A Randomized Linearly Convergent Frank-Wolfe-type Method for Smooth Convex Minimization over the Spectrahedron

Abstract

We consider the problem of minimizing a smooth and convex function over the $n$-dimensional spectrahedron -- the set of real symmetric $n\times n$ positive semidefinite matrices with unit trace, which underlies numerous applications in statistics, machine learning and additional domains. Standard first-order methods often require high-rank matrix computations which are prohibitive when the dimension $n$ is large. The well-known Frank-Wolfe method on the other hand only requires efficient rank-one matrix computations, however, suffers from worst-case slow convergence, even under conditions that enable linear convergence rates for standard methods. In this work we present the first Frank-Wolfe-based algorithm that only applies efficient rank-one matrix computations and, assuming quadratic growth and strict complementarity conditions, is guaranteed, after a finite number of iterations, to converge linearly, in expectation, and independently of the ambient dimension.