Title: DEPTH-ADAPTIVE TRANSFORMER

Abstract: State of the art sequence-to-sequence models for large scale tasks perform a fixed number of computations for each input sequence regardless of whether it is easy or hard to process. In this paper, we train Transformer models which can make output predictions at different stages of the network and we investigate different ways to predict how much computation is required for a particular sequence. Unlike dynamic computation in Universal Transformers, which applies the same set of layers iteratively, we apply different layers at every step to adjust both the amount of computation as well as the model capacity. On IWSLT German-English translation our approach matches the accuracy of a well tuned baseline Transformer while using less than a quarter of the decoder layers.

1. INTRODUCTION:

The size of modern neural sequence models (Gehring et al., 2017; Vaswani et al., 2017; Devlin et al., 2019) can amount to billions of parameters (Radford et al., 2019) . For example, the winning entry of the WMT'19 news machine translation task in English-German used an ensemble totaling two billion parameters (Ng et al., 2019) . While large models are required to do better on hard examples, small models are likely to perform as well on easy ones, e.g., the aforementioned ensemble is probably not required to translate a short phrase such as "Thank you". However, current models apply the same amount of computation regardless of whether the input is easy or hard.

In this paper, we propose Transformers which adapt the number of layers to each input in order to achieve a good speed-accuracy trade off at inference time. We extend Graves (2016; ACT) who introduced dynamic computation to recurrent neural networks in several ways: we apply different layers at each stage, we investigate a range of designs and training targets for the halting module and we explicitly supervise through simple oracles to achieve good performance on large-scale tasks.

Universal Transformers (UT) rely on ACT for dynamic computation and repeatedly apply the same layer (Dehghani et al., 2018) . Our work considers a variety of mechanisms to estimate the network depth and applies a different layer at each step. Moreover, Dehghani et al. (2018) fix the number of steps for large-scale machine translation whereas we vary the number of steps to demonstrate substantial improvements in speed at no loss in accuracy. UT uses a layer which contains as many weights as an entire standard Transformer and this layer is applied several times which impacts speed. Our approach does not increase the size of individual layers. We also extend the resource efficient object classification work of Huang et al. (2017) and Bolukbasi et al. (2017) to structured prediction where dynamic computation decisions impact future computation. Related work from computer vision includes Teerapittayanon et al. (2016) ; Figurnov et al. (2017) and Wang et al. (2018) who explored the idea of dynamic routing either by exiting early or by skipping layers.

We encode the input sequence using a standard Transformer encoder to generate the output sequence with a varying amount of computation in the decoder network. Dynamic computation poses a challenge for self-attention because omitted layers in prior time-steps may be required in the future. We experiment with two approaches to address this and show that a simple approach works well ( §2). Next, we investigate different mechanisms to control the amount of computation in the decoder network, either for the entire sequence or on a per-token basis. This includes multinomial and binomial classifiers supervised by the model likelihood or whether the argmax is already correct as well as simply thresholding the model score ( §3). Experiments on IWSLT14 German-English translation (Cettolo et al., 2014) as well as WMT'14 English-French translation show that we can match the performance of well tuned baseline models at up to 76% less computation ( §4).

2. ANYTIME STRUCTURED PREDICTION:

We first present a model that can make predictions at different layers. This is known as anytime prediction for computer vision models (Huang et al., 2017) and we extend it to structured prediction.

2.1. TRANSFORMER WITH MULTIPLE OUTPUT CLASSIFIERS:

We base our approach on the Transformer sequence-to-sequence model (Vaswani et al., 2017) . Both encoder and decoder networks contain N stacked blocks where each has several sub-blocks surrounded by residual skip-connections. The first sub-block is a multi-head dot-product self-attention and the second a position-wise fully connected feed-forward network. For the decoder, there is an additional sub-block after the self-attention to add source context via another multi-head attention.

Given a pair of source-target sequences (x, y), x is processed with the encoder to give representations s = (s 1 , . . . , s |x| ). Next, the decoder generates y step-by-step. For every new token y t input to the decoder at time t, the N decoder blocks process it to yield hidden states (h n t ) 1≤n≤N :

EQUATION

where block n is the mapping associated with the n th block and embed is a lookup table.

The output distribution for predicting the next token is computed by feeding the activations of the last decoder layer h N t into a softmax normalized output classifier W :

EQUATION

Standard Transformers have a single output classifier attached to the top of the decoder network. However, for dynamic computation we need to be able to make predictions at different stages of the network. To achieve this, we attach output classifiers C n parameterized by W n to the output h n t of each of the N decoder blocks: ∀n, p(y t+1 |h n t ) = softmax(W n h n t )

(3) The classifiers can be parameterized independently or we can share the weights across the N blocks.

2.2. TRAINING MULTIPLE OUTPUT CLASSIFIERS:

Dynamic computation enables the model to use any of the N exit classifiers instead of just the final one. Some of our models can choose a different output classifier at each time-step which results in an exponential number of possible output classifier combinations in the sequence length.

We consider two possible ways to train the decoder network (Figure 1 ). Aligned training optimizes all classifiers simultaneously and assumes all previous hidden states required by the self-attention are available. However, at test time this is often not the case when we choose a different exit for every token which leads to misaligned states. Instead, mixed training samples several sequences of exits for a given sentence and exposes the model to hidden states from different layers.

Generally, for a given output sequence y, we have a sequence of chosen exits (n 1 , . . . , n |y| ) and we denote the block at which we exit at time t as n t .

2.2.1. ALIGNED TRAINING:

Aligned training assumes all hidden states h n-1 1 , . . . , h n-1 t are available in order to compute selfattention and it optimizes N loss terms, one for each exit (Figure 1a ):

EQUATION

)

The compound loss L dec (x, y) is a weighted average of N terms w.r.t. to (ω 1 , . . . ω N ). We found that uniform weights achieve better BLEU compared to other weighing schemes (c.f . Appendix A). At inference time, not all time-steps will have hidden states for the current layer since the model exited early. In this case, we simply copy the last computed state to all upper layers, similar to mixed training ( §2.2.2). However, we do apply layer-specific key and value projections to the copied state.

2.2.2. MIXED TRAINING:

Aligned training assumes that all hidden states of the previous time-steps are available but this assumption is unrealistic since an early exit may have been chosen previously. and evaluate the following loss:

EQUATION

|y| ).

(5)

When n t < N , we copy the last evaluated hidden state h n t to the subsequent layers so that the self-attention of future time steps can function as usual (see Figure 1b ).

3. ADAPTIVE DEPTH ESTIMATION:

We present a variety of mechanisms to predict the decoder block at which the model will stop and output the next token, or when it should exit to achieve a good speed-accuracy trade-off. We consider two approaches: sequence-specific depth decodes all output tokens using the same block ( §3.1) while token-specific depth determines a separate exit for each individual token ( §3.2).

We model the distribution of exiting at time-step t with a parametric distribution q t where q t (n) is the probability of computing block 1 , . . . , block n and then emitting a prediction with C n . The parameters of q t are optimized to match an oracle distribution q * t with cross-entropy:

EQUATION

The exit loss (L exit ) is back-propagated to the encoder-decoder parameters. We simultaneously optimize the decoding loss (Eq. ( 4)) and the exit loss (Eq. ( 6)) balanced by a hyper-parameter α to ensure that the model maintains good generation accuracy. The final loss takes the form:

EQUATION

In the following we describe for each approach how the exit distribution q t is modeled (illustrated in Figure 2 ) and how the oracle distribution q * t is inferred. 

s 2 C 2 C 2 C 2 (a) Sequence-specific depth Decoder depth Decoding step 2 C 2 1 C 1 3 C 3 (b) Token-specific -Multinomial Decoder depth Decoding step C S C 2 S C 1 C C C 3 (c) Token-specific -Geometric-like

Figure 2 : Variants of the adaptive depth prediction classifiers. Sequence-specific depth uses a multinomial classifier to choose an exit for the entire output sequence based on the encoder output s (2a).

It then outputs a token at this depth with classifier C n . The token-specific multinomial classifier determines the exit after the first block and proceeds up to the predicted depth before outputting the next token (2b). The token geometric-like classifier (2c) makes a binary decision after every block to dictate whether to continue (C) to the next block or to stop (S) and emit an output distribution.

3.1. SEQUENCE-SPECIFIC DEPTH::

For sequence-specific depth, the exit distribution q and the oracle distribution q * are independent of the time-step so we drop subscript t. We condition the exit on the source sequence by feeding the average s of the encoder outputs to a multinomial classifier:

EQUATION

where W h and b h are the weights and biases of the halting mechanism. We consider two oracles to determine which of the N blocks should be chosen. The first is based on the sequence likelihood and the second looks at an aggregate of the correctly predicted tokens at each block.

Likelihood-based: This oracle is based on the likelihood of the entire sequence after each block and we optimize it with the Dirac delta centered around the exit with the highest sequence likelihood.

q * (x, y) = δ(arg max n LL n ).

We add a regularization term to encourage lower exits that achieve good likelihood:

EQUATION

Correctness-based: Likelihood ignores whether the model already assigns the highest score to the correct target. Instead, this oracle chooses the lowest block that assigns the largest score to the correct prediction. For each block, we count the number of correctly predicted tokens over the sequence and choose the block with the most number of correct tokens. A regularization term controls the trade-off between speed and accuracy.

EQUATION

)

Oracles based on test metrics such as BLEU are feasible but expensive to compute since we would need to decode every training sentence N times. We leave this for future work.

3.2. TOKEN-SPECIFIC DEPTH::

The token-specific approach can choose a different exit at every time-step. We consider two options for the exit distribution q t at time-step t: a multinomial with a classifier conditioned on the first decoder hidden state h 1 t and a geometric-like where an exit probability χ n t is estimated after each block based on the activations of the current block h n t .

Multinomial q t :

EQUATION

The most probable exit arg max q t (n|x, y <t ) is selected at inference.

EQUATION

EQUATION

where, d is the dimension of the decoder states, W h ∈ R N ×d and w h ∈ R d are the weights of the halting mechanisms, and b h their biases. During inference the decoder exits when the halting signal χ n t exceeds a threshold τ n which we tune on the valid set to achieve a better accuracy-speed trade-off. If thresholds (τ n ) 1≤n<N have not been exceeded, then we default to exiting at block N .

The two classifiers are trained to minimize the cross-entropy with respect to either one the following oracle distributions:

Likelihood-based: At each time-step t, we choose the block whose exit classifier has the highest likelihood plus a regularization term weighted by λ to encourage lower exits.

EQUATION

This oracle ignores the impact of the current decision on the future time-steps and we therefore consider smoothing the likelihoods with an RBF kernel.

EQUATION

where we control the size of the surrounding context with σ the kernel width. We refer to this oracle as LL(σ, λ) including the case where we only look at the likelihood of the current token with σ → 0.

Correctness-based: Similar to the likelihood-based oracle we can look at the correctness of the prediction at time-step t as well as surrounding positions. We define the target q * t as follows:

EQUATION

EQUATION

)

Confidence thresholding Finally, we consider thresholding the model predictions ( §2), i.e., exit when the maximum score of the current output classifier p(y t+1 |h n t ) exceeds a hyper-parameter threshold τ n . This does not require training and the thresholds τ = (τ 1 , . . . , τ N -1 ) are simply tuned on the valid set to maximize BLEU. Concretely, for 10k iterations, we sample a sequence of thresholds τ ∼ U(0, 1) N -1 , decode the valid set with the sampled thresholds and then evaluate the BLEU score and computational cost achieved with this choice of τ . After 10k evaluations we pick the best performing thresholds, that is τ with the highest BLEU in each cost segment.

4. EXPERIMENTS:



4.1. EXPERIMENTAL SETUP:

We evaluate on several benchmarks and measure tokenized BLEU (Papineni et al., 2002) :

IWSLT'14 German to English (De-En). We use the setup of Edunov et al. (2018) and train on 160K sentence pairs. We use N = 6 blocks, a feed-forward network (ffn) of intermediate-dimension .6] for all tokens, as well as the average BLEU over the fixed exits. As baseline we show six standard Transformer models with 1-6 blocks.

Uniform n = 1 n = 2 n = 3 n = 4 n = 5 n =

1024, 4 heads, dropout 0.3, embedding dimension d enc = 512 for the encoder and d dec = 256 for the decoder. Embeddings are untied with 6 different output classifiers. We evaluate with a single checkpoint and a beam of width 5.

WMT'14 English to French (En-Fr). We also experiment on the much larger WMT'14 English-French task comprising 35.5m training sentence pairs. We develop on 26k held out pairs and test on newstest14. The vocabulary consists of 44k joint BPE types (Sennrich et al., 2016) . We use a Transformer big architecture and tie the embeddings of the encoder, the decoder and the output classifiers ((W n ) 1≤n≤6 ; §2.1). We average the last ten checkpoints and use a beam of width 4.

Models are implemented in fairseq (Ott et al., 2019) and are trained with Adam (Kingma & Ba, 2015) . We train for 50k updates on 128 GPUs with a batch size of 460k tokens for WMT'14 En-Fr and on 2 GPUs with 8k tokens per batch for IWSLT'14 De-En. To stabilize training, we re-normalize the gradients if the norm exceeds g clip = 3.

For models with adaptive exits, we first train without exit prediction (α = 0 in Eq. ( 7)) using the aligned mode (c.f . §2.2.1) for 50k updates and then continue training with α = 0 until convergence. The exit prediction classifiers are parameterized by a single linear layer (Eq. ( 8)) with the same input dimension as the embedding dimension, e.g., 1024 for a big Transformer; the output dimension is N for a multinomial classifier or one for geometric-like. We exit when χ t,n > 0.5 for geometric-like classifiers.

We first compare the two training regimes for our model ( §2.2). Aligned training performs selfattention on aligned states ( §2.2.1) and mixed training exposes self-attention to hidden states from different blocks ( §2.2.2).

We compare the two training modes when choosing either a uniformly sampled exit or a fixed exit n = 1, . . . , 6 at inference time for every time-step. The sampled exit experiment tests the robustness to mixed hidden states and the fixed exit setup simulates an ideal setting where all previous states are available. As baselines we show six separate standard Transformers with N ∈ [1..6] decoder blocks. All models are trained with an equal number of updates and mixed training with M =6 paths is most comparable to aligned training since the number of losses per sample is identical.

Table 1 shows that aligned training outperforms mixed training both for fixed exits as well as for randomly sampled exits. The latter is surprising since aligned training never exposes the self-attention mechanism to hidden states from other blocks. We suspect that this is due to the residual connections which copy features from lower blocks to subsequent layers and which are ubiquitous in Transformer models ( §2). Aligned training also performs very competitively to the individual baseline models.

Aligned training is conceptually simple and fast. We can process a training example with N exits in a single forward/backward pass while M passes are needed for mixed training. In the remaining paper, we use the aligned mode to train our models. Appendix A reports experiments with weighing the various output classifiers differently but we found that a uniform weighting scheme worked well. On our largest setup, WMT'14 English-French, the training time of an aligned model with six output classifiers increases only marginally by about 1% compared to a baseline with a single output classifier keeping everything else equal. 

Next, we train models with aligned states and compare adaptive depth classifiers in terms of BLEU as well as computational effort. We measure the latter as the average exit per output token (AE).

As baselines we use again six separate standard Transformers with N ∈ [1..6] with a single output classifier. We also measure the performance of the aligned mode trained model for fixed exits n ∈ [1..6]. For the adaptive depth token-specific models (Tok), we train four combinations: likelihoodbased oracle (LL) + geometric-like, likelihood-based oracle (LL) + multinomial, correctness based oracle (C) + geometric-like and correctness-based oracle (C) + multinomial. Sequence-specific models (Seq) are trained with the correctness oracle (C) and the likelihood oracle (LL) with different values for the regularization weight λ. All parameters are tuned on the valid set and we report results on the test set for a range of average exits.

Figure 3 shows that the aligned model (blue line) can match the accuracy of a standard 6-block Transformer (black line) at half the number of layers (n = 3) by always exiting at the third block. The aligned model outperforms the baseline for n = 2, . . . , 6.

For token specific halting mechanisms (Figure 3a ) the geometric-like classifiers achieves a better speed-accuracy trade-off than the multinomial classifiers (filled vs. empty triangles). For geometriclike classifiers, the correctness oracle outperforms the likelihood oracle (Tok-C geometric-like vs. Tok-LL geometric-like) but the trend is less clear for multinomial classifiers. At the sequence-level, likelihood is the better oracle (Figure 3b ).

The rightmost Tok-C geometric-like point (σ = 0, λ = 0.1) achieves 34.73 BLEU at AE = 1.42 which corresponds to similar accuracy as the N = 6 baseline at 76% fewer decoding blocks.

Published as a conference paper at ICLR 2020 Figure 5 : Speed and accuracy on the WMT'14 English-French benchmark (c.f . Figure 3 ).

The best accuracy of the aligned model is 34.95 BLEU at exit 5 and the best comparable Tok-C geometric-like configuration achieves 34.99 BLEU at AE = 1.97, or 61% fewer decoding blocks. When fixing the budget to two decoder blocks, Tok-C geometric-like with AE = 1.97 achieves BLEU 35, a 0.64 BLEU improvement over the baseline (N = 2) and aligned which both achieve BLEU 34.35.

Confidence thresholding (Figure 3c ) performs very well but cannot outperform Tok-C geometriclike.

Ablation of hyper-parameters In this section, we look at the effect of the two main hyperparameters on IWSLT'14 De-En: λ the regularization scale (c.f . Eq. ( 9)), and the RBF kernel width σ used to smooth the scores (c.f . Eq. ( 15)). We train Tok-LL Geometric-like models and evaluate them with their default thresholds (exit if χ n t > 0.5). Figure 4a shows that higher values of λ lead to lower exits. Figure 4b shows the effect of σ for two values of λ. In both curves, we see that wider kernels favor higher exits.

4.4. SCALING THE ADAPTIVE-DEPTH MODELS:

Finally, we take the best performing models form the IWSLT benchmark and test them on the large WMT'14 English-French benchmark. Results on the test set (Figure 5a ) show that adaptive depth still shows improvements but that they are diminished in this very large-scale setup. Confidence thresholding works very well and sequence-specific depth approaches improve only marginally over the baseline. Tok-LL geometric-like can match the best baseline result of BLEU 43.4 (N = 6) by using only AE = 2.40 which corresponds to 40% of the decoder blocks; the best aligned result of BLEU 43.6 can be matched with AE = 3.25. In this setup, Tok-LL geometric-like slightly outperforms the Tok-C counterpart.

Confidence thresholding matches the accuracy of the N =6 baseline with AE 2.5 or 59% fewer decoding blocks. However, confidence thresholding requires computing the output classifier at each block to determine whether to halt or continue. This is a large overhead since output classifiers predict 44k types for this benchmark ( §4.1). To better account for this, we measure the average number of FLOPs per output token (details in Appendix B). Figure 5b shows that the Tok-LL geometric-like approach provides a better trade-off when the overhead of the output classifiers is considered.

4.5. QUALITATIVE RESULTS:

The exit distribution for a given sample can give insights into what a Depth-Adaptive Transformer decoder considers to be a difficult task. In this section, for each hypothesis y, we will look at the sequence of selected exits (n 1 , . . . , n | y| ) and the probability scores (p 1 , . . . p | y| ) with p t = p( y t |h nt t-1 ) i.e. the confidence of the model in the sampled token at the selected exit. Figures 6 and 7 show hypotheses from the WMT'14 En-Fr and IWSLT'14 De-En test sets, respectively. For each hypothesis we state the exits and the probability scores. In Figure 6a 

Score:

(a) Src: diesen trick können sie ihren freunden und nachbarn vor@@führen . danke .

Ref: there is a trick you can do for your friends and neighb@@ors . thanks . 'présent' (meaning 'present') is hard. A straightforward translation is 'était là' but the model chooses 'present' which is also appropriate. In Figure 6b , the model uses more computation to predict the definite article 'les' since the source has omitted the article for 'passengers'.

A clear trend in both benchmarks is that the model requires less computation near the end of decoding to generate the end of sequence marker 
 and the preceding full-stop when relevant. In Figure 8 , we show the distribution of the exits at the beginning and near the end of test set hypotheses. We consider the beginning of a sequence to be the first 10% of tokens and the end as the last 10% of tokens. The exit distributions are shown for three models on WMT'14 En-Fr: Model 1 has an average exit of AE = 2.53, Model 2 exits at AE = 3.79 on average and Model 3 with AE = 4.68. Within the same models, deep exits late are used at the beginning of the sequence and early exits are selected near the end. For heavily regularized models such as Model 1 with AE = 2.53, the disparity between beginning and end is less severe as the model exits early most of the time. Model 2 and Model 3 are less regularized (higher AE) and tend to use late exits at the beginning of the sequence and early exits near the end. On the other hand, the more regularized Model 1 with AE = 2.53 exits and near the end (rpos>0.9) of the hypotheses of three models.

1 2 3 4 5 6 0.0-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1.0 early most of the time. There is also a correlation between the model probability and the amount of computation, particularly in models with low AE . Figure 9 shows the joint histogram of the scores and the selected exit. For both Model 1 and Model 2 , low exits (n ≤ 2) are used in the high confidence range [0.8 -1] and high exits (n ≥ 4) are used in the low-confidence range [0 -0.5].

Model 3 has a high average exit (AE = 4.68) so most tokens exit late, however, in low confidence ranges the model does not exit earlier than n = 5.

5. CONCLUSION:

We extended anytime prediction to the structured prediction setting and introduced simple but effective methods to equip sequence models to make predictions at different points in the network. We compared a number of different mechanisms to predict the required network depth and find that a simple correctness based geometric-like classifier obtains the best trade-off between speed and accuracy. Results show that the number of decoder layers can be reduced by more than three quarters at no loss in accuracy compared to a well tuned Transformer baseline.

Algorithm 1 Pseudo-code for gradient scaling (illustrated for a single step t) 

:



Figures and Tables:

Figure 1: Training regimes for decoder networks able to emit outputs at any layer. Aligned training optimizes all output classifiers C n simultaneously assuming all previous hidden states for the current layer are available. Mixed training samples M paths of random exits at which the model is assumed to have exited; missing previous hidden states are copied from below.

Figure 3: Trade-off between speed (average exit or AE) and accuracy (BLEU) for depth-adaptive methods on the IWSLT14 De-En test set.

Figure 7: Example from the IWSLT'14 De-En test set with Tok-LL geometric-like depth estimation. See Figure 6 for more details.

Figure8: WMT'14 En-Fr test set: exit distributions in the beginning (relative-position: rpos<0.1) and near the end (rpos>0.9) of the hypotheses of three models.

Figure 9: Joint histogram of the exits and the confidence scores for 3 Tok-LL geometric-like models on newstest14.

Aligned vs. mixed training on IWSLT De-En. We report valid BLEU for a uniformly sampled exit n ∼ U([1..6]) at each token, a fixed exit n ∈ [1.

Table content: <table><tr><td>6 Average</td></tr></table>

, predicting Examples from the WMT'14 En-Fr test set (newstest14) with Tok-LL geometric-like depth estimation. Token exits are in blue and confidence scores are in gray. The '@@' are due to BPE or subword tokenization. For each example the source (Src) and the reference (Ref) are provided in the caption.

Table content: <table><tr><td/><td>6</td><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>1</td><td/><td/><td>6</td><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>1</td></tr><tr><td>Exit</td><td>1 2 3 4 5</td><td>Chi@@</td><td>rac</td><td>,</td><td>le</td><td>Premier</td><td colspan="2">ministre</td><td>,</td><td>était</td><td>présent</td><td>.</td><td>&lt;/s&gt;</td><td/><td>0.2 0.4 0.6 0.8</td><td/><td/><td>5 1 2 3 4</td><td>Mais</td><td>les</td><td>passagers</td><td>ne</td><td>devraient</td><td>pas</td><td>s'</td><td>attendre</td><td>à</td><td>des</td><td>changements</td><td>immédiats</td><td>.</td><td>&lt;/s&gt;</td><td>0.8 0.2 0.4 0.6</td><td>Score</td></tr><tr><td colspan="17">(a) Src: Chi@@rac , the Prime Minister , was there .</td><td colspan="15">(b) Src: But passengers shoul@@dn't expect changes to</td></tr><tr><td/><td colspan="14">Ref: Chi@@rac , Premier ministre , est là .</td><td/><td/><td/><td colspan="7">happen immediately .</td><td/><td/><td/><td/><td/><td/></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td colspan="14">Ref: Mais les passagers ne devraient pas s' attendre à</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td colspan="9">des changements immédiats .</td><td/><td/><td/><td/></tr><tr><td colspan="8">1 2 3 4 5 Figure 6: you Exit 6</td><td colspan="2">can</td><td>per@@</td><td>form</td><td>this</td><td>trick</td><td>to</td><td>your</td><td colspan="2">friends</td><td>and</td><td>neighb@@</td><td>ors</td><td>.</td><td>thank</td><td>you</td><td>.</td><td>&lt; /s&gt;</td><td/><td>0.2 0.4 0.6 0.8 1</td><td/><td/><td/></tr></table>

in PyTorch with x.detach(). 10: end functionUniform n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 Average

Table content: <table><tr><td>2: 3: 4: 5:</td><td colspan="6">h n t = block n (h n-1 t p(y t+1 |h n t ) = softmax(W n h n ) t ) p(y t+1 |h n t ) = SCALE_GRADIENT(p(y t+1 |h n t ), γ n ) if n &lt; N then h n t = SCALE_GRADIENT(h n t , 1 γ n+1</td><td>)</td><td/></tr><tr><td colspan="2">6: end for</td><td/><td/><td/><td/><td/><td/><td/></tr><tr><td colspan="6">7: function SCALE_GRADIENT(Tensor x, scale γ)</td><td/><td/><td/></tr><tr><td>8:</td><td colspan="5">return γx + (1 -γ)STOP_GRADIENT(x)</td><td/><td/><td/></tr><tr><td>9:</td><td/><td/><td/><td/><td/><td/><td/><td/></tr><tr><td/><td>Baseline</td><td>-</td><td>34.2</td><td>35.3</td><td>35.6</td><td colspan="2">35.7</td><td>35.6</td><td>35.9</td><td>35.4</td></tr><tr><td/><td>∅</td><td>35.5</td><td>34.1</td><td>35.5</td><td>35.8</td><td colspan="2">36.1</td><td>36.1</td><td>36.2</td><td>35.6</td></tr><tr><td/><td>γ = 0.3</td><td>35.1</td><td>33.7</td><td>34.7</td><td>35.3</td><td colspan="2">35.7</td><td>35.8</td><td>36.0</td><td>35.2</td></tr><tr><td/><td>γ = 0.5</td><td>35.4</td><td>34.8</td><td>35.4</td><td>35.6</td><td colspan="2">35.6</td><td>35.7</td><td>35.6</td><td>35.4</td></tr><tr><td/><td>γ = 0.7</td><td>34.9</td><td>34.6</td><td>35.1</td><td>35.1</td><td colspan="2">35.2</td><td>35.4</td><td>35.3</td><td>35.1</td></tr><tr><td/><td>γ = 0.9</td><td>34.9</td><td>34.8</td><td>35.3</td><td>35.3</td><td colspan="2">35.3</td><td>35.4</td><td>35.5</td><td>35.3</td></tr><tr><td/><td>γ = 1.1</td><td>35.1</td><td>34.9</td><td>35.2</td><td>35.3</td><td colspan="2">35.3</td><td>35.3</td><td>35.3</td><td>35.2</td></tr><tr><td/><td>Baseline</td><td>-</td><td>33.7</td><td>34.6</td><td>34.6</td><td colspan="2">34.6</td><td>34.6</td><td>34.8</td><td>34.5</td></tr><tr><td/><td>∅</td><td>34.4</td><td>33.2</td><td>34.4</td><td>34.8</td><td colspan="2">34.9</td><td>35.0</td><td>34.9</td><td>34.5</td></tr><tr><td/><td>γ = 0.3</td><td>34.2</td><td>32.8</td><td>33.9</td><td>34.3</td><td colspan="2">34.6</td><td>34.8</td><td>35.0</td><td>34.2</td></tr><tr><td/><td>γ = 0.5</td><td>34.5</td><td>33.8</td><td>34.2</td><td>34.6</td><td colspan="2">34.5</td><td>34.7</td><td>34.7</td><td>34.6</td></tr><tr><td/><td>γ = 0.7</td><td>34.0</td><td>33.7</td><td>34.2</td><td>34.3</td><td colspan="2">34.3</td><td>34.3</td><td>34.3</td><td>34.2</td></tr><tr><td/><td>γ = 0.9</td><td>34.1</td><td>34.0</td><td>34.2</td><td>34.3</td><td colspan="2">34.4</td><td>34.4</td><td>34.4</td><td>34.3</td></tr><tr><td/><td>γ = 1.1</td><td>34.2</td><td>34.0</td><td>34.3</td><td>34.3</td><td colspan="2">34.3</td><td>34.3</td><td>34.2</td><td>34.2</td></tr></table>

Aligned training with different gradient scaling ratios γ : 1 on IWSLT'14 De-En. For each model we report the BLEU4 score evaluated with a uniformly sampled exit n ∼ U([1..6]) for each token and a fixed exit n ∈ [1..6]. The average corresponds to the average BLEU4 of all fixed exits.

Table content: <table/>

Peer review (iclr2020):
[Post-reponse EDIT]: After reading the authors* response, and taking into account the other reviewers* opinions, I am willing to suggest acceptance of this paper on the condition that ***the authors will add the experimental results to address issue #7 and #8 of my original review (as they promised in their rebuttal)***. Before that, I*m keeping the current score, while noting that a more accurate rating should be *5 - Marginally below the acceptance threshold* (which is not available in ICLR this year). I want to thank the authors for addressing the other issues/questions I raised. ----------------------------------------- Original Review Below ----------------------------------------- Summary: The author proposes a new way to improve the supervision of training encoder-decoder Transformers so that it can make flexible, depth-adaptive predictions at inference time. The paper compares multiple dynamic computation schemes to investigate the effectiveness of these different approaches (e.g., at both sequence and token levels). Relatively extensive experiments were conducted in different settings (on a small and a large MT dataset). While the dynamic halting on Transformers has been previously explored in the Universal Transformers paper, this technique has not been applied to standard (encoder-)decoder Transformers. I think this is an interesting and exciting topic to pursue, as the authors seem to demonstrate (at least on the small IWSLT*14 DE-EN dataset) that we may not need as many layers at the decoding phase as we thought (and thus # of FLOPs). However, I think the paper can be further improved in terms of both its organization/clarity and its experimental study (see details following). -------------------------------------------- Questions/comments: 1. Please add a Related Work section. While using dynamic halting on standard Transformers may be novel, general dynamic routing on deep networks (which also aims to reduce the computation) is not. For example, SkipNet [1] uses an input-based dynamic skipping to bypass certain layers of a DNN. It*s a good scholarship to give proper credit to prior related work(s). 2. I think Poisson binomial (PB) may be a misnomer for the second qt modeling technique. PB is a probability distribution on the __sum__ of N (unequal) Bernoulli trials [2], which doesn*t make sense here. What you are looking for, I think, is a learned *coin flip* at every layer of the network to decide whether to continue to the next layer (i.e., you are looking for the first *head* in the coin flips, instead of the sum of heads). And indeed, in Eq. (9), the definition for qt(n|...) is a *geometric-alike* distribution with success probability chi_{t,n} at the n^th flip. (Which brings to a side point: for the *otherwise* case in Eq. (9), I think you mean prod_n (1-chi_{t, n}); otherwise you can*t guarantee sum_n qt(n|...) = 1 :-) ) 3. Why and how did you pick the exit threshold chi_{t, n} > 0.5 for your experiments? According to your definition of chi_{t,n} and q_t in Eq. (9), isn*t the correct thing to do to sample from chi_{t, n}? My major concern is that, given different sigma (which you didn*t specify, but I guess is something like a sigmoid), your chi_{t, n} may land in very different ranges. What if you have the following case: chi_{t, 1} = 0.501, chi_{t, 2} = 0.1, chi_{t,3} = 0.999, chi_{t,4} = 0.95? Doesn*t this mean it may worth it to stop at the 3rd layer instead? (Using the coin flip analogy, having a coin with head probability 0.501 doesn*t necessarily mean you *must* get a *head* at the first flip.) Moreover, if I use , won*t I always get a chi > 0.5? 4. In the token-specific likelihood-based method, you used an RBF kernel to model the influence of a time step t on its neighboring time steps. Two questions: 1) When you do sum_{t*}, did you also include t*<t? 2) Have you tried any other kernels, and how does this choice affect the performance? 5. The aligned training is actually a type of intermediate auxiliary loss (or deep supervision, as the computer vision community probably more often calls it), which makes it not surprising that the unaligned Transformer could perform slightly better than the baseline. I find it interesting that the mixed training strategy doesn*t work. 6. Why did you use different training schemes for WMT*14 and IWSLT*14 (e.g., freezing the parameters, etc.)? 7. One of the major concerns I have is that the empirical results don*t seem that impressive. First, to demonstrate the effectiveness of the proposed adaptive depth estimation methodology, besides comparing to the baseline (black line) in Figures 3 & 4, you should also compare with the blue line (which adds these auxiliary losses to the baseline, but doesn*t use the adaptive strategy that involves qt(n|...) at all) for fairness. From Figures 3 & 4, it seems that the adaptive-depth predictions are usually in the close neighborhood of the blue lines when at the same value of AE (instead of substantial, consistent improvement). In addition, while Tok-LL Poisson did well in Figure 4(b), it didn*t seem to make a difference in Figure 4(a), which is where one is supposed to tune the hyperparameters on (such as lambda, sigma; see #8 below). This makes me a bit dubious about how much help the adaptive module brought, compared to just using the aligned model. 8. For the WMT*14 experiments (Figure 4), why is there only one run of the *Tok-LL Poisson*? My understanding is that, if you are to use the proposed approach on a new dataset, you would want to try different settings of (sigma, lambda) on the validation set and pick the best one to use for testing. In Figure 4(a), for instance, for Seq-LL I would probably pick the (sigma, lambda) setting corresponding to the top-right yellow square--- which turns out to be slightly worse than the blue line on the test set (Figure 4(b)). It*d be useful to plot more *red triangles* in Figure 4 to evaluate how much the adaptive-depth methodology contributes to the performance. ======================== Some minor errors that don*t have much impact on the score: 9. The beginning sentence of the 4th paragraph of Section 1 is a bit strange (grammatically). 10. Not all equations are numbered! See section 2. 11. In the first equation of Section 2, do you mean h_{leq t}^{n-1} rather than h_{< t}^{n-1}? I think h_t from the previous layer is also used? 12. In the 3rd equation of Eq. (12), you should have m somewhere within the summation. 13. Equation (7) missing a right parenthesis. 14. In Eq. (8) W_h is a matrix, whereas in Eq. (9) W_h is a vector (if I*m not mistaken)? Maybe use a lowercase w. 15. Inconsistent notations. For instance, as I described in #3 above, you didn*t say in the paper what sigma means in Eq. (9), but later re-used the same letter for a different meaning in Eq. (11) for the RBF kernel. Another example is the usage of 	heta_n in the *Confidence thresholding* paragraph for the threshold value; you used the same letter again in the *Gradient scaling* section of the appendix but with a different meaning (learnable parameters). 16. In the last sentence of the Likelihood-based token specific depth, I*d suggest sigma ightarrow 0 instead of sigma=0, which would otherwise make frac{...}{sigma} in Eq. (11) undefined. 17. I don*t think you specified the 	heta_n used in your experiments (and how you tune them). 18. What is the FS in Appendix B? Also, for *ffn* in Table 4, it*s best to write its full name before referring to it with acronyms. 19. Just curious: how did you implement the gradient scaling described in Appendix A (e.g., ensuring the gamma_n only applies to block n, but not blocks < n)? ======================== I think the current shape of the paper is marginally below the acceptance threshold. (Note that while the rating is *3 - Weak reject*, I don*t mean the score, but only the *weak reject* part. I*m still excited about the idea of applying dynamic computation in Transformer.) But I*m happy to consider adjusting the score if my concerns above can be satisfactorily addressed. [1] https://arxiv.org/abs/1711.09485 [2] https://en.wikipedia.org/wiki/Poisson_binomial_distribution