  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.
