  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.
