Multiplicative Updates for Online Convex Optimization over Symmetric Cones
This work provides a unifying framework for optimization models like linear, second-order cone, and semidefinite optimization, which is incremental but broadens applicability in machine learning and optimization domains.
The paper tackles online convex optimization over symmetric cones, a generalization of experts and quantum settings, by introducing the Symmetric-Cone Multiplicative Weights Update (SCMWU) algorithm. The result shows SCMWU is a no-regret algorithm, unifying and generalizing previous multiplicative weights methods.
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.