We study online convex optimization where the possible actions are trace-one elements in a symmetric cone, generalizing the extensively-studied experts setup and its quantum counterpart. Symmetric cones provide a unifying framework for some of the most important optimization models, including linear, second-order cone, and semidefinite optimization. Using tools from the field of Euclidean Jordan Algebras, we introduce the Symmetric-Cone Multiplicative Weights Update (SCMWU), a projection-free algorithm for online optimization over the trace-one slice of an arbitrary symmetric cone. We show that SCMWU is equivalent to Follow-the-Regularized-Leader and Online Mirror Descent with symmetric-cone negative entropy as regularizer. Using this structural result we show that SCMWU is a no-regret algorithm, and verify our theoretical results with extensive experiments. Our results unify and generalize the analysis for the Multiplicative Weights Update method over the probability simplex and the Matrix Multiplicative Weights Update method over the set of density matrices.

翻译：本研究探讨在线凸优化问题，其中可行动作为对称锥中的迹一元素，这推广了广泛研究的专家设置及其量子对应问题。对称锥为一些最重要的优化模型（包括线性优化、二阶锥优化和半定优化）提供了统一框架。利用欧几里得若当代数的工具，我们提出了对称锥乘法权重更新（SCMWU），一种针对任意对称锥迹一片上的在线优化的无投影算法。我们证明SCMWU等价于带正则项的跟随正则化领导者算法和在线镜像下降法，其中正则项为对称锥负熵。利用这一结构结果，我们证明SCMWU是一种无悔算法，并通过大量实验验证了我们的理论结果。我们的结果统一并推广了概率单纯形上的乘法权重更新方法和密度矩阵集上的矩阵乘法权重更新方法的分析。