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 includes 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 Poincar\'e 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 the results of Piliouras and Shamma (SODA, 2014) and Mertikopoulos, Papadimitriou and Piliouras (SODA, 2018).

通过控制理论中能量守恒的概念，将在线优化与游戏学习结合起来，证明了所有连续时间的Follow-the-Regularized-Leader动态都是无损耗的，这启发我们构建了一族具有简单梯度结构的无损耗学习动态，并将其拓展到了图形常和游戏等多种游戏动力学中。

控制理论在游戏中的在线优化：连接遗憾、被动性和庞加莱循环