Chaos, Extremism and Optimism: Volume Analysis of Learning in Games

Part of Advances in Neural Information Processing Systems 33 pre-proceedings (NeurIPS 2020)

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Yun Kuen Cheung, Georgios Piliouras


<p>We perform volume analysis of Multiplicative Weights Updates (MWU) and its optimistic variant (OMWU) in zero-sum as well as coordination games. Our analysis provides new insights into these game/dynamical systems, which seem hard to achieve via the classical techniques within Computer Science and Machine Learning.</p> <p>First, we examine these dynamics not in their original space (simplex of actions) but in a dual space (aggregate payoffs of actions). Second, we explore how the volume of a set of initial conditions evolves over time when it is pushed forward according to the algorithm. This is reminiscent of approaches in evolutionary game theory where replicator dynamics, the continuous-time analogue of MWU, is known to preserve volume in all games. Interestingly, when we examine discrete-time dynamics, the choices of the game and the algorithm both play a critical role. So whereas MWU expands volume in zero-sum games and is thus Lyapunov chaotic, we show that OMWU contracts volume, providing an alternative understanding for its known convergent behavior. Yet, we also prove a no-free-lunch type of theorem, in the sense that when examining coordination games the roles are reversed.</p> <p>Using these tools, we prove two novel, rather negative properties of MWU in zero-sum games. (1) Extremism: even in games with a unique fully-mixed Nash equilibrium, the system recurrently gets stuck near pure-strategy profiles, despite them being clearly unstable from game-theoretic perspective. (2) Unavoidability: given any set of good states (with a rather relaxed interpretation of “good” states), the system cannot avoid bad states indefinitely.</p>