# Multiplicative Updates for Online Convex Optimization over Symmetric Cones

arXiv: https://arxiv.org/abs/2307.03136 · HTML: https://mlsanity.com/paper/2307.03136/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 37 references that are in the map.
- First posted on arXiv 6 Jul 2023.

## Builds on (most cited first, 6 of 6)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Variance Reduction for Matrix Games](https://mlsanity.com/paper/1907.02056/) (2019)
- [Online mirror descent and dual averaging: keeping pace in the dynamic case](https://mlsanity.com/paper/2006.02585/) (2020)
- [Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates](https://mlsanity.com/paper/1506.04838/) (2015)

## Built on by (most cited first, 1 of 1)

- [Optimistic Online Learning in Symmetric Cone Games](https://mlsanity.com/paper/2504.03592/) (2025)

## Abstract (arXiv)

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.