Benign overfitting, the phenomenon where interpolating models generalize well in the presence of noisy data, was first observed in neural network models trained with gradient descent. To better understand this empirical observation, we consider the generalization error of two-layer neural networks trained to interpolation by gradient descent on the logistic loss following random initialization. We assume the data comes from well-separated class-conditional log-concave distributions and allow for a constant fraction of the training labels to be corrupted by an adversary. We show that in this setting, neural networks exhibit benign overfitting: they can be driven to zero training error, perfectly fitting any noisy training labels, and simultaneously achieve minimax optimal test error. In contrast to previous work on benign overfitting that require linear or kernel-based predictors, our analysis holds in a setting where both the model and learning dynamics are fundamentally nonlinear.
In the nonparametric regression setting, we construct an estimator which is a continuous function interpolating the data points with high probability, while attaining minimax optimal rates under mean squared risk on the scale of Hölder classes adaptively to the unknown smoothness.
UC Berkeley（加州大学伯克利分校）；Google（谷歌）；Pompeu Fabra University（庞培法布拉大学）；ICREA（加泰罗尼亚研究与高等教育学院）；Barcelona Graduate School of Economics（巴塞罗那经济学研究生院）
The phenomenon of benign overfitting is one of the key mysteries uncovered by deep learning methodology: deep neural networks seem to predict well, even with a perfect fit to noisy training data. Motivated by this phenomenon, we consider when a perfect fit to training data in linear regression is compatible with accurate prediction. We give a characterization of linear regression problems for which the minimum norm interpolating prediction rule has near-optimal prediction accuracy. The characterization is in terms of two notions of the effective rank of the data covariance. It shows that overp…
MIT（麻省理工学院）；University of Chicago（芝加哥大学）；UBC（不列颠哥伦比亚大学）；Amii（阿尔伯塔机器智能研究所）；TTI-Chicago（芝加哥丰田技术研究所）
We consider interpolation learning in high-dimensional linear regression with Gaussian data, and prove a generic uniform convergence guarantee on the generalization error of interpolators in an arbitrary hypothesis class in terms of the class's Gaussian width. Applying the generic bound to Euclidean norm balls recovers the consistency result of Bartlett et al. (2020) for minimum-norm interpolators, and confirms a prediction of Zhou et al. (2020) for near-minimal-norm interpolators in the special case of Gaussian data. We demonstrate the generality of the bound by applying it to the simplex, ob…
The University of Hong Kong（香港大学）；University of California, Los Angeles（加州大学洛杉矶分校）；University of California, San Diego（加州大学圣地亚哥分校）
Modern neural networks often have great expressive power and can be trained to overfit the training data, while still achieving a good test performance. This phenomenon is referred to as "benign overfitting". Recently, there emerges a line of works studying "benign overfitting" from the theoretical perspective. However, they are limited to linear models or kernel/random feature models, and there is still a lack of theoretical understanding about when and how benign overfitting occurs in neural networks. In this paper, we study the benign overfitting phenomenon in training a two-layer convoluti…
University College London（伦敦大学学院）；Nanjing University（南京大学）
Modern machine learning models often employ a huge number of parameters and are typically optimized to have zero training loss; yet surprisingly, they possess near-optimal prediction performance, contradicting classical learning theory. We examine how these benign overfitting phenomena occur in a two-layer neural network setting where sample covariates are corrupted with noise. We address the high dimensional regime, where the data dimension $d$ grows with the number $n$ of data points. Our analysis combines an upper bound on the bias with matching upper and lower bounds on the variance of the…
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；University of British Columbia（不列颠哥伦比亚大学）
Deep neural networks generalize well despite being exceedingly overparameterized and being trained without explicit regularization. This curious phenomenon has inspired extensive research activity in establishing its statistical principles: Under what conditions is it observed? How do these depend on the data and on the training algorithm? When does regularization benefit generalization? While such questions remain wide open for deep neural nets, recent works have attempted gaining insights by studying simpler, often linear, models. Our paper contributes to this growing line of work by examini…
Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias
Department of Computer Science and Engineering（计算机科学与工程系）；The Hong Kong University of Science and Technology（香港科学与技术大学）；School of Mathematical Sciences（数学科学学院）；Peking University（北京大学）；Department of Computer Science（计算机科学系）；University of Warwick（沃里克大学）；Industrial Engineering & Management Sciences（工业工程与管理科学）；Northwestern University（西北大学）
University of California, Los Angeles（加州大学洛杉矶分校）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of Washington（华盛顿大学）；Microsoft Research（微软研究院）
There is an increasing realization that algorithmic inductive biases are central in preventing overfitting; empirically, we often see a benign overfitting phenomenon in overparameterized settings for natural learning algorithms, such as stochastic gradient descent (SGD), where little to no explicit regularization has been employed. This work considers this issue in arguably the most basic setting: constant-stepsize SGD (with iterate averaging or tail averaging) for linear regression in the overparameterized regime. Our main result provides a sharp excess risk bound, stated in terms of the full…
The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that…
Classical Statistical (In-Sample) Intuitions Don't Generalize Well: A Note on Bias-Variance Tradeoffs, Overfitting and Moving from Fixed to Random Designs
Stanford University（斯坦福大学）；Google（谷歌）；University of California, Berkeley（加州大学伯克利分校）
The recent success of neural network models has shone light on a rather surprising statistical phenomenon: statistical models that perfectly fit noisy data can generalize well to unseen test data. Understanding this phenomenon of $\textit{benign overfitting}$ has attracted intense theoretical and empirical study. In this paper, we consider interpolating two-layer linear neural networks trained with gradient flow on the squared loss and derive bounds on the excess risk when the covariates satisfy sub-Gaussianity and anti-concentration properties, and the noise is independent and sub-Gaussian. B…
We prove a lower bound on the excess risk of sparse interpolating procedures for linear regression with Gaussian data in the overparameterized regime. We apply this result to obtain a lower bound for basis pursuit (the minimum $\ell_1$-norm interpolant) that implies that its excess risk can converge at an exponentially slower rate than OLS (the minimum $\ell_2$-norm interpolant), even when the ground truth is sparse. Our analysis exposes the benefit of an effect analogous to the "wisdom of the crowd", except here the harm arising from fitting the $\textit{noise}$ is ameliorated by spreading it…
Learning Curves and Benign Overfitting of Spectral Algorithms in Large Dimensions
Department of Statistics and Data Science, National University of Singapore（新加坡国立大学统计与数据科学系）；Department of Statistics and Data Science, Tsinghua University（清华大学统计与数据科学系）
Generalisation and benign over-fitting for linear regression onto random functional covariates
Andrew Jones School of Mathematics, University of Edinburgh（安德鲁·琼斯 数学学院，爱丁堡大学）；Nick Whiteley School of Mathematics, University of Bristol（尼克·怀特利 数学学院，布里斯托尔大学）
The University of Tokyo（东京大学）；CyberAgent, Inc.（CyberAgent公司）
We study the benign overfitting theory in the prediction of the conditional average treatment effect (CATE), with linear regression models. As the development of machine learning for causal inference, a wide range of large-scale models for causality are gaining attention. One problem is that suspicions have been raised that the large-scale models are prone to overfitting to observations with sample selection, hence the large models may not be suitable for causal prediction. In this study, to resolve the suspicious, we investigate on the validity of causal inference methods for overparameterize…
Modern machine learning often operates in the regime where the number of parameters is much higher than the number of data points, with zero training loss and yet good generalization, thereby contradicting the classical bias-variance trade-off. This \textit{benign overfitting} phenomenon has recently been characterized using so called \textit{double descent} curves where the risk undergoes another descent (in addition to the classical U-shaped learning curve when the number of parameters is small) as we increase the number of parameters beyond a certain threshold. In this paper, we examine the…
Pennsylvania State University（宾夕法尼亚州立大学）；The University of Hong Kong（香港大学）；University of California, Los Angeles（加州大学洛杉矶分校）
"Benign overfitting", where classifiers memorize noisy training data yet still achieve a good generalization performance, has drawn great attention in the machine learning community. To explain this surprising phenomenon, a series of works have provided theoretical justification in over-parameterized linear regression, classification, and kernel methods. However, it is not clear if benign overfitting still occurs in the presence of adversarial examples, i.e., examples with tiny and intentional perturbations to fool the classifiers. In this paper, we show that benign overfitting indeed occurs i…
Meta learning has demonstrated tremendous success in few-shot learning with limited supervised data. In those settings, the meta model is usually overparameterized. While the conventional statistical learning theory suggests that overparameterized models tend to overfit, empirical evidence reveals that overparameterized meta learning methods still work well -- a phenomenon often called "benign overfitting." To understand this phenomenon, we focus on the meta learning settings with a challenging bilevel structure that we term the gradient-based meta learning, and analyze its generalization perf…
Linear classifiers and leaky ReLU networks trained by gradient flow on the logistic loss have an implicit bias towards solutions which satisfy the Karush--Kuhn--Tucker (KKT) conditions for margin maximization. In this work we establish a number of settings where the satisfaction of these KKT conditions implies benign overfitting in linear classifiers and in two-layer leaky ReLU networks: the estimators interpolate noisy training data and simultaneously generalize well to test data. The settings include variants of the noisy class-conditional Gaussians considered in previous work as well as new…
Stern School of Business, New York University（纽约大学斯特恩商学院）；National University of Singapore（新加坡国立大学）
One classical canon of statistics is that large models are prone to overfitting, and model selection procedures are necessary for high dimensional data. However, many overparameterized models, such as neural networks, perform very well in practice, although they are often trained with simple online methods and regularization. The empirical success of overparameterized models, which is often known as benign overfitting, motivates us to have a new look at the statistical generalization theory for online optimization. In particular, we present a general theory on the generalization error of stoch…
University of Oxford（牛津大学）；The Alan Turing Institute（艾伦图灵研究所）；Five AI Ltd.（Five AI有限公司）
We investigate two causes for adversarial vulnerability in deep neural networks: bad data and (poorly) trained models. When trained with SGD, deep neural networks essentially achieve zero training error, even in the presence of label noise, while also exhibiting good generalization on natural test data, something referred to as benign overfitting [2, 10]. However, these models are vulnerable to adversarial attacks. We identify label noise as one of the causes for adversarial vulnerability, and provide theoretical and empirical evidence in support of this. Surprisingly, we find several instance…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Catastrophic overfitting is a phenomenon observed during Adversarial Training (AT) with the Fast Gradient Sign Method (FGSM) where the test robustness steeply declines over just one epoch in the training stage. Prior work has attributed this loss in robustness to a sharp decrease in $\textit{local linearity}$ of the neural network with respect to the input space, and has demonstrated that introducing a local linearity measure as a regularization term prevents catastrophic overfitting. Using a simple neural network architecture, we experimentally demonstrate that maintaining high local linearit…
Understanding Generalization in Transformers: Error Bounds and Training Dynamics Under Benign and Harmful Overfitting
Zhongtai Securities Institute for Financial Studies, Shandong University（山东大学中泰证券金融研究院）；School of Artificial Intelligence, Beijing Normal University（北京师范大学人工智能学院）；Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学高瓴人工智能学院）
