# Optimistic Online Learning in Symmetric Cone Games

arXiv: https://arxiv.org/abs/2504.03592 · HTML: https://mlsanity.com/paper/2504.03592/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 71 references that are in the map.
- First posted on arXiv 4 Apr 2025.

## Builds on (most cited first, 23 of 23)

- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Fast Convergence of Regularized Learning in Games](https://mlsanity.com/paper/1507.00407/) (2015)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Near-Optimal No-Regret Learning in General Games](https://mlsanity.com/paper/2108.06924/) (2021)
- [Scalable Semidefinite Programming](https://mlsanity.com/paper/1912.02949/) (2019)
- [Quantum Wasserstein Generative Adversarial Networks](https://mlsanity.com/paper/1911.00111/) (2019)
- [Near-Optimal No-Regret Learning Dynamics for General Convex Games](https://mlsanity.com/paper/2206.08742/) (2022)
- [A unified stochastic approximation framework for learning in games](https://mlsanity.com/paper/2206.03922/) (2022)
- [Beyond Time-Average Convergence: Near-Optimal Uncoupled Online Learning via Clairvoyant Multiplicative Weights Update](https://mlsanity.com/paper/2111.14737/) (2021)
- [Kernelized Multiplicative Weights for 0/1-Polyhedral Games: Bridging the Gap Between Learning in Extensive-Form and Normal-Form Games](https://mlsanity.com/paper/2202.00237/) (2022)
- [Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates](https://mlsanity.com/paper/1506.04838/) (2015)
- [Matrix Multiplicative Weights Updates in Quantum Zero-Sum Games: Conservation Laws & Recurrence](https://mlsanity.com/paper/2211.01681/) (2022)
- [No-Regret Learning and Equilibrium Computation in Quantum Games](https://mlsanity.com/paper/2310.08473/) (2023)
- [A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games](https://mlsanity.com/paper/2311.10859/) (2023)
- [Revisiting Area Convexity: Faster Box-Simplex Games and Spectrahedral Generalizations](https://mlsanity.com/paper/2303.15627/) (2023)
- [A Rank-1 Sketch for Matrix Multiplicative Weights](https://mlsanity.com/paper/1903.02675/) (2019)
- [A Primal-Dual Framework for Symmetric Cone Programming](https://mlsanity.com/paper/2405.09157/) (2024)
- [A First-order Method for Monotone Stochastic Variational Inequalities on Semidefinite Matrix Spaces](https://mlsanity.com/paper/1809.09155/) (2018)
- [Convergence Rate Analysis of the Multiplicative Gradient Method for PET-Type Problems](https://mlsanity.com/paper/2109.05601/) (2021)
- [Multiplicative Updates for Online Convex Optimization over Symmetric Cones](https://mlsanity.com/paper/2307.03136/) (2023)
- [Approximation Algorithms for Smallest Intersecting Balls](https://mlsanity.com/paper/2406.11369/) (2024)
- [Fixed-budget and Multiple-issue Quadratic Voting](https://mlsanity.com/paper/2409.06614/) (2024)

## Abstract (arXiv)

We introduce symmetric cone games (SCGs), a broad class of multi-player games where each player's strategy lies in a generalized simplex (the trace-one slice of a symmetric cone). This framework unifies a wide spectrum of settings, including normal-form games (simplex strategies), quantum games (density matrices), and continuous games with ball-constrained strategies. It also captures several structured machine learning and optimization problems, such as distance metric learning and Fermat-Weber facility location, as two-player zero-sum SCGs. To compute approximate Nash equilibria in two-player zero-sum SCGs, we propose a single online learning algorithm: Optimistic Symmetric Cone Multiplicative Weights Updates (OSCMWU). Unlike prior methods tailored to specific geometries, OSCMWU provides closed-form updates over any symmetric cone and achieves a tildeO(1/ε) iteration complexity for computing ε-saddle points. Our analysis builds on the Optimistic Follow-the-Regularized-Leader framework and hinges on a key technical contribution: We prove that the symmetric cone negative entropy is strongly convex with respect to the trace-one norm. This result extends known results for the simplex and spectraplex to all symmetric cones, and may be of independent interest.