Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes

TL;DR

This paper establishes linear convergence rates for the Frank-Wolfe algorithm over product polytopes, linking convergence to condition numbers derived from polytope components, with practical applications in high-dimensional feasibility problems.

Contribution

It introduces new condition numbers for product polytopes and proves linear convergence of Frank-Wolfe algorithms under these conditions for -Polyak-{}iewicz convex functions.

Findings

Linear convergence rates depend on pyramidal width and vertex-facet distance.

Theoretical results apply to high-dimensional polytope intersection problems.

Empirical results demonstrate practical efficiency of the proposed algorithms.

Abstract

We study the linear convergence of Frank-Wolfe algorithms over product polytopes. We analyze two condition numbers for the product polytope, namely the \emph{pyramidal width} and the \emph{vertex-facet distance}, based on the condition numbers of individual polytope components. As a result, for convex objectives that are $\mu$-Polyak-{\L}ojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers. We apply our results to the problem of approximately finding a feasible point in a polytope intersection in high-dimensions, and demonstrate the practical efficiency of our algorithms through empirical results.