We consider minimizing the sum of three convex functions, where the first one F is smooth, the second one is nonsmooth and proximable and the third one is the composition of a nonsmooth proximable function with a linear operator L. This template problem has many applications, for instance, in image processing and machine learning. First, we propose a new primal-dual algorithm, which we call PDDY, for this problem. It is constructed by applying Davis-Yin splitting to a monotone inclusion in a primal-dual product space, where the operators are monotone under a specific metric depending on L. We show that three existing algorithms (the two forms of the Condat-Vu algorithm and the PD3O algorithm) have the same structure, so that PDDY is the fourth missing link in this self-consistent class of primal-dual algorithms. This representation eases the convergence analysis: it allows us to derive sublinear convergence rates in general, and linear convergence results in presence of strong convexity. Moreover, within our broad and flexible analysis framework, we propose new stochastic generalizations of the algorithms, in which a variance-reduced random estimate of the gradient of F is used, instead of the true gradient. Furthermore, we obtain, as a special case of PDDY, a linearly converging algorithm for the minimization of a strongly convex function F under a linear constraint; we discuss its important application to decentralized optimization.
National University of Singapore（新加坡国立大学）；Dept. of Industrial Systems Engineering and Management（工业系统工程与管理系（此处按要求粒度，该系属于新加坡国立大学，机构级别为大学，原部门为系，非独立机构，按要求只提取到大学/学院/公司/研究院级别，所以主要机构为新加坡国立大学，部门系不提取）。修正：根据规则，机构粒度提取到大学/学院/公司/研究院级别即可，不要细拆到系。所以这里的机构是National University of Singapore（新加坡国立大学），而Dept. of Industrial Systems Engineering and Management是系，属于新加坡国立大学内部的系，不应提取。因此正确输出应为{）
In this paper, we consider the problem of minimizing the average of a large number of nonsmooth and convex functions. Such problems often arise in typical machine learning problems as empirical risk minimization, but are computationally very challenging. We develop and analyze a new algorithm that achieves robust linear convergence rate, and both its time complexity and gradient complexity are superior than state-of-art nonsmooth algorithms and subgradient-based schemes. Besides, our algorithm works without any extra error bound conditions on the objective function as well as the common strong…
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）
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…
UC Berkeley（加州大学伯克利分校）；Pontificia Universidad Católica de Chile（智利天主教大学）；Millennium Nucleus Center for the Discovery of Structures in Complex Data（千年复杂数据结构发现核心中心）
We study the question of whether parallelization in the exploration of the feasible set can be used to speed up convex optimization, in the local oracle model of computation. We show that the answer is negative for both deterministic and randomized algorithms applied to essentially any of the interesting geometries and nonsmooth, weakly-smooth, or smooth objective functions. In particular, we show that it is not possible to obtain a polylogarithmic (in the sequential complexity of the problem) number of parallel rounds with a polynomial (in the dimension) number of queries per round. In the ma…
Block coordinate descent methods and stochastic subgradient methods have been extensively studied in optimization and machine learning. By combining randomized block sampling with stochastic subgradient methods based on dual averaging, we present stochastic block dual averaging (SBDA)---a novel class of block subgradient methods for convex nonsmooth and stochastic optimization. SBDA requires only a block of subgradients and updates blocks of variables and hence has significantly lower iteration cost than traditional subgradient methods. We show that the SBDA-based methods exhibit the optimal c…
Permutation Randomization on Nonsmooth Nonconvex Optimization: A Theoretical and Experimental Study
School of Computer and Cyber Sciences（计算机与电子科学学院）；Augusta University（奥古斯塔大学）；Department of Graduate Psychology（研究生心理学系）；James Madison University（詹姆斯·麦迪逊大学）；School of Computing（计算学院）；University of Georgia（佐治亚大学）
Leveraging Randomized Smoothing for Optimal Control of Nonsmooth Dynamical Systems
Inria（法国国家信息与自动化研究所）；Département d’Informatique de l’Ecole Normale Supérieure, PSL Research University（巴黎高等师范学院计算机科学系，PSL研究大学）；Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University（捷克理工大学信息学、机器人与控制论研究所）
We propose a new randomized coordinate descent method for a convex optimization template with broad applications. Our analysis relies on a novel combination of four ideas applied to the primal-dual gap function: smoothing, acceleration, homotopy, and coordinate descent with non-uniform sampling. As a result, our method features the first convergence rate guarantees among the coordinate descent methods, that are the best-known under a variety of common structure assumptions on the template. We provide numerical evidence to support the theoretical results with a comparison to state-of-the-art al…
Exploiting partial first-order information in a cyclic way is arguably the most natural strategy to obtain scalable first-order methods. However, despite their wide use in practice, cyclic schemes are far less understood from a theoretical perspective than their randomized counterparts. Motivated by a recent success in analyzing an extrapolated cyclic scheme for generalized variational inequalities, we propose an Accelerated Cyclic Coordinate Dual Averaging with Extrapolation (A-CODER) method for composite convex optimization, where the objective function can be expressed as the sum of a smoot…
Consider the problem of minimizing the sum of a smooth (possibly non-convex) and a convex (possibly nonsmooth) function involving a large number of variables. A popular approach to solve this problem is the block coordinate descent (BCD) method whereby at each iteration only one variable block is updated while the remaining variables are held fixed. With the recent advances in the developments of the multi-core parallel processing technology, it is desirable to parallelize the BCD method by allowing multiple blocks to be updated simultaneously at each iteration of the algorithm. In this work, …
We develop randomized (block) coordinate descent (CD) methods for linearly constrained convex optimization. Unlike most CD methods, we do not assume the constraints to be separable, but let them be coupled linearly. To our knowledge, ours is the first CD method that allows linear coupling constraints, without making the global iteration complexity have an exponential dependence on the number of constraints. We present algorithms and analysis for four key problem scenarios: (i) smooth; (ii) smooth + nonsmooth separable; (iii) asynchronous parallel; and (iv) stochastic. We illustrate empirical b…
Proximal splitting algorithms are well suited to solving large-scale nonsmooth optimization problems, in particular those arising in machine learning. We propose a new primal-dual algorithm, in which the dual update is randomized; equivalently, the proximity operator of one of the function in the problem is replaced by a stochastic oracle. For instance, some randomly chosen dual variables, instead of all, are updated at each iteration. Or, the proximity operator of a function is called with some small probability only. A nonsmooth variance-reduction technique is implemented so that the algorit…
We provide the first non-asymptotic analysis for finding stationary points of nonsmooth, nonconvex functions. In particular, we study the class of Hadamard semi-differentiable functions, perhaps the largest class of nonsmooth functions for which the chain rule of calculus holds. This class contains examples such as ReLU neural networks and others with non-differentiable activation functions. We first show that finding an $ε$-stationary point with first-order methods is impossible in finite time. We then introduce the notion of $(δ, ε)$-stationarity, which allows for an $ε$-approximate gradient…
We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a black-box algorithm of Zhang et al. (2020) that approximates an $ε$-stationary point of any directionally differentiable Lipschitz objective using $O(ε^{-4})$ calls to a specialized subgradient oracle and a randomized line search. Our simple black-box deterministic version, achieves $O(ε^{-5})$ for any difference-of-convex objective, and $O(ε^{-4})$ for the w…
It is well-known that given a smooth, bounded-from-below, and possibly nonconvex function, standard gradient-based methods can find $ε$-stationary points (with gradient norm less than $ε$) in $\mathcal{O}(1/ε^2)$ iterations. However, many important nonconvex optimization problems, such as those associated with training modern neural networks, are inherently not smooth, making these results inapplicable. In this paper, we study nonsmooth nonconvex optimization from an oracle complexity viewpoint, where the algorithm is assumed to be given access only to local information about the function at v…
Motivated by a class of applied problems arising from physical layer based security in a digital communication system, in particular, by a secrecy sum-rate maximization problem, this paper studies a nonsmooth, difference-of-convex (dc) minimization problem. The contributions of this paper are: (i) clarify several kinds of stationary solutions and their relations; (ii) develop and establish the convergence of a novel algorithm for computing a d-stationary solution of a problem with a convex feasible set that is arguably the sharpest kind among the various stationary solutions; (iii) extend the …
The Stochastic Primal-Dual Hybrid Gradient (SPDHG) was proposed by Chambolle et al. (2018) and is an efficient algorithm to solve some nonsmooth large-scale optimization problems. In this paper we prove its almost sure convergence for convex but not necessarily strongly convex functionals. We also look into its application to parallel Magnetic Resonance Imaging reconstruction in order to test performance of SPDHG. Our numerical results show that for a range of settings SPDHG converges significantly faster than its deterministic counterpart.
We study the performance of a family of randomized parallel coordinate descent methods for minimizing the sum of a nonsmooth and separable convex functions. The problem class includes as a special case L1-regularized L1 regression and the minimization of the exponential loss ("AdaBoost problem"). We assume the input data defining the loss function is contained in a sparse $m\times n$ matrix $A$ with at most $ω$ nonzeros in each row. Our methods need $O(n β/τ)$ iterations to find an approximate solution with high probability, where $τ$ is the number of processors and $β= 1 + (ω-1)(τ-1)/(n-1)$ f…
We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove that our algorithm achieves optimal $\mathcal{O}(n/k)$ and $\mathcal{O}(n^2/k^2)$ convergence rates (up to a constant factor) in two cases: general convexity and strong convexity, respectively, where $k$ is the iteration counter and n is the number of block-coordinates. Our convergence rates are obtained through three criteria: primal objective residual and p…
We consider randomized block coordinate stochastic mirror descent (RBSMD) methods for solving high-dimensional stochastic optimization problems with strongly convex objective functions. Our goal is to develop RBSMD schemes that achieve a rate of convergence with a minimum constant factor with respect to the choice of the stepsize sequence. To this end, we consider both subgradient and gradient RBSMD methods addressing nonsmooth and smooth problems, respectively. For each scheme, (i) we develop self-tuned stepsize rules characterized in terms of problem parameters and algorithm settings; (ii) w…
This paper proposes TriPD, a new primal-dual algorithm for minimizing the sum of a Lipschitz-differentiable convex function and two possibly nonsmooth convex functions, one of which is composed with a linear mapping. We devise a randomized block-coordinate version of the algorithm which converges under the same stepsize conditions as the full algorithm. It is shown that both the original as well as the block-coordinate scheme feature linear convergence rate when the functions involved are either piecewise linear-quadratic, or when they satisfy a certain quadratic growth condition (which is wea…
State University of New York at Buffalo（纽约州立大学布法罗分校）；Boston University（波士顿大学）
Minimizing a convex risk function is the main step in many basic learning algorithms. We study protocols for convex optimization which provably leak very little about the individual data points that constitute the loss function. Specifically, we consider differentially private algorithms that operate in the local model, where each data record is stored on a separate user device and randomization is performed locally by those devices. We give new protocols for \emph{noninteractive} LDP convex optimization---i.e., protocols that require only a single randomized report from each user to an untrus…
In this paper, we present several new results on minimizing a nonsmooth and nonconvex function under a Lipschitz condition. Recent work shows that while the classical notion of Clarke stationarity is computationally intractable up to some sufficiently small constant tolerance, the randomized first-order algorithms find a $(δ, ε)$-Goldstein stationary point with the complexity bound of $\tilde{O}(δ^{-1}ε^{-3})$, which is independent of dimension $d \geq 1$~\citep{Zhang-2020-Complexity, Davis-2022-Gradient, Tian-2022-Finite}. However, the deterministic algorithms have not been fully explored, le…
In this paper we introduce a class of novel distributed algorithms for solving stochastic big-data convex optimization problems over directed graphs. In the addressed set-up, the dimension of the decision variable can be extremely high and the objective function can be nonsmooth. The general algorithm consists of two main steps: a consensus step and an update on a single block of the optimization variable, which is then broadcast to neighbors. Three special instances of the proposed method, involving particular problem structures, are then presented. In the general case, the convergence of a d…
This paper suggests two novel ideas to develop new proximal variable-metric methods for solving a class of composite convex optimization problems. The first idea is a new parameterization of the optimality condition which allows us to develop a class of homotopy proximal variable-metric methods. We show that under appropriate assumptions such as strong convexity-type and smoothness, or self-concordance, our new schemes can achieve finite global iteration-complexity bounds. Our second idea is a primal-dual-primal framework for proximal-Newton methods which can lead to some useful computational …
Dual first-order methods are essential techniques for large-scale constrained convex optimization. However, when recovering the primal solutions, we need $T(ε^{-2})$ iterations to achieve an $ε$-optimal primal solution when we apply an algorithm to the non-strongly convex dual problem with $T(ε^{-1})$ iterations to achieve an $ε$-optimal dual solution, where $T(x)$ can be $x$ or $\sqrt{x}$. In this paper, we prove that the iteration complexity of the primal solutions and dual solutions have the same $O\left(\frac{1}{\sqrtε}\right)$ order of magnitude for the accelerated randomized dual coordin…
We consider algorithms for solving structured convex optimization problems over a network of agents with communication delays. It is assumed that each agent performs its local updates by using possibly outdated information from its neighbors under the assumption that the delay with respect to each neighbor is bounded but otherwise arbitrary. The private objective of each agent is represented by the sum of two possibly nonsmooth functions, one of which is composed with a linear mapping. The global optimization problem is the aggregate of the local cost functions and a common Lipschitz-different…
