On the von Neumann and Frank-Wolfe Algorithms with Away Steps

TL;DR

This paper extends the von Neumann and Frank-Wolfe algorithms with away steps, demonstrating linear convergence under weaker conditions and introducing geometric parameters that influence convergence rates.

Contribution

It introduces a variant of the von Neumann algorithm that converges linearly under weaker conditions and extends these results to Frank-Wolfe with away steps.

Findings

Linear convergence of the new von Neumann variant under boundary conditions

Convergence rate depends on a new geometric parameter of the polytope

Results apply to Frank-Wolfe algorithm with away steps for strongly convex functions

Abstract

The von Neumann algorithm is a simple coordinate-descent algorithm to determine whether the origin belongs to a polytope generated by a finite set of points. When the origin is in the of the polytope, the algorithm generates a sequence of points in the polytope that converges linearly to zero. The algorithm's rate of convergence depends on the radius of the largest ball around the origin contained in the polytope. We show that under the weaker condition that the origin is in the polytope, possibly on its boundary, a variant of the von Neumann algorithm that includes generates a sequence of points in the polytope that converges linearly to zero. The new algorithm's rate of convergence depends on a certain geometric parameter of the polytope that extends the above radius but is always positive. Our linear convergence result and geometric insights also extend to a variant of the Frank-Wolfe algorithm with away steps for minimizing a strongly convex function over a polytope.