Online Optimization in Games via Control Theory: Connecting Regret, Passivity and Poincaré Recurrence

We present a novel control-theoretic understanding of online optimization and learning in games, via the notion of passivity. Passivity is a fundamental concept in control theory, which abstracts energy conservation and dissipation in physical systems. It has become a standard tool in analysis of general feedback systems, to which game dynamics belong. Our starting point is to show that all continuous-time Follow-the-Regularized-Leader (FTRL) dynamics, which include the well-known Replicator Dyn...