  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.
