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 discuss further improvements to these calculations and comment on possible implications for stochastic optimization.
Duke University（杜克大学）；The Ohio State University（俄亥俄州立大学）；Microsoft Research, Asia（微软研究院亚洲研究院（微软亚洲研究院））
The success of deep learning has led to a rising interest in the generalization property of the stochastic gradient descent (SGD) method, and stability is one popular approach to study it. Existing works based on stability have studied nonconvex loss functions, but only considered the generalization error of the SGD in expectation. In this paper, we establish various generalization error bounds with probabilistic guarantee for the SGD. Specifically, for both general nonconvex loss functions and gradient dominant loss functions, we characterize the on-average stability of the iterates generated…
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…
CNRS（法国国家科学研究中心）；Ecole Polytechnique（巴黎综合理工学院）；Université Pierre et Marie Curie（皮埃尔和玛丽·居里大学）
We introduce a doubly stochastic proximal gradient algorithm for optimizing a finite average of smooth convex functions, whose gradients depend on numerically expensive expectations. Our main motivation is the acceleration of the optimization of the regularized Cox partial-likelihood (the core model used in survival analysis), but our algorithm can be used in different settings as well. The proposed algorithm is doubly stochastic in the sense that gradient steps are done using stochastic gradient descent (SGD) with variance reduction, where the inner expectations are approximated by a Monte-Ca…
In overparametrized models, the noise in stochastic gradient descent (SGD) implicitly regularizes the optimization trajectory and determines which local minimum SGD converges to. Motivated by empirical studies that demonstrate that training with noisy labels improves generalization, we study the implicit regularization effect of SGD with label noise. We show that SGD with label noise converges to a stationary point of a regularized loss $L(θ) +λR(θ)$, where $L(θ)$ is the training loss, $λ$ is an effective regularization parameter depending on the step size, strength of the label noise, and the…
Stochastic Gradient Descent (SGD) has become popular for solving large scale supervised machine learning optimization problems such as SVM, due to their strong theoretical guarantees. While the closely related Dual Coordinate Ascent (DCA) method has been implemented in various software packages, it has so far lacked good convergence analysis. This paper presents a new analysis of Stochastic Dual Coordinate Ascent (SDCA) showing that this class of methods enjoy strong theoretical guarantees that are comparable or better than SGD. This analysis justifies the effectiveness of SDCA for practical a…
The multiplicative structure of parameters and input data in the first layer of neural networks is explored to build connection between the landscape of the loss function with respect to parameters and the landscape of the model function with respect to input data. By this connection, it is shown that flat minima regularize the gradient of the model function, which explains the good generalization performance of flat minima. Then, we go beyond the flatness and consider high-order moments of the gradient noise, and show that Stochastic Gradient Descent (SGD) tends to impose constraints on these…
Despite being highly over-parametrized, and having the ability to fully interpolate the training data, deep networks are known to generalize well to unseen data. It is now understood that part of the reason for this is that the training algorithms used have certain implicit regularization properties that ensure interpolating solutions with "good" properties are found. This is best understood in linear over-parametrized models where it has been shown that the celebrated stochastic gradient descent (SGD) algorithm finds an interpolating solution that is closest in Euclidean distance to the initi…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；University of Toronto（多伦多大学）
We revisit the choice of SGD for training deep neural networks by reconsidering the appropriate geometry in which to optimize the weights. We argue for a geometry invariant to rescaling of weights that does not affect the output of the network, and suggest Path-SGD, which is an approximate steepest descent method with respect to a path-wise regularizer related to max-norm regularization. Path-SGD is easy and efficient to implement and leads to empirical gains over SGD and AdaGrad.
Stability and Generalization of lp-Regularized Stochastic Learning for GCN
University of Electronic Science and Technology of China（电子科技大学）；Northwestern Polytechnical University（西北工业大学）；Nanjing Audit University（南京审计大学）；Zhejiang Normal University（浙江师范大学）
Johns Hopkins University（约翰斯·霍普金斯大学）；Missouri University of Science and Technology（密苏里科技大学）；Baidu Research（百度研究院）；Styling.AI Inc.（Styling.AI公司）；Peking University（北京大学）
The gradient noise of SGD is considered to play a central role in the observed strong generalization abilities of deep learning. While past studies confirm that the magnitude and the covariance structure of gradient noise are critical for regularization, it remains unclear whether or not the class of noise distributions is important. In this work we provide negative results by showing that noises in classes different from the SGD noise can also effectively regularize gradient descent. Our finding is based on a novel observation on the structure of the SGD noise: it is the multiplication of the…
Computer Science, Wayne State University（韦恩州立大学计算机科学（系））
Deep neural networks (DNNs) for supervised learning can be viewed as a pipeline of a feature extractor (i.e. last hidden layer) and a linear classifier (i.e. output layer) that is trained jointly with stochastic gradient descent (SGD). In each iteration of SGD, a mini-batch from the training data is sampled and the true gradient of the loss function is estimated as the noisy gradient calculated on this mini-batch. From the feature learning perspective, the feature extractor should be updated to learn meaningful features with respect to the entire data, and reduce the accommodation to noise in …
Quebec Artificial Intelligence Institute (Mila)（魁北克人工智能研究所（米拉））；Université de Montréal（蒙特利尔大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Athens（雅典大学）
Several works have aimed to explain why overparameterized neural networks generalize well when trained by Stochastic Gradient Descent (SGD). The consensus explanation that has emerged credits the randomized nature of SGD for the bias of the training process towards low-complexity models and, thus, for implicit regularization. We take a careful look at this explanation in the context of image classification with common deep neural network architectures. We find that if we do not regularize \emph{explicitly}, then SGD can be easily made to converge to poorly-generalizing, high-complexity models:…
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…
Johns Hopkins University（约翰霍普金斯大学）；University of California, Los Angeles（加州大学洛杉矶分校）
Regularization for optimization is a crucial technique to avoid overfitting in machine learning. In order to obtain the best performance, we usually train a model by tuning the regularization parameters. It becomes costly, however, when a single round of training takes significant amount of time. Very recently, Neu and Rosasco show that if we run stochastic gradient descent (SGD) on linear regression problems, then by averaging the SGD iterates properly, we obtain a regularized solution. It left open whether the same phenomenon can be achieved for other optimization problems and algorithms. In…
UC San Diego（加州大学圣地亚哥分校）；Princeton University（普林斯顿大学）；Carnegie Mellon University（卡内基梅隆大学）
A number of competing hypotheses have been proposed to explain why small-batch Stochastic Gradient Descent (SGD)leads to improved generalization over the full-batch regime, with recent work crediting the implicit regularization of various quantities throughout training. However, to date, empirical evidence assessing the explanatory power of these hypotheses is lacking. In this paper, we conduct an extensive empirical evaluation, focusing on the ability of various theorized mechanisms to close the small-to-large batch generalization gap. Additionally, we characterize how the quantities that SGD…
Yahoo Research（雅虎研究院）；The University of Iowa（爱荷华大学）；Nanjing University（南京大学）
We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent~(SGD) methods. To reduce the computation costs pertaining to the projections, we propose an Epoch-Projection Stochastic Gradient Descent~(Epro-SGD) method. The proposed Epro-SGD method consists of a sequence of epochs; it applies SGD to an augmented objective function at each iteration within the epoch, and then performs a projection at the end of each epoch. Gi…
Multi-epoch, small-batch, Stochastic Gradient Descent (SGD) has been the method of choice for learning with large over-parameterized models. A popular theory for explaining why SGD works well in practice is that the algorithm has an implicit regularization that biases its output towards a good solution. Perhaps the theoretically most well understood learning setting for SGD is that of Stochastic Convex Optimization (SCO), where it is well known that SGD learns at a rate of $O(1/\sqrt{n})$, where $n$ is the number of samples. In this paper, we consider the problem of SCO and explore the role of…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
The notion of implicit bias, or implicit regularization, has been suggested as a means to explain the surprising generalization ability of modern-days overparameterized learning algorithms. This notion refers to the tendency of the optimization algorithm towards a certain structured solution that often generalizes well. Recently, several papers have studied implicit regularization and were able to identify this phenomenon in various scenarios. We revisit this paradigm in arguably the simplest non-trivial setup, and study the implicit bias of Stochastic Gradient Descent (SGD) in the context of …
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…
Understanding the implicit bias of Stochastic Gradient Descent (SGD) is one of the key challenges in deep learning, especially for overparametrized models, where the local minimizers of the loss function $L$ can form a manifold. Intuitively, with a sufficiently small learning rate $η$, SGD tracks Gradient Descent (GD) until it gets close to such manifold, where the gradient noise prevents further convergence. In such a regime, Blanc et al. (2020) proved that SGD with label noise locally decreases a regularizer-like term, the sharpness of loss, $\mathrm{tr}[\nabla^2 L]$. The current paper gives…
Centre Borelli（博雷利中心）；ENS Paris Saclay（巴黎-萨克雷高等师范学校）；Télécom Paris（巴黎电信学院）；Institut Polytechnique de Paris（巴黎综合理工学院）
In this paper, we investigate the limiting behavior of a continuous-time counterpart of the Stochastic Gradient Descent (SGD) algorithm applied to two-layer overparameterized neural networks, as the number or neurons (ie, the size of the hidden layer) $N \to +\infty$. Following a probabilistic approach, we show 'propagation of chaos' for the particle system defined by this continuous-time dynamics under different scenarios, indicating that the statistical interaction between the particles asymptotically vanishes. In particular, we establish quantitative convergence with respect to $N$ of any p…
Optimizers that further adjust the scale of gradient, such as Adam, Natural Gradient (NG), etc., despite widely concerned and used by the community, are often found poor generalization performance, compared with Stochastic Gradient Descent (SGD). They tend to converge excellently at the beginning of training but are weak at the end. An immediate idea is to complement the strengths of these algorithms with SGD. However, a truncated replacement of optimizer often leads to a crash of the update pattern, and new algorithms often require many iterations to stabilize their search direction. Driven b…
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 …
University of Connecticut（康涅狄格大学）；IBM Research（IBM研究院）
We study Stochastic Gradient Descent (SGD) with diminishing step sizes for convex objective functions. We introduce a definitional framework and theory that defines and characterizes a core property, called curvature, of convex objective functions. In terms of curvature we can derive a new inequality that can be used to compute an optimal sequence of diminishing step sizes by solving a differential equation. Our exact solutions confirm known results in literature and allows us to fully characterize a new regularizer with its corresponding expected convergence rates.
Lehigh University（理海大学）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）；University of Connecticut（康涅狄格大学）；KAUST（阿卜杜拉国王科技大学）；MIPT（莫斯科物理技术学院）
Stochastic gradient descent (SGD) is the optimization algorithm of choice in many machine learning applications such as regularized empirical risk minimization and training deep neural networks. The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is always violated for cases where the objective function is strongly convex. In (Bottou et al.,2016), a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded wi…
Previous work has examined the ability of larger capacity neural networks to generalize better than smaller ones, even without explicit regularizers, by analyzing gradient based algorithms such as GD and SGD. The presence of noise and its effect on robustness to parameter perturbations has been linked to generalization. We examine a property of GD and SGD, namely that instead of iterating through all scalar weights in the network and updating them one by one, GD (and SGD) updates all the parameters at the same time. As a result, each parameter $w^i$ calculates its partial derivative at the sta…
Despite the non-convex nature of their loss functions, deep neural networks are known to generalize well when optimized with stochastic gradient descent (SGD). Recent work conjectures that SGD with proper configuration is able to find wide and flat local minima, which have been proposed to be associated with good generalization performance. In this paper, we observe that local minima of modern deep networks are more than being flat or sharp. Specifically, at a local minimum there exist many asymmetric directions such that the loss increases abruptly along one side, and slowly along the opposit…
Decentralized SGD and Average-direction SAM are Asymptotically Equivalent
College of Computer Science and Technology, Zhejiang University（浙江大学计算机科学与技术学院）；JD Explore Academy, JD.com, Inc.（京东探索研究院，京东集团）；Artificial Intelligence and its Applications Institute, School of Informatics, University of Edinburgh（爱丁堡大学信息学院人工智能及其应用研究所）；The University of Sydney（悉尼大学）
IST Austria（奥地利科学技术研究所）；Czech Technical University in Prague（布拉格捷克理工大学）
Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean-field view, and consider a two-layer ReLU network trained via SGD for a univariate regularized regression problem. Our main result is that SGD is biased towards a simple solution: at convergence, the ReLU network implements a piecewise linear map of the inputs, and the number of "knot" points - i.e., points where the tangent of the ReLU network estimator changes - between two consecutive training inputs is at most three. In p…
Mitsubishi Electric Research Laboratories (MERL)（三菱电机研究实验室）
By lifting the ReLU function into a higher dimensional space, we develop a smooth multi-convex formulation for training feed-forward deep neural networks (DNNs). This allows us to develop a block coordinate descent (BCD) training algorithm consisting of a sequence of numerically well-behaved convex optimizations. Using ideas from proximal point methods in convex analysis, we prove that this BCD algorithm will converge globally to a stationary point with R-linear convergence rate of order one. In experiments with the MNIST database, DNNs trained with this BCD algorithm consistently yielded bett…
Two different approaches exist to handle missing values for prediction: either imputation, prior to fitting any predictive algorithms, or dedicated methods able to natively incorporate missing values. While imputation is widely (and easily) use, it is unfortunately biased when low-capacity predictors (such as linear models) are applied afterward. However, in practice, naive imputation exhibits good predictive performance. In this paper, we study the impact of imputation in a high-dimensional linear model with MCAR missing data. We prove that zero imputation performs an implicit regularization …
SOFIM: Stochastic Optimization Using Regularized Fisher Information Matrix
Swinburne University of Technology（斯威本科技大学）；Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）；Mahindra University（马恒达大学）；Ritsumeikan University（立命馆大学）；Indian Institute of Technology Roorkee（印度理工学院鲁基分校）
New York University（纽约大学）；Jagiellonian University（雅盖隆大学）；Stanford University（斯坦福大学）；Salesforce Research（Salesforce研究院）；Facebook AI Research（脸书人工智能研究院）
The early phase of training of deep neural networks is critical for their final performance. In this work, we study how the hyperparameters of stochastic gradient descent (SGD) used in the early phase of training affect the rest of the optimization trajectory. We argue for the existence of the "break-even" point on this trajectory, beyond which the curvature of the loss surface and noise in the gradient are implicitly regularized by SGD. In particular, we demonstrate on multiple classification tasks that using a large learning rate in the initial phase of training reduces the variance of the g…
Large-batch SGD is important for scaling training of deep neural networks. However, without fine-tuning hyperparameter schedules, the generalization of the model may be hampered. We propose to use batch augmentation: replicating instances of samples within the same batch with different data augmentations. Batch augmentation acts as a regularizer and an accelerator, increasing both generalization and performance scaling. We analyze the effect of batch augmentation on gradient variance and show that it empirically improves convergence for a wide variety of deep neural networks and datasets. Our …
In this paper, we propose a simple variant of the original stochastic variance reduction gradient (SVRG), where hereafter we refer to as the variance reduced stochastic gradient descent (VR-SGD). Different from the choices of the snapshot point and starting point in SVRG and its proximal variant, Prox-SVRG, the two vectors of each epoch in VR-SGD are set to the average and last iterate of the previous epoch, respectively. This setting allows us to use much larger learning rates or step sizes than SVRG, e.g., 3/(7L) for VR-SGD vs 1/(10L) for SVRG, and also makes our convergence analysis more ch…
Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reduction
University of Tokyo（东京大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；University of Toronto（多伦多大学）；Vector Institute for Artificial Intelligence（矢量人工智能研究所）；Kyushu Institute of Technology（九州工业大学）
Rutgers University（罗格斯大学）；Queens College of CUNY（纽约市立大学皇后学院）；Indian Institute of Science（印度科学学院）；Stony Brook University（石溪大学）
Stochastic Gradient Descent (SGD) based methods have been widely used for training large-scale machine learning models that also generalize well in practice. Several explanations have been offered for this generalization performance, a prominent one being algorithmic stability [18]. However, there are no known examples of smooth loss functions for which the analysis can be shown to be tight. Furthermore, apart from the properties of the loss function, data distribution has also been shown to be an important factor in generalization performance. This raises the question: is the stability analys…
Noise Balance and Stationary Distribution of Stochastic Gradient Descent
Research Laboratory of Electronics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA；Physics \& Informatics Laboratories, NTT Research, 940 Stewart Drive Sunnyvale, CA 94085, USA；Department of Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan；RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan；Institute for Physics of Intelligence, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan
Two types of low cost-per-iteration gradient descent methods have been extensively studied in parallel. One is online or stochastic gradient descent (OGD/SGD), and the other is randomzied coordinate descent (RBCD). In this paper, we combine the two types of methods together and propose online randomized block coordinate descent (ORBCD). At each iteration, ORBCD only computes the partial gradient of one block coordinate of one mini-batch samples. ORBCD is well suited for the composite minimization problem where one function is the average of the losses of a large number of samples and the other…
We study the performance of Stochastic Cubic Regularized Newton (SCRN) on a class of functions satisfying gradient dominance property with $1\leα\le2$ which holds in a wide range of applications in machine learning and signal processing. This condition ensures that any first-order stationary point is a global optimum. We prove that the total sample complexity of SCRN in achieving $ε$-global optimum is $\mathcal{O}(ε^{-7/(2α)+1})$ for $1\leα< 3/2$ and $\mathcal{\tilde{O}}(ε^{-2/(α)})$ for $3/2\leα\le 2$. SCRN improves the best-known sample complexity of stochastic gradient descent. Even under a…
