  Reinforcement learning is a general technique that allows an agent to learn
an optimal policy and interact with an environment in sequential decision
making problems. The goodness of a policy is measured by its value function
starting from some initial state. The focus of this paper is to construct
confidence intervals (CIs) for a policy's value in infinite horizon settings
where the number of decision points diverges to infinity. We propose to model
the action-value state function (Q-function) associated with a policy based on
series/sieve method to derive its confidence interval. When the target policy
depends on the observed data as well, we propose a SequentiAl Value Evaluation
(SAVE) method to recursively update the estimated policy and its value
estimator. As long as either the number of trajectories or the number of
decision points diverges to infinity, we show that the proposed CI achieves
nominal coverage even in cases where the optimal policy is not unique.
Simulation studies are conducted to back up our theoretical findings. We apply
the proposed method to a dataset from mobile health studies and find that
reinforcement learning algorithms could help improve patient's health status. A
Python implementation of the proposed procedure is available at
https://github.com/shengzhang37/SAVE.
