In this work, we consider the popular tree-based search strategy within the framework of reinforcement learning, the Monte Carlo tree search (MCTS), in the context of the infinite-horizon discounted cost Markov decision process (MDP). Although MCTS is believed to provide an approximate value function for a given state with enough simulations, the claimed proof of this property is incomplete. This is because the variant of MCTS, the upper confidence bound for trees (UCT), analyzed in prior works, uses “logarithmic” bonus term for balancing exploration and exploitation within the tree-based search, following the insights from stochastic multiarm bandit (MAB) literature. In effect, such an approach assumes that the regret of the underlying recursively dependent nonstationary MABs concentrates around their mean exponentially in the number of steps, which is unlikely to hold, even for stationary MABs. As the key contribution of this work, we establish polynomial concentration property of regret for a class of nonstationary MABs. This in turn establishes that the MCTS with appropriate polynomial rather than logarithmic bonus term in UCB has a claimed property. Interestingly enough, empirically successful approaches use a similar polynomial form of MCTS as suggested by our result. Using this as a building block, we argue that MCTS, combined with nearest neighbor supervised learning, acts as a “policy improvement” operator; that is, it iteratively improves value function approximation for all states because of combining with supervised learning, despite evaluating at only finitely many states. In effect, we establish that to learn an ε approximation of the value function with respect to [Formula: see text] norm, MCTS combined with nearest neighbor requires a sample size scaling as [Formula: see text], where d is the dimension of the state space. This is nearly optimal because of a minimax lower bound of [Formula: see text], suggesting the strength of the variant of MCTS we propose here and our resulting analysis.
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Statistical Inference of the Value Function for Reinforcement Learning in Infinite-Horizon Settings
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 was 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.
more » « less- NSF-PAR ID:
- 10398628
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- Volume:
- 84
- Issue:
- 3
- ISSN:
- 1369-7412
- Format(s):
- Medium: X Size: p. 765-793
- Size(s):
- ["p. 765-793"]
- Sponsoring Org:
- National Science Foundation
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