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

arXiv: https://arxiv.org/abs/2306.13233 · HTML: https://mlsanity.com/paper/2306.13233/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2023.

## Built on by (most cited first, 8 of 8)

- [Instance-Dependent Regret Bounds for Learning Two-Player Zero-Sum Games with Bandit Feedback](https://mlsanity.com/paper/2502.17625/) (2025)
- [Near-Optimal Pure Exploration in Matrix Games: A Generalization of Stochastic Bandits & Dueling Bandits](https://mlsanity.com/paper/2310.16252/) (2023)
- [Adversarial Learning in Games with Bandit Feedback: Logarithmic Pure-Strategy Maximin Regret](https://mlsanity.com/paper/2602.06348/) (2026)
- [Near-Optimal Last-Iterate Convergence for Zero-Sum Games with Bandit Feedback and Opponent Actions](https://mlsanity.com/paper/2605.09363/) (2026)
- [Two-Player Zero-Sum Games with Bandit Feedback](https://mlsanity.com/paper/2506.14518/) (2025)
- [Prudent-Banker: No Extra Fees for Baseline Safety in Adversarial Bandits With and Without Delays](https://mlsanity.com/paper/2605.23351/) (2026)
- [Polylogarithmic Nash Regret in Matrix Games with Bandit Feedback](https://mlsanity.com/paper/2609.34812/) (2026)
- [Thompson Sampling for Learning in Zero-Sum Games](https://mlsanity.com/paper/2610.18899/) (2026)

## Abstract (arXiv)

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.