Optimistic Online Learning in Symmetric Cone Games

TMLR Paper6554 Authors

Abstract: We introduce symmetric cone games (SCGs), a broad class of multi-player games where each player's strategy lies in a generalized simplex (the trace-one slice of a symmetric cone). This framework unifies a wide spectrum of settings, including normal-form games (simplex strategies), quantum games (density matrices), and continuous games with ball-constrained strategies. It also captures several structured machine learning and optimization problems, such as distance metric learning and Fermat–Weber facility location, as two-player zero-sum SCGs. To compute approximate Nash equilibria in two-player zero-sum SCGs, we propose a single online learning algorithm: Optimistic Symmetric Cone Multiplicative Weights Updates (OSCMWU). Unlike prior methods tailored to specific geometries, OSCMWU provides closed-form, projection-free updates over any symmetric cone and achieves a~$\tilde{\mathcal{O}}(1/\epsilon)$ iteration complexity for computing $\epsilon$-saddle points. Our analysis builds on the Optimistic Follow-the-Regularized-Leader framework and hinges on a key technical contribution: We prove that the symmetric cone negative entropy is strongly convex with respect to the trace-one norm. This result extends known results for the simplex and spectraplex to all symmetric cones, and may be of independent interest.

Submission Type: Regular submission (no more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=8zLtEaiYRV

Changes Since Last Submission: The initial version was desk rejected because of a TMLR template formatting issue, the page header “Under review as submission to TMLR” was missing. We did use the TMLR template but the simultaneous use of another package inadvertently removed the required header. We have now corrected the format, ensuring that the “Under review as submission to TMLR” header appears as required.

Assigned Action Editor: ~Franck_Iutzeler1

Submission Number: 6554