Randomized Matvec Lower Bounds for Simplex-Based Matrix Games

2026-10-01

AI Summary

This study investigates randomized query complexity lower bounds for simplex matrix games. Focusing on games over Euclidean balls and probability bisimplexes, it proposes a novel method that extracts Gaussian kernels following adaptive queries, leveraging the uncertainty of minimum singular values to establish the hardness of solving the resulting linear systems. Furthermore, through reduction techniques, the full saddle-point gap is transformed into small residuals, and the theoretical analysis is completed via Gaussian process methods. Ultimately, this work derives an Ω(ε⁻²/³) lower bound for randomized matrix-vector queries, matching known deterministic upper bounds up to logarithmic factors. These results bridge a critical theoretical gap concerning randomized algorithms under specific geometric structures.

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.

Problem

• matrix games
• query complexity
• lower bounds
• probability simplex
• saddle-point gap

Innovation

• Randomized Matvec Lower Bounds
• Matrix Games
• Probability Simplex
• Query Complexity
• Saddle-point Gap