Geometric last-iterate convergence of unregularized OMMWU

Establish whether unregularized optimistic matrix multiplicative weights updates (OMMWU) achieve geometric last-iterate convergence for every quantum zero-sum game with a unique Nash equilibrium.

Background

The paper discusses prior work transferring classical optimistic-learning results to diagonal quantum games. That work conjectured that unregularized OMMWU should converge geometrically at the last iterate whenever the Nash equilibrium is unique.

The diagonal lower-bound constructions described in the paper do not resolve this conjecture because they do not establish a failure of geometric convergence in the full class of quantum games. The present paper subsequently gives a non-diagonal game with a unique, strictly complementary equilibrium exhibiting polynomial OMMWU convergence, thereby addressing and refuting the stated conjecture in the setting studied.

References

Su et al. transferred these examples to diagonal quantum games. In the same work, they pursued geometric convergence through semidefinite error bounds and conjectured geometric last-iterate convergence of unregularized OMMWU whenever the Nash equilibrium is unique. The diagonal lower bounds do not settle this conjecture.

— Average-and Last-Iterate Lower Bounds for Optimistic Matrix Mirror-Prox in Quantum Zero-Sum Games  (2609.38835 - Su et al., 30 Sep 2026) in Section 1, subsection “Prior works and a motivating question”

Since these results concern non-optimistic continuous-time dynamics, they leave open how optimism affects this eigenvector motion in discrete time.

— Average-and Last-Iterate Lower Bounds for Optimistic Matrix Mirror-Prox in Quantum Zero-Sum Games  (2609.38835 - Su et al., 30 Sep 2026) in Section 1, subsection “Prior works and a motivating question”