Revisiting Area Convexity: Faster Box-Simplex Games and Spectrahedral Generalizations

Abstract

We investigate different aspects of area convexity [She17], a mysterious tool introduced to tackle optimization problems under the challenging ∞ geometry. We develop a deeper understanding of its relationship with more conventional analyses of extragradient methods [Nem04,Nes07]. We also give improved solvers for the subproblems required by variants of the [She17] algorithm, designed through the lens of relative smoothness [BBT17, LFN18].Leveraging these new tools, we give a state-of-the-art first-order algorithm for solving boxsimplex games (a primal-dual formulation of ∞ regression) in a d×n matrix with bounded rows, using O(log d• −1 ) matrix-vector queries. As a consequence, we obtain improved complexities for approximate maximum flow, optimal transport, min-mean-cycle, and other basic combinatorial optimization problems. We also develop a near-linear time algorithm for a matrix generalization of box-simplex games, capturing a family of problems closely related to semidefinite programs recently used as subroutines in robust statistics and numerical linear algebra.