Multiplicative Updates for Online Convex Optimization over Symmetric Cones

TL;DR

This paper introduces the SCMWU algorithm for online convex optimization over symmetric cones, unifying and generalizing multiplicative weights methods across various optimization models using Jordan algebra tools.

Contribution

The paper presents SCMWU, a projection-free algorithm for symmetric cone optimization, and establishes its equivalence to existing methods like Follow-the-Regularized-Leader.

Findings

SCMWU is a no-regret algorithm for symmetric cone optimization.

Theoretical analysis unifies multiplicative weights methods across different models.

Experimental results confirm the effectiveness of SCMWU.

Abstract

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.