Asymptotic Analysis of Conditioned Stochastic Gradient Descent
École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；ENSAI（法国国立统计与信息分析学院）；École Nationale de la Statistique et de l’Analyse de l’Information（法国国立统计与信息分析学院）
In this paper, we investigate a general class of stochastic gradient descent (SGD) algorithms, called Conditioned SGD, based on a preconditioning of the gradient direction. Using a discrete-time approach with martingale tools, we establish under mild assumptions the weak convergence of the rescaled sequence of iterates for a broad class of conditioning matrices including stochastic first-order and second-order methods. Almost sure convergence results, which may be of independent interest, are also presented. Interestingly, the asymptotic normality result consists in a stochastic equicontinuity property so when the conditioning matrix is an estimate of the inverse Hessian, the algorithm is asymptotically optimal.
Low-rank matrix estimation is a canonical problem that finds numerous applications in signal processing, machine learning and imaging science. A popular approach in practice is to factorize the matrix into two compact low-rank factors, and then optimize these factors directly via simple iterative methods such as gradient descent and alternating minimization. Despite nonconvexity, recent literatures have shown that these simple heuristics in fact achieve linear convergence when initialized properly for a growing number of problems of interest. However, upon closer examination, existing approach…
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 …
Accelerating Ill-conditioned Hankel Matrix Recovery via Structured Newton-like Descent
Department of Statistics and Data Science and Department of Computer Science, University of Central Florida（统计与数据科学系和计算机科学系，中央佛罗里达大学）；Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University（计算数学、科学与工程系和数学系，密歇根州立大学）；School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University（数学与统计学系，和湖北计算科学重点实验室，武汉大学）；School of Artificial Intelligence, Wuhan University, National Center for Applied Mathematics in Hubei, and Hubei Key Laboratory of Computational Science, Wuhan, China（人工智能学院，武汉大学，湖北应用数学中心，和湖北计算科学重点实验室，中国武汉）；Institute for Advanced Study, Shenzhen University（先进研究学院，深圳大学）
Sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, the exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. However, it is only very recently that a mathematical theory providing non-asymptotic guarantees for approximate sampling problem in the high-dimensional settings started to be developed. In this paper we introduce a n…
This work addresses distributed optimization, where a network of agents wants to minimize a global strongly convex objective function. The global function can be written as a sum of local convex functions, each of which is associated with an agent. We propose a continuous-time distributed mirror descent algorithm that uses purely local information to converge to the global optimum. Unlike previous work on distributed mirror descent, we incorporate an integral feedback in the update, allowing the algorithm to converge with a constant step-size when discretized. We establish the asymptotic conve…
University College London（伦敦大学学院）；KAUST（阿卜杜拉国王科技大学）
We study the Stein Variational Gradient Descent (SVGD) algorithm, which optimises a set of particles to approximate a target probability distribution $π\propto e^{-V}$ on $\mathbb{R}^d$. In the population limit, SVGD performs gradient descent in the space of probability distributions on the KL divergence with respect to $π$, where the gradient is smoothed through a kernel integral operator. In this paper, we provide a novel finite time analysis for the SVGD algorithm. We provide a descent lemma establishing that the algorithm decreases the objective at each iteration, and rates of convergence …
Many recent problems in signal processing and machine learning such as compressed sensing, image restoration, matrix/tensor recovery, and non-negative matrix factorization can be cast as constrained optimization. Projected gradient descent is a simple yet efficient method for solving such constrained optimization problems. Local convergence analysis furthers our understanding of its asymptotic behavior near the solution, offering sharper bounds on the convergence rate compared to global convergence analysis. However, local guarantees often appear scattered in problem-specific areas of machine …
University of British Columbia（不列颠哥伦比亚大学）；Cornell University（康奈尔大学）；Amii（阿尔伯塔机器智能研究所）
Expectation maximization (EM) is the default algorithm for fitting probabilistic models with missing or latent variables, yet we lack a full understanding of its non-asymptotic convergence properties. Previous works show results along the lines of "EM converges at least as fast as gradient descent" by assuming the conditions for the convergence of gradient descent apply to EM. This approach is not only loose, in that it does not capture that EM can make more progress than a gradient step, but the assumptions fail to hold for textbook examples of EM like Gaussian mixtures. In this work we first…
Northwestern University（西北大学）；Columbia Business School（哥伦比亚商学院）
In this paper, we study a simple algorithm to construct asymptotically valid confidence regions for model parameters using the batch means method. The main idea is to cancel out the covariance matrix which is hard/costly to estimate. In the process of developing the algorithm, we establish process-level functional central limit theorem for Polyak-Ruppert averaging based stochastic gradient descent estimators. We also extend the batch means method to accommodate more general batch size specifications.
In this work, we investigate the idea of variance reduction by studying its properties with general adaptive mirror descent algorithms in nonsmooth nonconvex finite-sum optimization problems. We propose a simple yet generalized framework for variance reduced adaptive mirror descent algorithms named SVRAMD and provide its convergence analysis in both the nonsmooth nonconvex problem and the P-L conditioned problem. We prove that variance reduction reduces the SFO complexity of adaptive mirror descent algorithms and thus accelerates their convergence. In particular, our general theory implies tha…
National University of Defense Technology（国防科技大学）；Dongguan University of Technology（东莞理工学院）
We provide a new theoretical analysis framework to investigate online gradient descent in the dynamic environment. Comparing with the previous work, the new framework recovers the state-of-the-art dynamic regret, but does not require extra gradient queries for every iteration. Specifically, when functions are $α$ strongly convex and $β$ smooth, to achieve the state-of-the-art dynamic regret, the previous work requires $O(κ)$ with $κ= \fracβα$ queries of gradients at every iteration. But, our framework shows that the query complexity can be improved to be $O(1)$, which does not depend on $κ$. T…
Acceleration of stochastic gradient descent with momentum by averaging: finite-sample rates and asymptotic normality
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Daniels School of Business, Purdue University（普渡大学丹尼尔斯商学院）；Stern School of Business, New York University（纽约大学斯特恩商学院）
Asymptotic Behavior of Multi--Task Learning: Implicit Regularization and Double Descent Effects
Department of Electrical Engineering, College of Engineering, University of Ha’il（胡赛尔大学电气工程系，工程学院）；John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保罗森工程与应用科学学院）；Department of Electrical and Electronic Engineering, King’s College London – Strand（伦敦国王学院-街分校电气与电子工程系）
Institute of Mathematics, Universität Potsdam（波茨坦大学数学研究所）；WIAS Berlin（柏林应用分析与随机学研究所）
Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: variational inference and Markov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent …
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
We analyze the inductive bias of gradient descent for weight normalized smooth homogeneous neural nets, when trained on exponential or cross-entropy loss. We analyse both standard weight normalization (SWN) and exponential weight normalization (EWN), and show that the gradient flow path with EWN is equivalent to gradient flow on standard networks with an adaptive learning rate. We extend these results to gradient descent, and establish asymptotic relations between weights and gradients for both SWN and EWN. We also show that EWN causes weights to be updated in a way that prefers asymptotic rel…
This paper investigates asymptotic behaviors of gradient descent algorithms (particularly accelerated gradient descent and stochastic gradient descent) in the context of stochastic optimization arising in statistics and machine learning where objective functions are estimated from available data. We show that these algorithms can be computationally modeled by continuous-time ordinary or stochastic differential equations. We establish gradient flow central limit theorems to describe the limiting dynamic behaviors of these computational algorithms and the large-sample performances of the related…
We introduce a simple algorithm, True Asymptotic Natural Gradient Optimization (TANGO), that converges to a true natural gradient descent in the limit of small learning rates, without explicit Fisher matrix estimation. For quadratic models the algorithm is also an instance of averaged stochastic gradient, where the parameter is a moving average of a "fast", constant-rate gradient descent. TANGO appears as a particular de-linearization of averaged SGD, and is sometimes quite different on non-quadratic models. This further connects averaged SGD and natural gradient, both of which are arguably op…
Gradient descent dynamics for deep equilibrium models
Carnegie Mellon University, Department of Statistics（卡内基梅隆大学统计系）；Carnegie Mellon University, Department of Statistics and Machine Learning（卡内基梅隆大学统计与机器学习系）
Deep learning methods operate in regimes that defy the traditional statistical mindset. Neural network architectures often contain more parameters than training samples, and are so rich that they can interpolate the observed labels, even if the latter are replaced by pure noise. Despite their huge complexity, the same architectures achieve small generalization error on real data. This phenomenon has been rationalized in terms of a so-called `double descent' curve. As the model complexity increases, the test error follows the usual U-shaped curve at the beginning, first decreasing and then peak…
Shenzhen Research Institute of Big Data（深圳大数据研究院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Boston University（波士顿大学）
This paper is concerned with minimizing the average of $n$ cost functions over a network in which agents may communicate and exchange information with each other. We consider the setting where only noisy gradient information is available. To solve the problem, we study the distributed stochastic gradient descent (DSGD) method and perform a non-asymptotic convergence analysis. For strongly convex and smooth objective functions, DSGD asymptotically achieves the optimal network independent convergence rate compared to centralized stochastic gradient descent (SGD). Our main contribution is to char…
Columbia University（哥伦比亚大学）；Rutgers University（罗格斯大学）；Seoul National University（首尔国立大学）；McMaster University（麦克马斯特大学）
We develop a new method of online inference for a vector of parameters estimated by the Polyak-Ruppert averaging procedure of stochastic gradient descent (SGD) algorithms. We leverage insights from time series regression in econometrics and construct asymptotically pivotal statistics via random scaling. Our approach is fully operational with online data and is rigorously underpinned by a functional central limit theorem. Our proposed inference method has a couple of key advantages over the existing methods. First, the test statistic is computed in an online fashion with only SGD iterates and t…
Motivated by the problem of online canonical correlation analysis, we propose the \emph{Stochastic Scaled-Gradient Descent} (SSGD) algorithm for minimizing the expectation of a stochastic function over a generic Riemannian manifold. SSGD generalizes the idea of projected stochastic gradient descent and allows the use of scaled stochastic gradients instead of stochastic gradients. In the special case of a spherical constraint, which arises in generalized eigenvector problems, we establish a nonasymptotic finite-sample bound of $\sqrt{1/T}$, and show that this rate is minimax optimal, up to a po…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper establishes risk convergence and asymptotic weight matrix alignment --- a form of implicit regularization --- of gradient flow and gradient descent when applied to deep linear networks on linearly separable data. In more detail, for gradient flow applied to strictly decreasing loss functions (with similar results for gradient descent with particular decreasing step sizes): (i) the risk converges to 0; (ii) the normalized i-th weight matrix asymptotically equals its rank-1 approximation $u_iv_i^{\top}$; (iii) these rank-1 matrices are aligned across layers, meaning $|v_{i+1}^{\top}u_…
Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Recent evidence has shown the existence of a so-called double-descent and even triple-descent behavior for the generalization error of deep-learning models. This important phenomenon commonly appears in implemented neural network architectures, and also seems to emerge in epoch-wise curves during the training process. A recent line of research has highlighted that random matrix tools can be used to obtain precise analytical asymptotics of the generalization (and training) errors of the random feature model. In this contribution, we analyze the whole temporal behavior of the generalization and …
Stochastic gradient descent (SGD) is almost ubiquitously used for training non-convex optimization tasks. Recently, a hypothesis proposed by Keskar et al. [2017] that large batch methods tend to converge to sharp minimizers has received increasing attention. We theoretically justify this hypothesis by providing new properties of SGD in both finite-time and asymptotic regimes. In particular, we give an explicit escaping time of SGD from a local minimum in the finite-time regime and prove that SGD tends to converge to flatter minima in the asymptotic regime (although may take exponential time to…
University of Chicago（芝加哥大学）；New York University（纽约大学）
The stochastic gradient descent (SGD) algorithm is widely used for parameter estimation, especially for huge data sets and online learning. While this recursive algorithm is popular for computation and memory efficiency, quantifying variability and randomness of the solutions has been rarely studied. This paper aims at conducting statistical inference of SGD-based estimates in an online setting. In particular, we propose a fully online estimator for the covariance matrix of averaged SGD iterates (ASGD) only using the iterates from SGD. We formally establish our online estimator's consistency a…
In this paper, we establish the almost sure convergence of two-timescale stochastic gradient descent algorithms in continuous time under general noise and stability conditions, extending well known results in discrete time. We analyse algorithms with additive noise and those with non-additive noise. In the non-additive case, our analysis is carried out under the assumption that the noise is a continuous-time Markov process, controlled by the algorithm states. The algorithms we consider can be applied to a broad class of bilevel optimisation problems. We study one such problem in detail, namely…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）
Current theoretical results on optimization trajectories of neural networks trained by gradient descent typically have the form of rigorous but potentially loose bounds on the loss values. In the present work we take a different approach and show that the learning trajectory can be characterized by an explicit asymptotic at large training times. Specifically, the leading term in the asymptotic expansion of the loss behaves as a power law $L(t) \sim t^{-ξ}$ with exponent $ξ$ expressed only through the data dimension, the smoothness of the activation function, and the class of function being app…
University of Maryland（马里兰大学）；Georgetown University（乔治敦大学）
This paper considers the problem of multi-agent distributed optimization. In this problem, there are multiple agents in the system, and each agent only knows its local cost function. The objective for the agents is to collectively compute a common minimum of the aggregate of all their local cost functions. In principle, this problem is solvable using a distributed variant of the traditional gradient-descent method, which is an iterative method. However, the speed of convergence of the traditional gradient-descent method is highly influenced by the conditioning of the optimization problem being…
We provide (high probability) bounds on the condition number of random feature matrices. In particular, we show that if the complexity ratio $\frac{N}{m}$ where $N$ is the number of neurons and $m$ is the number of data samples scales like $\log^{-1}(N)$ or $\log(m)$, then the random feature matrix is well-conditioned. This result holds without the need of regularization and relies on establishing various concentration bounds between dependent components of the random feature matrix. Additionally, we derive bounds on the restricted isometry constant of the random feature matrix. We prove that …
Stopping criteria for Stochastic Gradient Descent (SGD) methods play important roles from enabling adaptive step size schemes to providing rigor for downstream analyses such as asymptotic inference. Unfortunately, current stopping criteria for SGD methods are often heuristics that rely on asymptotic normality results or convergence to stationary distributions, which may fail to exist for nonconvex functions and, thereby, limit the applicability of such stopping criteria. To address this issue, in this work, we rigorously develop two stopping criteria for SGD that can be applied to a broad clas…
Ill-posed linear inverse problems appear in many scientific setups, and are typically addressed by solving optimization problems, which are composed of data fidelity and prior terms. Recently, several works have considered a back-projection (BP) based fidelity term as an alternative to the common least squares (LS), and demonstrated excellent results for popular inverse problems. These works have also empirically shown that using the BP term, rather than the LS term, requires fewer iterations of optimization algorithms. In this paper, we examine the convergence rate of the projected gradient d…
VinAI Research（VinAI研究院）；Monash University（莫纳什大学）；University of Texas, Austin（德克萨斯大学奥斯汀分校）
Sampling from an unnormalized target distribution is an essential problem with many applications in probabilistic inference. Stein Variational Gradient Descent (SVGD) has been shown to be a powerful method that iteratively updates a set of particles to approximate the distribution of interest. Furthermore, when analysing its asymptotic properties, SVGD reduces exactly to a single-objective optimization problem and can be viewed as a probabilistic version of this single-objective optimization problem. A natural question then arises: "Can we derive a probabilistic version of the multi-objective …
Statistical Inference for Differentially Private Stochastic Gradient Descent
Center for Data Science, Zhejiang University（浙江大学数据科学中心）；Department of Statistics, Rutgers University（罗格斯大学统计系）；Faculty of Business and Economics, The University of Hong Kong（香港大学商学院）
