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

Abstract

This paper proposes a fast decentralized algorithm for solving a consensus optimization problem defined in a directed networked multi-agent system, where the local objective functions have the smooth+nonsmooth composite form, and are possibly nonconvex. Examples of such problems include decentralized compressed sensing and constrained quadratic programming problems, as well as many decentralized regularization problems. We extend the existing algorithms PG-EXTRA and ExtraPush to a new algorithm PG-ExtraPush for composite consensus optimization over a directed network. This algorithm takes advantage of the proximity operator like in PG-EXTRA to deal with the nonsmooth term, and employs the push-sum protocol like in ExtraPush to tackle the bias introduced by the directed network. With a proper step size, we show that PGExtraPush converges to an optimal solution at a linear rate 1 under some regular assumptions.We conduct a series of numerical experiments to show the effectiveness of the proposed algorithm. Specifically, with a proper step size, PG-ExtraPush performs linear rates in most of cases, even in some nonconvex cases, and is significantly faster than Subgradient-Push, even if the latter uses a hand-optimized step size. The established theoretical results are also verified by the numerical results.Keywords: Decentralized optimization; directed network; composite objective; nonconvex; consensus.✩ The work of J. Zeng is supported in part by the National Natural Science Foundation of China (Grants No. 61603162, 11401462).1 In this paper, we use the notion of R-linear rate, i.e., a sequence {x t } converging to x * at an R-linear rate means that x t − x * ≤ Cρ t for some constants C > 0 and ρ ∈ (0, 1).

Distributed Mirror Descent for Online Composite Optimization

“…where the last inequality follows from the fact that K has finite diameter D K . Combining the inequalities in (16) and (18), we get…”

“…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.…”

Distributed and Inexact Proximal Gradient Method for Online Convex Optimization

“…A different approach is employed by PG-EXTRA [20], which combines the proximal gradient method with a gradient tracking scheme. PG-ExtraPush, proposed in [24], further combines PG-EXTRA with a push-sum (or ratio) consensus scheme. A drawback of [20], and [24] is that in order to choose the step size, the agents need to compute the minimum eigenvalue of the consensus matrix.…”

“…PG-ExtraPush, proposed in [24], further combines PG-EXTRA with a push-sum (or ratio) consensus scheme. A drawback of [20], and [24] is that in order to choose the step size, the agents need to compute the minimum eigenvalue of the consensus matrix. To resolve this, a modification of PG-EXTRA, called NIDS, was proposed in [21], which allows each node to choose the step-size independently (provided that there is agreement on an auxiliary parameter, that however does not depend on the topology of the network).…”