Recently, optimization on the Riemannian manifold have provided valuable insights to the optimization community. In this regard, extending these methods to to the Wasserstein space is of particular interest, since optimization on Wasserstein space is closely connected to practical sampling processes. Generally, the standard (continuous) optimization method on Wasserstein space is Riemannian gradient flow (i.e., Langevin dynamics when minimizing KL divergence). In this paper, we aim to enrich the family of continuous optimization methods in the Wasserstein space, by extending the gradient flow on it into the stochastic gradient descent (SGD) flow and stochastic variance reduction gradient (SVRG) flow.
By leveraging the property of Wasserstein space, we construct stochastic differential equations (SDEs) to approximate the corresponding discrete Euclidean dynamics of the desired Riemannian stochastic methods. Then, we obtain the flows in Wasserstein space by Fokker-Planck equation. Finally, we establish convergence rates of the proposed stochastic flows, which align with those known in the Euclidean setting.
Stochastic variance-reduced gradient (SVRG) is a classical optimization method. Although it is theoretically proved to have better convergence performance than stochastic gradient descent (SGD), the generalization performance of SVRG remains open. In this paper we investigate the effects of some training techniques, mini-batching and learning rate decay, on the generalization performance of SVRG, and verify the generalization performance of Batch-SVRG (B-SVRG). In terms of the relationship between optimization and generalization, we believe that the average norm of gradients on each training s…
Barzilai-Borwein Step Size for Stochastic Gradient Descent
The Chinese University of Hong Kong（香港中文大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；The University of Hong Kong（香港大学）
Xidian University（西安电子科技大学）；The Chinese University of Hong Kong（香港中文大学）；University of Technology Sydney（悉尼科技大学）；Nanjing University（南京大学）；University of Sydney（悉尼大学）
In this paper, we propose a simple variant of the original SVRG, called variance reduced stochastic gradient descent (VR-SGD). Unlike the choices of snapshot and starting points in SVRG and its proximal variant, Prox-SVRG, the two vectors of VR-SGD are set to the average and last iterate of the previous epoch, respectively. The settings allow us to use much larger learning rates, and also make our convergence analysis more challenging. We also design two different update rules for smooth and non-smooth objective functions, respectively, which means that VR-SGD can tackle non-smooth and/or non-…
Variance reduced stochastic gradient (SGD) methods converge significantly faster than the vanilla SGD counterpart. However, these methods are not very practical on large scale problems, as they either i) require frequent passes over the full data to recompute gradients---without making any progress during this time (like for SVRG), or ii)~they require additional memory that can surpass the size of the input problem (like for SAGA). In this work, we propose $k$-SVRG that addresses these issues by making best use of the \emph{available} memory and minimizes the stalling phases without progress. …
The Ohio State University（俄亥俄州立大学）；Duke University（杜克大学）
Two types of zeroth-order stochastic algorithms have recently been designed for nonconvex optimization respectively based on the first-order techniques SVRG and SARAH/SPIDER. This paper addresses several important issues that are still open in these methods. First, all existing SVRG-type zeroth-order algorithms suffer from worse function query complexities than either zeroth-order gradient descent (ZO-GD) or stochastic gradient descent (ZO-SGD). In this paper, we propose a new algorithm ZO-SVRG-Coord-Rand and develop a new analysis for an existing ZO-SVRG-Coord algorithm proposed in Liu et al.…
Fudan University（复旦大学）；SUNY at Buffalo（纽约州立大学布法罗分校）
Stochastic particle-optimization sampling (SPOS) is a recently-developed scalable Bayesian sampling framework that unifies stochastic gradient MCMC (SG-MCMC) and Stein variational gradient descent (SVGD) algorithms based on Wasserstein gradient flows. With a rigorous non-asymptotic convergence theory developed recently, SPOS avoids the particle-collapsing pitfall of SVGD. Nevertheless, variance reduction in SPOS has never been studied. In this paper, we bridge the gap by presenting several variance-reduction techniques for SPOS. Specifically, we propose three variants of variance-reduced SPOS,…
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）
Virtually all state-of-the-art methods for training supervised machine learning models are variants of SGD enhanced with a number of additional tricks, such as minibatching, momentum, and adaptive stepsizes. One of the tricks that works so well in practice that it is used as default in virtually all widely used machine learning software is {\em random reshuffling (RR)}. However, the practical benefits of RR have until very recently been eluding attempts at being satisfactorily explained using theory. Motivated by recent development due to Mishchenko, Khaled and Richtárik (2020), in this work w…
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）
Zhejiang University（浙江大学）；The Pennsylvania State University（宾夕法尼亚州立大学）
We develop a general framework unifying several gradient-based stochastic optimization methods for empirical risk minimization problems both in centralized and distributed scenarios. The framework hinges on the introduction of an augmented graph consisting of nodes modeling the samples and edges modeling both the inter-device communication and intra-device stochastic gradient computation. By designing properly the topology of the augmented graph, we are able to recover as special cases the renowned Local-SGD and DSGD algorithms, and provide a unified perspective for variance-reduction (VR) and…
Unbiased and Biased Variance-Reduced Forward-Reflected-Backward Splitting Methods for Stochastic Composite Inclusions
Department of Statistics and Operations Research（统计与运筹学系）；The University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；Hanes Hall, UNC-Chapel Hill, NC 27599-3260（UNC-教堂山分校318号哈尼斯楼，NC 27599-3260）
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Google（谷歌）
Based on the concepts of Wasserstein barycenter (WB) and Gromov-Wasserstein barycenter (GWB), we propose a unified mathematical framework for neural network (NN) model fusion and utilize it to reveal new insights about the linear mode connectivity of SGD solutions. In our framework, the fusion occurs in a layer-wise manner and builds on an interpretation of a node in a network as a function of the layer preceding it. The versatility of our mathematical framework allows us to talk about model fusion and linear mode connectivity for a broad class of NNs, including fully connected NN, CNN, ResNet…
TRSVR: An Adaptive Stochastic Trust-Region Method with Variance Reduction
Department of Mathematics, University of California, Berkeley, CA, USA（加州大学伯克利分校数学系）；Boston University, MA, USA（波士顿大学）；Department of IEOR, University of California, Berkeley, CA, USA（加州大学伯克利分校工业工程与运筹学系）
University of Texas at Austin（得克萨斯大学奥斯汀分校）；Princeton University（普林斯顿大学）；Massachusetts Institute of Technology（麻省理工学院）
Generative adversarial networks (GANs) are a widely used framework for learning generative models. Wasserstein GANs (WGANs), one of the most successful variants of GANs, require solving a minmax optimization problem to global optimality, but are in practice successfully trained using stochastic gradient descent-ascent. In this paper, we show that, when the generator is a one-layer network, stochastic gradient descent-ascent converges to a global solution with polynomial time and sample complexity.
We study distributed optimization algorithms for minimizing the average of \emph{heterogeneous} functions distributed across several machines with a focus on communication efficiency. In such settings, naively using the classical stochastic gradient descent (SGD) or its variants (e.g., SVRG) with a uniform sampling of machines typically yields poor performance. It often leads to the dependence of convergence rate on maximum Lipschitz constant of gradients across the devices. In this paper, we propose a novel \emph{adaptive} sampling of machines specially catered to these settings. Our method r…
Generalization Error Bounds for Optimization Algorithms via Stability
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing Jiaotong University（北京交通大学）；Microsoft Research（微软研究院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
The Chinese University of Hong Kong（香港中文大学）；Tencent AI Lab（腾讯人工智能实验室）；University of California, Davis（加州大学戴维斯分校）；University of Rochester（罗切斯特大学）
Regularized empirical risk minimization problem with linear predictor appears frequently in machine learning. In this paper, we propose a new stochastic primal-dual method to solve this class of problems. Different from existing methods, our proposed methods only require O(1) operations in each iteration. We also develop a variance-reduction variant of the algorithm that converges linearly. Numerical experiments suggest that our methods are faster than existing ones such as proximal SGD, SVRG and SAGA on high-dimensional problems.
Innopolis University（Innopolis 大学）；HSE University（俄罗斯国立高等经济大学）；Yandex（Yandex 公司）；MBZUAI（穆罕默德·本·扎耶德人工智能大学）；Mila（米拉研究所）；DIRO, Université de Montréal（蒙特利尔大学 DIRO 学院）；AMS and MINDS, Johns Hopkins University（约翰斯·霍普金斯大学 AMS 与 MINDS 研究中心）
Stochastic Gradient Descent-Ascent (SGDA) is one of the most prominent algorithms for solving min-max optimization and variational inequalities problems (VIP) appearing in various machine learning tasks. The success of the method led to several advanced extensions of the classical SGDA, including variants with arbitrary sampling, variance reduction, coordinate randomization, and distributed variants with compression, which were extensively studied in the literature, especially during the last few years. In this paper, we propose a unified convergence analysis that covers a large variety of sto…
Approximate Heavy Tails in Offline (Multi-Pass) Stochastic Gradient Descent
Inria Paris（巴黎国立信息与自动化研究所）；CNRS（法国国家科学研究中心）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；CMAP（应用数学中心）；Ecole Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）
Error estimates between SGD with momentum and underdamped Langevin diffusion
Laboratoire de Mathematiques Blaise Pascal, CNRS-UMR 6620, Universite Clermont-Auvergne（克莱蒙奥弗涅大学）；University of Macau（澳门大学）；UM Zhuhai Research Institute（澳门大学珠海研究院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing International Center for Mathematical Research (BICMR), Peking University（北京大学北京国际数学研究中心）
Purdue University（普渡大学）；University of Minnesota（明尼苏达大学）
Past research has indicated that the covariance of the Stochastic Gradient Descent (SGD) error done via minibatching plays a critical role in determining its regularization and escape from low potential points. Motivated by some new research in this area, we prove universality results by showing that noise classes that have the same mean and covariance structure of SGD via minibatching have similar properties. We mainly consider the Multiplicative Stochastic Gradient Descent (M-SGD) algorithm as introduced in previous work, which has a much more general noise class than the SGD algorithm done …
Microsoft Research（微软研究院）；Massachusetts Institute of Technology（麻省理工学院）
We study stochastic gradient descent {\em without replacement} (\sgdwor) for smooth convex functions. \sgdwor is widely observed to converge faster than true \sgd where each sample is drawn independently {\em with replacement} \cite{bottou2009curiously} and hence, is more popular in practice. But it's convergence properties are not well understood as sampling without replacement leads to coupling between iterates and gradients. By using method of exchangeable pairs to bound Wasserstein distance, we provide the first non-asymptotic results for \sgdwor when applied to {\em general smooth, strong…
This paper underlines a subtle property of batch-normalization (BN): Successive batch normalizations with random linear transformations make hidden representations increasingly orthogonal across layers of a deep neural network. We establish a non-asymptotic characterization of the interplay between depth, width, and the orthogonality of deep representations. More precisely, under a mild assumption, we prove that the deviation of the representations from orthogonality rapidly decays with depth up to a term inversely proportional to the network width. This result has two main implications: 1) Th…
University of Vienna（维也纳大学）；Universidad de Chile（智利大学）；NoiseGrasp SpA（NoiseGrasp公司）
We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the posterior law over models. We first show how this estimator, termed Bayesian Wasserstein barycenter (BWB), arises naturally in a general, parameter-free Bayesian model-selection framework, when the considered Bayesian risk is the Wasserstein distance. Examples are given, illustrating how the BWB extends some classic parametric and non-parametric selection strategies. Furthermore, we also provi…
Computational Science Initiative（计算科学计划）；Brookhaven National Laboratory（布鲁克海文国家实验室）
This work examines the convergence of stochastic gradient-based optimization algorithms that use early stopping based on a validation function. The form of early stopping we consider is that optimization terminates when the norm of the gradient of a validation function falls below a threshold. We derive conditions that guarantee this stopping rule is well-defined, and provide bounds on the expected number of iterations and gradient evaluations needed to meet this criterion. The guarantee accounts for the distance between the training and validation sets, measured with the Wasserstein distance.…
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；Florida State University（佛罗里达州立大学）；Rutgers University（罗格斯大学）；Princeton University（普林斯顿大学）；CNRS（法国国家科学研究中心）
Heavy-tail phenomena in stochastic gradient descent (SGD) have been reported in several empirical studies. Experimental evidence in previous works suggests a strong interplay between the heaviness of the tails and generalization behavior of SGD. To address this empirical phenomena theoretically, several works have made strong topological and statistical assumptions to link the generalization error to heavy tails. Very recently, new generalization bounds have been proven, indicating a non-monotonic relationship between the generalization error and heavy tails, which is more pertinent to the rep…
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…
KAUST（阿卜杜拉国王科技大学）；Cairo University（开罗大学）；Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；Facebook AI Research（脸书人工智能研究院）
We present a unified theorem for the convergence analysis of stochastic gradient algorithms for minimizing a smooth and convex loss plus a convex regularizer. We do this by extending the unified analysis of Gorbunov, Hanzely \& Richtárik (2020) and dropping the requirement that the loss function be strongly convex. Instead, we only rely on convexity of the loss function. Our unified analysis applies to a host of existing algorithms such as proximal SGD, variance reduced methods, quantization and some coordinate descent type methods. For the variance reduced methods, we recover the best known c…
Uniform-in-Time Wasserstein Stability Bounds for (Noisy) Stochastic Gradient Descent
Florida State University（佛罗里达州立大学）；Rutgers Business School（罗格斯商学院）；Princeton University（普林斯顿大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Despite the strong theoretical guarantees that variance-reduced finite-sum optimization algorithms enjoy, their applicability remains limited to cases where the memory overhead they introduce (SAG/SAGA), or the periodic full gradient computation they require (SVRG/SARAH) are manageable. A promising approach to achieving variance reduction while avoiding these drawbacks is the use of importance sampling instead of control variates. While many such methods have been proposed in the literature, directly proving that they improve the convergence of the resulting optimization algorithm has remained…
Centre de Mathématiques Appliquées（应用数学中心）；Ecole Polytechnique（巴黎综合理工学院）；Ecole Normale Supérieure（巴黎高等师范学院）
Stochastic Gradient Langevin Dynamics (SGLD) has emerged as a key MCMC algorithm for Bayesian learning from large scale datasets. While SGLD with decreasing step sizes converges weakly to the posterior distribution, the algorithm is often used with a constant step size in practice and has demonstrated successes in machine learning tasks. The current practice is to set the step size inversely proportional to $N$ where $N$ is the number of training samples. As $N$ becomes large, we show that the SGLD algorithm has an invariant probability measure which significantly departs from the target poste…
Nowadays, the major challenge in machine learning is the Big Data challenge. The big data problems due to large number of data points or large number of features in each data point, or both, the training of models have become very slow. The training time has two major components: Time to access the data and time to process (learn from) the data. So far, the research has focused only on the second part, i.e., learning from the data. In this paper, we have proposed one possible solution to handle the big data problems in machine learning. The idea is to reduce the training time through reducing …
Nonasymptotic Analysis of Stochastic Gradient Descent with the Richardson-Romberg Extrapolation
HSE University（俄罗斯高等经济大学）；Duisburg-Essen University（杜伊斯堡-埃森大学）；CMAP, UMR 7641, Ecole Polytechnique（巴黎高等理工学院CMAP研究所）；Mohamed Bin Zayed University of AI（马尔代夫人工智能大学）；Skolkovo Institute of Science and Technology（斯克尔科沃科技研究院）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯克洛夫数学研究所）
INRIA（法国国家信息与自动化研究所）；École Normale Supérieure, Paris（巴黎高等师范学院）；Université de Montréal（蒙特利尔大学）
As datasets continue to increase in size and multi-core computer architectures are developed, asynchronous parallel optimization algorithms become more and more essential to the field of Machine Learning. Unfortunately, conducting the theoretical analysis asynchronous methods is difficult, notably due to the introduction of delay and inconsistency in inherently sequential algorithms. Handling these issues often requires resorting to simplifying but unrealistic assumptions. Through a novel perspective, we revisit and clarify a subtle but important technical issue present in a large fraction of …
