  Monte-Carlo planning and Reinforcement Learning (RL) are essential to
sequential decision making. The recent AlphaGo and AlphaZero algorithms have
shown how to successfully combine these two paradigms in order to solve large
scale sequential decision problems. These methodologies exploit a variant of
the well-known UCT algorithm to trade off exploitation of good actions and
exploration of unvisited states, but their empirical success comes at the cost
of poor sample-efficiency and high computation time. In this paper, we overcome
these limitations by considering convex regularization in Monte-Carlo Tree
Search (MCTS), which has been successfully used in RL to efficiently drive
exploration. First, we introduce a unifying theory on the use of generic convex
regularizers in MCTS, deriving the regret analysis and providing guarantees of
exponential convergence rate. Second, we exploit our theoretical framework to
introduce novel regularized backup operators for MCTS, based on the relative
entropy of the policy update, and on the Tsallis entropy of the policy.
Finally, we empirically evaluate the proposed operators in AlphaGo and
AlphaZero on problems of increasing dimensionality and branching factor, from a
toy problem to several Atari games, showing their superiority w.r.t.
representative baselines.
