Affine invariant convergence rates of the conditional gradient method

TL;DR

This paper establishes affine invariant convergence rates for the conditional gradient method applied to convex composite problems, showing that the duality gap converges at rates from sublinear to linear depending on a growth property.

Contribution

It introduces an affine invariant analysis of the convergence rates of the conditional gradient method based on a growth property, independent of norms.

Findings

Duality gap converges to zero under certain growth conditions.

Convergence rate varies from sublinear to linear depending on the growth property.

Analysis is affine invariant and norm-independent.

Abstract

We show that the conditional gradient method for the convex composite problem \[\min_x\{f(x) + \Psi(x)\}\] generates primal and dual iterates with a duality gap converging to zero provided a suitable {\em growth property} holds and the algorithm makes a judicious choice of stepsizes. The rate of convergence of the duality gap to zero ranges from sublinear to linear depending on the degree of the growth property. The growth property and convergence results depend on the pair $(f,\Psi)$ in an affine invariant and norm-independent fashion.