Avoiding bad steps in Frank Wolfe variants

TL;DR

This paper introduces the Short Step Chain procedure to simplify convergence analysis of Frank Wolfe variants, achieving improved rates and a unified approach in non-convex settings, including linear convergence under certain conditions.

Contribution

It proposes a new SSC technique that removes step distinctions, enabling unified convergence analysis and rates for Frank Wolfe variants in non-convex optimization.

Findings

Unified convergence analysis for FW variants

Linear convergence under KL property

Angle condition satisfied on polytopes

Abstract

The analysis of Frank Wolfe (FW) variants is often complicated by the presence of different kinds of "good" and "bad" steps. In this article we aim to simplify the convergence analysis of some of these variants by getting rid of such a distinction between steps, and to improve existing rates by ensuring a sizable decrease of the objective at each iteration. In order to do this, we define the Short Step Chain (SSC) procedure, which skips gradient computations in consecutive short steps until proper stopping conditions are satisfied. This technique allows us to give a unified analysis and converge rates in the general smooth non convex setting, as well as a linear convergence rate under a Kurdyka-Lojasiewicz (KL) property. While this setting has been widely studied for proximal gradient type methods, to our knowledge, it has not been analyzed before for the Frank Wolfe variants under study. An angle condition, ensuring that the directions selected by the methods have the steepest slope possible up to a constant, is used to carry out our analysis. We prove that this condition is satisfied on polytopes by the away step Frank-Wolfe (AFW), the pairwise Frank-Wolfe (PFW), and the Frank-Wolfe method with in face directions (FDFW).