Linear Convergence of Stochastic Frank Wolfe Variants

TL;DR

This paper proves linear convergence in expectation and almost surely for stochastic Frank-Wolfe variants, introducing a novel proof technique and demonstrating competitive performance in large-scale experiments.

Contribution

It establishes the first linear convergence results for ASFW and PSFW algorithms using a new proof approach based on empirical processes.

Findings

ASFW and PSFW converge linearly in expectation.

Linear convergence implies almost sure convergence.

Algorithms perform well in large-scale numerical experiments.

Abstract

In this paper, we show that the Away-step Stochastic Frank-Wolfe Algorithm (ASFW) and Pairwise Stochastic Frank-Wolfe algorithm (PSFW) converge linearly in expectation. We also show that if an algorithm convergences linearly in expectation then it converges linearly almost surely. In order to prove these results, we develop a novel proof technique based on concepts of empirical processes and concentration inequalities. Such a technique has rarely been used to derive the convergence rates of stochastic optimization algorithms. In large-scale numerical experiments, ASFW and PSFW perform as well as or better than their stochastic competitors in actual CPU time.