Randomized Matvec Lower Bounds for Simplex-Based Matrix Games

Abstract

We prove randomized matrix-vector query lower bounds for two normalized matrix-game geometries: a Euclidean unit ball against a probability simplex, with row norms at most one, and two probability simplices, with entries of absolute value at most one. Each query returns $(Ax,A^\top y)$ for arbitrary real vectors. The algorithm must return a feasible pair with full saddle-point gap at most $\varepsilon$, with probability at least $2/3$ on every admissible matrix. For sufficiently small $\varepsilon$, the worst-case query complexities are $Ω(\varepsilon^{-2/3}/(\log^2(1/\varepsilon)\log\log(1/\varepsilon)))$ for ball-simplex games and $Ω(\varepsilon^{-2/3}/(\log^{7/3}(1/\varepsilon)\log\log(1/\varepsilon)))$ for simplex-simplex games. The hard instances have dimensions of order $\varepsilon^{-2/3}$ and $\varepsilon^{-2/3}/\log^{1/3}(1/\varepsilon)$, respectively, and the bounds extend to larger dimensions. These lower bounds match the deterministic upper bounds of Karmarkar, O'Carroll, and Sidford up to logarithmic factors. The proof extracts a fresh Gaussian core after adaptive two-sided queries and uses uncertainty in its smallest singular value to establish linear-system solve hardness. Two reductions transfer this hardness to matrix games by converting a small full gap into a small residual, with an additional logarithmic normalization loss only for simplex-simplex games.