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

TL;DR

This work analyzes Nash regret minimization in two-player zero-sum matrix games with noisy feedback and unknown payoff matrices. It shows that conventional online-learning methods (e.g., Hedge, FTRL, OMD) and even strategies that play the empirical Nash equilibrium cannot beat $\Omega(\sqrt{T})$ Nash regret, including in the simple $2\times2$ case and under full-information feedback; bandit extensions like UCB likewise suffer $\Omega(\sqrt{T})$ regret. The authors then present a general full-information algorithm for $n\times m$ games achieving instance-dependent polylogarithmic Nash regret, together with a bandit-enabled polylogarithmic guarantee for the $2\times2$ case, and provide empirical evidence supporting the theory. The results illuminate a fundamental separation between external and Nash regret and contribute a constructive path toward polylogarithmic regret in matrix games with noisy feedback. These insights advance understanding of regret in strategic, partially observable environments and offer practical algorithms for efficient equilibrium-driven performance.

Abstract

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 $A_{i,j}$, 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 $Ω(\sqrt{T})$ 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 $Ω(\sqrt{T})$ Nash regret under bandit feedback. Notably, these lower bounds hold even in the simplest case of $2 \times 2$ matrix games, where the instance-dependent matrix parameters are constant. We next ask whether instance-dependent $\text{polylog}(T)$ Nash regret is achievable against adversarial opponents. We answer this affirmatively. In the full-information setting, we present the first algorithm for general $n \times m$ matrix games that achieves instance-dependent $\text{polylog}(T)$ Nash regret. In the bandit feedback setting, we design an algorithm with similar guarantees for the special case of $2 \times 2$ game--the same regime in which existing algorithms provably suffer $Ω(\sqrt{T})$ regret despite the simplicity of the instance. Finally, we validate our theoretical results with empirical evidence.