Despite its groundbreaking success in Go and computer games, Monte Carlo Tree Search (MCTS) is computationally expensive as it requires a substantial number of rollouts to construct the search tree, which calls for effective parallelization. However, how to design effective parallel MCTS algorithms has not been systematically studied and remains poorly understood. In this paper, we seek to lay its first theoretical foundation, by examining the potential performance loss caused by parallelization when achieving a desired speedup. In particular, we discover the necessary conditions of achieving a desirable parallelization performance, and highlight two of their practical benefits. First, by examining whether existing parallel MCTS algorithms satisfy these conditions, we identify key design principles that should be inherited by future algorithms, for example tracking the unobserved samples (used in WU-UCT (Liu et al., 2020)). We theoretically establish this essential design facilitates $mathcal{O} ( ln n + M / sqrt{ln n} )$ cumulative regret when the maximum tree depth is 2, where $n$ is the number of rollouts and $M$ is the number of workers. A regret of this form is highly desirable, as compared to $mathcal{O} ( ln n )$ regret incurred by a sequential counterpart, its excess part approaches zero as $n$ increases. Second, and more importantly, we demonstrate how the proposed necessary conditions can be adopted to design more effective parallel MCTS algorithms. To illustrate this, we propose a new parallel MCTS algorithm, called BU-UCT, by following our theoretical guidelines. The newly proposed algorithm, albeit preliminary, out-performs four competitive baselines on 11 out of 15 Atari games. We hope our theoretical results could inspire future work of more effective parallel MCTS.
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