An Efficient Algorithm for Thresholding Monte Carlo Tree Search

We introduce the Thresholding Monte Carlo Tree Search problem, in which, given a tree $\mathcal{T}$ and a threshold $\theta$, a player must answer whether the root node value of $\mathcal{T}$ is at least $\theta$ or not. In the given tree, 'MAX' or 'MIN' is labeled on each internal node, and the value of a 'MAX'-labeled ('MIN'-labeled) internal node is the maximum (minimum) of its child values. The value of a leaf node is the mean reward of an unknown distribution, from which the player can sample rewards. For this problem, we develop a $\delta$-correct sequential sampling algorithm based on the Track-and-Stop strategy that has asymptotically optimal sample complexity. We show that a ratio-based modification of the D-Tracking arm-pulling strategy leads to a substantial improvement in empirical sample complexity, as well as reducing the per-round computational cost from linear to logarithmic in the number of arms.