The ODE Method for Stochastic Approximation and Reinforcement Learning with Markovian Noise

Stochastic approximation is a class of algorithms that update a vector iteratively, incrementally, and stochastically, including, e.g., stochastic gradient descent and temporal difference learning. One fundamental challenge in analyzing a stochastic approximation algorithm is to establish its stability, i.e., to show that the stochastic vector iterates are bounded almost surely. In this paper, we extend the celebrated Borkar-Meyn theorem for stability from the Martingale difference noise setting to the Markovian noise setting, which greatly improves its applicability in reinforcement learning, especially in those off-policy reinforcement learning algorithms with linear function approximation and eligibility traces. Central to our analysis is the diminishing asymptotic rate of change of a few functions, which is implied by both a form of strong law of large numbers and a commonly used V4 Lyapunov drift condition and trivially holds if the Markov chain is finite and irreducible.

Improving Monte Carlo Evaluation with Offline Data

Monte Carlo (MC) methods are the most widely used methods to estimate the performance of a policy. Given an interested policy, MC methods give estimates by repeatedly running this policy to collect samples and taking the average of the outcomes. Samples collected during this process are called online samples. To get an accurate estimate, MC methods consume massive online samples. When online samples are expensive, e.g., online recommendations and inventory management, we want to reduce the number of online samples while achieving the same estimate accuracy. To this end, we use off-policy MC methods that evaluate the interested policy by running a different policy called behavior policy. We design a tailored behavior policy such that the variance of the off-policy MC estimator is provably smaller than the ordinary MC estimator. Importantly, this tailored behavior policy can be efficiently learned from existing offline data, i,e., previously logged data, which are much cheaper than online samples. With reduced variance, our off-policy MC method requires fewer online samples to evaluate the performance of a policy compared with the ordinary MC method. Moreover, our off-policy MC estimator is always unbiased.