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Optimistic Online Learning in Symmetric Cone Games

Anas Barakat, Wayne Lin, John Lazarsfeld, Antonios Varvitsiotis

TL;DR

It is proved that the symmetric cone negative entropy is strongly convex with respect to the trace-one norm, which extends known results for the simplex and spectraplex to all symmetric cones, and may be of independent interest.

Abstract

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 $\tilde{\mathcal{O}}(1/ε)$ 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.

Optimistic Online Learning in Symmetric Cone Games

TL;DR

It is proved that the symmetric cone negative entropy is strongly convex with respect to the trace-one norm, which extends known results for the simplex and spectraplex to all symmetric cones, and may be of independent interest.

Abstract

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 iteration complexity for computing -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.

Paper Structure

This paper contains 34 sections, 18 theorems, 79 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries: Euclidean Jordan Algebras, Symmetric Cones, and Entropy
  3. Symmetric Cone Games
  4. Examples of SCGs
  5. Representative Applications of SCGs
  6. Optimistic Symmetric Cone Multiplicative Weights Updates
  7. Regret Analysis and Average Iterate Convergence Guarantees
  8. Individual Regret Bound and Strong Convexity of Symmetric Cone Negative Entropy
  9. Sum of Regrets Bound in Symmetric Cone Games
  10. Application to Two-Player Zero-Sum Symmetric Cone Games
  11. Applications to Min-Max Problems: Metric Learning and Facility Location
  12. Simplex-Spectraplex Games (e.g. Metric Learning)
  13. Second-Order Cone Min-Max Games (e.g. Fermat-Weber Problem)
  14. Conclusion
  15. Related Work
  16. ...and 19 more sections

Key Result

Proposition 4

For any symmetric cone $\mathcal{K}$, the iterates of OSCMWU coincide with the iterates of OFTRL with the symmetric cone negative entropy regularizer ($\Phi = \Phi_{\text{ent}}$).

Figures (3)

  • Figure 1: Duality gap of the average iterates versus number of iterations for OSCMWU and SCMWU for a distance metric-learning application on the Iris dataset (mean $\pm$ std over 5 seeds).
  • Figure 2: (Left) Objective function values of the average iterates (Right) Duality gap of the average iterates for OSCMWU and SCMWU for a facility location problem (mean $\pm$ std over 5 seeds).
  • Figure 3: Time-scaled sum of regrets for OSCMWU and SCMWU for an online facility location problem (mean $\pm$ std over 5 seeds).

Theorems & Definitions (36)

  • Remark 2
  • Remark 3
  • Proposition 4
  • Proposition 5
  • Remark 6
  • Theorem 7
  • Remark 8
  • Remark 9
  • Theorem 10
  • Theorem 11
  • ...and 26 more