Consider $n$ points $x_1,\ldots,x_n$ in finite-dimensional euclidean space, each having one of two colors. Suppose there exists a separating hyperplane (identified with its unit normal vector $w)$ for the points, i.e a hyperplane such that points of same color lie on the same side of the hyperplane. We measure the quality of such a hyperplane by its margin $γ(w)$, defined as minimum distance between any of the points $x_i$ and the hyperplane. In this paper, we prove that the margin function $γ$ satisfies a nonsmooth Kurdyka-Lojasiewicz inequality with exponent $1/2$. This result has far-reaching consequences. For example, let $γ^{opt}$ be the maximum possible margin for the problem and let $w^{opt}$ be the parameter for the hyperplane which attains this value. Given any other separating hyperplane with parameter $w$, let $d(w):=\|w-w^{opt}\|$ be the euclidean distance between $w$ and $w^{opt}$, also called the bias of $w$. From the previous KL-inequality, we deduce that $(γ^{opt}-γ(w)) / R \le d(w) \le 2\sqrt{(γ^{opt}-γ(w))/γ^{opt}}$, where $R:=\max_i \|x_i\|$ is the maximum distance of the points $x_i$ from the origin. Consequently, for any optimization algorithm (gradient-descent or not), the bias of the iterates converges at least as fast as the square-root of the rate of their convergence of the margin. Thus, our work provides a generic tool for analyzing the implicit bias of any algorithm in terms of its margin, in situations where a specialized analysis might not be available: it is sufficient to establish a good rate for converge of the margin, a task which is usually much easier.
Univ Lyon, UJM-Saint-Etienne, CNRS, Institut d Optique Graduate School, Laboratoire Hubert Curien UMR 5516（里昂大学，圣艾蒂安大学，法国国家科学研究中心，光学研究所研究生院，于贝尔·居里安实验室）；Université Laval（拉瓦尔大学）
In the PAC-Bayesian literature, the C-Bound refers to an insightful relation between the risk of a majority vote classifier (under the zero-one loss) and the first two moments of its margin (i.e., the expected margin and the voters' diversity). Until now, learning algorithms developed in this framework minimize the empirical version of the C-Bound, instead of explicit PAC-Bayesian generalization bounds. In this paper, by directly optimizing PAC-Bayesian guarantees on the C-Bound, we derive self-bounding majority vote learning algorithms. Moreover, our algorithms based on gradient descent are s…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper shows that the implicit bias of gradient descent on linearly separable data is exactly characterized by the optimal solution of a dual optimization problem given by a smoothed margin, even for general losses. This is in contrast to prior results, which are often tailored to exponentially-tailed losses. For the exponential loss specifically, with $n$ training examples and $t$ gradient descent steps, our dual analysis further allows us to prove an $O(\ln(n)/\ln(t))$ convergence rate to the $\ell_2$ maximum margin direction, when a constant step size is used. This rate is tight in both…
We study the implicit bias of gradient descent methods in solving a binary classification problem over a linearly separable dataset. The classifier is described by a nonlinear ReLU model and the objective function adopts the exponential loss function. We first characterize the landscape of the loss function and show that there can exist spurious asymptotic local minima besides asymptotic global minima. We then show that gradient descent (GD) can converge to either a global or a local max-margin direction, or may diverge from the desired max-margin direction in a general context. For stochastic…
UC Berkeley（加州大学伯克利分校）；TTI Chicago（芝加哥丰田技术研究所）；Hebrew University（希伯来大学）；Google（谷歌）；University of Michigan（密歇根大学）
The implicit biases of gradient-based optimization algorithms are conjectured to be a major factor in the success of modern deep learning. In this work, we investigate the implicit bias of gradient flow and gradient descent in two-layer fully-connected neural networks with leaky ReLU activations when the training data are nearly-orthogonal, a common property of high-dimensional data. For gradient flow, we leverage recent work on the implicit bias for homogeneous neural networks to show that asymptotically, gradient flow produces a neural network with rank at most two. Moreover, this network is…
TTI at Chicago（芝加哥丰田理工学院）；USC（南加利福尼亚大学）；Technion（以色列理工学院）
We show that gradient descent on full-width linear convolutional networks of depth $L$ converges to a linear predictor related to the $\ell_{2/L}$ bridge penalty in the frequency domain. This is in contrast to linearly fully connected networks, where gradient descent converges to the hard margin linear support vector machine solution, regardless of depth.
Faster Margin Maximization Rates for Generic and Adversarially Robust Optimization Methods
College of Computing, Georgia Institute of Technology（佐治亚理工学院计算学院）；Google Research（谷歌研究院）；School of Electrical and Computer Engineering, Georgia Institute of Technology（佐治亚理工学院电气与计算机工程学院）；School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院工业与系统工程学院）
Despite their overwhelming capacity to overfit, deep neural networks trained by specific optimization algorithms tend to generalize well to unseen data. Recently, researchers explained it by investigating the implicit regularization effect of optimization algorithms. A remarkable progress is the work (Lyu&Li, 2019), which proves gradient descent (GD) maximizes the margin of homogeneous deep neural networks. Except GD, adaptive algorithms such as AdaGrad, RMSProp and Adam are popular owing to their rapid training process. However, theoretical guarantee for the generalization of adaptive optimiz…
A common method in training neural networks is to initialize all the weights to be independent Gaussian vectors. We observe that by instead initializing the weights into independent pairs, where each pair consists of two identical Gaussian vectors, we can significantly improve the convergence analysis. While a similar technique has been studied for random inputs [Daniely, NeurIPS 2020], it has not been analyzed with arbitrary inputs. Using this technique, we show how to significantly reduce the number of neurons required for two-layer ReLU networks, both in the under-parameterized setting with…
University of Science & Technology of China（中国科学技术大学）；Microsoft Research Asia（微软亚洲研究院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
The momentum acceleration technique is widely adopted in many optimization algorithms. However, there is no theoretical answer on how the momentum affects the generalization performance of the optimization algorithms. This paper studies this problem by analyzing the implicit regularization of momentum-based optimization. We prove that on the linear classification problem with separable data and exponential-tailed loss, gradient descent with momentum (GDM) converges to the L2 max-margin solution, which is the same as vanilla gradient descent. That means gradient descent with momentum accelerati…
Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎萨克雷大学）；INRIA（法国国家信息与自动化研究所）；ENS（巴黎高等师范学院）；PSL Research University（PSL研究大学）
Neural networks trained to minimize the logistic (a.k.a. cross-entropy) loss with gradient-based methods are observed to perform well in many supervised classification tasks. Towards understanding this phenomenon, we analyze the training and generalization behavior of infinitely wide two-layer neural networks with homogeneous activations. We show that the limits of the gradient flow on exponentially tailed losses can be fully characterized as a max-margin classifier in a certain non-Hilbertian space of functions. In presence of hidden low-dimensional structures, the resulting margin is indepen…
The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient
School of Statistics, Renmin University of China（中国人民大学统计学院）；School of Computing and Data Science, The University of Hong Kong（香港大学计算与数据科学学院）；School of Computing and Data Science & Institute of Data Science, The University of Hong Kong（香港大学计算与数据科学学院及数据科学研究所）
Università di Milano（米兰大学）；Boston University（波士顿大学）
In the setting of online learning, Implicit algorithms turn out to be highly successful from a practical standpoint. However, the tightest regret analyses only show marginal improvements over Online Mirror Descent. In this work, we shed light on this behavior carrying out a careful regret analysis. We prove a novel static regret bound that depends on the temporal variability of the sequence of loss functions, a quantity which is often encountered when considering dynamic competitors. We show, for example, that the regret can be constant if the temporal variability is constant and the learning …
Indian Institute of Technology, Guwahati（印度理工学院古瓦哈提分校）
Generalization error defines the discriminability and the representation power of a deep model. In this work, we claim that feature space design using deep compositional function plays a significant role in generalization along with explicit and implicit regularizations. Our claims are being established with several image classification experiments. We show that the information loss due to convolution and max pooling can be marginalized with the compositional design, improving generalization performance. Also, we will show that learning rate decay acts as an implicit regularizer in deep model …
Institute for Interdisciplinary Information Sciences（交叉信息研究院）；Tsinghua University（清华大学）
In this paper, we study the implicit regularization of the gradient descent algorithm in homogeneous neural networks, including fully-connected and convolutional neural networks with ReLU or LeakyReLU activations. In particular, we study the gradient descent or gradient flow (i.e., gradient descent with infinitesimal step size) optimizing the logistic loss or cross-entropy loss of any homogeneous model (possibly non-smooth), and show that if the training loss decreases below a certain threshold, then we can define a smoothed version of the normalized margin which increases over time. We also f…
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）
Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially…
Deep ReLU networks trained with the square loss have been observed to perform well in classification tasks. We provide here a theoretical justification based on analysis of the associated gradient flow. We show that convergence to a solution with the absolute minimum norm is expected when normalization techniques such as Batch Normalization (BN) or Weight Normalization (WN) are used together with Weight Decay (WD). The main property of the minimizers that bounds their expected error is the norm: we prove that among all the close-to-interpolating solutions, the ones associated with smaller Frob…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Weizmann Institute of Science（魏茨曼科学研究所）
The implicit bias of neural networks has been extensively studied in recent years. Lyu and Li [2019] showed that in homogeneous networks trained with the exponential or the logistic loss, gradient flow converges to a KKT point of the max margin problem in the parameter space. However, that leaves open the question of whether this point will generally be an actual optimum of the max margin problem. In this paper, we study this question in detail, for several neural network architectures involving linear and ReLU activations. Perhaps surprisingly, we show that in many cases, the KKT point is not…
To improve the sample efficiency of policy-gradient based reinforcement learning algorithms, we propose implicit distributional actor-critic (IDAC) that consists of a distributional critic, built on two deep generator networks (DGNs), and a semi-implicit actor (SIA), powered by a flexible policy distribution. We adopt a distributional perspective on the discounted cumulative return and model it with a state-action-dependent implicit distribution, which is approximated by the DGNs that take state-action pairs and random noises as their input. Moreover, we use the SIA to provide a semi-implicit …
