Solving Matrix Games with Near-Optimal Matvec Complexity

We study the problem of computing an ϵ-approximate Nash equilibrium of a two-player, bilinear game with a bounded payoff matrix A ∈ R m×n , when the players' strategies are constrained to lie in simple sets. We provide algorithms which solve this problem in Õ(ϵ -2/3 ) matrix-vector multiplies (matvecs) in two well-studied cases: ℓ 1 -ℓ 1 (or zero-sum) games, where the players' strategies are both in the probability simplex, and ℓ 2 -ℓ 1 games (encompassing hard-margin SVMs), where the players' strategies are in the unit Euclidean ball and probability simplex respectively. These results improve upon the previous state-of-the-art complexities of Õ(ϵ -8/9 ) for ℓ 1 -ℓ 1 and Õ(ϵ -7/9 ) for ℓ 2 -ℓ 1 due to [KOS '25]. In both settings our results are nearly-optimal as they match lower bounds of [KS '25] up to polylogarithmic factors.