Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes

Gabriele Iommazzo, David Martínez-Rubio, Francisco Criado, Elias Samuel Wirth, Sebastian Pokutta

Abstract:

We study the linear convergence of Frank-Wolfe algorithms over product polytopes. We analyze two condition numbers for the product polytope, namely the pyramidal width and the vertex-facet distance, based on the condition numbers of individual polytope components.
As a result, for convex objectives that are $\mu$-Polyak-Łojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers.
We apply our results to the problem of approximately finding a feasible point in a polytope intersection in high-dimensions, and demonstrate the practical efficiency of our algorithms through empirical results.

Submission Number: 476