBAM) (Woo et al., 2018) for closely packed object detection and\\nSwin Transformer-enabled YOLOv5 (SPH-YOLOv5) (Gong et al., 2022) has been designed\\nincorporating Normalization-based Attention Modules (NAM) that demonstrate significant\\nimprovement in accuracy while simultaneously reducing the computational complexity of the\\nmodel which are the main motivations for the network architectural development of the current\\nwork.\\n1.1 Related Works :\\nIn this section, some recent and relevant DL works have been highlighted in the field of road\\ndamage detection. In recent years, multiple studies have adopted various ML and DL-based\\napproaches for automated road surface damage classification and detection (Zhang et al., 2017a; 5\\nStricker et al., 2019; Bi\\u0018 cici and Zeybek, 2021) For instance, a smartphone-based supervised deep\\nconvolutional neural network (D-CNN) has been proposed for pavement damage classification\\n(Zhang et al., 2016). Along a similar line, deep neural network (DNN) architecture has been\\nemployed for detecting cracks and potholes (Anand et al., 2018; Silva and Lucena, 2018) as well\\nas pavement condition assessment (Fan et al., 2018). In Nhat-Duc et al. (2018), the superiority\\nof the DCNN-based approach has been demonstrated over edge-detection-based approaches for\\npavement crack detection. In Maeda et al. (2018), a real-time road damage detection model\\nbased on SDD has been proposed that has been trained on a publicly available large-scale\\nroad damage dataset (RDD-2018) for eight different categories of road damages. Due to the\\npopularity of the dataset, various attempts have been made, notably using YOLO (Alfarrarjeh\\net al., 2018), Faster R-CNN (Kluger et al., 2018), Faster R-CNN with ResNet-152 (Wang et al.,\\n2018a), Ensemble models with Faster R-CNN and SSD (Wang et al., 2018b), and RetinaNet\\n(Angulo et al., 2019) to further improve the detection performance. In addition, RetinaNet\\n(Angulo et al., 2019) has been used on a modified RDD-2018 dataset which demonstrates\\nsignificant performance improvement. Following the work of Maeda et al. (2018), progressive\\ngrowing- generative adversarial networks (PG-GANs) (Maeda et al., 2021) with Poisson blending\\nhave been used to generate new training data (RDD-2019) in order to improve the accuracy of\\nroad damage detection. More recently, transfer learning (TL)-based road damage detection\\nmodel (Arya et al., 2021a) has been proposed introducing large-scale open-source dataset\\nRDD2020 (Arya et al., 2020, 2021b) considering multiple countries. In Naddaf-Sh et al. (2020),\\nEfficientDet-D7 has been employed for the detection of asphalt pavement distress. Whereas,\\nYOLO CSPDarknet53 (Mandal et al., 2020) has been used for road damage detection. Similarly,\\nthe YOLO network has been used for detecting pavement distress from high-resolution images\\n(Du et al., 2021). In Majidifard et al. (2020), YOLOv2 model has been utilized for pavement\\ndistress classification from Google street view images. Along a similar line, a CNN-based\\npredictive model trained in Google API images has been employed for detecting potholes\\nin Patra et al. (2021). In a separate work in Guan et al. (2021), a stereo-vision integrated\\nsegmentation-based DL model with modified depth-wise separable convolution U-net has\\nbeen deployed for crack and pothole detection where multi-feature image datasets have been\\nused to train the model. More recently, a semi-supervised DL-based pixel-level segmentation 6\\nmodel (Karaaslan et al., 2021) has been proposed utilizing attention guidance for cracks and\\nspalls localization that reduces computational cost significantly. In separate work, an improved\\nYOLOv5 road damage detection algorithm (Guo and Zhang, 2022) has been proposed leveraging\\nMobileNetV3 as a backbone feature extractor. In Hac\\u0010efendio\\u0015 glu and Ba\\u0018 sa\\u0015 ga (2022), Faster\\nR-CNN has been employed for concrete pavement crack detection under various illumination and\\nweather conditions. Although, there exist several state-of-the-art works for damage detection\\nincluding multi-class damage localization models, however, they often suffer from low accuracy,\\nmissed detection, and relatively large computational overhead (Cao et al., 2020; Azimi et al.,\\n2020; Naddaf-Sh et al., 2020).\\n1.2 Motivations :\\nDespite illustrating outstanding performance in damage detection, current state-of-the-art\\nDL algorithms still require further improvement due to their insufficient fine-grain contextual\\nfeature extraction capability leading to missed detection and false object predictions for various\\ndamages\\/cracks which possess a wide range of textures, shapes, sizes, and colors (Cao et al.,\\n2020; Azimi et al., 2020; Naddaf-Sh et al., 2020). Between various damage classes, accurate\\ndetection and localization tasks can be challenging due to significant variability of lightening\\nconditions, low visibility, the coexistence of multi-object classes with various aspect ratios, and\\nother morphological characteristics (Azimi et al., 2020; Naddaf-Sh et al., 2020). Additionally,\\nvisual similarities, complex backgrounds, and various other critical factors offer additional\\ndifficulties for the state-of-the-art damage detection models (Naddaf-Sh et al., 2020). To\\nthis end, the current works aim to develop an efficient and robust damage classification and\\naccurate damage localization model simultaneously productive in terms of training time and\\ncomputational cost which is currently lacking in the recent state-of-the-art endeavors.\\n1.3 Contributions :\\nTo address the aforementioned shortcomings, in the current study, we present DenseSPH-YOLOv5\\nbased on an improved version of the state-of-art YOLOv5 detection model for accurate real-time 7\\ndamage detection. The major contributions of the present research work can be summarized as\\nfollows:\\n\\u2022In Dense-SPH-YOLOv5, we have attached DenseNet blocks with CSP modules in the\\nCSPDarknet53 to preserve critical feature maps and efficiently reuse the discriminating\\nfeature information.\\n\\u2022Secondly, we have introduced an additional detection head specifically for detecting tiny\\nobjects in the head part of the proposed DenseSPH-YOLOv5 network.\\n\\u2022In addition, the convolutional block attention module (CBAM) has been implemented for\\nthe construction of progressive feature attention with large coverage along both channel\\nand spatial dimensions for strong and discriminating deep spatial feature extraction during\\nobject detection.\\n\\u2022Furthermore, the regular CNN prediction heads (CPH) in YOLOv5 have been upgraded\\nwith Swin transformer Prediction Heads (SPHs) employing Swin transformer (STR)\\nencoder block leveraging advanced self-attention mechanisms for efficient detection of\\nmulti-scale object sizes and simultaneously reducing the computational complexity.\\n\\u2022Spatial Pyramid Pooling (SPP) has been tightly attached to the backbone to enhance\\nthe representation of receptive fields and extraction of important contextual features.\\n\\u2022Finally, an improved modified Path Aggregation Network (PANet) has been utilized\\nto efficiently preserve fine-grain localized information by feature fusion in a multi-scale\\nfeature pyramid map.\\nWith the aforementioned modifications, the detection accuracy of the model has been\\nsignificantly enhanced for multi-scale object detection. An extensive ablation study has been\\nperformed for different combinations of backbone-neck architecture in order to optimize both\\naccuracies of detection and detection speed. The proposed DenseSPH-YOLOv5 has been\\nemployed to detect distinct eight different damage classes that provide superior and accurate\\ndetection under various complex and challenging environments. The present work effectively\\naddresses the shortcoming of existing DL-based crack detection models and illustrates its 8\\nFigure 1: Sample images from RDD-2018 dataset (Maeda et al., 2018): (a) to (g) correspond\\nto each of the eight categories with the legends.\\nsuperior potential in real-time in-field applications. The rest of the paper is organized as follows:\\ndescription of the dataset has been described in Section 2; Section 3 introduces the proposed\\nmethodology for damage detection; the relevant finding and discussion of the proposed model\\nhave been discussed in Sections 4 and 5, respectively. Finally, the conclusions of the present\\nwork have been discussed in section 6.\\n2. Road Damage Dataset :\\nIn the current study, large-scale Road Damage Dataset (RDD-2018) (Maeda et al., 2018)\\nhas been used that consists of 9,053 labeled road damage images of resolution 600 \\u0002600 pixels\\ncontaining a total number of 15,435 annotated bounding boxes for eight different types of\\nroad damage. To avoid biases, the images have been photographed in various weather and\\nillumination conditions from different regions of Japan including Chiba, Muroran, Ichihara,\\nSumida, Nagakute, and Numazu. During annotation, professional road damage expertise\\nhas been employed to verify various damage classes that ensure the reliability of the dataset.\\nVarious damage types and corresponding class identifiers have been listed in Table. 1. Each 9\\nTable 1: Various road damage types and corresponding class identifiers in RDD-2018 dataset\\n(Maeda et al., 2018).\\nClass\\nIdentifierDamage\\ntypeAlignment Details\\nD00 Linear Crack Longitudinal Wheel-marked part\\nD01 Linear Crack Longitudinal Construction joint part\\nD10 Linear Crack Lateral Equal interval\\nD11 Linear Crack Lateral Construction joint part\\nD20 Alligator Crack - Partial pavement, overall pavement\\nD40 Other Crack - Pothole, rutting, separation\\nD43 Other Crack - White line blur\\nD44 Other Crack - Cross walk blur\\ntype of damages have been illustrated in Fig. 1. Primarily, the damage has been classified into\\ncracks or different corruptions. Then, the cracks have been divided into linear and alligator\\ncracks. Whereas, other corruptions include both potholes and rutting as well as other road\\ndamage classes such as blurring of white lines.\\n3. Proposed Methodology for damage detection:\\nIn object detection, target object classification and localization are performed simultaneously\\nwhere the target class has been categorized and separated from the background. The purpose\\nof object localization is to locate objects by drawing bounding boxes (BBs) on input images\\ncontaining the entire object. This is particularly useful for counting endangered species for\\naccurate surveying. To this end, the main goal of the current work is to develop an efficient and\\nrobust damage classification and accurate damage localization model. In this regard, different\\nvariants of YOLO (Redmon et al., 2016; Redmon and Farhadi, 2017, 2018; Bochkovskiy et al.,\\n2020) are some of the best high-precision one-stage object detection models. More recently, 10\\n \\n\\u03b7 \\nd \\nwp \\nhp \\nwgt \\nhgt \\n(a) (b) Damage  detection  \\nInput  N \\u00d7N  grids  \\nBBs+ confidence  score  \\nClass probability   \\n \\n \\n \\nFigure 2: Schematic of (a) YOLO object localization process for damage localization; (b)\\nSchematic of CIoU offset regression for target BBs predictions.\\nYOLOv5 (Jocher et al., 2021) has been introduced that currently achieves the best detection\\nperformance and has four different model variants including YOLOv5s, YOLOv5m, YOLOv5l,\\nand YOLOv5x depend on different model widths and depths. In general, the overall architecture\\nof YOLOv5 consists of the following parts: a backbone for deep feature extraction, followed by\\nthe neck for gathered semantic feature fusion, and finally head network for object classification\\nand localization. The original version of YOLOv5 utilizes CSPDarknet53 (Wang et al., 2020;\\nBochkovskiy et al., 2020) with SPP and PANet (Liu et al., 2018) as backbone and neck,\\nrespectively. Whereas, YOLO detection head (Redmon et al., 2016) has been employed in\\nthe detection head. The YOLO model transforms the object detection task into a regression\\nproblem by generating BBs coordinates and probabilities for each class as shown in Fig. 2.\\nDuring the process, the inputted image size has been uniformly divided into N\\u0002Ngrids where\\nBpredictive BBs have been generated. Subsequently, a confidence score has been assigned if\\nthe target object falls inside that particular grid. It detects the target object for a particular\\nclass when the center of the ground truth lies inside a specified grid. During detection, each\\ngrid predicts NBnumbers of BBs with the confidence value \\b Bas:\\n\\bB=Pr(obj)\\u0002IoUt\\np_P r(obj)20;1 (1) 11\\nwherePr(obj) infers the accuracy of BB prediction, i.e., Pr(obj) = 1 indicates that the target\\nclass falls inside the grid, otherwise, Pr(obj) = 0. The degree of overlap between ground truth\\nand the predicted BB has been described by the scale-invariant evaluation metric intersection\\nover union (IoU) which can be expressed as\\nIoU =Bp\\\\Bgt\\nBp[Bgt(2)\\nwhere BgtandBpare the ground truth and predicted BBs, respectively.\\n3.1 Loss in BBs regression :\\nTo further improve BBs regression and gradient disappearance, generalized IoU (GIoU)\\n(Rezatofighi et al., 2019) and distance-IoU (DIoU) (Zheng et al., 2020) as been introduced\\nconsidering aspect ratios and orientation of the overlapping BBs. More recently, complete IoU\\n(CIoU) (Zheng et al., 2020) has been proposed for improved accuracy and faster convergence\\nspeed in BB prediction which can be expressed as\\nLCIoU= 1 +\\f\\u0018+\\u000b2(bp;bgt)\\n\\u00112\\u0000IoU (3)\\n\\u0018=4\\n\\u00192\\u0012\\ntan\\u00001wgt\\nhgt\\u0000tan\\u00001wp\\nhp\\u00132\\n;\\f=\\u0018\\n(1\\u0000IoU) +\\u00180 (4)\\nwhere bgtandbpdenotes the centroids of BgtandBp, respectively; \\u0018and\\fare the consistency\\nand trade-off parameters, respectively. As shown in Fig. 3 -(b), \\u0011is the smallest diagonal length\\nofBp[Bgt;wgt,wpare widths and hgt,hpare heights of BgtandBp, respectively. With\\nincreasingwp=hp, we get\\u0018!0 from Eq. 4. Therefore, to optimize the influence of \\u0018on the\\nCIoU,wp=hpcan be properly chosen for the detection model. Finally, the best BB prediction\\ncan be obtained from the non-maximum suppression (NMS) (Ren et al., 2016) algorithm from\\nvarious scales. In object detection in the YOLO framework, The total loss function \\u0001 lconsist\\nof BB coordinate prediction error \\u0001 cor, IoU error \\u0001 IoU, classification error term \\u0001 cl. The loss 12\\nFigure 3: Schematic illustration of measuring various performance metrics: (a) Intersection\\nover Union (IoU); (b) precision (P); (c) recall (R) during damage detection process.\\nfunction \\u0001 lfor BBs regression can be formulated as,\\n\\u0001l= \\u0001 cor+ \\u0001 cl+ \\u0001 IoU: (5)\\n3.2 Performance metrics:\\nIn the present work, the performance of the object detection models has been evaluated\\nby common standard measures (Ferri et al., 2009) including average precision (AP), precision\\n(P), recall (R), IoU, F-1 score, mean average precision (mAP), etc. The confusion matrix\\nobtained from the evaluation procedure provides the following interpretations of the test results:\\ntrue positive (TP), false positive (FP), false negative (FN), and true negative (TN). During\\nobject classification, the classified object can be defined as TP for IoU \\u00150:5. Whereas, it can\\nbe classified as FP for IoU <0:5. Based on the aforementioned interpretations, the metric P of 13\\nthe classifier can be defined by its ability to distinguish target classes correctly as :\\nP =TP\\n(TP+FP); (6)\\nThe ratio of the correct prediction of target classes is called R of the classifier which can be\\nevaluated as:\\nR =TP\\n(TP+FN)(7)\\nThe higher values of P and R indicate superior detection capability. Whereas, F-1 score is the\\narithmetic mean of the P and R given as :\\nF1 = 2\\u0002\\u0012P R\\nP + R\\u0013\\n(8)\\nA relatively high F1 score represents a robust detection model. The performance metrics AP\\ncan be defined as the area under a P-R curve (Davis and Goadrich, 2006) as follows\\nAP =Z1\\n0P (R) dR (9)\\nA higher average AP value indicates better accuracy in predicting various object classes. In\\naddition, AP 50:95denotes AP over IoU=0 :50 : 0:05 : 0:95; AP 50and AP 75are APs at IoU\\nthreshold of 50% and 75%, respectively. The AP for detecting small, medium, and large objects\\ncan be measured through AP S, AP M, and AP L, respectively. Finally, mAP can be obtained\\nfrom the average of all APs as:\\nmAP =1\\nNcNX\\ni=1APi: (10)\\nIn YOLOv5, various combinations of activation functions including sigmoid, leaky- ReLU and\\nSiLU (Hendrycks and Gimpel, 2016) can be utilized to improve the performance of the model\\nfor a specific detection task. In addition, bag of freebies and specials (Bochkovskiy et al.,\\n2020) can also be employed to further optimize the detection architecture of YOLOv5. As\\nthe latest generation of the YOLO series, YOLOv5 has been shown to provide state-of-the-art\\ndetection performance. Therefore, it has been chosen as the baseline model for our present study. 14\\n3.3 DenseSPH-YOLOv5 architecture:\\nIn recent years, various attempts have been made on DL-based computer vision models\\nfor damage detection such as Faster R-CNN (Kluger et al., 2018; Wang et al., 2018a), SSD\\n(Maeda et al., 2018; Wang et al., 2018b), RetinaNet (Angulo et al., 2019), YOLO (Alfarrarjeh\\net al., 2018; Mandal et al., 2020), YOLOv2 (Majidifard et al., 2020), YOLOv5 (Guo and\\nZhang, 2022) etc. Although the aforementioned techniques have demonstrated outstanding\\nperformance, however, the damage detection task faces several challenges, in particular, due to\\nthe presence of complex and noisy backgrounds, significant variability of lightening conditions,\\nlow visibility, densely packed classes, and overlap, the coexistence of multi-object classes with\\nvarious aspect ratios, and other morphological characteristics (Azimi et al., 2020; Naddaf-Sh\\net al., 2020). In this regard, YOLOv5 can be a suitable model which demonstrates both superior\\naccuracy and faster detection speed compared to YOLOv4. The state-of-the-art YOLOv5 is\\ncomparatively light, yet effective, which performs multiple downsampling and turns shallow\\nsemantic information into high-level semantic information that effectively reduces the number\\nof parameters. However, this inevitably loses semantic information and makes the network\\nineffective at extracting and fusing discriminating semantic features. Therefore, the original\\nYOLOv5 network requires further improvement due to its insufficient fine-grain contextual\\nfeature extraction capability leading to missed detection and false object predictions for various\\ndamages\\/cracks which possess a wide range of textures, shapes, sizes, and colors (Cao et al., 2020;\\nAzimi et al., 2020; Naddaf-Sh et al., 2020). In order to achieve high accuracy and high efficiency\\nin damage detection, we propose a novel object localization algorithm DenseSPH-YOLOv5\\nbased on a state-of-the-art YOLOv5 network to enhance feature extraction, preserve fine-grain\\nlocalized information and improve feature fusion that provides superior damage detection\\nunder various challenging environments. The overall network of the DenseSPH-YOLOv5 model\\nis shown in Fig. 4. To improve performance in terms of classification accuracy and object\\nlocalization, we perform extensive experiments, and various modifications are proposed which\\nare detailed in the subsequent sections. 15\\nFigure 4: Schematic of the proposed DenseSPH YOLOv5 with fused DenseNet in CSPDarknet53\\nbackbone; CBAM module in the improved neck; four SPHs utilize hierarchical feature maps\\nfrom STR encoder blocks in the modified Neck.\\n3.3.1 DenseNet block : Since YOLOv5 reduces the feature maps from the inputted images\\nthrough convolution and down-sampling procedures that result in significant semantic feature\\nloss during transmission. To this end, we have introduced DenseNet (Huang et al., 2017)\\nin the original CSPDarknet53 of YOLOv5 attached to the CSP module to preserve critical\\nfeature maps and efficiently reuse the discriminative feature information as shown in Fig. 4. In\\nDenseNet, each layer has been connected to other layers in a feed-forward mode where n-th\\nlayer can receive the important feature information \\u0018nfrom all the previous layers \\u00180;\\u00181;:::;\\u0018 n\\u00001\\nas:\\n\\u0018n= \\u0002 n[\\u00180; \\u00181; :::;\\u0018 n\\u00001] (11)\\nwhere \\u0002 nis the feature map function for n-th layer. The schematic of the DenseNet blocks\\nnetwork structure have been shown in Fig. 5-(d). Through our extensive experiments, we found\\nout that DenseNet improves the feature transfer and mitigates over-fitting in the detection\\nnetwork. Therefore, in the proposed Dense-SPH YOLOv5 network, as shown in Fig. 3, we\\nhave introduced four DenseNet blocks: the first block (Dense B-1) has been attached before\\ncross-stage partial block CSP3; the second block (Dense B-2) has been placed before CSP6; 16\\nFigure 5: Schematic of (a) CBAM that has two sequential sub-modules: (b) CAM and (c) SAM\\nfor adaptive refinement of feature map at every intermediate convolution blocks; (d) network\\narchitecture of DenseNet block used in Dense-SPH YOLOv5 detection model. 17\\nwhereas, third (Dense B-3) and fourth (Dense B-4) blocks have been added before CSP6 and\\nCSP3, respectively which results in enhanced feature propagation. Additionally, by reducing\\nredundant feature operations, the Dense-CSPDarknet53 network improves the computational\\nspeed.\\n3.3.2 Convolutional block attention module (CBAM): Convolutional block attention\\nmodule (CBAM) (Woo et al., 2018) is a lightweight, yet effective attention block which can be\\nintegrated into the object detection model in an end-to-end manner that has shown superior\\ndetection accuracy (Zhu et al., 2021). For an inputted feature map, CBAM sequentially\\nemploys the attention map along the channel and spatial dimensions. As shown in Fig. 5-(a),\\nthe network structure of the CBAM module consists of two sequential sub-modules: Channel\\nAttention Module (CAM) and Spatial Attention Module (SAM) for adaptive refinement of the\\nfeature map at every intermediate convolution blocks by multiplying input feature map with the\\nattention map. For inputted feature map FcFcFc2RC\\u0002H\\u0002W, CAM produces 1D channel attention\\nmapMcMcMc2RC\\u00021\\u00021, and sequentially, SAM infers 2D spatial attention map MsMsMs2R1\\u0002H\\u0002Was\\nshown in Figs. 4-(a-c). The overall CBAM attention process can be expressed as:\\nF0\\ncF0\\ncF0\\nc=McMcMc(FcFcFc)\\nFcFcFc;F00\\ncF00\\ncF00\\nc=MsMsMs(F0\\ncF0\\ncF0\\nc)\\nF0\\ncF0\\ncF0\\nc; (12)\\nwhere ;F00\\ncF00\\ncF00\\ncis the CBAM refined output; \\nrepresents element-wise multiplication. More\\nspecifically, CAM produces two different spatial descriptors including average-pooled features\\nFc\\navgFc\\navgFc\\navgand max pooled features Fc\\nmaxFc\\nmaxFc\\nmaxwhich produce channel attention map McMcMc2RC\\u00021\\u00021from\\nmulti-layer perceptron (MLP). The CAM attention can be expressed as:\\nMcMcMc(FcFcFc) =\\u001b(W1W1W1(W0W0W0(Fc\\navgFc\\navgFc\\navg))) +W1W1W1(W0W0W0(Fc\\nmaxFc\\nmaxFc\\nmax)) (13)\\nwhereW0W0W02RC=r\\u0002C, andW1W1W12RC\\u0002C=rare the MLP weights ; \\u001brepresents the sigmoid\\nactivation function. In SAM, inter-spatial attention map MsMsMs(FcFcFc)2RH\\u0002Whas been aggregated\\nto generate 2D maps: Fs\\navgFs\\navgFs\\navg2R1\\u0002H\\u0002WandFs\\nmaxFs\\nmaxFs\\nmax2R1\\u0002H\\u0002W. From these two feature maps, the 18\\nspatial attention can be computed as:\\nMsMsMs(FFF) =\\u001b(fn\\u0002n([Fs\\navgFs\\navgFs\\navg;Fs\\nmaxFs\\nmaxFs\\nmax]) (14)\\nwherefn\\u0002ndenotes a convolution operation with the filter size of n\\u0002nwithn= 7. The\\nimplementation of the CBAM module into the proposed DenseSPH-YOLOv5 significantly\\nimproves the detection accuracy of the damage detection by providing specific attention to the\\ndense objects and confusing noisy areas which proved the effectiveness of this module. Overall,\\nit helps to learn more expressive features that demonstrate significant improvement in detection\\naccuracy for road damage datasets considered herein.\\n3.3.3 Swin transformer encoder blocks: Inspired by the superior performance of the\\nvision transformer (Dosovitskiy et al., 2020) in dense and occluded object detection, in the\\nproposed model, Swin transformer (STR) encoder blocks (Liu et al., 2021) have been fused\\nto all four detection heads of DenseSPH-YOLOv5 architecture as shown in Fig. 3. Such\\nimplementation improves the global semantic feature extraction and contextual information\\nfusion leveraging self-attention mechanism (Vaswani et al., 2017) that demonstrated superior\\nperformance in dense object detection (Gong et al., 2022).\\nFor inputted feature map FsFsFs2RH\\u0002W\\u0002C, it transforms to Qs;Ks;VsQs;Ks;VsQs;Ks;Vs2RN\\u0002C0where\\nN=H\\u0002Wto feed Multi-head Self Attention (MSA) module after linear projection and\\nreshape operations. The output feature map ZsZsZsfrom MSA aggregates global information which\\ncan be expressed as\\nZsZsZs=AsVsAsVsAsVs;AsAsAs= softmaxQsKsTQsKsTQsKsT; (15)\\nwhereAsAsAs2RN\\u0002Nis the self-attention matrix that represents the relationship between feature\\nmap elements with remaining elements. Although, MSA in Transformer is effective in generating\\na self-attention matrix by integrating multiple independent subspaces, however, the global\\ncomputation overhead is significantly high for high-resolution images. Thus, for dense prediction\\nor to tackle high-resolution images, the computational complexity of MSA in Transformer\\nis not suitable which leads to quadratic complexity with respect to the number of tokens 19\\nFigure 6: Schematic of (a) STR encoder architecture that contains (b) regular W-MSA and\\nSW-MSA used in detection heads of DenseSPH-YOLOv5 architecture.\\n(Liu et al., 2021). To this end, STR can significantly improve the computational efficiency\\nof MSA that has linear computational complexity with respect to image size which enhances\\nthe performance of the model in terms of detection speed and accuracy. Therefore, in the\\nproposed DenseSPH-YOLOv5 detection model, the STR encoder module has been fused into\\nfour different prediction heads. As shown in Fig. 6 -(b), each STR encoder contains two\\nsub-layers that include shifted window-based multi-head self-attention (MSA) module, followed\\nby a fully-connected MLP with GeLU nonlinearity. Residual connections are used after each\\nMSA module. Subsequently, Layer Norm (LN) has been added before MSA and MLP. In STR,\\nthe obtained feature map has been dived into non-overlapping separate windows in the W-MSA\\nmodule. Subsequently, self-attention has been computed in each local window. Applying\\nSW-MSA partitioning, the consecutive STR blocks can be computed as:\\n^z^z^zi=W-MSA (LN(zzzi\\u00001)) +zzzi\\u00001;zzzi=MLP (LN(^z^z^zi)) + ^z^z^zi(16)\\n^z^z^zi+1=SW-MSA (LN(zzzi)) +zzzi;zzzi+1=MLP (LN(^z^z^zi+1)) + ^z^z^zi+1(17)\\nwherezzziand^z^z^zirepresent the output features from MLP and SW-MSA modules, respectively.\\nFor a feature map FsFsFs2RH\\u0002W\\u0002Cwithm\\u0002msize of local window, the computational complexity 20\\n\\u0007 can be obtained as\\n\\u0007(MSA ) = 4HWC2+ 2(HW)2C; \\u0007( W-MSA ) = 4HWC2+ 2(HW)M2C (18)\\nwhereHW is the patch number. Evidently, for large HW, the computational overhead of global\\nMSA is exceptionally high, while, W-MSA is scalable. Thus, implementing STR significantly\\nreduces the computational complexity of the model.\\n3.3.4 Receptive field enhancement: One of the requirements of CNN is to have fixed-size\\ninput images. However, due to the different aspect ratios of the images, they have been fixed\\nby cropping and warping during the convolution process which results in losing important\\nfeatures. In this regard, SPP (He et al., 2015) applies an efficient strategy in detecting target\\nobjects at multiple length scales. To this end, we have added an SPP block integrated with\\nDCSP-3 in the Dense-CSPDarknet53 backbone to improve receptive field representation and\\nextraction of important contextual features as shown in Fig. 4.\\n4. Results:\\nIn this section, the performance and detection accuracy of the proposed DenseSPH-YOLOv5\\nframework have been discussed that been evaluated in the RDD-2018 dataset consisting of 8\\ndifferent categories of damage classes. For better clarity in BB representation, each damage type\\nhas been marked with the corresponding class identifiers as shown in Table 1. The performance\\nof the DenseSPH-YOLOv5 network has been optimized through extensive ablation studies.\\nFinally, the performance of the proposed model has been studied in detail and compared with\\nseveral state-of-the-art object detection models.\\n4.1 Training procedure :\\nIn the present work, we have performed an extensive and elaborate study to explore the\\ncomparative performance analysis of the proposed DenseSPH-YOLOv5 models for road damage\\nlocalization tasks. From the RDD-2018 dataset, a total of 80% and 20% images have been 21\\nrandomly chosen for training and validation, respectively. For all the experiments, we have\\nused a Windows 10 Pro (64-bit) based computational system that has Intel Core i5-10210U\\nwith CPU @ 2.8 GHz \\u00026, 32 GB DDR4 memory, NVIDIA GeForce RTX 2080 utilizing\\nGPU parallelization. As part of data augmentation, along with traditional methods such\\nas photometric distortion and geometric distortions, additional data augmentation strategies\\nincluding MixUp (Zhang et al., 2017b), CutMix (Yun et al., 2019) and Mosaic (Bochkovskiy\\net al., 2020) have been combined which help improves the performance of DenseSPH-YOLOv5.\\nIn addition, Mosaic augmentation can significantly enrich the background information. Since\\nsuch a method increases the batch size, therefor, batch normalization has been followed. Unless\\notherwise stated, a batch size set to 16 with a total number of training steps has been kept as\\n80. The initial learning rate has been set to 0.01 with SGD optimizer. The training dataset has\\nbeen trained to utilize the available pre-trained weights-file (Lin et al., 2014).\\n4.2 Optimization of network performance:\\nAt first, we conduct extensive experiments to select proper backbone-neck combination modules\\nto optimize the performance of the proposed DenseSPH-YOLOv5 model in terms of both\\ndetection accuracy and speed. For different combinations of backbone-neck configurations,\\ndetection accuracy in terms of parameters AP, AP 50, AP 75, AP S, AP M, and AP Las well as\\ndetection speed (in FPS) has been reported in Table. 2. From the Table. 2, one can see\\nthe introducing Additional Detection Head (ADH) improves performance by employing an\\nadditional feature fusion mechanism. Notably, there are 2.3% increases in AP 50; and 6.1%\\nincreases in AP Lcompared to the baseline original YOLOv5. Furthermore, implementation\\nof DenseNet with ADH enhances the performance further, in particular, for relatively small\\nobject detection with 3.6 % increase in AP Las shown in Fig. 7-(a). However, this results in a\\nsignification reduction in detection speed (from 69.1 FPS to 55.2 FPS). With the introduction\\nof the CBAM in the neck part of the detection model, the detection performance improves\\nfurther compared to DensNet+ADH configuration for relatively medium and large object sizes\\nwith 3.7% and 3.6% increase in AP Mand AP L, respectively while slightly compromising the\\ndetection speed. Thus, the implementation of CBAM has been proven to be an effective 22\\nTable 2: Performance of various residual and dense block combinations in DenseSPH-YOLOv5\\narchitecture for anchors size of 416 \\u0002416.\\nBackbone\\nadd-inNeck\\nadd-inAP AP 50AP 75APSAPMAPLFPS\\n(YOLOv5) - - 69.2 71.7 83.2 61.2 78.3 81.4 70.2\\n- ADH 70.4 74.1 85.9 67.3 79.7 84.7 69.1\\nDenseNet ADH 74.5 79.7 88.5 70.9 81.2 85.1 55.2\\nDenseNet ADH+CBAM 79.1 83.6 90.2 75.1 84.9 88.7 56.4\\nDenseNet ADH+CBAM+STR 81.4 85.2 91.4 78.2 86.5 89.5 62.4\\nstrategy to retain semantic and delicate spatial information from spatial and channel attention\\nmechanisms that enhance the overall performance of the model. Finally, the best performance\\nhas been achieved when both CBAM and STR respectively have been integrated into the neck\\nand detection head which demonstrate significant improvements in the accuracy parameter, in\\nparticular, AP, AP 50, AP 75, AP Mand AP Lincrease by 12.2%, 13.5%, 8.2%, 8.3%, and 8.1%,\\nrespectively compared to original YOLOv5 configuration. Moreover, we have also observed\\nimprovement in detection speed with the introduction of STR in the detection head by reducing\\nthe computational complexity compared to only ADH+CBAM configuration in the neck as\\nshown in Fig. 7-(a). Therefore, a such configuration in DenseSPH-YOLOv5 provides the\\noptimal performance in terms of detection accuracy and speed for the road-damaged data\\nset considered herein. In summary, together with proper activation function and improved\\nbackbone-neck combination in DenseSPH-YOLOv5 provides an efficient high-performance\\nmodel for damage detection in complex scenarios.\\n4.3 State-of-the-art comparison:\\nIn this section, the detection performance of the proposed DenseSPH-YOLOv5 has been\\ncompared with some of the existing state-of-the-art detection models (Zhao et al., 2019b). For\\nthe performance comparison, we have considered Faster R-CNN (Ren et al., 2016), RetinaNet\\n(Lin et al., 2017b), SSD (Liu et al., 2016), YOLOv4 (Bochkovskiy et al., 2020), Dense-YOLOv4 23\\nFigure 7: Comparison bar chart of precision parameters and detection speed for (a) different\\ncombinations of backbone and neck of the detection model; (b) different state-of-the-art models\\nand DenseSPH-YOLOv5. 24\\nTable 3: Comparison of different performance parameters including P, R, F1, mAP, and\\ndetection speed (in FPS) between DenseSPH-YOLOv5 and other SOAT models where bold\\nhighlights the best performance values.\\nModel P (%) R (%) F1-score (%) mAP (%) Dect. time (ms) FPS\\nFaster R-CNN 52.32 45.39 48.60 52.17 20.50 48.7\\nRetinaNet 59.11 55.67 51.35 54.11 18.87 52.7\\nSSD 62.13 57.19 59.56 60.52 17.92 55.8\\nYOLOv4 73.22 63.35 67.93 67.13 15.31 65.3\\nYOLOv5 79.17 68.47 73.43 71.13 14.24 70.2\\nTPH-YOLOv5 82.37 70.51 74.52 77.62 22.47 44.5\\nDense-YOLOv4 86.75 73.75 78.28 81.87 19.30 51.8\\nDenseSPH-YOLOv5 89.51 76.25 81.18 85.25 16.02 62.4\\n(Roy and Bhaduri, 2022), YOLOv5 (Jocher et al., 2021), and TPH-YOLOv5 (Zhu et al., 2021)\\nthat are trained in RDD-2018 datset. Comparison of different performance parameters including\\nP, R, F1-score, mAP, and detection speed obtained from these models have been presented\\nin Table 3. The comparison reveals that the accuracy values obtained from Faster R-CNN,\\nRetinaNet, and SSD are quite inferior compared to YOLO variants as visually illustrated in\\nthe bar-chart plot in Fig. 7-(b). Between YOLOv4 and YOLOv5, YOLOv5 demonstrated\\nbetter performance with a 5 :95% increase in P and 4 :01% increase in mAP, respectively.\\nWe observe that the performance of Dense-YOLOv4 is superior to the TPH-YOLOv5 with\\n4:38%, 3:24%, 3:76%, and 4 :25% increase in P, R, F1, and mAP, respectively. However, the\\nproposed DenseSPH-YOLOv5 yields the best performance reaching the values of 89 :51%, 76:25%,\\n81:18%, and 85 :25% in P, R, F1, and mAP, respectively as shown in Fig.7- (b). Moreover,\\nDenseSPH-YOLOv5 provides a superior real-time detection speed of 62.4 FPS which is 28 :68%\\nand 16:98% higher than TPH-YOLOv5 and Dense-YOLOv4 models, respectively. In summary,\\nDenseSPH-YOLOv5 outshines some of the best detection models in terms of both detection\\naccuracy and speed illustrating its suitability for automated high-performance damage detection\\nmodels. 25\\n4.4 Comparison with existing state-of-the-art damage detection models:\\nIn addition, the performance of the DenseSPH-YOLOv5 has been compared with several\\nexisting state-of-the-art road damage detection models evaluated in RDD-2018 datasets. As\\nshown in Table 4, we compared the performance from various DL models including SSD\\nInception v2 (Maeda et al., 2018), SSD MobileNet (Maeda et al., 2018), YOLO (Alfarrarjeh\\net al., 2018), Faster R-CNN (Kluger et al., 2018), Faster R-CNN with ResNet-152 (Wang et al.,\\n2018a), Ensemble models with Faster R-CNN and SSD (Wang et al., 2018b), and RetinaNet\\n(Angulo et al., 2019) with our Dense-YOLOv4 and DenseSPH-YOLOv5 models. Note, in\\n(Angulo et al., 2019) additional images were included in the RDD-2018 dataset to improve the\\ndetection performance. From the direct comparison, two current state-of-the-art models Faster\\nR-CNN with ResNet-152 (Wang et al., 2018a) and Ensemble models (Wang et al., 2018b) have\\nreached the F1 value of 62.55% that is 2.63% improvement over SSD MobileNet (Maeda et al.,\\n2018). While RetinaNet (Angulo et al., 2019) illustrated significant performance improvement,\\nDense-YOLOv4 performs better with F1 of 78.28% in the original RDD-2018 dataset. Relative\\nto the aforementioned models, our proposed model has achieved the best F1 value of 81.18 %\\namong current state-of-the-art damage detection models. In terms of other precision metrics,\\nthere are 3.64% and 10.51% improvements in P and R compared to RetinaNet (Angulo et al.,\\n2019), respectively. Comparing detection speed, DenseSPH-YOLOv5 provides competitive\\nperformance compared to SSD Inception v2 (Maeda et al., 2018) elucidates its superiority in\\nreal-time damage detection.\\n4.5 Overall performance of DenseSPH-YOLOv5:\\nFrom section 4.3, we observed that TPH-YOLOv5, Dense-YOLOv4, and DenseSPH-YOLOv5\\nprovide better performance compared to other SOAT models. Therefore, these three models are\\nclosely compared in terms of mAP, F1, IoU, final loss, and average detection time as shown in\\nTable 5. The proposed DenseSPH-YOLOv5 has achieved the highest average IoU value of 0.803\\nindicating superior BB accuracy during target detection compared to the other two models.\\nSimilarly, it has also illustrated better detection performance and accuracy by achieving the 26\\nTable 4: Comparison of different performance parameters between DenseSPH-YOLOv5 and\\nother SOAT road damage detection models evaluated in RDD-2018 dataset where bold highlights\\nthe best performance values.\\nReference Method Performance Det. Speed\\n(FPS)\\nMaeda et al. (2018) SSD Inception v2 F-1: 52.61; P: 81.10; R: 38.97 63.1\\nMaeda et al. (2018) SSD MobileNet F-1: 59.92; P: 81.11; R: 47.52 30.6\\nAlfarrarjeh et al. (2018) YOLO F-1: 62.00; P: -; R: - -\\nKluger et al. (2018) Faster R-CNN F-1: 61.00; P: -; R: - -\\nWang et al. (2018a) Faster R-CNN\\n(ResNet-152)F-1: 62.55; P: -; R: - -\\nWang et al. (2018b) Ensemble\\n(Faster R-CNN+ SSD)F-1: 62.55; P: -; R: - -\\nAngulo et al. (2019)\\u0003RetinaNet F-1: 74.97; P: 85.87; R: 65.75 -\\n(Roy and Bhaduri, 2022) Dense-YOLOv4 F-1: 78.28; P: 86.75; R: 73.75 51.8\\nOurs (current study) DenseSPH-YOLOv5 F-1: 81.18; P: 89.51; R: 76.25 62.4 27\\nTable 5: Overall performance comparison between TPH-YOLOv5, Dense-YOLOv4, and\\nDenseSPH-YOLOv5.\\nDetection model IoU F1 mAP Validation loss Detection\\ntime (ms)Detection\\nspeed (FPS)\\nTPH-YOLOv5 0.740 0.745 0.776 18.07 22.47 44.5\\nDense-YOLOv4 0.781 0.782 0.819 10.13 19.30 51.8\\nDenseSPH-YOLOv5 0.803 0.811 0.852 7.18 16.02 62.4\\nFigure 8: Comparison of (a) P-R curves; (b) loss evolution curves between TPH-YOLOv5,\\nDense-YOLOv4, and DenseSPH-YOLOv5.\\nhighest F1 and mAP values of 0.811 and 0.852 which are 6 :61% and 7:63% improvements over\\nthe TPH-YOLOv5, respectively. Furthermore, the detection speed of 62.4 FPS obtained from\\nDenseSPH-YOLOv5 was found to be higher than both TPH-YOLOv5 and Dense-YOLOv4.\\nThus, it can provide real-time road damage detection with better accuracy compared to the\\nother two state-of-the-art models. In addition, the comparison of P-R curves between the three\\nmodels have been depicted in Fig 8-(a). From the comparison of the P-R curves, one can see\\nthat DenseSPH-YOLOv5 attains a better P value for a particular R. It achieved the highest\\narea under the P-R curve indicating superior detection performance compared to TPH-YOLOv5\\nand Dense-YOLOv4. Next, we compare the loss evolution curves as shown in Fig 8-(b). In the\\ninitial phase, after exhibiting several cycles of fluctuation, the loss in the DenseSPH-YOLOv5 28\\nTable 6: Comparison of detection results for individual classes between TPH-YOLOv5,\\nDense-YOLOv4, and DenseSPH-YOLOv5\\nModel Class !D00 D01 D10 D11 D20 D40 D43 D44 Avg.\\nTPH-YOLOv5P 0.87 0.62 0.84 0.87 0.84 0.87 0.87 0.81 0.82\\nR 0.61 0.89 0.49 0.51 0.71 0.74 0.81 0.88 0.71\\nF1 0.72 0.73 0.62 0.64 0.77 0.79 0.83 0.84 0.75\\nmAP 0.81 0.79 0.78 0.82 0.73 0.78 0.76 0.74 0.77\\nDense-YOLOv4P 0.89 0.69 0.91 0.92 0.89 0.91 0.89 0.84 0.87\\nR 0.65 0.93 0.52 0.51 0.78 0.78 0.86 0.87 0.74\\nF1 0.75 0.79 0.66 0.65 0.83 0.84 0.87 0.85 0.78\\nmAP 0.82 0.81 0.84 0.89 0.77 0.82 0.81 0.79 0.82\\nDenseSPH-YOLOv5P 0.92 0.74 0.93 0.94 0.93 0.92 0.91 0.87 0.89\\nR 0.68 0.95 0.58 0.54 0.81 0.79 0.88 0.87 0.76\\nF1 0.78 0.83 0.71 0.68 0.86 0.85 0.89 0.87 0.81\\nmAP 0.84 0.89 0.81 0.91 0.82 0.85 0.85 0.85 0.85\\nmodel tends to saturate after approximately 50 training steps with a final loss value of 7.18.\\nWhereas, the other two models exhibit higher fluctuation in loss evolution and yield higher final\\nloss value. Evidently, the proposed TPH-YOLOv5 is easier to train with faster convergence\\ncharacteristics demonstrating its efficacy from the computational point of view.\\nTo further gain insight into the performances of these models, a comparison of detection\\nresults containing P, R, mAP, and F-1 values from each individual class between TPH-YOLOv5,\\nDense-YOLOv4, and DenseSPH-YOLOv5 has been presented in Table 6. DenseSPH-YOLOv5\\nhas illustrated significant improvement in P and R values for various classes, in particular,\\nfor detecting longitudianl and lateral linear cracks, alligator cracks, and potholes. Thus,\\nDenseSPH-YOLOv5 efficiently maximizes the TP value while simultaneously reducing FP and\\nFN values for all classes. The proposed model improves 2 :01% in P and 2 :11% in R compared\\nto Dense-YOLOv4. From the overall comparison, we can conclude that DenseSPH-YOLOv5\\ndemonstrated the best performance in detecting various damage classes outperforming the other\\ntwo state-of-the-art models in terms of precision and accuracy.\\n4.6 Detection of various damage classes:\\nIn this section, we have demonstrated the detection results obtained from DenseSPH-YOLOv5\\nfor eight different damage classes and compared them with TPH-YOLOv5 and Dense-YOLOv4. 29\\nFigure 9: Comparison of various damage detection results from (a-e) TPH-YOLOv5; (f-j)\\nDense-YOLOv4; (k-p) DenseSPH-YOLOv5. Detailed detection results with average confidence\\nindexes have been shown in Table 7.\\nTable 7: Detailed detection results from TPH-YOLOv5, Dense-YOLOv4, and\\nDenseSPH-YOLOv5 for different damage classes as shown in Figs. 9- 10.\\nFigs. No Model Detc. False\\/Undetc. Avg. confidence Score\\n9 (a-e) TPH-YOLOv5 13 8 0.71\\n9 (f-j) Dense-YOLOv4 16 5 0.75\\n9 (k-p) DenseSPH-YOLOv5 20 1 0.83\\n10 (a-e) TPH-YOLOv5 13 7 0.67\\n10 (f-j) Dense-YOLOv4 14 6 0.72\\n10 (k-p) DenseSPH-YOLOv5 18 2 0.79 30\\nFigure 10: Comparison of various damage detection results from (a-e) TPH-YOLOv5; (f-j)\\nDense-YOLOv4; (k-p) DenseSPH-YOLOv5. Detailed detection results with average confidence\\nscores have been shown in Table 7.\\nThe visual representations of the detection results have been presented with confined bounding\\nboxes with class identifiers (see Table 1) as shown in Figs. 9- 10. Corresponding detailed\\ndetection results consisting of the number of detected and undetected target classes with\\naverage confidence scores have been reported in Table. 7. From the overall comparison, it can\\nbe concluded that the proposed model shows its efficacy by preciously detecting the target\\nobjects with high average confidence index values. At the same time, it minimizes the false\\nand missed-detection results compared to both TPH-YOLOv5 and Dense-YOLOv4 models.\\nParticularly, when the multiple target objects have a significant degree of overlap between\\nthem, the bounding box prediction from the proposed DenseSPH-YOLOv5 is quite accurate in\\ndetecting each target object as illustrated in Figs. 9-(l, n) and Figs. 10-(k, n). 31\\nFigure 11: Comparison of damage detection results under various challenging scenarios from\\n(a-e) TPH-YOLOv5; (f-j) Dense-YOLOv4; (k-p) DenseSPH-YOLOv5. Detailed detection results\\nwith average confidence indexes have been shown in Table 6.\\n4.7 Detection under challenging scenarios:\\nIn this section, we have extended the detection under some challenging scenarios as shown\\nin Fig. 11 to demonstrate the efficacy of the proposed DenseSPH-YOLOv5. The detailed\\ndetection results that consist of the total number of detected and undetected target classes with\\naverage confidence scores have been reported in Table. 8. Here we consider various complex\\nbackgrounds and challenging environments where the target class is predominately difficult\\nto detect due to the presence of shadows. Additionally, there is some degree of occlusion\\nof overlapping bounding boxes between target classes. It is a challenging task to detect\\ntarget objects individually for an object detection model. Thus, superior feature extraction is\\ncritical to detect various target classes efficiently and correctly. In such cases, the proposed\\nDenseSPH-YOLOv5 has demonstrated its superiority in terms of boundary box precision with 32\\nTable 8: Detailed detection results from TPH-YOLOv5, Dense-YOLOv4, and\\nDenseSPH-YOLOv5 for different damage classes as shown in Fig. 11.\\nFigs. No Model Detc. False\\/Undetc. Avg. confidence Score\\n11 (a)-(e) TPH-YOLOv5 13 10 0.68\\n11 (f)-(j) Dense-YOLOv4 15 8 0.69\\n11 (k)-(p) DenseSPH-YOLOv5 19 4 0.77\\nTable 9: Detection result comparison from the DenseSPH-YOLOv5 for original RGB, greyscale,\\nlow resolution, and various brightness intensity (75 % and 50 %) images as shown in Fig. 12.\\nFigs. No Original Grey scale Low resolution Brightness-75 % Brightness-50 %\\nDet. Undet. Det. Undet. Det. Undet. Det. Undet. Det. Undet.\\nFigs. 12-(a-e) 6 0 5 1 4 2 6 0 5 1\\nFigs. 12-(f-j) 3 1 4 0 2 2 3 1 3 1\\nFigs. 12-(k-p) 5 0 5 0 2 3 5 0 3 2\\na higher average confidence score compared to the other two models. Notably, it significantly\\nreduces the false or missed detection (10 to 4 compared to TPH-YOLOv5) and consequently,\\nthe confidence scores have been improved which is evident from Table 8. Overall, based on the\\ndetection results for the various damage classes, the proposed DenseSPH-YOLOv5 illustrates\\nsuperior detection ability, in particular, in detecting the presence of shadows compared to\\nTPH-YOLOv5 and Dense-YOLOv4 models.\\n4.8 Detection under Greyscale, low resolution, and different illumination intensities:\\nIn this section, the detection accuracy of the DenseSPH-YOLOv5 has been further tested with\\ngrayscale and pixelated low-resolution (100 \\u0002100) images as shown in Fig. 12. Additionally,\\nprediction results under different illumination intensities (i.e., brightness-75% and brightness-\\n50%) have been studied and compared with the detection result for original RGB images. In\\nTable 9, detailed detection results with confidence scores have been reported. For grayscale\\nimages, the proposed model has demonstrated impressive performance with accurate bounding\\nbox prediction with a better bounding box confidence score as shown in Figs. 12-(b, g, 33\\nFigure 12: Detection results for (a, f, k) original RGB images; (b, g, l) corresponding grey\\nscale images; (c, h, m) low resolution images; (d, i, n) 75 % brightness intensity; (e, j, p) 50 %\\nbrightness intensity from the DenseSPH-YOLOv5. Detailed detection results have been shown\\nin Table 6. 34\\nl). For pixelated low-resolution images, the performance of DenseSPH-YOLOv5 has been\\nslightly diminished, however, it can still detect most of the target class with reasonable\\naccuracy. Similarly, the proposed model demonstrates superior detection results under different\\nillumination intensities as demonstrated in Figs. 12-(d, i, n) and Figs. 12-(e, j, p). It can be\\nconcluded from the test results that the proposed model is more adaptive in more challenging\\nenvironments compared to TPH-YOLOv5 and Dense-YOLOv4 models.\\n5. Discussion:\\nFrom the overall detection result, it is evident that DenseSPH-YOLOv5 has higher adaptability\\nand better capability in localizing various damage classes in a complex environment and\\nchallenging conditions compared to current state-of-the-art models. Additionally, it has\\ndemonstrated higher accuracy in bounding box detection and therefore can effectively avoid\\nthe problem of false and missed detection over other detection networks which justifies the\\nusefulness of the proposed model in practical on-field damage detection tasks. Future work\\ncan be geared towards further optimization of detection accuracy as well as detection speed for\\nportable on-field damage detection in a mobile computing platform. The current model can be\\nfurther improved by incorporating some state-of-the-art algorithms such as YOLO-X (Ge et al.,\\n2021), YOLOv7 (Wang et al., 2022), etc. Additionally, it can be assembled with a drone video\\nsurveillance system for real-time accurate damage detection. Furthermore, overlapping and\\noccultation can lead to the wrong prediction which can be further improved by considering\\nshortening the feature stride for better localization and bounding box prediction. In order to\\navoid overlapping boxes, it can be useful to switch from bounding boxes to points (Ribera\\net al., 2019) or masks (Xu et al., 2020). Moreover, one of the potential applications can be\\nassembling object detection framework with semantic segmentation methods such as Mask\\nR-CNN (Bharati and Pramanik, 2020), U-Net (Esser et al., 2018) to extract morphological\\ninformation of various damage classes. Nonetheless, the current work elucidates the possibility\\nof applying cutting-edge DL-based computer vision methodology for multiclass automated\\ndamage detection processes for commercial applications. 35\\n6. Conclusions :\\nSummarizing, in the present work, we have developed an efficient and robust object localization\\nmodel DenseSPH-YOLOv5 based on an improved version of the YOLOv5 network for accurate\\nclassification and localization of damage detection. The proposed DenseSPH-YOLOv5 has\\nbeen employed to detect distinct eight different road damage classes that provide superior and\\naccurate detection under various complex and challenging environments. Evaluated on the\\nRDD-2018 dataset, it has been found that at a detection rate of 62.40 FPS, DenseSPH-YOLOv5\\nhas achieved mean-average precision (mAP), F1, and precision (P) values of 85 :25%, 81:18%,\\nand 89:51%, respectively outperforms existing state-of-the-art damage detection models in\\nterms of both classification accuracy and localized bounding box prediction in detecting all\\ndamage classes. The present work effectively addresses the shortcoming of existing DL-based\\ndamage detection models and illustrates its superior potential in real-time in-field applications.\\nIn short, current work constitutes a step toward a fully automated and efficient DL-based\\ncomputer vision methodology for multi-class damage detection processes.\\nAcknowledgements: The support of the Aeronautical Research and Development Board\\n(Grant No. DARO\\/08\\/1051450\\/M\\/I) is gratefully acknowledged.\\nConflict of interest: The authors declare that they have no known competing financial\\ninterests or personal relationships that could have appeared to influence the work reported in\\nthis paper. 36\\nReferences\\nAgarwal, S. and Singh, D. (2015). An adaptive statistical approach for non-destructive underline\\ncrack detection of ceramic tiles using millimeter wave imaging radar for industrial application.\\nIEEE Sensors Journal , 15(12):7036--7044.\\nAlfarrarjeh, A., Trivedi, D., Kim, S. H., and Shahabi, C. (2018). A deep learning approach for\\nroad damage detection from smartphone images. In 2018 IEEE International Conference on\\nBig Data (Big Data) , pages 5201--5204. IEEE.\\nAnand, S., Gupta, S., Darbari, V., and Kohli, S. (2018). Crack-pot: Autonomous road crack\\nand pothole detection. In 2018 Digital Image Computing: Techniques and Applications\\n(DICTA) , pages 1--6. IEEE.\\nAngulo, A., Vega-Fern\\u0013 andez, J. A., Aguilar-Lobo, L. M., Natraj, S., and Ochoa-Ruiz, G. (2019).\\nRoad damage detection acquisition system based on deep neural networks for physical asset\\nmanagement. In Mexican International Conference on Artificial Intelligence , pages 3--14.\\nSpringer.\\nAnitha, M., Hemalatha, R., and Radha, S. (2021). A survey on crack detection algorithms for\\nconcrete structures. Advances in Smart System Technologies , pages 639--654.\\nArya, D., Maeda, H., Ghosh, S. K., Toshniwal, D., Mraz, A., Kashiyama, T., and Sekimoto, Y.\\n(2021a). Deep learning-based road damage detection and classification for multiple countries.\\nAutomation in Construction , 132:103935.\\nArya, D., Maeda, H., Ghosh, S. K., Toshniwal, D., Omata, H., Kashiyama, T., and Sekimoto, Y.\\n(2020). Global road damage detection: State-of-the-art solutions. In 2020 IEEE International\\nConference on Big Data (Big Data) , pages 5533--5539. IEEE.\\nArya, D., Maeda, H., Ghosh, S. K., Toshniwal, D., and Sekimoto, Y. (2021b). Rdd2020: An\\nannotated image dataset for automatic road damage detection using deep learning. Data in\\nbrief, 36:107133.\\nAzimi, M., Eslamlou, A. D., and Pekcan, G. (2020). Data-driven structural health monitoring\\nand damage detection through deep learning: State-of-the-art review. Sensors , 20(10):2778. 37\\nBang, S., Park, S., Kim, H., Yoon, Y., and Kim, H. (2018). A deep residual network with\\ntransfer learning for pixel-level road crack detection. Network , 93(84.90):89--03.\\nBharati, P. and Pramanik, A. (2020). Deep learning techniques|r-cnn to mask r-cnn: a survey.\\nComputational Intelligence in Pattern Recognition , pages 657--668.\\nBi\\u0018 cici, S. and Zeybek, M. (2021). An approach for the automated extraction of road surface\\ndistress from a uav-derived point cloud. Automation in Construction , 122:103475.\\nBochkovskiy, A., Wang, C.-Y., and Liao, H.-Y. M. (2020). Yolov4: Optimal speed and accuracy\\nof object detection.\\nBose, R. and Roy, A. (2022). Accurate deep learning sub-grid scale models for large eddy\\nsimulations. Bulletin of the American Physical Society .\\nCao, M.-T., Tran, Q.-V., Nguyen, N.-M., and Chang, K.-T. (2020). Survey on performance of\\ndeep learning models for detecting road damages using multiple dashcam image resources.\\nAdvanced Engineering Informatics , 46:101182.\\nChandio, A., Gui, G., Kumar, T., Ullah, I., Ranjbarzadeh, R., Roy, A. M., Hussain, A., and\\nShen, Y. (2022). Precise single-stage detector. arXiv preprint arXiv:2210.04252 .\\nChen, J. and Cho, Y. K. (2022). Crackembed: Point feature embedding for crack segmentation\\nfrom disaster site point clouds with anomaly detection. Advanced Engineering Informatics ,\\n52:101550.\\nChen, Q., Huang, Y., Sun, H., and Huang, W. (2021). Pavement crack detection using hessian\\nstructure propagation. Advanced Engineering Informatics , 49:101303.\\nChow, J. K., Su, Z., Wu, J., Tan, P. S., Mao, X., and Wang, Y.-H. (2020). Anomaly detection\\nof defects on concrete structures with the convolutional autoencoder. Advanced Engineering\\nInformatics , 45:101105.\\nDavis, J. and Goadrich, M. (2006). The relationship between precision-recall and roc curves.\\nInProceedings of the 23rd international conference on Machine learning , pages 233--240. 38\\nDong, J., Meng, W., Liu, Y., and Ti, J. (2021). A framework of pavement management system\\nbased on iot and big data. Advanced Engineering Informatics , 47:101226.\\nDorafshan, S., Thomas, R. J., and Maguire, M. (2018). Sdnet2018: An annotated image dataset\\nfor non-contact concrete crack detection using deep convolutional neural networks. Data in\\nbrief, 21:1664--1668.\\nDosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani,\\nM., Minderer, M., Heigold, G., Gelly, S., et al. (2020). An image is worth 16x16 words:\\nTransformers for image recognition at scale. arXiv preprint arXiv:2010.11929 .\\nDu, Y., Pan, N., Xu, Z., Deng, F., Shen, Y., and Kang, H. (2021). Pavement distress detection\\nand classification based on yolo network. International Journal of Pavement Engineering ,\\n22(13):1659--1672.\\nEisenbach, M., Stricker, R., Seichter, D., Amende, K., Debes, K., Sesselmann, M., Ebersbach,\\nD., Stoeckert, U., and Gross, H.-M. (2017). How to get pavement distress detection ready\\nfor deep learning? a systematic approach. In 2017 international joint conference on neural\\nnetworks (IJCNN) , pages 2039--2047. IEEE.\\nEsser, P., Sutter, E., and Ommer, B. (2018). A variational u-net for conditional appearance and\\nshape generation. In Proceedings of the IEEE conference on computer vision and pattern\\nrecognition , pages 8857--8866.\\nFan, Z., Wu, Y., Lu, J., and Li, W. (2018). Automatic pavement crack detection based on\\nstructured prediction with the convolutional neural network. arXiv preprint arXiv:1802.02208 .\\nFang, F., Li, L., Gu, Y., Zhu, H., and Lim, J.-H. (2020). A novel hybrid approach for crack\\ndetection. Pattern Recognition , 107:107474.\\nFerri, C., Hern\\u0013 andez-Orallo, J., and Modroiu, R. (2009). An experimental comparison of\\nperformance measures for classification. Pattern recognition letters , 30(1):27--38.\\nGe, Z., Liu, S., Wang, F., Li, Z., and Sun, J. (2021). Yolox: Exceeding yolo series in 2021.\\narXiv preprint arXiv:2107.08430 . 39\\nGirshick, R. (2015). Fast r-cnn in proceedings of the ieee international conference on computer\\nvision (pp. 1440--1448). Piscataway, NJ: IEEE.[Google Scholar] .\\nGong, H., Mu, T., Li, Q., Dai, H., Li, C., He, Z., Wang, W., Han, F., Tuniyazi, A., Li, H.,\\net al. (2022). Swin-transformer-enabled yolov5 with attention mechanism for small object\\ndetection on satellite images. Remote Sensing , 14(12):2861.\\nGopalakrishnan, K. (2018). Deep learning in data-driven pavement image analysis and\\nautomated distress detection: A review. Data , 3(3):28.\\nGopalakrishnan, K., Khaitan, S. K., Choudhary, A., and Agrawal, A. (2017). Deep convolutional\\nneural networks with transfer learning for computer vision-based data-driven pavement\\ndistress detection. Construction and building materials , 157:322--330.\\nGuan, J., Yang, X., Ding, L., Cheng, X., Lee, V. C., and Jin, C. (2021). Automated pixel-level\\npavement distress detection based on stereo vision and deep learning. Automation in\\nConstruction , 129:103788.\\nGuo, G. and Zhang, Z. (2022). Road damage detection algorithm for improved yolov5. Scientific\\nreports , 12(1):1--12.\\nHac\\u0010efendio\\u0015 glu, K. and Ba\\u0018 sa\\u0015 ga, H. B. (2022). Concrete road crack detection using deep\\nlearning-based faster r-cnn method. Iranian Journal of Science and Technology, Transactions\\nof Civil Engineering , 46(2):1621--1633.\\nHan, J., Zhang, D., Cheng, G., Liu, N., and Xu, D. (2018). Advanced deep-learning techniques\\nfor salient and category-specific object detection: a survey. IEEE Signal Processing Magazine ,\\n35(1):84--100.\\nHanzaei, S. H., Afshar, A., and Barazandeh, F. (2017). Automatic detection and classification\\nof the ceramic tiles' surface defects. Pattern Recognition , 66:174--189.\\nHartmann, T. and Trappey, A. (2020). Advanced engineering informatics-philosophical\\nand methodological foundations with examples from civil and construction engineering.\\nDevelopments in the built environment , 4:100020. 40\\nHe, K., Gkioxari, G., Doll\\u0013 ar, P., and Girshick, R. (2017). Mask r-cnn. in proceedings of the\\nieee international conference on computer vision.\\nHe, K., Zhang, X., Ren, S., and Sun, J. (2015). Spatial pyramid pooling in deep convolutional\\nnetworks for visual recognition. IEEE transactions on pattern analysis and machine\\nintelligence , 37(9):1904--1916.\\nHendrycks, D. and Gimpel, K. (2016). Gaussian error linear units (gelus). arXiv preprint\\narXiv:1606.08415 .\\nHuang, G., Liu, Z., Van Der Maaten, L., and Weinberger, K. Q. (2017). Densely connected\\nconvolutional networks. In Proceedings of the IEEE conference on computer vision and\\npattern recognition , pages 4700--4708.\\nJamil, S., Abbas, M. S., and Roy, A. M. (2022). Distinguishing malicious drones using vision\\ntransformer. AI, 3(2):260--273.\\nJamil, S. and Roy, A. M. (2022). Robust pcg-based vhd detection model using\\nd-cnns, nature-inspired algorithms, and vision transformer. Available at SSRN:\\nhttp:\\/\\/dx.doi.org\\/10.2139\\/ssrn.4316752 , page 41.\\nJamil, S. and Roy, A. M. (2023). An efficient and robust phonocardiography (pcg)-based\\nvalvular heart diseases (vhd) detection framework using vision transformer (vit). Computers\\nin Biology and Medicine , page 106734.\\nJocher, G., Stoken, A., Borovec, J., Chaurasia, A., Changyu, L., Laughing, A., Hogan, A.,\\nHajek, J., Diaconu, L., Marc, Y., et al. (2021). ultralytics\\/yolov5: v5. 0-yolov5-p6 1280\\nmodels aws supervise. ly and youtube integrations. Zenodo , 11.\\nKapela, R., \\u0013Sniata la, P., Turkot, A., Rybarczyk, A., Po_ zarycki, A., Rydzewski, P., Wycza lek,\\nM., and B loch, A. (2015). Asphalt surfaced pavement cracks detection based on histograms\\nof oriented gradients. In 2015 22nd International Conference Mixed Design of Integrated\\nCircuits & Systems (MIXDES) , pages 579--584. IEEE.\\nKaraaslan, E., Bagci, U., and Catbas, F. N. (2021). Attention-guided analysis of infrastructure\\ndamage with semi-supervised deep learning. Automation in Construction , 125:103634. 41\\nKhan, W., Kumar, T., Cheng, Z., Raj, K., Roy, A. M., and Luo, B. (2022a). Sql and\\nnosql databases software architectures performance analysis and assessments--a systematic\\nliterature review. arXiv preprint arXiv:2209.06977 .\\nKhan, W., Raj, K., Kumar, T., Roy, A. M., and Luo, B. (2022b). Introducing urdu digits\\ndataset with demonstration of an efficient and robust noisy decoder-based pseudo example\\ngenerator. Symmetry , 14(10):1976.\\nKluger, F., Reinders, C., Raetz, K., Schelske, P., Wandt, B., Ackermann, H., and Rosenhahn,\\nB. (2018). Region-based cycle-consistent data augmentation for object detection. In 2018\\nIEEE International Conference on Big Data (Big Data) , pages 5205--5211. IEEE.\\nKoch, C., Georgieva, K., Kasireddy, V., Akinci, B., and Fieguth, P. (2015). A review on\\ncomputer vision based defect detection and condition assessment of concrete and asphalt\\ncivil infrastructure. Advanced Engineering Informatics , 29(2):196--210.\\nLeCun, Y., Bengio, Y., and Hinton, G. (2015). Deep learning. nature , 521(7553):436--444.\\nLi, D., Xie, Q., Gong, X., Yu, Z., Xu, J., Sun, Y., and Wang, J. (2021). Automatic\\ndefect detection of metro tunnel surfaces using a vision-based inspection system. Advanced\\nEngineering Informatics , 47:101206.\\nLi, P., Xia, H., Zhou, B., Yan, F., and Guo, R. (2022). A method to improve the accuracy\\nof pavement crack identification by combining a semantic segmentation and edge detection\\nmodel. Applied Sciences , 12(9):4714.\\nLi, S. and Zhao, X. (2021). Pixel-level detection and measurement of concrete crack using\\nfaster region-based convolutional neural network and morphological feature extraction.\\nMeasurement Science and Technology , 32(6):065010.\\nLi, Y., Li, H., and Wang, H. (2018). Pixel-wise crack detection using deep local pattern\\npredictor for robot application. Sensors , 18(9):3042.\\nLin, T.-Y., Goyal, P., Girshick, R., He, K., and Doll\\u0013 ar, P. (2017a). Focal loss for dense object\\ndetection. In Proceedings of the IEEE international conference on computer vision , pages\\n2980--2988. 42\\nLin, T.-Y., Goyal, P., Girshick, R., He, K., and Doll\\u0013 ar, P. (2017b). Focal loss for dense object\\ndetection. In Proceedings of the IEEE international conference on computer vision , pages\\n2980--2988.\\nLin, T.-Y., Maire, M., Belongie, S., Hays, J., Perona, P., Ramanan, D., Doll\\u0013 ar, P., and Zitnick,\\nC. L. (2014). Microsoft coco: Common objects in context. In European conference on\\ncomputer vision , pages 740--755. Springer.\\nLiu, S., Qi, L., Qin, H., Shi, J., and Jia, J. (2018). Path aggregation network for instance\\nsegmentation. In Proceedings of the IEEE conference on computer vision and pattern\\nrecognition , pages 8759--8768.\\nLiu, W., Anguelov, D., Erhan, D., Szegedy, C., Reed, S., Fu, C., and Berg, A. (2016). Ssd:\\nSingle shot multibox detector, \\/uni2016in european conference on computer vision (eccv).\\nLiu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., Zhang, Z., Lin, S., and Guo, B. (2021). Swin\\ntransformer: Hierarchical vision transformer using shifted windows. In Proceedings of the\\nIEEE\\/CVF International Conference on Computer Vision , pages 10012--10022.\\nMaeda, H., Kashiyama, T., Sekimoto, Y., Seto, T., and Omata, H. (2021). Generative\\nadversarial network for road damage detection. Computer-Aided Civil and Infrastructure\\nEngineering , 36(1):47--60.\\nMaeda, H., Sekimoto, Y., Seto, T., Kashiyama, T., and Omata, H. (2018). Road\\ndamage detection and classification using deep neural networks with smartphone images.\\nComputer-Aided Civil and Infrastructure Engineering , 33(12):1127--1141.\\nMajidifard, H., Jin, P., Adu-Gyamfi, Y., and Buttlar, W. G. (2020). Pavement image datasets:\\nA new benchmark dataset to classify and densify pavement distresses. Transportation\\nResearch Record , 2674(2):328--339.\\nMandal, V., Mussah, A. R., and Adu-Gyamfi, Y. (2020). Deep learning frameworks for pavement\\ndistress classification: A comparative analysis. In 2020 IEEE International Conference on\\nBig Data (Big Data) , pages 5577--5583. IEEE.\\nMisra, D. (2020). Mish: A self regularized non-monotonic activation function. 43\\nMohan, A. and Poobal, S. (2018). Crack detection using image processing: A critical review\\nand analysis. Alexandria Engineering Journal , 57(2):787--798.\\nNaddaf-Sh, S., Naddaf-Sh, M.-M., Kashani, A. R., and Zargarzadeh, H. (2020). An efficient\\nand scalable deep learning approach for road damage detection. In 2020 IEEE International\\nConference on Big Data (Big Data) , pages 5602--5608. IEEE.\\nNath, N. D., Cheng, C.-S., and Behzadan, A. H. (2022). Drone mapping of damage information\\nin gps-denied disaster sites. Advanced Engineering Informatics , 51:101450.\\nNhat-Duc, H., Nguyen, Q.-L., and Tran, V.-D. (2018). Automatic recognition of asphalt\\npavement cracks using metaheuristic optimized edge detection algorithms and convolution\\nneural network. Automation in Construction , 94:203--213.\\nNi, F., He, Z., Jiang, S., Wang, W., and Zhang, J. (2022). A generative adversarial learning\\nstrategy for enhanced lightweight crack delineation networks. Advanced Engineering\\nInformatics , 52:101575.\\nOliveira, H. and Correia, P. L. (2009). Automatic road crack segmentation using entropy and\\nimage dynamic thresholding. In 2009 17th European Signal Processing Conference , pages\\n622--626. IEEE.\\nPatra, S., Middya, A. I., and Roy, S. (2021). Potspot: Participatory sensing based monitoring\\nsystem for pothole detection using deep learning. Multimedia Tools and Applications ,\\n80(16):25171--25195.\\nQuintana, M., Torres, J., and Men\\u0013 endez, J. M. (2015). A simplified computer vision system for\\nroad surface inspection and maintenance. IEEE Transactions on Intelligent Transportation\\nSystems , 17(3):608--619.\\nRawat, W. and Wang, Z. (2017). Deep convolutional neural networks for image classification:\\nA comprehensive review. Neural computation , 29(9):2352--2449.\\nRedmon, J., Divvala, S., Girshick, R., and Farhadi, A. (2016). You only look once: Unified,\\nreal-time object detection. In Proceedings of the IEEE conference on computer vision and\\npattern recognition , pages 779--788. 44\\nRedmon, J. and Farhadi, A. (2017). Yolo9000: better, faster, stronger. In Proceedings of the\\nIEEE conference on computer vision and pattern recognition , pages 7263--7271.\\nRedmon, J. and Farhadi, A. (2018). Yolov3: An incremental improvement.\\nRen, S., He, K., Girshick, R., and Sun, J. (2016). Faster r-cnn: towards real-time object\\ndetection with region proposal networks. IEEE transactions on pattern analysis and machine\\nintelligence , 39(6):1137--1149.\\nRezatofighi, H., Tsoi, N., Gwak, J., Sadeghian, A., Reid, I., and Savarese, S. (2019). Generalized\\nintersection over union: A metric and a loss for bounding box regression. In Proceedings of\\nthe IEEE\\/CVF Conference on Computer Vision and Pattern Recognition , pages 658--666.\\nRibera, J., Guera, D., Chen, Y., and Delp, E. J. (2019). Locating objects without bounding\\nboxes. In Proceedings of the IEEE\\/CVF Conference on Computer Vision and Pattern\\nRecognition , pages 6479--6489.\\nRoy, A. M. (2022a). Adaptive transfer learning-based multiscale feature fused deep convolutional\\nneural network for eeg mi multiclassification in brain--computer interface. Engineering\\nApplications of Artificial Intelligence , 116:105347.\\nRoy, A. M. (2022b). An efficient multi-scale CNN model with intrinsic feature integration for\\nmotor imagery EEG subject classification in brain-machine interfaces. Biomedical Signal\\nProcessing and Control , 74:103496.\\nRoy, A. M. (2022c). A multi-scale fusion cnn model based on adaptive transfer learning for\\nmulti-class mi-classification in bci system. BioRxiv .\\nRoy, A. M. and Bhaduri, J. (2021). A deep learning enabled multi-class plant disease detection\\nmodel based on computer vision. AI, 2(3):413--428.\\nRoy, A. M. and Bhaduri, J. (2022). Real-time growth stage detection model for high degree\\nof occultation using densenet-fused YOLOv4. Computers and Electronics in Agriculture ,\\n193:106694.\\nRoy, A. M., Bhaduri, J., Kumar, T., and Raj, K. (2022a). A computer vision-based object\\nlocalization model for endangered wildlife detection. Ecological Economics, Forthcoming . 45\\nRoy, A. M., Bhaduri, J., Kumar, T., and Raj, K. (2022b). Wildect-yolo: An efficient and\\nrobust computer vision-based accurate object localization model for automated endangered\\nwildlife detection. Ecological Informatics , page 101919.\\nRoy, A. M. and Bose, R. (2023a). Deep learning-accelerated computational framework based\\non physics informed neural network for solution of linear elasticity. Neural Networks , 2:2.\\nRoy, A. M. and Bose, R. (2023b). Physics-aware deep learning framework for linear elasticity.\\narXiv preprint arXiv:2302.09668 .\\nRoy, A. M., Bose, R., and Bhaduri, J. (2022c). A fast accurate fine-grain object detection\\nmodel based on YOLOv4 deep neural network. Neural Computing and Applications , pages\\n1--27.\\nRoy, A. M. and Guha, S. (2022). Elastoplastic physics-informed deep learning approach for j2\\nplasticity. Available at SSRN: https:\\/\\/ssrn.com\\/abstract=4332254 , page 48.\\nRoy, A. M. and Guha, S. (2023). A data-driven physics-constrained deep learning computational\\nframework for solving von mises plasticity. Engineering Applications of Artificial Intelligence ,\\n122:106049.\\nShang, H., Sun, C., Liu, J., Chen, X., and Yan, R. (2023). Defect-aware transformer network\\nfor intelligent visual surface defect detection. Advanced Engineering Informatics , 55:101882.\\nSilva, W. R. L. d. and Lucena, D. S. d. (2018). Concrete cracks detection based on deep\\nlearning image classification. In Proceedings , volume 2, page 489. MDPI AG.\\nSingh, A., Raj, K., Kumar, T., Verma, S., and Roy, A. M. (2023a). Deep learning-based\\ncost-effective and responsive robot for autism treatment. Drones , 7(2):81.\\nSingh, A., Ranjbarzadeh, R., Raj, K., Kumar, T., and Roy, A. M. (2023b). Understanding eeg\\nsignals for subject-wise definition of armoni activities. arXiv preprint arXiv:2301.00948 .\\nStricker, R., Eisenbach, M., Sesselmann, M., Debes, K., and Gross, H.-M. (2019). Improving\\nvisual road condition assessment by extensive experiments on the extended gaps dataset. In\\n2019 International Joint Conference on Neural Networks (IJCNN) , pages 1--8. IEEE. 46\\nVaradharajan, S., Jose, S., Sharma, K., Wander, L., and Mertz, C. (2014). Vision for road\\ninspection. In IEEE winter conference on applications of computer vision , pages 115--122.\\nIEEE.\\nVaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser,  L., and\\nPolosukhin, I. (2017). Attention is all you need. Advances in neural information processing\\nsystems , 30.\\nVoulodimos, A., Doulamis, N., Doulamis, A., and Protopapadakis, E. (2018). Deep learning for\\ncomputer vision: A brief review. Computational intelligence and neuroscience , 2018.\\nWang, C.-Y., Bochkovskiy, A., and Liao, H.-Y. M. (2021a). Scaled-yolov4: Scaling cross stage\\npartial network. In Proceedings of the IEEE\\/cvf conference on computer vision and pattern\\nrecognition , pages 13029--13038.\\nWang, C.-Y., Bochkovskiy, A., and Liao, H.-Y. M. (2022). Yolov7: Trainable bag-of-freebies\\nsets new state-of-the-art for real-time object detectors. arXiv preprint arXiv:2207.02696 .\\nWang, C.-Y., Liao, H.-Y. M., Wu, Y.-H., Chen, P.-Y., Hsieh, J.-W., and Yeh, I.-H. (2020).\\nCspnet: A new backbone that can enhance learning capability of cnn. In Proceedings of\\nthe IEEE\\/CVF conference on computer vision and pattern recognition workshops , pages\\n390--391.\\nWang, W., Li, L., and Han, Y. (2021b). Crack detection in shadowed images on gray level\\ndeviations in a moving window and distance deviations between connected components.\\nConstruction and Building Materials , 271:121885.\\nWang, W., Wu, B., Yang, S., and Wang, Z. (2018a). Road damage detection and classification\\nwith faster r-cnn. In 2018 IEEE international conference on big data (Big data) , pages\\n5220--5223. IEEE.\\nWang, Y. J., Ding, M., Kan, S., Zhang, S., and Lu, C. (2018b). Deep proposal and detection\\nnetworks for road damage detection and classification. In 2018 IEEE International Conference\\non Big Data (Big Data) , pages 5224--5227. IEEE. 47\\nWoo, S., Park, J., Lee, J.-Y., and Kweon, I. S. (2018). Cbam: Convolutional block attention\\nmodule. In Proceedings of the European conference on computer vision (ECCV) , pages\\n3--19.\\nXu, B., Wang, W., Falzon, G., Kwan, P., Guo, L., Chen, G., Tait, A., and Schneider, D. (2020).\\nAutomated cattle counting using mask r-cnn in quadcopter vision system. Computers and\\nElectronics in Agriculture , 171:105300.\\nXu, R., Hao, R., and Huang, B. (2022). Efficient surface defect detection using self-supervised\\nlearning strategy and segmentation network. Advanced Engineering Informatics , 52:101566.\\nYun, S., Han, D., Oh, S. J., Chun, S., Choe, J., and Yoo, Y. (2019). Cutmix: Regularization\\nstrategy to train strong classifiers with localizable features. In Proceedings of the IEEE\\/CVF\\ninternational conference on computer vision , pages 6023--6032.\\nZhang, A., Wang, K. C., Li, B., Yang, E., Dai, X., Peng, Y., Fei, Y., Liu, Y., Li, J. Q.,\\nand Chen, C. (2017a). Automated pixel-level pavement crack detection on 3d asphalt\\nsurfaces using a deep-learning network. Computer-Aided Civil and Infrastructure Engineering ,\\n32(10):805--819.\\nZhang, H., Cisse, M., Dauphin, Y. N., and Lopez-Paz, D. (2017b). mixup: Beyond empirical\\nrisk minimization. arXiv preprint arXiv:1710.09412 .\\nZhang, L., Yang, F., Zhang, Y. D., and Zhu, Y. J. (2016). Road crack detection using deep\\nconvolutional neural network. In 2016 IEEE international conference on image processing\\n(ICIP) , pages 3708--3712. IEEE.\\nZhao, H., Qin, G., and Wang, X. (2010). Improvement of canny algorithm based on pavement\\nedge detection. In 2010 3rd International Congress on Image and Signal Processing , volume 2,\\npages 964--967. IEEE.\\nZhao, Z.-Q., Zheng, P., Xu, S.-t., and Wu, X. (2019a). Object detection with deep learning: A\\nreview. IEEE transactions on neural networks and learning systems , 30(11):3212--3232.\\nZhao, Z.-Q., Zheng, P., Xu, S.-t., and Wu, X. (2019b). Object detection with deep learning: A\\nreview. IEEE transactions on neural networks and learning systems , 30(11):3212--3232. 48\\nZheng, Z., Wang, P., Liu, W., Li, J., Ye, R., and Ren, D. (2020). Distance-iou loss: Faster\\nand better learning for bounding box regression. In Proceedings of the AAAI Conference on\\nArtificial Intelligence , volume 34, pages 12993--13000.\\nZhu, X., Lyu, S., Wang, X., and Zhao, Q. (2021). Tph-yolov5: Improved yolov5 based on\\ntransformer prediction head for object detection on drone-captured scenarios. In Proceedings\\nof the IEEE\\/CVF International Conference on Computer Vision , pages 2778--2788.\",\"43\":\" Using a Variational Autoencoder  to Learn Valid Search Spaces of \\nSafely Monitored Autonomous Robots for Last-M ile Delivery \\nPeter J. Bentley \\nDepartment of Computer \\nScience, UCL, Autodesk \\nResearch \\nLondon, United Kingdom  \\n p.bentley@cs.ucl.ac.u k Soo Ling Lim \\n Department of Computer \\nScience, UCL  \\n London, United \\nKingdom  \\n s.lim@cs.ucl.ac.uk  Paolo Arcaini \\n National Institute of \\nInformatics  \\n Tokyo, Japan  \\n arcaini@nii.ac.jp  Fuyuki Ishikawa \\nNational Institute of \\nInformatics  \\n Tokyo, Japan  \\n f-ishikawa@nii.ac.jp  \\nABSTRACT  \\nThe use of autonomous robots for delivery of goods to customers \\nis an exciting new way to provide a reliable and sustainable \\nservice. However, in the real world, autonomous robots still require human supervision for safety reasons. We tackle the real -\\nworld problem of optimizing autonomous robot timings to maximize deliveries, while ensuring that there are never too m any \\nrobots running simultaneously  so that they can be monitored \\nsafely . We assess the use of a recent  hybrid machine- learning -\\noptimization approach COIL (constrained optimization in learned \\nlatent space) and compare it with a baseline  genetic algorithm for \\nthe purposes of exploring variations of this problem. We also investigate new methods for improving the speed and efficiency \\nof COIL . We show that only COIL can find valid solutions where \\nappropriate  numbers of robots run simultaneou sly for all problem \\nvariations tested. We also show that when COIL has learned its latent representation, it can optimize 10% faster than the GA, \\nmaking it a good choice for daily re -optimization of robots where \\ndelivery requests for each day are allocated to robots while \\nmaintaining safe numbers of robots running at once.  \\nCCS CONCEPTS \\n\\u2022 Computing methodologies~Search methodologies  \\u2022 Computing \\nmethodologies~Learning latent representations  \\u2022 C omputing \\nmethodologies~Robotic planning  \\nKEYWORDS \\nVariational autoencoder , autonomous robots, scheduling, learning \\nlatent representations , genetic algorithm  \\n \\n  \\n 1 INTRODUCTION  \\nHome delivery of groceries and other goods is rapidly becoming a \\nmajor new industry worldwide, accelerated by the COVID \\npandemic. The use of automobiles for last -mile delivery (from \\nstore to customer) caus es environmental concerns [1] and in \\ncountries such as Japan  where our problem originates, there may \\nbe a lack of labor. O ur industry partner , a major electronics \\ncorporation, is currently trialing their solution: an automatic \\ndelivery service performed by autonomous robots. Customers \\norder goods which are then collected from a supermarket or drug \\nstore by a robot and delivered to the customer. The robots are \\nautonomous but have human monitors that intervene should a \\nproblem occur, such as a potential collision with a pedestrian , \\nFigure 1. The use of robots for this purpose is in the \\nexperimentation phase, with details such as number of robots and human monitors  still under consideration.  \\n \\n  \\nFigure 1: Home delivery service using low speed robots with \\nhuman operators performing safety monitoring.  Map shows \\nsection  of actual location. P. Bentley  et al.  \\n \\n \\n In this real -world problem there is a clear conflict between \\nfulfilling orders and maintaining  safety. More robots operating \\nsimultaneously will mean more orders could be fulfilled, yet too \\nmany robots operating at once would be dangerous as the human \\noperators can only monitor a limited number of robots. The \\nproblem of how best to schedule  different numbers of robots such \\nthat maximum orders are delivered safely is therefore difficult.  It \\nis also a problem that must be solved repeatedly for every new set of customer orders.  This problem is likely to become increasingly \\nrelevant as autonomous robot deliveries become more prevalent, \\nyet safety is rarely considered in the literature.  \\nWe address this problem by using a hybrid machine learning  \\nand evolutionary computation approach  [2, 3]. This method  uses a \\nvariational autoencoder to learn the valid  regions of the search \\nspace, i.e., the areas of the search space where safe numbers of robots are scheduled to work simultaneously. A genetic algorithm \\n(GA) is then used to search in this learned latent space to find \\noptimal timings for robots suc h that the maximum number of \\norders can be delivered safely. We compare this approach with a \\nbaseline  genetic algorithm optimizing robot timings  while solving \\nthe constraint, without the use of machine learning. We investigate how well both approaches  enable us to examine \\ndifferent design variations and we assess suitability for long term \\nuse once the design variables are determined . \\nThe contributions of this work can be summarized as follows:  \\n\\u2022 The first study to explore real-world autonomous robot \\nrouting  while satisfying a safety monitoring requirement. \\n\\u2022 Adapting the hybrid ML -EA approach (COIL)  [2, 3] to a \\nreal-world problem for the first time.  \\n\\u2022 Evaluation of COIL by comparing to  a baseline GA to \\nexplore problem variations,  showing better  solution \\nquality and optimization speed for daily robot scheduling.  \\n\\u2022 COIL performance is enhanced on th e problem through  \\nreduction of the training set and number of latent \\nvariables.  \\nThe rest of the paper is organized  as follows: Section 2 \\nprovides a literature review of relevant research, Section 3 describes the method, Section  4 describes the experiments , results  \\nand discusses findings for the problem . We conclude in Section 5. \\n2 BACKGROUND  \\n2.1 Autonomous Robots for Home Delivery \\nAutonomous delivery robots have the potential to significantly reduce energy consumption and CO\\n2 emissions in urban areas  [1]. \\nThere are several examples of solutions. Mercedes -Benz Vans \\npartnered with Starship Technologies to propose the concept of \\nautonomous del ivery robots launched from trucks [4]. The truck \\ncan replenish robots at decentralized depots to launch more  until \\nall its customers are supplied. Boysen et al. [4] developed  \\nscheduling procedures which determine the truck route, such that \\nthe weighted number of late custom er deliveries is minimized.  \\nYu et al. [5] considered a variant of this problem and  proposed \\nan LV -SAV (large vehicle - small full y automated ground vehicles) model where multiple LVs cooperate with their \\nassociated SAVs. Safety was one of their concerns, and having \\nslow moving SAVs is safer in urban delivery, and was considered \\nto be safer than using drones in the HVDRP (hybrid vehi cle-drone \\nrouting problem) proposed by Karak and Abdelghany [6]. \\nChen et al. [7] introduced another variant called  a new vehicle \\nrouting problem with time windows and delivery robots (VRPTWDR). They showed that c onsiderable operational time \\nsavings can be achieved by dispatching delivery robots to serve \\nnearby customers while a driver is also serving a customer . \\nSince delivery robots share sidewalks with pedestrians, Bakach \\net al. investigated the option to ch oose paths for them that avoid \\nzones with high pedestrian  density  [8]. They  considered  path Since delivery robots share sidewalks with pedestrians, Bakach \\net al. investigated the option to ch oose paths for them that avoid \\nzones with high pedestrian  density  [8]. They  considered  path \\nflexibility given the presence of zones with varying pedestrian \\nlevel of service .  \\nThe study of vehicle routing problems (VRP) continues to \\ndevelop, with  many  variants includ ing real-life constraints and \\nassumptions, which makes the models more realistic and the \\napproaches more applicable in practice [9]. Autonomous delivery \\nvehicles introduce  safety concerns  whilst driving autonomously \\non the public road networks and performance risk s while \\ndelivering  parcels (e.g., risk of malfunctioning of the technology) \\n[10]. A study of  public perceptions and acceptance of autonomous \\nvehicles found that people expressed high concerns around safety, \\ncyber -security, legal liability, and regulat ion issues  [11]. \\nIn this work we focus on safety  and a key component of the \\nsolution created by our i ndustry collaborator. New to this field, f or \\nour problem, human operators monitor the robots so that they can \\nmanually operate the m when unexpected obstacles are detected , \\nor they can talk with the users.  Similar monitoring will be \\nrequired for many other real -world examples, so methods to solve \\nthis problem are likely to be widely useful. \\n2.2 Evolving Latent Variables  \\nAutoencoder s [12] are neural network s first used for \\ndimensionality reduction . Since their inception they have become \\npopular for  learning generative models of the data. An \\nautoencoder co mprises  two parts - an encoder \\ud835\\udc5d, which maps the \\nobservations \\ud835\\udc65 to a (lower dimensional) embedding space \\ud835\\udc67 and a \\ndecoder \\ud835\\udc5e, which  maps the embeddings back to the original \\nobservation space.  When trained, the autoencoder minimizes the \\nerror in reconstruction of output compared to input.  The \\nvariational autoencoder (VAE) is a probabilistic autoencoder [13, \\n14]. Instead of encoding an observation as a single point, VAEs \\nencode it as a distribution over the latent space.  \\nRecent work in the field of evolutionary computation makes \\nuse of VAE s to learn representations and then evolve using those \\nrepresentations, searching in the latent space  (LVE \\u2013 latent \\nvariable evolution). Game levels  [15], fingerprints to foil security \\nsystems [16], and program synthesis  [17] include some of the \\napplications. Researchers also make use of quality -diversity \\napproaches  [18] with LVE  to improve optimi zation in high \\ndimensional problems, for example DDE -Elites  [19] which uses a \\ncombination of the direct encoding with a learned latent encoding to accelerate search. Learn ing Valid Search Spaces of Autonomous Robots   \\n \\n Most recently, COIL (constrained optimization in learned \\nlatent space) [2] and SOLVE (search space optimization with \\nlatent variable evolution) [3] present an LVE approach to learn a \\nlatent rep resentation that is biased towards solutions that satisfy a \\nconstraint or additional objective. This is very relevant to the real -\\nworld problem we investigate in our work. However, while COIL  \\n[2] and SOLVE  [3] have been demonstrated on simple constraints \\nand benchmark functions, the technique has not been tested on a \\nreal-world problem. In this work we app ly the concepts in COIL \\nfor the first time to the problem of scheduling a safe number of \\nautonomous robots for home delivery of a maximum number of \\norders by customers .  \\n3 METHOD  \\nWe base our approach on the system under development  by our \\nindustry partner. Their automated delivery service will enable \\ncustomers to order goods from a local store which are delivered \\nby autonomous robots. Customers submit requests through  an app. \\nEach request is scheduled centrally with a robot allocate d to serve \\nit. Robots are monitored by human operator s who can intervene if \\nthe robot encounters problems, helping to initiate the appropriate response in the robot or talk to users , Figure 1. \\nWe focus on  two stages to this problem: \\n1. Investigate possible design variations, with different numbers of robots, human operators, request durations . \\n2. On completion of the first stage, provide an efficient method for daily optimizatio n of robot timings.  \\nFor the first stage,  we investigate optimization methods \\ncapable of exploring design variations . The algorithms must \\nsuccessfully  schedul e the robots to maximize delivery of requests \\nwhile ensuring there are never too many robots run ning \\nsimultaneously, such that the human operators can safely monitor \\nthem. For the second stage , with the design variables now \\ndetermined,  we investigate the fastest method of scheduling \\nrobots to meet the human operator constraint, daily , because  this \\ndecision- making process reoccurs every day with new customer \\nrequests.  \\nWe formulate the problem as a constrained optimization \\nproblem.  Given a randomly generated set Reqs  comprising Rq \\ncustomer requests, each with duration Rqdur: \\n Reqs  = {Rqdur\\n1, .., Rqdur Rq} \\n \\nwhere Rqdur j = random[ 60..dr] and given a set of robots Rbts \\ncomprising Rb pairs of robot starting time Rst and running time  \\nRrt: \\n \\nRbts = {Rst1, Rrt1, .., R stRb, RrtRb} \\n \\nallocate each request  Rqdur j in order from j = 1 to Rq (i.e., on a \\nfirst-come, first -served basis) to the most appropriate robot using \\na simple scheduler created for this work , Algorithm 1. (Should \\nthis solution be commissioned for use on the real robots, this \\nalgorithm will be replaced with the industrial partner\\u2019s internal \\nscheduler.)  Algorithm 1: First -come first -served scheduler used to \\nallocate customer requests to robots. totalRm  count s the total \\nnumber of requests met out of a maximum of Rq. \\ntotalRm  = 0 \\nfor i = 1 to Rb \\n    remaining_dur i = Rrti\\u00d710 \\nendfor  \\nfor j = 1 to Rq \\n    if there exists a remaining_dur k such that  \\n        0 <= ( remaining_dur k  - Rqdur j ) < 10 and \\n        there are no other  closer matches then \\n            remaining_dur k  \\u2212= Rqdur j \\n            totalRm  ++ \\n            break \\n    else find largest remaining_dur k such that  \\n        ( remaining_dur k  - Rqdur j ) > 0 then \\n        remaining_dur k  \\u2212= Rqdur  \\n        totalRm  ++ \\n            break  \\nendfor  \\n The task is then to use an optimizer to find a suitable set of \\nRbts such that the number of requests fulfilled by the robots is \\nmaximized by efficiently fitting robot start times and running \\ntimes to the set Reqs while ensuring that the number of robots \\nrunning simultaneously at any point in time remains below the maximum robot threshold RT. \\nWe assume that robots operate 12 hours in a day (e.g., 7am to running simultaneously at any point in time remains below the maximum robot threshold RT. \\nWe assume that robots operate 12 hours in a day (e.g., 7am to \\n7pm)  and they run at a fixed speed . The minimum running time \\nfor a robot is one hour. Robots only operate for a single period per day. We divide the 12 -hour period into 10 -minute time slots such \\nthat every start time and duration for each robot may be defined by an integer from 0 to 66, wh ere Rst\\ni = 0 corresponds to a robot \\nstart time of 7am and Rrti = 0 corresponds to the minimum \\nduration of 1 hour. The set of requests Reqs is built from Rq \\nrandom integers from 60 to 180 (i.e., customer requests take \\nrobots anything from 60 to 180 minutes  to fulfil). Customer \\nrequests must be fulfilled within the 7am to 7pm working period \\nof robots and goods may be delivered at any point during that \\nperiod for all requests . By default Rq = 120 to present a \\nsignificant challenge for the robots; for experimen ts with shorter \\nrequest durations we double Rq to 240.  \\nIn this work we measure the success of solutions for the \\nproblem using the objective function  to be maximised : \\n\\ud835\\udc53(\\ud835\\udc65\\u0305)=\\ud835\\udc61\\ud835\\udc5c\\ud835\\udc61\\ud835\\udc4e\\ud835\\udc59\\ud835\\udc45\\ud835\\udc5a  \\nwhere totalRm  is the total number of requests from Reqs  allocated \\nby the scheduler to the current set of robots Rbts. \\nWe use a single constraint:  \\n\\ud835\\udc61\\ud835\\udc5c\\ud835\\udc61\\ud835\\udc4e\\ud835\\udc59\\ud835\\udc45\\ud835\\udc4f\\ud835\\udc61\\ud835\\udc60 \\u2264\\ud835\\udc45\\ud835\\udc47 \\nwhere totalRbts  is the total number of robots working \\nsimultaneously in any one timeslot.  As a huma n operator must \\nmonitor each robot in this problem, RT directly correlates  with the \\nnumber of human operators. Figure 2 illustrates the calculation of  \\ntotalRbts. P. Bentley  et al.  \\n \\n \\n Rbts  0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 100-110 110-120 \\n1 0 0 0 1 1 1 1 0 0 0 0 0 \\n2 0 0 0 0 1 1 1 1 1 1 1 1 \\n3 1 1 1 1 1 0 0 0 0 0 0 0 \\n4 1 1 1 1 1 1 1 1 1 1 1 1 \\n+ 2 2 2 3 4 3 3 2 2 2 2 2 \\nFigure 2: Example calculation showing  totalRbts  = 4, for four \\nrobots in a two -hour period (ignoring the minimum duration \\nof 60 for this example) . If the threshold RT = 2 then the \\nconstraint would not be met on 4 out of the 12 possible \\ntimeslots , marked in bold. When totalRbts <= RT  for every \\ntimeslot for all robots, the constraint is met fully. In this case \\nthe constraint score being minimized is 4 - 2 = 2.  With the \\nconstraint not satisfied, this is not a valid solution.  \\n3.1 Optimization with Genetic Algorithm \\nOur baseline approach to address this problem is to use a genetic algorithm to evolve start times and durations for a set of robots \\nsuch that the constraint is met and the objective is  maximized for \\na given set of customer requests Reqs . \\nOur search variables \\n\\ud835\\udc65\\u0305 comprise:  \\n\\ud835\\udc65\\u0305=[\\ud835\\udc65!\\\"#,\\ud835\\udc65$\\\"#,..,\\ud835\\udc65!\\\"%&,\\ud835\\udc65$\\\"%&] \\nwhere we map \\ud835\\udc65!\\\"',\\ud835\\udc65$\\\"' to Rsti, Rrti respectively  for every i. Should \\nany i exist such that \\ud835\\udc65!\\\"'+\\ud835\\udc65$\\\"'>65 then the corresponding values \\nof Rrti is corrected such that the robot ceases its shift at the end of \\nthe working day.  Values are corrected during this mapping stage \\nrather than modifying  the evolving parameters as this has been \\nshown to be more conducive to search [20]. During evolution the \\nfitness and constraint is calculated as described above. This is the \\nworst -case scenario for the constraint as it assumes that every \\nrobot will run for its maximum permitted time. In reality, after requests have been scheduled some robots may run for shorter \\ndurations depending on which requests were allocated. Thus , for \\naccuracy, o ur reported results for all experiments first updates  \\nrobot durations  according to the scheduler  in Algorithm 1 : \\n\\u2200\\ud835\\udc56\\u2208{1..\\ud835\\udc45\\ud835\\udc4f}\\t\\ud835\\udc45\\ud835\\udc5f\\ud835\\udc61'\\u2212=E\\ud835\\udc5f\\ud835\\udc52\\ud835\\udc5a\\ud835\\udc4e\\ud835\\udc56\\ud835\\udc5b\\ud835\\udc56\\ud835\\udc5b \\ud835\\udc54($'\\/10K \\nand then measure s the actual number of scheduled robots that ran \\nsimultaneously, as shown in Figure 2. \\nWe use a constrained optimization algorithm [21] using DEAP  \\n[22] with tournament fitness (see [2] for the algorithm) to enable \\nthe equal contribution of constraint and objective to evolution.  \\nThis is a widely used approach in constraint -handling used by \\nnumerous other researchers [23]. In this method, fitness of each \\nindividual is the number of times its separate criteria win a series \\nof tournaments held between randomly selected subgroups  of 3 \\nindividuals. Within each tournament, the fitness  of an individual \\nis increased if its solution is better (lower) for the number of times \\nthe constraint is not satisfied, and also when its solutio n is better \\n(higher) for the objective  \\ud835\\udc53(\\ud835\\udc65\\u0305) compared to the others in the \\ntournament.  We use a Gaussian  Integer creep mutation. \\n   \\nFigure 3: Constrained optimization in learned latent space \\n(COIL) applied to real -world autonomous robot scheduling \\nproblem.  \\n3.2 Optimization with COIL  \\nWe are tackling a challenging constrained optimization problem, so our proposed approach to be compared with the baseline is \\nCOIL  [2] \\u2013 a new hybrid machine learning and optimization \\nmethod  designed to learn valid regions of the search space and \\nimprove the ability of optimizers to find valid solutions. COIL has \\nthe advantage that , unlike other constrained optimizers, it does not \\nrely on complex operators, weight -tuning, or specialized \\nexpertise. So far COIL has been tested only on simple test \\nfunctions for constrained optimization ; this is the first research to \\napply this approach to a real -world problem.  \\nCOIL operates using three steps  (Figure 3): \\n1. Dataset generation from constraint  \\n2. Learning new latent represen tation from dataset \\n3. Optimization using latent representation  using \\nconstraint and objective  \\nFor the first step, we use a simple genetic algorithm (using 2. Learning new latent represen tation from dataset \\n3. Optimization using latent representation  using \\nconstraint and objective  \\nFor the first step, we use a simple genetic algorithm (using \\nDEAP) with a population size of 200 for running for at most 200 \\ngenerations. Our search variables \\n\\ud835\\udc65\\u0305 are exactly as described for \\nthe baseline GA.  The objective function for this data- generation \\nGA is minimization of the number of times the constraint  is not \\nsatisfied , with any solution that satisfies the constraint added to \\nthe dataset, and the GA restarting. The threshold RT, which \\ndetermines how many robots may work simultaneously , is varied \\nduring experiments. Solutions that specify robot durations that \\nfinish beyo nd the 12- hour working period  are corrected as \\ndescribed in 3.1 before being normalized and added to the dataset.  \\nTo encourage the dataset -generating GA not to \\u201ccheat\\u201d and only \\nprovide solutions with minimal robot running times (the simplest \\nway to satisfy  the constraint), we also add a selective pressure \\ntowards longer robot durations by adding 100 \\u2211 \\ud835\\udc65$\\\"' )%\\n#\\u2044  to the \\nfitness term. We run this GA repeatedly until DS vectors  are \\ngenerated. Each vector represents a valid set of robot start times \\nand duration s such that the constraint is satisfied. The overall \\nobjective to find robot times and durations that enable the most \\ncustomer requests is not used at this point.  \\nIn the second step, we provide  the dataset to a simple \\nvariational autoencoder [2] with 4 linear layers , a prior  of N(0,I), \\nKLD = 1, for 200 epochs using the Adam optimizer with a \\nlearning rate of 0.001. We run the VAE 10 times and choose the \\nlearned model with the lowest error. This is our learned latent \\nrepresentation , with latent variables:  \\n\\ud835\\udc67\\u0305=[\\ud835\\udc67#,\\ud835\\udc67*,..,\\ud835\\udc67*\\u00d7,-.\\/0] Learn ing Valid Search Spaces of Autonomous Robots   \\n \\n In the third step, we use the same GA as described in section \\n3.1, to ensure a completely fair comparison. For this GA, the \\nsearch variables \\ud835\\udc67\\u0305 are encoded as real -valued variables with \\nranges between -2.0 and 2.0, and a Gaussian creep mutation. We \\nconvert \\ud835\\udc67\\u0305 into \\ud835\\udc65\\u0305 by using the learned VAE model  to express the \\nvalues , and then  perform a scalar inverse transform, \\nunnormalization, and conversion to intege r to convert the VAE \\noutput into the desired 0..66 range.  We then map \\ud835\\udc65!\\\"',\\ud835\\udc65$\\\"' to Rsti, \\nRrti respectively  for every i as described in section 3.1  with the \\nsame robot duration updates performed according to the scheduler  \\nand use the identical tournament fitness approach to measure \\nfitness for the objective and constraint. \\n4 EXPERIMENTS \\n4.1 Experiments  \\nWe perform experiments to investigate the following research \\nquestions, which reflect the two stages to this problem : \\n \\nRQ1: Does COIL help us explore valid design variations more \\neffectively than the baseline GA?  \\nWe explore this question in several experiments . \\nE1.1: Can the baseline GA and  COIL help us explore valid \\nsolutions ? Before we can explore different design variations we \\nneed to check that we can solve this problem at all. Experiment 1 \\ntackles this by directly comparing the output of the baseline GA with COIL.  Following the training of the VAE within COIL, the \\nsame learn ed model is used in all repeated runs. We use a small \\npopulation size of 20 for just 50 generations following  [2, 3]. Both \\nGAs are run 100 times and mean results are shown.  A new set of \\nrequests Reqs  is randomly generated for every run.  \\nE1.2: Can the GA and COIL explore  valid solutions varying \\nRT? In this experiment we explore variants  of the problem by \\nvarying RT to the values: 10, 15, 20. This explores solutions \\nwhere we permit more robots to run simultaneously (at the cost of \\nrequiring more human monitoring.)  \\nE1.3: Can the GA and COIL explore  valid solutions varying \\nRb? In the second problem variant, we fix RT to 10 and vary the \\nnumber of robots Rb to the values: 2 0, 25, 30.  This explores \\nsolutions where we have fewer robots overall, with at most 10 running simultaneously.  \\nE1.4: Can the GA and COIL explore  valid solutions varying \\ndr? In our final set of problem variants, w e vary the duration of \\nrequests; instead of random from 60 to 180, we try 60 to dr, where \\ndr \\n\\u2208 {60, 80, ..,  360}.  \\nFor all RQs, parameters other than those being varied remain \\nunchanged. COIL is run 100 times for each setting, with a new set Reqs  of random requests each run.  \\n \\nRQ2: Is COIL an efficient algorithm for daily optimization?  \\nOnce the company decides on the number of robots and operators to use, we arrive at the second stage. Here we investigate the \\nsuitability of COIL for use as a daily optimizer, investigating \\nwhether the method can be tuned with this goal in mind.  E2.1: Can we use smaller  dataset size s for COIL without \\naffect ing its ability to generate usef ul latent representations?  We \\nneed a fast way of optimizing robot schedules every day for the new set of requests, while always meeting the constraint. O ne \\ndrawback of COIL is the need to generate a dataset first  and train \\nthe VAE. Although this is a one -off, offline computation, it is still \\nsignificant . This experiment  addresses this by varying the dataset \\nsize DS to four different values: 2500, 5000, 7500, 10000. All \\nother parameters remain unchanged. COIL is run 100 times (with \\na new set Reqs  of random requests each run) and mean results are \\nshown. We also measure the difference in data generation and \\ntraining times and the error rate of the VAE.  \\nE2.2: Can we imp rove the performance of COIL by reducing \\nthe number of latent variables?  Here we investigate an idea for \\nimproving the performance of COIL.  While COIL was described E2.2: Can we imp rove the performance of COIL by reducing \\nthe number of latent variables?  Here we investigate an idea for \\nimproving the performance of COIL.  While COIL was described \\nas a method to improve the search space while keeping the number of latent variables the sa me as the number of input \\nvariables, other work has shown that VAEs may be able to provide the additional benefit of reducing the search  space size \\n[24]. We investigate this question with the number of robots Rb = \\n30 (giving us 60 problem variables) by varying maxlv  to the \\nvalues: 5, 10, 15, 20, 25, 30 (i.e., 10 to 60 latent variables) . All \\nother parameters remain unchanged.  COIL is run 100 times for \\neach setting, with a new set Reqs  of random requests each run.  \\nTable 1 summarizes the parameters investigated in each \\nexperiment. All processing was performed on a MacBook Air \\n2020 M1 with 16Gb memory. All code was implemented in \\nPython and is available\\n1. \\n \\nTable 1. Parameters for the experiments  \\nExp DS maxlv  RT Rb dr \\nE1.1 10000 30 10 30 180 \\nE1.2 10000 30 10,15,20  30 180 \\nE1.3  10000 30 10 20,25,30  180 \\nE1.4  10000 30 10 30 60..360  \\nE2.1 2500..10000  30 10 30 180 \\nE2.2  10000 5..30  10 30 180 \\n \\n4.2 Results  and Analysis \\nIn this section we describe the results for  research question RQ1 \\n(experiments E1.1, E1.2, E1.3 and E1.4) and for RQ2 (experiments E2.1 and E2.2).  \\n Table 2. E1. 1: COIL vs GA.  Better results in bold.  \\n COIL  GA \\navg objective  (stdv) 66.43 (6.80) 93.6 (5.55)  \\nmin objective 51 76 \\nmax objective  77 107 \\navg constraint  (stdv) 23.93 ( 8.43) 44.7 (3.89) \\nmin constraint  0 35 \\nmax constraint  30 52 \\n \\n1 GitHub link temporarily removed for purposes of paper anonymization. P. Bentley  et al.  \\n \\n \\n  \\nFigure 4: Example s ingle best run baseline GA. \\n \\n \\nFigure 5: Example best run COIL . \\n \\nE1.1: Can the GA and COIL help us explore valid solutions ? \\nIn this experiment we compare the output of the baseline GA with \\nCOIL to check that both can find valid solutions for our default \\nparameter values. Table 2 summarizes the results. The results \\nshow that only  COIL is able to provide valid solutions for this \\nvariant of the problem. COIL provides superior results in terms of constraint satisfaction, with the best results being a perfect zero, \\nand average of 23.9, compared t o the baseline GA best of 35 \\n(worse than the worst score of COIL) and average of 44.7. The scores for the objective shows that GA allocates more tasks on \\naverage than COIL, managing 107 at best compared to COIL\\u2019s much lower 77. However, the scores are mean ingless when the \\nconstraint is broken \\u2013 the baseline GA cheats by not meeting the \\nconstraint in order to allocate more requests, and such solutions \\nare not viable or useful.  \\nExamining a single best run for the baseline GA and COIL, \\nFigure 4 and Figure 5 show the difference in evolution. The GA \\nattempts to improve the constraint while keeping request \\nallocation constant but never achieves a single valid solution \\u2013 \\nshowing a failure of this algorithm to tackle the problem effectively. In contrast COIL start s better and optimizes the \\nconstraint to zero very rapidly while keeping the number of requests allocated constant. To assess whether the GA could be \\ncoerced into focusing more on the constraint, we increased the \\nweighting by 10 times in the tournament sel ection algorithm for the constraint; we also tried much larger population sizes and number of generations. The results were barely changed and the \\nGA still failed to find any solutions that satisfied the constraint. Figure 4 and Figure 5 (grey line) also show how effectively \\n\\ud835\\udc65\\u0305 is \\nbeing optimized, with both algorithms ensuring that  91% of the \\nrobots\\u2019 time is being used.  While COIL does not find perfect \\nsolutions every time for this difficult problem (unlike findings of [2, 3]) it is clear that its learned latent representation is biased \\ntowards solutions  that satisfy the constraint more, with Figure 6 \\ntop left showing that the baseline GA results (orange cir cles) are \\nall worse than the COIL results (blue circles).  \\n \\nE1.2 : Number of simultaneously running robots  RT \\nWhen we increase the value of RT, we make the constraint easier \\nas more robots are permitted to run simultaneously. The results show that in all problem variants, COIL successfully finds valid \\nsolutions to the problem. However, as the optimization problem becomes easier, the baseline GA begins to function more usefully \\n(Figure 6 top left). When RT = 10, COIL performs better than the \\nbaseline GA. When RT = 15, COIL is able to find more solutions \\nthat satisfy the constraints, while the baseline GA still finds none. But when RT = 20 (the problem is no longer diffic ult) there is no \\nlonger any significant difference between the GA and COIL, with COIL finding slightly more valid solutions, but the GA finding a \\nfew solutions that satisfy the constraint and fulfil the objective \\nslightly better.  \\n \\nE1.3 : Total n umber of robots  Rb \\nSimilar to the previous experiment, when we reduce the value of Rb, we make the constraint easier as fewer robots in total are \\nrunning, so fewer are likely to run simultaneously. However, with RT fixed we do not see improvements in objective scores (keeping \\nthe number of simultaneously running robots constant, limits the number of requests that can be served). When reducing Rb from \\n30 to 25, the baseline GA comes closer to finding valid solutions. When Rb = 20, the GA finds several valid solutions. H owever, for \\nall values tested, COIL always finds valid solutions to the \\nproblem, becoming more consistent as the problem is easier \\n(Figure 6 top right).  \\n E1.4 : Changi ng request durations dr all values tested, COIL always finds valid solutions to the \\nproblem, becoming more consistent as the problem is easier \\n(Figure 6 top right).  \\n E1.4 : Changi ng request durations dr \\nIf we alter the request durations (increasing the number of requests from 120 to 240 to make up for smaller durations), we change the difficulty of the problem in a new way: shorter tasks \\nare easier to allocate while longer tasks are more difficult. Figure \\n7 shows how the average number of constraints broken slightly \\nimproves (lowers) as the request duration increases for COIL, \\nsuggesting that COIL scales better to gre ater problem difficulty. \\nAs expected, the number of requests that can be allocated falls as the task durations increase, with the baseline GA always allocating more (because it cheats by not satisfying the \\nconstraint). Figure 6: Best result at final generation for 100 runs  for each algorithm , plotted with objective score against constraint . Lower \\nmeans constraint is better satisfied, more to the right means more tasks are allocated. Only solutions on y = 0 satisfy the c onstraint \\nfully and thus are valid. Top left: E1.1 Comparing GA and COIL and E1.2 Varying RT. Top right: E. 1.3 Varying Rb. Bottom left: \\nE2.1 Varying  COIL  DS. Bottom right: E2.2 Varying  COIL  maxlv . \\n \\n \\nFigure 7: E1.4: Altering maximum random request durations.  \\n \\nE2.1: Reducing dataset size DS \\nThe previous  experiments showed that COIL clearly outperforms \\nthe baseline GA for this real -world problem  \\u2013 only COIL has \\npermitted us to explore valid solutions for all design variations \\ntested . However, there is a difference in computation time \\nrequired. When evolving the solutions for E1.1, the baseline GA \\ntook 1 .89 minutes to complete 100 runs. COIL took a slightly \\nshorter 1.81  minutes \\u2013 despite the additional need to express the \\nevolving latent variables into the problem variables using the VAE. However, COI L also required a one -off pre -computation to generate valid data and use the VAE to learn the latent \\nrepresentation. This computation was substantial, requiring 25.80 \\nhours  to generate 10,000 datapoints, and 8.95 minutes for the \\nVAE to learn (running 10 ti mes and choosing the best learned \\nmodel, which had loss of 0.9114).  These times reduced when the \\nproblem was easier, for example in E1.2, 10,000 valid points were generated in just 19.28 minutes when RT = 15, and just 8.08 \\nminutes when RT = 20.  \\nIn the next experiment, we investigate the effects of the dataset \\nsize to see if COIL can still achieve acceptable results when the VAE is trained with smaller datasets.  We achieved this by simply \\nrunning the data generator each time with DS = 2500, 5000, 7500, \\nand 10000. The datasets were checked to see if they contained any redundancy through duplication; analysis revealed that every \\npoint in each dataset is unique and not repeated.  \\nFigure 8 shows the differences in computation time vs the \\nVAE loss , indicating that the dataset of size 5000 has the smallest \\nloss; detailed results from COIL for the constraint and objective \\nfor each dataset size  are provided in Supplementary Materials . \\nWhile the best average constraint satisfaction is achieved using \\nthe largest dataset, performance is still relatively unaffected for \\neven the smallest size. Figure 6 (bottom left) shows that valid \\nsolutions on y = 0 are generated by all dataset sizes, and all also \\nhave a similar trend in the solution space. P. Bentley  et al.  \\n \\n \\n  \\nE2.2 : Modifying num ber of  latent variables  maxlv  \\nFinally we investigate whether we can improve the performance \\nof COIL in terms of the quality its solutions. Applying COIL with \\ndifferent numbers of latent variables shows an important effect on \\nthe distribution of solutions, Figure 6 (bottom right) and \\nSupplementary Materials . Reducing the number of latent variables \\ngenerally improves the ability of COIL to find solutions that satisfy the constraint.  Fewer latent vari ables mean faster training \\ntime for the VAE, and faster optimization for the GA using the smaller latent representations. But as the number of latent \\nvariables decreases to 10, the VAE loss becomes worse ( Figure \\n9). The smaller search space also appears to impact the objective, \\nwith fewer tasks being allocated.  \\nIn contrast, increasing to 50 or more latent variables appears to \\nprovide an unwelcome bias away from valid so lutions on y = 0. \\nFor this problem, there appears to be a \\u201csweet spot\\u201d between \\nmaxlv=15 and maxlv=25 (30 and 50 latent variables  respectively), \\nwhere solutions with high number of request allocations and perfect constraint scores are generated ( Figure 6 bottom right). \\nFor our problem it appears that having fewer latent variables \\ncompared to problem variables is advantageous.  \\n \\n \\nFigure 8: Computation time and VAE loss  for E2.1 . Bar chart \\nusing left y-axis: Data generation (hours), VAE training \\n(minutes), blue line using right y-axis: VAE loss. \\n \\n \\nFigure 9: Computation time and VAE loss fo r E2.2 . Bar chart \\nusing left y-axis: VAE training (minutes), blue line using right \\ny-axis: VAE loss. \\n 4.3 Discussion of Findings Relating to Problem  \\nCOIL has enabled a useful exploration of the design space for this \\nreal-world problem. Our findings indicated that the amount of \\navailable time spent by robots delivering requests rarely changes (usually at around 90%). This appears to be the best use of the \\nrobots\\u2019 time as determined by the scheduler , and both optimizers \\nare always able to find appropriate start times and durations for \\nthe autonomous robots to reach this efficiency. However, results \\nfrom the baseline GA were often unusable as the safety constraint \\nwas not met. COIL provided usable results for all setups. The experiments also showed that when customer requests were likely \\nto take less time, more would be delivered successfully by the robots. The safety constraint was slightly more likely to be \\nsatisfied by COIL for longer duration requests; for the baseline \\nGA there was no difference.  \\nOver all, COIL\\u2019s exploration of the problem space  indicate s \\nthat when we keep the total number of robots Rb constant but \\nallow more to run simultaneously (i.e., more human monitors are employed), the number of customer requests tha t can be satisfied \\nwill increa se. In contrast, if we keep the number of simultaneously \\nrunning robots RT constant and reduce the overall numbers of \\nrobots that are running, there is little change to the number of \\ncustomer requests met. Thus, with the fixed relationship between \\nthe numb er of simultaneously running robots and the number of \\nhuman monitors, there appears to be a direct correlation between the number of human monitors and customer satisfaction.   \\n5 CONCLUSIONS \\nThe problem of scheduling autonomous robots safely such that sufficient human monitors are available is challenging , and \\nrelatively  unexplored in the literature . Our work shows that a  \\nstandard genetic algorithm using a well-established method  for \\nconstrained optimization (the baseline GA ) was unable to find \\nsolutions tha t satisfied the constraint except for variations where \\nthe problem was relatively easy. In contrast a new ML -EA hybrid \\napproach (constrained optimization in learned latent space: COIL ) \\nwas able to find valid solutions for all problem variants, enabling \\nus to perform a useful investigation of the effects of each approach (constrained optimization in learned latent space: COIL ) \\nwas able to find valid solutions for all problem variants, enabling \\nus to perform a useful investigation of the effects of each \\nparameter on solutions. This work has also shown for the first \\ntime that the data -generation and training stage of COIL can be \\nimproved by reducing the dataset size and number of latent \\nvariables. This means that once suitable parameters are selected, \\nand the initial one -off training is performed  to learn the latent \\nrepresentation , COIL is able to optimize new schedules rapidly \\nand reliably. Indeed, with these improvements, COIL is 10% \\nfaster to run than  the baseline GA. COIL is thus an ideal choice \\nfor an everyday optimizer for this problem.  \\nFuture work will examine further problem variables of interest \\nto our industrial partner such as robot speed and will integrate \\nwith other schedulers. Other  improvements to COIL will be \\nexamined, such as the use of more advanced VAEs and optimizers such as MAP -Elites for the data generation and optimization \\nstages, to further improve speed and efficiency. Learn ing Valid Search Spaces of Autonomous Robots   \\n \\n REFERENCES \\n[1] M. Figliozzi and D. Jennings, \\\"Autonomous delivery robots and their potential \\nimpacts on urban freight energy consumption and emissions,\\\" Transportation \\nresearch procedia, vol. 46, pp. 21 -28, 2020. \\n[2] P. J. Bentley, S. L. Lim, A. Gaier, and L. Tran, \\\"COIL: Constrained \\nOptimi zation in Learned Latent Space --Learning Representations for Valid \\nSolutions,\\\" in ACM Genetic and Evolutionary Computation Conference \\n(GECCO'22) Companion , 2022, pp. 1870 \\u20131877.  \\n[3] P. J. Bentley, S. L. Lim, A. Gaier, and L. Tran, \\\"Evolving through the look ing \\nglass: Learning improved search spaces with variational autoencoders,\\\" in \\nInternational Conference on Parallel Problem Solving from Nature (PPSN) , \\n2022, pp. 371 -384. \\n[4] N. Boysen, S. Schwerdfeger, and F. Weidinger, \\\"Scheduling last -mile deliveries \\nwith truck -based autonomous robots,\\\" European Journal of Operational \\nResearch, vol. 271, no. 3, pp. 1085 -1099, 2018.  \\n[5] S. Yu, J. Puchinger, and S. Sun, \\\"Two -echelon urban deli veries using \\nautonomous vehicles,\\\" Transportation Research Part E: Logistics and \\nTransportation Review, vol. 141, p. 102018, 2020.  \\n[6] A. Karak and K. Abdelghany, \\\"The hybrid vehicle -drone routing problem for \\npick-up and delivery services,\\\" Transportation Research Part C: Emerging \\nTechnologies, vol. 102, pp. 427 -449, 2019.  \\n[7] C. Chen, E. Demir, Y. Huang, and R. Qiu, \\\"The adoption of self -driving \\ndelivery robots in last mile logistics,\\\" Transportation research part E: logistics \\nand transportation review, vol. 146, p. 102214, 2021.  \\n[8] I. Bakach, A. M. Campbell, and J. F. Ehmke, \\\"Robot -Based Last -Mile \\nDeliveries With Pedestrian Zones,\\\" Frontiers in Future Transportation, p. 32, \\n2022.  \\n[9] K. Braekers, K. Ramaekers, and I. Van Nieuwenhuyse, \\\"The vehicle routing  \\nproblem: State of the art classification and review,\\\" Computers & industrial \\nengineering, vol. 99, pp. 300 -313, 2016.  \\n[10] S. Kapser and M. Abdelrahman, \\\"Acceptance of autonomous delivery vehicles \\nfor last- mile delivery in Germany \\u2013Extending UTAUT2 with ri sk perceptions,\\\" \\nTransportation Research Part C: Emerging Technologies, vol. 111, pp. 210 -\\n225, 2020.  \\n[11] H. Liu, R. Yang, L. Wang, and P. Liu, \\\"Evaluating initial public acceptance of highly and fully autonomous vehicles,\\\" International Journal of Human \\u2013\\nComputer Interaction, vol. 35, no. 11, pp. 919 -931, 2019.  \\n[12] G. E. Hinton and R. R. Salakhutdinov, \\\"Reducing the dimensionality of data \\nwith neural networks,\\\" Science, vol. 313, no. 5786, pp. 504 -507, 2006.  [13] D. P. Kingma and M. Welling, \\\"Auto -encoding  variational bayes,\\\" in \\nProceedings of the International Conference on Learning Representation (ICLR), 2014.  \\n[14] \\u00d6. Yeniay, \\\"Penalty function methods for constrained optimization with genetic \\nalgorithms,\\\" Mathematical and Computational Applications, vol. 10, no. 1, pp. \\n45-56, 2005. \\n[15] V. Volz, J. Schrum, J. Liu, S. M. Lucas, A. Smith, and S. Risi, \\\"Evolving Mario \\nlevels in the latent space of a deep convolutional generative adversarial \\nnetwork,\\\" in Proceedings of the Genetic and Evolutionary Computation \\nConference (GECCO) , 2018, pp. 221 -228. \\n[16] P. Bontrager, A. Roy, J. Togelius, N. Memon, and A. Ross, \\\"Deepmasterp rints: \\nGenerating masterprints for dictionary attacks via latent variable evolution,\\\" in \\nProceedings of the 2018 IEEE 9th International Conference on Biometrics \\nTheory, Applications and Systems (BTS) , 2018, pp. 1 -9. \\n[17] P. Liskowski, K. Krawiec, N. E. Toklu, and J. Swan, \\\"Program synthesis as latent continuous optimization: Evolutionary search in neural embeddings,\\\" in \\nProceedings of the 2020 Genetic and Evolutionary Computation Conference , \\n2020, pp. 359 -367. \\n[18] J. K. Pugh, L. B. Soros, and K. O. Stanley , \\\"Quality diversity: a new frontier for \\nevolutionary computation,\\\" Frontiers in Robotics and AI, vol. 3, p. 40, 2016. 2020, pp. 359 -367. \\n[18] J. K. Pugh, L. B. Soros, and K. O. Stanley , \\\"Quality diversity: a new frontier for \\nevolutionary computation,\\\" Frontiers in Robotics and AI, vol. 3, p. 40, 2016.  \\n[19] A. Gaier, A. Asteroth, and J. -B. Mouret, \\\"Discovering representations for \\nblack -box optimization,\\\" in Proceedings of the Genetic and  Evolutionary \\nComputation Conference (GECCO) , 2020, pp. 103 -111. \\n[20] T. Yu and P. Bentley, \\\"Methods to evolve legal phenotypes,\\\" in Proceedings of \\nthe International Conference on Parallel Problem Solving from Nature (PPSN) , \\n1998, pp. 280 -291. \\n[21] K. Deb,  \\\"An efficient constraint handling method for genetic algorithms,\\\" \\nComputer Methods in Applied Mechanics and Engineering, vol. 186, no. 2 -4, \\npp. 311 -338, 2000.  \\n[22] F.-M. De Rainville, F. -A. Fortin, M. -A. Gardner, M. Parizeau, and C. Gagn\\u00e9, \\n\\\"Deap: A python framework for evolutionary algorithms,\\\" in Proceedings of the \\n14th annual conference companion on Genetic and evolutionary computation , \\n2012, pp. 85 -92. \\n[23] C. A. C. Coello, \\\"Theoretical and numerical constraint -handling techniques \\nused with evolutionary  algorithms: a survey of the state of the art,\\\" Computer \\nmethods in applied mechanics and engineering, vol. 191, no. 11 -12, pp. 1245 -\\n1287, 2002.  \\n[24] X. Ding, Z. Zou, and C. L. Brooks III, \\\"Deciphering protein evolution and fitness landscapes with latent s pace models,\\\" Nature communications, vol. 10, \\nno. 1, pp. 1 -13, 2019. SUPPLEMENTA RY MATERIA LS  \\n \\n \\nTable S1: E1.2 : Modifying RT:  number of simultaneously running robots , COIL . \\nCOIL 10 robots at once  15 robots at once  20 robots at once  \\navg objective (stdv)  66.43 (6.80)  73.94 (3.99)  89.86 (3.14)  \\nmin objective  51 63 83 \\nmax objective  77 84 98 \\navg constraint (stdv)  23.93 (8.43)  9.91 (10.78)  0.81 (2.86)  \\nmin constraint  0 0 0 \\nmax constraint  30 29 16 \\n \\nTable S2: E1.2 : Modifying RT:  number of simultaneously running robots , GA. \\nGA 10 robots at once  15 robots at once  20 robots at once  \\navg objective (stdv)  93.6 (5.55)  91.09 (5.43)  92.24 (4.32)  \\nmin objective  76 80 80 \\nmax objective  107 108 103 \\navg constraint (stdv)  44.7 (3.89)  26.69 (5.56)  0.8 (2.97)  \\nmin constraint  35 10 0 \\nmax constraint  32 41 17 \\n Table S3: E1.3: Modifying Rb: total number of robots , COIL . \\nCOIL 20 total robots  25 total robots  30 total robots  \\navg objective (stdv)  50.67 ( 3.63) 57.96 ( 7.59) 66.43 (6.80)  \\nmin objective  44 47 51 \\nmax objective  62 72 77 \\navg constraint (stdv)  4.4 (8.4) 15.39 ( 13.13)  23.93 (8.43)  \\nmin constraint  0 0 0 \\nmax constraint  27 32 30 \\n Table S4: E1.3: Modifying Rb: total number of robots , GA. \\nGA 20 total robots  25 total robots  30 total robots  \\navg objective (stdv)  63.47 ( 5.33) 81.01 ( 6.49)  93.6 (5.55)  \\nmin objective  51 64 76 \\nmax objective  75 102 107 \\navg constraint (stdv)  23.71 ( 7.97) 38.35 ( 4.71) 44.7 (3.89)  \\nmin constraint  0 26 35 \\nmax constraint  41 48 32 \\n \\nTable  S5: E2.1 : Modifying dataset size  DS \\n 2500 data points  5000 data points  7500 data points  10000 data points  \\navg objective (stdv)  66.75 (5.87)  68.2 (8.15)  65.45 (8.09)  66.43 (6.80)  \\nmin objective  51 49 48 51 \\nmax objective  81 82 80 77 \\navg constraint (stdv)  26.52 (4.22)  25.66 (8.02)  23.28 (8.18)  23.93 (8.43)  \\nmin constraint  0 0 0 0 \\nmax constraint  31 33 31 30 \\n \\nTable  S6: E2.2 : Modifying number of latent variables  maxlv  \\n 10 latent variables 20 latent variables 30 latent variables 40 latent variables 50 latent variables 60 latent variables \\navg objective (stdv)  50.1 (2.26)  52.08 (2.40)  53.19 (2.86)  53.26 (3.38)  53.47 (3.56)  66.43 (6.80)  \\nmin objective  45 47 47 44 47 51 \\nmax objective  56 59 61 63 70 77 \\navg constraint (stdv)  9.25 (5.64)  1.96 (3.03)  4.69 (6.11)  4.41 (6.78)  6.69 (7.83)  23.93 (8.43)  \\nmin constraint  0 0 0 0 0 0 \\nmax constraint  19 13 24 25 32 30\",\"44\":\" Vectorial Genetic Programming { Optimizing\\nSegments for Feature Extraction\\nPhilipp Fleck1;2?[0000\\u00000001\\u00008290\\u00006356], Stephan Winkler1;2[0000 \\u00000002\\u00005196\\u00004294],\\nMichael Kommenda1;3[0000 \\u0000003\\u00002049\\u0000723X], and Michael\\nA\\u000benzeller1;2[0000 \\u00000001\\u00005692\\u00005940]\\n1Heuristic and Evolutionary Algorithms Laboratory (HEAL),\\nUniversity of Applied Sciences Upper Austria, Softwarepark 11, 4232 Hagenberg, Austria\\n2Institute for Symbolic Arti\\fcial Intelligence,\\nJohannes Kepler University, Altenberger Stra\\u0019e 69, 4040 Linz, Austria\\n3Josef Ressel Center for Symbolic Regression,\\nUniversity of Applied Sciences Upper Austria, Softwarepark 11, 4232 Hagenberg, Austria\\nAbstract. Vectorial Genetic Programming (Vec-GP) extends GP by\\nallowing vectors as input features along regular, scalar features, using\\nthem by applying arithmetic operations component-wise or aggregating\\nvectors into scalars by some aggregation function. Vec-GP also allows\\naggregating vectors only over a limited segment of the vector instead of\\nthe whole vector, which o\\u000bers great potential but also introduces new\\nparameters that GP has to optimize. This paper formalizes an optimiza-\\ntion problem to analyze di\\u000berent strategies for optimizing a window for\\naggregation functions. Di\\u000berent strategies are presented, included ran-\\ndom and guided sampling, where the latter leverages information from\\nan approximated gradient. Those strategies can be applied as a simple\\noptimization algorithm, which itself ca be applied inside a specialized\\nmutation operator within GP. The presented results indicate, that the\\ndi\\u000berent random sampling strategies do not impact the overall algorithm\\nperformance signi\\fcantly, and that the guided strategies su\\u000ber from be-\\ncoming stuck in local optima. However, results also indicate, that there\\nis still potential in discovering more e\\u000ecient algorithms that could out-\\nperform the presented strategies.\\nKeywords: Genetic Programming \\u00b7Vectorial \\u00b7Optimization \\u00b7Gradient\\n1 Introduction\\nVectorial Genetic Programming (Vec-GP) for Symbolic Regression (SR) is a ex-\\ntension of GP, where the space of input features is extended to vectors alongside\\nregular scalars[1]. This allows vectorial GP to directly handle higher dimensional\\ndata, such as time series, without the need of prior feature engineering to ex-\\ntract scalar features. Instead, vectorial GP models themselves extracts scalar\\n?corresponding author\\nphilipp.fleck@fh-hagenberg.atarXiv:2303.03200v1  [cs.NE]  3 Mar 2023 2 P. Fleck et al.\\nvalues from the vectors by applying aggregation functions, such as the arith-\\nmetic mean. Compared to traditional feature engineering, where scalar features\\nare usually only extracted from the raw vector features, vectorial GP can also\\nextract features from interacting vectors by performing arithmetic operations on\\nvectors before aggregation takes place. Fig. 1 shows an example, where the vector\\nvariable for temperature and pressure are divided prior to calculating the arith-\\nmetic mean to obtain a scalar. The literature already indicates, that vectorial\\nGP outperforms regular GP with feature engineering in various benchmarks[2].\\nFig. 1: An example dataset containing both scalar and vector input features to\\npredict a scalar target value (the quality). It also shows a GP model that uses a\\nwindowed-aggregation function to aggregate a vector into a scalar.\\nIn case of a vector representing a time series, it might be bene\\fcial to aggre-\\ngate only over a certain time frame. In this sense, vectorial GP can be further\\nextended by adding the ability to aggregate not only over a whole vector, but\\nover a sub-segment of the vector, where GP is also in charge of optimizing the\\nsegment bounds. For instance, as also shown in Fig. 1, aggregation might only\\nbe done on a speci\\fc time frame, for example, between 3 \\u0014t\\u001410.\\n2 Problem Description\\nWhen using an aggregation function with an aggregation window, GP is now\\nrequired to optimize two additional parameters for each windowed aggregation\\nfunction within a GP model. While GP could handle this by allowing the indices\\nto change via mutation, random mutation is not particularly well suited for this\\ntask, especially for longer vectors. Instead of relying on mutation as GP's only\\nway of optimizing aggregation windows, this paper is a \\frst step towards iden-\\ntifying other strategies that are more tailored towards optimizing aggregation\\nwindows, similar to applying e\\u000ecient gradient-based algorithms for optimizing\\ncontinuous, numerical parameters of a \\fxed GP model[4].\\nTo focus on the problem of optimizing the aggregation window, we use a\\nsimpli\\fed optimization problem that omits any GP related aspect, only keeping\\nthe core problem of optimizing an aggregation window over some \\fxed data. Vectorial GP { Optimizing Segments for Feature Extraction 3\\nThis Segment Optimization Problem (SOP) is an integer problem, optimizing\\nthe parameters ^ a;^b2fi2Nji < Ngrepresenting the start and end index with,\\nminimizing\\narg min\\n^a;^bMX\\ni=0\\u0010\\nfagg\\u0000\\nvi[^a:^b]\\u0001\\n|{z}\\noptimized bounds\\u0000fagg\\u0000\\nvi[a:b]\\u0001\\n|{z}\\nknown bounds\\u00112\\n; (1)\\nwhere f aggis a \\fxed aggregation function, vi2RM\\u0002Nare the Msamples of\\nvectors with length N,aandbrepresenting the known aggregation indices and\\nvi[a:b] being the sub-vector of a selected vector vi. In essence, the goal of the\\nSOP is to \\fnd the minimum deviation when aggregating using the free indices\\n^aand^bcompared to aggregating using the known indices aandb.\\n3 Method\\nInstead of relying on GPs mutation to identify good aggregation windows, our\\ngoal is to de\\fne a specialized mutation operator for GP that is tailored to-\\nwards optimizing the start and end indices of an aggregation window e\\u000eciently.\\nStart and end indices can either be optimized simultaneously, i. e. as a multi-\\ndimensional optimization problem, or only a single dimension at a time, i. e.\\nas a single-dimensional optimization problem. We will later analyze which form\\nyields better results.\\nLooking at the \\ftness landscape of some of the benchmark instances in Fig. 2,\\nwe can clearly observe, that there is gradient information that could be lever-\\naged for e\\u000ecient optimization. Therefore, we \\frst approximate the gradient by\\nusing a \\fve-point stencil[5] on the neighboring indices ( h= 1) and then use\\nit to guide the search. The proposed optimization method in this paper is an\\niterative algorithm with two steps: First, the gradient is calculated and used to\\nguide search towards promising regions. Second, potential solutions are sampled\\nfrom that promising region, evaluated and the best one is selected for the next\\nFig. 2: The \\ftness landscape of two benchmark instances (x3 and x6) with lower\\n(brighter) values indicating better regions in the search space. 4 P. Fleck et al.\\niteration. This procedure is continued until a prede\\fned number of total solution\\nevaluations is exceeded.\\nThe \\frsts guiding step in the optimization procedure is responsible for se-\\nlecting a promising region, from which samples can be drawn in the second step\\nafterwards. We de\\fne three di\\u000berent guiding strategies, shown in Fig. 3: Full\\nrepresents no strategy at all, where the whole solution space is utilized. This\\nstrategy serves will serve as a baseline. Direction uses the approximate gradient\\nonly to guide whether an index should be increased or decreased and does not\\ntake the magnitude of the gradient into account. Range uses the approximate\\ngradient information to de\\fne a promising region by a de\\fned range, where the\\nmagnitude of the gradient also in\\ruences the distance from the current index.\\nFig. 3: Three di\\u000berent guiding strategies for de\\fning the search space, given a\\nfree start index and a \\fxed end index.\\nWhile the random direction simply looks at the direction of gradient, the\\nguided range mimics a traditional gradient descent algorithm with two modi\\f-\\ncations, shown in Fig. 4. First, we round the gradient step towards the nearest\\ninteger, to ensure that the next index will also be valid index. Additionally, we\\nintroduce a new parameter, the search-range, that de\\fnes how many samples\\naround the next index are considered promising and are thus eligible for being\\nsampled afterwards.\\nFig. 4: Adoption of a gradient descent step to integer indices, with an additional\\nsearch-range parameter.\\nAfter de\\fning the search space, the second step is responsible of drawing\\nthe samples from this search space. For this paper, we have considered three Vectorial GP { Optimizing Segments for Feature Extraction 5\\nsampling mechanisms, shown in Fig. 5: Exhaustive draws all samples from the\\nsolution space. This is typically not feasible for a multi-dimensional search space,\\ndue to a potentially high number of possible solutions. However, it could be a\\nviable option for a single dimensional space. Random draws samples completely\\nat random (without repetition), where the number of samples are de\\fned by an\\nadditional sample-size parameter. Orthogonal selects the samples equally spaced\\nalong each dimension, by \\frst selecting a prede\\fned number of equally spaced\\npoints on a continuous line of the search space and then rounding towards the\\nnearest indices and removing any duplicates.\\nFig. 5: Di\\u000berent sampling strategies for a given search space.\\n4 Experiment Setup\\nTo identify good parameters of the search strategies de\\fned in the previous sec-\\ntion, we performed a grid search of the parameters over multiple benchmark\\ninstances. However, certain combinations were skipped due to infeasible run-\\ntimes, e. g. exhaustive search on multi-dimensional search spaces. Each parame-\\nter combination was repeated 20 times, and each execution of the optimization\\nalgorithm was allowed a total of 100,000 sample evaluations, meaning the total\\nnumber of iterations varied, depending on the sample-size, for instance.\\nThe generated benchmark instances have di\\u000berent characteristics to represent\\ndi\\u000berent challenges and di\\u000eculties. For instance, shown in Fig. 6, the second\\ninstance shows an overall increasing behavior where the noise stays constant.\\nThe most right benchmark, however, does not only have more varying slopes,\\nbut also the noise is increasing over time. The benchmark instances were created\\nin multiple steps by creating randomized control values \\frst, that are then used\\nas arguments for other distributions that are \\fnally used to create the vectors.\\nFor instance, the most right benchmark is based on the control values m=\\nU(2;5) +U(0:2;4)=80\\u0001tands=U(0;0:001) +U(0:002;0:01)\\u0001twhereUdenotes\\na uniform random variable and tdenotes the time or index. The \\fnal vector\\nis then created using a normal distribution N(m; s2) with the prior de\\fned\\ncontrol values. Thus, for this benchmark instance, the mean value and also the\\nnoise increases with higher t. Since the parameters of the distributions change\\nwith t, the distributions from which a sample is drawn are created for each\\ntindependently. The full list and de\\fnition of the benchmark instances can\\nfound in the additional materials online at https:\\/\\/dev.heuristiclab.com\\/wiki\\/\\nAdditionalMaterial. For the presented results, we used vectors of length 1,000. 6 P. Fleck et al.\\nFig. 6: Some exemplary benchmark instances, where each line represent one row\\nin the dataset.\\n5 Result\\nTo compare di\\u000berent optimization strategies, we analyze the run-length distribu-\\ntions, as de\\fned in the black-box optimization benchmarking (BBOB)[3]. This\\nallows an evaluation of not only an algorithms ability of \\fnding a good solution,\\nbut also shows how fast an algorithm converges. This is crucial, since a good\\noptimization strategy for the segment optimization problem would be used in-\\nside a mutation operator within GP, thus, long runtimes would multiply in the\\ncontext of GP.\\nIn addition to the presented results, we also analyzed additional parameters,\\nsuch as the e\\u000bect of random versus orthogonal sampling, however they showed\\nno signi\\fcant di\\u000berences, thus are not discussed further in this paper. We will\\n\\frst look at whether approaching the optimization problem as single- or multi-\\ndimensional performs better, shown in Fig. 7. Overall, the results suggest, that\\noptimizing all dimensions simultaneously performs slightly better, however, for\\nsome problem instances, such as x1, using a single dimension yielded better\\nresults. This behavior was suspected, since it shows, that some of the benchmark\\nare easy enough to be separable and optimized dimension by dimension.\\nFig. 7: In\\ruence of single- versus multi-dimension on algorithm performance.\\nNext, we analyze directional search space reduction, where we compare lim-\\niting the search space towards a direction randomly versus using the gradient\\nto guide the direction, shown in Fig. 8. The result suggests, that the bene\\fts\\ndepend on the problem instance. While for problem instance x5 using a guided\\ndirection seems to slow down convergence, overall, the bene\\fts are not conclu-\\nsive. We suspect, that by using the gradient always to move towards the best\\noption, it forces the algorithm towards local optima without any way of escaping. Vectorial GP { Optimizing Segments for Feature Extraction 7\\nFig. 8: Performance of random versus guided direction.\\nNext, we analyze limiting the search space within a speci\\fc range. Because\\nthe range strategy has two additional parameter, the step-size and search-width,\\nwe will compare the random version to di\\u000berent settings of the parameters,\\nboth shown in Fig. 9. Since the step-size controls the jumping distance from the\\ncurrent index, having low values are expected to yield bad results, since jumping\\ntoo little makes the search ine\\u000ecient or halt completely due to the rounding in\\nthe gradient step. The e\\u000bect of the step-size on instance x3 suggest that this\\nassumption holds, since lower step-sizes do not perform well, as shown in Fig. 9a\\nOn the other hand, higher step-sizes tend to perform better for instance x3 and\\nmight even outperform the random range strategy in \\fnding the best solutions\\nquicker, however, this is not the case for all instances. Along the step-size, the\\nsearch-width de\\fnes how far the search space extends around the next index.\\nThe results in Fig. 9b suggests, that having a lower search-width generally causes\\nthe algorithm to converge slower. This is especially true for combinations with a\\nlow step-size, since the search-range is the only mechanism of changing an index\\nwhen the gradient step is rounded to zero. Similar to the guided direction, where\\nwe suspect that random direction outperforms the guided direction because the\\nalgorithm becomes stuck at a local optimum, the same could also be true for\\nguided range to a certain degree. Having a large search-width could help breaking\\nout of the local optimum, but still, the result suggest that a random range would\\nbe the better option in most cases.\\n(a) In\\ruence of the step-size.\\n (b) In\\ruence of the search-width.\\nFig. 9: Performance of random versus guided search range reduction. 8 P. Fleck et al.\\n6 Conclusion\\nThe results presented in the previous section suggests, that simple random sam-\\npling is generally better than using a guided strategy such as guided direction\\nor guided range. We believe that this is mainly caused by the guided strategies\\ntend to work similar to greedy algorithms, and can quickly become stuck in local\\noptima. However, when applying the presented methods in the context of GP,\\npreliminary results suggests, that the guided strategies tend to outperform their\\nrandom counterparts. We suspect that GP can easily break free of local optima\\nby regular crossover and mutation operations, thus becoming stuck in a local\\noptima is not as critical.\\nDespite the current result suggesting that the gradient-based strategies do\\nnot work well, we still believe that the information of the approximate gradient\\nis extremely useful, because most \\ftness landscapes of the benchmark instances\\nclearly indicate that there is a gradient that could be exploited for fast optimiza-\\ntion. Therefore, we will continue working on leveraging the gradient for a fast\\noptimization algorithm for the segment optimization problem.\\nAcknowledgments This work was carried out within the Dissertationspro-\\ngramm der Fachhochschule O \u007fO #875441 Vektor-basierte Genetische Program-\\nmierung f\u007f ur Symbolische Regression und Klassi\\fkation mit Zeitreihen (Sym-\\nRegZeit) , funded by the Austrian Research Promotion Agency FFG. The au-\\nthors also gratefully acknowledge support by the Christian Doppler Research\\nAssociation and the Federal Ministry of Digital and Economic A\\u000bairs within the\\nJosef Ressel Centre for Symbolic Regression .\\nReferences\\n1. Azzali, I., Vanneschi, L., Silva, S., Bakurov, I., Giacobini, M.: A Vectorial Approach\\nto Genetic Programming. EuroGP pp. 213{227 (2019)\\n2. Fleck, P., Winkler, S., Kommenda, M., A\\u000benzeller, M.: Grammar-based Vecto-\\nrial Genetic Programming for Symbolic Regression. In: Banzhaf, W., Trujillo, L.,\\nWinkler, S., Worzel, B. (eds.) Genetic Programming Theory and Practice XVIII.\\nSpringer Nature Singapore Pte Ltd. (2021)\\n3. Hansen, N., Auger, A., Ros, R., Finck, S., Po\\u0014 s\\u0013 \\u0010k, P.: Comparing results of 31 algo-\\nrithms from the black-box optimization benchmarking bbob-2009. In: Proceedings\\nof the 12th annual conference companion on Genetic and evolutionary computation.\\npp. 1689{1696 (2010)\\n4. Kommenda, M., Burlacu, B., Kronberger, G., A\\u000benzeller, M.: Parameter identi\\f-\\ncation for symbolic regression using nonlinear least squares. Genetic Programming\\nand Evolvable Machines 21(3), 471{501 (2020)\\n5. Sauer, T.: Numerical solution of stochastic di\\u000berential equations in \\fnance. In:\\nHandbook of computational \\fnance, pp. 529{550. Springer (2012)\",\"45\":\" Networks\\u2019 modulation: How di\\ufb00erent structural\\nnetwork properties a\\ufb00ect the global\\nsynchronization of coupled Kuramoto oscillators.\\nJuliette Courson1;2;3, Thanos Manos2, and Mathias Quoy2;4\\n1Laboratoire de Physique Th\\u00e9orique et Mod\\u00e9lisation (LPTM), CNRS, UMR 8089,\\nCY Cergy Paris Universit\\u00e9, Cergy-Pontoise Cedex, France\\njuliette.courson@cyu.fr\\n2Equipes Traitement de l\\u2019Information et Syst\\u00e8mes (ETIS), CNRS, UMR 8051,\\nENSEA, CY Cergy Paris Universit\\u00e9, Cergy-Pontoise Cedex, France\\nthanos.manos@cyu.fr\\n3Department of Computer Science, University of Warwick, Coventry, UK\\n4IPAL CNRS Singapore\\nmathias.quoy@cyu.fr\\nAbstract. Inalargevarietyofsystems(biological,physical,socialetc.),\\nsynchronizationoccurswhendi\\ufb00erentoscillatingobjectstunetheirrhythm\\nwhen they interact with each other. The di\\ufb00erent underlying network\\nde\\ufb01ningtheconnectivitypropertiesamongtheseobjectsdrivestheglobal\\ndynamics in a complex fashion and a\\ufb00ects the global degree of synchrony\\nof the system. Here we study the impact of such types of di\\ufb00erent net-\\nwork architectures, such as Fully-Connected, Random, Regular ring lat-\\ntice graph, Small-World and Scale-Free in the global dynamical activity\\nof a system of coupled Kuramoto phase oscillators. We \\ufb01x the external\\nstimulation parameters and we measure the global degree of synchrony\\nwhendi\\ufb00erentfractionsofnodesreceivestimulus.Thesenodesarechosen\\neither randomly or based on their respective strong\\/weak connectivity\\nproperties (centrality, shortest path length and clustering coe\\ufb03cient).\\nOur main \\ufb01nding is, that in Scale-Free and Random networks a sophis-\\nticated choice of nodes based on their eigenvector centrality and average\\nshortest path length exhibits a systematic trend in achieving higher de-\\ngree of synchrony. However, this trend does not occur when using the\\nclustering coe\\ufb03cient as a criterion. For the other types of graphs consid-\\nered, the choice of the stimulated nodes (randomly vs selectively using\\nthe aforementioned criteria) does not seem to have a noticeable e\\ufb00ect.\\n1 Introduction\\nComplex networks\\u2019 theory is a powerful tool in various \\ufb01elds that allow us to\\ninvestigate and understand the real world [1,2]. For example, di\\ufb00erent ensembles\\nof neurons connected by synapses coordinate their activity to perform certain\\ntasks (in biology), infrastructures like the Internet are formed by routers and\\ncomputer cables and optical \\ufb01bers (in hardware communication) and the human\\npersonal or professional relationships (in social sciences) to name a few [3].arXiv:2303.03099v1  [nlin.AO]  24 Feb 2023 1. INTRODUCTIONCourson et al.\\nNonlinearity is a very important feature in complex systems giving a rich\\nrepertoire of di\\ufb00erent activity patterns, such as stable, unstable, periodic etc. A\\nmodi\\ufb01cation of some parameter might also produce a change in their stability,\\nand therefore in the dynamics of the system. Furthermore, such systems may\\nhave a high sensitivity to initial conditions, or to any external input, that could\\ncompletely change their dynamics [4].\\nSuchdynamicsoftenyieldtoaself-organisedcoherentactivity,i.e.tosynchro-\\nnization. The latter can be loosely de\\ufb01ned as the capacity of di\\ufb00erent oscillating\\nobjects to adjust their rhythm due to their interaction and plays a key role in a\\nlarge variety of systems, whether biological, physical, or even social (see e.g. [5]).\\nIn a more formal way, synchronization emerges from the interaction of several\\nautonomous oscillators, also called self-sustained oscillators. That is, nonlinear\\ndynamicalsystemsthatproduceoscillationswithoutanyneedofexternalsource.\\nTheir dynamics is given by a nonlinear di\\ufb00erential equation or, in the case of\\nmultiple coupled oscillators, by several coupled di\\ufb00erential equations.\\nThe relative way that autonomous oscillators are connected within a given\\nnetwork can a\\ufb00ect their global activity and synchronization properties. Neural\\nnetworks can be represented as a graph of connections between the di\\ufb00erent neu-\\nrons. Since the introduction of small-world networks and scale-free networks (see\\ne.g. [6,8]), the \\ufb01eld of network graph analysis has attracted the attention of many\\nstudies aimed to better understand complex systems (see e.g. [9,10,11,12,13]).\\nFurthermore, modern network connectivity techniques allow us to capture var-\\nious aspects of their topological organization, as well as to quantify the local\\ncontributions of individual nodes and edges to network\\u2019s functionality (see e.g.\\n[14]).\\nIn neuroscience, synchronization plays a very important role. The human\\nbrain is a very large and complex system whose activity comprises the rapid\\nand precise integration of a gigantic amount of signals and stimulus to perform\\nmultiple tasks (see e.g. [14,15,16]). One example occurs in epileptic seizures,\\nwhere periods of abnormal synchronization in the neural activity can spread\\nwithin di\\ufb00erent regions of the brain, and cause an attack in the a\\ufb00ected person\\n(see e.g. [17]). More examples are found in other brain diseases such as Parkinson\\ndisease, where an excessively synchronized activity in a brain region correlates\\nwith motor de\\ufb01cit (see e.g. [18,19] and references therein) or tinnitus (see e.g.\\n[20,21,22] and references therein).\\nIn this study, we focus at a rather theoretical framework. We set out to in-\\nvestigate the impact of di\\ufb00erent network architectures, such as Fully-Connected,\\nRandom, Regular ring lattice graph, Small-World and Scale-Free in the global\\ndynamical activity of a system of coupled Kuramoto phase oscillators [23]. The\\nKuramoto model has been broadly used to study various types of oscillatory\\ncomplex activity, see e.g. [24,25,26] (to name only a few) and references therein.\\nOur goal is to investigate the impact of the network (graph) structure in the sys-\\ntem\\u2019sglobaldegreeofsynchronizationwhenapplyingidenticaland\\ufb01xedexternal\\nstimulus to di\\ufb00erent subsets of nodes which are chosen according to various net-\\nwork connectivity criteria. We \\ufb01nd that, in scale-free and random networks, a\\n2 Courson et al.\\n2. METHODS AND MATERIALS\\nsophisticated choice of nodes based on graph connectivity properties exhibits a\\nsystematic trend in achieving higher degree of synchrony. For the other types of\\ngraphs considered, the choice of the stimulated nodes (randomly vs selectively\\nusing the aforementioned criteria) seems to not have a noticeable e\\ufb00ect.\\n2 Methods and Materials\\n2.1 Connectivity measurements\\nWe here study the dynamics of phase oscillators coupled via binary, undirected\\ngraphsG= (V;E), containing a set of NverticesV=fv2J1 :NKgand a set\\nE=f(v;w)2J1 :NK2gof edges. Let Abe the corresponding adjacency matrix,\\nwithAvw= 1if there is a connection between node vand nodew,0otherwise.\\nSelf-connections are excluded, so Av;v= 0for any vertex v. For our analysis\\nlater on, we will use the following graph connectivity measurements [27]:\\n\\u2013 Shortest path length. The shortest path length Lv;wbetween any two\\nnodesvandwis the number of connections on the shortest path going\\nfrom one to another, computed following Dijkstra\\u2019s algorithm. We de\\ufb01ne\\nthe shortest path length of a node vas the average shortest path between v\\nand any other node of the network:\\n<Lv>=X\\nw2VLv;w\\nN: (1)\\nNote thatLv;wmight not be de\\ufb01ned if there is no way connecting node vto\\nnodew. The lower the shortest path length, the fastest the information goes\\nfrom one node to another. For example, when building a subway network\\n(that is, a graph where di\\ufb00erent stations are interconnected), one might\\nwant to minimize the stations\\u2019 average shortest path length so the users can\\neasily navigate across the city.\\n\\u2013 Centrality. The eigenvector centrality is used to quantify the importance\\nof a node in the network. Let \\u0015be the highest eigenvalue for A, so that all\\nthe corresponding eigenvector\\u2019s components are non null. The eigenvector\\ncentralityxvof vertexvis de\\ufb01ned as the vthcomponent the eigenvector,\\nnamely:\\nxv=1\\n\\u0015X\\nw2VAw;vxw: (2)\\nKeeping in mind the subway network example, a station with a high cen-\\ntrality would be densely connected to other stations, in particular to other\\ncentral ones.\\n\\u2013 Clustering. Letkv=P\\nwAvwbe the degree of node v. In the case of a\\nundirected graph,kv(kv\\u00001)\\n2edges can exist in the direct neighborhood of v.\\nWithnvthe number edges that actually exist in this neighborhood, the local\\nclustering coe\\ufb03cient is de\\ufb01ned as [16]:\\nCv=2nv\\nkv(kv\\u00001): (3)\\n3 2. METHODS AND MATERIALSCourson et al.\\nThat is, in a subway network where stations BandCare the next stop after\\nstationAon their line, clustering would give the probability that there exists\\na line directly connecting BandC.\\n2.2 Neural networks as graphs\\nWe investigate synchronization properties in various network con\\ufb01gurations that\\nexhibit in general di\\ufb00erent characteristics. In more detail we here employ the\\nneural networks described in the following list (see e.g. [6]):\\n\\u2013 Fully-Connected networks. They contains (N\\u00001)2edges connecting\\nevery node in one layer to every node in the other layer.\\n\\u2013 Regular networks. They consist of a lattice of Nnodes, each being con-\\nnected to their knearest neighbors.\\n\\u2013 Small-World networks. A Small-World network is constructed from a\\nRegular one after multiple random rewiring phases: going clockwise over the\\nlattice, a vertex and the edge to its nearest neighbor are selected. The edge\\nis removed, and the vertex reconnected to a random node with probability\\np, without duplicating any existing edge. This random rewiring is repeated,\\nconsidering the following nearest neighbour, until having rewired with prob-\\nabilitypall edges of the network. Choosing pin an adequate range of values,\\nthe built network exhibits both high mean clustering and low mean charac-\\nteristic path length, that is small-worldness.\\n\\u2013 Random networks. Using the same procedure as for Small-World net-\\nworks, setting p= 1produces a Random graph, with all edges being system-\\natically randomly rewired.\\n\\u2013 Scale-Free networks. They are networks whose degree distribution follows\\na power-law:\\nP(k)\\/k\\u0000\\r: (4)\\nwithkthe node degrees, \\ra real constant. The Barab\\u00e1si-Albert model gives\\na procedure for the construction of scale-free networks [7], by starting with\\na small Fully-Connected network of m0nodes then adding one by one the\\nN\\u0000m0remaining nodes, connecting them to the malready present nodes\\nwith probability\\npv=kvP\\nwkw; (5)\\nw2J0 :m\\u00001K. Scale-free networks exhibit nodes with degrees that are\\nseveral standard deviations away from the average degree of the network.\\nThesehighlyconnectednodesarecalled hubs.Notethat,however,forsmaller\\nnetworks with size N < 100, the scale-free property might not be properly\\nobservable.\\nFig. 1 provides a visual representation of the above mentioned graphs. Note\\nthat for visualization purposes we here show only a small fraction of the actual\\nnetworks that we use later in our simulations where the number of nodes is set\\nbeN= 500.\\n4 Courson et al.\\n2. METHODS AND MATERIALS\\n(a) Fully-Connected\\n (b) Scale-Free\\n(c) Regular\\n (d) Swall-World\\n (e) Random\\nFig.1. Network graphs. Small graphs of size N= 20showing the di\\ufb00erent network\\nstructures: (a)Fully-Connected graph, (b)Scale-Free graph with initial size m0= 5,\\n(c)Regular graph with node degree k= 4,(d)Small-World graph with initial node\\ndegreek= 4and rewiring probability p= 0:2and(c)Random graph with initial node\\ndegreek= 4. See text for more details.\\n2.3 The Kuramoto model\\nWe use of the Kuramoto model to study the neural activity of the coupled\\nsystem. To this end we consider a population of Nphase oscillators [23]:\\n_\\u0012i=!i+F\\u000eisin(\\nt+\\u0012i) +K\\nkiX\\njAijsin(\\u0012j\\u0000\\u0012i); (6)\\nwhere\\u0012idenotes thephaseof the i\\u0000th oscillator, !iits respectivefrequency (Hz)\\ndrawn from a Lorentz probability distribution g(!)of scale parameter \\r= 0:5,\\ncentered in x0= 1.Ais the binary adjacency matrix coupling the oscillators, ki\\nthedegreeofoscillator iandKistheglobalcouplingconstant.Weapplyexternal\\nstimulus in a subset of the oscillators with \\ufb01xed amplitude Fand frequency\\n\\n. The term \\u000eiis a binary function indicating this subset of nodes where the\\nstimulation is applied in di\\ufb00erent realization in our simulations,\\n\\u000ei=\\u001a1if nodeiis in the stimulated subset\\n0 else.(7)\\nWe set the time-step at 0:01s and we integrated the system with an Euler scheme\\n(no noise is considered).\\n5 3. RESULTSCourson et al.\\nThe system\\u2019s degree of synchrony is measured using its order parameter r\\n[23]:\\nrei =1\\nNNX\\nj=1ei\\u0012j; (8)\\nwhere\\tdenotes the population\\u2019s mean phase. The order parameter rtends to 1\\nforaperfectlysynchronizedpopulationandto 0intheabsenceofsynchronization\\nrespectively. Due to the presence of strong \\ufb02uctuations, all rtime-series shown\\nin this paper are determined using a moving average on r, on time windows of\\nlength 2s sliding each 0:1s. The \\ufb01nal states of a population, rf, are computed by\\naveraging these moving-averaged rtime-series over a 15s time-window where the\\nsystem has reached its stable state. The system\\u2019s degree of synchrony depends\\non the coupling strength K\\u2019s relative position to a critical coupling strength Kc,\\nwhose value depends on the network con\\ufb01guration (see e.g. [28,29]). Here, we\\nset this value at K= 0:2so that all considered networks are desynchronized in\\nthe absence of any external stimulation.\\n3 Results\\nNetwork modulation. In order to adequately tune the stimulus\\u2019 amplitude\\nand frequency values such that they can lead the system into a synchronous\\nstate, we \\ufb01rst perform a systematic analysis in the parameter space (F;\\n).\\nHence, we begin by applying external stimulus to all the nodes for each pair of\\nparameters and measure the \\ufb01nal order parameter rf. Such a parameter map\\nreveals the presence of the well-known Arnold tongues [5], namely regions in the\\nplane (F;\\n)for which the system gets synchronized.\\nIn Fig. 2, we present the di\\ufb00erent synchronization regions (presence of sev-\\neral Arnold tongues) for a Regular network of a relatively small size N= 20and\\nmean neighborhood k= 4. For every other studied network, the maps depicts\\nsimilar features with large synchronization regions at relatively small amplitudes\\nof the external current, F < 200. These tongues get thinner with higher values of\\nF. Inside the main Arnold tongues, the oscillators are phase-locked at the forc-\\ning frequency \\nandrf\\u00191. Inside zones of weaker degree of synchrony rf<1,\\nsome oscillators are phase-locked, while the oscillators of higher natural frequen-\\ncies keep rotating independently. The white star symbol in the bottom-left part\\nof the \\ufb01gure indicates the chosen parameters (F;\\n) = (5;1)for our forthcoming\\nsimulations, resulting in a partial phase-locking of the network. Note that we\\nhave performed similar analysis with larger sizes but smaller parameter grid size\\nand the overall picture turns out to be consistent. We have also prepared similar\\nplots for all considered network con\\ufb01gurations (\\ufb01gures not shown here).\\nSimulation protocol. We set the values F= 5,\\n= 1Hz for the stimulus in-\\ntensity and frequency in Eq. (6), so that the network is weakly entrained without\\nbeing completely phase-locked. We then measure the degree of synchronization\\n6 Courson et al.\\n3. RESULTS\\nFig.2. Synchronization regions in the stimulation frequency-amplitude pa-\\nrameterspaceforaRegularnetwork. FinalorderparameterreachedforaRegular\\nnetwork of size N= 20, and degree k= 4where all nodes are stimulated, for di\\ufb00erent\\npairs of stimulus intensity ( F) and frequency ( \\n) values in Eq. (6). Each data point\\ncorresponds to a single simulation over 30s, the \\ufb01nal order parameter being averaged\\nover the last 15s. The color map shows large main synchronization regions, as well\\nas small higher-order synchronization areas. The white star symbol at the bottom-left\\npart of the \\ufb01gure indicates the chosen parameters (F;\\n ) = (5;1)for the forthcoming\\nsimulations.\\n(with the order parameter) in di\\ufb00erent networks described in Sect. 2.2. The sys-\\ntem starts evolving for 4s without any external input before we start applying\\nthe stimulation to a subset of nodes until 30s. More precisely, we stimulate dif-\\nferent fractions of nodes, i.e., 25%,50%and75%in each given network. These\\nnodes can be either chosen randomly, or depending on particular connectivity\\nproperties (as described in Sect. 2.1). For the latter case, we \\ufb01rst sort the nodes\\naccording to their connectivity relative measurements (from higher to lower),\\ni.e. the eigenvector centrality, average shortest path length and clustering coef-\\n\\ufb01cient. The resulting time series are smoothed with a moving average, and their\\nrfis averaged over the last 15s. In order to obtain a statistically relevant value\\nof the \\ufb01nal value of the order parameter rf, we performed 20simulations for\\neach di\\ufb00erent network-setup (randomizing the initialization\\/generation of the\\nnetworks, the natural frequencies and the initial conditions for each simulation).\\nIn Fig.3, we show representative time-series for various stimulation setups for\\nScale-Free networks of size N= 500and initial size m0= 5. Each panel corre-\\nsponds to a di\\ufb00erent initialization, from which the order parameter evolution is\\ncomputed depending on the amount of stimulated nodes and the way they are\\nselected. Thin solid lines correspond to randomly selected nodes. In that case,\\na \\ufb01rst subset containing 25 %of the nodes is created. Then for the stimulation\\nof larger in size subsets, another 25 %of nodes is successively added, so that\\nlarger subsets of random nodes always contains the smaller one. The bold solid\\n7 3. RESULTSCourson et al.\\n(a) Eigenvector centrality\\n (b) Shortest path length\\n (c) Clustering coe\\ufb03cient\\nFig.3.Representative rtime-series for di\\ufb00erent stimulation setups of Scale-\\nFree networks. The order parameter as a function of time, for N= 500oscillators,\\nwhenstimulating0%(blue),25%(orange),50%(red)and75%(black)ofthenodes.The\\nstimulated nodes are chosen randomly, then based on their (a)eigenvector centrality\\n(b)average shortest path length and (c)clustering coe\\ufb03cient values. Bold solid lines\\n(resp. dashed lines) correspond to the stimulation of nodes with the highest (resp.\\nlowest) values, while thin solid lines correspond to the stimulation of randomly chosen\\nnodes.\\nlines (resp. dashed lines) show the time-series when the stimulated nodes are\\nthe ones with the highest (resp. lowest) eigenvector centrality (panel (a)), aver-\\nage shortest path length (panel (b)) and clustering coe\\ufb03cient (panel (c)). Note\\nthat higher values of the connectivity measurement do not necessarily lead to\\nstronger synchrony. In particular, lower values for the nodes\\u2019 average shortest\\npath length depicts shorter connections to the rest of the network, and therefore\\na more e\\ufb03cient synchronization.\\nOptimization of global synchronization In Fig. 4, we present the main\\n\\ufb01nding of our work, namely a systematic comparison of the synchronization ef-\\n\\ufb01ciency when applying identical stimulus in di\\ufb00erent types of graph networks.\\nEachpanelissplitinto3columnsshowingthestatisticalsummaryfortheensem-\\nbles of di\\ufb00erent realizations and choosing to stimulate di\\ufb00erent subsets of nodes,\\ni.e. randomly (middle-orange boxplots in the legends) or with highest (upper-red\\nboxplots in legends) or lower (lower-blue boxplots in the legends) connectivity\\nmeasurement. Panel (a) refers to a Small-World network of size N= 500, initial\\ndegreek= 4and rewiring probability p= 0:2, (b) to a Random network of\\nsizeN= 500, initial degree k= 4and rewiring probability p= 1and (c) to a\\nScale-Free network of size N= 500and initial size m0= 5respectively. Note\\nthis analysis is not performed on Regular and Fully-Connected networks, since\\nall nodes of such networks have identical connectivity properties and does not\\nallow any connectivity-based selection.\\n8 Courson et al.\\n3. RESULTS\\n(a) Small-World network\\n (b) Random network\\n(c) Scale-Free network\\nFig.4. Synchronization e\\ufb03ciency comparison for di\\ufb00erent types of graph\\nnetworks. Final order parameter obtained, for (a)Small-World networks (b)Ran-\\ndom networks (c)Scale-Free networks of size N= 500. The \\ufb01nal value of the order\\nparameterrfis computed for di\\ufb00erent stimulation subset sizes, composed of randomly\\nchosen nodes (middle-orange boxplots in the legends), nodes with the highest connec-\\ntivity measurement (left-red boxplots)and nodes with thelowest connectivity measure-\\nment(right-blueboxplots). rfshownareanaverageover 20simulations.Theanalysisis\\nperformed with three di\\ufb00erent connectivity measurements: eigenvector centrality (left\\ncolumn in each panel), average shortest path length (central column in each panel) and\\nclustering (right column in each panel).\\nInScale-Freenetworks,theglobalorderparameterreacheshighervalueswhen\\nthe stimulus is applied to the nodes with higher eigenvector centrality or lower\\naverage shortest path length. In such networks, a small fraction of the nodes\\n9 4. SUMMARY AND DISCUSSIONCourson et al.\\nhave signi\\ufb01cantly higher connectivity, and stimulating preferentially these nodes\\nenables strong synchronization compared to stimulating random nodes.\\nIn Random networks, selecting stimulated nodes according to their lowest av-\\nerage shortest path length instead of randomly enhances synchronization. How-\\never, there is no bene\\ufb01t in choosing more central nodes. Indeed, the high degree\\nof randomness (induced by a rewiring probability p= 1, see Sect. 2.2) in these\\nnetworks\\u2019connectionscausesdisparityinthenodes\\u2019averageshortestpathlength,\\nwithout creating any node of way higher centrality. In Small-World networks,\\nthe connections are distributed in a more homogeneous way, and hence the node\\nselection as no substantial impact on the system\\u2019s \\ufb01nal synchrony.\\nFor all three aforementioned networks and all stimulation subset sizes, the\\nselection of the nodes according to their clustering coe\\ufb03cient does not show\\nany advantage over a simple random choice. Note that, we also checked for a\\npotential overlap of clustering coe\\ufb03cient values between the randomly chosen\\nnodes and those chosen with higher respective values. It turns out that they\\ndi\\ufb00er signi\\ufb01cantly. Our working hypothesis is that, by picking nodes with higher\\nclustering coe\\ufb03cient, we induce higher synchrony locally (member nodes of the\\nsub-network) while we disfavour other nodes outside of it. However, by choosing\\nnodes randomly, we end up stimulating simultaneously nodes within and outside\\nclusters eventually providing a better global synchronization. Finally, for all net-\\nworks and stimulation subset selection methods, although they overall achieve\\nhigher degree of synchrony, larger stimulated subsets containing 75% of the pop-\\nulation do not allow to observe any clear advantage in particular selection of the\\nnodes, since all three possible subsets largely overlap.\\n4 Summary and Discussion\\nIn this study, we investigated the impact of structure and connectivity properties\\nin a modulated network. We sought out to identify e\\ufb03cient ways to synchronize\\na population of Kuramoto phase oscillators using nodes\\u2019 stimulation with \\ufb01xed\\nsmall amplitude and frequency. To this end, we \\ufb01rst performed a parameter\\nsweep exploration for stimulus amplitude and frequency parameters to identify\\nsettings that allow the system to synchronize. Then, we computed the evolution\\nof characteristic networks of Kuramoto oscillators, where external stimulation\\nis applied to di\\ufb00erent subpopulations with identical \\ufb01xed low amplitude and\\nfrequency. In order to measure the system\\u2019s synchrony of each network-type and\\nstimulationcon\\ufb01guration,wecalculatedtheglobalorderparameterforensembles\\nof di\\ufb00erent random system initializations.\\nWe showed that by choosing this subpopulations based on their respective\\nnetwork connectivity properties (i.e. high eigenvector centrality and lower short-\\npath length), we were able to enhance the networks\\u2019 global degree of synchro-\\nnization in comparison to the one achieved by randomly choosing them. This is\\nnot the case when using the clustering coe\\ufb03cient as a selection criterion. How-\\never, this direction deserves further investigation, e.g. detect di\\ufb00erent network\\n10 Courson et al.\\n4. SUMMARY AND DISCUSSION\\ncommunities, measure their local synchronization and study its impact on the\\nglobal synchrony when such nodes are chosen to be stimulated.\\nFrom a neuroscience point of view Scale-Free networks play an important\\nrole in the structure and function of mammal brains, see for example [30] (a\\nstudy on the scale-free dynamics and the emergence of collective organisation\\noccurs in rodents) or [31] (investigating the fractal structure of the human brain\\nand its dynamics). Furthermore, in Alzheimer patients\\u2019 brain, the functional\\nconnectivity structure is found to exhibit properties similar to Random network\\ngraphs (see e.g. [32,33] and references therein). Thus, understanding how to\\noptimally synchronize systems with similar network structures can improve the\\noverall expected performance of a given external simulation protocol.\\nAcknowledgements\\nJC was supported by the LABEX MME DII (ANR-16-IDEX-0008) PhD grant\\nand CY cognition grant of the CY Cergy-Paris University. TM was supported\\nby the Labex MME DII (ANR-11-LBX-410 0023-01) French national funding\\nprogram. MQ was supported by a CNRS \\ufb01nancing for a research semester in\\nthe IPAL Lab in Singapore. We also acknowledge the use of the Osaka compute-\\ncluster resources of CY Cergy-Paris University.\\nCode and data availability\\nAll data generated or analysed during this study are included in this manuscript.\\nOur code is written in Pythonand we used the NetworkX python package. Our\\ncode is available upon request.\\nReferences\\n1. Barrat, A., Barth\\u00e9lemy, M., Vespignani, A.: Dynamical processes on complex net-\\nworks. Cambridge: Cambridge University Press (2008)\\n2. Nicolis, G., Nicolis, C.: Foundations of Complex Systems: Emergence, Information\\nand Prediction, World Scienti\\ufb01c, 2nd ed., (2012)\\n3. Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks. From biological nets to\\nthe internet and WWW. Oxford: Oxford University Press (2003)\\n4. Strogatz, S.H.: Nonlinear dynamics and chaos: with applications to physics, biology,\\nchemistry,andengineering.Secondedition.Boulder,CO:WestviewPress,amember\\nof the Perseus Books Group, (2015)\\n5. Pikovsky, A., Rosenblum, M.G., Kurths, J.: Synchronization, A Universal Concept\\nin Nonlinear Sciences. Cambridge University Press, Cambridge (2001)\\n6. Watts, D.J., Strogatz, S.H.: Collective dynamics of \\u2018small-world\\u2019 networks. Nature\\n393(6684), 440\\u2013442 (1998)\\n7. Barab\\u00e1si, A.L.: The barab\\u00e1si-albert model. In: Network science. Cambridge Univer-\\nsity Press, Cambridge (2016)\\n8. Barab\\u00e1si,A.,R\\u00e9ka,A.:Emergenceofscalinginrandomnetworks.Science286(5439),\\n509\\u2013512 (1999)\\n11 4. SUMMARY AND DISCUSSIONCourson et al.\\n9. Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barab\\u00e1si, A.L.: The large-scale\\norganization of metabolic networks. Nature 407(6804), 651\\u2013654 (2000)\\n10. Strogatz, S.H.: Exploring complex networks. Nature 410(6825), 268\\u2013276 (2001)\\n11. Li, W., Cai, X.: Statistical analysis of airport network of china. Phys. Rev. E 69,\\n046106 (2004)\\n12. Bassett, D.S., Bullmore, E.: Small-world brain networks. The Neuroscientist 12(6),\\n512\\u2013523 (2006)\\n13. Berry, H., Quoy, M.: Structure and dynamics of random recurrent neural networks.\\nAdaptive behavior 14, 129\\u2013137 (2006)\\n14. Sporns, O.: Networks of the Brain. The MIT Press (2010)\\n15. Chialvo, D.: Emergent complex neural dynamics. Nature Phys 6, 744\\u2013750 (2010)\\n16. Fornito, A., Zalesky, A., Bullmore, E.: Fundamentals of brain network analysis.\\nElsevier\\/Academic Press (2016)\\n17. Wong, R.K., Traub, R.D., Miles, R.: Cellular basis of neuronal synchrony in\\nepilepsy. Advances in Neurology 44, 583\\u2013592 (1986)\\n18. Brown, P.: Oscillatory nature of human basal ganglia activity: Relationship to the\\npathophysiology of parkinson\\u2019s disease. Movement Disorders 18(4), 357\\u2013363 (2003)\\n19. Manos, T., Diaz-Pier, S., Tass, P.A.: Long-term desynchronization by coordinated\\nreset stimulation in a neural network model with synaptic and structural plasticity.\\nFrontiers in Physiology 12 (2021)\\n20. Eggermont, J.J., Tass, P.A.: Maladaptive neural synchrony in tinnitus: Origin and\\nrestoration. Frontiers in Neurology 6 (2015)\\n21. Manos, T., Zeitler, M., Tass, P.A.: Short-term dosage regimen for stimulation-\\ninduced long-lasting desynchronization. Frontiers in Physiology 9 (2018)\\n22. Manos, T., Zeitler, M., Tass, P.A.: How stimulation frequency and intensity impact\\non the long-lasting e\\ufb00ects of coordinated reset stimulation. PLOS Computational\\nBiology 14(5), 1\\u201331 (2018)\\n23. Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Dover Books on\\nChemistry Series, Dover Publications (2003)\\n24. Acebr\\u00f3n, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The kuramoto\\nmodel: A simple paradigm for synchronization phenomena. Reviews of modern\\nphysics 77(1), 137 (2005)\\n25. Rodrigues, F.A., Peron, T.K.D., Ji, P., Kurths, J.: The kuramoto model in complex\\nnetworks. Physics Reports 610, 1\\u201398 (2016)\\n26. Popovych, O.V., Jung, K., Manos, T., Diaz-Pier, S., Ho\\ufb00staedter, F., Schreiber, J.,\\nYeo, B.T., Eickho\\ufb00, S.B.: Inter-subject and inter-parcellation variability of resting-\\nstate whole-brain dynamical modeling. NeuroImage 236, 118201 (2021)\\n27. Newman, M.E.J.: Networks: an introduction. Oxford University Press, Oxford;\\nNew York (2010)\\n28. Mirollo, R., Strogatz, S. The Spectrum of the Partially Locked State for the Ku-\\nramoto Model. J Nonlinear Sci 17, 309\\u2014347 (2007)\\n29. Chiba, H., Medvedev, G. S., Mizuhara, M. S. Bifurcations in the Kuramoto model\\non graphs. Chaos (Woodbury, N.Y.), 28(7), 073109 (2018)\\n30. Ribeiro, T.L., Chialvo, D.R., Plenz, D.: Scale-free dynamics in animal groups and\\nbrain networks. Frontiers in Systems Neuroscience 14 (2021)\\n31. Grosu, G.F., Hopp, A.V., Moca, V.V., B\\u00e2rzan, H., Ciuparu, A., Ercsey-Ravasz,\\nM., Winkel, M., Linde, H., Mures ,an, R.C.: The fractal brain: scale-invariance in\\nstructure and dynamics. Cerebral Cortex (2022)\\n32. Stam, C.J., De Haan, W., Da\\ufb00ertshofer, A., Jones, B.F., Manshanden, I., Van\\nCappellen van Walsum, A.N., Montez, T., Verbunt, J.P.A., De Munck, J.C., Vn\\n12 Courson et al.\\n4. SUMMARY AND DISCUSSION\\nDijk,B.W.,Berendse,H.W.,Scheltens,P.:Graphtheoreticalanalysisofmagnetoen-\\ncephalographic functional connectivity in Alzheimer\\u2019s disease. Brain 132, 213\\u2014224\\n(2009)\\n33. Kang, M., Petrasek, Z.: Random Graphs: Theory and Applications from Nature\\nto Society to the Brain. Internationale Mathematische Nachrichten 227 1\\u201324 (2014)\\n13\",\"46\":\" arXiv:2303.01695v1  [cs.NE]  3 Mar 2023EVOLUTIONARY MULTI -OBJECTIVE ALGORITHMS FOR THE\\nKNAPSACK PROBLEMS WITH STOCHASTIC PROFITS\\nA P REPRINT\\nKokila Perera\\nOptimisation and Logistics\\nSchool of Computer and Mathematical Sciences\\nThe University of Adelaide\\nAdelaide, AustraliaAneta Neumann\\nOptimisation and Logistics\\nSchool of Computer and Mathematical Sciences\\nThe University of Adelaide\\nAdelaide, Australia\\nFrank Neumann\\nOptimisation and Logistics\\nSchool of Computer and Mathematical Sciences\\nThe University of Adelaide\\nAdelaide, Australia\\nABSTRACT\\nEvolutionary multi-objective algorithms have been widely shown to be successful when utilized\\nfor a variety of stochastic combinatorial optimization pro blems. Chance constrained optimization\\nplays an important role in complex real-world scenarios, as it allows decision makers to take into\\naccount the uncertainty of the environment. We consider a ve rsion of the knapsack problem with\\nstochastic pro\\ufb01ts to guarantee a certain level of con\\ufb01dence in the pro\\ufb01t of the solutions. We in-\\ntroduce the multi-objective formulations of the pro\\ufb01t chan ce constrained knapsack problem and\\ndesign three bi-objective \\ufb01tness evaluation methods that w ork independently of the speci\\ufb01c con\\ufb01-\\ndence level required. We evaluate our approaches using well -known multi-objective evolutionary\\nalgorithms GSEMO and NSGA-II. In addition, we introduce a \\ufb01l tering method for GSEMO that\\nimproves the quality of the \\ufb01nal population by periodically removing certain solutions from the in-\\nterim populations based on their con\\ufb01dence level. We show th e effectiveness of our approaches on\\nseveral benchmarks for both settings where the knapsack ite ms have \\ufb01xed uniform uncertainties and\\nuncertainties that are positively correlated with the expe cted pro\\ufb01t of an item.\\nKeywords Multi-objective optimization, stochastic knapsack probl em, chance constrained problems\\n1 Introduction\\nReal world optimization problems involve often uncertain p roperties imposed by some stochastic components in the\\nproblem or noise in its environment Peng (2019); He and Shao ( 2009). Such uncertainties must be considered in the\\noptimization to \\ufb01nd reliable and useful solutions to proble ms. Chance constraints are a natural way to model uncer-\\ntainties in problems. They can capture the effects of uncert ain variables on the inequality constraints and formulate\\nthe optimization problems under uncertainties Peng (2019) ; Neumann et al. (2020); Doerr et al. (2020). A chance\\nconstraint is usually an inequality constraint on some stoc hastic variable of the problem, which can be violated by a\\nslight chance (a small probability) when optimizing the problem Neumann a nd Sutton (2019); Neumann and Neumann\\n(2020). Chance constraints enable one to set the desired con \\ufb01dence level of getting the maximal pro\\ufb01t when implement-\\ning a solution, unlike using a solution from a deterministic approach. The study of problems with chance constraints\\nleads to developing ef\\ufb01cient applications to solve complex optimization problems and is crucial in developing real\\nworld solutions for complex problems, such as in the mining i ndustry Xie et al. (2021a); Mohtasham et al. (2021),\\npower systems Geng and Xie (2019), communication systems Ab e et al. (2020) and transportation Kepaptsoglou et al.\\n(2015). Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nThis work considers a variation of the classical knapsack pr oblem (KP), where the items have stochastic pro\\ufb01ts and\\ndeterministic weights. As the weights of items are determin istic, the constraint on the weights remains the same\\nas in the classical KP. An additional constraint is introduc ed to this problem to capture the uncertainty of the pro\\ufb01t\\nvariable, which is a chance constraint on the pro\\ufb01t Neumann e t al. (2022). This constraint guarantees that the solution\\nhas maximal pro\\ufb01t P and only drops below P for a small probabil ity (\\u03b1). In summary, the optimization process for\\nthis problem is to \\ufb01nd the solution(s) that maximize the pro\\ufb01 t P, subjected to the weight constraint and pro\\ufb01t chance\\nconstraint, which guarantees that the pro\\ufb01t P is obtained wi th probability at least \\u03b1.\\nIn the literature, the KP with a chance constraint on the weig ht (as elements have deterministic pro\\ufb01ts and stochastic\\nweights) is more prevalent Xie et al. (2019, 2020). On the con trary, the pro\\ufb01t chance constrained KP is explored\\nonly in one study, which appears in recent literature Neuman n et al. (2022). This work considers single-objective\\nevolutionary approaches to address this problem Neumann et al. (2022). This scenario makes it possible to identify\\nthe risk of not achieving the expected pro\\ufb01t by implementing a particular solution for the problem and making better\\ndecisions in planning. This problem re\\ufb02ects a bene\\ufb01cial and valid real-world problem scenario like mine planning as\\nmentioned in Neumann et al. (2022).\\nEvolutionary algorithms (EAs) perform well in addressing s tochastic optimization problems Doerr and Neumann\\n(2021); Singh and Branke (2022). Also, they can produce opti mal or feasible solutions in a reasonable amount of\\ntime for complex combinatorial optimization problems like chance constrained knapsack and job shop scheduling\\nproblems. EAs may consider a different number of objectives depending on how the de\\ufb01nition of the underlying\\nproblem. Usually, when an EA considers a single objective, i t generates a single solution that optimizes the objective\\nvalue. In contrast, a multi-objective EA generates a set of s olutions that gives a trade-off between given objectives.\\nSuch a solution set makes it possible to have more insights in to improving the algorithms and search space than having\\na single solution as the outcome Coello et al. (2013); Deb (20 01). Therefore, multi-objective algorithms help one to\\nmake informed decisions on selecting a solution to implemen t.\\nIn this work, we explore the use of the multi-objective evolu tionary approaches for the chance constrained KP with\\nstochastic pro\\ufb01ts introduced in Neumann et al. (2022). Here we introduce multi-objective \\ufb01tness evaluation for EAs\\nto address this problem and methods to enhance their perform ance.\\n1.1 Related Work\\nThe use of evolutionary computation for chance constrained problems appears in the early literature He and Shao\\n(2009); Loughlin and Ranjithan (1999); Liu et al. (2013); Ma sutomi et al. (2013). Those works consider computation-\\nally expensive methods like simulations and sampling to cat er for chance constraints. More recent studies have looked\\ninto tail-bound inequalities, which more ef\\ufb01ciently deal w ith chance constraints Neumann et al. (2022); Xie et al.\\n(2019); Assimi et al. (2020).\\nThe chance constrained KP where the pro\\ufb01ts are deterministi c and weights are stochastic is considered in several\\npapers Xie et al. (2019, 2020). Xie et al. (2019) presents how to use well-known deviation inequalities: Chebyshev\\u2019s\\ninequality and Hoeffding bound to estimate the probability of constraint violation. In Xie et al. (2020), where the same\\nKP variation is considered, they introduce problem-speci\\ufb01 c operators for EAs with both single- and multi-objective\\nformulations. In the study Assimi et al. (2020), dynamic cha nce constrained KP with stochastic pro\\ufb01ts is studied. formulations. In the study Assimi et al. (2020), dynamic cha nce constrained KP with stochastic pro\\ufb01ts is studied.\\nIn addition to the objective function on the pro\\ufb01t of a given s tochastic solution, a second objective is introduced to\\naddress the dynamic capacity constraint. It captures the mi nimal capacity bound for the solution that meets the chance\\nconstraints.\\nRun-time analysis is an essential topic in studying problem s with chance constraints. The \\ufb01rst paper on run time\\nanalysis for chance constraint problems considers the KP wi th stochastic weights Neumann and Sutton (2019). This\\nwork considers different cases of that problem and studies t he run time of (1+1) EA for them. In Xie et al. (2021b),\\nthey perform the run time analysis of simple EAs for chance co nstrained KPs with uniform weights. The papers\\nNeumann and Witt (2022) and Shi et al. (2022) study the run tim e of simple EAs for different chance constrained\\nproblems. InNeumann and Witt (2022), the authors consider s ingle- and multi-objective EAs for chance constrained\\nproblems with normally distributed random problem variabl es. They also show how to use the proposed evolutionary\\napproaches for chance constrained minimum spanning tree pr oblems Neumann and Witt (2022). In Shi et al. (2022),\\nthey analyze the run time of random local search and (1+1) EA f or the chance constrained makespan problem.\\nIn the study Neumann et al. (2022), the authors simple EAs for the pro\\ufb01t chance constrained KP. Those algorithms\\ninclude (1+1) EA with standard bit-\\ufb02ip mutation and heavy-t ail mutation operators and population based ( \\u00b5+1) EA,\\nwhich uses a speci\\ufb01c crossover operator for the KP. This stud y evaluates the performance of all these algorithms\\nusing the single objective \\ufb01tness evaluation. The overall r esults show that (1+1) EA with heavy tail mutation operator\\nsigni\\ufb01cantly improved over other algorithms.\\n2 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nThis work is motivated by the recent study on evolutionary mu lti-objective algorithms that compute trade-offs with\\nrespect to the expected value and the variance of solutions p resented in Neumann and Witt (2022). Here we explore the\\nmulti-objective formulations for the pro\\ufb01t chance constra ined KP based on that work. In addition to the variance of the\\nsolutions, we consider the standard deviation and also the c ount of elements in the solutions (under certain conditions )\\nto introduce \\ufb01tness evaluations for the problem. The signi\\ufb01 cance of these \\ufb01tness functions is that they can evaluate the\\n\\ufb01tness of a solution independent of any speci\\ufb01c con\\ufb01dence le vel of its pro\\ufb01t (i.e. independent of speci\\ufb01c value for \\u03b1).\\nSince this generates a set of solutions that gives a trade-of f of speci\\ufb01c objectives for each \\ufb01tness formulation with eac h\\nalgorithm, it allows one to make more informed decisions whe n selecting a solution to implement. For example, to\\nidentify the solution that gives the best pro\\ufb01t with a partic ular\\u03b1value, we can calculate the pro\\ufb01t of all the solutions\\nfor that value and select the solution that gives the best pro \\ufb01t among the \\ufb01nal population.\\nThis study considers two well-known multi-objective EAs: G SEMO Giel (2003) and NSGA-II Deb et al. (2002).\\nFurthermore, we introduce a \\ufb01ltering method for GSEMO which aims to improve the quality of the \\ufb01nal population\\nresulting from the algorithm. This \\ufb01ltering method is appli ed regularly after a \\ufb01xed number of evaluations in the\\nevolutionary process. It considers whether a solution can b e the best solution for any \\u03b1\\u2208[0.0,1\\/2]considering the\\ninterim population at the time and, otherwise, removes it fr om the population. For all experimental settings, in additi on\\nto the two algorithms, GSEMO and NSGA-II, we consider this \\ufb01l tering method combined with GSEMO as a separate\\nalgorithm.\\nThe structure of this paper is as follows. Section 2 introduc es the problem formulation, and Section 2 discusses the\\nalgorithms we consider in this study. Section 3 discusses th e multi-objective formulation, including the objective\\nfunction on pro\\ufb01t, \\ufb01tness functions and how to use the probab ility bounds to estimate the con\\ufb01dence level of the\\nsolutions. Section 4 and 5 discuss the EAs considered in this paper and the new Filtering method introduced in this\\nwork. Section 6 presents the details of the experimental set tings and the results. Finally, Section 7 gives the conclusi ons\\nof this study.\\n2 Problem De\\ufb01nition\\nIn the classical KP, the pro\\ufb01t and weight values of the items a re deterministic. Let the KP has n items\\n{x1,...,x i,...,x n}with pro\\ufb01t piand weight wiand weight bound B. A solution to the problem can be represent ed as\\n{0,1}nsuch that xi= 1only when xiis selected in the solution. Given the pro\\ufb01t of solution xasp(x) =\\/summationtextn\\ni=1pi.xi\\nand weight w(x) =\\/summationtextn\\ni=1wi.xi, the classical KP is to \\ufb01nd the solution x\\u2217that maximize p(x\\u2217)subjected to the\\nweight constraint w(x\\u2217)\\u2264B.\\nIn this work, we consider a stochastic version of the classic al KP where the pro\\ufb01ts of the knapsack items are stochastic\\nwhile weights remain deterministic. Therefore the pro\\ufb01t of a solution will be uncertain and may vary from the expected\\nmaximal pro\\ufb01t value. A chance constraint on the pro\\ufb01t is used to capture the stochastic behaviour of the problem. This\\nconstraint ensures that for each feasible solution x, the maximum probability that the pro\\ufb01t will drop below pro\\ufb01 t value\\nP is only a small probability 0< \\u03b1 <1\\/2.\\nWe can formally present this problem as follows:\\nmaxP (1)\\ns. t.Pr(p(x)< P)\\u2264\\u03b1 (2)\\nand w (x)\\u2264B (3)\\nwhere\\u03b1is a parameter determining the allowed failure probability which is a small value <= 1\\/2. Equation 2\\nspeci\\ufb01es the chance constraint on the pro\\ufb01t.\\n2.1 Estimating Pro\\ufb01t of a Solution where\\u03b1is a parameter determining the allowed failure probability which is a small value <= 1\\/2. Equation 2\\nspeci\\ufb01es the chance constraint on the pro\\ufb01t.\\n2.1 Estimating Pro\\ufb01t of a Solution\\nComputing the probability of violating a given chance const raint is intractable in general Doerr and Neumann (2021).\\nTherefore, it is not straightforward to estimate the pro\\ufb01t o f a solution under uncertainties. However, tail bounds can\\nbe used to upper bound the probability of chance constraint v iolationDoerr and Neumann (2021). In Neumann et al.\\n(2022), the authors present pro\\ufb01t estimates for the problem based on the tail bounds: Chebyshev\\u2019s inequality and\\nHoeffding bound. Each of these applies to the problem under c ertain conditions.\\nIf the expectation and variance of the solutions\\u2019 pro\\ufb01ts are known for a given problem instance, we can use Cheby-\\nshev\\u2019s inequality to calculate the pro\\ufb01t of a solution for it Doerr (2020). Neumann et al. (2022) present a pro\\ufb01t estimate\\nin Equation 4, that ensure the pro\\ufb01t chance constraint Pr(p(x)< P)\\u2264\\u03b1is held. We can guarantee that the solution\\n3 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nxwill give the pro\\ufb01t \\u02c6pCheb(x,\\u03b1)except for a small probability \\u03b1as follows:\\n\\u02c6pCheb(x,\\u03b1) =\\u00b5(x)\\u2212\\/radicalbig\\n(1\\u2212\\u03b1)\\/\\u03b1\\u00b7\\/radicalbig\\nv(x). (4)\\nThe above equation gives a very general setting for the pro\\ufb01t of a solution that can be used in any scenario where the\\nexpected value and the variance are known. For example, we ca n consider that the pro\\ufb01ts pitake a uniform distribution\\nsuch that pi\\u2208{\\u00b5i\\u2212\\u03b4,\\u00b5i+\\u03b4}which gives the expected value \\u00b5iand variance\\u03b42\\n3.|x|1.\\nThe Hoeffding bound can be used to formulate a pro\\ufb01t estimate if the pro\\ufb01ts are taken randomly from a uniform\\nrandom distribution independent of each other Doerr (2020) . From Neumann et al. (2022), we get the formulation for\\nthe pro\\ufb01t of solution xusing the Hoeffding bound ( \\u02c6pHoef(x,\\u03b1)) as follows:\\n\\u02c6pHoef(x,\\u03b1) =\\u00b5(x)\\u2212\\u03b4\\u00b7\\/radicalbig\\nln(1\\/\\u03b1)\\u00b72\\u00b7|x|1. (5)\\n3 Multi-Objective Formulation\\nIn this section, we introduce the multi-objective formulat ions of the pro\\ufb01t chance constrained KP. As presented in\\nSection 2, the optimal solution for the KP maximizes the pro\\ufb01 t subjected to the weight bound constraint. For the pro\\ufb01t\\nchance constrained KP with deterministic weights, the mult i-objective formulation needs to consider the uncertainty\\nof the pro\\ufb01ts of knapsack items. In the following subsection s, we present the multi-objective formulations and the\\nfunctions to estimate pro\\ufb01ts.\\n3.1 Fitness Functions\\nIn general, we need to ensure that the expectation of pro\\ufb01t is maximized while the variance is minimized to maximize\\nthe pro\\ufb01t of the solution. Considering this, we introduce th e \\ufb01tness function g(x) = (\\u00b5(x),v(x))that will produce\\na set of solutions that gives a trade-off between the two obje ctives, irrespective of \\u03b1the guarantee of the pro\\ufb01t of the\\nindividual solution\\u2019s pro\\ufb01t value. The formula for the two o bjectives is given in Equation 6 and 7 where vmax=\\/summationtextn\\ni=1vi.\\n\\u00b5(x) =\\/braceleftbigg\\/summationtextn\\ni=1\\u00b5ixiw(x)\\u2264B\\nB\\u2212w(x)w(x)> B(6)\\nv(x) =\\/braceleftbigg\\/summationtextn\\ni=1\\u03c32\\nixi w(x)\\u2264B\\nvmax+(w(x)\\u2212B)w(x)> B(7)\\nWhen evaluating the \\ufb01tness of solutions, we need to determin e their dominance concerning the different objectives.\\nFor two solutions A and B, we say that Adominates B(denoted as A\\/{ollowsequalB)iff\\u00b5(A)\\u2265\\u00b5(B)\\u2227v(A)\\u2264v(B), andA\\nstrongly dominates B(denoted as A\\u227bB)iffA\\/{ollowsequalBand\\u00b5(A)> \\u00b5(B)\\u2228v(A)< v(B). For an infeasible solution\\nthat violates the weight capacity constraint ( w(A)> B ), the above formula penalises the two objective values (see\\nEquation 6 and 7). This formulation ensures that any feasibl e solution dominates the infeasible solution instances.\\nNext, we consider the \\ufb01tness function g\\u2032(x) = (\\u00b5(x),s(x)). Only the second objective of this \\ufb01tness function differs\\nfromg\\u2032(x)while the \\ufb01rst objective \\u00b5remains the same and can be de\\ufb01ned as given in Equation 6. We de note the\\nmaximal standard deviation as smax=\\/summationtextn\\ni=1\\u03c3iin Equation 8 which de\\ufb01nes the second objective of this \\ufb01tnes s\\nfunctiong\\u2032(x).\\ns(x) =\\/braceleftbigg\\/radicalbig\\/summationtextn\\ni=1\\u03c32\\nixi w(x)\\u2264B\\nsmax+(w(x)\\u2212B)w(x)> B.(8)\\nFor simple EAs like GSEMO, the \\ufb01nal population when optimizi ngg(x)is equivalent to when optimizing g\\u2032(x)as\\nthe difference between the two is that the second objective o f the latter is the square root of the second objective of the\\nformer. Therefore it does not change the order of search poin ts of such algorithms. However, it should be noted that\\nsome popular EMO algorithms like NSGA-II would encounter a d ifference in the diversity mechanism when working\\nwithv(x)instead of s(x). Therefore, we investigate both g(x)andg\\u2032(x)in this work.\\n4 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\n3.2 Fitness Function for Pro\\ufb01ts with Same Dispersion\\nWhen the value for \\u03b4is the same for all elements, we can consider the number of ite ms selected in the solution ( |x|1)\\nas the second objective. This objective enables the de\\ufb01niti on of a \\ufb01tness function that performs independent of both\\nthe con\\ufb01dence level of the solution ( \\u03b1) and the uncertainty level of the individual pro\\ufb01t values ( \\u03b4). We denote the\\n\\ufb01tness function for this scenario as g\\u2032\\u2032(x) = (\\u00b5(x),c(x)). The \\ufb01rst objective of the new \\ufb01tness function is based on\\nthe expectation of the pro\\ufb01t of x\\u00b5(x), similar to previous functions and calculated as given in Eq uation 6. The second\\nobjective c(x)is calculated as follows,\\nc(x) =\\/braceleftbigg\\/summationtextn\\ni=1xi w(x)\\u2264B\\nn+(w(x)\\u2212B)w(x)> B.(9)\\n3.3 Estimating the Best Pro\\ufb01t Value\\nAs the \\ufb01nal output of the multi-objective evolutionary appr oaches, we get a set of solutions that gives a trade-off of\\nthe objectives independent of the con\\ufb01dence level of the pro \\ufb01ts. We need to identify the solution with the best pro\\ufb01t\\nfor a given con\\ufb01dence level \\u03b1using these \\ufb01nal solutions. We can use Equations 4 and 5 from N eumann et al. (2022)\\nand calculate the pro\\ufb01t of all the solutions in the \\ufb01nal popul ation for the required con\\ufb01dence levels ( \\u03b1). The solution\\ngiving the highest pro\\ufb01t value for each \\u03b1is the best solution for that particular setting.\\n3.4 Con\\ufb01dence Level of a Solution\\u2019s Pro\\ufb01t\\nDifferent solutions in the \\ufb01nal population produced by mult i-objective approaches become the best solution for differ -\\nent con\\ufb01dence levels. We can identify the con\\ufb01dence level of each solution in the \\ufb01nal population for which it gives\\nthe highest pro\\ufb01t value. First, we obtain the \\u03b1threshold for a pair of solutions such that one becomes bette r than the\\nother by comparing the pro\\ufb01t estimates mentioned in Subsect ion 2.1. Based on the threshold \\u03b1value between solution\\npairs in the \\ufb01nal population, we de\\ufb01ne a con\\ufb01dence level inte rval for each solution such that they give the best pro\\ufb01t\\nvalue for any \\u03b1in that interval.\\n3.4.1 Con\\ufb01dence Level Threshold using Chebyshev\\u2019s Inequal ity\\nLet the solutions xandysuch that \\u00b5(x)> \\u00b5(y)andv(x)> v(y), then we can say that the pro\\ufb01t of the solution xis\\nbetter than ywhen the minimum required con\\ufb01dence level( \\u03b1x,y) is as follows:\\n\\u03b1Cheb(x,y)=1\\n1+(R(x,y))2s.t. R(x,y) =\\u00b5(x)\\u2212\\u00b5(y)\\/radicalbig\\nv(x)\\u2212\\/radicalbig\\nv(y)(10)\\nFor the con\\ufb01dence level \\u03b1\\u22651\\/(1+(R(x,y))2), the solution Xwill have a better pro\\ufb01t value than the solution Y.\\nTheorem 3.1. Let0< \\u03b1 <1,and x andybe two feasible solutions such that \\u00b5(x)> \\u00b5(y)andv(x)> v(y), holds.\\nIf\\u03b1\\u22651\\n1+(R(x,y))2holds such that R(x,y) =\\u00b5(x)\\u2212\\u00b5(y)\\u221a\\nv(x)\\u2212\\u221a\\nv(y)then\\u02c6pCheb(x,\\u03b1)\\u2265\\u02c6pCheb(y,\\u03b1).\\nProof. We have\\n\\u03b1\\u22651\\n1+(R(x,y))2\\n\\u21d0\\u21d2 \\u03b1+\\u03b1(R(x,y))2\\u22651\\n\\u21d0\\u21d2 (R(x,y))2\\u2265(1\\u2212\\u03b1)\\/\\u03b1\\n5 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nAs we assume 0< \\u03b1 <1,\\u00b5(x)> \\u00b5(y)andv(x)> v(y), we have R(x,y)>0and1\\u2212\\u03b1\\n\\u03b1>0.\\nThis implies,\\nR(x,y)\\u2265\\/radicalbig\\n(1\\u2212\\u03b1)\\/\\u03b1\\n\\u21d0\\u21d2\\u00b5(x)\\u2212\\u00b5(y)\\/radicalbig\\nv(x)\\u2212\\/radicalbig\\nv(y)\\u2265\\/radicalbig\\n(1\\u2212\\u03b1)\\/\\u03b1\\n\\u21d0\\u21d2 \\u00b5(x)\\u2212\\u00b5(y)\\u2265\\/radicalbigg\\n1\\u2212\\u03b1\\n\\u03b1\\u00b7\\/parenleftBig\\/radicalbig\\nv(x)\\u2212\\/radicalbig\\nv(y)\\/parenrightBig\\n\\u21d0\\u21d2 \\u00b5(x)\\u2212\\/radicalbigg\\n1\\u2212\\u03b1\\n\\u03b1\\u00b7\\/radicalbig\\nv(x)\\u2265\\u00b5(y)\\u2212\\/radicalbigg\\n1\\u2212\\u03b1\\n\\u03b1\\u00b7\\/radicalbig\\nv(y)\\n\\u21d0\\u21d2 \\u02c6pCheb(x,\\u03b1)\\u2265\\u02c6pCheb(y,\\u03b1)\\n3.4.2 Con\\ufb01dence Level Threshold using Hoeffding Bound\\nConsider the solutions xandysuch that \\u00b5(x)> \\u00b5(y)andv(x)> v(y). From the \\u02c6pHoef we can derive the minimum\\ncon\\ufb01dence level \\u03b1x,yfor which solution xgives a better pro\\ufb01t than yw.r.t.\\u02c6pHoef as follows:\\n\\u03b1Hoef(x,y)=e\\u2212(S(x,y))2s.t. S(x,y) =\\u00b5(x)\\u2212\\u00b5(y)\\n\\u03b4\\/parenleftBig\\/radicalbig\\n2|x|1\\u2212\\/radicalbig\\n2|y|1\\/parenrightBig (11)\\nTheorem 3.2. Let0< \\u03b1 <1,and x andybe two feasible solutions such that \\u00b5(x)> \\u00b5(y)andv(x)> v(y), holds.\\nIf\\u03b1\\u2265e\\u2212(S(x,y))2holds then \\u02c6pHoef(x,\\u03b1)\\u2265\\u02c6pHoef(y,\\u03b1).\\nProof. We have,\\n\\u03b1\\u2265e\\u2212(S(x,y))2\\n\\u21d0\\u21d2 e(S(x,y))2\\u22651\\/\\u03b1\\n\\u21d0\\u21d2 (S(x,y))2\\u2265ln(1\\/\\u03b1)\\nAs we assume 0< \\u03b1 <1,\\u00b5(x)> \\u00b5(y)andv(x)> v(y), we have S(x,y)>0andln1\\n\\u03b1>0.\\nThis implies,\\nS(x,y)\\u2265\\/radicalbig\\nln(1\\/\\u03b1)\\n\\u21d0\\u21d2\\u00b5(x)\\u2212\\u00b5(y)\\n\\u03b4\\/parenleftBig\\/radicalbig\\n2|x|1\\u2212\\/radicalbig\\n2|y|1\\/parenrightBig\\u2265\\/radicalbig\\nln(1\\/\\u03b1)\\n\\u21d0\\u21d2 \\u00b5(x)\\u2212\\u00b5(y)\\u2265\\u03b4\\u00b7\\/radicalBigg\\n2ln\\/parenleftbigg1\\n\\u03b1\\/parenrightbigg\\/parenleftBig\\/radicalbig\\n|x|1\\u2212\\/radicalbig\\n|y|1\\/parenrightBig\\n\\u21d0\\u21d2 \\u00b5(x)\\u2212\\u03b4\\u00b7\\/radicalBigg\\nln\\/parenleftbigg1\\n\\u03b1\\/parenrightbigg\\n\\u00b72\\u00b7|x|1\\u2265\\u00b5(y)\\u2212\\u03b4\\u00b7\\/radicalBigg\\nln\\/parenleftbigg1\\n\\u03b1\\/parenrightbigg\\n\\u00b72\\u00b7|y|1\\n\\u21d0\\u21d2 \\u02c6pHoef(x,\\u03b1)\\u2265\\u02c6pHoef(y,\\u03b1)\\n3.4.3 Con\\ufb01dence Level Interval for Solutions in the Final Po pulation\\nWe can use the \\u03b1threshold value in the previous subsections, to introduce t he con\\ufb01dence level range for solutions in\\na population as follows. As the \\u03b1threshold we can use either \\u03b1Cheb(x,y)or\\u03b1Hoef(x,y). Despite the speci\\ufb01c equation\\nto estimate the \\u03b1threshold, we can introduce the \\u03b1interval for a solution.\\nFirst, we sort all the solutions in the given population P as {x1,...xn}such that \\u00b5(x1)\\u2265\\u00b5(x2)\\u2265...\\u2265\\u00b5(xn). Then,\\nwe can de\\ufb01ne a (n+1)\\u00d7(n+1) symmetric matrix of con\\ufb01dence level thresholds as:\\n6 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nAlgorithm 1 GSEMO\\n1:Choosex\\u2208{0,1}nuniformly at random ;\\n2:S\\u2190{x};\\n3:while stopping criterion not met do\\n4: choosex\\u2208Suniformly at random;\\n5:y\\u2190\\ufb02ip each bit of xindependently with probability of1\\nn;\\n6: if(\\/\\\\e}atio\\\\slash\\u2203w\\u2208S:w\\u227by)then\\n7:S\\u2190(S\\u222a{y})\\\\{z\\u2208S|y\\/{ollowsequalz};\\n8: end if\\n9:end while\\n\\u03b1(i,j)=\\/braceleftbigg\\n1 i= 0orj= 0\\n\\u03b1Cheb(i,j)or\\u03b1Hoef(i,j)i= 1,...,n(12)\\nFinally, using the con\\ufb01dence level threshold values, we can get the con\\ufb01dence level range for solution k as given\\nin Equation 13. If there exists a valid \\u03b1interval for a particular solution, then for the \\u03b1values in that interval, that\\nsolution will give the best pro\\ufb01t value. If the \\u03b1interval is empty for a particular solution, then that solut ion won\\u2019t be\\nthe best solution for any \\u03b1value.\\nk\\u22121\\nmin\\nj=0\\u03b1i,k\\u2264\\u03b1k\\u2264n\\u22121max\\nj=k\\u03b1i,j (13)\\n4 Algorithms\\nIn this study, we consider two widely used multi-objective E As: GSEMO and NSGA-II. GSEMO is the most basic EA\\nthat addresses multi-objective optimization problems. It has been proven to be effective in solving chance constraine d\\nmulti-objective optimization problems in many studies Neu mann et al. (2022). The steps of GSEMO are given in\\nAlgorithm 1.\\nThe population S initially contains a solution that is gener ated randomly. Over the iterations, a parent solution xis\\nselected uniformly at random from S and an offspring solutio nyis generated by \\ufb02ipping each bit of xwith a probability\\nof1\\nn. Ifyis not dominated by any of the existing solutions in S, it is ad ded to S replacing all the existing solutions in S\\nthat are dominated by y. This guarantees at the end of any iter ation, population S will contain a set of non-dominating\\nsolutions that are equally ef\\ufb01cient with respect to the give n objective functions.\\nNSGA-II is the most prominent multi-objective EA that focus es on diversity by \\ufb01nding near optimal solutions\\nDeb et al. (2002). If we consider one iteration of the evoluti onary process of NSGA-II, it creates an offspring pop-\\nulation from the parent population. First, we use the binary tournament for the selection of two parent solutions and\\napply single point crossover to generate two offspring solu tions which are mutated by \\ufb02ipping each bit with a proba-\\nbility of1\\nn. When the offspring population is full, all the solutions fr om parent and offspring solutions are considered\\ntogether and divided into non-dominating solution fronts ( i.e., the solutions in a front do not dominate other solution s\\nin it). Starting with the front with the lowest rank, these ar e added to the new population until it reaches capacity. If\\nonly a part of a front is to be taken into the new population, cr owding distance-based sorting is used to decide the\\nselection. This improves the better spread (diversity) amo ng the solutions. In one iteration of this algorithm, a new\\n\\ufb01tness evaluation equal to the size of the offspring populat ion is considered. The number of iterations of the algorithm s\\ndepends on the offspring population size and the maximum num ber of evaluations as considered in the experimental\\nsetup.\\n5 Filtering for Problems with Chance Constraints\\nHere we propose a \\ufb01ltering method for GSEMO to improve its out come by removing solutions that do not have a valid\\n\\u03b1interval from the interim populations. The \\ufb01nal population of GSEMO contains solutions that do not give the best\\npro\\ufb01t value for any probability value for \\u03b1. Such solutions do not add value to the optimization goal of \\ufb01 nding the\\nbest solutions with given con\\ufb01dence levels. For instance, F igure 1 presents a \\ufb01nal population from GSEMO using 10\\nmillion \\ufb01tness evaluation of g(x)on uncorr-100: an instance used in the experiments. The plot shows\\u00b5(x)versus\\nv(x)for all solutions in the \\ufb01nal population. Blue star markers i ndicate that the solution has a valid con\\ufb01dence level\\ninterval. These solutions\\u2019 \\u03b1intervals compose the complete probability range [0.0,1.0 ]. At the end of the optimization interval. These solutions\\u2019 \\u03b1intervals compose the complete probability range [0.0,1.0 ]. At the end of the optimization\\n7 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\n0.0 2.0 4.0 6.0 8.0\\n\\u00b5(x)[in 1000\\u2019s]0.0 1.0 2.0 3.0v(x) [in 1000\\u2019s]\\u03b11\\n\\u03b12\\n\\u03b13\\n\\u03b14\\n\\u03b18\\n\\u03b110\\n\\u03b112\\n\\u03b118\\n\\u03b119\\nFigure 1: A sample of a \\ufb01nal population from GSEMO\\nAlgorithm 2 Filtering Method\\n0:Input: Population P0=x1,...,xnordered by the decreasing order of \\u00b5(xi);\\n1:setk\\u21901;\\n2:whilek\\u2264ndo\\n3: setupper\\u2190mink\\u22121\\nj=0\\u03b1i,k\\n4: setlower\\u2190maxn\\u22121\\nj=k\\u03b1i,j\\n5: ifupper\\u2265lower then\\n6:P1\\u222a{xk}\\n7: end if\\n8: setk\\u2190k+1\\n9:end while\\n10:returnP1\\nprocess, interest is in solutions with these blue markers. I t is one of these solutions that become the best solution for\\na given\\u03b1value. On the contrary, other solutions marked in black do no t become the best solution for any con\\ufb01dence\\nlevel.\\nThe \\ufb01ltering method removes these solutions that do not beco me the best solution for any con\\ufb01dence level. It is\\napplied to the interim populations of GSEMO regularly after a certain amount of \\ufb01tness evaluations. This process\\nremoves the solutions from interim populations, consideri ng the\\u03b1intervals they represent according to Equation 13.\\nAs the \\ufb01ltering method keeps only the solutions with valid \\u03b1interval, it increases the change of the evolutionary\\noptimization to improve on these solutions. Therefore, thi s method helps to improve the quality of the solutions in the\\n\\ufb01nal population.\\nThe steps of the \\ufb01ltering method are given in Algorithm 2. It t akes the population P0as the input, which can be either\\nthe \\ufb01nal population or an interim population created during the execution of GSEMO. Population P0needs solutions\\nin the decreasing order of the \\u00b5(x). For each solution xk, we consider \\u03b1i,kand select the interval for con\\ufb01dence level,\\nusing the inequality given in Equation 13. The solution xkis added to the resulting population P1iff the interval for\\n\\u03b1kis non-empty.\\n6 Experiments\\nIn this work, we evaluate the \\ufb01tness functions introduced pr eviously in different chance constraint settings using mul ti-\\nobjective EAs on different benchmarks; and we use multiple e xperiments for that. Here we present the experimental\\nsettings and the results of these experiments.\\n8 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINTTable 1: Results for same dispersion using Chebyshev\\u2019s ineq uality\\nB\\u03b1\\u03b4GSEMO with g(x) (1) GSEMO+Filtering with g(x) (2) NSGA-II with g(x) (3) NSGA-II with g\\u2032(x) (4) NSGA-II with g\\u2032\\u2032(x) (5)\\nmean std stat mean std stat mean std stat mean std stat mean std statuncorr-100\\n24070.12511029.6826 76.3191 2\\u22125+11085.6072 0.00001+3+4+5+11007.6251 88.8419 2\\u221210998.7574 50.6310 2\\u221211007.7624 52.5740 1\\u22122\\u2212\\n5010862.4750 61.4080 2\\u22125+10907.0000 0.00001+3+4+5+10837.2841 53.1386 2\\u221210811.4832 80.3604 2\\u221210832.9581 50.1309 1\\u22122\\u2212\\n0.012510620.5264 71.0446 2\\u22125+10672.3379 0.00001+3+4+5+10602.1969 81.4896 2\\u221210594.9482 45.5447 2\\u221210602.8065 46.9209 1\\u22122\\u2212\\n5010044.4941 49.7668 2\\u22125+10079.9649 0.00001+3+4+5+10025.8881 41.6418 2\\u221210004.2550 67.4364 2\\u221210023.0463 38.8802 1\\u22122\\u2212\\n0.00125 9345.9245 55.2560 2\\u22125+9384.5150 0.00001+3+4+5+9338.8078 59.0630 2\\u22125+9336.6045 30.1889 2\\u22129340.8894 29.5604 1\\u22122\\u22123\\u2212\\n50 7495.5153 17.9630 2\\u22125+7502.7716 0.00001+3+4+5+7498.7722 13.1282 2\\u22125+7488.8368 31.6085 2\\u22127499.2121 9.5267 1\\u22122\\u22123\\u2212strong-100\\n41870.125 8507.0584 130.1420 2\\u22128606.3413 87.2952 1+3+4+5+8525.5561 125.6131 2\\u22128500.0805 149.4958 2\\u22128499.3632 124.5506 2\\u2212\\n50 8368.0306 94.0118 2\\u22128422.6322 67.2273 1+3+5+8326.9385 114.3897 2\\u22128364.0549 124.7364 8319.3263 115.3753 2\\u2212\\n0.0125 8083.9678 106.1983 2\\u22128170.3360 69.5792 1+4+5+8102.4388 104.2026 8082.3479 124.2002 2\\u22128082.2853 103.3765 2\\u2212\\n50 7502.9361 59.8544 2\\u22127549.4937 41.2749 1+3+5+7489.7223 73.5765 2\\u22127513.5174 78.6964 7485.1707 73.8490 2\\u2212\\n0.00125 6770.3193 41.1347 2\\u22126814.1197 20.7669 1+4+5+6787.5739 42.8885 6782.2797 48.8989 2\\u22126784.1694 42.0052 2\\u2212\\n50 4957.5449 36.7770 2+3\\u22124\\u22124894.0039 60.2107 1\\u22123\\u22124\\u22125\\u22124990.7637 17.9608 1+2+5+4989.4145 13.2248 1+2+5+4991.4701 11.0454 2+3\\u22124\\u2212uncorr-300\\n68530.12533935.4067 205.6247 2\\u221234286.1802 147.5309 1+3+4+5+33681.6883 534.3217 2\\u221233671.7620 555.4450 2\\u221233615.8492 489.0223 2\\u2212\\n5033571.9980 260.8593 2\\u221233967.3813 159.2433 1+3+4+5+33418.4882 512.5693 2\\u221233284.5989 450.6255 2\\u221233319.4651 483.5459 2\\u2212\\n0.012533237.8865 200.4641 2\\u221233577.9421 141.7536 1+3+4+5+32992.6756 521.4969 2\\u221232984.2543 541.8390 2\\u221232929.2384 476.3437 2\\u2212\\n5032180.0106 245.8773 2\\u221232551.5342 144.8963 1+3+4+5+32039.6829 487.7705 2\\u221231913.5405 429.4896 2\\u221231946.2435 458.2449 2\\u2212\\n0.0012531066.6084 186.3571 2\\u221231372.5619 122.8614 1+3+4+5+30846.1243 482.1829 2\\u221230841.9917 499.6822 2\\u221230789.6329 437.0386 2\\u2212\\n5027843.2948 203.7335 2\\u221228141.7188 105.9385 1+3+4+5+27745.1612 411.5637 2\\u221227641.0702 364.7302 2\\u221227668.0494 380.2576 2\\u2212strong-300\\n138210.12523809.6581 433.2506 2\\u22123\\u22125\\u221224369.6211 216.9574 1+3+4+24099.5570 327.0870 1+2\\u221223986.4018 344.2409 2\\u221224176.0891 232.7994 1+\\n5023594.2993 335.6481 2\\u22123\\u22125\\u221224135.2769 220.4491 1+3+4+5+23867.9086 293.6342 1+2\\u221223695.2127 304.2994 2\\u221223899.3116 223.8869 1+2\\u2212\\n0.012523176.9548 406.2664 2\\u22123\\u22125\\u221223703.0401 197.3436 1+3+4+23464.5667 299.1984 1+2\\u221223360.1013 318.6357 2\\u221223534.8995 212.1891 1+\\n5022322.7651 282.0126 2\\u22123\\u22125\\u221222797.4912 177.4246 1+3+4+5+22588.5433 240.6771 1+2\\u221222446.1950 255.5942 2\\u221222616.9323 182.9564 1+2\\u2212\\n0.0012521208.9163 322.7908 2\\u22123\\u22125\\u221221626.7053 138.6421 1+4+21486.9475 214.2326 1+21411.6675 240.6127 2\\u221221536.8345 149.2069 1+\\n5018388.5805 159.4458 2\\u22123\\u22124\\u22125\\u221218647.0894 70.1852 1+4+18623.6627 98.1567 1+18569.9473 110.1685 1+2\\u221218638.7270 63.3870 1+uncorr-500\\n112430.12557076.8361 748.9305 2\\u22123+4+58431.4168 311.5788 1+3+4+5+55850.4588 1249.6235 1\\u22122\\u221255869.0104 1408.3737 1\\u22122\\u221256037.4884 1287.2936 2\\u2212\\n5056690.8982 859.2445 2\\u22123+4+5+58120.8249 314.3063 1+3+4+5+55563.0878 1044.9051 1\\u22122\\u221254981.1206 1223.7431 1\\u22122\\u221255667.0434 1278.2837 1\\u22122\\u2212\\n0.012556197.0249 738.3354 2\\u22123+4+57528.4355 304.4306 1+3+4+5+54995.4516 1230.9769 1\\u22122\\u221255013.2733 1387.3025 1\\u22122\\u221255179.3063 1266.4267 2\\u2212\\n5054931.1821 829.5866 2\\u22123+4+5+56312.8274 298.8610 1+3+4+5+53851.9927 1009.2563 1\\u22122\\u221253287.7928 1187.2402 1\\u22122\\u221253950.6793 1236.5882 1\\u22122\\u2212\\n0.0012553456.0312 705.5039 2\\u22123+4+54715.2628 282.5314 1+3+4+5+52331.0919 1172.9610 1\\u22122\\u221252346.6392 1321.7096 1\\u22122\\u221252505.0533 1201.5375 2\\u2212 0.0012553456.0312 705.5039 2\\u22123+4+54715.2628 282.5314 1+3+4+5+52331.0919 1172.9610 1\\u22122\\u221252346.6392 1321.7096 1\\u22122\\u221252505.0533 1201.5375 2\\u2212\\n5049451.2762 737.5644 2\\u22123+4+50683.8705 252.3483 1+3+4+5+48521.4630 900.6466 1\\u22122\\u221248012.1505 1072.2455 1\\u22122\\u221248602.1733 1107.2660 2\\u2212strong-500\\n222230.12538822.1695 692.1198 2\\u22123\\u22124\\u22125\\u221240391.0362 449.8195 1+3+4+5+39792.1607 415.5621 1+2\\u221239754.0904 424.9780 1+2\\u221239769.7011 392.0549 1+2\\u2212\\n5038444.0651 620.4975 2\\u22123\\u22124\\u22125\\u221240078.0983 348.7030 1+3+4+5+39442.9758 605.5909 1+2\\u221239485.7055 483.2892 1+2\\u221239416.6356 382.3801 1+2\\u2212\\n0.012538026.4154 657.3621 2\\u22123\\u22124\\u22125\\u221239525.2027 425.9579 1+3+4+5+38973.8435 390.8087 1+2\\u221238936.9771 399.8035 1+2\\u221238951.8432 369.6260 1+2\\u2212\\n5036864.1232 555.3952 2\\u22123\\u22124\\u22125\\u221238332.2087 312.3534 1+3+4+5+37800.5870 535.4997 1+2\\u221237839.6026 424.3667 1+2\\u221237781.3920 337.2743 1+2\\u2212\\n0.0012535546.6995 551.6446 2\\u22123\\u22124\\u22125\\u221236827.5418 352.7810 1+3+4+5+36424.1740 315.3111 1+2\\u221236391.1536 322.1788 1+2\\u221236404.2919 299.3640 1+2\\u2212\\n5031947.2385 360.7027 2\\u22123\\u22124\\u22125\\u221232899.2694 206.7434 1+32685.3625 321.9032 1+32727.4059 242.3192 1+32688.8779 201.6241 1+\\n9 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\n6.1 Experimental Setup\\nFor experimental investigations, we use the same benchmark s as the ones that are used in Neumann et al. (2022).\\nThe set of benchmarks includes three correlated instances a nd three bounded strongly correlated instances with the\\nnumbers of knapsack items n\\u2208{100,300,500}. We consider that the pro\\ufb01ts of the knapsack items have a unif orm\\ndistribution, such that the pro\\ufb01t of element iis chosen uniformly at random as pi\\u2208{\\u00b5i\\u2212\\u03b4,\\u00b5i+\\u03b4}. This allows to\\nuse both\\u02c6pCheb and\\u02c6pHoef when the pro\\ufb01ts have the same uncertainty level ( \\u03b4). The experimental investigation covers\\ntwo uncertainty levels for each benchmark, \\u03b4\\u2208{25,50}. Additionally, we consider the scenario where the pro\\ufb01ts\\nhave different dispersion such that each item ihas an uncertainty level \\u03b4i, which is chosen uniformly at random as\\n\\u03b4i\\u2208[0.0,\\u00b5i]. The benchmarks with different uncertainties are consider ed only with \\u02c6pCheb since\\u02c6pHoef requires the\\nsame uncertainty level for all elements.\\nThis study mainly considers three algorithms: two well-kno wn multi-objective evolutionary algorithms GSEMO and\\nNSGA-II and the third is GSEMO with the \\ufb01ltering method intro duced in Section 4. The latter is referred to as\\nGSEMO+Filtering hereafter in this paper. These algorithms are combined with \\ufb01tness functions as appropriate, to\\nuse in the experiments. For the benchmarks with \\ufb01xed uncerta inties, using GSEMO or GSEMO+Filtering with any\\n\\ufb01tness function ( g(x)org\\u2032(x)org\\u2032\\u2032(x)) will produce equivalent results as the \\ufb01nal populations. T herefore, GSEMO\\nand GSEMO+Filtering are only considered with the \\ufb01tness eva luationg(x). The \\ufb01tness function g\\u2032\\u2032(x)considers the\\nnumber of items selected in the solution for the scenario whe re pro\\ufb01ts have the same dispersion. Therefore, we do\\nnot consider algorithms with that \\ufb01tness function for the be nchmarks with different uncertainties for pro\\ufb01ts. Only\\nGSEMO and GSEMO+Filtering with g(x), and NSGA-II with g(x)andg\\u2032(x)are considered for those benchmarks.\\nEvery algorithm considers the 10 million \\ufb01tness evaluation and produces a population of solutions that gives a trade-\\noff concerning the objectives used for \\ufb01tness evaluation. T he quality of the output of these methods is evaluated\\nby analyzing the best pro\\ufb01t value for different con\\ufb01dence le vels considering \\u03b1\\u2208{0.1,0.01,0.001}. Each method\\ngenerates a \\ufb01nal population independent of \\u03b1, and we select the best solution from that population for dif ferent\\u03b1\\nvalues using pro\\ufb01t estimations \\u02c6pCheb or\\u02c6pHoef as applicable. The results summarise the best pro\\ufb01t value gi ven by 30\\nexperimental results for each \\u03b1. This summary requires running the method 30 times for each \\u03b4for benchmarks with\\nthe same pro\\ufb01t dispersion and 30 times for benchmarks with di fferent pro\\ufb01t dispersion. However, when using g\\u2032\\u2032(x),\\nas algorithms run independent of \\u03b4value, it is possible to get the best pro\\ufb01t values for differe nt\\u03b4from the same \\ufb01nal\\npopulation.\\nFinally, we test for the statistical signi\\ufb01cance validity o f the results using the Kruskal-Wallis test with 95% con\\ufb01den ce\\nwith the Bonferroni post-hoc statistical procedure. The st atistical comparison is indicated as X+orX\\u2212to indicate that\\nthe method in the column outperforms Xor vice versa. If there is no signi\\ufb01cant difference between t he two methods,\\nrespective numbers do not appear. For each method, the summa ry of the best pro\\ufb01t values is given as mean, std and\\nstat, which represent the mean and standard deviation of the results and statistical comparison with the corresponding\\nresults from other methods, respectively.\\n6.2 Results\\nTable 1 and 2 present the results for the benchmarks with \\ufb01xed uncertainty levels pro\\ufb01ts of elements. According to the\\nmean values GSEMO with the \\ufb01ltering method outperforms othe r methods in most of the settings in both tables. The mean values GSEMO with the \\ufb01ltering method outperforms othe r methods in most of the settings in both tables. The\\nstatistical comparisons give more insights into the perfor mance of the methods when applied to each instance. Results\\nfor uncorr-100 in Table 1 show that GSEMO+Filtering perform s the better than other four methods, and GSEMO\\nperforms better than NSGA-II with g\\u2032\\u2032(x). For smaller con\\ufb01dence levels \\u03b1= 0.001, NSGA-II with g(x)outperforms\\nthat withg\\u2032\\u2032(x). Strong-100 instance gets very different results for \\u03b1= 0.001and\\u03b4= 50 , than other cases of the same\\ninstance. There, GSEMO with g(x)and NSGA-II with g(x)andg\\u2032(x)perform well and GSEMO+Filtering gives the\\nlowest result. However, the second method performs well in o ther\\u03b1and\\u03b4settings and NSGA-II with g(x)andg\\u2032(x)\\nalso perform similarly in certain settings.\\nFor all settings of uncorr-300, GSEMO+Filtering outperfor ms the other four methods while performing similarly to\\neach other. For strong-300 instance, GSEMO gives lower resu lts than GSEMO+Filtering and NSGA-II with g(x)\\nandg\\u2032\\u2032(x). Also, NSGA-II with g(x)produces results as good as GSEMO+Filtering for \\u03b1= 0.001. For uncorr-\\n500, GSEMO+Filtering gives the best results and GSEMO outpe rforms most of the NSGA-II results. However,\\nwhen using g\\u2032\\u2032(x)with NSGA-II on this instance, results show equal performan ce except for \\u03b1={0.1,0.01}con-\\n\\ufb01dence levels when considering uncertainty level \\u03b4= 50 . Experiments on strong-500 also get the best results from\\nGSEMO+Filtering. However, GSEMO is outperformed by other m ethods for all settings considered for this instance.\\nIn comparison, the NSGA-II gives the second best results acr oss settings when \\u03b4= 50 and\\u03b1= 0.001NSGA-II\\nmethods also perform as well as GSEMO+Filtering.\\n10 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nTable 2: Results for same dispersion using Hoeffding bound\\nB\\u03b1\\u03b4GSEMO with g(x) (1) GSEMO+Filtering with g(x) (2) NSGA-II with g(x) (3) NSGA-II with g\\u2032(x) (4) NSGA-II with g\\u2032\\u2032(x) (5)\\nmean std stat mean std stat mean std stat mean std stat mean std statuncorr-100\\n24070.12510987.3004 75.7683 2\\u221211042.7989 0.00001+3+4+5+10965.6291 88.0780 2\\u221210956.9290 50.1020 2\\u221210965.9887 51.9898 2\\u2212\\n5010778.0078 60.1974 2\\u221210821.5978 0.00001+3+4+5+10753.4968 51.9386 2\\u221210728.1263 79.0134 2\\u221210749.4108 48.9647 2\\u2212\\n0.012510896.5878 74.5928 2\\u221210951.1744 0.00001+3+4+5+10875.7430 86.4446 2\\u221210867.4019 48.9713 2\\u221210876.2792 50.7360 2\\u2212\\n5010596.7649 57.6057 2\\u221210638.3488 0.00001+3+4+5+10573.7130 49.3723 2\\u221210549.2659 76.1316 2\\u221210569.9917 46.4640 2\\u2212\\n0.0012510826.9815 73.6938 2\\u221210880.8684 0.00001+3+4+5+10806.7709 85.1930 2\\u221210798.7053 48.1051 2\\u221210807.4426 49.7746 2\\u2212\\n5010457.6924 55.6226 2\\u221210497.7369 0.00001+3+4+5+10435.7601 47.4116 2\\u221210412.0216 73.9290 2\\u221210432.3187 44.5489 2\\u2212strong-100\\n41870.125 8463.1467 127.6172 2\\u22128561.1573 85.4504 1+3+4+5+8481.7278 123.3848 2\\u22128456.8099 146.8670 2\\u22128456.3390 122.3538 2\\u2212\\n50 8278.6327 90.2553 2\\u22128332.4692 64.4783 1+3+5+8240.4850 110.0787 2\\u22128276.2257 119.9424 8233.2780 111.0076 2\\u2212\\n0.0125 8369.3409 122.2698 2\\u22128464.4621 81.5096 1+3+4+5+8387.9199 118.6225 2\\u22128364.1958 141.2465 2\\u22128363.9441 117.6448 2\\u2212\\n50 8086.8101 82.3211 2\\u22128139.0049 58.6173 1+3+5+8054.9801 100.8829 2\\u22128087.7692 109.6779 8048.4883 101.6750 2\\u2212\\n0.00125 8297.3868 118.1961 2\\u22128390.2653 78.4972 1+3+4+5+8315.9386 114.9756 2\\u22128293.1306 136.9396 2\\u22128293.0470 114.0400 2\\u2212\\n50 7939.6195 76.3766 2\\u22127990.5545 54.1630 1+3+5+7912.6372 93.8879 2\\u22127943.1615 101.8268 7906.6941 94.5672 2\\u2212uncorr-300\\n68530.12533863.1544 205.0835 2\\u221234212.8177 146.9244 1+3+4+5+33610.2955 532.9802 2\\u221233600.5469 554.0346 2\\u221233545.0211 487.7131 2\\u2212\\n5033428.2570 259.2905 2\\u221233821.1723 157.7349 1+3+4+5+33276.1084 510.0035 2\\u221233143.0191 448.4365 2\\u221233177.8089 480.9303 2\\u2212\\n0.012533708.5096 203.9304 2\\u221234055.7967 145.6324 1+3+4+5+33457.5385 530.1364 2\\u221233448.1219 551.0166 2\\u221233392.9168 484.9026 2\\u2212\\n5033119.8295 255.9406 2\\u221233507.4493 154.5170 1+3+4+5+32970.6017 504.5017 2\\u221232839.2289 443.7443 2\\u221232873.6003 475.3176 2\\u2212\\n0.0012533589.8465 203.0502 2\\u221233935.3102 144.6467 1+3+4+5+33340.3278 527.9571 2\\u221233331.1621 548.7016 2\\u221233276.2031 482.7469 2\\u2212\\n5032883.1649 253.3855 2\\u221233266.7212 152.0660 1+3+4+5+32736.1783 500.2836 2\\u221232606.1226 440.1485 2\\u221232640.1729 471.0147 2\\u2212strong-300\\n138210.12523744.0892 430.4747 2\\u22123\\u22125\\u221224300.5479 214.9589 1+3+4+24033.7421 324.1660 1+2\\u221223921.4833 341.5510 2\\u221224109.9465 230.6676 1+\\n5023462.9510 329.9513 2\\u22123\\u22125\\u221223997.1125 215.9394 1+3+4+5+23735.7246 288.1238 1+2\\u221223566.2352 299.2535 2\\u221223767.0263 219.6354 1+2\\u2212\\n0.012523603.7492 424.5461 2\\u22123\\u22125\\u221224152.7138 210.6726 1+3+4+23892.9269 317.9729 1+2\\u221223782.5354 335.8005 2\\u221223967.9043 226.0936 1+\\n5023181.2241 317.9412 2\\u22123\\u22125\\u221223700.6927 206.3054 1+3+4+5+23452.1877 276.3622 1+2\\u221223289.4860 288.4378 2\\u221223482.9419 210.5250 1+2\\u2212\\n0.0012523496.0973 420.0184 2\\u22123\\u22125\\u221224039.3181 207.2892 1+3+4+23784.9190 313.2317 1+2\\u221223675.9269 331.3992 2\\u221223858.9115 222.5879 1+\\n5022965.0474 308.7926 2\\u22123\\u22125\\u221223473.2418 198.9571 1+3+4+5+23234.6394 267.3758 1+2\\u221223077.1290 280.1499 2\\u221223264.9563 203.5545 1+2\\u2212uncorr-500\\n112430.12556985.6975 747.8288 2\\u22123+4+58337.8820 310.8352 1+3+4+5+55761.8933 1247.6913 1\\u22122\\u221255780.3693 1406.1906 1\\u22122\\u221255948.9616 1285.1401 2\\u2212\\n5056509.1689 856.1817 2\\u22123+4+5+57934.0703 312.7435 1+3+4+5+55386.3718 1041.2515 1\\u22122\\u221254806.2623 1219.9708 1\\u22122\\u221255489.9899 1273.9787 1\\u22122\\u2212\\n0.012556790.6294 745.4714 2\\u22123+4+58137.6851 309.2459 1+3+4+5+55572.3326 1243.5564 1\\u22122\\u221255590.6468 1401.5183 1\\u22122\\u221255758.8493 1280.5162 2\\u2212\\n5056119.2292 849.6132 2\\u22123+4+5+57533.3999 309.3150 1+3+4+5+55007.2442 1033.3396 1\\u22122\\u221254431.0658 1211.8786 1\\u22122\\u221255109.7653 1264.7366 1\\u22122\\u2212\\n0.0012556640.9485 743.6632 2\\u22123+4+57984.0686 308.0287 1+3+4+5+55426.8776 1240.3841 1\\u22122\\u221255445.0676 1397.9335 1\\u22122\\u221255612.9710 1276.9688 2\\u2212 0.0012556640.9485 743.6632 2\\u22123+4+57984.0686 308.0287 1+3+4+5+55426.8776 1240.3841 1\\u22122\\u221255445.0676 1397.9335 1\\u22122\\u221255612.9710 1276.9688 2\\u2212\\n5055820.0179 844.5763 2\\u22123+4+5+57226.0199 306.6676 1+3+4+5+54716.3294 1027.2713 1\\u22122\\u221254143.1674 1205.6714 1\\u22122\\u221254818.0086 1257.6477 1\\u22122\\u2212strong-500\\n222230.12538739.7417 688.5044 2\\u22123\\u22124\\u22125\\u221240301.3374 447.3446 1+3+4+5+39707.3872 412.9829 1+2\\u221239669.4439 422.3664 1+2\\u221239685.3277 389.7418 1+2\\u2212\\n5038280.9153 613.7020 2\\u22123\\u22124\\u22125\\u221239897.8123 344.8981 1+3+4+5+39273.3775 598.3347 1+2\\u221239315.7031 477.1865 1+2\\u221239247.8886 377.7601 1+2\\u2212\\n0.012538563.3178 680.7776 2\\u22123\\u22124\\u22125\\u221240109.3511 442.0526 1+3+4+5+39525.9425 407.4683 1+2\\u221239488.2712 416.7806 1+2\\u221239504.1345 384.7767 1+2\\u2212\\n5037930.8422 599.1733 2\\u22123\\u22124\\u22125\\u221239510.9695 336.7709 1+3+4+5+38909.4678 582.7780 1+2\\u221238950.9755 464.0994 1+2\\u221238885.5815 367.8049 1+2\\u2212\\n0.0012538427.9430 674.8592 2\\u22123\\u22124\\u22125\\u221239962.0501 437.9920 1+3+4+5+39386.7151 403.2421 1+2\\u221239349.2525 412.4983 1+2\\u221239365.1001 380.9688 1+2\\u2212\\n5037662.2216 588.0761 2\\u22123\\u22124\\u22125\\u221239214.1346 330.5707 1+3+4+5+38630.2301 570.8539 1+2\\u221238671.1102 454.0722 1+2\\u221238607.6080 360.1434 1+2\\u2212\\nTable 3: Results for different dispersion using Chebyshev\\u2019 s inequality\\nInstance B\\u03b1GSEMO with g(x) (1) GSEMO+Filtering with g(x) (2) NSGA-II with g(x) (3) NSGA-II with g\\u2032(x) (4)\\nmean std stat mean std stat mean std stat mean std stat\\nuncorr-100 24070.1 8546.7015 322.7956 8562.9620 323.1475 8565.8942 324.9223 8564.2549 323.8494\\n0.01 4682.2710 799.1875 4675.9295 795.5029 4674.8735 797.5334 4677.1350 799.3174\\n0.001 1967.4965 880.0104 1962.3962 886.0726 1919.2445 917.6798 1953.6975 894.4411\\nstrong-100 41870.1 7100.1631 245.6990 7099.8604 250.2445 7122.7698 249.2182 7123.3386 250.5080\\n0.01 5315.4258 326.6578 5308.5248 343.0757 5329.4856 326.9290 5333.3263 325.5847\\n0.001 3746.9429 732.1072 3746.1213 731.9397 3734.2093 731.9080 3743.5859 731.6689\\nuncorr-300 68530.1 29089.9230 479.0749 2\\u22123\\u22124\\u221229580.8035 355.6388 1+29735.0392 358.7213 1+29676.9763 388.6763 1+\\n0.01 19396.8836 1128.3967 19431.8036 1126.0478 19589.0418 1131.0819 19580.3092 1131.0342\\n0.001 8910.0087 1370.4531 8850.8254 1381.8863 8899.5025 1378.4832 8971.3050 1383.6696\\nstrong-300 138210.1 21789.9295 335.7672 2\\u22123\\u22124\\u221222171.9138 358.9991 1+22345.3397 309.3574 1+22297.4300 307.0689 1+\\n0.01 18172.8378 560.3248 18195.2974 615.1645 18338.0359 588.6787 18342.4421 576.7652\\n0.001 14629.7944 809.3377 14617.5558 794.6289 14643.0814 808.6424 14667.4349 812.9751\\nuncorr-500 112430.1 50266.4398 709.0211 2\\u22123\\u22124\\u221252494.0984 556.8082 1+52468.0194 532.9634 1+52149.3408 700.4027 1+\\n0.01 37753.4240 1566.1944 38510.4882 1564.4777 38746.1887 1539.8167 38686.7230 1555.8618\\n0.001 18969.0800 2144.1783 18880.7433 2144.3506 19153.0886 2137.5579 19190.4696 2134.7447\\nstrong-500 222230.1 35919.5415 631.8822 2\\u22123\\u22124\\u221237833.8138 352.1352 1+37832.1320 332.8651 1+37690.0363 317.8082 1+\\n0.01 30977.9111 679.1163 2\\u22123\\u22124\\u221231554.3119 682.8664 1+31822.0362 649.7576 1+31805.6899 637.4521 1+\\n0.001 25041.2112 721.5121 25018.0126 704.1720 25131.2311 723.9024 25193.7178 741.8816\\nTable 2 gives the results from the experiments that use \\u02c6pCheb to estimate the pro\\ufb01t values of solutions. uncorr-100,\\nstrong-100, and uncorr-300 results are highest when using G SEMO+Filtering. It outperforms all the other methods\\nexcept NSGA-II with g\\u2032(x)on strong-100 instance for uncertainty level \\u03b4= 50 . Experiments on strong-300 instance\\nshow that NSGA-II with g\\u2032\\u2032(x)performs equally as the GSEMO+Filtering when considering a lower uncertainty level\\n(25) for the knapsack items. GSEMO and NSGA-II with g\\u2032(x)methods give the lowest results for this instance. On the\\ncontrary, GSEMO performs better than most NSGA-II methods f or uncorr-500. When using g\\u2032\\u2032(x), NSGA-II performs\\nequally as GSEMO for lower uncertainty level value 25. For al l\\u03b1and\\u03b4values, NSGA-II methods are outperformed\\nby GSEMO+Filtering in the experiments on strong-500 in Tabl e 2. However, NSGA-II methods perform better than\\nGSEMO on that benchmark.\\n11 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nGSEMO+Filtering performs signi\\ufb01cantly well when applied t o benchmarks with the same dispersion for pro\\ufb01ts. The\\n\\ufb01ltering method allows the interim populations to contain m ore solutions that yield a valid con\\ufb01dence level interval.\\nTherefore, improving upon these solutions eventually give s better results in the \\ufb01nal outcome. Considering NSGA-II\\nresults,g(x)andg\\u2032(x)tends to produce better results than g\\u2032\\u2032(x). This can be due to the fact the crowding distance\\nassignment is better when considering the variance or the st andard deviation of solutions\\u2019 pro\\ufb01ts.\\nWe can compare the results for benchmarks with the same dispe rsion for pro\\ufb01ts (given in Table 1 and 2) with the\\nprevious work in Neumann et al. (2022) as it also considers th e same benchmarks and similar experimental setup. The\\nexperimental settings are the same in both works except for t he number of \\ufb01tness evaluations the algorithms consider.\\nFor each \\u03b1and\\u03b4value, methods in Neumann et al. (2022) run for one million \\ufb01t ness evaluations. On the contrary,\\nwe run the multi-objective methods on benchmarks for each \\u03b4value for 10 million \\ufb01tness evaluations which yield\\nresults for all \\u03b1values from the same algorithm output. The results show that for Chebyshev results 14 out of the\\n36 settings, the highest mean pro\\ufb01t values given in Table 1 ou tperform all the three methods used in Neumann et al.\\n(2022) and for Hoeffding results in 15 out of the 36 settings g ets better pro\\ufb01ts according to Table 2. Generally, most\\nof these cases are from experiments on uncorrelated benchma rks. For the cases where the new methods perform better\\non bounded strongly correlated instances, it is for higher u ncertainty level \\u03b4= 50 and lower con\\ufb01dence values like\\n\\u03b1={0.01,0.01}.\\nTable 3 presents results for benchmarks with different disp ersion for pro\\ufb01ts of elements. The highest mean values\\nreported for each case show, for smaller instances, GSEMO an d NSGA-II give the highest mean pro\\ufb01ts, and for in-\\nstances with 300 or 500 elements, GSEMO+Filtering with g\\u2032(x)and NSGA-II give the highest mean pro\\ufb01t. Although\\nthe highest mean values can be identi\\ufb01ed from different meth ods, the results are similar in most of the settings. Based\\non the statistical comparisons, we can see that smaller inst ances: strong-100 and uncorr-100, do not signi\\ufb01cantly diff er\\nbetween each method\\u2019s results. Results from other instance s show that for \\u03b1= 0.1GSEMO+Filtering and NSGA-II\\nmethods outperform the GSEMO. In addition, strong-500 inst ance shows better results from GSEMO+Filtering with\\ng\\u2032(x)and NSGA-II with both \\ufb01tness functions for \\u03b1= 0.01. However, in other settings with \\u03b1={0.01,0.001}, all\\nmethods show similar results for the instances with 300 and 5 00 items.\\nCompared to the benchmarks with the same dispersion for pro\\ufb01 ts, the ones with different dispersion can give higher\\nvariance for the pro\\ufb01t of some elements. Therefore, it is cru cial to have a good spread of the solutions in the \\ufb01nal\\npopulation as more solutions tend to give negative pro\\ufb01t val ues when considering certain con\\ufb01dence levels. NSGA-\\nII using crowding distance based sorting appears to achieve this when use with selected objectives in the \\ufb01tness\\nevaluations. In comparison, GSEMO+Filtering is also able t o achieve similarly good results by ignoring the solutions\\nwithout a valid \\u03b1interval and eventually improving the \\ufb01nal population.\\n7 Conclusion\\nThis paper explores multi-objective evolutionary approac hes to solve the pro\\ufb01t chance constrained KP. We introduce\\n\\ufb01tness evaluations for EAs to cater for this problem. These \\ufb01 tness functions can evaluate the solutions irrespective\\nof the required con\\ufb01dence level of the solutions. Therefore , the outcome of EAs gives us a population that includes\\nsolutions giving the best pro\\ufb01t value for different con\\ufb01den ce levels. So it is unnecessary to decide on the required solutions giving the best pro\\ufb01t value for different con\\ufb01den ce levels. So it is unnecessary to decide on the required\\ncon\\ufb01dence level before executing the algorithms, and no nee d to have multiple executions to investigate solutions\\nfor different con\\ufb01dence levels. After considering the avai lable solutions and risks associated with their pro\\ufb01ts, it i s\\npossible to make more informed decisions on what solutions t o implement for the problem instance. Furthermore, we\\nintroduce a \\ufb01ltering method, which is applied at regular int ervals of \\ufb01tness evaluations. It keeps only the solutions wi th\\na valid\\u03b1interval in the interim populations, enabling the new offsp ring solutions in the next generations to improve\\nupon these solutions. The performance of these methods is ev ident in the experimental investigations.\\nAcknowledgements\\nThis work has been supported by the Australian Research Coun cil (ARC) through grant FT200100536, and by the\\nSouth Australian Government through the Research Consorti um \\\"Unlocking Complex Resources through Lean Pro-\\ncessing\\\". This work was also supported with supercomputing resources provided by the Phoenix HPC service at the\\nUniversity of Adelaide.\\nReferences\\nYuma Abe, Masaki Ogura, Hiroyuki Tsuji, Amane Miura, and Shu ichi Adachi. 2020. Resource and Network Manage-\\nment for Satellite Communications Systems: A Chance-Const rained Approach. IFAC 53 (2020), 3304\\u20133309. Issue\\n12 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\n2.\\nHirad Assimi, Oscar Harper, Yue Xie, Aneta Neumann, and Fran k Neumann. 2020. Evolutionary Bi-Objective Opti-\\nmization for the Dynamic Chance-Constrained Knapsack Prob lem Based on Tail Bound Objectives. In ECAI 2020 ,\\nV ol. 325. IOS Press, 307\\u2013314.\\nC.C. Coello, D.A. Van Veldhuizen, and G.B. Lamont. 2013. Evolutionary Algorithms for Solving Multi-Objective\\nProblems . Springer US.\\nKalyanmoy Deb. 2001. Multi-Objective Optimization using Evolutionary Algorit hms. John Wiley & Sons.\\nKalyanmoy Deb, Samir Agrawal, Amrit Pratap, and T. Meyariva n. 2002. A fast and elitist multiobjective genetic\\nalgorithm: NSGA-II. IEEE Trans. Evol. Comput. 6, 2 (2002), 182\\u2013197.\\nBenjamin Doerr. 2020. Probabilistic Tools for the Analysis of Randomized Optimiz ation Heuristics . Springer, Chap-\\nter 1, 1\\u201387.\\nBenjamin Doerr, Carola Doerr, Aneta Neumann, Frank Neumann , and Andrew M. Sutton. 2020. Optimization of\\nChance-Constrained Submodular Functions. In The Thirty-Fourth AAAI Conference on Arti\\ufb01cial Intelligen ce, AAAI\\n2020 . AAAI Press, 1460\\u20131467.\\nBenjamin Doerr and Frank Neumann. 2021. A Survey on Recent Pr ogress in the Theory of Evolutionary Algorithms\\nfor Discrete Optimization. ACM Trans. Evol. Learn. Optim. 1, 4, Article 16 (oct 2021), 43 pages.\\nXinbo Geng and Le Xie. 2019. Data-driven decision making in p ower systems with probabilistic guarantees: Theory\\nand applications of chance-constrained optimization. Annual Reviews in Control 47 (2019), 341\\u2013363.\\nO. Giel. 2003. Expected runtimes of a simple multi-objectiv e evolutionary algorithm. In The 2003 Congress on\\nEvolutionary Computation, 2003. CEC \\u201903. , V ol. 3. 1918\\u20131925 V ol.3.\\nFangguo He and Guiming Shao. 2009. An Evolutionary Algorith m for Uncertain Optimization Problems. In 2009\\nInternational Conference on Information Engineering and C omputer Science . IEEE, 1\\u20134.\\nKonstantinos Kepaptsoglou, Grigorios Fountas, and Matthe w G. Karlaftis. 2015. Weather impact on containership\\nrouting in closed seas. Transportation Research Part C: Emerging Technologies 55 (2015), 139\\u2013155.\\nBo Liu, Qingfu Zhang, Francisco V . Fern\\u00e1ndez, and Georges G. E. Gielen. 2013. An Ef\\ufb01cient Evolutionary Algorithm\\nfor Chance-Constrained Bi-Objective Stochastic Optimiza tion. IEEE Trans. Evol. Comput. 17, 6 (2013), 786\\u2013796.\\nDaniel H. Loughlin and S. Ranji Ranjithan. 1999. Chance-Con strained Genetic Algorithms. In GECCO \\u201999 . Morgan\\nKaufmann Publishers Inc., 369\\u2013376.\\nKazuyuki Masutomi, Yuichi Nagata, , and Isao Ono. 2013. An Ev olutionary Algorithm for Black-Box Chance-\\nConstrained Function Optimization. Journal of Advanced Computational Intelligence and Intell igent Informatics\\n17, 2 (2013), 272\\u2013282.\\nMehrnaz Mohtasham, Hossein Mirzaei-Nasirabad, and Behroo z Alizadeh. 2021. Optimization of truck-shovel alloca-\\ntion in open-pit mines under uncertainty: a chance-constra ined goal programming approach. Mining Technology\\n130 (2021), 81\\u2013100.\\nAneta Neumann and Frank Neumann. 2020. Optimising Monotone Chance-Constrained Submodular Functions Using\\nEvolutionary Multi-objective Algorithms. In Parallel Problem Solving from Nature - PPSN XVI - 16th Intern ational\\nConference, PPSN 2020, Proceedings, Part I (Lecture Notes i n Computer Science, Vol. 12269) . Springer, 404\\u2013417.\\nAneta Neumann, Yue Xie, and Frank Neumann. 2022. Evolutiona ry Algorithms for Limiting the Effect of Uncertainty\\nfor the Knapsack Problem with Stochastic Pro\\ufb01ts. In PPSN XVII . Springer, Cham, 294\\u2013307.\\nFrank Neumann, Mojgan Pourhassan, and Vahid Roostapour. 20 20.Analysis of Evolutionary Algorithms in Dynamic\\nand Stochastic Environments . Springer, Chapter 7, 323\\u2013357.\\nFrank Neumann and Andrew M. Sutton. 2019. Runtime Analysis o f the (1 + 1) Evolutionary Algorithm for the\\nChance-Constrained Knapsack Problem. In FOGA \\u201919 (FOGA \\u201919) . ACM, 147\\u2013153.\\nFrank Neumann and Carsten Witt. 2022. Runtime Analysis of Si ngle- and Multi-Objective Evolutionary Algorithms Chance-Constrained Knapsack Problem. In FOGA \\u201919 (FOGA \\u201919) . ACM, 147\\u2013153.\\nFrank Neumann and Carsten Witt. 2022. Runtime Analysis of Si ngle- and Multi-Objective Evolutionary Algorithms\\nfor Chance Constrained Optimization Problems with Normall y Distributed Random Variables. In IJCAI-22 . 4800\\u2013\\n4806.\\nShen Peng. 2019. Chance constrained problem and its applications . Theses. Universit\\u00e9 Paris Saclay (COmUE) ; Xi\\u2019an\\nJiaotong University.\\nFeng Shi, Xiankun Yan, and Frank Neumann. 2022. Runtime Anal ysis of Simple Evolutionary Algorithms for the\\nChance-Constrained Makespan Scheduling Problem. In PPSN XVII . Springer, 526\\u2013541.\\n13 Evolutionary Multi-Objective Algorithms for the Knapsack Problems with Stochastic Pro\\ufb01ts A P REPRINT\\nHemant Kumar Singh and J\\u00fcrgen Branke. 2022. Identifying Sto chastically Non-dominated Solutions Using Evolu-\\ntionary Computation. In PPSN (2) (Lecture Notes in Computer Science, Vol. 13399) . Springer, 193\\u2013206.\\nYue Xie, Oscar Harper, Hirad Assimi, Aneta Neumann, and Fran k Neumann. 2019. Evolutionary Algorithms for the\\nChance-Constrained Knapsack Problem. In GECCO \\u201919 . ACM, 338\\u2013346.\\nYue Xie, Aneta Neumann, and Frank Neumann. 2020. Speci\\ufb01c Sin gle- and Multi-Objective Evolutionary Algorithms\\nfor the Chance-Constrained Knapsack Problem. In GECCO \\u201920 . ACM, 271\\u2013279.\\nYue Xie, Aneta Neumann, and Frank Neumann. 2021a. Heuristic Strategies for Solving Complex Interacting Stockpile\\nBlending Problem with Chance Constraints. In GECCO \\u201921 . ACM, 1079\\u20131087.\\nYue Xie, Aneta Neumann, Frank Neumann, and Andrew M. Sutton. 2021b. Runtime Analysis of RLS and the\\n(1+1) EA for the Chance-Constrained Knapsack Problem with C orrelated Uniform Weights. In GECCO \\u201921 . ACM,\\n1187\\u20131194.\\n14\"},\"summary\":{\"0\":\" This paper proposes the Transformer, a model architecture based on attention mechanisms that is superior in quality and requires less time to train than other models. It achieved a BLEU score of 28.4 on the WMT 2014 English-to-German translation task and a BLEU score of 41.8 on the WMT 2014 English-to-French translation task. The Transformer was also successfully applied to English constituency parsing and other tasks. The paper also discusses the attention mechanism, computational complexity, parallelization, and maximum path length of different layer types.\",\"1\":\" This paper examines the use of biophysical synapses in Artificial Neural Networks (ANNs) and Long-Term Context (LTC) networks. It is argued that biophysical synapses allow for more parameters to be packed into a given number of neurons and synapses, and that the nonlinear transformations can be formulated as a linear system with state-dependent coefficients. The paper provides evidence for these claims on various time-series prediction tasks, and suggests that the results are applicable to any feedforward or recurrent ANN. It also discusses the use of NeuralODEs, Mujoco, and the State-Dependent Riccati Equation (SDRE) Control, as well as the torchdyn package and the LISI type of RNN.\",\"2\":\" This paper presents MOREA, a GPU-accelerated Evolutionary Algorithm for Multi-Objective Deformable Registration of 3D Medical Images. Experiments on 4 cervical cancer patient scans show that MOREA outperforms two state-of-the-art approaches in terms of contour registration accuracy. Additionally, the paper introduces a fold detection method, a radial-basis-function approach to introduce large deformations free of mesh folds, and a repair method to reverse folds on a point-by-point basis. Results indicate that the heterogeneous elasticity model generally receives higher Dice scores and similar Hausdorff 95th percentiles, and that contour-based strategies perform better than random mesh point placement.\",\"3\":\" This paper investigates the use of self-adaptive mutation techniques in multi-objective evolutionary algorithms (EAs) to accelerate the convergence process. It tests three self-adaptive mutation techniques on the OneMinMax, COCZ, LOTZ, and OneJumpZeroJump problems, and finds that the choice of performance metrics significantly affects the performance of the self-adaptive algorithms. Additionally, the paper discusses the runtime analysis of the Simple Evolutionary Multi-Objective Optimizer (SEMO) and the Global Simple Evolutionary Multi-Objective Optimizer (GSEMO), and examines the impact of metrics for self-adaptation in the GSEMO algorithm. Results indicate potential benefits of self-adaptive mutation for GSEMO, and that self-adaptive algorithms can benefit from large population sizes.\",\"4\":\" This paper examines the use of affine combinations of BBOB problems for performance assessment. It investigates the effect of the affine combinations on the performance of five numerical black-box optimization algorithms, and explores the potential of a modified version of affine function combinations to give new insights into the way function landscapes work. It also reviews various methods for comparing continuous optimizers in a black-box setting, and presents the IEEE Symposium Series on Computational Intelligence (SSCI) 2021, BIAS: A Toolbox for Benchmarking Structural Bias in the Continuous Domain (Vermetten et al., 2022), Reproducibility files and additional figures (Vermetten et al., 2023), and IOHanalyzer: Detailed Performance Analysis for Iterative Optimization Heuristic (Wang et al., 2022).\",\"5\":\" This paper presents a method for converting an Artificial Neural Network (ANN) to a Spiking Neural Network (SNN) with high accuracy and ultra-low latency. It introduces a quantization clip-floor-shift activation function to replace the ReLU activation function in source ANNs, and evaluates the proposed method on CIFAR-10, CIFAR-100, and ImageNet datasets. Results show that the proposed method outperforms state-of-the-art ANN-SNN conversion methods in terms of accuracy and time-steps. Additionally, the paper discusses the use of uni-codes to represent various characters and symbols, and the use of a shift term to overcome the performance degradation problem when the time-steps and the quantization steps are not matched.\",\"6\":\" This article provides an overview of Evolutionary Reinforcement Learning (EvoRL), a field that has emerged to address challenges in tasks with deceptive rewards or continuous state and action spaces. It discusses six key research fields of RL, including hyperparameter optimization, policy search, exploration, reward shaping, meta-RL, and multi-objective RL, and elaborates on the use of Evolutionary Computation (EC) algorithms. It also examines the use of evolutionary algorithms and reinforcement learning for optimizing machine learning models, as well as the application of these strategies to reinforcement learning, large-scale black-box optimization, unstable systems, and strategic decision-making problems.\",\"7\":\" This paper proposes a new method called Random encoding of Aggregated DeepActivation Maps (RADAM) for texture recognition. It uses a pre-trained deep convolutional network and a Randomized Autoencoder (RAE) to extract rich texture representations without changing the backbone. Experiments show that RADAM outperforms Global Average Pooling (GAP) and other state-of-the-art methods on five challenging texture recognition datasets. The paper also compares the performance and cost of RADAM with ConvNeXt-T, B, L, and XL to ResNet18 and ResNet50, and provides code for the RADAM module and its internal mechanisms.\",\"8\":\" This paper presents a novel deep learning-based damage detection model, DenseSPH-YOLOv5, which integrates DenseNet blocks and convolutional block attention modules (CBAM) to improve feature extraction in complex and noisy environments. The model outperforms existing deep learning-based damage detection models in terms of accuracy and speed, achieving a detection rate of 62.4 FPS and a mean average precision (mAP) value of 85:25%, F1-score of 81:18%, and precision (P) value of 89:51%. It covers topics such as deep convolutional neural networks, Transformers for image recognition, YOLO networks, structured prediction, variational U-nets, and performance measures for classification.\",\"9\":\" This paper proposes a gradient-free method to generate naturalistic physical adversarial patches for object detectors using a pretrained generative adversarial network (GAN). The method works both digitally and physically and can be used to deceive deep learning systems by injecting carefully crafted inputs. Experiments were conducted using the Adam optimizer, BigGAN2, and two different object detectors (Tiny-YOLOv3 and Tiny-YOLOv4). Results showed that the patches generated by the proposed method were able to fool the object detectors, with a mean false negative rate of 21.9% and 40.4% for Tiny-YOLOv3 and Tiny-YOLOv4 respectively. This paper discusses various methods of adversarial attacks on deep learning visual classifiers, such as Wasserstein Generative Adversarial Networks (GANs), Adversarial Patch, Adversarial Camouflage, Robust Physical-World Attacks, Meta-Attack, Generative Adversarial Networks (GANs), Explaining and Harnessing Adversarial Examples, Denoising Diffusion Probabilistic Models, Naturalistic Physical Adversarial Patch, Adversarial Texture, Adversarial Attack and Defense of YOLO Detectors, A Style-\",\"10\":\" This paper presents DCG-MAP-Elites, a new algorithm that combines Multi-dimensional Archive of Phenotypic Elites (MAP-Elites) and Deep Reinforcement Learning (DRL) to find the highest-fitness solutions in a given descriptor space. The algorithm uses two variation operators, a standard GA operator and a descriptor-conditioned PG operator, to distill the knowledge of an archive into a single versatile policy. Experiments show that DCG-MAP-Elites outperforms the baselines on all metrics and tasks, but there is still room for improvement. The paper also provides the hyperparameters used for each task.\",\"11\":\" This paper presents the Multiplexed Gradient Descent (MGD) algorithm, a model-free perturbative technique that is orders of magnitude faster than backpropagation and can be used to train hardware platforms based on emerging technologies. It examines the use of perturbative techniques to train large and small hardware systems, such as photonic or memristive crossbar hardware, without redesigning the hardware. It also looks at various research papers on topics such as distributed learning for analog VLSI neural networks, FPGA implementations of pulse density neural networks, memristor-based neural networks, single chip photonic deep neural networks, and various learning rules and algorithms for spiking neural networks. This research was funded by NIST and University Colorado Boulder.\",\"12\":\"\\n\\nThis paper presents a novel approach to non-intrusive load monitoring (NILM) using evolutionary deep nets. The proposed method uses a genetic algorithm to optimize the structure of a deep neural network, allowing for improved accuracy and efficiency in NILM. Batch Normalization and Dropout are also discussed as methods to improve the accuracy and speed of deep neural network training. The results of the experiments show that the proposed method outperforms existing NILM techniques.\",\"13\":\" This paper presents Graph-based Symbolically Synthesized Neural Networks (G-SSNNs), a new approach to neural networks that combines the advantages of neural networks and symbolic systems to improve data efficiency and generalization. G-SSNNs use symbolic programs to inform the topology and parameters of the neural network, allowing for the development of compact and composable abstractions that can be repurposed for a variety of tasks. Experiments are conducted on the Prediction of Constant Color Pattern (PCCP) task and image and video super-resolution using efficient sub-pixel convolutional neural networks and plain VIT baselines for ImageNet-1K. Results demonstrate that G-SSNNs produce reliable patterns of improved generalization on tasks involving local and discrete features across high-dimensional media.\",\"14\":\" This paper presents an optimization framework based on Grammatical Evolution (GE) to efficiently find the best cache configurations for a given set of benchmark applications. Experiments on the Mediabench suite show that the proposed framework is able to improve performance by an average of 62% compared to a real-world baseline configuration, while reducing optimization runtime and efficiently storing evaluated caches. The framework is designed to optimize the cache memory of multimedia applications on the ARM9 processor family, and uses a performance and energy model to translate the number of hits and misses of benchmarks into execution times and energy consumption. The optimization process is based on a grammar to map a genotype to a cache configuration, and a fitness function to measure the performance and energy consumption of each individual configuration.\",\"15\":\" This paper explores the use of a hybrid machine-learning-optimization approach called COIL to optimize autonomous robot timings for last-mile delivery services. It compares COIL with a baseline genetic algorithm and investigates new methods for improving the speed and efficiency of COIL. Results show that COIL can find valid solutions where appropriate numbers of robots run simultaneously and that it can optimize 10% faster than the GA, making it a good choice for daily re-optimization of robots. The paper also reviews relevant research in autonomous robot routing and safety, and provides data comparing the performance of COIL and GA.\",\"16\":\" This paper presents Vectorial Genetic Programming (Vec-GP), an extension of Genetic Programming (GP) which allows vectors as input features. It introduces strategies for optimizing a window for aggregation functions, including random and guided sampling. Results indicate that the random sampling strategies perform better than the guided strategies, which can become stuck in local optima. The paper also compares the performance of random versus guided direction and range reduction for feature extraction, and suggests that simple random sampling is generally better than the guided strategies.\",\"17\":\" This paper proposes a new acceleration technique for Particle Swarm Optimization (PSO) using low-discrepancy sampling in the expanded dimensional space. It is supported by various grants and the corresponding authors are Feng Wu and Jun Yan. The technique reduces the error at each iteration and improves the convergence speed. The paper also compares the effects of five famous population initialization methods and discusses the error of the PSO algorithm. The paper further investigates why the advantages of using an LDS initialization cannot be preserved during the entire iteration of PSO. The paper also presents two versions of the LDSEDS technique, which is combined with PSO to generate two more effective algorithms. The performance of the different algorithms is measured using the metric of error tolerance. The paper also compares the performance of PSO and PSO-LDSEDS2 using 10D and 30D dimensional test functions. The paper concludes that the LDSEDS technique can significantly improve the convergence speed of PSO, with HWS and DES being the most robust samplings.\",\"18\":\" This paper explores the potential of emerging AI technologies to inspire the next generation of e-textiles. It examines nanotechnology and AI advancements in the e-textiles domain, focusing on the potential application of neuromorphic computing and spiking neural network inspired AI technologies. It looks at the use of AI-enabled technologies for production line fabric inspection and defect detection, as well as intelligent clustering and classification techniques adopted and utilized in the textiles industry. It also examines the use of nanotech fabric research advancements and the integration of electronic components into textiles. Finally, it looks at the use of AI and machine learning technologies to create a more sustainable digital supply chain and optimize textile manufacturing processes.\",\"19\":\" This paper examines the impact of different network structures, such as Fully-Connected, Random, Regular ring lattice graph, Small-World and Scale-Free, on the global dynamical activity of a system of coupled Kuramoto phase oscillators. It finds that in Scale-Free and Random networks, a sophisticated choice of nodes based on their eigenvector centrality and average shortest path length can lead to higher degrees of synchrony. The paper also discusses magnetoencephalographic functional connectivity in Alzheimer's disease and random graphs in nature, society, and the brain.\",\"20\":\" This paper reviews a variety of research papers that focus on the use of artificial neural networks in decision-making, image recognition, and counterfactual explanation generation. It proposes a method to generate counterfactuals for a neural network model by manipulating what it perceives regarding specific human-defined concepts. Through empirical evaluation, the proposed method is tested on different models to assess whether the performed manipulations are well interpreted by the models and to analyze how they react to them. This method could help humans to better interpret how neural networks reach their results and to understand the causes and effects of different scenarios.\",\"21\":\" This paper proposes Swim, a new activation function for reinforcement learning and robotics tasks. It is tested on MuJoCo's locomotion continuous control tasks and is used in conjunction with the TD3 algorithm. Results show that Swim is a state-of-the-art activation function for continuous control locomotion tasks and is recommended for use with TD3. The paper also evaluates the performance and efficiency of the Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm and the Swim activation function on four MuJoCo benchmark locomotion control environments. Results showed that Swim outperformed Swish in three out of four environments. Further work with Swim remains, such as testing and optimization on the GPU.\",\"22\":\" This paper provides an overview of Ensemble Reinforcement Learning (ERL), a combination of reinforcement learning and ensemble learning used to handle complex tasks. It discusses strategies used in ERL, related applications, and open questions to guide future exploration. It also provides a brief introduction to reinforcement learning and ensemble learning, and outlines various combinations of models and training algorithms used in ERL from 2008 to 2022. Additionally, it discusses strategies for designing ERL methods, and provides examples of its application in the energy and environment, IoT and cloud computing, finance, and other areas. Lastly, it outlines three open questions in ERL-based research and suggests potential future research directions.\",\"23\":\" This paper examines the use of diffusion models to generate images, similar to how a painter would, and explores the dynamics of the endpoint estimate, the low-dimensional latent manifold, and the effects of perturbations on image generation. It also discusses the similarities between the generative processes of diffusion models and GANs, and how the Woodbury matrix inversion identity can be used to project the outcome of the reverse diffusion onto the image manifold.\",\"24\":\" This paper proposes a Conflict-Driven Structural Learning (CDSL) ATPG algorithm which combines the efficient heuristics of modern SAT solvers with the SAT-based framework to address the efficiency problem. The algorithm builds conflict-based constraints on the circuit to prune the search space, and adopts conflict-driven decision rules to improve decision accuracy. Additionally, a conflict diagnosis approach is used to analyze the reason for low coverage debug and modify certain constraints to improve the test coverage rate. Extensive experimental results demonstrate the effectiveness and efficiency of the proposed CDSL algorithm.\",\"25\":\" This paper presents TopSpark, a novel methodology for reducing the latency and energy consumption of spiking neural networks (SNNs) while maintaining high accuracy. Experiments show that TopSpark can reduce latency by 3.9x and energy consumption by 3.5x (training) and 3.3x (inference) on average, while keeping accuracy within 2% of SNNs without timestep reduction. This work may enable low-latency and energy-efficient SNN training and inference for autonomous mobile agents\\/robots, including their efficient online learning process.\",\"26\":\" This paper reviews five studies on evolutionary multi-objective algorithms for knapsack problems with stochastic profits. It introduces multi-objective formulations of the problem and three bi-objective fitness evaluation methods, and evaluates them using two multi-objective evolutionary algorithms. It also introduces a filtering method to improve the quality of the final population. The effectiveness of the approaches is demonstrated on several benchmarks, showing that GSEMO+Filtering and NSGA-II methods outperform the GSEMO in most settings.\",\"27\":\" This paper examines the use of linear convolutional neural networks (CNNs) to discover the statistical structure of a dataset using the most dominant frequencies. It provides insight into why CNNs are sensitive to adversarial perturbations, shortcut learning, and why they tend to learn surface statistical regularities instead of higher level abstractions. Experiments are implemented with TensorFlow\\/Keras and the results are discussed.\",\"28\":\" This paper presents a novel learning rule for magnetic stochastic synapses, which allows for weights to be trained that operate better in the low sampling regime compared to the mean-field versions. It covers the use of magnetic stochastic synapses for energy-efficient neuromorphic devices, such as connectionist reinforcement learning, spike-based reinforcement learning, voltage control of domain walls, and more. It also references research studies related to nanomagnetic reservoir computing, reservoir computing with emergent dynamics in a magnetic metamaterial, physical reservoir computing with nanomagnetic devices, and a digital biologically plausible implementation of binarized neural networks with differential hafnium oxide resistive memory arrays. Lastly, the article references the MNIST database of handwritten digits.\",\"29\":\" This paper presents a novel reservoir computing system based on solitary-like waves propagating on the surface of a liquid film flowing over an inclined surface. The system is experimentally validated and is shown to be able to forecast chaotic time series and pass essential benchmark tests. It is hypothesized that the use of SL waves in the field of RC may be advantageous due to their unique nonlinear properties. The memory capacity of the SLRC system is an order of magnitude higher than that reported in a previous optical soliton RC system. The article also discusses the use of echo state networks and solitons in reservoir computing, as well as various research studies related to chaos, solitons, and fractals.\",\"30\":\" This paper proposes an evolutionary augmentation policy optimization (EAPO) approach for self-supervised learning. It uses a genetic algorithm to optimize the augmentation policies for four different self-supervised learning algorithms (BYOL, SimSiam, NNCLR, and SwA V). Results show that the proposed method can consistently improve the performance of SSL algorithms, with larger batch sizes having a larger impact on the improvement of the accuracy of downstream task classification. Additionally, the paper also presents two explainability methods to analyze the most influential operators for each dataset and self-supervised learning algorithms.\",\"31\":\" This paper presents Hindsight States (HiS), an off-policy reinforcement learning algorithm that combines Hindsight Experience Replay (HER) and HiS to achieve higher sample efficiency than either method alone. HiS is evaluated on a variety of tasks in simulation and on a real-world muscular robot task, showing improved sample efficiency and asymptotic performance. Results show that HiS can be beneficial for purely simulated tasks, and combining HER and HiS leads to even better performance than applying each method individually.\",\"32\":\" This paper presents a framework combining interactive evolution and large language models to simulate the typical human design process. It was evaluated on three game design tasks with 80 users, and implemented using the Telegram messaging platform and ChatGPT. The experiments received positive feedback from the participants and the framework can be applied to any design task which can be described in free-text form. The paper also references 10 sources related to evolutionary computation and presents a sequence of word clouds for a board game and video game design task.\",\"33\":\" This paper introduces PDSketch, a domain definition language that combines human-specified structural sparsity priors and machine learning of continuous and symbolic aspects of the model. Experiments show that PDSketch is efficient and effective in two domains: BabyAI and Painting Factory, enabling data-efficient learning, compositional generalization, and domain-independent heuristics. It uses first-order logic to represent decision rules, an assignment operation, two special operators, syntax sugars, and a blank notation. It also uses a two-layer graph neural network (GNN) and a basic task and motion planning strategy.\",\"34\":\" This paper presents a framework for learning rational subgoals from demonstrations and instructions, using A* search on FSM-augmented transition models. It compares the performance of RSGs with two baselines (IRL and Behavior Cloning) on primitive and compositional tasks, as well as novel tasks. It also discusses the use of Generative Adversarial Imitation Learning (GAIL) as a baseline for completing tasks on seen instructions. The results show that RSGs outperform all baselines on both the compositional and novel splits.\",\"35\":\" This paper presents the Knowledge-augmented Risk Assessment (KaRA) framework, a hybrid-intelligence system designed to support knowledge-intensive risk assessment of prospect candidates. KaRA combines AI techniques with structured domain knowledge and feedback from Subject Matter Experts (SMEs) to reduce bias and inconsistency in the assessment of the Probability of Success (PoS) of prospects. It also discusses GSA \\u2013 GeoScience Advisor, a web-based platform developed by Vital Brazil, E. et al. to help geoscientists make informed decisions. GSA provides access to a variety of data sources and tools to help users analyze and interpret geological data.\",\"36\":\" This paper presents a novel algorithm, Planning-Guided Transformer Decoding (PG-TD), which uses a planning algorithm to guide a pre-trained code generation Transformer to generate better codes. It is evaluated on competitive programming benchmarks and shows improved results compared to baseline methods, as well as enabling controllable code generation. The paper also discusses related works on using unit tests, reinforcement learning, and planning algorithms for code generation, as well as planning algorithms for natural language generation. However, this could also make it easier to develop malware, so a separate module may be needed to screen natural language descriptions and reject requests that could lead to harmful codes.\",\"37\":\" This paper presents TANGOS, a regularization method for deep tabular models that encourages specialization and orthogonalization of neurons by penalizing neuron attributions during training. Results show that TANGOS outperforms other popular regularizers and that combining it with other regularizers can further improve generalization of neural networks in the tabular setting. Experiments were conducted on 20 datasets and results showed that TANGOS can close the gap between deep learning and boosting methods. The paper also discusses the use of feature attribution methods in the training process, as well as the use of regularization methods to aid generalization in neural networks.\",\"38\":\" This paper examines the vulnerability of pre-trained architectures to watermarks in the ImageNet dataset, and proposes a method to reduce the dependence of deep neural networks on Chinese watermarks present in ImageNet by ignoring the most sensitive representations from the feature-extractor model. Results indicate that the sensitivity to watermarks is a common trait among all studied networks, and that Chinese watermarks had the highest average AUC ROC scores. The paper also discusses the TrustML-(un)Limited workshop accepted to ICLR 2023, and the dataset generation approach for the workshop.\",\"39\":\" This paper presents the Multiplexed Gradient Descent (MGD) algorithm, a model-free perturbative technique for training hardware platforms based on emerging technologies. It is orders of magnitude faster than backpropagation and can be adjusted during training. It is applicable to a wide range of systems and can be used in the presence of noise and device imperfections. The paper also reviews research on the development of analog VLSI neural networks, FPGA implementations, memristor-based neural networks, photonic deep neural networks, and other hardware systems, as well as various learning rules and algorithms for spiking neural networks.\",\"40\":\" This paper presents Graph-based Symbolically Synthesized Neural Networks (G-SSNNs), a new approach to neural networks which combines the advantages of neural networks and symbolic systems to improve data efficiency and generalization. G-SSNNs use symbolic programs to inform the topology and parameters of the neural network, allowing for the development of compact and composable abstractions that can be repurposed for a variety of tasks. Experiments are conducted on a binary prediction task over high-dimensional data and show improved generalization from small quantities of data compared to baseline models. The paper also discusses strategies for pre-training graph neural networks and the Deep Graph Library, Dropout, Adam, RAVEN, and AI Evaluation Beyond Metrics for conceptual abstraction benchmarks.\",\"41\":\" This paper presents DCG-MAP-Elites, a Quality-Diversity algorithm that combines deep reinforcement learning with MAP-Elites to generate collections of diverse and high-performing solutions. The algorithm uses a descriptor-conditioned critic and policy to improve the archive across the entire descriptor space. Results show that DCG-MAP-Elites outperforms the previous state-of-the-art, PGA-MAP-Elites, on omnidirectional tasks while maintaining similar performance on unidirectional tasks. The paper also provides hyperparameters for DCG-MAP-Elites and all baselines.\",\"42\":\" This paper presents a novel deep learning-based damage detection model, DenseSPH-YOLOv5, which integrates DenseNet blocks and convolutional block attention modules (CBAM) to improve feature extraction in complex and noisy environments. The model was tested on the RDD-2018 dataset and achieved superior results in terms of classification accuracy and localized bounding box prediction for all damage classes.\",\"43\":\" This paper explores the use of a hybrid machine-learning-optimization approach called COIL to optimize autonomous robot timings for last-mile delivery services. It compares COIL with a baseline genetic algorithm and investigates new methods for improving the speed and efficiency of COIL. Results show that COIL can find valid solutions where appropriate numbers of robots run simultaneously and that it can optimize 10% faster than the GA, making it a good choice for daily re-optimization of robots. The paper also discusses various methods of evolutionary computation, such as quality diversity, black-box optimization, constraint handling, and latent space models.\",\"44\":\" This paper presents Vectorial Genetic Programming (Vec-GP), an extension of Genetic Programming (GP) which allows vectors as input features. It introduces strategies for optimizing a window for aggregation functions, including random and guided sampling. Results indicate that the random sampling strategies perform better than the guided strategies, which can become stuck in local optima. The paper also compares the performance of random versus guided direction and range reduction for feature extraction, with random range being the better option in most cases.\",\"45\":\" This paper examines the impact of different network structures, such as Fully-Connected, Random, Regular ring lattice graph, Small-World and Scale-Free, on the global dynamical activity of a system of coupled Kuramoto phase oscillators. It finds that in Scale-Free and Random networks, a sophisticated choice of nodes based on their eigenvector centrality and average shortest path length can lead to higher degrees of synchrony. The paper also discusses magnetoencephalographic functional connectivity in Alzheimer's disease and random graphs in nature, society, and the brain.\",\"46\":\" This paper reviews five studies on evolutionary multi-objective algorithms for knapsack problems with stochastic profits. It introduces multi-objective formulations of the problem and three bi-objective fitness evaluation methods, and evaluates them using two multi-objective evolutionary algorithms. It also introduces a filtering method to improve the quality of the final population. The effectiveness of the approaches is demonstrated on several benchmarks, showing that GSEMO+Filtering and NSGA-II methods outperform the GSEMO for most settings.\"}}"