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State-of-the-art performances on language comprehension tasks are achieved by huge language models pre-trained on massive unlabeled text corpora, with very light subsequent fine-tuning in a task-specific supervised manner.
It seems the pre-training procedure learns a very good common initialization for further training on various natural language understanding tasks, such that only few steps need to be taken in the parameter space to learn each task.
In this work, using Bidirectional Encoder Representations from Transformers (BERT) as an example, we verify this hypothesis by showing that task-specific fine-tuned language models are highly close in parameter space to the pre-trained one.
Taking advantage of such observations, we further show that the fine-tuned versions of these huge models, having on the order of $10^8$ floating-point parameters, can be made very computationally efficient.
First, fine-tuning only a fraction of critical layers suffices.
Second, fine-tuning can be adequately performed by learning a binary multiplicative mask on pre-trained weights, \textit{i.e.} by parameter-sparsification.
As a result, with a single effort, we achieve three desired outcomes: (1) learning to perform specific tasks, (2) saving memory by storing only binary masks of certain layers for each task, and (3) saving compute on appropriate hardware by performing sparse operations with model parameters.
One very puzzling fact about overparameterized deep neural networks is that sheer increases in dimensionality of the parameter space seldom make stochastic gradient-based optimization more difficult.
Given an effective network architecture reflecting proper inductive biases, deeper and/or wider networks take just about the same, if not a lower, number of training iterations to converge, a number often by orders of magnitude smaller than the dimensionality of the parameter space.
For example, ResNet-18 (parameter count 11.7M) and ResNet-152 (parameter count 60.2M) both train to converge, at similar convergence rates, in no more than 600K iterations on Imagenet (He et al., 2015) .
Meaningful optimization seems to happen in only a very low-dimensional parameter subspace, viz.
the span of those relatively few weight updates, with its dimensionality not ostensibly scaling with the model size.
In other words, the network seems already perfectly converged along most of the parameter dimensions at initialization, suggesting that training only marginally alters a high-dimensional parameter configuration.
This phenomenon is epitomized in fine-tuning of pre-trained models.
Pre-training is a, often unsupervised, learning procedure that yields a good common initialization for further supervised learning of various downstream tasks.
The better a pre-trained model is, the fewer iterations are required on average to fine-tune it to perform specific tasks, resulting in fine-tuned models hypothetically closer 1 to the pre-trained one in parameter space.
However, better pre-trained models are, almost always, larger models (Hestness et al., 2017) , and nowhere is this trend more prominent than recent pretrained language models that achieved state-of-the-art natural language understanding performance, e.g. GPT-2 (Radford et al., 2019 ) has 1.5B parameters.
Thus, a problem naturally arises hand-in-hand with an obvious hint to its solution: as pre-trained models get larger, on the one hand, computation of each fine-tuned model becomes more expensive in terms of both memory and compute for inference, while on the other hand, greater closeness between the pre-trained and fine-tuned models in the parameter space prescribes a higher degree of computational redundancy that could be potentially avoided.
Additionally, there might exist more computationally efficient fine-tuned networks that are not necessarily close to, but cheaply attainable from, the pre-trained parameters, which are shared across all tasks.
In this study, we seek to address these questions, using Bidirectional Encoder Representations from Transformers (BERT) (Devlin et al., 2018) and the General Language Understanding Evaluation (GLUE) benchmark tasks (Wang et al., 2018) as a working example.
We first found that the fine-tuned and pre-trained parameters are both L 1 -close and angular-close in parameter space, consistent with the small number of fine-tuning iterations separating them.
Next, we demonstrated that there also exist good fine-tuned models that are L 0 -close (i.e. having a small number of different components) to the pre-trained one.
Further, we showed that there exist good fine-tuned parameters that are L 0 -small (i.e. sparse, or having a large fraction of zero components).
Finally, we successfully found fine-tuned language models that are both L 0 -small and L 0 -close to the pre-trained models.
We remark the practical implications of these constraints.
By forcing fine-tuned parameters to be L 0 -close to the pre-trained ones, one only needs to store a small number of different weights per task, in addition to the common pre-trained weights, substantially saving parameter memory.
By forcing fine-tuned parameters to be sparse, one potentially saves memory and compute, provided proper hardware acceleration of sparse linear algebraic operations.
Surprisingly, our findings also reveal an abundance of good task-specific parameter configurations within a sparse L 0 -vicinity of large pre-trained language models like BERT: a specific task can be learned by simply masking anywhere between 1% to 40% of the pre-trained weights to zero.
See Figure 1 for an explanation of the L 0 -and sparse L 0 -vicinities.
Figure 1: An illustration of the L 0 -vicinity and the sparse L 0 -vicinity of a pre-trained parameter in a three-dimensional parameter space.
The L 0 -vicinity is continuous and contains parameters that are L 0 -close, whereas the sparse L 0 -vicinity is a discrete subset of L 0 -close parameters that are also L 0 -small.
Our search for L 0 -close fine-tuning solutions is motivated by the observation that sensitivities of the optimization objective to different layers in a network are highly variable (Zhang et al., 2019) .
Zhou et al. (2019) trained fine-grain sparse connectivity patterns over randomly initialized network parameters, termed supermasks, suggesting a similar and complementary role model sparsification plays to gradient-based learning of the objective.
This is also related to network architecture search (NAS).
The most similar study to ours is piggyback and its variants (Mallya et al., 2018; Mancini et al., 2019) , where in a multi-task visual object classification scenario, the authors trained taskspecific binary masks on top of a shared set of pre-trained parameters.
In this work, we not only applied similar techniques further to larger pre-trained language models, but also studied the trade- off between L 0 -closeness and sparseness in a systematic way.
Also, note that randomly generated high-dimensional masks can also support multi-task learning, e.g. Cheung et al. (2019) .
A large body of literature is concerned with sparsification of large networks for efficient inference.
Here, we employed iterative pruning (Zhu & Gupta, 2017) during fine-tuning to produce highperformance fine-grain sparse models.
To enforce parameter sparsity differentiably in combination with the L 0 -closeness constraint, instead of principled approaches to imposing L 0 -regularization (Louizos et al., 2017) , we used simpler straight-through estimator, much like binary quantization techniques (Courbariaux et al., 2015; Courbariaux & Bengio, 2016) ; note that this is also used by Mallya et al. (2018) and Zhou et al. (2019) .
Consider a pre-trained network F θ : x → F (x; θ), parameterized by θ, noted as subscript for convenience.
The fine-tuning procedure to perform a task t ∈ T can be described as a supervised training procedure of model G
task-specific last layer unique to task t, and • denotes function composition.
In the case of BERT, we have a stack of modules
among which E is the embedding layer, P a final pooling layer and each B is a transformer block
where
collects all the learnable parameter matrices in the block.
A(·, ·, ·) represents scaled dot-product attention (Vaswani et al., 2017) , DO(·) dropout, LN (·) layer normalization and GeLU(·) the Gaussian error linear unit activation function (Hendrycks & Gimpel, 2016) .
We experimented with the BERT BASE model (Devlin et al., 2018) , for which L = 12, with total parameter count of 109M 2 .
See Table 1 for additional taskspecific parameter counts, all 5 orders of magnitude smaller than the total parameter count.
Optimization of them alone fails to fine-tune (see Appendix A).
The GLUE benchmark is a collection of diverse natural language understanding tasks (Wang et al., 2018) .
We fine-tune on these tasks and report the evaluation performances.
We exclude the problematic WNLI set 3 .
F1 scores are reported for QQP and MRPC, Spearman correlations are reported for STS-B, and accuracy scores are reported for all other tasks.
For all fine-tuning procedures, we use the exact hyperparameters as described in the original paper (Devlin et al., 2018) unless specified otherwise, with additional constraints described as follows.
No constraints are imposed on task-specific last layers G (t) .
L 0 -close fine-tuning To search for fine-tuned solutions that are L 0 -close to the pre-trained parameters, we selectively fix certain parameter matrices at pre-trained values and perform fine-tuning optimization on a lower-dimensional parameter space.
Sparse (L 0 -small) fine-tuning We use iterative pruning (Zhu & Gupta, 2017) during fine-tuning to produce sparse models.
Pruning is based on weight magnitudes in each layer and is performed periodically during fine-tuning with sparsity gradually increasing from 0% to a final level according to a cubic schedule.
Iterative pruning successfully ensures that parameters are L 0 -small (see Appendix B).
Supermask training as fine-tuning (sparse and L 0 -close) In order to search for fine-tuned networks that are both sparse and L 0 -close to the pre-trained one, we reparameterize the model by a multiplicative binary mask θ =θ µ,
whereθ is the pre-trained parameters, and µ ∈ {0, 1} N the mask, N being the dimensionality of the parameter space and the Hadamard product.
If learning is purely through optimizing the mask µ while holdingθ constant, the mask is called a supermask (Zhou et al., 2019) .
Since µ is discrete-valued and thus not differentiable, we reparame-
where Bern(p) denotes an element-wise independent Bernoulli sampler with probability p, and σ(·) the sigmoid function, applied element-wise on ν ∈ R N , the continuous mask parameter.
We treat gradient backpropagation through µ as a straight-through estimator, similar to the techniques used in Mallya et al. (2018) ; Zhou et al. (2019) .
Same fine-tuning hyperparameters were used except for the learning rate (see Appendix C).
Control over the final sparsity is exerted by initialization of µ for fine-tuning.
We initialize ν according to a soft magnitude-based pruning mask: a fraction of small-magnitude values are initialized to ν = −5 and the rest to ν = 5.
We found that the initial sparsity directly controls the final sparsity (see Appendix D), allowing us to produce masks with sparsity levels ranging from 1% to 89%.
We observe that the original fine-tuning procedures for GLUE tasks all take 10 2 to 10 4 parameter update steps (Table 1) , negligible compared to the dimensionality of the parameter space, viz.
10 8 .
Thus, we first questioned whether fine-tuned parameters are indeed close to the pre-trained ones in parameter space.
We measured the L 1 -distances, i.e. L 1 -norm of parameter difference, and angular distances (Table 2) .
Specifically, we inspect the weight matrices in self-attention layers, of size 768× 768 where 768 is the hidden state dimension.
We report the minimum and maximum values across GLUE tasks: MNLI showed the largest values, and RTE showed the smallest values.
Evidently, we see a significantly higher L 1 -and angular-closeness between fine-tuned and pre-trained parameters as compared to the expected distance between two independent random initializations.
This suggests that, during the course of fine-tuning, the very few model parameter updates traversed a very short distance in the parameter space.
Comparing the parameter distance across GLUE tasks, we find that it scales with the number of fine-tuning iterations (see Appendix E).
Further, we inspect the closeness in parameter subspaces for each layer.
We found that, though all layers change very little during fine-tuning, there is nevertheless a high degree of variability across different parameter matrices (Figure 2 ).
Blocks deeper in the encoder stack are less L 1 -close but more angular-close than shallower ones.
In all self-attention modules, value and output projection
Next, inspired by the high degree of variability in each layer's parameter change during fine-tuning, we ask whether effective fine-tuning can be achieved by optimizing only a fraction of layers while having others fixed at pre-trained values, resulting in fine-tuned models L 0 -close in parameter space.
Our results suggest this is indeed feasible (Table 3) .
Informed by different layers' sensitivity to finetuning, we performed fine-tuning experiments by progressively excluding: (1) key projection layers in self-attention across all encoder stack layers, (2) the penultimate and ultimate encoder stacks, and (3) the word embedding layer.
Each of these exclusions independently or all three combined do not significantly degrade performance, while reducing the number of parameters to fine-tune by up to 40% (from 109M to 66M).
Encouraged by these results, we ask whether more aggressive constraints can be imposed to the finetuning process to further cut computational cost.
Though L 0 -close fine-tuning obviates optimization of a large fraction of parameters, avoiding full storage of all parameters for each fine-tuned task, all operations still need to be performed at inference time.
In order to reduce operations, we seek to sparsify parameters.
Combined with strict L 0 -closeness, this amounts to a search over a binary mask in a high-dimensional parameter space.
We adopt supermask training (see Section 3) to this end.
Figure 3 shows fine-tuned model performance across GLUE tasks obtained by supermask training.
Final sparsity level of the supermask is controlled by its initialization (see Section 3 and Appendix D).
We note that there is little task performance degradation between 1% and 40% of parameter sparsity.
Layer-wise sparsity levels of supermasks also demonstrate systematic trends ( Figure 4 ): (1) across GLUE tasks, W Q , W K and W I tend to be sparser than W V , W D and W O , and (2) shallower encoder stack layers are sparser than deeper ones.
Moreover, we show that supermask fine-tuning of only a fraction of sensitive layers could also achieve high performance (Table 3) .
Figure 4: Supermask sparsity levels across layers.
Shown is the low-sparsity MNLI supermask with a global sparsity level of 12.9%, but similar patterns are observed across all GLUE tasks.
How does the learning of a supermask actually work?
Does a supermask trivially learn to prune away the weights with smallest magnitudes?
To address these questions, we inspect the magnitudes of the pre-trained weights zeroed by the supermasks ( Figure 5 , Table 4 ).
These weights turn out to have remarkably higher magnitudes than the next smallest entries, suggesting the learning of supermaks is mechanistically distinct from trivial magnitude-based pruning.
Table 4 : Comparison between supermask pruned weights and magnitude-based pruned weights.
Specifically, we compare the weights pruned with low-sparsity supermasks (initialized at 0% sparsity) and weights pruned with one-shot magnitude-based pruning at the same final sparsity.
We report the maximum and mean magnitude of the pruned weights.
The last row shows percentages of the overlap between the supermask and the magnitude-based pruning mask, i.e. the percentages of weights zeroed by the supermask that are also the smallest weights.
One surprising finding of this study is the many occurrences of good fine-tuned parameters among the 2 N configurations in the set θ : θ =θ µ, µ ∈ {0, 1} N , viz.
vertices of an N -dimensional hypercube.
First, there exist supermasks up to 40% sparse without significant performance degrada- Table 5 : Low-sparsity supermask performance.
We report the sparsity levels achieved when the supermasks were initialized at 0% sparsity.
For several tasks, fine-tuning is achieved with less than 3% of pre-trained weights pruned.
For the supermask evaluation results, we include the mean and standard deviation of 10 Bernoulli samplings of a single run.
tion for all GLUE tasks, for some tasks even sparser (Figure 3 ).
It is natural that performance drops as the mask becomes sparser; however, it is rather counterintuitive that there exist good supermasks at the dense end (Figure 3 ), because we know that the pre-trained model with only the task-specific last layer fine-tuned utterly fails to learn any task (Appendix A).
Figure 6 : Low-sparsity supermask performance, i.e. task performance of super-masks initialized at 0% sparsity, compared against baseline.
To shed more light on this phenomenon, we study the supermasks trained with all-dense initialization ( Figure 6) .
Surprisingly, these low-sparsity supermasks successfully learned to perform all the tasks without significant degradation from baseline.
Essentially, complicated tasks like MNLI and QQP can be learned by clamping 12 − 13% of the pre-trained weights to zero, whereas for simple tasks like MRPC and RTE, setting only 1 − 2% of the pre-trained weight entries to zero suffices to learn the task (Table 5) .
Fine-tuning can indeed be very fine, suggesting relative frequent occurrences of good solutions within a sparse L 0 -neighborhood of the pre-trained parameters.
Finally, we question whether the supermasks learned to perform different tasks share commonalities.
Specifically, we quantify the amount of overlapping zeros in learned supermasks across different tasks (Figure 7) .
It seems the overlaps are not substantially larger than what would have been caused by chance, suggesting that, even though there seem to be many good supermasks for each task, these masks are largely distinct from each other, each unique to the task it learns.
Figure 7: Fractions of overlap of zero elements in supermasks across GLUE tasks, compared to randomly generated masks.
Each value in the grid shows the fraction of pruned elements in one task (horizontal axis) that are also pruned in the other (vertical axis).
Here, we show low-sparsity supermasks (initialized at 0% sparsity) and compare the masks in the value layer of the first encoder, which is one of the most sparse layers in the entire model.
We show that, due to surprisingly frequent occurrences of good parameter configurations in the sparse L 0 -vicinity of large pre-trained language models, two techniques are highly effective in producing efficient fine-tuned networks to perform specific language understanding tasks: (1) optimizing only the most sensitive layers and (2) learning to sparsify parameters.
In contrast to commonly employed post-training compression methods that have to trade off with performance degradation, our procedure of generating sparse networks is by itself an optimization process that learns specific tasks.
Optimization of only task-specific layers does not lead to successful fine-tuning.
For instance, for the MRPC task, freezing parameter weights in the pre-trained model and optimizing the task-specific last layer alone yields a low-performing model.
Across 10 independent runs, the model consistently predicts all 1's for the paraphrase classification task, yielding an F1 score of 81.2.
This is a significant degradation compared to the baseline performance of 89.4 ± 0.7 across multiple runs (Table  3) .
Thus, it is critical to fine-tune layers in the pre-trained model and not just the task-specific layers alone.
Iterative pruning during fine-tuning (Figure 8 ) outperforms supermask training (Figure 3 ) at higher sparsity levels.
While supermask training remains successful up to 40% sparsity, iterative pruning produces binary masks up to 50% sparse and for some tasks even sparser without significant performance degradation.
However, while iterative pruning saves compute on appropriate hardware by performing sparse operations with model parameters, supermask training further saves memory by storing only binary masks of certain layers for each task.
Figure 8 : Iterative pruning during fine-tuning.
We plot the evaluation performance at sparsity levels from 10% to 90% across GLUE tasks.
Note the baseline performance for each task marked by the leftmost end of each curve (0% sparsity).
Supermask training requires a much larger learning rate compared to typical training (Zhang et al., 2019) .
While a learning rate of 2 × 10 −5 is used for optimizing weights, a learning rate of 2 × 10 −1 is used for optimizing masks.
We notice a degradation in performance at smaller learning rates for supermask training (Table 6 ).
Table 6 : MRPC low-sparsity supermask performance at learning rates from 2 × 10 −5 and 2 × 10 −1 .
We note that supermask training requires a much higher learning rate than typical parameter training.
At low learning rates, the model significantly degrades in performance and predicts all 0's for the paraphrase classification task, yielding an F1 score of 0.0.
This pattern holds true across GLUE tasks.
There is no straightforward to control the amount of weights pruned when training supermasks (Zhang et al., 2019) .
We find that setting the initial sparsity through a soft magnitude-based pruning mask controls the final sparsity level.
Figure 9 shows this correlation between initial and final sparsities of supermasks across GLUE tasks.
The supermasks are initialized to a soft magnitude-based pruning mask and the sparsity level shifts during supermask training.
This figure shows the initial sparsity level plotted against the final sparsity level.
We note that, at lower initial sparsity levels, the supermask is pushed to a greater sparsity level, whereas at higher sparsity levels, the supermask is pushed to a lower sparsity level.
This pattern is similar across GLUE tasks but is most prominent in the MNLI task, scaling with the number of fine-tuning steps (Table 1) .
We find that parameter distance scales with the number of fine-tuning steps (Figure 10 ).
Figure 10 : Scaling of parameter distance with the number of fine-tuning iterations.
We find that angular distance correlates with the amount of fine-tuning (Table 1) .
Each data point corresponds to a different GLUE task.

assistant: text: Sparsification as fine-tuning of language models