We consider the problem of recovering a real-valued $n$-dimensional signal from $m$ phaseless, linear measurements and analyze the amplitude-based non-smooth least squares objective. We establish local convergence of subgradient descent with optimal sample complexity based on the uniform concentration of a random, discontinuous matrix-valued operator arising from the objective's gradient dynamics. While common techniques to establish uniform concentration of random functions exploit Lipschitz continuity, we prove that the discontinuous matrix-valued operator satisfies a uniform matrix concentration inequality when the measurement vectors are Gaussian as soon as $m = Ω(n)$ with high probability. We then show that satisfaction of this inequality is sufficient for subgradient descent with proper initialization to converge linearly to the true solution up to the global sign ambiguity. As a consequence, this guarantees local convergence for Gaussian measurements at optimal sample complexity. The concentration methods in the present work have previously been used to establish recovery guarantees for a variety of inverse problems under generative neural network priors. This paper demonstrates the applicability of these techniques to more traditional inverse problems and serves as a pedagogical introduction to those results.
Cornell University（康奈尔大学）；Simons Institute for the Theory of Computing（西蒙斯理论计算研究所）；DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization（DIMACS/西蒙斯连续与离散优化桥接合作项目）
We extend the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions. For the deterministic projected subgradient method, we present a global $O(1/\sqrt{T})$ convergence rate for any convex function which is locally Lipschitz around its minimizers. This approach is based on Shor's classic subgradient analysis and implies generalizations of the standard convergence rates for gradient descent on functions with Lipschitz or Hölder continuous gradients. Further, we show a $O(1/\sqrt{T})$ convergence rate for the stochastic projected subgradient method on convex…
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…
Given a convex optimization problem and its dual, there are many possible first-order algorithms. In this paper, we show the equivalence between mirror descent algorithms and algorithms generalizing the conditional gradient method. This is done through convex duality, and implies notably that for certain problems, such as for supervised machine learning problems with non-smooth losses or problems regularized by non-smooth regularizers, the primal subgradient method and the dual conditional gradient method are formally equivalent. The dual interpretation leads to a form of line search for mirro…
Randomized Coordinate Subgradient Method for Nonsmooth Composite Optimization
Institute of Translational Medicine（转化医学研究院）；National Center for Translational Medicine（国家转化医学中心）；Shanghai Jiao Tong University（上海交通大学）；Alibaba Group（阿里巴巴集团）；Antai College of Economics and Management（安泰经济与管理学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；School of Data Science（数据科学学院）；Xiangfu Laboratory（祥符实验室）
In this paper we analyze boosting algorithms in linear regression from a new perspective: that of modern first-order methods in convex optimization. We show that classic boosting algorithms in linear regression, namely the incremental forward stagewise algorithm (FS$_\varepsilon$) and least squares boosting (LS-Boost($\varepsilon$)), can be viewed as subgradient descent to minimize the loss function defined as the maximum absolute correlation between the features and residuals. We also propose a modification of FS$_\varepsilon$ that yields an algorithm for the Lasso, and that may be easily ext…
De-singularity Subgradient for the $q$-th-Powered $\ell_p$-Norm Weber Location Problem
College of Cyber Security, Jinan University（暨南大学网络空间安全学院）；College of Information Science and Technology, Jinan University（暨南大学信息科学技术学院）；Pazhou Laboratory（琶洲实验室）；Research Institute of Multiple Agents and Embodied Intelligence, Peng Cheng Laboratory（鹏城实验室多智能体与具身智能研究院）
Convergence of the gradient descent algorithm has been attracting renewed interest due to its utility in deep learning applications. Even as multiple variants of gradient descent were proposed, the assumption that the gradient of the objective is Lipschitz continuous remained an integral part of the analysis until recently. In this work, we look at convergence analysis by focusing on a property that we term as concavifiability, instead of Lipschitz continuity of gradients. We show that concavifiability is a necessary and sufficient condition to satisfy the upper quadratic approximation which i…
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）；University of Chinese Academy of Sciences（中国科学院大学）
For first-order smooth optimization, the research on the acceleration phenomenon has a long-time history. Until recently, the mechanism leading to acceleration was not successfully uncovered by the gradient correction term and its equivalent implicit-velocity form. Furthermore, based on the high-resolution differential equation framework with the corresponding emerging techniques, phase-space representation and Lyapunov function, the squared gradient norm of Nesterov's accelerated gradient descent (\texttt{NAG}) method at an inverse cubic rate is discovered. However, this result cannot be dire…
Shanghai University of Finance and Economics（上海财经大学）
Novel coordinate descent (CD) methods are proposed for minimizing nonconvex functions consisting of three terms: (i) a continuously differentiable term, (ii) a simple convex term, and (iii) a concave and continuous term. First, by extending randomized CD to nonsmooth nonconvex settings, we develop a coordinate subgradient method that randomly updates block-coordinate variables by using block composite subgradient mapping. This method converges asymptotically to critical points with proven sublinear convergence rate for certain optimality measures. Second, we develop a randomly permuted CD meth…
Iowa State University（爱荷华州立大学）；IBM Research AI（IBM人工智能研究院）
We propose a new \textit{randomized Bregman (block) coordinate descent} (RBCD) method for minimizing a composite problem, where the objective function could be either convex or nonconvex, and the smooth part are freed from the global Lipschitz-continuous (partial) gradient assumption. Under the notion of relative smoothness based on the Bregman distance, we prove that every limit point of the generated sequence is a stationary point. Further, we show that the iteration complexity of the proposed method is $O(n\varepsilon^{-2})$ to achieve $ε$-stationary point, where $n$ is the number of blocks…
Stochastic convex optimization is a basic and well studied primitive in machine learning. It is well known that convex and Lipschitz functions can be minimized efficiently using Stochastic Gradient Descent (SGD). The Normalized Gradient Descent (NGD) algorithm, is an adaptation of Gradient Descent, which updates according to the direction of the gradients, rather than the gradients themselves. In this paper we analyze a stochastic version of NGD and prove its convergence to a global minimum for a wider class of functions: we require the functions to be quasi-convex and locally-Lipschitz. Quasi…
We study three families of online convex optimization algorithms: follow-the-proximally-regularized-leader (FTRL-Proximal), regularized dual averaging (RDA), and composite-objective mirror descent. We first prove equivalence theorems that show all of these algorithms are instantiations of a general FTRL update. This provides theoretical insight on previous experimental observations. In particular, even though the FOBOS composite mirror descent algorithm handles L1 regularization explicitly, it has been observed that RDA is even more effective at producing sparsity. Our results demonstrate that…
University of British Columbia（不列颠哥伦比亚大学）；Amii（阿尔伯塔机器智能研究所）
In online convex optimization (OCO), Lipschitz continuity of the functions is commonly assumed in order to obtain sublinear regret. Moreover, many algorithms have only logarithmic regret when these functions are also strongly convex. Recently, researchers from convex optimization proposed the notions of "relative Lipschitz continuity" and "relative strong convexity". Both of the notions are generalizations of their classical counterparts. It has been shown that subgradient methods in the relative setting have performance analogous to their performance in the classical setting. In this work, we…
A Unified Analysis on the Subgradient Upper Bounds for the Subgradient Methods Minimizing Composite Nonconvex, Nonsmooth and Non-Lipschitz Functions
Antai College of Economics and Management, Shanghai Jiao Tong University（上海交通大学安泰经济管理学院）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）；Institute of Translational Medicine and National Center for Translational Medicine, Shanghai Jiao Tong University（上海交通大学转化医学研究院和转化医学中心）；Department of Industrial and Systems Engineering, University of Minnesota（明尼苏达大学工业与系统工程系）
We study non-convex subgradient flows for training two-layer ReLU neural networks from a convex geometry and duality perspective. We characterize the implicit bias of unregularized non-convex gradient flow as convex regularization of an equivalent convex model. We then show that the limit points of non-convex subgradient flows can be identified via primal-dual correspondence in this convex optimization problem. Moreover, we derive a sufficient condition on the dual variables which ensures that the stationary points of the non-convex objective are the KKT points of the convex objective, thus pr…
This paper concerns dictionary learning, i.e., sparse coding, a fundamental representation learning problem. We show that a subgradient descent algorithm, with random initialization, can provably recover orthogonal dictionaries on a natural nonsmooth, nonconvex $\ell_1$ minimization formulation of the problem, under mild statistical assumptions on the data. This is in contrast to previous provable methods that require either expensive computation or delicate initialization schemes. Our analysis develops several tools for characterizing landscapes of nonsmooth functions, which might be of indep…
In this work, we consider the resilience of distributed algorithms based on stochastic gradient descent (SGD) in distributed learning with potentially Byzantine attackers, who could send arbitrary information to the parameter server to disrupt the training process. Toward this end, we propose a new Lipschitz-inspired coordinate-wise median approach (LICM-SGD) to mitigate Byzantine attacks. We show that our LICM-SGD algorithm can resist up to half of the workers being Byzantine attackers, while still converging almost surely to a stationary region in non-convex settings. Also, our LICM-SGD meth…
Iowa State University（爱荷华州立大学）；IBM Research AI（IBM研究院人工智能部门）；IBM Thomas J. Waston Research Center（IBM托马斯·J·沃森研究中心）
In the applications of signal processing and data analytics, there is a wide class of non-convex problems whose objective function is freed from the common global Lipschitz continuous gradient assumption (e.g., the nonnegative matrix factorization (NMF) problem). Recently, this type of problem with some certain special structures has been solved by Bregman proximal gradient (BPG). This inspires us to propose a new Block-wise two-references Bregman proximal gradient (B2B) method, which adopts two reference functions so that a closed-form solution in the Bregman projection is obtained. Based on …
Step by Step: Adaptive Gradient Descent for Training L-Lipschitz Neural Networks
Department of Mathematics and Statistics, McMaster University（数学与统计学系，麦斯特大学）；Vector Institute（向量研究所）；The Ennio De Giorgi Mathematical Research Centre（埃尼奥·德·吉奥吉数学研究中心）；Scuola Normale Superiore di Pisa（比萨高等学院）
Stochastic gradient algorithms are often unstable when applied to functions that do not have Lipschitz-continuous and/or bounded gradients. Gradient clipping is a simple and effective technique to stabilize the training process for problems that are prone to the exploding gradient problem. Despite its widespread popularity, the convergence properties of the gradient clipping heuristic are poorly understood, especially for stochastic problems. This paper establishes both qualitative and quantitative convergence results of the clipped stochastic (sub)gradient method (SGD) for non-smooth convex f…
We consider the problem of unconstrained online convex optimization (OCO) with sub-exponential noise, a strictly more general problem than the standard OCO. In this setting, the learner receives a subgradient of the loss functions corrupted by sub-exponential noise and strives to achieve optimal regret guarantee, without knowledge of the competitor norm, i.e., in a parameter-free way. Recently, Cutkosky and Boahen (COLT 2017) proved that, given unbounded subgradients, it is impossible to guarantee a sublinear regret due to an exponential penalty. This paper shows that it is possible to go arou…
Revisiting Subgradient Method: Complexity and Convergence Beyond Lipschitz Continuity
School of Data Science（数据科学学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Institute of Translational Medicine（转化医学研究所）；National Center for Translational Medicine（国家转化医学中心）；Shanghai Jiao Tong University（上海交通大学）；Antai College of Economics and Management（安泰经济与管理学院）
Coordinate descent with random coordinate selection is the current state of the art for many large scale optimization problems. However, greedy selection of the steepest coordinate on smooth problems can yield convergence rates independent of the dimension $n$, and requiring upto $n$ times fewer iterations. In this paper, we consider greedy updates that are based on subgradients for a class of non-smooth composite problems, which includes $L1$-regularized problems, SVMs and related applications. For these problems we provide (i) the first linear rates of convergence independent of $n$, and sho…
The University of Chicago（芝加哥大学）；Singapore Management University（新加坡管理大学）；Institute for Infocomm Research, A*STAR（新加坡科技研究局资讯通信研究院）；Nanyang Technological University（南洋理工大学）
Learning for maximizing AUC performance is an important research problem in Machine Learning and Artificial Intelligence. Unlike traditional batch learning methods for maximizing AUC which often suffer from poor scalability, recent years have witnessed some emerging studies that attempt to maximize AUC by single-pass online learning approaches. Despite their encouraging results reported, the existing online AUC maximization algorithms often adopt simple online gradient descent approaches that fail to exploit the geometrical knowledge of the data observed during the online learning process, and…
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL, United States（工程科学与应用数学系，西北大学，埃文斯顿，伊利诺伊州，美国）；Department of Industrial Engineering（工业工程系）；Applied Mathematics, Northwestern University, Evanston, IL, United States（应用数学，西北大学，埃文斯顿，伊利诺伊州，美国）
Nonmonotone subgradient methods based on a local descent lemma
Department of Mathematics, University of Alicante（数学系，阿利坎特大学）；Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (CNRS IRL2807), Universidad de Chile（数学工程系和数学建模中心（CNRS IRL2807），智利大学）
North Carolina State University（北卡罗来纳州立大学）；BITS Pilani K K Birla Goa Campus（比尔拉理工学院皮拉尼 K K 比尔拉果阿校区）；PES University（PES大学）
Optimizing deep neural networks is largely thought to be an empirical process, requiring manual tuning of several hyper-parameters, such as learning rate, weight decay, and dropout rate. Arguably, the learning rate is the most important of these to tune, and this has gained more attention in recent works. In this paper, we propose a novel method to compute the learning rate for training deep neural networks with stochastic gradient descent. We first derive a theoretical framework to compute learning rates dynamically based on the Lipschitz constant of the loss function. We then extend this fra…
University of Arizona（亚利桑那大学）；Boston University（波士顿大学）
We consider the problem of minimizing a convex risk with stochastic subgradients guaranteeing $ε$-locally differentially private ($ε$-LDP). While it has been shown that stochastic optimization is possible with $ε$-LDP via the standard SGD (Song et al., 2013), its convergence rate largely depends on the learning rate, which must be tuned via repeated runs. Further, tuning is detrimental to privacy loss since it significantly increases the number of gradient requests. In this work, we propose BANCO (Betting Algorithm for Noisy COins), the first $ε$-LDP SGD algorithm that essentially matches the …
We describe novel subgradient methods for a broad class of matrix optimization problems involving nuclear norm regularization. Unlike existing approaches, our method executes very cheap iterations by combining low-rank stochastic subgradients with efficient incremental SVD updates, made possible by highly optimized and parallelizable dense linear algebra operations on small matrices. Our practical algorithms always maintain a low-rank factorization of iterates that can be conveniently held in memory and efficiently multiplied to generate predictions in matrix completion settings. Empirical com…
Cornell University（康奈尔大学）；University of Washington（华盛顿大学）
We investigate the stochastic optimization problem of minimizing population risk, where the loss defining the risk is assumed to be weakly convex. Compositions of Lipschitz convex functions with smooth maps are the primary examples of such losses. We analyze the estimation quality of such nonsmooth and nonconvex problems by their sample average approximations. Our main results establish dimension-dependent rates on subgradient estimation in full generality and dimension-independent rates when the loss is a generalized linear model. As an application of the developed techniques, we analyze the …
Beijing Jiaotong University（北京交通大学）；The Hong Kong Polytechnic University（香港理工大学）；University of California, Davis（加利福尼亚大学戴维斯分校）
In this paper, we propose a Riemannian smoothing steepest descent method to minimize a nonconvex and non-Lipschitz function on submanifolds. The generalized subdifferentials on Riemannian manifold and the Riemannian gradient sub-consistency are defined and discussed. We prove that any accumulation point of the sequence generated by the Riemannian smoothing steepest descent method is a stationary point associated with the smoothing function employed in the method, which is necessary for the local optimality of the original non-Lipschitz problem. Under the Riemannian gradient sub-consistency con…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Microsoft Research, India（微软研究院印度分院）；University of Washington, Seattle（华盛顿大学西雅图分校）
We consider the classical setting of optimizing a nonsmooth Lipschitz continuous convex function over a convex constraint set, when having access to a (stochastic) first-order oracle (FO) for the function and a projection oracle (PO) for the constraint set. It is well known that to achieve $ε$-suboptimality in high-dimensions, $Θ(ε^{-2})$ FO calls are necessary. This is achieved by the projected subgradient method (PGD). However, PGD also entails $O(ε^{-2})$ PO calls, which may be computationally costlier than FO calls (e.g. nuclear norm constraints). Improving this PO calls complexity of PGD …
Many problems in data science can be treated as estimating a low-rank matrix from highly incomplete, sometimes even corrupted, observations. One popular approach is to resort to matrix factorization, where the low-rank matrix factors are optimized via first-order methods over a smooth loss function, such as the residual sum of squares. While tremendous progresses have been made in recent years, the natural smooth formulation suffers from two sources of ill-conditioning, where the iteration complexity of gradient descent scales poorly both with the dimension as well as the condition number of t…
We study a variant of the median problem for a collection of point sets in high dimensions. This generalizes the geometric median as well as the (probabilistic) smallest enclosing ball (pSEB) problems. Our main objective and motivation is to improve the previously best algorithm for the pSEB problem by reducing its exponential dependence on the dimension to linear. This is achieved via a novel combination of sampling techniques for clustering problems in metric spaces with the framework of stochastic subgradient descent. As a result, the algorithm becomes applicable to shape fitting problems i…
VinAI Research（VinAI 研究院）；Hong Kong Polytechnic University（香港理工大学）
Principal component analysis is a simple yet useful dimensionality reduction technique in modern machine learning pipelines. In consequential domains such as college admission, healthcare and credit approval, it is imperative to take into account emerging criteria such as the fairness and the robustness of the learned projection. In this paper, we propose a distributionally robust optimization problem for principal component analysis which internalizes a fairness criterion in the objective function. The learned projection thus balances the trade-off between the total reconstruction error and t…
University of Luxembourg（卢森堡大学）；Technische Universität Darmstadt（达姆施塔特工业大学）
In this paper, we propose a convergent parallel best-response algorithm with the exact line search for the nondifferentiable nonconvex sparsity-regularized rank minimization problem. On the one hand, it exhibits a faster convergence than subgradient algorithms and block coordinate descent algorithms. On the other hand, its convergence to a stationary point is guaranteed, while ADMM algorithms only converge for convex problems. Furthermore, the exact line search procedure in the proposed algorithm is performed efficiently in closed-form to avoid the meticulous choice of stepsizes, which is howe…
Mathematical Institute for Data Science（数据科学数学研究所）；Johns Hopkins University（约翰斯·霍普金斯大学）
Robust subspace recovery (RSR) is a fundamental problem in robust representation learning. Here we focus on a recently proposed RSR method termed Dual Principal Component Pursuit (DPCP) approach, which aims to recover a basis of the orthogonal complement of the subspace and is amenable to handling subspaces of high relative dimension. Prior work has shown that DPCP can provably recover the correct subspace in the presence of outliers, as long as the true dimension of the subspace is known. We show that DPCP can provably solve RSR problems in the {\it unknown} subspace dimension regime, as long…
