The design of a calendar should not  deviate from calendar interfaces that users are familiar with ( DG2 ) . This was observed in various aspects of the design such as layout, medication entries, and icons used to annotate entries such as those which should be taken with food. Over 80% of the participants preferred Design B because of its prob- able similitude to already existing calendars. This was surprising to us because Design A was the design that was intended to resemble ex- isting calendars. The sidelining of Design A can be attributed mainly to the height of medication entries which spanned the entire allowed administration period. This, according to participants, introduced too much clutter. Results indicate that the preferred layout should be vertically oriented with days of the week at the top and times of the day on the left. The dosage used on the medication entry should be one that users are familiar with. The unit used should also be consistent with the ones used in the prescriptions. It should show the actual quantity (e.g., milligrams) as opposed to relative classifications such as number of pills or tablets. This observation also applies to the icon used to denote  medication that should be taken with food.Realistic icons that are related to food should be used instead of custom designed icons. In this case, bananas are more effective in indicating the take-with-food action than an icon resembling a utensil such as spoon. Such icon should be positioned together with the entry and not as part of medication summaries.

The design of a calendar should not radically deviate from calendar interfaces that users are familiar with ( DG2 ) . This was observed in various aspects of the design such as layout, medication entries, and icons used to annotate entries such as those which should be taken with food. Over 80% of the participants preferred Design B because of its probable similarity to already existing calendars. This was surprising to us because Design A was the design that was intended to resemble existing calendars. The sidelining of Design A is attributed mainly to the height of medication entries which spanned the entire allowed administration period and introduced too much clutter. The results indicate that the preferred layout should be vertically oriented with days of the week at the top and times of the day on the left. The dosage used on the medication entry should be one that users are familiar with. The unit used should be consistent with the one used in the prescription. It should show the actual quantity (e.g., milligrams) as opposed to relative classifications such as number of pills or tablets. Similarly, realistic food-related icons (e.g., a banana) should be used to denote that medication that should be taken with food. Such icon should be positioned together with the entry and not as part of medication summaries.

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It is necessary to emphasize our difference with emerging  causal inference approaches based on optimal transport. Dunipace (2021) augments the IPS method via a propensity score estimator based on optimal transport; however, it is limited by the aforementioned high variance issue. Torous et al. (2021) uses the push forward operator to improve change-in-change models; however, they are designed for multi-phase data which is not available in our case. Li et al. (2022) shares similar settings to us, while they focuses on variable decomposition in latent space and is identical to Uri et al. (2017) in terms of alignment technology. Our contribution lies in investigating the role and flexibility of optimal transport to augment CFR, mitigating the MSE and UCE issues that have been long circumvented in the literature as recent as this year.

It is necessary to distinguish ourselves from emerging OT-based causal inference approaches. Dunipace (2021) augments the IPS method with a propensity score estimator based on OT; however, it is limited by the aforementioned high variance issue. Torous et al. (2021) uses the push-forward operator to improve change-in-change models; however, they are designed for multi-phase data which is not available in our case. Li et al. (2022) has a similar setup to to us, while it focuses on the decomposition of latent variables and is identical to Shalit et al. (2017) in terms of alignment technology. Our work is a new take on OT under the CFR framework, alleviating the MSE and UCE issues that have been long neglected by the causal inference community until this year.

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Training protocol. A fully connected neural network with two  hidden layers as 60-60 has been selected to realize the representation map ψ and the factual outcome map ϕ for ESCFR and other neural network based baselines. For fair comparison, all neural models are trained for 400 epochs with Adam Kingma & Ba (2015) optimizer, where both learning rate and weight decay are set to 1 × e − 3 . Other settings of optimizers follow Kingma & Ba (2015). We search hyperparameters within the range in Figure 5, checkpoint the validation performance every 2 epochs, and export the best model to evaluate its performance on the test dataset.

Training protocol. A fully connected neural network with two 60-dimensional hidden layers is selected to instantiate the representation mapping ψ and the factual outcome mapping ϕ for ESCFR and other neural network based baselines. To ensure a fair comparison, all neural models are trained for 400 epochs with Adam optimizer, with the learning rate and weight decay both set to 0.001. Other settings of optimizers follow Kingma & Ba (2015). We fine-tune hyperparameters within the range in Figure 5, validate performance every two epochs, and save the optimal model for test.

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The shortcomings of the discussed methods led us to consider the use of an entropy-based objective. In principle, the proposal entropy objective could be optimized for the L2HMC sampler with variational inference (Poole et al., 2019; Song & Ermon, 2019), but our preliminary experiments using this idea were not promising. Therefore, we take inspiration from the Normalizing Flow model and investigated tractable ways to optimize the proposal entropy directly.

The shortcomings of the discussed methods led us to consider the use of an entropy-based objective. However, L2HMC does not have tractable proposal probability p ( x (cid:48) | x ) , preventing the direct application of the entropy-based objective. In principle, the proposal entropy objective could be optimized for the L2HMC sampler with variational inference (Poole et al., 2019; Song & Ermon, 2019), but our preliminary experiments using this idea were not promising. Therefore, we designed our sampler that possess tractable proposal probability and investigated tractable optimization of the proposal entropy objective.


Whereas the superposition stack can be viewed as calculating expectations over individual stack elements, the Nondeterministic Stack RNN (NS-RNN) model of DuSell & Chiang (2020) maintains a probability distribution over whole stacks, using a weighted PDA. It has cubic time complexity and quadratic space complexity with respect to input length, leading to higher wall-clock run time than other stack RNNs but often better task performance.

The stack module in the Nondeterministic Stack RNN (NS-RNN) model of DuSell & Chiang (2020) maintains a probability distribution over whole stacks by simulating a weighted PDA. It has cubic time complexity and quadratic space complexity with respect to input length, leading to higher wall-clock run time than other stack RNNs, but often better task performance.

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Study design For each task introduced in §5.1,  we construct a dataset where we duplicate one of the modalities as the two  input modalities, in addition to the original dataset. For example,  we predict the digit class using two identical gray-scale images in the case of MNIST. We train a multi-modal DNN on each dataset of each task as explained below:

Study design For each task introduced in §5.1, in addition to the original dataset, we construct a dataset with two identical input modalities by copying one of the modalities. For example, when using the colored-and-gray-MNIST dataset, we predict the digit class using two identical gray-scale images. We train a multi-modal DNN on these dataset as explained below for each task:

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PDEs of the same family), and a challenging 2D mesh-based simulation of paper folding. In 1D, we show that our model outperforms state-of-the-art deep learning-based surrogate models in terms of long-term evolution error by up to 39.3%, and can adaptively tradeoff computation to improve long-term prediction error. On a 2D mesh-based simulation, our modelcan strategically choose appropriate edges to refine or coarsen, and outperforms a baseline without remeshing.

PDEs of the same family), and a challenging 2D mesh-based simulation of paper folding. In 1D, we show that our model outperforms state-of-the-art deep learning-based surrogate models in terms of long-term evolution error by 33.7%, and can adaptively tradeoff computation to improve long-term prediction error. On a 2D mesh-based simulation, our model outperforms state-of-the-art MeshGraphNets + classical Adaptive Mesh Refinement (AMR) in 2D mesh-based simulations.


We range the measures by their ARI score on every graph of the dataset. The rank is deﬁned as the position of the measure in this list, averaged over the dataset (see Table 2). It is important to note that the global leadership does not give a comprehensive advice on which measure is better to use, because for a particular graph, the global leader can perform worse than the others. Here we consider the entire LFR space, not just its zone corresponding to real graphs, so the ranking may differ from similar works.

We range the measures by their ARI score on every graph of the dataset. The rank is deﬁned as the position of the measure in this list, averaged over the dataset (see Table 2). It is important to note that the global leadership does not give a comprehensive advice on which measure is better to use, because for a particular graph, the global leader can perform worse than the others. Here we consider the entire LFR space, not just its zone corresponding to common real-world graphs, so the ranking may differ from those obtained for restricted settings.


The point cloud completion network is trained with Gridding Loss [44], which is a L1 distancebetween predicted G p = < V p , W p > and ground truth G gt = < V gt , W gt > 3D grids in N Gresolution. Gridding Loss bypasses the un-orderedness of point clouds and is evaluated on the 3D grid. The depth completion network is trained using log L 1 pair-wise loss which forces the pairs of pixels in the predicted depth to regress to similar values as the corresponding pairs in the ground truth depth [47]. Let G describe the set of pixels where the ground truth depth is non-zero, i and j are the pixel pairs, and y and y ∗ denote the ground truth and predicted depths, respectively. We express these two loss functions as:

The point cloud completion network is trained with Gridding Loss [44], which is a L1 distancebetween predicted G p = < V p , W p > and ground truth G gt = < V gt , W gt > 3D grids in N Gresolution. V = { v i } N 3 G i =1 is collection of all vertices in 3D grid and W = { w i } N 3 G i =1 is the weightscorresponding to each vertex. Gridding Loss bypasses the un-orderedness of point clouds and is evaluated on the 3D grid. The depth completion network is trained using log L 1 pair-wise loss which forces the pairs of pixels in the predicted depth to regress to similar values as the corresponding pairs in the ground truth depth [47]. Let G describe the set of pixels where the ground truth depth is non-zero, i and j are the pixel pairs, and y and y ∗ denote the ground truth and predicted depths, respectively. We express these two loss functions as:


This simple sampler tends to produce diverse batches similar to a k -DPP. As shown in Figure 1, switching between the two samplers does not affect the active learner’s statistical performance while greatly improving the computational performance. A thorough comparison on the running times and test accuracies of k - MEANS ++ and k -DPP basedgradient embedding samplingcan be found in Appendix G.

This simple sampler tends to produce diverse batches similar to a k -DPP. As shown in Figure 1, switching between the two samplers does not affect the active learner’s statistical performance while greatly improving the computational performance. Appendix G compares run time and test accuracy for both k - MEANS ++ and k -DPP based sampling in our proposed gradient space.

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Here, γ ∈ [0 , 1] is a parameter that controls how much our ReGAL’s RPN balances between the coverage of a token (i.e. how often it occurs) and its instance level importance. Low values of γ favor tokens with high coverage while high values of γ favor rules with high impact tokens without regard for coverage. We use γ = 0 . 6 for our experiments, which works well in practice.

Here, γ ∈ [0 , 1] is a parameter that controls how much our ReGAL’s RPN balances between the coverage of a token (i.e. how often it occurs) and its instance level importance. Low values of γ favor tokens with high coverage while high values of γ favor rules with high impact tokens without regard for coverage. Since the types of rules needed may differ as training progresses, we allow users to choose γ for each round of proposed rules. In practice, we ﬁnd that γ ∈ [0 . 5 , 0 . 9] tend to produce good rules.


Let { s k | s k ∈ Z 3 } Mk =0 denote a series of window sizes, where s 0 is the size of the query windowand s 1 ,...,M are the sizes of M successively larger key windows. Let V = { v i | v i = ( x i , f i ) } |V| i =1 bethe input voxel set, with xyz coordinates x i ∈ Z 3 and feature vector f i ∈ R C for voxel i . We firstpartition the voxel set into non-overlapping 3D windowseach of size s 0, and find the non-emptyones as query windows with their centers denoted by { c i | c i ∈ Z 3 } Li =0 , where L is the total numberof query windows. To get query  voxels V c i , s 0for the query window centered on c i, one can simplygather all the non-empty voxels within the window  as the queries. While keeping efficiency in mind,we present a novel chessboard samplingstrategy, which will be detailed in Section 3.1.2.

Given a series of window sizes { s k | s k ∈ Z 3 } Mk =0 , where s 0 denotes the query window size ands 1 ,...,M denote M different key window sizes from small to large. Let V = { v i | v i = ( x i , f i ) } |V| i = 1be the input voxel set, where x i ∈ Z 3 denotes voxel coordinates and f i ∈ R C denotes voxel features. We first partition the voxel set into non-overlap 3D windows s 0 by finding non-empty window centers{ c i | c i ∈ Z 3 } Li =0 , where L is the number of non-empty windows. To get query voxels, we can simplygather all the non-empty voxels V c i , s 0 centered on c i within query window s 0 to guarantee that everyvoxel can serve as a query and be updated after one attention layer. We also prsent another novelchessboard sampling strategy for the query voxel sampling which will be discussed in Section 3.2.

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• Mask-Output (No  representations, No input action set): Assuming a ﬁxed action space in the output, q-values or policy probabilities are masked out for unavailable actions. It represents prior SAS-MDP works - Boutilier et al. ; Chandak et al. (2020a), Huang & Ontañón (2020). • Mask-Input-Output (No  representations): Augments the binary availability mask of given actions, 1s for available  indices and 0s for unavailable, to the state input of Mask-Output . • Utility-Policy (No input action set): Proposed by Jain et al. (2020), action representations are used independently for computing each action’s utility, ignoring any  interdependence. • Simple DQN (No cascade, No input action set): For listwise RL speciﬁcally, we include the DQN baseline  that simply selects top-K items, instead of reasoning about the overall list. Thus, it ignores both action interdependences: (i) on other available actions, (ii) on other items in the list.

• Mask-Output (No action representations, No input action set): Assumes a ﬁxed action space output. Q-values or policy probabilities are masked out for unavailable actions. It represents prior work on SAS-MDP: Boutilier et al. ; Chandak et al. (2020a) ; Huang & Ontañón (2020). • Mask-Input-Output (No action representations): Augments Mask-Output with an extra input of action set via a binary availability vector: having 1s at available action indices and 0s otherwise. • Utility-Policy (No input action set): Jain et al. (2020) propose a parallel architecture to compute each action’s utility using action representations. But, it ignores any action interdependence. • Simple DQN (No cascade, No input action set): A non-cascaded DQN baseline for listwise RL that selects top-K items instead of reasoning about the entire list. Thus, it ignores two action interdependences: (i) on other items in the list and (ii) on other available actions.

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• No modiﬁcation ( p ). This is a baseline experiment where we keep the original strings as are • Text Truncation ( p short ). We truncate texts to 1 / 3 of their original length. • Article Removal ( p no art ). We remove all articles (i e., ‘a’, ‘an’ and ‘the’). • Stopwords Removal ( p no stop ). We remove all stopwords (e g., ‘that’ or ‘so’).• Sentence-level Permutation ( p swap ). We permute the ﬁrst halves of texts (as delineated by sentences) across the entire corpus (i

• No modiﬁcation ( p ). This is a baseline experiment where we keep the original strings as are. • Text Truncation ( p short ). We truncate texts to 1 / 3 of their original length. This allows us to understand whether the divergence metrics pick up on differences in dataset length statistics. • Article Removal ( p no art ). We remove all articles (‘a’, ‘an’ and ‘the’) in the text. This allows us to understand whether the divergence metrics can distinguish between texts with or without basic levels of ﬂuency and grammaticality. • Stopwords Removal ( p no stop ). We remove all stopwords (e.g., ‘that’ or ‘so’) in the text. This allows us to understand whether the divergence metrics can detect differing levels of syntactic coherence, rather than just focusing on content words.• Sentence-level Permutation ( p swap ). We permute the ﬁrst halves of texts (as delineated by sentences) across the entire corpus (i


STL [7] offers a formalism for expressing and reasoning among a rich set of rules. The rules aredeﬁned over predicates (inequalities of real-valued functions). In standard STL, the formulas areforward-looking, meaning that the formulas are evaluated by looking at future trajectories. In thispaper, we need to evaluate formulas using past trajectories (trajectories of road agents we havetracked over the past). Therefore, our rules will be encoded as a set of past time STL (ptSTL) [8] formulas with the following syntax

STL [7] offers a formalism for expressing and reasoning among a rich set of rules. The rules aredeﬁned over predicates (inequalities of real-valued functions). In standard STL, the formulas areforward-looking, meaning that the formulas are evaluated by looking at future trajectories. In thiswork, we need to evaluate formulas using past trajectories (trajectories of road agents we havetracked over the past), using this past information to compute a risk of violating these rules in thefuture. Therefore, our rules will be encoded as a set of past time STL (ptSTL) [8] formulas with the following syntax


GBDT (Chen and Guestrin, 2016) is the widely-used tree model. DeepFM (Guo et al., 2017) is theinner-product interaction-based model. FATE (Wu et al., 2021) and TabGNN (Guo et al., 2021b) arethe recent graph models. DIN (Zhou et al., 2018) and DIEN (Zhou et al., 2019) are the attentionbased sequential models. SIM (Pi et al., 2020), UBR (Qin et al., 2020), and RIM (Qin et al., 2021) are the retrieval-based models. On top-n recommendation tasks, we compare PET with six strongrecommendation models, including factorization-based FPMC (Rendle et al., 2010) and TransRec (He et al., 2017), and recently proposed DNN models NARM (Li et al., 2017), GRU4Rec (Hidasiet al., 2016), SASRec (Kang and McAuley, 2018), and RIM (Qin et al., 2021).

GBDT (Chen and Guestrin, 2016) is the widely-used tree model. DeepFM (Guo et al., 2017) is theinner-product interaction-based model. DIN (Zhou et al., 2018) and DIEN (Zhou et al., 2019) arethe attention-based sequential models. SIM (Pi et al., 2020), UBR (Qin et al., 2020), and RIM (Qin et al., 2021) are the retrieval-based models. On top-n recommendation tasks, we compare PET withsix strong recommendation models, including factorization-based FPMC (Rendle et al., 2010) and TransRec (He et al., 2017), and recently proposed DNN models NARM (Li et al., 2017), GRU4Rec(Hidasi et al., 2016), SASRec (Kang and McAuley, 2018), and RIM (Qin et al., 2021).


In fact, this biological implausibility is also an important issue for machine intelligence. For their impressive performance, ANNs trade off other desired properties, which are present in biological systems. For example, ANN training often demands very large and labelled datasets. When labels are unavailable, self-supervised learning schemes exist, where supervisory error signals generated by the network itself are exploited and backpropagated from the output towards the input to update the network’s parameters (Goodfellow et al., 2014; Devlin et al., 2018; Chen et al., 2020). However, this global propagation of signals in deep networks introduces another limitation. Namely, it prevents the implementation of efﬁcient distributed computing hardware that would be based on only local signals from neighbouring physical nodes in the network, and is in contrast to the local synaptic plasticity rules that are believed to govern biological learning. Several pieces of work have been addressing parts of the biological implausibility and drawbacks of backpropagation in ANNs (Bengio et al., 2015; Lillicrap et al., 2016; Guerguiev et al., 2017; Pfeiffer & Pfeil, 2018; Illing et al., 2019; Pogodin & Latham, 2020; Millidge et al., 2020; Pogodin et al., 2021). Recently, an approximation to backpropagation that is mostly Hebbian, i.e. relies on mostly pre- and post-synaptic activity of each synapse, has been achieved by reducing the global error requirements to 1-bit information (Pogodin & Latham, 2020). Two schemes that further localize the signal that is required for a weight update are Equilibrium Propagation (Scellier & Bengio, 2017) and Predictive Coding (Millidge et al., 2020).

In fact, this biological implausibility is also an important issue for machine intelligence. For their impressive performance, ANNs trade off other desired properties, which are present in biological systems. For example, ANN training often demands very large and labelled datasets. When labels are unavailable, self-supervised learning schemes exist, where supervisory error signals generated by the network itself are exploited and backpropagated from the output towards the input to update the network’s parameters (Goodfellow et al., 2014; Devlin et al., 2018; Chen et al., 2020). However, this global propagation of signals in deep networks introduces another limitation. Namely, it prevents the implementation of efﬁcient distributed computing hardware that would be based on only local signals from neighbouring physical nodes in the network, and is in contrast to local synaptic plasticity rules that partly govern biological learning. Several pieces of work have been addressing parts of the biological implausibility and hardware-inefﬁciency of backpropagation in ANNs (Bengio et al., 2015; Lillicrap et al., 2016; Guerguiev et al., 2017; Pfeiffer & Pfeil, 2018; Illing et al., 2019; Pogodin & Latham, 2020; Millidge et al., 2020; Pogodin et al., 2021). such as the need for exactly symmetric forward and backward weights or the waiting time caused by the network’s forward-backward pass between two training updates in a layer (weight transport and update-locking problems). Recently, an approximation to backpropagation that is mostly Hebbian, i.e. relies on mostly pre- and post- synaptic activity of each synapse, has been achieved by reducing the global error requirements to 1-bit information (Pogodin & Latham, 2020). Two schemes that further localize the signal that is required for a weight update are Equilibrium Propagation (Scellier & Bengio, 2017) and Predictive Coding (Millidge et al., 2020).


Due to space limitations, we can only summarize the results here. The full results can be found in Table 2  in Sec. C of the appendix. The code (both Java and Python) to reproduce these results is publicly available 1 .

Due to space limitations, Fig. 2 only summarizes the results very superﬁcially through ﬁnal ranks. More detailed results can be found in Table 2 and Fig. 4 in Sec. C of the appendix. The code (both Java and Python) to reproduce these results is publicly available 1 .


Our goal is to design a policy framework that is optimal for any given action space by addressing the challenges in Sec. 3.2. To work with action representations, we build on the utility network proposed by Jain et al. Our key insight is to use graph neural networks for both, summarizing the list of given action representations  and learning inter-action interdependence.

Our goal is to design a policy framework that is optimal for any given action set by addressing the challenges in Sec. 3.2. We build on the utility network proposed by Jain et al. (2020) that acts in parallel on each action’s representation. Our central insight is to use graph neural networks for summarizing the set of action representations as a state component and learning action relations.

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Proactive Motion Capture Few previous works studied proactive motion capture with a single mobile camera (Zhou et al., 2018; Cheng et al., 2018; Kiciroglu et al., 2019). In comparison, more works studied the control of a multi-camera team. Among them, many are based on optimization with various system designs, including marker-based (Nägeli et al., 2018), RGBD-based (Xu et al., 2017), two-stage system (Saini et al., 2019; Tallamraju et al., 2019), hierarchical system (Ho et al., 2021), etc . It is important to note that all the above methods deal with static occlusion sources or clean landscapes. Also, the common similarity shared by these works is the use of hand-crafted optimization objectives and some fixed-form camera formations. These factors resulted in poor adaptability to dynamic scenes saturated with uncertainties. Recently, RL-based methods received more attention due to their potential for dynamic formation adjustments. These works focused on active 3D HPE in the Gazebo simulation (Tallamraju et al., 2020) or Panoptic dome (Joo et al., 2015; Pirinen et al., 2019; Gärtner et al., 2020) for active view selection. Among them, AirCapRL (Tallamraju et al., 2020) shares similarities with our work. However, it is restricted to coordinating between two cameras in clean landscapes without occlusion. We study the collaboration between more cameras  and resolve the credit assignment issue with  novel CTCR incentives. Meanwhile, we consider a more challenging scenario with multiple distracting humans  as  dynamic occlusions, which requires dedicated algorithms to handle.

Proactive Motion Capture Few previous works studied proactive motion capture with a single mobile camera (Zhou et al., 2018; Cheng et al., 2018; Kiciroglu et al., 2019). In comparison, more works studied the control of a multi-camera team. Among them, many are based on optimization with various system designs, including marker-based (Nägeli et al., 2018), RGBD-based (Xu et al., 2017), two-stage system (Saini et al., 2019; Tallamraju et al., 2019), hierarchical system (Ho et al., 2021), etc . It is important to note that all the above methods deal with static occlusion sources or clean landscapes. Also, the majority of these works adopt hand-crafted optimization objectives and some forms of fixed camera formations. These factors resulted in poor adaptability to dynamic scenes saturated with uncertainties. Recently, RL-based methods receive more attentions due to their potentials on dynamic formation adjustments. These works studied active 3D HPE in the Gazebo simulation (Tallamraju et al., 2020) or Panoptic dome (Joo et al., 2015; Pirinen et al., 2019; Gärtner et al., 2020) for active view selection. Among them, AirCapRL (Tallamraju et al., 2020) shares similarities with our work. However, it is restricted to coordinating between two cameras in clean landscapes without occlusions. We study the collaborations between multiple cameras ( n ≥ 3) and resolve the credit assignment issue with our novel reward design (CTCR). Meanwhile, we study a more challenging scenario with multiple distracting humans served as the sources of dynamic occlusions, which requires a more sophisticated algorithms to handle.

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Visual quality comparison and Ablation study. Fig.shows that our results are well aligned with image boundary and visually close to fully-supervised counterpart. Fig. 7 shows signiﬁcant improvement of our results by additively introducing different relationships for more regularization. Please refer to the Appendix for more details and ablation studies.

Visual quality and ablation study. Fig. 6 shows that our results are better aligned with region boundaries and visually closer to fully-supervised counterparts. Fig. 7 shows that our results improve signiﬁcantly with different relationships for more regularization. See Appendix for more details and ablation studies.

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Self-supervised Learning for Point Clouds. 3D representation learning without annotations has been widely studied in recent years. Mainstream methods mainly build the pretext tasks to reconstruct the transformed input point cloud based on the encoded latent vectors, such as rotation [27], deformation [1], rearranged parts [35] and occlusion [39]. From another perspective, PointContrast [44]utilizes contrastive learning between features of the same points from different views to learn discriminative 3D representations. DepthContrast [50] further extends the contrast for depth maps of different augmentations. CrossPoint [2] conducts cross-modality contrastive learning between point clouds and their corresponding rendering images to acquire rich self-supervised signals. Point-BERT [48]first introduces BERT-style  pre-training  for 3D point clouds witha standard transformer network andperforms competitively on various downstream tasks. In this paper, we propose  an MAE-style [18]pre-training framework, Point-M2AE, which reconstructs the highly masked 3D coordinates of theinput point cloud for self-supervised learning. Point-M2AE with a hierarchical architecture achievesstate-of-the-art downstream performance by learning the multi-scale representation of point clouds.

Self-supervised Learning for Point Clouds. 3D representation learning without annotations has been widely studied in recent years. Mainstream methods mainly build the pretext tasks to reconstruct the transformed input point cloud based on the encoded latent vectors, such as rotation [34], deformation [1], rearranged parts [42] and occlusion [49]. From another perspective, PointContrast [55] utilizes contrastive learning between features of the same points from different views to learn discriminative 3D representations. DepthContrast [63] further extends the contrast for depth maps of different augmentations. CrossPoint [2] conducts cross-modality contrastive learning between point clouds and their corresponding rendering images to acquire rich self-supervised signals. Point-BERT [60] and Point-MAE [33] respectively introduce BERT-style [11] and MAE-style [20] pre-training schemes for 3D point clouds with standard transformer networks and performs competitively on various downstream tasks, but both of them can only encode point clouds with a single resolution and ignores the local-global relations between 3D shapes. In this paper, we propose Point-M2AE, an MAE-style framework with a hierarchical transformer for multi-scale point cloud pre-training. We achieve state-of-the-art downstream performance by learning the multi-scale representation of point clouds.


In this research, we focus on i) visualizing a single patient’s data, and ii) visualizing events on a calendar – a two-dimensional chart where one dimension shows days of a particular year and the other dimension shows  time of day [43]. Therefore we then turn to work on on-calendar visualization.

In this research, we focus on i) visualizing a single patient’s data, and ii) visualizing events on typical calendar layouts, which often consist of two-dimensional charts where one dimension shows days and the other dimension shows the time of day [43]. Alternative layouts exist and we discuss them in the next subsection.


We use the stochastic method to estimate the entropy. First, we sample a set of rotamers from the distribution using the inversion of the flows (Eq.1, 4). Then, we compute the negative log probability of the samples and take the average as an estimation of the entropy. These two steps can be efficiently done thanks to the capability of computing the exact likelihood of normalizing flows.

To estimate the entropy, we use a stochastic method: First, we sample a set of rotamers from the distribution using the inverted flows (Eq.1, 4). Then, we compute the negative log probability of the samples and take their average as an estimate of the entropy. Computing these steps is efficient thanks to the ability to compute the exact likelihood of normalizing flows.

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We have developed a novel influence analysis to understand the effects of graph elements on the parameter changes of GCNs without needing to retrain the GCNs. We chose Simple Graph Convolution due to its convexity and its competitive performance to non-linear GNNs on a variety of tasks. Our influence functions can be used to approximate the changes in model parameters caused by edge or node removals from an attributed graph. Moreover, we provided theoretical bounds on the estimation error of the edge and node influence on model parameters. We experimentally validated the accuracy and effectiveness of our influence functions by comparing its estimation with the actual influence obtained by model retraining. We showed in our experiments that our influence functions could be used to reliably identify edge and node with negative and positive influences on model performance. Finally, we demonstrated that our influence function could be applied to graph rectification and model attacks. A P ROOFS Lemma 3.1.

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DETR [6] applies ﬁxed positional encodings [26, 4] to the input of the transformer architecturefor object detection. Swin Transformer [24] adds relative position biases in similarity computation of self-attention, improving performance over models without these biases. LoFTR [35] uses the2D extension of the position encoding to produce position-dependent features for image matching.

DETR [ ﬁxed positional encodings [ the input of the transformer architecture position of self-attention, improving performance over models without these biases. LoFTR [35] uses the2D extension of the position encoding to produce position-dependent features for image matching.


The movement distance ( A ) from the starting position to the center of the target was 100 and 200 mm . We set the A values such that the interval between ID s in Eq. 1 was approximately constant. The interval ( I ) between the notch and the target was  0, 3.41, 6.28, 11.9, and ∞ mm . We set the I values with reference to the  previous study [23]. Note that  ∞ mm indicated a condition of no notch.

The movement amplitude ( A ) from the starting position to the center of the target was 100 and 200 mm (364 and 729 pixels). We set the A values such that the interval between ID s in Eq. 1 was approximately constant. The interval ( I ) between the notch and the target was set as 0, 3.41, 6.28, 11.9, and ∞ mm (0, 12, 23, 44, and ∞ pixels). We set the I values with reference to the values of the previous study that investigated the interval between the distractor and the target [25]. Note that an interval of ∞ mm indicates a condition with no notch.


In addition, these results only hold asymptotically, where the function of interest can be exactly identified in a pointwise manner. Such an overly strong guarantee is unrealistic in the finite-sample regime, where one can only hope to approximate the function well in an average sense under some distribution , i.e., finite-sample performance guarantees should ideally bound ∥ (cid:98) q − q π ∥ 2 ,ν for the learned (cid:98) q , where ∥·∥ 2 ,ν is ν -weighted 2-norm. Such fine-grained analyses are non-existent in MIS. Even in the broader literature, such results not only require Bellman-completeness-type assumptions (Uehara et al., 2021), they also come with some fixed ν (which is not necessarily d D ; see Section 2) and the user has no freedom in choosing ν . This creates a gap in the literature, as downstream learning algorithms that use off-policy function estimation as a subroutine often assume the estimation to be accurate under certain specific distributions (Kakade & Langford, 2002; Abbasi-Yadkori et al., 2019).

In addition, these results only hold asymptotically, where the function of interest can be exactly identified in a pointwise manner. Such an overly strong guarantee is unrealistic in the finite-sample regime, where one can only hope to approximate the function well in an average sense under some distribution , i.e., finite-sample performance guarantees should ideally bound ∥ (cid:98) q − q π ∥ 2 ,ν for the learned (cid:98) q , where ∥·∥ 2 ,ν is ν -weighted 2-norm. Such fine-grained analyses are non-existent in MIS. Even in the broader literature, such results not only require Bellman-completeness-type assumptions (Uehara et al., 2021), they also come with some fixed ν (which is not necessarily d D ; see Section 2) and the user has no freedom in choosing ν . This creates a gap in the literature, as downstream learning algorithms that use offpolicy function estimation as a subroutine often assume the estimation to be accurate under certain specific distributions. For example, in the setting of online policy optimization, Abbasi-Yadkori et al. (2019) require value estimates to be accurate on the occupancy of the (unknown) optimal policy, and (Kakade & Langford, 2002) require them to be accurate on the occupancy of the learning policy at each iteration.


Social network models have been applied for many purposes to include: modeling an individual’s behavioral patterns to predict future nodal attributes (e.g., connections) over time McConnell et al.[2018] McAvoy et al. [2020], modeling interactions and cluster formation within online communities Fortunato and Hric [2016] Xu et al. Liu et al. [2018], and modeling how network characteristics (e.g., centrality) influences its users Qiu et al. Overall, these models attempt to characterize the relationship amongst network structure and information diffusion, decision making, and individual behavior. Jackson et al.

Social network models have been applied for many purposes to include: modeling an individual’s behavioral patterns to predict future nodal attributes (e.g., connections) over time [McConnell et al., 2018] [McAvoy et al., 2020], modeling interactions and cluster formation within online communities [Fortunato and Hric, 2016] [Xu et al., 2020] [Liu et al., 2018], and modeling how network characteristics (e.g., centrality) influences its users [Qiu et al., 2017]. Overall, these models attempt to characterize the relationship amongst network structure and information diffusion, decision making, and individual behavior. [Jackson et al., 2017].

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Intuitively, ˆΘ( ϑ ) measures the trigger generator’s trainability with respect to a randomly initialized target model. The generator’s trainability indicates the easiness of effectively generating input-aware triggers, implying the model’s vulnerability to input-aware backdoor attacks. To verify the hypothesis, on the CIFAR10 dataset with the generator configured as in Appendix § A, we measure ˆΘ( ϑ ) with respect to 900 randomly generated arches as well as the model accuracy (ACC) on clean inputs and the attack success rate (ASR) on trigger-embedded inputs, with  the  results shown in Figure 2. Observe that the conditional number of ˆΘ( ϑ ) has a strong negative correlation with ASR, with a smaller value indicating higher attack vulnerability; meanwhile, it has a limited correlation with ACC, with most of the arches having ACC within the range from 80% to 95%.

Intuitively, ˆΘ( ϑ ) measures the trigger generator’s trainability with respect to a randomly initialized target model. The generator’s trainability indicates the easiness of effectively generating input-aware triggers, implying the model’s vulnerability to input-aware backdoor attacks. To verify the hypothesis, on the CIFAR10 dataset with the generator configured as in Appendix § A, we measure ˆΘ( ϑ ) with respect to 900 randomly generated arches as well as the model accuracy (ACC) on clean inputs and the attack success rate (ASR) on trigger-embedded inputs. Specifically, for each arch α , we first train the model f α to measure ACC and then train the trigger generator g with respect to f α on the same dataset to measure ASR, with results shown in Figure 2. Observe that the conditional number of ˆΘ( ϑ ) has a strong negative correlation with ASR, with a smaller value indicating higher attack vulnerability; meanwhile, it has a limited correlation with ACC, with most of the arches having ACC within the range from 80% to 95%.


[-Similarity − 0 .0 .0 .0 .0 .1 .
 P e r c en t-] ML-GCN Positive/Negative Link Scores PositiveLinks NegativeLinks Similarity We can see that while they behave similarly, the ML-GCN does a better job of ensuring that positive/negative links are well separated. These are computed on Amazon however, BGRL outperforms the ML-GCN (the strongest contrastive baseline) on 3/6 of the datasets and performs similarly on 1 other ( Cora ). It also outperforms GRACE across all of the datasets.

[-Similarity − 0 .0 .0 .0 .0 .1 .
 P e r c en t-]We can see that while they behave similarly, the ML-GCN does a better job of ensuring that positive/negative links are well separated. These scores are computed on Amazon-Photos . perform poorly relative to the other methods. This is intuitive, as neither method was designed for link prediction and were only evaluated for node classification in their respective papers. Surprisingly, however, BGRL outperforms the ML-GCN (the strongest contrastive baseline) on 3/6 of the datasets and performs similarly on 1 other ( Cora ). It also outperforms GRACE across all of the datasets.


Interpretability – a universal boon for ML? In the PCC domain, the model interpretability is linked to the wealth of domain knowledges. By distilling a blackbox neural network into a white boxsymbolic rule, the congestion rule is made easier for the networkcongestion practitioners to locatethe bug and modify/improve manually.

Interpretability – a universal boon for ML? In the PCC domain, the model interpretability is linked to the wealth of domain knowledge. By distilling a black-box neural network into white-box symbolic rules, the resulting rules are easier for the network practitioners to digest and improve.

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Taken together, these results have several implications. First, we ﬁnd that a highly expressive nonlinear decoder does not yield any increase in decoding accuracy, even as we scale up in MDFA complexity. From this, and the fact that we did extensive hyperparameter search in training decoders, we can conclude that the decoder models we chose are expressive enough for the decoding task. Second, we ﬁnd that decoding accuracy for MDFA states is in general not very high. These two observations suggest  the need for a different interpretation of the internal representation of the trained RNN.

Taken together, these results have several implications. First, we ﬁnd that a highly expressive nonlinear decoder does not yield any increase in decoding accuracy, even as we scale up in MDFA complexity. We can conclude from this ﬁnding and our extensive hyperparameter search for each decoder model that the decoder models we chose are expressive enough for the decoding task. Second, we ﬁnd that decoding accuracy for MDFA states is in general not very high. These two observations suggest linear decoders are sufﬁcient for the decoding task, but also suggests the need for a different interpretation of the internal representation of the trained RNN.


The existing work based on iPPM framework either require compactness, e.g., [39], or some special structure on L , e.g., [35]. This is also true for VR-based methods, e.g.,[15, 26, 37]. To our knowledge, ours is the ﬁrst one to overcome this difﬁculty and strictly improve the  complexity bounds for  WCSC and WCMC settings without compactness assumption; moreover, the same idea also workssimultaneously with a variance reduction technique that will be discussed later (see section 3). Finally,same trick for removing compactness assumption for  WCSC setting also helps removing compactnessassumption for the primal domain in WCMC setting  as well (see section 4).

The existing work based on iPPM framework either require compactness, e.g., [41], or some special structure on L , e.g., [37]. This is also true for VR-based methods, e.g.,[16, 27, 39]. To our knowledge, ours is the ﬁrst one to overcome this difﬁculty and strictly improve the best known complexity bound for the WCSC setting without compactness assumption; moreover, the same idea also works simultaneously with a variance reduction technique that will be discussed later (see section 4). Finally, the same trick for removing compactness assumption for the WCSC setting also helps removing the compactness assumption for the primal domain in WCMC setting and we still improve the best known complexity for this setting as well (see section 5).


It has been observed that a cyclical learning rate  schedule leads to better generalization (Smith, 2015). In Sec. 4.2 we demonstrated that one can exchange cyclic learning rate schedule (CLR) with batch size (CBS) and approximately preserve the practical beneﬁt of CLR. This inspired us to hypothesize that CLR is related to switching between  noise levels ( ηS ), switching between sharp/deep and wide/shallow minima. To validate that, we run VGG-11 on CIFAR10 using 4 training schedules: CLR with stepsize of 4 and 15 epochs in each stage, CBS with stepsize 4 and 15 epochs in each stage. Each run is repeated 8 times and for each variant we track evolution of sharpness and accuracy. We observe that CBS and CLR with longer stages lead to 89 . 9 ± 0 . 20% and 90 . 2 ± 0 . 10% test accuracy in both cases 0 . 2 − 0 . 3% above variance with shorter range. CBS and CLR with long stages times seem to be promising schedules for training DNNs. Finally, we validate that CBS and CLR switch between sharp/deep and wide/shallow minima, suggesting that CLR improves mixing time. Plot included in appendix.

It has been observed that a cyclic learning rate (CLR) schedule leads to better generalization (Smith, 2015). In Sec. 4.2 we demonstrated that one can exchange cyclic learning rate schedule (CLR) with batch size (CBS) and approximately preserve the practical beneﬁt of CLR. This inspired us to hypothesize that by changing between controllable noise levels ( ηS ) CLR switches between sharp/deep and wide/shallow minima. To validate that, we run VGG-11 on CIFAR10 using 4 training schedules: CLR with stepsize of 4 and 15 epochs in each stage, CBS with stepsize 4 and 15 epochs in each stage. Each run is repeated 8 times and for each variant we track evolution of sharpness and accuracy. We observe that CBS and CLR with longer stages lead to 89 . 9 ± 0 . 20% and 90 . 2 ± 0 . 10% test accuracy, in both cases 0 . 2 − 0 . 3% above variance with shorter range. CBS and CLR with long stepsize each seem to be promising schedules for training DNNs. Finally, we validate that CBS and CLR switch between sharp/deep and wide/shallow minima, suggesting that CLR improves convergence time to stationary distribution, as seen in Fig. 13 of Appendix G.4.


In this section, we  investigate the influence of activation functions in PINNs for solving PDE/ODE systems. We evaluate and compare the effectiveness of several common activation functions on some simple problems with analytical solutions. The results show that the choice of activation functions is crucial for PINNs and depends on the problem. Motivated by this observation, we propose to learn specialized activation function for different PDE systems.

In this section, we first investigate the influence of activation functions in PINNs for solving simple ODE systems with analytical solutions. The results show that the choice of activation functions is crucial for PINNs and depends on the problem. Motivated by this observation, we propose to learn specialized activation functions for different PDE systems.

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The above example demonstrates the importance of providing parts of an explanation in an online fashion. Mark gradually reveals the reasoning to maintain his plan as the execution unfolds so that it also becomes  acceptable (and understandable) to Emma. The key point here is to explain minimally and only when necessary, as long as the next action becomes understandable. In this way, the information to be conveyed is spread out into the future so that there is less cognitive requirement at the current step–from Emma’s perspective, the interaction with Mark is simpler and requires less thought.

The above example demonstrates the importance of providing an explanation in an online fashion. Mark gradually reveals the reasoning to maintain his plan as the execution unfolds so that it also becomes both acceptable and understandable to Emma, even though being subject to different values due to model differences (e.g., Mark values lunch break more than Emma thinks he does). The key point here is to explain minimally and only when necessary. In this way, the information to be conveyed is spread out throughout the plan execution, potentially with even a reduced amount of information, so that there is less mental workload requirement at the current step–from Emma’s perspective, the interaction with Mark is more straightforward.


Implementation Details For all the experiments, we use the same architecture for the meta attacker A , which consists of four convolutional layers and four deconvolutional layers. We use Reptile (Nichol et al., 2018) with 0 . 01 learning rate to train meta attacker. Fine-tuning parameters are set as m = 5 for MNIST and CIFAR10, and m = 3 for tiny-Imagenet. Top q = 128 coordinates are selected as part coordinates for attacker ﬁne-tuning and model attacking on MNIST; and q =on CIFAR10 and tiny-Imagenet.

Meta-training Details For all the experiments , we use the same architecture for the meta attacker A as shown in Table 6. We use Reptile (Nichol et al., 2018) with 0 . 01 learning rate to train meta attackers. We use 10000 randomly selected images from the training set to train the meta-attackers in three datasets. The proportion of the selected images to the whole training set are 16%, 20%, and 10% respectively. Fine-tuning parameters are set as m = 5 for MNIST and CIFAR10, and m =for tiny-Imagenet. Top q = 128 coordinates are selected as part coordinates for attacker ﬁne-tuning and model attacking on MNIST; and q = 500 on CIFAR10 and tiny-Imagenet.


We give some intuition as to why parameter-efficient methods seem to be more effective for private fine-tuning. For simplicity, we assume that the fine-tuning method is additive as in (3), such that the fine-tuned weights W FT = W PT + π ( θ ) . We can imagine that W FT lies on a manifold passing through W PT of very small dimension (equal to the dimension of θ ) compared to the dimension of W PT .

We give some intuition as to why parameter-efﬁcient methods can to be more effective for private ﬁne-tuning, especially on smaller datasets. For simplicity, we assume that the ﬁne-tuning method is additive as in (3), such that the ﬁne-tuned weights W FT = W PT + π ( θ ) . We can imagine that W FT lies on a manifold passing through W PT of very small dimension (equal to the dimension of θ ) compared to the dimension of W PT .


Regularization is known to impact the learning speed when training a deep neural net with SGD. Here, we investigate the effect of regularization on multi-modal learning. We demonstrate that, aspredicted in the last conjecture 3.2, strong regularization encourages the multi-modal learning process to be greedy.

We investigate L1 regularization’s impact on multi-modal DNNs and demonstrate that, as the second conjecture in §3.2 says, strong regularization encourages greediness in multi-modal learning.

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Jaccard index for segmentation. During perturbation generation, we set (cid:15) as 0.01 and  stop at the 10-th iteration for all the attack methods. The α in Eq. 4 is set as 1. We provide more results including using different Gaussian kernels W and show the pseudo code in the supplementary ﬁles.

Jaccard index for segmentation. During perturbation generation, we set (cid:15) as 0 . 05 , 0 . 01 , 5 × 10 −for lung segmentation, artefact detection and diabetic retinopathy grading respectively. We stop at the 10-th iteration for all the attack methods. The α in Eq. 4 is set as 1. We provide more results including using different Gaussian kernels W and show the pseudo code in the supplementary ﬁles.


MS COCO 2014 dataset: Table 2 illustrates the segmentation performance on MS COCO 2014 compared with other methods. Our method achieves 42.6% in terms of the mIoU values on validation set, surpassing 1.2% over the IRN [1],  regarded as our baseline and outperforming the other recentcompetitive methods [8, 56, 49, 1] by a large margin. In particular, we reproduce different resultsof IRN [1] with CONTA [56], in which we achieve 41.4% mIoU values. Hence, we compare therelative improvements for comparison: CONTA reaches a 0.8% mIoU improvement compared with IRN (32.6 to 33.4), while our method achieves 1.2% mIoU improvement (41.4 to 42.6).

MS COCO 2014 dataset: Table 2 illustrates the segmentation performance on MS COCO 2014 compared with other methods. Our method achieves 42.6% in terms of the mIoU values on validation set, surpassing 1.2% over the IRN [1], also regarded as our baseline and outperforming the other recent competitive methods [1, 9, 52, 61] by a large margin. We further compare the relative improvements for comparison: CONTA reaches a 0.8% mIoU improvement compared with IRN (32.6 to 33.4), while our method achieves 1.2% mIoU improvement (41.4% to 42.6%).

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S UMMARY We propose a novel universal weakly-supervised semantic segmentation method via Semisupervised Pixel-wise Metric Learning. Four common types of pixel-to-segment attraction and repulsion relationships can be derived, whether the partial annotation is coarse image tags and bounding boxes, or sparse keypoints and scribbles. Our results on PASCAL VOC and DensePose show consistent and substantial gains over SOTA, especially for the sparsest keypoint supervision.

Summary. We propose a novel weakly-supervised semantic segmentation method via Semisupervised Pixel-wise Metric Learning, based on four common types of pixel-to-segment attraction and repulsion relationships. It is universally applicable to various weak supervision settings, whether the training images are coarsely annotated by image tags or bounding boxes, or sparsely annotated by keypoints or scribbles. Our results on PASCAL VOC and DensePose show consistent and substantial gains over SOTA, especially for the sparsest keypoint supervision. Acknowledgements.

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We started the design process with an ideation session with eight researchers with background in human-computer interaction and visualization. The goal of the session was to explore layout variations and features that a calendar that integrates medication prescriptions should have. We asked the researchers to sketch calendars that show all the data (D1 - D5) and satisfy the given requirements. The ideation session  lasted  30 minutes. We identified several design dimensions and their variations from the different sketches that were produced: the layout of the calendar (linear or cyclic), the positioning of the days and times of the day (on the left or at the top), the shape of drug entries (rectangular, cylindrical, or circular), and the orientation of the calendar (vertical or horizontal). Various sizes, colors, and shapes were also used in the designs. Connecting lines were predominantly used to denote the presence (and absence) of conflicts. We created three designs ( Design A , Design B , Design C ) by con- sidering i) variations according to these design dimensions, ii) the constraint of compatibility with already existing calendars, and iii) the intention to remain as close as possible to the design of regular medication schedules. The three designs cover a range of design variations regarding layout, representation of medication entries, and representation of conflicts. This will allow us to assess the usefulness of design variations at a component level, rather than at an overall design level. Next, we structure the presentation of these three design variations according to the usability requirements.

We started the design process with an ideation session involving eight researchers with background in human-computer interaction and visualization. The goal of the session was to explore layout variations and features that a calendar with integrated medication prescriptions should have. We asked the researchers to sketch calen- dars that show all the data (D1 - D5) and satisfy the given requirements. One ideation session was held for this task and it lasted for 30 minutes. Participants used pens, colored markers, pencils, and regular printing paper for their designs. We identified several design dimensions and their variations from the different sketches that were produced: the layout of the calendar (linear or cyclic), the positioning of the days and times of the day (on the left or at the top), the shape of drug entries (rectangular, cylindrical, or circular), and the orientation of the calendar (vertical or horizontal). Various sizes, colors, and shapes were also used in the designs. Connecting lines were predominantly used to denote the presence (and absence) of conflicts. We created three designs ( Design A , Design B , Design C ) by considering i) variations according to these design dimensions, ii) the constraint of compatibility with already existing calendars, and iii) the intention to remain as close as possible to the design of regular medication schedules. The three designs cover a range of design variations regarding layout, representation of medication entries, and representation of conflicts. This allowed us to assess the usefulness of design variations at a component level, rather than at an overall design level. The following subsections discuss the resulting three design variations according to our usability requirements.


We choose for M a small constant number, since L D , [ X D , X M ] is a lower bound for the log marginal likelihood log p ( D ) for M > 1 , and we set r to a number of particle functions that can represent the posterior SP reasonably well. Thus, we are interested in estimating (cid:101) ∇ F L D , X ( Q [ T ] ) (cid:12)(cid:12) F =0 with mini-batches. In principle, an unbiased estimate of ℓ ( D , f iX D ) is n/s · ℓ ( D s , f i (cid:101) X ) , which suggests that λ = s/n. Although (in general) L D s , X is not a lower bound of log p ( D ) , we found in a practice setting that λ to s/n still results in reasonable performance. However, our theoretical framework gives the reassuring guarantee that if we use full-batch training, we would, in fact, maximize a lower bound of log p ( D ) . In the following, we present two algorithms, namely Stein functional variational NNs and Stein functional variational gradient boosting (A.3.1), based on the estimated Stein functional variational gradient – i.e., they depend on the score gradient of the functional prior evaluated at X . If there exists no analytical score gradient, we can use a score gradient estimator, as suggested in Sun et al. This only requires function samples of the prior process evaluated at X , but estimating the score gradient is usually computationally expensive (Zhou et al., 2020). Since our approach builds upon SVGD, there exists an additional approach in our framework based on a gradient-free SVGD (Han & Liu, 2018) that only requires the evaluation of the marginal densities of the prior process.

We choose for M a small constant number, since L D , [ X D , X M ] is a lower bound for the log marginal likelihood log p ( D ) for M > 1 , and we set r to a number of particle functions that can represent the posterior SP reasonably well. Thus, we are interested in estimating (cid:101) ∇ F L D , X ( Q [ T ] ) (cid:12)(cid:12) F =0 with mini-batches. In principle, an unbiased estimate of ℓ ( D , f iX D ) is n/s · ℓ ( D s , f i (cid:101) X ) , which suggests that λ = s/n. Although (in general) L D s , X is not a lower bound of log p ( D ) , we found in a practice setting that λ to s/n still results in reasonable performance. In the following, we present two algorithms based on the estimated Stein functional variational gradient – i.e., they depend on the score gradient of the functional prior evaluated at X . If there exists no analytical score gradient, we can use a score gradient estimator, as suggested in Sun et al. This only requires function samples of the prior process evaluated at X , but estimating the score gradient is usually computationally expensive (Zhou et al., 2020). Since our approach builds upon SVGD, there exists an additional approach in our framework based on a gradient-free SVGD (Han & Liu, 2018) that only requires the evaluation of the marginal densities of the prior process.

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Low-level image similarity : Positive and negative segments can be generated by selecting pixel i ’s own segment or all the other segments, respectively. Semantic annotation : Pseudo-labeled segments can be generated bytheir majority pixels’ annotations. Here, the annotations can be dense masks, scribbles, points, or the localization cues generated by Class Activation Maps (Zhou et al., 2016). Semantic co-occurrence : Positive (negative) segments can be generated by using other images that include (exclude) the same category asin pixel i ’s image. Feature afﬁnity : Pseudo-labeled segments can be generated by propagating labels from known pixels to their nearest neighbor segments  in the feature space, assuming that semantic clusters emerge during training and their territories are thus expanded.

Low-level image similarity : We impose a spatial smoothness prior on the pixel-wise feature to keep pixels together in visually coherent regions. The segment pixel i belongs to based on low-level image cues is a positive segment to pixel i ; any other segments are negative ones. Semantic annotation : We expand the semantics from labeled points and scribbles to pseudolabels inferred from image- or box-wise CAM. The label of a segment can be estimated by majority vote among pixels; if it is the same as pixel i ’s, the segment is a positive segment to i . Semantic co-occurrence : We expand the semantics by assuming that pixels in similar semantic contexts tend to be grouped together. If a segment appears in an image that shares the same semantic classes as pixel i ’s image, it is a positive segment to i and otherwise a negative one. Feature afﬁnity : We impose a featural smoothness prior assuming that pixels and segments of the same semantics form a cluster in the feature space. We propagate the semantics within and across images from pixel i to its closest segment s in the feature space.


Split-CIFAR10 and Split-CIFAR100 . In Split-CIFAR10, we split CIFAR10 (Krizhevsky & Hinton, 2009) into ﬁve tasks in the same manner as Split-MNIST. For Split-CIFAR100, we buildtasks, each containing ﬁve classes according to the pre-deﬁned superclasses in CIFAR100. The training sets of CIFAR10 and CIFAR100 consist of 50K examples each. To the best of our knowledge, we are ﬁrst to report Split-CIFAR100 performance without using task information at test time. In Split-CIFAR100 experiments of all previous works (Rebufﬁ et al., 2017; Zenke et al., 2017; Lopez-Paz & Ranzato, 2017; Aljundi et al., 2019c; Chaudhry et al., 2019a) a distinct output head is used for each task, and  the task information to select the corresponding output head is given at both training and test time. Knowing the right output head, however, the task reduces to 5-way classiﬁcation. Therefore, our setting is far more difﬁcult than the prior works since the model has to perform 100-way classiﬁcation only from the given input.

Split-CIFAR10 and Split-CIFAR100 . In Split-CIFAR10, we split CIFAR10 (Krizhevsky & Hinton, 2009) into ﬁve tasks in the same manner as Split-MNIST. For Split-CIFAR100, we buildtasks, each containing ﬁve classes according to the pre-deﬁned superclasses in CIFAR100. The training sets of CIFAR10 and CIFAR100 consist of 50K examples each. Note that most of the previous works (Rebufﬁ et al., 2017; Zenke et al., 2017; Lopez-Paz & Ranzato, 2017; Aljundi et al., 2019c; Chaudhry et al., 2019a), except Maltoni & Lomonaco (2019), use task information at test time in Split-CIFAR100 experiments. They assign distinct output heads for each task and utilize the task identity to choose the responsible output head at both training and test time. Knowing the right output head, however, the task reduces to 5-way classiﬁcation. Therefore, our setting is far more difﬁcult than the prior works since the model has to perform 100-way classiﬁcation only from the given input.


We also optimized LeNet-5, which is trained with the MNIST dataset. The optimization goal is togenerate an image that makes the LeNet-5 predict a target number with a maximum score (maximum [s = 0.999] 1 : [s = 0.959] 2 : [s = 0.999] 3 : [s = 0.999] 4 : [s = 0.999] prediction score = 1 . 0 ). The LeNet-5 is regarded as a non-differentiable black-box. After 50 , 000function calls, the final scores of generated images reach very close to the maximum score (Figure 6).

We also optimized LeNet-5, which is trained with the MNIST dataset. The optimization goal is togenerate an image that makes the LeNet-5 predict a target number with a maximum score (maximumprediction score = 1 . 0 ). The LeNet-5 is regarded as a non-differentiable black-box. After 50 , 000function calls, the final scores of generated images reach very close to the maximum score (Figure 6).


At every time step in training the RNN decoder, the probability of the next word is computed based on the left-context embedding, ground-truth current word embedding, and sentence representation. The existence of the ground-truth current word embedding potentially decreases the tendency for the decoder toexploit other information from the  sentence representation.

Finding II: The model with an autoregressive decoder works roughly the same as the model with a predict-all-words decoder. With Finding I, we noticed that the correct ground-truth input words to the autoregressive decoder is not necessary in terms of learning sentence representations.

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Li et al. (2022) show the performance of fine-tuning the full model can be significantly improved with proper configuration. In this section, we re-evaluate the tasks in Table 3 and 4 under the configuration in Li et al. and show such a configuration also improves the performance of our methods.

Li et al. (2022) show the performance of ﬁne-tuning the full model can be signiﬁcantly improved with proper conﬁguration. In this section, we re-evaluate the tasks in Table 3 and 4 under the conﬁguration in Li et al. and show such a conﬁguration also improves the performance of our methods.


To verify basic operations of our proposed method, we perform post-training weighted quantization using pre-trained models of BERT-base (Devlin et al., 2018) with a GLUE benchmark (Wang et al., 2018) on MNLI and MRPC dataset. In the case of  BERT models, we quantize all weights except those of a segment embedding layer and a classiﬁcation layer which show a tiny storage footprint. For conventional or weighted Alternating quantization methods, we conduct iterative reﬁnements of α and B values 20 times over which no further noticeable quantization error improvement is recognized. Given a weight matrix or tensor, α and B are computed for each row, independently. Due to the space limit, see Appendix for additional experimental results with various models not included in this section.

To verify basic operations of our proposed method, we perform post-training weighted quantization using ﬁne-tuned models of BERT-base (Devlin et al., 2018) on MNLI and MRPC dataset within a GLUE benchmark (Wang et al., 2018). In the case of ﬁne-tuned BERT models, we quantize all weights except those of a segment embedding layer and a classiﬁcation layer which show a tiny storage footprint. For conventional or weighted Alternating quantization methods, we conduct iterative reﬁnements of α and B values 20 times over which no further noticeable quantization error improvement is recognized. Given a weight matrix or tensor, α and B are computed for each row, independently (hence, we study row-wise quantization in this work). Due to the space limit, see Appendix for additional experimental results with various models not included in this section.


Matching methods, e . g ., PSM exhibit promising performance on ranking metric, which explains why they are favored by counterfactual ranking applications (Betlei et al., 2021) in practice. However, their poor performance on PEHE hinders their application in counterfactual estimation applications such as advertising systems where accuracy metrics are more critical.

Matching methods, e . g ., PSM exhibit compelling ranking performance, which explains why they are favored in counterfactual ranking practice (Betlei et al., 2021). However, their poor performance on PEHE hinders their application in counterfactual estimation applications such as advertising systems that place more emphasis on the accuracy of treatment effect estimation.

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ImageNet dataset and can further ensemble them to obtain 60 . 38 % top 1 accuracy. In this section,we replace the cosine distance between the global visual feature and prompt ensemble with the OT distance between the feature map and all 7 prompts. However, without any learning, the OTdistance only obtains 58 . 78 % accuracy. It is a limitation of the PLOT to still need few-shot datafor optimization, which cannot be directly applied in the zero-shot setting. We argue there are two reasons why the OT distance does not work without learning: 1) prompt engineering selects prompts based on the global feature and cosine distance, instead of OT distance with feature map; 2) all these selected prompts are closed to the global feature and lack the complementarity.

ImageNet dataset and can further ensemble them to obtain 60 . 38 % top 1 accuracy. In this section,we replace the cosine distance between the global visual feature and prompt ensemble with the OT distance between the feature map and all 7 prompts. However, without any learning, the OT distanceonly obtains 58 . 78 % accuracy. We argue there are two reasons why the OT distance does not work without learning: 1) prompt engineering selects prompts based on the global feature and cosine distance, instead of OT distance with feature map; 2) all these selected prompts are closed to the global feature and lack the complementarity.

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Highlight and free-text explanations are the most prominent explanation types for teaching NLP models (Wiegreffe and Marasovic, 2021). Highlight explanations ( HIGHLIGHT ) are subsets of input elements that are deemed relevant for a prediction. For text-based NLP tasks, they correspond to sets of words, phrases or sentences. Free-text explanations ( FREE - TEXT ) are texts in natural language that are not constrained to be grounded in the input elements. Some recent works rely on semistructured text explanations ( SEMI - STRUCTURED ) (Wiegreffe and Marasovic, 2021), which combine properties of both highlight and free-text explanations. They consist of text in natural language and contain an explicit indication of the input elements that the free-text applies to. If and how much a model can learn from such explanations depends on the amount of information contained in the explanation (§ 2.2), and to what extent this information can be integrated into the learning process (§ 2.1). User satisfaction is affected by the effort required to produce explanations and by the difficulty of the task, that might in turn affect explanation quality (§ 2.3). In the following, we discuss these factors in detail and where possible contrast them with respect to explanation type.

Highlight and free-text explanations are the most prominent explanation types used to improve NLP models (Wiegreffe and Marasovic, 2021). Highlight explanations ( HIGHLIGHT ) are subsets of input elements that are deemed relevant for a prediction. For text-based NLP tasks, they correspond to sets of words, phrases or sentences. Free-text explanations ( FREE - TEXT ) are texts in natural language that are not constrained to be grounded in the input elements and contain implicit or explicit information about why an instance is assigned a specific label. Some recent works rely on semistructured text explanations ( SEMI - STRUCTURED ) (Wiegreffe and Marasovic, 2021), which combine properties of both highlight and free-text explanations. They consist of text in natural language and contain an explicit indication of the input elements that the free-text applies to. If and how much a model can be improved based on such explanations depends on the amount of information contained in the explanation (§ 2.2), and to what extent this information can be integrated into the learning process (§ 2.1). User satisfaction is affected by the effort required to produce explanations and by the difficulty of the task, that might in turn affect explanation quality (§ 2.3). In the following, we discuss these factors in detail and where possible contrast them with respect to explanation type.


Intriguingly, over the course of training we do not see a consistent downward trend across all games and update objectives, suggesting that while effective dimension is indeed effective at identifying representation collapse it does not perfectly correlate with network capacity.

Unlike in target-ﬁtting capacity, over the course of training we do not see a consistent downward trend across all games and update objectives, suggesting that while effective dimension is indeed effective at identifying representation collapse, it is measuring a subtly different notion of capacity than simply an agent’s ability to ﬁt new target functions.


Fig 2 plots this proportion for different datasets and models. Each bar is an averageover three runs,injecting in each run 10% random data points from – respectively – one of the three other datasets. The biggest inﬂuence on how much uncertainty sampling selects outliers is the dataset, with the biggest proportion for newsgroup , where 80% of the selected data points are outliers on average (for

Fig 2 plots this proportion for different datasets and models. Each bar is the average of the three runs. The biggest inﬂuence on how much uncertainty sampling selects outliers is the dataset, with the biggest proportion for newsgroup , where 80% of the selected data points are outliers on average (for

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We present the training curves of Algorithm 1 on AKI dataset in Figure 3 and 4. • Policy improvement during RL training. In Figure 3, we showed the performance of the RL policies evaluated on train, validation and test sets. We note that the three curves closely match one another, confirming the generalizability of the learned dynamic classification policy. • SM-DDPO learns the disease model. In end-to-end training, the diagnostic classifier is trained from the scratch. It maps any partially-observed patient state to a diagnosis/prediction. We evaluate this classifier on static data distributions, in order to eliminate the effect of dynamic test selection and focus on classification quality. Figure 4 shows that the classifier learns to make the right prediction with improved quality during RL, via training only on data selected by the RL algorithm.

We present the training curves on AKI dataset in Figure 3. We refer more results to Appendix B. • SM-DDPO learns the disease model. In end-to-end training, the diagnostic classifier is trained from the scratch. It maps any partially-observed patient state to a diagnosis/prediction. We evaluate this classifier on static data distributions, in order to eliminate the effect of dynamic test selection and focus on classification quality. Figure 3 shows that the classifier learns to make the high-quality prediction with improved quality during RL, via training only on data selected by the RL algorithm.

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VT. Furthermore, Kahn et al. (2018) design a self-supervised approach to model environments by reinforcement learning. A Bayesian relational memory is introduced by Wu et al. (2019) to explore the spatial layout among rooms rather than steering an agent to desired objects with least steps. Meanwhile, Shen et al. (2019) employ multiple visual representations to generate multiple actions and then fuse those actions to produce an effective one. However, requesting such a large number of visual representations may restrict the transferring ability of a navigation system and increases the difﬁculty of data labeling. Note that Fang et al. (2019) propose a transformer to select the embedded scene memory slot, while our VT is designed to learn expressive visual representations correlated with directional signals.

VT. Furthermore, Kahn et al. (2018) design a self-supervised approach to model environments by reinforcement learning. Tang et al. (2021) customize a specialized network for visual navigation via an Auto-Navigator. A Bayesian relational memory is introduced by Wu et al. (2019) to explore the spatial layout among rooms rather than steering an agent to desired objects with least steps. Meanwhile, Shen et al. (2019) employ multiple visual representations to generate multiple actions and then fuse those actions to produce an effective one. However, requesting such a large number of visual representations may restrict the transferring ability of a navigation system and increases the difﬁculty of data labeling. Note that Fang et al. (2019) propose a transformer to select the embedded scene memory slot, while our VT is designed to learn expressive visual representations correlated with directional signals.


Moreover, recently, Angelini & Ricci-Tersenghi (2022) have shown that the learning-based method in (Schuetz et al., 2022) could not achieve comparable results with the degree-based greedy algorithm (DGA) (Angelini & Ricci-Tersenghi, 2019) in the max independent set (MIS) problem on large-scaled random-regular graphs (RRGs), which raises attentions from machine learning community. We observe the issues come from two aspects: (1) graph neural networks (GNNs) used to encode the regular graph suffer from the node ambiguity issue due to their limited expressive power ( ? ); (2) the model in (Schuetz et al., 2022) was not trained and learned from history. By addressing these two issues, Meta-EGN can consistently outperform DGA while maintaining the same time complexity to generate solutions. Fig. 1 show the results.

Moreover, recently, Angelini & Ricci-Tersenghi (2022) have shown that the learning-based method in (Schuetz et al., 2022) could not achieve comparable results with the degree-based greedy algorithm (DGA) (Angelini & Ricci-Tersenghi, 2019) in the max independent set (MIS) problem on large-scaled random-regular graphs (RRGs), which raises attentions from machine learning community. We observe the issues come from two aspects: (1) graph neural networks (GNNs) used to encode the regular graph suffer from the node ambiguity issue due to their limited expressive power (Xu et al., 2019); (2) the model in (Schuetz et al., 2022) did not learn from history but was directly optimized over each testing case, which tends to be trapped into a local optimum. By addressing these two issues, Meta-EGN can consistently outperform DGA while maintaining the same time complexity to generate solutions. Fig. 1 show the results.


Under the  supervised setting, we can deﬁne the positive and negative sets as the same- and differentcategory pixels with respect to pixel i , which are denoted as C + and C − . However, this idea is not applicable to weakly- or un-supervised setting, when the label at x i is unknown. Particularly for weakly-supervised segmentation, we only have sparsely labeled pixels in the image, resulting in much smaller sets of C + and C − and degraded learning efﬁciency.

In the fully supervised setting, we can deﬁne pixel i ’s positive and negative sets, denoted by C + and C − respectively, as pixels in the same (different) category. However, this idea is not applicable to weakly- or un-supervised settings where the label is not available on every pixel. In the labeled points setting, C + and C − would only contain a few exemplars according to the sparse pixel labels.

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NLP (Cherry et al., 2019). Much of the information in such sentence embeddings is irrelevant to the  target task, and it can be difficult to distinguish relevant from irrelevant information when fine-tuning the languagemodels with a large number of parameters on a small amount of target task data, resulting in over-fitting. For many real-world applications, it can be difficult and expensive tosolve this problem by collecting sufficient annotated data for these large neural models to excel.

If the amount of target task data is small, it can be hard for fine-tuning to distinguish relevant from irrelevant information, leading to overfitting on statistically spurious correlations between the irrelevant information and target labels. Learning low-resource tasks is an important topic in NLP (Cherry et al., 2019) because annotating more data can be very costly and time-consuming, and because in several tasks access to data is limited.

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Ours is the ﬁrst systematic method able to train  sparse models directly without an increased parameter footprint during the entire course of training, and still achieve performance  on par with post-training compression of  dense models,  the best result at a given sparsity.We described the ﬁrst dynamic reparameterization method for training convolutional networks. We showed that ourdynamic sparse reparameterization signiﬁcantly outperformed static ones. Our method not only outperformed existing dynamic sparse reparameterization techniques, but also incurred much lower computational costs.

We showed that it is possible to train a small sparse network directly without a larger than inference-time parameter footprint at any stages of training, yet still achieving generalization performance at least on par with post-training iterative pruning of large dense models, yielding the most parameter efﬁcient model at a given sparsity. We showed that our method is more scalable and efﬁcient, and leads to signiﬁcantly better accuracy than existing dynamic sparse reparameterization training techniques.

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The synthesis of realistic human motion has recently seen increased interest Holden et al. Fragkiadaki et al. ; Jain et al. ; Bütepage et al. ; Martinez et al. (2017) with applications beyond animation and video games. The simulation of human looking virtual agents is likely to become mainstream with the dramatic advancement of Artiﬁcial Intelligence and the democratization of Virtual Reality. A challenge for human motion synthesis is to automatically generate new variations of motions while preserving a certain style, e.g., generating large numbers of different Bollywood dances for hundreds of characters in an animated scene of an Indian party. Aided by the availability of large human-motion capture databases, many database-driven frameworks have been employed to this end, including motion graphs Kovar et al. ; Safonova & Hodgins (2007); Min & Chai (2012), as well as linear Safonova et al. ; Chai & Hodgins (2005); Tautges et al. and kernel methods Mukai (2011); Park et al. ; Levine et al. (2012); Grochow et al. ; Moeslund et al. ; Wang et al. (2008), which blend key-frame motions from a database. It is hard for these methods, however, to add new variations to existing motions in the database while keeping the style consistent. This is especially true for motions with a complex style such as dancing and martial arts. More recently, with the rapid development in deep learning, people have started to use neural networks to accomplish this task Holden et al. These works have shown promising results, demonstrating the ability of using high-level parameters (such as a walking-path) to synthesize locomotion tasks such as jumping, running, walking, balancing, etc. These networks do not generate new variations of complex motion, however, being instead limited to speciﬁc use cases.

The synthesis of realistic human motion has recently seen increased interest (Holden et al., 2016; 2017; Fragkiadaki et al., 2015; Jain et al., 2016; Bütepage et al., 2017; Martinez et al., 2017) with applications beyond animation and video games. The simulation of human looking virtual agents is likely to become mainstream with the dramatic advancement of Artiﬁcial Intelligence and the democratization of Virtual Reality. A challenge for human motion synthesis is to automatically generate new variations of motions while preserving a certain style, e.g., generating large numbers of different Bollywood dances for hundreds of characters in an animated scene of an Indian party. Aided by the availability of large human-motion capture databases, many database-driven frameworks have been employed to this end, including motion graphs (Kovar et al., 2002; Safonova & Hodgins, 2007; Min & Chai, 2012), as well as linear (Safonova et al., 2004; Chai & Hodgins, 2005; Tautges et al., 2011) and kernel methods (Mukai, 2011; Park et al., 2002; Levine et al., 2012; Grochow et al., 2004; Moeslund et al., 2006; Wang et al., 2008), which blend key-frame motions from a database. It is hard for these methods, however, to add new variations to existing motions in the database while keeping the style consistent. This is especially true for motions with a complex style such as dancing and martial arts. More recently, with the rapid development in deep learning, people have started to use neural networks to accomplish this task (Holden et al., 2017; 2016; 2015). These works have shown promising results, demonstrating the ability of using high-level parameters (such as a walking-path) to synthesize locomotion tasks such as jumping, running, walking, balancing, etc. These networks do not generate new variations of complex motion, however, being instead limited to speciﬁc use cases.


In this subsection, we conduct ablation study on the superiority of thelow-dimensional contrastive embedding ( i.e. , our method) over the traditional contrastive embedding ( i.e. , thebaseline method). We use the STL-10and CIFAR-10 datasets to train the baseline SimCLR [7] and two implementations of CLLR, i.e. , the ℓ 2 , 1 -normbased regularization and nuclear-normbased regularization. We train all models with 100 and 400 epochs with the same batch size and learning rate, respectively, and we record the test accuracy of all methods by finetuning a linear softmax . The baseline method learns contrastive embeddings in the high-dimensionalspace (dimension = 2048 , 3072 , and 4096 ) and the simply fixed low-dimensional space (dimension =256 and 512 ). We also include the baseline results that do not use the ℓ 2 , 1 -norm and nuclear normconstraints ( i.e. , α = 0 ). Our method learns embeddings in low-dimensional space, where we use theregularizer to maintain the corresponding non-zero columns in the projection matrix L .

In this subsection, we conduct ablation study on the superiority of thelow-dimensional contrastive embedding ( i.e. , our method) over the traditional contrastive embedding ( i.e. , thebaseline method). We use the STL-10and CIFAR-10 datasets to train the baseline SimCLR [7] and two implementations of CLLR, i.e. , the (cid:96) 2 , 1 -norm basedregularization and nuclear-norm basedregularization. We train all models with 100 and 400 epochs with the same batch size and learning rate, respectively, and we record the test accuracy of all methods by ﬁne-tuning a linear softmax . The baseline method learns contrastive embeddings in the high-dimensional space (where a commonsetting is 2048 -dimension) and the simply ﬁxed low-dimensional space ( 256 -dimension and 512dimension). Our method learns embeddings in low-dimensional space, where we use the regularizerto maintain the corresponding non-zero columns in the projection matrix L .

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We compare the version of AGILE where the GAT only receives action features as input and no state. Thus, the decision-choice is still aware of other actions, but the learned relations are ﬁxed, not dependent on the state. Figure 7 shows a drop in performance for Grid World and CREATE, where the relevant action relations change based on the state. However, there is no drop in RecSim because CPR task only requires  knowing the most common category, which is independent of user state.

We evaluate a version of AGILE where the GAT only receives action representations as input and no state. Thus, the action relations are inferred independently of the state. Figure 7 shows a drop in performance for Grid World and CREATE, where the relevant action relations change based on the state. However, this effect is less apparent on RecSim because CPR requires only knowing the most common category, independent of user state.

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Therefore, to optimize the performance of SREDA further, we tune q, m from a grid search over { 10 , 100 , 200 } . For methods without variance reduction, i.e., for SAPD, SMDA and PASGDA, wetune the batch size from { 10 , 100 , 200 } as well. For SAPD and SAPD-VR, we tune the momentum θ from { 0 . 8 , 0 . 85 , 0 . 9 } and let the inner iteration numbers N = ln(265)ln( 1 θ ) according to eq.

Therefore, to optimize the performance of SREDA further, we tune q, m from a grid search over { 10 , 100 , 200 } . For methods without variance reduction, i.e., for SAPD+ , SMDA and PASGDA , we also use mini-batch to estimate the gradients and tune the batch size from { 10 , 100 , 200 } as well. For SAPD+ and SAPD+VR , we tune the momentum θ from { 0 . 8 , 0 . 85 , 0 . 9 } and the inner iteration number from N = { 10 , 50 , 100 } .


Vectors and scalars are not the only kind of geometric features that can be inputs and outputs of a GEM-CNN layer. In general, the coefﬁcients of a geometric feature of C dimensions changes by a linear transformation ρ ( − g ) ∈ R C × C if the gauge is rotated by angle g . The map ρ : [0 , 2 π ) → R C × C is called the type of the geometric quantity and is formally known as a group representation of the planar rotation group SO(2) .  From the theory of group representations, we know that any feature type can be composed from “irreducible representations” (irreps). For SO(2) , these are the one dimensional invariant scalar representation ρ 0 and for all n ∈ N > 0 , a two dimensional representation ρ n ,

Vectors and scalars are not the only type of geometric features that can be inputs and outputs of a GEM-CNN layer. In general, the coefﬁcients of a geometric feature of C dimensions changes by an invertible linear transformation ρ ( − g ) ∈ R C × C if the gauge is rotated by angle g . The map ρ : [0 , 2 π ) → R C × C is called the type of the geometric quantity and is formally known as a group representation of the planar rotation group SO(2) . Group representations have the property that ρ ( g + h ) = ρ ( g ) ρ ( h ) (they are group homomorphisms), which implies in particular that ρ (0) = and ρ ( − g ) = ρ ( g ) − 1 . For more background on group representation theory, we refer the reader to (Serre, 1977) and, speciﬁcally in the context of equivariant deep learning, to (Lang & Weiler, 2020). From the theory of group representations, we know that any feature type can be composed from “irreducible representations” (irreps). For SO(2) , these are the one dimensional invariant scalar representation ρ 0 and for all n ∈ N > 0 , a two dimensional representation ρ n ,


BERT in natural settings. They can provide the appropriate prompts and tasks to answer questions about linguistic mechanisms underlying predictive responses. This paper adopted psycholinguistic datasets to probe language models’ commonsense reasoning. Findings suggest that DistillBERT had some understanding of the (implied) intent that’s shared among most people. Such intent is implicitly reflected in the usage of conversational implicatures and presuppositions. Whether or not fine-tuning improved its performance to human-level depends on the type of commonsense reasoning.

BERT in natural settings. They can provide the appropriate prompts and tasks to answer questions about linguistic mechanisms underlying predictive responses. This paper adopted psycholinguistic datasets to probe language models’ commonsense reasoning. Findings suggest that GPT-3’s performance was mostly at chance in the psycholinguistic tasks. We also showed that DistillBERT had some understanding of the (implied) intent that’s shared among most people. Such intent is implicitly reflected in the usage of conversational implicatures and presuppositions. Whether or not fine-tuning improved its performance to human-level depends on the type of commonsense reasoning.


Hofbauer & Weibull (1996) showed that in a class of learning dynamics which includes replicator dynamics — the continuous-time variant of FTRL, all iteratively strictly dominated actions vanish over time, while Mertikopoulos & Moustakas (2010) proved similar results for stochastic replicator dynamics; however, neither work provides finite-time guarantees. Cohen et al. (2017) proved that Hedge eliminates dominated actions in finite time, but did not extend their results to the more challenging case of iteratively dominated actions.

For this equivalence to hold, we need to allow dominance by mixed strategies, and correlated beliefs when there are more than two players. These conditions are met in the setting of this work. strictly dominated actions vanish over time, while Mertikopoulos & Moustakas (2010) proved similar results for stochastic replicator dynamics; however, neither work provides finite-time guarantees. Cohen et al. (2017) proved that Hedge eliminates dominated actions in finite time, but did not extend their results to the more challenging case of iteratively dominated actions.