Nonasymptotic Analysis of Monte Carlo Tree Search

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 ${\ell }_{\infty }$ norm, MCTS combined with nearest neighbor requires a sample size scaling as $\tilde{O}({\epsilon }^{-(d+4)})$, where d is the dimension of the state space. This is nearly optimal because of a minimax lower bound of $\tilde{\mathrm{\Omega }}({\epsilon }^{-(d+2)})$, suggesting the strength of the variant of MCTS we propose here and our resulting analysis.

Funding: This work was supported by the National Science Foundation [Grant CNS-1955997 and TRIPODS Phase II Grant] and MIT-IBM project on “Representation Learning as a Tool for causal Discovery,” Siemens Futuremakers Fellowship.