Ethan S Dyer
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Catastrophic forgetting presents a challenge in developing deep learning models capable of continual learning, i.e. learning tasks sequentially. Recently, both computer vision and natural-language processing have witnessed great progress through the use of large-scale pretrained models. In this work, we present an empirical study of catastrophic forgetting in this pretraining paradigm.
Our experiments indicate that large, pretrained ResNets and Transformers are significantly more resistant to forgetting than randomly-initialized, trained-from-scratch models; this robustness systematically improves with scale of both model and pretraining dataset size.
We take initial steps towards characterizing what aspect of model representations allows them to perform continual learning so well, finding that in the pretrained models, distinct class representations grow more orthogonal with scale. Our results suggest that, when possible, scale and a diverse pretraining dataset can be useful ingredients in mitigating catastrophic forgetting.
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Beyond the Imitation Game: Quantifying and extrapolating the capabilities of language models
Aitor Lewkowycz
Daniel Freeman
Guy Gur-Ari
Jascha Sohl-dickstein
Liam B. Fedus
TBD (2022)
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Language models demonstrate both quantitative improvement and new qualitative capabilities with increasing scale. Despite their potentially transformative impact, these new capabilities are as yet poorly characterized. In order to direct future research, prepare for disruptive new model capabilities, and ameliorate socially harmful effects, it is vital that we understand the present and near-future capabilities and limitations of language models.
To address this challenge, we introduce the Beyond the Imitation Game benchmark (BIG-bench). BIG-bench consists of 207 tasks, contributed by over 400 authors across 132 institutions. Task topics are diverse, drawing problems from linguistics, childhood development, math, common sense reasoning, biology, physics, social bias, software development, and beyond. BIG-bench focuses on capabilities that are believed to be beyond current language models. We evaluate the behavior of OpenAI's GPT models, Google-internal dense transformer architectures, and Switch-style sparse transformers on BIG-bench, across model sizes spanning millions to hundreds of billions of parameters. A team of human experts further performed all tasks, to provide a strong baseline. Findings include: model performance and calibration both improve with scale, but are poor in absolute terms (and when compared with human performance); model performance is remarkably similar across model classes; tasks that improve gradually and predictably commonly involve a large knowledge or memorization component, whereas tasks that exhibit ``breakthrough'' behavior at a critical scale often involve a significant reasoning or algorithmic component; social bias typically increases with scale in settings with ambiguous context, but this can be improved with prompting.
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Exploring Length Generalization in Large Language Models
Cem Anil
Yuhuai Wu
Aitor Lewkowycz
Guy Gur-Ari
NeurIPS Oral (2022)
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The ability to extrapolate from short problem instances to longer ones is an important form of out-of-distribution generalization in reasoning tasks, and is crucial when learning from datasets where longer problem instances are rare. These include theorem proving, solving quantitative mathematics problems, and reading/summarizing novels. In this paper, we run careful empirical studies exploring the length generalization capabilities of transformer-based language models. We first establish that naively finetuning transformers on length generalization tasks shows significant generalization deficiencies independent of model scale. We then show that combining pretrained large language models' in-context learning abilities with scratchpad prompting (asking the model to output solution steps before producing an answer) results in a dramatic improvement in length generalization. We run careful failure analyses on each of the learning modalities and identify common sources of mistakes that highlight opportunities in equipping language models with the ability to generalize to longer problems.
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Solving Quantitative Reasoning Problems with Language Models
Aitor Lewkowycz
David Martin Dohan
Henryk Michalewski
Cem Anil
Imanol Schlag
Theo Gutman-Solo
Yuhuai Wu
Guy Gur-Ari
NeurIPS (2022)
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Language models have achieved remarkable performance on a wide range of tasks that require natural language understanding. Nevertheless, state-of-the-art models have generally struggled with tasks that require quantitative reasoning, such as solving mathematics, science, and engineering problems at the college level. To help close this gap, we introduce Minerva, a large language model pretrained on general natural language data and further trained on technical content. The model achieves state-of-the-art performance on technical benchmarks without the use of external tools. We also evaluate our model on over two hundred undergraduate-level problems in physics, biology, chemistry, economics, and other sciences that require quantitative reasoning, and find that the model can correctly answer nearly a third of them.
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We introduce the Block-Recurrent Transformer, which applies a transformer layer in a recurrent fashion along a sequence, and has linear complexity with respect to sequence length. Our recurrent cell operates on blocks of tokens rather than single tokens, and leverages parallel computation within a block in order to make efficient use of accelerator hardware. The cell itself is strikingly simple. It is merely a transformer layer: it uses self-attention and cross-attention to efficiently compute a recurrent function over a large set of state vectors and tokens. Our design was inspired in part by LSTM cells, and it uses LSTM-style gates, but it scales the typical LSTM cell up by several orders of magnitude.
Our implementation of recurrence has the same cost in both computation time and parameter count as a conventional transformer layer, but offers dramatically improved perplexity in language modeling tasks over very long sequences. Our model out-performs a long-range Transformer XL baseline by a wide margin, while running twice as fast. We demonstrate its effectiveness on PG19 (books), arXiv papers, and GitHub source code.
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Inspired by human learning, researchers have proposed ordering examples during training based on their difficulty. Both curriculum learning, exposing a network to easier examples early in training, and anti-curriculum learning, showing the most difficult examples first, have been suggested as improvements to the standard i.i.d. training. In this work, we set out to investigate the relative benefits of ordered learning. We first investigate the implicit curricula resulting from architectural and optimization bias and find that samples are learned in a highly consistent order. Next, to quantify the benefit of explicit curricula, we conduct extensive experiments over thousands of orderings spanning three kinds of learning: curriculum, anti-curriculum, and random-curriculum -- in which the size of the training dataset is dynamically increased over time, but the examples are randomly ordered. We find that for standard benchmark datasets, curricula have only marginal benefits, and that randomly ordered samples perform as well or better than curricula and anti-curricula, suggesting that any benefit is entirely due to the dynamic training set size. Inspired by common use cases of curriculum learning in practice, we investigate the role of limited training time budget and noisy data in the success of curriculum learning. Our experiments demonstrate that curriculum, but not anti-curriculum can indeed improve the performance either with limited training time budget or in existence of noisy data.
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Though data augmentation has become a standard component of deep neural network training, the underlying mechanism behind the effectiveness of these techniques remains poorly understood. In practice, augmentation policies are often chosen using heuristics of distribution shift or augmentation diversity. Inspired by these, we conduct an empirical study to quantify how data augmentation improves model generalization. We introduce two interpretable and easy-to-compute measures: Affinity and Diversity. We find that augmentation performance is predicted not by either of these alone but by jointly optimizing the two.
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The test loss of well-trained neural networks often follows precise power-law scaling relations with either the size of the training dataset or the number of parameters in the network. We propose a theory that explains and connects these scaling laws. We identify variance-limited and resolution-limited scaling behavior for both model and dataset size, for a total of four scaling regimes. The variance-limited scaling follows simply from the existence of a well-behaved infinite data or infinite width limit, while the resolution-limited regime can be explained by positing that models are effectively resolving a smooth data manifold. In the large width limit, this can be equivalently obtained from the spectrum of certain kernels, and we present evidence that large width and large dataset resolution-limited scaling exponents are related by a duality. We exhibit all four scaling regimes in the controlled setting of large random feature and pre-trained models and test the predictions empirically on a range of standard architectures and datasets. We also observe several empirical relationships between datasets and scaling exponents: super-classing image classifiers does not change exponents, while changing input distribution (via changing datasets or adding noise) has a strong effect. We further explore the effect of architecture aspect ratio on scaling exponents.
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Whitening and second order optimization both destroy information about the dataset, and can make generalization impossible
Jascha Sohl-dickstein
Neha Wadia
Sam S. Schoenholz
ICML Spotlight (2021) (to appear)
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We show theoretically and experimentally that both data whitening and second order optimization erase information about the training dataset, and can prevent any generalization for high dimensional datasets. First we show that if the input layer of a model is a dense linear layer, then the datapoint-datapoint second moment matrix contains all information which can be used to make predictions. Second, we show that for high dimensional datasets, where the number of features is at least as large as the number of datapoints, and where the whitening transform is computed on the full (train+test) dataset, whitening erases all information in this datapoint-datapoint second moment matrix. Generalization is thus completely impossible for models trained on high dimensional whitened datasets. Second order optimization of a linear model is identical to first order optimization of the same model after data whitening. Second order optimization can thus also prevent any generalization in similar situations. We experimentally verify these predictions for models trained on whitened data, and for linear models trained with an online Newton optimizer. We further experimentally demonstrate that generalization continues to be harmed even when the theoretical constraints on input dimensionality (for whitening), or linearity of the model (for second order optimization) are relaxed.
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Wide neural networks have proven to be a rich class of architectures for both theory and practice. Motivated by the observation that finite width convolutional networks appear to outperform infinite width networks, we study scaling laws for wide CNNs and networks with skip connections. Following the approach of (Dyer & Gur-Ari, 2019), we present a simple diagrammatic recipe to derive the asymptotic width dependence for many quantities of interest. These scaling relationships provide a solvable description for the training dynamics of wide convolutional networks. We test these relations across a broad range of architectures. In particular, we find that the difference in performance between finite and infinite width models vanishes at a definite rate with respect to model width. Nonetheless, this relation is consistent with finite width models generalizing either better or worse than their infinite width counterparts, and we provide examples where the relative performance depends on the optimization details.
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A central challenge in developing versatile machine learning systems is catastrophic forgetting: a model trained on tasks in sequence will suffer significant performance drops on earlier tasks. Despite the ubiquity of catastrophic forgetting, there is limited understanding of the underlying process and its causes. In this paper, we address this important knowledge gap, investigating how forgetting affects representations in neural network models. Through representational analysis techniques, we find that deeper layers are disproportionately the source of forgetting. Supporting this, a study of methods to mitigate forgetting illustrates that they act to stabilize deeper layers. These insights enable the development of an analytic argument and empirical picture relating the degree of forgetting to representational similarity between tasks. Consistent with this picture, we observe maximal forgetting occurs for task sequences with intermediate similarity. We perform empirical studies on the standard split CIFAR-10 setup and also introduce a novel CIFAR-100 based task approximating realistic input distribution shift.
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The choice of initial learning rate can have a profound effect on the performance of deep networks. We present a class of neural networks with solvable training dynamics that exhibit sharply distinct behaviors at small and large learning rates. The two regimes are separated by a phase transition. In the small learning rate phase training can be understood using the existing theory of infinitely wide neural networks. At large learning rates the model captures qualitatively distinct phenomena, including the convergence of gradient descent dynamics to flatter minima. One key prediction of our model is a narrow range of large stable learning rates. We find good agreement between our model's predictions and training dynamics in realistic deep learning settings. Furthermore, we find that the optimal performance in such settings is often found in the large learning rate phase. We believe our results shed light on characteristics of models trained at different learning rates. In particular, they fill a gap between existing wide neural network theory, and the nonlinear, large learning rate, training dynamics relevant to practice.
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Understanding the asymptotic behavior of wide networks is of considerable interest. In this work, we present a general method for analyzing this large width behavior. The method is an adaptation of Feynman diagrams, a standard tool for computing multivariate Gaussian integrals. We apply our method to study training dynamics, improving existing bounds and deriving new results on wide network evolution during stochastic gradient descent. Going beyond the strict large width limit, we present closed-form expressions for higher-order terms governing wide network training, and test these predictions empirically.
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We show that in a variety of large-scale deep learning scenarios the gradient dynamically converges to a very small subspace after a short period of training. The subspace is spanned by a few top eigenvectors of the Hessian (equal to the number of classes in the dataset), and is mostly preserved over long periods of training. A simple argument then suggests that gradient descent may happen mostly in this subspace. We give an example of this effect in a solvable model of classification, and we comment on possible implications for optimization and learning.
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https://arxiv.org/abs/1812.07579
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