On the Limitations and Possibilities of Nash Regret Minimization in Zero-Sum Matrix Games under Noisy Feedback

This paper studies a variant of two-player zero-sum matrix games, where, at each timestep, the row player selects row i, the column player selects column j, and the row player receives a noisy reward with expected value Ai,j, along with noisy feedback on the input matrix A. The row player's goal is to maximize their total reward against an adversarial column player. Nash regret, defined as the difference between the player's total reward and the game's Nash equilibrium value scaled by the time horizon T, is often used to evaluate algorithmic performance in zero-sum games. We begin by studying the limitations of existing algorithms for minimizing Nash regret. We show that standard algorithm--including Hedge, FTRL, and OMD--as well as the strategy of playing the Nash equilibrium of the empirical matrix--all incur Omega(√T) Nash regret, even when the row player receives noisy feedback on the entire matrix A. Furthermore, we show that UCB for matrix games, a natural adaptation of the well-known bandit algorithm, also suffers Omega(√T) Nash regret under bandit feedback. Notably, these lower bounds hold even in the simplest case of 2 × 2 matrix games, where the instance-dependent matrix parameters are constant. We next ask whether instance-dependent polylog(T) Nash regret is achievable against adversarial opponents. We answer this affirmatively. In the full-information setting, we present the first algorithm for general n × m matrix games that achieves instance-dependent polylog(T) Nash regret. In the bandit feedback setting, we design an algorithm with similar guarantees for the special case of 2 × 2 game--the same regime in which existing algorithms provably suffer Omega(√T) regret despite the simplicity of the instance. Finally, we validate our theoretical results with empirical evidence.