A Proximal Gradient Algorithm for Decentralized Composite Optimization

Abstract

This paper proposes a decentralized algorithm for solving a consensus optimization problem defined in a static networked multi-agent system, where the local objective functions have the smooth+nonsmooth composite form. Examples of such problems include decentralized constrained quadratic programming and compressed sensing problems, as well as many regularization problems arising in inverse problems, signal processing, and machine learning, which have decentralized applications. This paper addresses the need for efficient decentralized algorithms that take advantages of proximal operations for the nonsmooth terms.We propose a proximal gradient exact first-order algorithm (PG-EXTRA) that utilizes the composite structure and has the best known convergence rate. It is a nontrivial extension to the recent algorithm EXTRA. At each iteration, each agent locally computes a gradient of the smooth part of its objective and a proximal map of the nonsmooth part, as well as exchanges information with its neighbors. The algorithm is "exact" in the sense that an exact consensus minimizer can be obtained with a fixed step size, whereas most previous methods must use diminishing step sizes. When the smooth part has Lipschitz gradients, PG-EXTRA has an ergodic convergence rate of O 1 k in terms of the first-order optimality residual. When the smooth part vanishes, PG-EXTRA reduces to P-EXTRA, an algorithm without the gradients (so no "G" in the name), which has a slightly improved convergence rate at o 1 k in a standard (non-ergodic) sense. Numerical experiments demonstrate effectiveness of PG-EXTRA and validate our convergence results.

A fast proximal gradient algorithm for decentralized composite optimization over directed networks

“…Moreover, we can observe from the numerical results that P-ExtraPush can generally accept a larger range of step size than PG-ExtraPush. Similar phenomenon between P-EXTRA and PG-EXTRA is also observed and verified in [16]. Moreover, we show the potential of PG-ExtraPush for solving the decentralized nonconvex regularized optimization problems.…”

“…Therefore, PG-ExtraPush reduces to PG-EXTRA [16], a recent algorithm for composite consensus optimization over undirected networks.…”

“…Clearly, PG-ExtraPush extends ExtraPush to handle nonsmooth objective terms. This extension is not the same as the extension from the gradient method to the proximal-gradient method, as well as the extension from EXTRA [15] to PG-EXTRA [16]. As the reader will see, PG-ExtraPush will have three interlaced sequences of iterates, whereas the proximal-gradient method just inherits of the sequence of iterates in the gradient method; and PG-ExtraPush uses the proximal maps of a sequence of transformed functions of r i associated with a positive weight sequence {w t } essentially introduced by the directed graph, while PG-EXTRA utilizes the proximity operator of r i .…”

Distributed Mirror Descent for Online Composite Optimization

“…The works [27]- [29] developed a class of D. Yuan distributed optimization algorithms that are built on mirror descent, which generalize the projection step by using the Bregman divergence. Different from the aforementioned works that deal only with non-composite objective functions, the authors in [16], [31] considered a decentralized composite optimization problem where the local objective function of every node is composed of a smooth function and a nonsmooth regularizer. This problem naturally arises in many real applications including distributed estimation in sensor networks [4], [10], distributed quadratic programming [31], and distributed machine learning [30], [32], to name a few.…”

“…Moreover, this rate of convergence is the same as that of [11], where the objective function is non-composite and the algorithm is gradient descent based. Different from the works [3], [16], [31], where the objective functions are time-invariant and the algorithm is gradient descent based, algorithm ODCMD is online and based on mirror descent.…”