As deep neural networks (DNNs) achieve tremendous success across many application domains, researchers tried to explore in many aspects on why they generalize well. In this paper, we provide a novel perspective on these issues using the gradient signal to noise ratio (GSNR) of parameters during training process of DNNs. The GSNR of a parameter is defined as the ratio between its gradient's squared mean and variance, over the data distribution. Based on several approximations, we establish a quantitative relationship between model parameters' GSNR and the generalization gap. This relationship indicates that larger GSNR during training process leads to better generalization performance. Moreover, we show that, different from that of shallow models (e.g. logistic regression, support vector machines), the gradient descent optimization dynamics of DNNs naturally produces large GSNR during training, which is probably the key to DNNs' remarkable generalization ability.
The Blavatnik School of Computer Science, Tel Aviv University（布拉瓦尼克计算机科学学院，特拉维夫大学）
Empirical evidence suggests that neural networks with ReLU activations generalize better with over-parameterization. However, there is currently no theoretical analysis that explains this observation. In this work, we provide theoretical and empirical evidence that, in certain cases, overparameterized convolutional networks generalize better than small networks because of an interplay between weight clustering and feature exploration at initialization. We demonstrate this theoretically for a 3-layer convolutional neural network with max-pooling, in a novel setting which extends the XOR problem…
Residual connections significantly boost the performance of deep neural networks. However, there are few theoretical results that address the influence of residuals on the hypothesis complexity and the generalization ability of deep neural networks. This paper studies the influence of residual connections on the hypothesis complexity of the neural network in terms of the covering number of its hypothesis space. We prove that the upper bound of the covering number is the same as chain-like neural networks, if the total numbers of the weight matrices and nonlinearities are fixed, no matter wheth…
Very large deep learning models trained using gradient descent are remarkably resistant to memorization given their huge capacity, but are at the same time capable of fitting large datasets of pure noise. Here methods are introduced by which models may be trained to memorize datasets that normally are generalized. We find that memorization is difficult relative to generalization, but that adding noise makes memorization easier. Increasing the dataset size exaggerates the characteristics of that dataset: model access to more training samples makes overfitting easier for random data, but somewha…
Local SGD is a communication-efficient variant of SGD for large-scale training, where multiple GPUs perform SGD independently and average the model parameters periodically. It has been recently observed that Local SGD can not only achieve the design goal of reducing the communication overhead but also lead to higher test accuracy than the corresponding SGD baseline (Lin et al., 2020b), though the training regimes for this to happen are still in debate (Ortiz et al., 2021). This paper aims to understand why (and when) Local SGD generalizes better based on Stochastic Differential Equation (SDE) …
Salesforce Research（Salesforce研究院）；National University of Singapore（新加坡国立大学）；Princeton University（普林斯顿大学）
It is not clear yet why ADAM-alike adaptive gradient algorithms suffer from worse generalization performance than SGD despite their faster training speed. This work aims to provide understandings on this generalization gap by analyzing their local convergence behaviors. Specifically, we observe the heavy tails of gradient noise in these algorithms. This motivates us to analyze these algorithms through their Levy-driven stochastic differential equations (SDEs) because of the similar convergence behaviors of an algorithm and its SDE. Then we establish the escaping time of these SDEs from a local…
Convolutional neural networks often dominate fully-connected counterparts in generalization performance, especially on image classification tasks. This is often explained in terms of 'better inductive bias'. However, this has not been made mathematically rigorous, and the hurdle is that the fully connected net can always simulate the convolutional net (for a fixed task). Thus the training algorithm plays a role. The current work describes a natural task on which a provable sample complexity gap can be shown, for standard training algorithms. We construct a single natural distribution on $\math…
The generalization mystery in deep learning is the following: Why do over-parameterized neural networks trained with gradient descent (GD) generalize well on real datasets even though they are capable of fitting random datasets of comparable size? Furthermore, from among all solutions that fit the training data, how does GD find one that generalizes well (when such a well-generalizing solution exists)? We argue that the answer to both questions lies in the interaction of the gradients of different examples during training. Intuitively, if the per-example gradients are well-aligned, that is, if…
We show that parametric models trained by a stochastic gradient method (SGM) with few iterations have vanishing generalization error. We prove our results by arguing that SGM is algorithmically stable in the sense of Bousquet and Elisseeff. Our analysis only employs elementary tools from convex and continuous optimization. We derive stability bounds for both convex and non-convex optimization under standard Lipschitz and smoothness assumptions. Applying our results to the convex case, we provide new insights for why multiple epochs of stochastic gradient methods generalize well in practice. In…
Deep neural networks (DNNs) generalize remarkably well without explicit regularization even in the strongly over-parametrized regime where classical learning theory would instead predict that they would severely overfit. While many proposals for some kind of implicit regularization have been made to rationalise this success, there is no consensus for the fundamental reason why DNNs do not strongly overfit. In this paper, we provide a new explanation. By applying a very general probability-complexity bound recently derived from algorithmic information theory (AIT), we argue that the parameter-f…
Hanoi University of Science and Technology（河内科技大学）
Generative adversarial networks (GANs) are so complex that the existing learning theories do not provide a satisfactory explanation for why GANs have great success in practice. The same situation also remains largely open for deep neural networks. To fill this gap, we introduce a Lipschitz theory to analyze generalization. We demonstrate its simplicity by analyzing generalization and consistency of overparameterized neural networks. We then use this theory to derive Lipschitz-based generalization bounds for GANs. Our bounds show that penalizing the Lipschitz constant of the GAN loss can improv…
Peking University（北京大学）；Georgia Institute of Technology（佐治亚理工学院）
Deep residual networks (ResNets) have demonstrated better generalization performance than deep feedforward networks (FFNets). However, the theory behind such a phenomenon is still largely unknown. This paper studies this fundamental problem in deep learning from a so-called "neural tangent kernel" perspective. Specifically, we first show that under proper conditions, as the width goes to infinity, training deep ResNets can be viewed as learning reproducing kernel functions with some kernel function. We then compare the kernel of deep ResNets with that of deep FFNets and discover that the class…
The Wharton School of Business, University of Pennsylvania（宾夕法尼亚大学沃顿商学院）；University of Pennsylvania（宾夕法尼亚大学）
Many important questions about a model cannot be answered just by explaining how much each feature contributes to its output. To answer a broader set of questions, we generalize a popular, mathematically well-grounded explanation technique, Shapley Additive Explanations (SHAP). Our new method - Generalized Shapley Additive Explanations (G-SHAP) - produces many additional types of explanations, including: 1) General classification explanations; Why is this sample more likely to belong to one class rather than another? 2) Intergroup differences; Why do our model's predictions differ between grou…
Empirical risk minimization (ERM) is known in practice to be non-robust to distributional shift where the training and the test distributions are different. A suite of approaches, such as importance weighting, and variants of distributionally robust optimization (DRO), have been proposed to solve this problem. But a line of recent work has empirically shown that these approaches do not significantly improve over ERM in real applications with distribution shift. The goal of this work is to obtain a comprehensive theoretical understanding of this intriguing phenomenon. We first posit the class o…
Of theories for why large-scale machine learning models generalize despite being vastly overparameterized, which of their assumptions are needed to capture the qualitative phenomena of generalization in the real world? On one hand, we find that most theoretical analyses fall short of capturing these qualitative phenomena even for kernel regression, when applied to kernels derived from large-scale neural networks (e.g., ResNet-50) and real data (e.g., CIFAR-100). On the other hand, we find that the classical GCV estimator (Craven and Wahba, 1978) accurately predicts generalization risk even in …
South China Normal University（华南师范大学）；University of the Witwatersrand（威特沃特斯兰德大学）；National Institute for Theoretical Physics（国家理论物理研究所）
Promising resolutions of the generalization puzzle observe that the actual number of parameters in a deep network is much smaller than naive estimates suggest. The renormalization group is a compelling example of a problem which has very few parameters, despite the fact that naive estimates suggest otherwise. Our central hypothesis is that the mechanisms behind the renormalization group are also at work in deep learning, and that this leads to a resolution of the generalization puzzle. We show detailed quantitative evidence that proves the hypothesis for an RBM, by showing that the trained RBM…
University of Maryland（马里兰大学）；University of Colorado（科罗拉多大学）；Google Research Zürich（谷歌苏黎世研究中心）
Cross-lingual word embeddings (CLWE) are often evaluated on bilingual lexicon induction (BLI). Recent CLWE methods use linear projections, which underfit the training dictionary, to generalize on BLI. However, underfitting can hinder generalization to other downstream tasks that rely on words from the training dictionary. We address this limitation by retrofitting CLWE to the training dictionary, which pulls training translation pairs closer in the embedding space and overfits the training dictionary. This simple post-processing step often improves accuracy on two downstream tasks, despite low…
Curse of Attention: A Kernel-Based Perspective for Why Transformers Fail to Generalize on Time Series Forecasting and Beyond
The University of Hong Kong（香港大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；The Simons Institute for the Theory of Computing（西蒙斯理论计算研究所）；UC Berkeley（加州大学伯克利分校）；Sun Yat-sen University（中山大学）
Gradient-based deep-learning algorithms exhibit remarkable performance in practice, but it is not well-understood why they are able to generalize despite having more parameters than training examples. It is believed that implicit bias is a key factor in their ability to generalize, and hence it was widely studied in recent years. In this short survey, we explain the notion of implicit bias, review main results and discuss their implications.
How and Why LLMs Generalize: A Fine-Grained Analysis of LLM Reasoning from Cognitive Behaviors to Low-Level Patterns
University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of California, Berkeley（加州大学伯克利分校）；University of Pennsylvania（宾夕法尼亚大学）；California Institute of Technology（加州理工学院）
Why ML-based cough models do not generalize: a systematic cross-dataset evaluation for tuberculosis screening
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；University of the Basque Country (EHU)（巴斯克大学）
Why Does RL Generalize Better Than SFT? A Data-Centric Perspective on VLM Post-Training
College of Computer Science, Sichuan University（四川大学计算机科学学院）；Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）；College of Computer Science and Technology, Zhejiang University（浙江大学计算机科学与技术学院）
UC Berkeley（加州大学伯克利分校）；Princeton University（普林斯顿大学）；Facebook AI Research（脸书人工智能研究院）
Generalization is a central challenge for the deployment of reinforcement learning (RL) systems in the real world. In this paper, we show that the sequential structure of the RL problem necessitates new approaches to generalization beyond the well-studied techniques used in supervised learning. While supervised learning methods can generalize effectively without explicitly accounting for epistemic uncertainty, we show that, perhaps surprisingly, this is not the case in RL. We show that generalization to unseen test conditions from a limited number of training conditions induces implicit partia…
INESC-ID（葡萄牙系统与计算机工程研究所）；Higher Technical Institute, University of Lisbon（里斯本大学高等技术学院）
When several models have similar training scores, classical model selection heuristics follow Occam's razor and advise choosing the ones with least capacity. Yet, modern practice with large neural networks has often led to situations where two networks with exactly the same number of parameters score similar on the training set, but the deeper one generalizes better to unseen examples. With this in mind, it is well accepted that deep networks are superior to shallow wide ones. However, theoretically there is no difference between the two. In fact, they are both universal approximators. In this…
Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
Several works have proposed Simplicity Bias (SB)---the tendency of standard training procedures such as Stochastic Gradient Descent (SGD) to find simple models---to justify why neural networks generalize well [Arpit et al. 2017, Nakkiran et al. 2019, Soudry et al. 2018]. However, the precise notion of simplicity remains vague. Furthermore, previous settings that use SB to theoretically justify why neural networks generalize well do not simultaneously capture the non-robustness of neural networks---a widely observed phenomenon in practice [Goodfellow et al. 2014, Jo and Bengio 2017]. We attempt…
In recent years, neural networks (NNs) have made giant leaps in a wide variety of domains. NNs are often referred to as black box algorithms due to how little we can explain their empirical success. Our foundational research seeks to explain why neural networks generalize. A recent advancement derived a mutual information measure for explaining the performance of deep NNs through a sequence of increasingly complex functions. We show deep NNs learn a series of boosted classifiers whose generalization is popularly attributed to self-averaging over an increasing number of interpolating sub-classi…
Why Machine Learning Models Systematically Underestimate Extreme Values II: How to Fix It with LatentNN
Department of Astronomy, The Ohio State University（俄亥俄州立大学天文学系）；Center for Cosmology and AstroParticle Physics (CCAPP), The Ohio State University（俄亥俄州立大学宇宙学与天体粒子物理中心（CCAPP））
Yale Center for Medical Informatics（耶鲁医学信息学中心）；VA Connecticut Healthcare System（康涅狄格州退伍军人医疗系统）
We study the generalization of over-parameterized deep networks (for image classification) in relation to the convex hull of their training sets. Despite their great success, generalization of deep networks is considered a mystery. These models have orders of magnitude more parameters than their training samples, and they can achieve perfect accuracy on their training sets, even when training images are randomly labeled, or the contents of images are replaced with random noise. The training loss function of these models has infinite number of near zero minimizers, where only a small subset of …
