Prediction Accuracy of Learning in Games : Follow-the-Regularized-Leader meets Heisenberg

TL;DR

This paper analyzes the prediction accuracy of the Follow-the-Regularized-Leader algorithm in zero-sum games, connecting it with Heisenberg's uncertainty principles and demonstrating that symplectic discretization improves predictive performance.

Contribution

It introduces a Heisenberg-type inequality for FTRL in zero-sum games and compares discretization methods, highlighting the benefits of symplectic discretization for prediction accuracy.

Findings

Symplectic discretization improves prediction accuracy.

Growth rates of covariance information are characterized.

A Heisenberg-type inequality for FTRL is established.

Abstract

We investigate the accuracy of prediction in deterministic learning dynamics of zero-sum games with random initializations, specifically focusing on observer uncertainty and its relationship to the evolution of covariances. Zero-sum games are a prominent field of interest in machine learning due to their various applications. Concurrently, the accuracy of prediction in dynamical systems from mechanics has long been a classic subject of investigation since the discovery of the Heisenberg Uncertainty Principle. This principle employs covariance and standard deviation of particle states to measure prediction accuracy. In this study, we bring these two approaches together to analyze the Follow-the-Regularized-Leader (FTRL) algorithm in two-player zero-sum games. We provide growth rates of covariance information for continuous-time FTRL, as well as its two canonical discretization methods (Euler and Symplectic). A Heisenberg-type inequality is established for FTRL. Our analysis and experiments also show that employing Symplectic discretization enhances the accuracy of prediction in learning dynamics.