Two-Sample Testing in Reinforcement Learning

Value-based reinforcement-learning algorithms have shown strong performances in games, robotics, and other real-world applications. The most popular sample-based method is $Q$-Learning. It subsequently performs updates by adjusting the current $Q$-estimate towards the observed reward and the maximum of the $Q$-estimates of the next state. The procedure introduces maximization bias with approaches like Double $Q$-Learning. We frame the bias problem statistically and consider it an instance of estimating the maximum expected value (MEV) of a set of random variables. We propose the $T$-Estimator (TE) based on two-sample testing for the mean, that flexibly interpolates between over- and underestimation by adjusting the significance level of the underlying hypothesis tests. A generalization, termed $K$-Estimator (KE), obeys the same bias and variance bounds as the TE while relying on a nearly arbitrary kernel function. We introduce modifications of $Q$-Learning and the Bootstrapped Deep $Q$-Network (BDQN) using the TE and the KE. Furthermore, we propose an adaptive variant of the TE-based BDQN that dynamically adjusts the significance level to minimize the absolute estimation bias. All proposed estimators and algorithms are thoroughly tested and validated on diverse tasks and environments, illustrating the bias control and performance potential of the TE and KE.