Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss
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）
Lenaic Chizat, Francis Bach
发表信息：
Conference on Learning Theory, Jul 2020, Graz, Austria
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 independent of the ambiant dimension, which leads to strong generalization bounds. In contrast, training only the output layer implicitly solves a kernel support vector machine, which a priori does not enjoy such an adaptivity. Our analysis of training is non-quantitative in terms of running time but we prove computational guarantees in simplified settings by showing equivalences with online mirror descent. Finally, numerical experiments suggest that our analysis describes well the practical behavior of two-layer neural networks with ReLU activation and confirm the statistical benefits of this implicit bias.
University of California, Berkeley（加州大学伯克利分校）；Google（谷歌）
We study the training of finite-width two-layer smoothed ReLU networks for binary classification using the logistic loss. We show that gradient descent drives the training loss to zero if the initial loss is small enough. When the data satisfies certain cluster and separation conditions and the network is wide enough, we show that one step of gradient descent reduces the loss sufficiently that the first result applies.
University of California, Berkeley（加州大学伯克利分校）；Google（谷歌）
We establish conditions under which gradient descent applied to fixed-width deep networks drives the logistic loss to zero, and prove bounds on the rate of convergence. Our analysis applies for smoothed approximations to the ReLU, such as Swish and the Huberized ReLU, proposed in previous applied work. We provide two sufficient conditions for convergence. The first is simply a bound on the loss at initialization. The second is a data separation condition used in prior analyses.
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Gradient descent, when applied to the task of logistic regression, outputs iterates which are biased to follow a unique ray defined by the data. The direction of this ray is the maximum margin predictor of a maximal linearly separable subset of the data; the gradient descent iterates converge to this ray in direction at the rate $\mathcal{O}(\ln\ln t / \ln t)$. The ray does not pass through the origin in general, and its offset is the bounded global optimum of the risk over the remaining data; gradient descent recovers this offset at a rate $\mathcal{O}((\ln t)^2 / \sqrt{t})$.
Exponential Convergence of (Stochastic) Gradient Descent for Separable Logistic Regression
Undergraduate Programme Indian Institute of Science, Bangalore（印度科学研究院布达佩斯分校本科项目）；Dept. of Computer Science and Automation Indian Institute of Science, Bangalore（计算机科学与自动化系印度科学研究院布达佩斯分校）；Inria, Ecole Normale Supérieure PSL Research University, Paris, France（法国巴黎PSL研究大学埃cole Normale Supérieure与Inria）
Research in statistical relational learning has produced a number of methods for learning relational models from large-scale network data. While these methods have been successfully applied in various domains, they have been developed under the unrealistic assumption of full data access. In practice, however, the data are often collected by crawling the network, due to proprietary access, limited resources, and privacy concerns. Recently, we showed that the parameter estimates for relational Bayes classifiers computed from network samples collected by existing network crawlers can be quite ina…
We study gradient descent (GD) with a constant stepsize for $\ell_2$-regularized logistic regression with linearly separable data. Classical theory suggests small stepsizes to ensure monotonic reduction of the optimization objective, achieving exponential convergence in $\widetilde{\mathcal{O}}(κ)$ steps with $κ$ being the condition number. Surprisingly, we show that this can be accelerated to $\widetilde{\mathcal{O}}(\sqrtκ)$ by simply using a large stepsize -- for which the objective evolves nonmonotonically. The acceleration brought by large stepsizes extends to minimizing the population ri…
Implicit deep learning has recently become popular in the machine learning community since these implicit models can achieve competitive performance with state-of-the-art deep networks while using significantly less memory and computational resources. However, our theoretical understanding of when and how first-order methods such as gradient descent (GD) converge on \textit{nonlinear} implicit networks is limited. Although this type of problem has been studied in standard feed-forward networks, the case of implicit models is still intriguing because implicit networks have \textit{infinitely} m…
McGill University（麦吉尔大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
In this paper, we propose an implicit gradient descent algorithm for the classic $k$-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is carried over to the next gradient step. We draw connections between the proposed stochastic backward Euler and the recent entropy stochastic gradient descent (Entropy-SGD) for improving the training of deep neural networks. Numerical experiments on various synthetic and real datas…
We propose graph-dependent implicit regularisation strategies for distributed stochastic subgradient descent (Distributed SGD) for convex problems in multi-agent learning. Under the standard assumptions of convexity, Lipschitz continuity, and smoothness, we establish statistical learning rates that retain, up to logarithmic terms, centralised statistical guarantees through implicit regularisation (step size tuning and early stopping) with appropriate dependence on the graph topology. Our approach avoids the need for explicit regularisation in decentralised learning problems, such as adding con…
As part of the effort to understand implicit bias of gradient descent in overparametrized models, several results have shown how the training trajectory on the overparametrized model can be understood as mirror descent on a different objective. The main result here is a characterization of this phenomenon under a notion termed commuting parametrization, which encompasses all the previous results in this setting. It is shown that gradient flow with any commuting parametrization is equivalent to continuous mirror descent with a related Legendre function. Conversely, continuous mirror descent wit…
Understanding the implicit bias of training algorithms is of crucial importance in order to explain the success of overparametrised neural networks. In this paper, we study the role of the label noise in the training dynamics of a quadratically parametrised model through its continuous time version. We explicitly characterise the solution chosen by the stochastic flow and prove that it implicitly solves a Lasso program. To fully complete our analysis, we provide nonasymptotic convergence guarantees for the dynamics as well as conditions for support recovery. We also give experimental results w…
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.
MIT Sloan School of Management（麻省理工学院斯隆管理学院）；University of California, Berkeley（加州大学伯克利分校）
Logistic regression is one of the most popular methods in binary classification, wherein estimation of model parameters is carried out by solving the maximum likelihood (ML) optimization problem, and the ML estimator is defined to be the optimal solution of this problem. It is well known that the ML estimator exists when the data is non-separable, but fails to exist when the data is separable. First-order methods are the algorithms of choice for solving large-scale instances of the logistic regression problem. In this paper, we introduce a pair of condition numbers that measure the degree of n…
In this paper, we consider supervised learning problems such as logistic regression and study the stochastic gradient method with averaging, in the usual stochastic approximation setting where observations are used only once. We show that after $N$ iterations, with a constant step-size proportional to $1/R^2 \sqrt{N}$ where $N$ is the number of observations and $R$ is the maximum norm of the observations, the convergence rate is always of order $O(1/\sqrt{N})$, and improves to $O(R^2 / μN)$ where $μ$ is the lowest eigenvalue of the Hessian at the global optimum (when this eigenvalue is greater…
Gradient Descent on Logistic Regression: Do Large Step-Sizes Work with Data on the Sphere?
Inria Paris（巴黎国家信息与自动化研究所）；Inria Saclay Palaiseau（萨克莱研究所）；ELLIS Institute Tübingen MPI for Intelligent Systems Tübingen AI Center, Germany（图宾根人工智能研究所）；Department of Computer Science Cornell University（康奈尔大学计算机科学系）
It is well known that the finite step-size ($h$) in Gradient Descent (GD) implicitly regularizes solutions to flatter minima. A natural question to ask is "Does the momentum parameter $β$ play a role in implicit regularization in Heavy-ball (H.B) momentum accelerated gradient descent (GD+M)?". To answer this question, first, we show that the discrete H.B momentum update (GD+M) follows a continuous trajectory induced by a modified loss, which consists of an original loss and an implicit regularizer. Then, we show that this implicit regularizer for (GD+M) is stronger than that of (GD) by factor …
We study discrete-time mirror descent applied to the unregularized empirical risk in matrix sensing. In both the general case of rectangular matrices and the particular case of positive semidefinite matrices, a simple potential-based analysis in terms of the Bregman divergence allows us to establish convergence of mirror descent -- with different choices of the mirror maps -- to a matrix that, among all global minimizers of the empirical risk, minimizes a quantity explicitly related to the nuclear norm, the Frobenius norm, and the von Neumann entropy. In both cases, this characterization impli…
In machine learning and statistical data analysis, we often run into objective function that is a summation: the number of terms in the summation possibly is equal to the sample size, which can be enormous. In such a setting, the stochastic mirror descent (SMD) algorithm is a numerically efficient method -- each iteration involving a very small subset of the data. The variance reduction version of SMD (VRSMD) can further improve SMD by inducing faster convergence. On the other hand, algorithms such as gradient descent and stochastic gradient descent have the implicit regularization property th…
University of Oxford（牛津大学）；Instadeep Ltd（Instadeep有限公司）；University of Southern California（南加州大学）
We prove linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We show that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite $p-$variation with $p=2$. Proofs are based on non-asymptotic estimates for the loss function and for norms of the network weights along the gradient descent path. We illustrate the relevance of our theoretical results to practical settings using detailed numerical experiments on supervis…
Northeast Normal University（东北师范大学）；Beihua University（北华大学）
The study on the implicit regularization induced by gradient-based optimization is a longstanding pursuit. In the present paper, we characterize the implicit regularization of momentum gradient descent (MGD) with early stopping by comparing with the explicit $\ell_2$-regularization (ridge). In details, we study MGD in the continuous-time view, so-called momentum gradient flow (MGF), and show that its tendency is closer to ridge than the gradient descent (GD) [Ali et al., 2019] for least squares regression. Moreover, we prove that, under the calibration $t=\sqrt{2/λ}$, where $t$ is the time par…
Gradient descent can be surprisingly good at optimizing deep neural networks without overfitting and without explicit regularization. We find that the discrete steps of gradient descent implicitly regularize models by penalizing gradient descent trajectories that have large loss gradients. We call this Implicit Gradient Regularization (IGR) and we use backward error analysis to calculate the size of this regularization. We confirm empirically that implicit gradient regularization biases gradient descent toward flat minima, where test errors are small and solutions are robust to noisy parameter…
UC Berkeley（加州大学伯克利分校）；National University of Defense Technology（国防科技大学）；Stanford University（斯坦福大学）；National University of Singapore（新加坡国立大学）
We consider a wide range of regularized stochastic minimization problems with two regularization terms, one of which is composed with a linear function. This optimization model abstracts a number of important applications in artificial intelligence and machine learning, such as fused Lasso, fused logistic regression, and a class of graph-guided regularized minimization. The computational challenges of this model are in two folds. On one hand, the closed-form solution of the proximal mapping associated with the composed regularization term or the expected objective function is not available. On…
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…
For infinitesimal learning rates, stochastic gradient descent (SGD) follows the path of gradient flow on the full batch loss function. However moderately large learning rates can achieve higher test accuracies, and this generalization benefit is not explained by convergence bounds, since the learning rate which maximizes test accuracy is often larger than the learning rate which minimizes training loss. To interpret this phenomenon we prove that for SGD with random shuffling, the mean SGD iterate also stays close to the path of gradient flow if the learning rate is small and finite, but on a m…
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…
California Institute of Technology（加州理工学院）；Department of Computing and Mathematical Sciences（计算与数学科学系）；Department of Electrical Engineering（电气工程系）
Stochastic descent methods (of the gradient and mirror varieties) have become increasingly popular in optimization. In fact, it is now widely recognized that the success of deep learning is not only due to the special deep architecture of the models, but also due to the behavior of the stochastic descent methods used, which play a key role in reaching "good" solutions that generalize well to unseen data. In an attempt to shed some light on why this is the case, we revisit some minimax properties of stochastic gradient descent (SGD) for the square loss of linear models---originally developed in…
Columbia University（哥伦比亚大学）；Flatiron Institute（弗莱托恩研究所）
Understanding the asymptotic behavior of gradient-descent training of deep neural networks is essential for revealing inductive biases and improving network performance. We derive the infinite-time training limit of a mathematically tractable class of deep nonlinear neural networks, gated linear networks (GLNs), and generalize these results to gated networks described by general homogeneous polynomials. We study the implications of our results, focusing first on two-layer GLNs. We then apply our theoretical predictions to GLNs trained on MNIST and show how architectural constraints and the imp…
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 degli Studi di Genova（热那亚大学）；Machine Learning Genoa Center（热那亚机器学习中心）；Massachusetts Institute of Technology（麻省理工学院）；Istituto Italiano di Tecnologia（意大利技术研究院）
We study learning properties of accelerated gradient descent methods for linear least-squares in Hilbert spaces. We analyze the implicit regularization properties of Nesterov acceleration and a variant of heavy-ball in terms of corresponding learning error bounds. Our results show that acceleration can provides faster bias decay than gradient descent, but also suffers of a more unstable behavior. As a result acceleration cannot be in general expected to improve learning accuracy with respect to gradient descent, but rather to achieve the same accuracy with reduced computations. Our theoretical…
Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Université Paris-Est（巴黎东大学）；CNRS（法国国家科学研究中心）；ESIEE Paris（巴黎高等电子工程师学院）
In this paper, we propose a scalable algorithm for spectral embedding. The latter is a standard tool for graph clustering. However, its computational bottleneck is the eigendecomposition of the graph Laplacian matrix, which prevents its application to large-scale graphs. Our contribution consists of reformulating spectral embedding so that it can be solved via stochastic optimization. The idea is to replace the orthogonality constraint with an orthogonalization matrix injected directly into the criterion. As the gradient can be computed through a Cholesky factorization, our reformulation allow…
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）
We derive a simple and model-independent formula for the change in the generalization gap due to a gradient descent update. We then compare the change in the test error for stochastic gradient descent to the change in test error from an equivalent number of gradient descent updates and show explicitly that stochastic gradient descent acts to regularize generalization error by decorrelating nearby updates. These calculations depends on the details of the model only through the mean and covariance of the gradient distribution, which may be readily measured for particular models of interest. We d…
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…
Missouri University of Science and Technology（密苏里科技大学）；Peking University（北京大学）；Beijing Institute of Big Data Research（北京大数据研究院）；Baidu Inc.（百度公司）
We interpret the variational inference of the Stochastic Gradient Descent (SGD) as minimizing a new potential function named the \textit{quasi-potential}. We analytically construct the quasi-potential function in the case when the loss function is convex and admits only one global minimum point. We show in this case that the quasi-potential function is related to the noise covariance structure of SGD via a partial differential equation of Hamilton-Jacobi type. This relation helps us to show that anisotropic noise leads to faster escape than isotropic noise. We then consider the dynamics of SGD…
Texas A&M University（德克萨斯农工大学）；New York University（纽约大学）
We study the implicit regularization of gradient descent towards structured sparsity via a novel neural reparameterization, which we call a diagonally grouped linear neural network. We show the following intriguing property of our reparameterization: gradient descent over the squared regression loss, without any explicit regularization, biases towards solutions with a group sparsity structure. In contrast to many existing works in understanding implicit regularization, we prove that our training trajectory cannot be simulated by mirror descent. We analyze the gradient dynamics of the correspon…
Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. For vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in nonconvex op…
Université Paris-Saclay（巴黎-萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能与替代能源委员会）；University of Genova（热那亚大学）
Acceleration of first order methods is mainly obtained via inertial techniques à la Nesterov, or via nonlinear extrapolation. The latter has known a recent surge of interest, with successful applications to gradient and proximal gradient techniques. On multiple Machine Learning problems, coordinate descent achieves performance significantly superior to full-gradient methods. Speeding up coordinate descent in practice is not easy: inertially accelerated versions of coordinate descent are theoretically accelerated, but might not always lead to practical speed-ups. We propose an accelerated versi…
