[2509.21570] Breaking $1/ε$ Barrier in Quantum Zero-Sum Video games: Generalizing Metric Subregularity for Spectraplexes

View a PDF of the paper titled Breaking $1/epsilon$ Barrier in Quantum Zero-Sum Video games: Generalizing Metric Subregularity for Spectraplexes, by Yiheng Su and a couple of different authors

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Summary:Quantum zero-sum video games present a framework for non-local video games, quantum interactive proofs, and quantum machine studying, the place gamers optimize a bilinear payoff over quantum states. In distinction to classical bilinear video games over polyhedral domains, for which gradient strategies obtain linear last-iterate convergence, comparable ensures over spectraplexes have remained open. Latest work achieved solely an $O(1/varepsilon)$ average-iterate fee and steered that semidefinite geometry might preclude classical-style linear charges.
We refute this obstruction. We show that quantum zero-sum video games admit algorithms with $O(log(1/varepsilon))$ last-iterate convergence to Nash equilibrium. Particularly, matrix variants of Nesterov’s iterative smoothing and Optimistic Gradient Descent–Ascent match the asymptotic fee of the classical polyhedral case. The important thing technical ingredient is a brand new error-bound concept for semidefinite video games, establishing metric subregularity of the related monotone operator over spectrahedra regardless of the absence of polyhedral construction.
We additionally give a geometrical characterization of Nash equilibria by way of slack operators, classifying strategic instructions as important, impartial, or non-essential. Underneath strict complementarity or nondegeneracy, this reduces to a pointy classical-style dichotomy. Lastly, we revisit Optimistic Matrix Multiplicative Weights Replace. By extending the Quantal Response Equilibrium framework to spectraplex video games, we show an $widetilde O(1/varepsilon)$ last-iterate assure, whereas exhibiting that any $O(log(1/varepsilon))$ speedup for this technique should rely upon a pure, dimension-dependent situation quantity. Experiments assist the theoretical image, with Optimistic Gradient Descent–Ascent outperforming Optimistic Matrix Multiplicative Weights Replace within the regimes studied.