Optimistic Online Learning in Symmetric Cone Games

• URL: http://arxiv.org/abs/2504.03592v1
• Date: Fri, 04 Apr 2025 16:59:19 GMT
• Title: Optimistic Online Learning in Symmetric Cone Games
• Authors: Anas Barakat, Wayne Lin, John Lazarsfeld, Antonios Varvitsiotis,
• Abstract summary: We introduce the Optimistic Symmetric Cone Multiplicative Weights Update algorithm and establish an iteration complexity of $mathcalO (1/epsilon)$ to reach an $epsilon$-saddle point.<n>A key technical contribution is a new proof of the strong convexity of the symmetric cone negative entropy with respect to the trace-one norm.
• Score: 3.124884279860061
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Optimistic online learning algorithms have led to significant advances in equilibrium computation, particularly for two-player zero-sum games, achieving an iteration complexity of $\mathcal{O}(1/\epsilon)$ to reach an $\epsilon$-saddle point. These advances have been established in normal-form games, where strategies are simplex vectors, and quantum games, where strategies are trace-one positive semidefinite matrices. We extend optimistic learning to symmetric cone games (SCGs), a class of two-player zero-sum games where strategy spaces are generalized simplices (trace-one slices of symmetric cones). A symmetric cone is the cone of squares of a Euclidean Jordan Algebra; canonical examples include the nonnegative orthant, the second-order cone, the cone of positive semidefinite matrices, and their products, all fundamental to convex optimization. SCGs unify normal-form and quantum games and, as we show, offer significantly greater modeling flexibility, allowing us to model applications such as distance metric learning problems and the Fermat-Weber problem. To compute approximate saddle points in SCGs, we introduce the Optimistic Symmetric Cone Multiplicative Weights Update algorithm and establish an iteration complexity of $\mathcal{O}(1/\epsilon)$ to reach an $\epsilon$-saddle point. Our analysis builds on the Optimistic Follow-the-Regularized-Leader framework, with a key technical contribution being a new proof of the strong convexity of the symmetric cone negative entropy with respect to the trace-one norm, a result that may be of independent interest.

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