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

*Proceedings of the 38th International Conference on Machine Learning*, PMLR 139:1855-1865, 2021.

#### Abstract

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 Dynamic, are lossless, i.e. it is passive with no energy dissipation. Interestingly, we prove that passivity implies bounded regret, connecting two fundamental primitives of control theory and online optimization. The observation of energy conservation in FTRL inspires us to present a family of lossless learning dynamics, each of which has an underlying energy function with a simple gradient structure. This family is closed under convex combination; as an immediate corollary, any convex combination of FTRL dynamics is lossless and thus has bounded regret. This allows us to extend the framework of Fox & Shamma [Games 2013] to prove not just global asymptotic stability results for game dynamics, but Poincar{é} recurrence results as well. Intuitively, when a lossless game (e.g. graphical constant-sum game) is coupled with lossless learning dynamic, their interconnection is also lossless, which results in a pendulum-like energy-preserving recurrent behavior, generalizing Piliouras & Shamma [SODA 2014] and Mertikopoulos et al. [SODA 2018].