Decentralized Frank-Wolfe Algorithm for Convex and Non-convex Problems

TL;DR

This paper introduces a decentralized Frank-Wolfe algorithm for high-dimensional constrained optimization, providing convergence guarantees for convex and non-convex problems, and demonstrating its effectiveness in matrix completion and sparse learning.

Contribution

It develops a decentralized Frank-Wolfe algorithm with proven convergence rates for convex and non-convex problems, addressing computational challenges of projection steps.

Findings

Convergence rate of O(1/t) for convex objectives

O(1/t^2) convergence for strongly convex objectives

Effective in low-complexity matrix completion and sparse learning tasks

Abstract

Decentralized optimization algorithms have received much attention due to the recent advances in network information processing. However, conventional decentralized algorithms based on projected gradient descent are incapable of handling high dimensional constrained problems, as the projection step becomes computationally prohibitive to compute. To address this problem, this paper adopts a projection-free optimization approach, a.k.a.~the Frank-Wolfe (FW) or conditional gradient algorithm. We first develop a decentralized FW (DeFW) algorithm from the classical FW algorithm. The convergence of the proposed algorithm is studied by viewing the decentralized algorithm as an inexact FW algorithm. Using a diminishing step size rule and letting $t$ be the iteration number, we show that the DeFW algorithm's convergence rate is ${\cal O}(1/t)$ for convex objectives; is ${\cal O}(1/t^2)$ for strongly convex objectives with the optimal solution in the interior of the constraint set; and is ${\cal O}(1/\sqrt{t})$ towards a stationary point for smooth but non-convex objectives. We then show that a consensus-based DeFW algorithm meets the above guarantees with two communication rounds per iteration. Furthermore, we demonstrate the advantages of the proposed DeFW algorithm on low-complexity robust matrix completion and communication efficient sparse learning. Numerical results on synthetic and real data are presented to support our findings.