An Affine Invariant Linear Convergence Analysis for Frank-Wolfe Algorithms

TL;DR

This paper establishes affine-invariant linear convergence rates for Frank-Wolfe algorithms and its variants on strongly convex problems, relying solely on geometric properties without needing problem-specific parameters.

Contribution

It provides the first affine-invariant convergence analysis for Frank-Wolfe and away-steps variants, with constants depending only on domain geometry.

Findings

Standard Frank-Wolfe converges linearly when the solution is interior.

Away-steps Frank-Wolfe also converges linearly with geometry-dependent constants.

Algorithms do not require knowledge of problem-specific parameters.

Abstract

We study the linear convergence of variants of the Frank-Wolfe algorithms for some classes of strongly convex problems, using only affine-invariant quantities. As in Guelat & Marcotte (1986), we show the linear convergence of the standard Frank-Wolfe algorithm when the solution is in the interior of the domain, but with affine invariant constants. We also show the linear convergence of the away-steps variant of the Frank-Wolfe algorithm, but with constants which only depend on the geometry of the domain, and not any property of the location of the optimal solution. Running these algorithms does not require knowing any problem specific parameters.