Frank-Wolfe algorithm for star-convex functions

TL;DR

This paper extends the Frank-Wolfe algorithm's convergence guarantees to star-convex functions, showing it maintains optimal complexity bounds even without convexity, using various stepsize rules.

Contribution

It introduces iteration-complexity bounds for the Frank-Wolfe algorithm applied to star-convex functions, broadening its applicability beyond convex functions.

Findings

Achieves $ ext{O}(1/k)$ convergence for objective and duality gap.

Diminishing and Armijo stepsizes do not require Lipschitz constant knowledge.

Maintains optimal complexity bounds in non-convex star-convex setting.

Abstract

We study the Frank-Wolfe algorithm for minimizing a differentiable function with Lipschitz continuous gradient over a compact convex set. To extend classical complexity bounds to certain non-convex functions, we focus on the class of \emph{star-convex functions}, which retain essential geometric properties despite the lack of convexity. We establish iteration-complexity bounds of $\mathcal{O}(1/k)$ for both the objective values and the duality gap under star-convexity, using diminishing, Armijo-type, and Lipschitz-based stepsize rules. Notably, the diminishing and Armijo strategies do not require prior knowledge of Lipschitz or curvature constants. These results demonstrate that the Frank-Wolfe method preserves optimal complexity guarantees beyond the convex setting.