The Evolution of Uncertainty of Learning in Games

Learning in games has become an object of intense interest for ML due to its connections to numerous AI architectures. We study standard online learning in games but from a non-standard perspective. Instead of studying the behavior of a single initial condition and whether it converges to equilibrium or not, we study the behavior of a probability distribution/measure over a set of initial conditions. This initial uncertainty is well-motivated both from a standard game-theoretic perspective (e.g. a modeler's uncertainty about the agents' initial beliefs) as well as from a ML one (e.g. noisy measurements, system initialization from a dataset distribution). Despite this, little is formally known about whether and under what conditions uncertainty is amplified or reduced in these systems. We use the popular measure of differential entropy to quantify the evolution of uncertainty. We find that such analysis shares an intimate relationship with volume analysis, a technique which was recently used to demonstrate the occurrence of Lyapunov chaos when using Multiplicative Weights Update (MWU) or Follow-the-Regularized-Leader (FTRL) algorithms in zero-sum games. This allows us to show that the differential entropy of these learning-in-game systems increases linearly with time, formalizing their increased unpredictability over time. We showcase the power of the framework by applying it in the study of multiple related systems, including different standard online optimization algorithms in numerous games and dynamics of evolutionary game theory.