Online Optimization in Games via Control Theory: Connecting Regret, Passivity and Poincaré Recurrence
arXiv:2106.04748
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 and Shamma [Games, 2013] to prove not just global asymptotic stability results for game dynamics, but Poincaré recurrence results as well. Intuitively, when a lossless game (e.g. graphical constant-sum game) is coupled with lossless learning dynamics, their feedback interconnection is also lossless, which results in a pendulum-like energy-preserving recurrent behavior, generalizing the results of Piliouras and Shamma [SODA, 2014] and Mertikopoulos, Papadimitriou and Piliouras [SODA, 2018].
arXiv commentIn ICML 2021
References in corpus (8)
- Graphical Models for Game Theory
- The Mechanics of n-Player Differentiable Games
- Online Learning with Predictable Sequences
- Fast Convergence of Regularized Learning in Games
- Escort Evolutionary Game Theory
- Learning in Games: Robustness of Fast Convergence
- From Poincaré Recurrence to Convergence in Imperfect Information Games: Finding Equilibrium via Regularization
- Poincaré Recurrence, Cycles and Spurious Equilibria in Gradient-Descent-Ascent for Non-Convex Non-Concave Zero-Sum Games