This paper introduces a novel backup strategy for Monte-Carlo Tree Search (MCTS) designed for highly stochastic and partially observable Markov decision processes. We adopt a probabilistic approach, modeling both value and action-value nodes as Gaussian distributions. We introduce a novel backup operator that computes value nodes as the Wasserstein barycenter of their action-value children nodes; thus, propagating the uncertainty of the estimate across the tree to the root node. We study our novel backup operator when using a novel combination of $L^1$-Wasserstein barycenter with $\alpha$-divergence, by drawing a notable connection to the generalized mean backup operator. We complement our probabilistic backup operator with two sampling strategies, based on optimistic selection and Thompson sampling, obtaining our Wasserstein MCTS algorithm. We provide theoretical guarantees of asymptotic convergence to the optimal policy, and an empirical evaluation on several stochastic and partially observable environments, where our approach outperforms well-known related baselines.

Path planning is a crucial algorithmic approach for designing robot behaviors. Sampling-based approaches, like rapidly exploring random trees (RRTs) or probabilistic roadmaps, are prominent algorithmic solutions for path planning problems. Despite its exponential convergence rate, RRT can only find suboptimal paths. On the other hand, $\textrm{RRT}^*$, a widely-used extension to RRT, guarantees probabilistic completeness for finding optimal paths but suffers in practice from slow convergence in complex environments. Furthermore, real-world robotic environments are often partially observable or with poorly described dynamics, casting the application of $\textrm{RRT}^*$ in complex tasks suboptimal. This paper studies a novel algorithmic formulation of the popular Monte-Carlo tree search (MCTS) algorithm for robot path planning. Notably, we study Monte-Carlo Path Planning (MCPP) by analyzing and proving, on the one part, its exponential convergence rate to the optimal path in fully observable Markov decision processes (MDPs), and on the other part, its probabilistic completeness for finding feasible paths in partially observable MDPs (POMDPs) assuming limited distance observability (proof sketch). Our algorithmic contribution allows us to employ recently proposed variants of MCTS with different exploration strategies for robot path planning. Our experimental evaluations in simulated 2D and 3D environments with a 7 degrees of freedom (DOF) manipulator, as well as in a real-world robot path planning task, demonstrate the superiority of MCPP in POMDP tasks.

Monte-Carlo Tree Search (MCTS) is a class of methods for solving complex decision-making problems through the synergy of Monte-Carlo planning and Reinforcement Learning (RL). The highly combinatorial nature of the problems commonly addressed by MCTS requires the use of efficient exploration strategies for navigating the planning tree and quickly convergent value backup methods. These crucial problems are particularly evident in recent advances that combine MCTS with deep neural networks for function approximation. In this work, we propose two methods for improving the convergence rate and exploration based on a newly introduced backup operator and entropy regularization. We provide strong theoretical guarantees to bound convergence rate, approximation error, and regret of our methods. Moreover, we introduce a mathematical framework based on the use of the $\alpha$-divergence for backup and exploration in MCTS. We show that this theoretical formulation unifies different approaches, including our newly introduced ones, under the same mathematical framework, allowing to obtain different methods by simply changing the value of $\alpha$. In practice, our unified perspective offers a flexible way to balance between exploration and exploitation by tuning the single $\alpha$ parameter according to the problem at hand. We validate our methods through a rigorous empirical study from basic toy problems to the complex Atari games, and including both MDP and POMDP problems.

We consider Monte-Carlo Tree Search (MCTS) applied to Markov Decision Processes (MDPs) and Partially Observable MDPs (POMDPs), and the well-known Upper Confidence bound for Trees (UCT) algorithm. In UCT, a tree with nodes (states) and edges (actions) is incrementally built by the expansion of nodes, and the values of nodes are updated through a backup strategy based on the average value of child nodes. However, it has been shown that with enough samples the maximum operator yields more accurate node value estimates than averaging. Instead of settling for one of these value estimates, we go a step further proposing a novel backup strategy which uses the power mean operator, which computes a value between the average and maximum value. We call our new approach Power-UCT and argue how the use of the power mean operator helps to speed up the learning in MCTS. We theoretically analyze our method providing guarantees of convergence to the optimum. Moreover, we discuss a heuristic approach to balance the greediness of backups by tuning the power mean operator according to the number of visits to each node. Finally, we empirically demonstrate the effectiveness of our method in well-known MDP and POMDP benchmarks, showing significant improvement in performance and convergence speed w.r.t. UCT.