Average-and Last-Iterate Lower Bounds for Optimistic Matrix Mirror-Prox in Quantum Zero-Sum Games

2026-09-29

AI Summary

This study investigates whether the Online Mirror Multiplicative Weights Update (OMMWU) algorithm achieves geometric convergence in quantum zero-sum games. By constructing an explicit single-qubit game, we analyze the convergence properties of Optimistic Gradient Descent Ascent (OGDA) and OMMWU. We establish, for the first time, an Ω(1/ε) lower bound on the uniform average output, alongside strict lower bounds for both the average error and the last-iterate distance. Furthermore, our analysis reveals that the last iterate exhibits Θ(1/t) decay accompanied by arbitrarily long delay phenomena, confirming that OMMWU attains only polynomial rather than geometric convergence. This work resolves a longstanding theoretical question regarding these dynamics and establishes tight bounds on the associated algorithmic complexity.

Abstract

Optimistic matrix mirror-prox (OMMP) computes $ε$-approximate Nash equilibria in quantum zero-sum games with an $O(1/\varepsilon)$ average-iterate guarantee [arXiv:2311.10859]. We investigate whether this dependence on accuracy is tight and whether geometric last-iterate convergence can be guaranteed. We study these questions through explicit games with one qubit per player. First, we prove an $Ω(1/\varepsilon)$ lower bound for the uniform-average output that includes the maximally mixed initial state, independently of the regularizer and step size. Second, we construct a fixed game on which optimistic gradient descent-ascent (OGDA), initialized at the maximally mixed state, has last-iterate Frobenius distance to equilibrium $Θ(1/t)$ and duality gap $Θ(1/t^3)$ for every sufficiently small fixed step size. A separate fixed game exhibits arbitrarily long delays in reducing the initial error by a constant factor across a family of initial states. Finally, we give a fixed game with a unique, strictly complementary equilibrium on which optimistic matrix multiplicative weights updates (OMMWU) converge only polynomially from the maximally mixed state for every fixed positive step size. The last-iterate Frobenius distance and quantum relative entropy from the equilibrium to the iterates decay as $Θ(1/t)$, while the duality gap decays as $Θ(1/t^2)$.

Problem

• Quantum zero-sum games
• Optimistic matrix mirror-prox
• Nash equilibria
• Lower bounds
• Last-iterate convergence

Innovation

• Quantum Zero-Sum Games
• Optimistic Matrix Mirror-Prox
• Last-Iterate Convergence
• Lower Bounds
• Nash Equilibria