In this paper we propose a variant of the random coordinate descent method for solving linearly constrained convex optimization problems with composite objective functions. If the smooth part of the objective function has Lipschitz continuous gradient, then we prove that our method obtains an $ε$-optimal solution in ${\cal O}(N^2/ε)$ iterations, where $N$ is the number of blocks. For the class of problems with cheap coordinate derivatives we show that the new method is faster than methods based on full-gradient information. Analysis for the rate of convergence in probability is also provided. For strongly convex functions our method converges linearly. Extensive numerical tests confirm that on very large problems, our method is much more numerically efficient than methods based on full gradient information.
We present and analyse a backtracking strategy for a general Fast Iterative Shrinkage/Thresholding Algorithm which has been recently proposed in (Chambolle, Pock, 2016) for strongly convex objective functions. Differently from classical Armijo-type line searching, our backtracking rule allows for local increasing and decreasing of the descent step size (i.e. proximal parameter) along the iterations. For such strategy accelerated convergence rates are proved and numerical results are shown for some exemplar imaging problems.
Minimax problems, such as generative adversarial network, adversarial training, and fair training, are widely solved by a multi-step gradient descent ascent (MGDA) method in practice. However, its convergence guarantee is limited. In this paper, inspired by the primal-dual hybrid gradient method, we propose a new semi-anchoring (SA) technique for the MGDA method. This makes the MGDA method find a stationary point of a structured nonconvex-nonconcave composite minimax problem; its saddle-subdifferential operator satisfies the weak Minty variational inequality condition. The resulting method, na…
We consider coordinate descent methods on convex quadratic problems, in which exact line searches are performed at each iteration. (This algorithm is identical to Gauss-Seidel on the equivalent symmetric positive definite linear system.) We describe a class of convex quadratic problems for which the random-permutations version of cyclic coordinate descent (RPCD) outperforms the standard cyclic coordinate descent (CCD) approach, yielding convergence behavior similar to the fully-random variant (RCD). A convergence analysis is developed to explain the empirical observations.
We extend coordinate descent to manifold domains, and provide convergence analyses for geodesically convex and non-convex smooth objective functions. Our key insight is to draw an analogy between coordinate blocks in Euclidean space and tangent subspaces of a manifold. Hence, our method is called tangent subspace descent (TSD). The core principle behind ensuring convergence of TSD is the appropriate choice of subspace at each iteration. To this end, we propose two novel conditions, the gap ensuring and $C$-randomized norm conditions on deterministic and randomized modes of subspace selection r…
We consider descent methods for solving non-finite valued nonsmooth convex-composite optimization problems that employ Gauss-Newton subproblems to determine the iteration update. Specifically, we establish the global convergence properties for descent methods that use a backtracking line search, a weak Wolfe line search, or a trust-region update. All of these approaches are designed to exploit the structure associated with convex-composite problems.
This short report gives a non-asymptotic rate of convergence proof for solving a two-block coordinate descent problem. This non-asymptotic proof is a simple result that can be derived easily from available results in the literature. We give the results in this report because in this general form we have not seen being stated in the literature.
We study ways to accelerate greedy coordinate descent in theory and in practice, where "accelerate" refers either to $O(1/k^2)$ convergence in theory, in practice, or both. We introduce and study two algorithms: Accelerated Semi-Greedy Coordinate Descent (ASCD) and Accelerated Greedy Coordinate Descent (AGCD). While ASCD takes greedy steps in the $x$-updates and randomized steps in the $z$-updates, AGCD is a straightforward extension of standard greedy coordinate descent that only takes greedy steps. On the theory side, our main results are for ASCD: we show that ASCD achieves $O(1/k^2)$ conve…
We consider the problem of distributed optimal power flow (OPF) for multi-area electric power systems. A novel distributed algorithm is proposed, referred to as the rotated coordinate descent critical region exploration (RCDCRE). It allows each entity to independently update its boundary information and optimally solve its local OPF in an asynchronous fashion. RCDCRE method stitches coordinate descent and parametric programming using coordinate system rotation to reduce coordination, keep privacy and ensure convergence. The solution process does not require warm starts and can iterate from inf…
In this paper, we study randomized and cyclic coordinate descent for convex unconstrained optimization problems. We improve the known convergence rates in some cases by using the numerical semidefinite programming performance estimation method. As a spin-off we provide a method to analyse the worst-case performance of the Gauss-Seidel iterative method for linear systems where the coefficient matrix is positive semidefinite with a positive diagonal.
We analyze the coordinate descent method with a new coordinate selection strategy, called volume sampling. This strategy prescribes selecting subsets of variables of certain size proportionally to the determinants of principal submatrices of the matrix, that bounds the curvature of the objective function. In the particular case, when the size of the subsets equals one, volume sampling coincides with the well-known strategy of sampling coordinates proportionally to their Lipschitz constants. For the coordinate descent with volume sampling, we establish the convergence rates both for convex and …
We introduce a stochastic coordinate extension of the first-order primal-dual method studied by Cohen and Zhu (1984) and Zhao and Zhu (2018) to solve Composite Optimization with Composite Cone-constraints (COCC). In this method, we randomly choose a block of variables based on the uniform distribution. The linearization and Bregman-like function (core function) to that randomly selected block allow us to get simple parallel primal-dual decomposition for COCC. We obtain almost surely convergence and O(1/t) expected convergence rate in this work. The high probability complexity bound is also der…
This paper introduces a coordinate descent version of the Vũ-Condat algorithm. By coordinate descent, we mean that only a subset of the coordinates of the primal and dual iterates is updated at each iteration, the other coordinates being maintained to their past value. Our method allows us to solve optimization problems with a combination of differentiable functions, constraints as well as non-separable and non-differentiable regularizers. We show that the sequences generated by our algorithm converge to a saddle point of the problem at stake, for a wider range of parameter values than previou…
We present a novel randomized block coordinate descent method for the minimization of a convex composite objective function. The method uses (approximate) partial second-order (curvature) information, so that the algorithm performance is more robust when applied to highly nonseparable or ill conditioned problems. We call the method Flexible Coordinate Descent (FCD). At each iteration of FCD, a block of coordinates is sampled randomly, a quadratic model is formed about that block and the model is minimized \emph{approximately/inexactly} to determine the search direction. An inexpensive line sea…
In this paper, we consider an online distributed composite optimization problem over a time-varying multi-agent network that consists of multiple interacting nodes, where the objective function of each node consists of two parts: a loss function that changes over time and a regularization function. This problem naturally arises in many real-world applications ranging from wireless sensor networks to signal processing. We propose a class of online distributed optimization algorithms that are based on approximate mirror descent, which utilize the Bregman divergence as distance-measuring function…
Distributed Block Coordinate Descent for Minimizing Partially Separable Functions
在这项工作中，我们提出了一个分布式随机块坐标下降法最小化一个凸函数与大量的变量/坐标。我们在假设目标函数的光滑部分是部分块可分的条件下分析了它的复杂性，并表明了可分程度直接影响复杂性。这将[Richtarik，Takac：Parallel coordinate下降法for big data optimization]中的结果扩展到分布式环境。我们首先表明，部分块可分函数承认预期的可分过逼近（ESO）相对于分布采样，计算ESO参数，然后专门的复杂性结果，从最近的文献中，一般ESO假设下举行。我们描述了几种方法来分配和同步的计算在一个集群的多核计算机，并提供有前途的计算结果。
In this paper, it is established finite active-set identification of an almost cyclic 2-coordinate descent method for problems with one linear coupling constraint and simple bounds. First, general active-set identification results are stated for non-convex objective functions. Then, under convexity and a quadratic growth condition (satisfied by any strongly convex function), complexity results on the number of iterations required to identify the active set are given. In our analysis, a simple Armijo line search is used to compute the stepsize, thus not requiring exact minimizations or addition…
Discrete optimization is a central problem in mathematical optimization with a broad range of applications, among which binary optimization and sparse optimization are two common ones. However, these problems are NP-hard and thus difficult to solve in general. Combinatorial search methods such as branch-and-bound and exhaustive search find the global optimal solution but are confined to small-sized problems, while coordinate descent methods such as coordinate gradient descent are efficient but often suffer from poor local minima. In this paper, we consider a hybrid method that combines the eff…
This paper introduces the Fejér-monotone hybrid steepest descent method (FM-HSDM), a new member to the HSDM family of algorithms, for solving affinely constrained minimization tasks in real Hilbert spaces, where convex smooth and non-smooth losses compose the objective function. FM-HSDM offers sequences of estimates which converge weakly and, under certain hypotheses, strongly to solutions of the task at hand. Fixed-point theory, variational inequalities and affine-nonexpansive mappings are utilized to devise a scheme that accommodates affine constraints in a more versatile way than state-of-t…
We present a generic coordinate descent solver for the minimization of a nonsmooth convex objective with structure. The method can deal in particular with problems with linear constraints. The implementation makes use of efficient residual updates and automatically determines which dual variables should be duplicated. A list of basic functional atoms is pre-compiled for efficiency and a modelling language in Python allows the user to combine them at run time. So, the algorithm can be used to solve a large variety of problems including Lasso, sparse multinomial logistic regression, linear and q…
Large-scale nonconvex and nonsmooth problems have attracted considerable attention in the fields of compress sensing, big data optimization and machine learning. Exploring effective methods is still the main challenge of today's research. Stochastic coordinate descent type methods have been widely used to solve large-scale optimization problems. In this paper, we derive the convergence of variable Bregman stochastic coordinate descent (VBSCD) method for a broad class of nonsmooth and nonconvex optimization problems, i.e., any accumulation of the sequence generated by VBSCD is almost surely a c…
Block-coordinate descent algorithms and alternating minimization methods are fundamental optimization algorithms and an important primitive in large-scale optimization and machine learning. While various block-coordinate-descent-type methods have been studied extensively, only alternating minimization -- which applies to the setting of only two blocks -- is known to have convergence time that scales independently of the least smooth block. A natural question is then: is the setting of two blocks special? We show that the answer is "no" as long as the least smooth block can be optimized exactly…
Accelerated coordinate descent is a widely popular optimization algorithm due to its efficiency on large-dimensional problems. It achieves state-of-the-art complexity on an important class of empirical risk minimization problems. In this paper we design and analyze an accelerated coordinate descent (ACD) method which in each iteration updates a random subset of coordinates according to an arbitrary but fixed probability law, which is a parameter of the method. If all coordinates are updated in each iteration, our method reduces to the classical accelerated gradient descent method AGD of Nester…
This article is devoted to one particular case of using universal accelerated proximal envelopes to obtain computationally efficient accelerated versions of methods used to solve various optimization problem setups. In this paper, we propose a proximally accelerated coordinate descent method that achieves the efficient algorithmic complexity of iteration and allows one to take advantage of the problem sparseness. An example of applying the proposed approach to optimizing a SoftMax-like function considered, for which the described method allowing weaken the dependence of the computational compl…
The coordinate descent (CD) method has recently become popular for solving very large-scale problems, partly due to its simple update, low memory requirement, and fast convergence. In this paper, we explore the greedy CD on solving non-negative quadratic programming (NQP). The greedy CD generally has much more expensive per-update complexity than its cyclic and randomized counterparts. However, on the NQP, these three CDs have almost the same per-update cost, while the greedy CD can have significantly faster overall convergence speed. We also apply the proposed greedy CD as a subroutine to sol…
We propose new restarting strategies for the accelerated coordinate descent method. Our main contribution is to show that for a well chosen sequence of restarting times, the restarted method has a nearly geometric rate of convergence. A major feature of the method is that it can take profit of the local quadratic error bound of the objective function without knowing the actual value of the error bound. We also show that under the more restrictive assumption that the objective function is strongly convex, any fixed restart period leads to a geometric rate of convergence. Finally, we illustrate …
The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and gradually nonconvex composite functions have been adopted to obtain more desirable properties. The convergence properties have been investigated, but only when either of composite functions is nonconvex. There is no convergence property when both composite functions are nonconvex, which is named the \textit{doubly-nonconvex} case.To overcome this difficulty, we ass…
A Stochastic Riemannian Alternating Descent Ascent Method for Nonsmooth Composite Expectation Optimization on Riemannian Manifolds
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）；Nanjing Normal University（南京师范大学）；Beijing University of Posts and Telecommunications（北京邮电大学）；The Chinese University of Hong Kong（香港中文大学）
This paper is devoted to studying the first-order variational analysis of non-convex and non-differentiable functions that may not be subdifferentially regular. To achieve this goal, we entirely rely on two concepts of directional derivatives known as subderivative and semi-derivative. We establish the exact chain and sum rules for this class of functions via these directional derivatives. These calculus rules provide an implementable auto-differentiation process such as back-propagation in composite functions. The latter calculus rules can be used to identify the directional stationary points…
In this book we collect many different and useful facts around gradient descent method. First of all we consider gradient descent with inexact oracle. We build a general model of optimized function that include composite optimization approach, level's methods, proximal methods etc. Then we investigate primal-dual properties of the gradient descent in general model set-up. At the end we generalize method to universal one.
