Optimistic Online Learning in Symmetric Cone Games

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.

Multiplicative Updates for Online Convex Optimization over Symmetric Cones

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.