Recurrent Memory Transformer
Anonymous Author(s)
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Abstract
Transformer-based models show their effectiveness across multiple domains andtasks.
The self-attention allows to combine information from all sequence elements into context-aware representations.
However, global and local informationhas to be stored mostly in the same element-wise representations.
Moreover, thelength of an input sequence is limited by quadratic computational complexity ofself-attention.
In this work, we propose and study a memory-augmented segmentlevel recurrent Transformer (RMT).
Memory allows to store and process local andglobal information as well as to pass information between segments of the longsequence with the help of recurrence.
We implement a memory mechanism withno changes to Transformer model by adding special memory tokens to the inputor output sequence.
Then the model is trained to control both memory operationsand sequence representations processing.
Results of experiments show that RMTperforms on par with the Transformer-XL on language modeling for smaller memory sizes and outperforms it for tasks that require longer sequence processing. Weshow that adding memory tokens to Tr-XL is able to improve it performance.
Thismakes Recurrent Memory Transformer a promising architecture for applicationsthat require learning of long-term dependencies and general purpose in memoryprocessing, such as algorithmic tasks and reasoning.
Introduction
During training gradients flow from the current segment through memory to the previous segment
Transformers (Vaswani et al., 2017) have been widelyadopted across multiple domains and tasks (Radfordet al., 2018;
Dong et al., 2018; Devlin et al., 2019;
Dosovitskiy et al., 2021; Ramesh et al., 2021; Jaegleet al., 2021).
The key component of Transformerlayer is a self-attention.
Self-attention allows to update each sequence element representation with information from all other elements in the sequence.
As a result, rich contextual representation for everyelement is generated at the end of encoding.
Thisway, global sequence-level and local information arestored in a single representation.
However, this mixing of two types of information in a single representation has limitations.
Distributed storage of globalfeatures across all sequence elements results in global features "blurring" and makes it harder toaccess them.
Another well-known deficiency of Transformers is poor scaling of self-attention withinput sequence length that hurts its applications to long inputs (Child et al., 2019;
Guo et al., 2019; Dai et al., 2019; Beltagy et al., 2020; Ainslie et al., 2020; Zaheer et al., 2020; Wang et al., 2020;
Choromanski et al., 2020).
Our work introduces a memory-augmented segment-level recurrent Transformer named Recurrent
Memory Transformer (RMT).
RMT uses a memory mechanism based on special memory tokens (Burtsev et al., 2020) added to the input sequence.
Memory tokens provide additional reserved capacity tothe model that could be used to process information which is not directly representing any element inthe input sequence.
To process long sequences, we split them into segments and pass memory statesfrom a previous to a current segment.
This memory passing makes the model recurrent and removesthe input sequence length limitations.
RMT model can theoretically work with infinite lengths but, inpractice, it is limited by memory capacity and the efficiency of memory access/update operations.
Our implementation of both memory and recurrence in RMT requires no changes to the Transformermodel because modifications are made only to the input and output sequences of the model.
We tested RMT on the tasks that require a global information about the whole input sequence to besolved.
We use copy, reverse, and associative retrieval tasks in the setting where the input sequenceis split into segments.
RMT and Transformer-XL perfectly solve these tasks, but exceeding somevalue of sequence length, RMT starts to outperform Transformer-XL.
Also, we experimentally showthat the proposed Recurrent Memory Transformer requires less memory size to perform closely to
Transformer-XL on language modeling tasks.
RMT code and experiments are available 1 .
Contributions
In this study we augment Transformer with token based memory storage and segment-level
recurrence.
We experimentally evaluate proposed architecture as well as vanilla Transformer and TransformerXL on memory-intensive tasks such as copy, reverse, associative retrieval and language modeling.
We show that RMT outperforms Transformer-XL for sequence processing tasks and on par withTransformer-XL on language modeling but requires less memory. We show that Tr-XL cache could be combined with RMT leading to better performance on language modeling.
We analysed how the Transformer model learns to use memory.
Specific interpretable memoryread-write patterns of attention are shown.
Related work
[Figure]
In our study we add a memory to general purpose attention based neural architecture.
Memory isa recurrent topic in neural networks research.
It had started from the early works (McCulloch and
Pitts, 1943; Stephen, 1956) and significantly progressed in 90’s with introduction of Backpropagation
Through Time learning algorithm (Werbos, 1990) and Long-Short Term Memory (LSTM) (Hochreiterand Schmidhuber, 1997) neural architecture.
Today memory-augmented neural networks (MANNs)usually rely on some kind of recurrent external-memory which is separate from the model’s parameters.
Neural Turing Machines (NTMs) (Graves et al., 2014) and Memory Networks (Westonet al., 2014) are equipped with a storage for vector representations that can be accessed with anattention mechanism.
Memory Networks (Weston et al., 2014; Sukhbaatar et al., 2015) were designedto enable reasoning by sequential attention over to the content of a memory.
NTMs followed by
Differentiable Neural Computer (DNC) (Graves et al., 2016) and Sparse DNC (Rae et al.
, 2016)are implemented as recurrent neural networks able to write to memory storage over time.
All thesemodels are differentiable and can be trained via backpropagation through time (BPTT).
Parallel lineof research extends recurrent neural networks such as LSTM with data structures like stacks, lists,or queues (Joulin and Mikolov, 2015; Grefenstette et al., 2015).
MANN architectures with a moreadvanced addressing mechanisms such as address-content separation and multi-step addressing wereproposed in (Gulcehre et al., 2016, 2017; Meng and Rumshisky, 2018).
The Global Context Layermodel (Meng and Rumshisky, 2018) uses the idea of address-content separation to solve the difficultyof training content-based addressing in the canonical NTM.
Recent rise of Transformer models also resulted in introduction of a number of new memory architectures.
Transformer-XL (Dai et al., 2019) introduces a segment-level recurrence at the level ofhidden representations.
These representations of a sequence are computed and stored in the cache to be reused as an extended context for the next segment.
Compressive Transformer (Rae et al.
,2019) adds the second layer of memory to Transformer-XL.
This memory compresses and storesinformation from the cache.
∞ -former (Martins et al., 2021) utilizes continuous-space attention andrepresents input sequence as a continuous signal to make long-term memory unbounded.
Memory
Layers (Lample et al., 2019) model has a product key memory layer instead of a feed-forward layerwithin Transformer block to increase model capacity.
In many variations of Transformer different sorts of global representations are added.
Among themare Star-Transformer (Guo et al., 2019), Longformer (Beltagy et al., 2020), GMAT (Gupta and Berant,2020), Extended Transformer Construction (ETC) (Ainslie et al., 2020) and Big Bird (Zaheer et al.,2020).
All these architectures re-design self-attention mechanism to reduce it computational complexity with and ensure input coverage with the help of global representations.
Memory Transformer(Burtsev et al., 2020) keeps Transformer model intact and adds memory by extending input sequencewith special memory tokens.
Perceiver IO (Jaegle et al., 2021) maps an entire arbitrary input to thefixed number of latent representations.
Transformer layers do further processing over latent memoryrepresentations only.
Segment-level recurrence in Transformers is actively explored in a number of studies.
TransformerXL, Compressive Transformer keep previous states and re-use them in subsequent segments.
ErnieDoc (Ding et al., 2021) improves processing by using same-layer recurrence instead of attending toprevious layer outputs of a precedent segment.
Memformer (Wu et al., 2020) introduces a dedicatedmemory module to keep previous hidden states in summarized representations.
Memformer uses twospecial layers added to the Transformer model.
Memory cross-attention layer reads from memoryand memory slot attention layer updates it.
MART (Lei et al., 2020) has a similar approach as
Memformer but uses memory update rules analogous to LSTM (Hochreiter and Schmidhuber, 1997)and GRU (Cho et al., 2014).
FeedBack Transformer (Fan et al., 2020) goes further with full, and notsegment-level, recurrence.
FeedBack Memory merges past hidden representations from all layersinto a single vector and makes it accessible to the computations at any layer.
The disadvantage of fullrecurrence is that it is less parallelizable.
FeedBack Memory requires every sequence element to beprocessed sequentially.
In segment-level recurrent models, all elements of a segment are processed by
Transformer layers in parallel.
Only segments are processed sequentially.
Staircase Transformer (Juet al., 2021) combines segment-level recurrence and depth recurrence.
Staircase models use theoutput for previous segments and pass them as input for the next segment.
Our Recurrent Memory
Transformer is based on special memory tokens similar to Memory Transformer, segment-levelrecurrence as in Transformer-XL, and depth-recurrent mechanism for memory processing similar to
Staircase.
Recurrent Memory Transformer
Transformer-XL (Dai et al., 2019) extends Transformer model with state re-use cache mechanismfor segment-level recurrence and relative position encoding.
Input sequence is split on segmentsprocessed sequentially.
Hidden states computed for the previous segment M n are cached for eachtransformer layer n .
The input of the layer n consists of the last m states from the cached memoryand output of previous Transformer layer for the current segment τ :
[Equation]
here, SG stands for stop-gradient, ◦ denotes concatenation.
Cached states allow to increase effectivecontext size of Transformer model and save on compute operations.
Then, ˜ H n − 1 τ goes to Transformer layer TL to produce layer n outputs for segment τ :
[Equation]
In Transformer-XL, self-attention layers are modified to use relative position encodings to improvegeneralization to longer attention lengths.
The overall architecture is shown in the Figure 2.
Memory augmented Transformers such as GMAT, ETC, Memory Transformer (Gupta and Berant,2020; Ainslie et al., 2020; Burtsev et al., 2020) proposed to use special global tokens as storagefor representations.
Usually, memory tokens are added to the beginning of the input sequence.
However, in decoder-only architectures the causal attention mask makes impossible for memorytokens at the start of the sequence to collect information from the subsequent tokens.
On the other hand, if memory tokens are placed at the end of the sequence then preceding tokens unable toaccess their representations.
To solve this problem we add a recurrence to the sequence processing.
Representations of memory tokens placed at the end of the segment are used as an input memoryrepresentations at the start as well as at the end of the next segment.
In the Recurrent Memory Transformer input is augmented with special [mem] tokens, processed ina standard way along with the sequence of tokens.
Each memory token is a real-valued vector.
mmemory tokens are added at the beginning of the segment tokens representations H 0 τ and the same mtokens are added at the end:
[Equation]
here N is a number of Transformer layers.
The starting group of memory tokens functions as a read memory that allows sequence tokens toattend to memory states produced at the previous segment.
The ending group works as a writememory that can attend to all current segment tokens and update representation stored in the memory.
As result, H writeτ contains updated memory tokens for the segment τ .
Segments of the input sequence are processed sequentially.
To enable recurrent connection betweensegments, we pass outputs of the memory tokens from the current segment to the input of the nextsegment:
[Equation]
Both memory and recurrence in the RMT are based only on global memory tokens.
It allows tokeep the backbone Transformer unchanged and make RMT memory augmentation compatible withany model from the Transformer family.
Memory tokens operate only on the input and output ofthe model.
In this study we implement RMT on top of the original Transformer-XL code.
Botharchitectures are shown in Figure 2.
Recurrence in the RMT is different compared to the Transformer-XL because the former stores onlym memory vectors per segment.
On the other hand, the Transformer-XL stores m × N vectors persegment.
Also, in the RMT model memory representations from the previous segment are processedby Transformer layers together with the current segment tokens.
This makes memory part of RMTeffectively deeper in a number of applied Transformer layers τ × N .
Additionally, we allow allmemory tokens in the read/write block to access all other tokens in the same block.
The causalattention mask is applied only to tokens of the input sequence (Figure 6(d)).
We train the RMT with Backpropagation Through Time (BPTT).
During backward pass, unlike in
Transformer-XL, memory gradients are not stopped between segments.
The number of previoussegments to backpropagate is a hyperparameter of a training procedure.
We vary BPTT unroll in ourexperiments from 0 to 4 previous segments.
Increasing this parameter is computationally expensiveand requires a lot of GPU RAM.
However, such techniques as gradient checkpointing could be usedto alleviate this problem.
Experiments
We designed our experiments to evaluate the ability of Recurrent Memory Transformers to preservelong-term dependencies across multiple input segments.
The first set of experiments includes copy,reverse, associative retrieval and quadratic equations tasks.
The second one addresses languagemodeling task for word-level on WikiText-103 (Merity et al., 2017) and for character-level onenwik8 (Mahoney, 2006).
We compare Recurrent Memory Transformer with Transformer and Transformer-XL models.
Our RMT implementation is based on Transformer-XL repository 2 .
The full set of hyperparametersis available in our repository as well as in supplementary materials. Language modeling experimentsfollow the same model and training hyperparameters as Transformer-XL. WikiText-103 experimentsuse 16-layer Transformers (10 heads, 410 hidden size, 2100 intermediate FF), enwik8 – 12 layer Transformers (8 heads, 512 hidden size, 2048 intermediate FF).
We used Adam optimizer Kingma and Ba (2015) with linear schedule learning rate starting from 0 . 00025 for 200,000 steps for WikiText-103and 400,000 steps for enwik8. We refer to Transformer-XL with memory size equal to zero as a
Baseline. With this experimental setup we were able to reproduce results for the Transformer-XLmodel close to the original paper.
Algorithmic Tasks. We evaluate RMT on algorithmic tasks that require information about the wholeinput sequence to be solved successfully. In a recurrent setting, the model has to keep informationabout all previous segments to make predictions.
In the Copy task, an input sequence should be replicated twice after a special start-to-generate token.
In the Reverse task, an input sequence should be generated in a reverse order.
Input for the Associative Retrieval task consists of N key-value pairs.
Then one key is randomly selected, and the task is toproduce an appropriate value for the selected key.
Another task is to solve quadratic equations.
Oneexample consists of an equation, its solution with discriminant, and an answer.
The task is to generatea solution and answer, while only answer quality is evaluated.
For all tasks, input and output sequences are split into segments and processed by models sequentially.
Datasets for algorithmic tasks were randomly pre-generated, the same data was used in all experiments,and character-level tokenization was used.
Because Transformer-XL and RMT are decoder-only
Transformer models, we don’t compute loss over the input sequence before the start-to-generatetoken.
The loss is computed over target sequence segments only (see Appendix A.1 for details).
Language Modeling and NLP.
We use two standard benchmarks for language modeling: WikiText103 and enwik8.
[-WikiText
-]{+WikiText-
+}
(Merity et al., 2017) is used for word-level language modelingand contains 103M words from English Wikipedia articles.
Enwik8 (Mahoney, 2006) is used forcharacter-level and consists of 10 8 first bytes of XML text dump of the English Wikipedia.
Vocabularycontains 267735 words and 204 characters for Wikitext-103 and enwik8 tokenizers accordingly.
We compare Recurrent Memory Transformer with decoder-only Transformer and Transformer-XLas baselines.
Model size and training parameters are selected to match Transformer-XL paper.
For
Wikitext-103 an input context length was set to 150 tokens, and for enwik8 it was set to 512 characters.
Another set of experiments inspected how RMT handles long-term dependencies and recurrence.
Weincreased the number of segments and recurrent steps by making segments smaller (50 tokens for
WikiText-103, 128 characters for enwik8).
The increased number of recurrent steps makes languagemodeling tasks harder for RMT because information has to be stored in the same amount of memoryfor more steps.
As a testbed for the real life application scenario we select popular long text classification benchmark
Hyperpartisan news (Kiesel et al., 2019).
Instead of pretraining RMT from scratch we add recurrentmemory mechanism to the most widely adopted models from HuggingFace Transformers (Wolf et al.,2020).
Specifically, we augment 500 input tokens of already pretrained BERT-base, RoBERTa-base, DeBERTa-base and T5-base with the recurrent memory of size 10 and fine-tune on the target task.
Results
Baseline, Transformer-XL (Tr-XL) and RMT perform perfectly in the single segment setting oncopy and reverse tasks (Figure 3).
In this case the models do not need recurrence because the wholesequence is available.
When the number of segments is larger than one, non-recurrent baselinestruggles to solve tasks, but both memory models demonstrate ability to retain required informationfrom the previous segments in memory.
On Copy and Reverse tasks as a number of segments increases, RMT starts to outperform TransformerXL with memory sizes less than the number of all previous tokens.
With the number of segmentsup to 6 mean accuracy of Transformer-XL drops by up to 0.2 points, and with 9 segments plungesclose to the baseline without memory.
Associative Retrieval results are similar with the number ofsegments up to 4.
RMT manages to solve the task with Transformer-XL closely behind.
However,in the setting with 5 segments, RMT performance slightly decreases and Transformer-XL averageaccuracy rises higher.
We analyze how a number of segments, sequence length, a length of training context, and memorysize affect models’ performance on Copy task (Figure 4).
As we split a sequence into more segmentsit becomes more crucial to be able to pass information between segments.
We split 360 tokens ofsource + target sequence into multiple segments.
In Figure 4a we observe that Transformer-XLperformance starts to degrade and eventually falls to the baseline model performance as the number ofsegments increases. In contrast, RMT continues to solve the task perfectly. In a more extreme setting,when we keep memory size fixed, but increase the total length of a sequence to copy Transformer-XLfails shortly, while RMT starts to gradually degrade only after the length of 720 tokens (Figure 4b). On the Quadratic Equations task (Table 1) we have checked that it is possible to solve the task withthe Transformer baseline and no segmentation used.
The baseline in this case defines upper bound forthis task.
With multiple segments recurrency RMT solves the task perfectly, while Transformer-XLfinds the task challenging.
[Table]
Results of experiments on word-level languagemodeling on WikiText-103 are shown in Table 2.
In the first section with a segment lengthof 150, Tr-XL and RMT outperform the baseline and Memory Transformer (MemTr) by alarge margin.
It shows the significance of increased effective context length by Tr-XL cacheor RMT memory for language modeling.
RMTimproves over MemTr memory mechanism withread/write blocks.
The best RMT models withmemory size 10 and 25 show similar performance as Transformer-XL with a memory size equal to75.
RMT learns to use smaller memory more effectively than Transformer-XL.
Additionally, thesmaller memory size of RMT leads to reducing required GPU memory for running the model.
To force models to process longer recurrent dependencies the size of a segment is set to 50, so thenumber of recurrent steps increases.
RMT with memory size 1 shows similar results to Transformer- [Figure] XL with memory size 10.
It is worth noting that Transformer-XL memory consists of hiddenrepresentations from all layers (in this case, it is 10 × 16 vectors) when RMT memory is onlymemory_size vectors.
Transformer-XL with memory size 50 and RMT with memory size 5 showssimilar perplexity values (see Appendix A.5). RMT could be combined with Tr-XL cache. In this case Tr-XL cache could be seen as short-termmemory keeping the nearest context and RMT memory as long-term memory. Such combinationleads to the best results on WikiText-103 improving over Tr-XL.
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[Table] On enwik8 RMT models with memory size 5and Transformer-XL with memory size 40 showsimilar results. Confirming that RMT learns touse smaller amounts of memory representationmore effectively.
All results for enwik8 datasetare shown in Appendix A.4.
Recurrent Memory Transformer learns to makepredictions depending on #BPTT_unrollsover previous segments +1 current segment.
Transformer-XL does not use BPTT and reliesonly on memory_size cached states and current segment making in total: memory_size+ segment_length tokens.
In Figure 5a, wecompare RMT and Tr-XL according to the described value of visible context at training time.
RMT with a single memory vector couldbe trained to achieve lower perplexity as Transformer-XL with memory size 10.
It meansthat RMT can learn to compress informationfrom the previous observations better.
Anotherobservation is that RMT with memory sizes 10and 25 performs only a bit weaker comparedto Transformer-XL even when Transformer-XLhas access to more non-compressed states (50,100, 200) from previous segments.
In general, training RMT with unrolling gradients in earliersegments drastically improves scores thus showing the importance of BPTT training but, we observeinstabilities and out-of-memory issues during RMT training for a larger memory sizes with deeper BPTT unrolls.
RMT wins a lot when only one memory token is added but then the effect from increasing memorysize from 5 to 50 fades (Figure 5b).
Still, RMT with memory size 5 have performance on par with Transformer-XL with cache 50, confirming that RMT learns to store more compact representations.
The results suggest that there is some optimal memory size for RMT to solve the task, and furtherincrease does not add much.
Proposed recurrent memory mechanism affects only input and gradient flows of the augmented coremodel.
This might be an important advantage because the memory can be added to already pretrained
[Figure]
model. Evaluation results for four memory augmented language models fine tuned for long textclassification are presented in the Table 3.
Incorporation of 10 memory tokens in the input sequenceof 512 allows to encode longer stretches of a text up to 2000 tokens and significantly improve metricsfor the majority of models. Moreover, a combination of recurrent memory with RoBERTa-base resultsin state of the art performance for the Hyperpartisan news classification task (Kiesel et al., 2019). Interestingly, many competing models have input size of 4096 that at least twice longer compared to RMT extended counterparts but still lag behind.
[Figure] [Table]
To get an understanding ofmemory operations, learnedby RMT for algorithmictasks we visualise attention maps for copy and reverse tasks (Figure 6).
Ineach RMT attention map sequence tokens are precededby read memory, located atthe top left corner, and followed by write memory atthe bottom right.
Diagonal at the central part of thefig.6(a) (top) shows classicattention of token sequenceto itself, but the bottom diagonal represents the operation of writing of sequence tokens to memory instraight order.
When completing reverse (fig.6(a) bottom) the model learns to write the sequence tothe memory in the reversed order, which is in line with common sense.
When it comes to reproducing the target sequence, the model accesses memory (fig.6(b)) and writesto the output sequence.
Another operation (fig.6(c)) is rewriting from read memory to write memory.
It is commonly used by RMT in settings with larger number of segments to keep information aboutrecent segments longer.
Transformer-XL mechanism of accessing memory (fig.6(d)) does not allow straightforward writingto memory without changing sequence token representations.
Sequential reading from cache isrepresented by diagonals on Transformer-XL attention maps.
Using token representations as storageharms model performance in tasks with larger number of segments.
For reverse task with 4 segments Transformer-XL with limited memory size 6 (Appendix B Figure 9(b)) attempts to mix representationsof tokens and read multiple symbols from one cached state in the next segments giving averageaccuracy of 0.8 on the target task.
Despite having the same memory size, RMT manages to compressthe whole segment in memory tokens (Appendix B Figure 9(a)) and achieve mean accuracy 1.
Visualizations from Figure 6 and Appendix B Figure 9 provide evidence to support our hypothesesthat Tr-XL has to mix representations from previous and current segments in the same hidden statesto pass information between segments.
Also, visualizations show how memory tokens in RMT help mitigate such kind of mixing.
RMT ability of sequence compression to memory is illustrated in Appendix A.1 Figure 8.
For copy with 6 segments RMT compresses and then reads the sequenceof 12 tokens with just 6 memory tokens.
For Transformer-XL decreasing memory size harms theaccuracy score significantly with number of segments larger than 2.
Conclusions
In this paper we introduced Recurrent Memory Transformer a simple recurrent memory augmentationof Transformer model.
RMT is implemented by extension of an input sequence with special globalmemory tokens and segment-level recurrence.
Importantly, our method allows to learn more compactsequence representations and improve existing pretrained models without extensive additional compute, thus making practical machine learning applications more energy efficient and environmentallyfriendly.
In our experiments we compared RMT with Transformer baseline and Transformer-XL whichis a well-known modification of Transformer for long sequences.
RMT almost perfectly solves
Copy, Reverse as well as quadratic equations tasks for sequences consisting of multiple segmentsoutperforming Transformer-XL.
It also demonstrates quality for associative retrieval task on parwith Transformer-XL.
As expected, baseline Transformer fails to solve these tasks for multi-segmentsettings.
RMT trained as a language model performs significantly ahead of Transformer baseline and showsquality metrics similar to Transformer-XL but for up to 10 times smaller memory size.
Experimentalresults demonstrate that for fixed memory size backpropagating gradients for more segments improvesperformance of RMT.
Proposed approach to memory augmentation is quite universal and might beeasily applied to any pretrained transformer based model as demonstrated by achievement of state ofthe art results for long text classification task by fine tuning a combination of RoBERTa and RMT.
Analysis of attention maps suggests that better RMT performance can be related to more effectivestorage of input representations in dedicated memory tokens compared to mixing representationsstorage in Transformer-XL. RMT could be combined with Transformer-XL cache and improve theperformance of both models.
Overall, results of the study show that dedicated memory storage and recurrence provided by Recurrent Memory Transformer make it a promising architecture for applications that require learningof long-term dependencies and general purpose in-memory processing, such as algorithmic tasks andreasoning.
Furthermore, we believe that RMT could open the way for adding memory and recurrenceto other models in the Transformer family.
Checklist
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[Yes] We mention training instabilities and GPU RAM issues in Section 5.
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[No] The proposed model and method do not have any specific impacts.
All general negative
societal impacts applicable to the field could be potentially relative.
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(c) Did you report error bars (e.g., with respect to the random seed after running exper-
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[Yes] All the key experiments results are reported with std.
Furthermore, we provide raw experimental data in the supplementary materials.
(d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
[Yes] We used different GPUs depending on the task: 1080Ti, V100, A100.
We provide this information in Appendix A for each task.
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[No] Tr-XL licence is Apache 2.0 and available at its github repo.
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from the Wikipedia (Wikitext-103, enwik8).
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A Training details and additional results
A.1 Algorithmic tasks
Datasets were randomly generated by uniformly sampling tokens from dictionary into task sequencesand generating targets accordingly to the tasks.
After generation, datasets are fixed for all experiments.
Copy and reverse use sequences of sizes 24, 40, 120, 240, and 360, making total copy/reverseinput length 48/72, 80/120, 240/360, 480/720, 720/1080.
The associative retrieval task consists of 4key-value pairs and one randomly selected key; the answer consists of one value.
Train, validationand test sizes of copy 24, reverse 24 and associative retrieval datasets are 100000, 5000 and 10000.
Transformer-XL had the same cache size on training and validation to match RMT.
For training all models on copy and reverse, we used constant learning rate 1e-4 with reduction onplateau with decay factor of 0.5.
Copy and reverse were solved by models with 4 layers and 4 heads,associative retrieval models had 6 layers and 4 heads.
Models with the same context size and memorysize were trained for the same number of steps and the same training parameters.
Experiments with sequence length 24 were conducted on a single Nvidia GTX 1080 Ti GPU from 1hour to 2-3 days.
Copy and reverse on longer sequence lengths were done on more powerful Tesla
V100 using 1-3 devices with training time varying from 1 hour to 3-4 days.
A.2 Associative retrieval
We used code for the task dataset generation from (Ba et al., 2016) 3 .
A.3 Quadratic equations
This dataset consists of equations with integer coefficients and step-by-step solutions using thediscriminant.
Process of equation generation is started from uniformly sampling real roots x 1 , x 2from -100 to 100.
The answer of an equation is represented as x 1 , x 2 .
Next, we find the equation asmultiplication of two parentheses ( x − x 1 )( x − x 2 ) = 0 , which is expanded to x 2 − ( x 1 + x 2 ) x
+x 1 x 2 = 0 .
Next, we multiply all coefficients by a random natural number α from 1 to 10.
The finalequation form is αx 2 − α ( x 1 + x 2 ) x
+ αx 1 x 2 = 0 .
A dataset sample is made of these stages inreversed order.
We also provide a string with the discriminant calculation to help find the equationroots.
20 percent of equations in the dataset do not have real roots.
Example equation string:-4*xˆ2+392*x-2208=0 ,solution string:xˆ2-98*x+552=0;D=98ˆ2-4*1*552=7396=86ˆ2;x=(98-86)/2=6;x=(98+86)/2=92 ,and answer:6,
Each solution step is tokenized on char level and padded to the length of 30 tokens.
The total lengthof each training sample is 180, the dataset has 100000 training, 10000 validation and 20000 testsamples.
For this task we used models with 6 layers, 6 heads and segment sizes 180 and 30.
Trained wasperformed with the same schedule as copy and reverse on a single GTX 1080 ti for 1-2 days.
Memorysize for RMT and Transformer-XL was chosen equal to the segment length.
A.4 Enwik8
We verified our experimental setup by reproducing Transformer-XL results on enwik8 dataset(Table 4).
We used 12-layer Baseline (Transformer), Transformer-XL, RMT in all enwik8 experiments.
All results on enwik8 dataset are in Table 4.
We used 2 NVIDIA A100 80Gb GPUs, training timevaried from 10 to 30 hours depending on sequence length, memory size, and number of BPTT unrolls.
A.5 WikiText-103
We used 16-layer models in all experiments on WikiText-103 dataset.
Training hyperparameterswere used from (Dai et al., 2019) and authors PyTorch scripts 4 .
All results on WikiText-103 datasetare in Table 5.
In most of the WikiText-103 experiments, we used 2 NVIDIA A100 80Gb GPUs,training time varied from 10 to 30 hours depending on sequence length, memory size, and numberof BPTT unrolls.
All models except the ones noted with 2x steps were trained for 200k batches.
Transformer-XL did not benefit from longer training unlike the Tr-XL + RMT model.
For trainingthe combined model we used an auxiliary loss for memory tokens, it was added to the main loss witha multiplier of 0 .
We set a new fixed special token to be predicted from memory as target in theauxiliary loss.
[Table] [Table]