We prove bounds on the population risk of the maximum margin algorithm for two-class linear classification. For linearly separable training data, the maximum margin algorithm has been shown in previous work to be equivalent to a limit of training with logistic loss using gradient descent, as the training error is driven to zero. We analyze this algorithm applied to random data including misclassification noise. Our assumptions on the clean data include the case in which the class-conditional distributions are standard normal distributions. The misclassification noise may be chosen by an adversary, subject to a limit on the fraction of corrupted labels. Our bounds show that, with sufficient over-parameterization, the maximum margin algorithm trained on noisy data can achieve nearly optimal population risk.
Importance weighting is a classic technique to handle distribution shifts. However, prior work has presented strong empirical and theoretical evidence demonstrating that importance weights can have little to no effect on overparameterized neural networks. Is importance weighting truly incompatible with the training of overparameterized neural networks? Our paper answers this in the negative. We show that importance weighting fails not because of the overparameterization, but instead, as a result of using exponentially-tailed losses like the logistic or cross-entropy loss. As a remedy, we show …
The University of Chicago（芝加哥大学）；Northwestern University（西北大学）；The University of Chicago Booth School of Business（芝加哥大学布斯商学院）
Training a classifier under non-convex constraints has gotten increasing attention in the machine learning community thanks to its wide range of applications such as algorithmic fairness and class-imbalanced classification. However, several recent works addressing non-convex constraints have only focused on simple models such as logistic regression or support vector machines. Neural networks, one of the most popular models for classification nowadays, are precluded and lack theoretical guarantees. In this work, we show that overparameterized neural networks could achieve a near-optimal and nea…
A PAC-Bayesian Perspective on the Interpolating Information Criterion
University of Melbourne（墨尔本大学）；Queensland University of Technology（昆士兰理工大学）；University of Queensland（昆士兰大学）；University of California, Berkeley（加州大学伯克利分校）
Most modern learning problems are highly overparameterized, meaning that there are many more parameters than the number of training data points, and as a result, the training loss may have infinitely many global minima (parameter vectors that perfectly interpolate the training data). Therefore, it is important to understand which interpolating solutions we converge to, how they depend on the initialization point and the learning algorithm, and whether they lead to different generalization performances. In this paper, we study these questions for the family of stochastic mirror descent (SMD) al…
Carnegie Mellon University（卡内基梅隆大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）
Aimed at explaining the surprisingly good generalization behavior of overparameterized deep networks, recent works have developed a variety of generalization bounds for deep learning, all based on the fundamental learning-theoretic technique of uniform convergence. While it is well-known that many of these existing bounds are numerically large, through numerous experiments, we bring to light a more concerning aspect of these bounds: in practice, these bounds can {\em increase} with the training dataset size. Guided by our observations, we then present examples of overparameterized linear class…
University of California, Berkeley（加州大学伯克利分校）；Princeton University（普林斯顿大学）；ICSI（国际计算机科学研究所）
Within the machine learning community, the widely-used uniform convergence framework has been used to answer the question of how complex, over-parameterized models can generalize well to new data. This approach bounds the test error of the worst-case model one could have fit to the data, but it has fundamental limitations. Inspired by the statistical mechanics approach to learning, we formally define and develop a methodology to compute precisely the full distribution of test errors among interpolating classifiers from several model classes. We apply our method to compute this distribution for…
The Interpolating Information Criterion for Overparameterized Models
School of Mathematics and Statistics, University of Melbourne（墨尔本大学数学与统计学学院）；Department of Electrical and Electronic Engineering, University of Melbourne（墨尔本大学电子与电气工程系）；CIRES and School of Mathematics and Physics, University of Queensland（昆士兰大学海洋研究所和数学与物理学院）；ICSI, LBNL, and Department of Statistics, University of California, Berkeley（伯克利大学统计学系、伯克利国家安全研究所和劳伦斯伯克利国家实验室）
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…
The University of Tokyo（东京大学）；Preferred Networks（Preferred Networks（首选网络公司））
Data augmentation by mixing samples, such as Mixup, has widely been used typically for classification tasks. However, this strategy is not always effective due to the gap between augmented samples for training and original samples for testing. This gap may prevent a classifier from learning the optimal decision boundary and increase the generalization error. To overcome this problem, we propose an alternative framework called Data Interpolating Prediction (DIP). Unlike common data augmentations, we encapsulate the sample-mixing process in the hypothesis class of a classifier so that train and …
Robust Implicit Regularization via Weight Normalization
School of Computation, Information and Technology, Technical University of Munich（慕尼黑工业大学计算、信息与技术学院）；Mathematics Institute, Ludwig Maximilian University of Munich（慕尼黑大学数学研究所）；Oden Institute for Computational Engineering & Sciences, University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）
We investigate the generalization and optimization properties of shallow neural-network classifiers trained by gradient descent in the interpolating regime. Specifically, in a realizable scenario where model weights can achieve arbitrarily small training error $ε$ and their distance from initialization is $g(ε)$, we demonstrate that gradient descent with $n$ training data achieves training error $O(g(1/T)^2 /T)$ and generalization error $O(g(1/T)^2 /n)$ at iteration $T$, provided there are at least $m=Ω(g(1/T)^4)$ hidden neurons. We then show that our realizable setting encompasses a special c…
We propose to study the generalization error of a learned predictor $\hat h$ in terms of that of a surrogate (potentially randomized) predictor that is coupled to $\hat h$ and designed to trade empirical risk for control of generalization error. In the case where $\hat h$ interpolates the data, it is interesting to consider theoretical surrogate classifiers that are partially derandomized or rerandomized, e.g., fit to the training data but with modified label noise. We also show that replacing $\hat h$ by its conditional distribution with respect to an arbitrary $σ$-field is a convenient way t…
Massachusetts Institute of Technology（麻省理工学院）；California Institute of Technology（加州理工学院）
Despite perfectly interpolating the training data, deep neural networks (DNNs) can often generalize fairly well, in part due to the "implicit regularization" induced by the learning algorithm. Nonetheless, various forms of regularization, such as "explicit regularization" (via weight decay), are often used to avoid overfitting, especially when the data is corrupted. There are several challenges with explicit regularization, most notably unclear convergence properties. Inspired by convergence properties of stochastic mirror descent (SMD) algorithms, we propose a new method for training DNNs wit…
Via an overparameterized linear model with Gaussian features, we provide conditions for good generalization for multiclass classification of minimum-norm interpolating solutions in an asymptotic setting where both the number of underlying features and the number of classes scale with the number of training points. The survival/contamination analysis framework for understanding the behavior of overparameterized learning problems is adapted to this setting, revealing that multiclass classification qualitatively behaves like binary classification in that, as long as there are not too many classes…
University of Chicago（芝加哥大学）；University of California, Berkeley（加州大学伯克利分校）
This paper provides elementary analyses of the regret and generalization of minimum-norm interpolating classifiers (MNIC). The MNIC is the function of smallest Reproducing Kernel Hilbert Space norm that perfectly interpolates a label pattern on a finite data set. We derive a mistake bound for MNIC and a regularized variant that holds for all data sets. This bound follows from elementary properties of matrix inverses. Under the assumption that the data is independently and identically distributed, the mistake bound implies that MNIC generalizes at a rate proportional to the norm of the interpol…
Precise asymptotic analysis of Sobolev training for random feature models
Center for Computational Science and Engineering, Massachusetts Institute of Technology（麻省理工学院计算科学与工程中心）；Department of Mathematics and Statistics, University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校数学与统计系）；Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）
To assess generalization, machine learning scientists typically either (i) bound the generalization gap and then (after training) plug in the empirical risk to obtain a bound on the true risk; or (ii) validate empirically on holdout data. However, (i) typically yields vacuous guarantees for overparameterized models. Furthermore, (ii) shrinks the training set and its guarantee erodes with each re-use of the holdout set. In this paper, we introduce a method that leverages unlabeled data to produce generalization bounds. After augmenting our (labeled) training set with randomly labeled fresh exam…
Georgia Institute of Technology（佐治亚理工学院）；University of Washington（华盛顿大学）；University of California Berkeley（加州大学伯克利分校）；UC San Diego（加州大学圣地亚哥分校）；Columbia University（哥伦比亚大学）
We compare classification and regression tasks in an overparameterized linear model with Gaussian features. On the one hand, we show that with sufficient overparameterization all training points are support vectors: solutions obtained by least-squares minimum-norm interpolation, typically used for regression, are identical to those produced by the hard-margin support vector machine (SVM) that minimizes the hinge loss, typically used for training classifiers. On the other hand, we show that there exist regimes where these interpolating solutions generalize well when evaluated by the 0-1 test lo…
Sharp analysis of out-of-distribution error for "importance-weighted" estimators in the overparameterized regime
Georgia Institute of Technology（佐治亚理工学院）；School of Electrical & Computer Engineering（电气与计算机工程学院）；H. Milton School of Industrial & Systems Engineering（H. Milton工业与系统工程学院）
We examine the necessity of interpolation in overparameterized models, that is, when achieving optimal predictive risk in machine learning problems requires (nearly) interpolating the training data. In particular, we consider simple overparameterized linear regression $y = X θ+ w$ with random design $X \in \mathbb{R}^{n \times d}$ under the proportional asymptotics $d/n \to γ\in (1, \infty)$. We precisely characterize how prediction (test) error necessarily scales with training error in this setting. An implication of this characterization is that as the label noise variance $σ^2 \to 0$, any e…
A major challenge in modern machine learning is theoretically understanding the generalization properties of overparameterized models. Many existing tools rely on uniform convergence (UC), a property that, when it holds, guarantees that the test loss will be close to the training loss, uniformly over a class of candidate models. Nagarajan and Kolter (2019) show that in certain simple linear and neural-network settings, any uniform convergence bound will be vacuous, leaving open the question of how to prove generalization in settings where UC fails. Our main contribution is proving novel genera…
The Price of Robustness: Stable Classifiers Need Overparameterization
Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Aleph Alpha Research（Aleph Alpha研究公司）；University of Tromso（特罗姆瑟大学）；DLR-German Aerospace Center（德国航空航天中心）
Generalization performance of classifiers in deep learning has recently become a subject of intense study. Deep models, typically over-parametrized, tend to fit the training data exactly. Despite this "overfitting", they perform well on test data, a phenomenon not yet fully understood. The first point of our paper is that strong performance of overfitted classifiers is not a unique feature of deep learning. Using six real-world and two synthetic datasets, we establish experimentally that kernel machines trained to have zero classification or near zero regression error perform very well on test…
University of Pennsylvania（宾夕法尼亚大学）；Apple Inc.（苹果公司）
There is a large literature explaining why AdaBoost is a successful classifier. The literature on AdaBoost focuses on classifier margins and boosting's interpretation as the optimization of an exponential likelihood function. These existing explanations, however, have been pointed out to be incomplete. A random forest is another popular ensemble method for which there is substantially less explanation in the literature. We introduce a novel perspective on AdaBoost and random forests that proposes that the two algorithms work for similar reasons. While both classifiers achieve similar predictiv…
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…
A continuing mystery in understanding the empirical success of deep neural networks is their ability to achieve zero training error and generalize well, even when the training data is noisy and there are more parameters than data points. We investigate this overparameterized regime in linear regression, where all solutions that minimize training error interpolate the data, including noise. We characterize the fundamental generalization (mean-squared) error of any interpolating solution in the presence of noise, and show that this error decays to zero with the number of features. Thus, overpara…
University of Washington（华盛顿大学）；Georgia Institute of Technology（佐治亚理工学院）；UC Berkeley（加州大学伯克利分校）
State-of-the-art deep learning classifiers are heavily overparameterized with respect to the amount of training examples and observed to generalize well on "clean" data, but be highly susceptible to infinitesmal adversarial perturbations. In this paper, we identify an overparameterized linear ensemble, that uses the "lifted" Fourier feature map, that demonstrates both of these behaviors. The input is one-dimensional, and the adversary is only allowed to perturb these inputs and not the non-linear features directly. We find that the learned model is susceptible to adversaries in an intermediate…
Harvard University（哈佛大学）；Mobileye（Mobileye公司）；Hebrew University（希伯来大学）；University of California San Diego（加利福尼亚大学圣迭戈分校）
Large neural networks trained in the overparameterized regime are able to fit noise to zero train error. Recent work \citep{nakkiran2020distributional} has empirically observed that such networks behave as "conditional samplers" from the noisy distribution. That is, they replicate the noise in the train data to unseen examples. We give a theoretical framework for studying this conditional sampling behavior in the context of learning theory. We relate the notion of such samplers to knowledge distillation, where a student network imitates the outputs of a teacher on unlabeled data. We show that …
Recent research in the theory of overparametrized learning has sought to establish generalization guarantees in the interpolating regime. Such results have been established for a few common classes of methods, but so far not for ensemble methods. We devise an ensemble classification method that simultaneously interpolates the training data, and is consistent for a broad class of data distributions. To this end, we define the manifold-Hilbert kernel for data distributed on a Riemannian manifold. We prove that kernel smoothing regression using the manifold-Hilbert kernel is weakly consistent in …
Implicit Bias of Spectral Descent and Muon on Multiclass Separable Data
Department of Computer Science University of British Columbia（计算机科学系不列颠哥伦比亚大学）；University of British Columbia Canada（不列颠哥伦比亚大学加拿大）；CIFAR AI Chair (Amii) &（CIFAR人工智能主席（Amii））；Department of Electrical and Computer Engineering University of British Columbia（电气与计算机工程系不列颠哥伦比亚大学）
Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；SAIT AI Lab（SAIT人工智能实验室）；University of British Columbia（不列颠哥伦比亚大学）；Google Research - Brain Team（谷歌研究-大脑团队）
We study the implicit regularization of optimization methods for linear models interpolating the training data in the under-parametrized and over-parametrized regimes. Since it is difficult to determine whether an optimizer converges to solutions that minimize a known norm, we flip the problem and investigate what is the corresponding norm minimized by an interpolating solution. Using this reasoning, we prove that for over-parameterized linear regression, projections onto linear spans can be used to move between different interpolating solutions. For under-parameterized linear classification, …
ETH Zürich（苏黎世联邦理工学院）；ETH AI Center（苏黎世联邦理工学院人工智能中心）
Popular iterative algorithms such as boosting methods and coordinate descent on linear models converge to the maximum $\ell_1$-margin classifier, a.k.a. sparse hard-margin SVM, in high dimensional regimes where the data is linearly separable. Previous works consistently show that many estimators relying on the $\ell_1$-norm achieve improved statistical rates for hard sparse ground truths. We show that surprisingly, this adaptivity does not apply to the maximum $\ell_1$-margin classifier for a standard discriminative setting. In particular, for the noiseless setting, we prove tight upper and lo…
State-of-the-art machine learning models can be vulnerable to very small input perturbations that are adversarially constructed. Adversarial training is an effective approach to defend against such examples. It is formulated as a min-max problem, searching for the best solution when the training data was corrupted by the worst-case attacks. For linear regression problems, adversarial training can be formulated as a convex problem. We use this reformulation to make two technical contributions: First, we formulate the training problem as an instance of robust regression to reveal its connection …
Hong Kong University of Science and Technology（香港科技大学）
Margin enlargement over training data has been an important strategy since perceptrons in machine learning for the purpose of boosting the robustness of classifiers toward a good generalization ability. Yet Breiman (1999) showed a dilemma that a uniform improvement on margin distribution does NOT necessarily reduces generalization errors. In this paper, we revisit Breiman's dilemma in deep neural networks with recently proposed spectrally normalized margins, from a novel perspective based on phase transitions of normalized margin distributions in training dynamics. Normalized margin distributi…
