Curvature-Dependent Lower Bounds for Frank-Wolfe

TL;DR

This paper establishes tight lower bounds for the convergence rates of the Frank-Wolfe algorithm on curved convex domains, extending understanding of its performance beyond strongly convex sets.

Contribution

It provides matching lower bounds for Frank-Wolfe's convergence rates on p-uniformly convex sets and extends these bounds to objectives with H"olderian error bounds.

Findings

Proves a matching (T^{-p/(p-1)}) lower bound for p .

Extends lower bounds to objectives with HF6lderian error bounds.

Analyzes Frank-Wolfe dynamics on simple instances, not limited to high dimensions.

Abstract

The Frank-Wolfe algorithm achieves a convergence rate of $\mathcal{O}(1/T)$ for smooth convex optimization over compact convex domains, accelerating to $\mathcal{O}(1/T^2)$ when both the objective and the feasible set are strongly convex. This acceleration extends beyond strong convexity: Kerdreux et al. (2021a) proved rates of $\mathcal{O}(T^{-p/(p-1)})$ over $p$-uniformly convex feasible sets, a class that interpolates between strongly convex sets and more general curved domains such as $\ell_p$ balls. In this work, we establish a matching $\Omega(T^{-p/(p-1)})$ lower bound for every $p\ge 3$ under exact line search or short steps, and extend the lower bound to objectives satisfying a H\"olderian error bound. The proofs analyze the dynamics of Frank-Wolfe iterates on simple instances and hence are not limited to the high-dimensional setting, unlike information-theoretic lower bounds.