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.

翻译：本文引入对称锥博弈（SCGs）这一多玩家博弈的广泛类别，其中每个玩家的策略位于广义单纯形（对称锥的迹一截面）中。该框架统一了多种博弈设定，包括标准形式博弈（单纯形策略）、量子博弈（密度矩阵）以及具有球约束策略的连续博弈。同时，它将若干结构化机器学习与优化问题——如距离度量学习和费马-韦伯设施选址问题——建模为双人零和对称锥博弈。为计算双人零和对称锥博弈中的近似纳什均衡，我们提出了一种单一的在线学习算法：乐观对称锥乘性权重更新法（OSCMWU）。与以往针对特定几何结构设计的方法不同，OSCMWU可在任意对称锥上提供闭式更新，并在计算ε-鞍点时实现$\tilde{\mathcal{O}}(1/ε)$的迭代复杂度。我们的分析基于乐观跟随正则化领导者框架，并依赖于一项关键的技术贡献：我们证明对称锥负熵关于迹一范数是强凸的。该结果将单纯形和谱单纯形的已知结论推广至所有对称锥，可能具有独立的学术价值。