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 $$n\times n$$ n × 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.