Inthis paper, we consider Monte-Carlo planning in an environment with continuous state-action spaces, amuchlessunderstood problem withimportant applications in control and robotics.

This problem-dependent sample complexityresult is expressed in terms of the sub-optimality gapsof the state-action pairs that are visited during exploration.

Monte-Carlo planning, as exemplified by Monte-Carlo Tree Search (MCTS), has demonstrated remarkable performance in applications with finite spaces. In this paper, we consider Monte-Carlo planning in an environment with continuous state-action spaces, a much less understood problem with important applications in control and robotics. We introduce POLY-HOOT, an algorithm that augments MCTS with a continuous armed bandit strategy named Hierarchical Optimistic Optimization (HOO) (Bubeck et al., 2011). Specifically, we enhance HOO by using an appropriate polynomial, rather than logarithmic, bonus term in the upper confidence bounds. Such a polynomial bonus is motivated by its empirical successes in AlphaGo Zero (Silver et al., 2017b), as well as its significant role in achieving theoretical guarantees of finite space MCTS (Shah et al., 2019). We investigate, for the first time, the regret of the enhanced HOO algorithm in non-stationary bandit problems. Using this result as a building block, we establish non-asymptotic convergence guarantees for POLY-HOOT: the value estimate converges to an arbitrarily small neighborhood of the optimal value function at a polynomial rate. We further provide experimental results that corroborate our theoretical findings.

Y ou are a robot and you live in a Markov decision process (MDP) with a finite or an infinite number of transitions from state-action to next states. Y ou got brains and so you plan before you act.

This problem-dependent sample complexity result is expressed in terms of the sub-optimality gaps of the state-action pairs that are visited during exploration.

This problem-dependent sample complexity result is expressed in terms of the sub-optimality gaps of the state-action pairs that are visited during exploration.

Typically MCTS is just useful for discrete action settings and this paper studies the extension to continuous actions with the aim of theoretically justifying the approach taken. The approach is relevant to people interested in planning or people interested in continuous action control (e.g., robotics). The paper first extends an existing UCB-like algorithm for continuous-armed bandits, HOO, by using a polynomial exploration bonus instead of a logarithmic one. This approach is justified by a similar approach in the influential AlphaGo paper and prior work that justifies the approach theoretically for non-stationiary bandit problems. The paper then integrates this enhanced HOO into MCTS and calls the resulting algorithm Poly-HOOT. Theoretical results are given for convergence of approach to optimal action and empirical results show the method out-performs baselines. Overall, I liked the paper and think it clears the acceptance bar.

Reviewers all agreed that this paper presents a novel work towards MCTS in continuous action spaces, with its theoretical analysis making an important contribution.

Monte-Carlo planning, as exemplified by Monte-Carlo Tree Search (MCTS), has demonstrated remarkable performance in applications with finite spaces. In this paper, we consider Monte-Carlo planning in an environment with continuous state-action spaces, a much less understood problem with important applications in control and robotics. We introduce POLY-HOOT, an algorithm that augments MCTS with a continuous armed bandit strategy named Hierarchical Optimistic Optimization (HOO) (Bubeck et al., 2011). Specifically, we enhance HOO by using an appropriate polynomial, rather than logarithmic, bonus term in the upper confidence bounds. Such a polynomial bonus is motivated by its empirical successes in AlphaGo Zero (Silver et al., 2017b), as well as its significant role in achieving theoretical guarantees of finite space MCTS (Shah et al., 2019). We investigate, for the first time, the regret of the enhanced HOO algorithm in non-stationary bandit problems.