A framework based on iterative coordinate minimization (CM) is developed for stochastic convex optimization. Given that exact coordinate minimization is impossible due to the unknown stochastic nature of the objective function, the crux of the proposed optimization algorithm is an optimal control of the minimization precision in each iteration. We establish the optimal precision control and the resulting order-optimal regret performance for strongly convex and separably nonsmooth functions. An interesting finding is that the optimal progression of precision across iterations is independent of the low-dimensional CM routine employed, suggesting a general framework for extending low-dimensional optimization routines to high-dimensional problems. The proposed algorithm is amenable to online implementation and inherits the scalability and parallelizability properties of CM for large-scale optimization. Requiring only a sublinear order of message exchanges, it also lends itself well to distributed computing as compared with the alternative approach of coordinate gradient descent.
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…
University of Minnesota Twin Cities（明尼苏达大学双城分校）；Georgia Institute of Technology（佐治亚理工学院）；Johns Hopkins University（约翰斯·霍普金斯大学）；Northwestern University（西北大学）
The cyclic block coordinate descent-type (CBCD-type) methods, which performs iterative updates for a few coordinates (a block) simultaneously throughout the procedure, have shown remarkable computational performance for solving strongly convex minimization problems. Typical applications include many popular statistical machine learning methods such as elastic-net regression, ridge penalized logistic regression, and sparse additive regression. Existing optimization literature has shown that for strongly convex minimization, the CBCD-type methods attain iteration complexity of $\mathcal{O}(p\log…
Difference-of-Convex (DC) minimization, referring to the problem of minimizing the difference of two convex functions, has been found rich applications in statistical learning and studied extensively for decades. However, existing methods are primarily based on multi-stage convex relaxation, only leading to weak optimality of critical points. This paper proposes a coordinate descent method for minimizing a class of DC functions based on sequential nonconvex approximation. Our approach iteratively solves a nonconvex one-dimensional subproblem globally, and it is guaranteed to converge to a coor…
The so-called Burer-Monteiro method is a well-studied technique for solving large-scale semidefinite programs (SDPs) via low-rank factorization. The main idea is to solve rank-restricted, albeit non-convex, surrogates instead of the SDP. Recent works have shown that, in an important class of SDPs with elegant geometric structure, one can find globally optimal solutions to the SDP by finding rank-deficient second-order critical points of an unconstrained Riemannian optimization problem. Hence, in such problems, the Burer-Monteiro approach can provide a scalable and reliable alternative to inter…
We consider a class of structured fractional minimization problems, in which the numerator part of the objective is the sum of a differentiable convex function and a convex non-smooth function, while the denominator part is a convex or concave function. This problem is difficult to solve since it is non-convex. By exploiting the structure of the problem, we propose two Coordinate Descent (CD) methods for solving this problem. The proposed methods iteratively solve a one-dimensional subproblem \textit{globally}, and they are guaranteed to converge to coordinate-wise stationary points. In the ca…
We introduce a proximal version of dual coordinate ascent method. We demonstrate how the derived algorithmic framework can be used for numerous regularized loss minimization problems, including $\ell_1$ regularization and structured output SVM. The convergence rates we obtain match, and sometimes improve, state-of-the-art results.
Mean-Field is an efficient way to approximate a posterior distribution in complex graphical models and constitutes the most popular class of Bayesian variational approximation methods. In most applications, the mean field distribution parameters are computed using an alternate coordinate minimization. However, the convergence properties of this algorithm remain unclear. In this paper, we show how, by adding an appropriate penalization term, we can guarantee convergence to a critical point, while keeping a closed form update at each step. A convergence rate estimate can also be derived based on…
Stochastic dual coordinate ascent (SDCA) is an effective technique for solving regularized loss minimization problems in machine learning. This paper considers an extension of SDCA under the mini-batch setting that is often used in practice. Our main contribution is to introduce an accelerated mini-batch version of SDCA and prove a fast convergence rate for this method. We discuss an implementation of our method over a parallel computing system, and compare the results to both the vanilla stochastic dual coordinate ascent and to the accelerated deterministic gradient descent method of \cite{ne…
We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization. The problem is closely related to decomposable submodular function minimization and arises in many learning on graphs and hypergraphs settings, such as graph-based semi-supervised learning and PageRank. We approach the problem via a new dual strategy and describe an objective that may be optimized via random coordinate descent (RCD) methods and projections onto cones. We also establish the linear convergence rate of the RCD algorithm and develop efficient projection algorithms with p…
We introduce a new approach to decomposable submodular function minimization (DSFM) that exploits incidence relations. Incidence relations describe which variables effectively influence the component functions, and when properly utilized, they allow for improving the convergence rates of DSFM solvers. Our main results include the precise parametrization of the DSFM problem based on incidence relations, the development of new scalable alternative projections and parallel coordinate descent methods and an accompanying rigorous analysis of their convergence rates.
We consider a generic convex-concave saddle point problem with separable structure, a form that covers a wide-ranged machine learning applications. Under this problem structure, we follow the framework of primal-dual updates for saddle point problems, and incorporate stochastic block coordinate descent with adaptive stepsize into this framework. We theoretically show that our proposal of adaptive stepsize potentially achieves a sharper linear convergence rate compared with the existing methods. Additionally, since we can select "mini-batch" of block coordinates to update, our method is also am…
University of Michigan（密歇根大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
Large-scale L1-regularized loss minimization problems arise in high-dimensional applications such as compressed sensing and high-dimensional supervised learning, including classification and regression problems. High-performance algorithms and implementations are critical to efficiently solving these problems. Building upon previous work on coordinate descent algorithms for L1-regularized problems, we introduce a novel family of algorithms called block-greedy coordinate descent that includes, as special cases, several existing algorithms such as SCD, Greedy CD, Shotgun, and Thread-Greedy. We g…
Submodular function minimization is a fundamental optimization problem that arises in several applications in machine learning and computer vision. The problem is known to be solvable in polynomial time, but general purpose algorithms have high running times and are unsuitable for large-scale problems. Recent work have used convex optimization techniques to obtain very practical algorithms for minimizing functions that are sums of ``simple" functions. In this paper, we use random coordinate descent methods to obtain algorithms with faster linear convergence rates and cheaper iteration costs. C…
Simon Fraser University（西蒙弗雷泽大学）；Microsoft Research（微软研究院）
In this paper we analyze the randomized block-coordinate descent (RBCD) methods proposed in [8,11] for minimizing the sum of a smooth convex function and a block-separable convex function. In particular, we extend Nesterov's technique developed in [8] for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general problem and obtain a sharper expected-value type of convergence rate than the one implied in [11]. Also, we obtain a better high-probability type of iteration complexity, which improves upon the one in …
Purdue University（普渡大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization (QDSFM), which allows to model a number of learning tasks on graphs and hypergraphs. The problem exhibits close ties to decomposable submodular function minimization (DSFM), yet is much more challenging to solve. We approach the problem via a new dual strategy and formulate an objective that can be optimized through a number of double-loop algorithms. The outer-loop uses either random coordinate descent (RCD) or alternative projection (AP) methods, for both of which we prove linear co…
In this paper we consider the composite self-concordant (CSC) minimization problem, which minimizes the sum of a self-concordant function $f$ and a (possibly nonsmooth) proper closed convex function $g$. The CSC minimization is the cornerstone of the path-following interior point methods for solving a broad class of convex optimization problems. It has also found numerous applications in machine learning. The proximal damped Newton (PDN) methods have been well studied in the literature for solving this problem that enjoy a nice iteration complexity. Given that at each iteration these methods t…
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 consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a stochastic primal-dual coordinate (SPDC) method, which alternates between maximizing over a randomly chosen dual variable and minimizing over the primal variable. An extrapolation step on the primal variable is performed to obtain accelerated convergence rate. We also develop a mini-batch version of the SPDC method which facilitates parallel computing, and an exten…
NEC Labs America（美国NEC实验室）；Michigan State University（密歇根州立大学）
In \citep{Yangnips13}, the author presented distributed stochastic dual coordinate ascent (DisDCA) algorithms for solving large-scale regularized loss minimization. Extraordinary performances have been observed and reported for the well-motivated updates, as referred to the practical updates, compared to the naive updates. However, no serious analysis has been provided to understand the updates and therefore the convergence rates. In the paper, we bridge the gap by providing a theoretical analysis of the convergence rates of the practical DisDCA algorithm. Our analysis helped by empirical stud…
Stochastic Dual Coordinate Ascent is a popular method for solving regularized loss minimization for the case of convex losses. In this paper we show how a variant of SDCA can be applied for non-convex losses. We prove linear convergence rate even if individual loss functions are non-convex as long as the expected loss is convex.
University of Heidelberg（海德堡大学）；Czech Technical University in Prague（布拉格捷克理工大学）
We consider the maximum-a-posteriori inference problem in discrete graphical models and study solvers based on the dual block-coordinate ascent rule. We map all existing solvers in a single framework, allowing for a better understanding of their design principles. We theoretically show that some block-optimizing updates are sub-optimal and how to strictly improve them. On a wide range of problem instances of varying graph connectivity, we study the performance of existing solvers as well as new variants that can be obtained within the framework. As a result of this exploration we build a new s…
Univ. Lille（里尔大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）；Univ Montpellier（蒙彼利埃大学）；Institut Universitaire de France（法国大学研究院）
Machine learning models can leak information about the data used to train them. To mitigate this issue, Differentially Private (DP) variants of optimization algorithms like Stochastic Gradient Descent (DP-SGD) have been designed to trade-off utility for privacy in Empirical Risk Minimization (ERM) problems. In this paper, we propose Differentially Private proximal Coordinate Descent (DP-CD), a new method to solve composite DP-ERM problems. We derive utility guarantees through a novel theoretical analysis of inexact coordinate descent. Our results show that, thanks to larger step sizes, DP-CD c…
In modern large-scale machine learning applications, the training data are often partitioned and stored on multiple machines. It is customary to employ the "data parallelism" approach, where the aggregated training loss is minimized without moving data across machines. In this paper, we introduce a novel distributed dual formulation for regularized loss minimization problems that can directly handle data parallelism in the distributed setting. This formulation allows us to systematically derive dual coordinate optimization procedures, which we refer to as Distributed Alternating Dual Maximizat…
High-Dimensional Private Empirical Risk Minimization by Greedy Coordinate Descent
Univ. Lille（里尔大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）；Univ Montpellier（蒙彼利埃大学）；Institut Universitaire de France (IUF)（法兰西大学研究院）
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…
MPI for Informatics（马克斯·普朗克信息学研究所）；IST Austria（奥地利科学技术研究所）
We present a new proximal bundle method for Maximum-A-Posteriori (MAP) inference in structured energy minimization problems. The method optimizes a Lagrangean relaxation of the original energy minimization problem using a multi plane block-coordinate Frank-Wolfe method that takes advantage of the specific structure of the Lagrangean decomposition. We show empirically that our method outperforms state-of-the-art Lagrangean decomposition based algorithms on some challenging Markov Random Field, multi-label discrete tomography and graph matching problems.
We consider the problem of minimizing the sum of two convex functions: one is the average of a large number of smooth component functions, and the other is a general convex function that admits a simple proximal mapping. We assume the whole objective function is strongly convex. Such problems often arise in machine learning, known as regularized empirical risk minimization. We propose and analyze a new proximal stochastic gradient method, which uses a multi-stage scheme to progressively reduce the variance of the stochastic gradient. While each iteration of this algorithm has similar cost as t…
This paper introduces AdaSDCA: an adaptive variant of stochastic dual coordinate ascent (SDCA) for solving the regularized empirical risk minimization problems. Our modification consists in allowing the method adaptively change the probability distribution over the dual variables throughout the iterative process. AdaSDCA achieves provably better complexity bound than SDCA with the best fixed probability distribution, known as importance sampling. However, it is of a theoretical character as it is expensive to implement. We also propose AdaSDCA+: a practical variant which in our experiments out…
Carnegie Mellon University（卡内基梅隆大学）；The University of Iowa（爱荷华大学）
We propose a doubly stochastic primal-dual coordinate optimization algorithm for empirical risk minimization, which can be formulated as a bilinear saddle-point problem. In each iteration, our method randomly samples a block of coordinates of the primal and dual solutions to update. The linear convergence of our method could be established in terms of 1) the distance from the current iterate to the optimal solution and 2) the primal-dual objective gap. We show that the proposed method has a lower overall complexity than existing coordinate methods when either the data matrix has a factorized s…
We consider the "all-for-one" decentralized learning problem for generalized linear models. The features of each sample are partitioned among several collaborating agents in a connected network, but only one agent observes the response variables. To solve the regularized empirical risk minimization in this distributed setting, we apply the Chambolle--Pock primal--dual algorithm to an equivalent saddle-point formulation of the problem. The primal and dual iterations are either in closed-form or reduce to coordinate-wise minimization of scalar convex functions. We establish convergence rates for…
