提出对称锥博弈框架，统一多类博弈与优化问题，实现高效求解纳什均衡。
Optimistic Online Learning in Symmetric Cone Games
- 基于对称锥设计通用在线学习算法，支持多种策略空间的闭式更新。
- 在零和博弈中以近似最优复杂度 $\tilde{\mathcal{O}}(1/ε)$ 收敛到 ε-鞍点。
- 理论创新：证明对称锥负熵在迹一范数下强凸，适用于广泛场景。
我们引入对称锥博弈（SCGs），一种多玩家博弈框架，其中每个玩家的策略位于广义单纯形（对称锥的迹一截面）。该框架统一了多种场景，包括标准形式博弈（单纯形策略）、量子博弈（密度矩阵）以及球约束连续博弈。同时，它可将距离度量学习、Fermat-Weber设施定位等结构化机器学习与优化问题建模为双人零和SCG。针对此类问题，我们提出一种单一在线学习算法：乐观对称锥乘法权重更新（OSCMWU）。与以往依赖特定几何结构的方法不同，OSCMWU可在任意对称锥上实现闭式更新，并以 $\tilde{\mathcal{O}}(1/ε)$ 的迭代复杂度求解 $ε$-鞍点。分析基于乐观跟随正则化领袖框架，其关键技术贡献在于证明对称锥负熵关于迹一范数是强凸的。该结果扩展了单纯形与谱单纯形上的已知结论至所有对称锥，可能具有独立研究价值。
原文摘要 · Abstract (English)
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 updates over any symmetric cone and achieves a $\tilde{\mathcal{O}}(1/ε)$ iteration complexity for computing $ε$-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.
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