Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets

TL;DR

This paper proves that the Frank-Wolfe method achieves a quadratic convergence rate of 1/t^2 over strongly convex sets, significantly improving upon the standard 1/t rate, with applications to various norm balls.

Contribution

It establishes a faster convergence rate of 1/t^2 for the Frank-Wolfe method on strongly convex sets, a notable improvement over the general case.

Findings

Frank-Wolfe converges at rate 1/t^2 over strongly convex sets.

Linear optimization over these sets has closed-form solutions.

Previous fast-rate results follow from this analysis.

Abstract

The Frank-Wolfe method (a.k.a. conditional gradient algorithm) for smooth optimization has regained much interest in recent years in the context of large scale optimization and machine learning. A key advantage of the method is that it avoids projections - the computational bottleneck in many applications - replacing it by a linear optimization step. Despite this advantage, the known convergence rates of the FW method fall behind standard first order methods for most settings of interest. It is an active line of research to derive faster linear optimization-based algorithms for various settings of convex optimization. In this paper we consider the special case of optimization over strongly convex sets, for which we prove that the vanila FW method converges at a rate of $\frac{1}{t^2}$. This gives a quadratic improvement in convergence rate compared to the general case, in which convergence is of the order $\frac{1}{t}$, and known to be tight. We show that various balls induced by $\ell_p$ norms, Schatten norms and group norms are strongly convex on one hand and on the other hand, linear optimization over these sets is straightforward and admits a closed-form solution. We further show how several previous fast-rate results for the FW method follow easily from our analysis.