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1 CMAP - Centre de Mathématiques Appliquées 2 SEQUEL - Sequential Learning LIFL - Laboratoire d-Informatique Fondamentale de Lille, LAGIS - Laboratoire d-Automatique, Génie Informatique et Signal, Inria Lille - Nord Europe

Abstract : Bandit based methods for tree search have recently gained popularity when applied to huge trees, e.g. in the game of go Gelly et al., 2006. The UCT algorithm Kocsis and Szepesvari, 2006, a tree search method based on Upper Confidence Bounds UCB Auer et al., 2002, is believed to adapt locally to the effective smoothness of the tree. However, we show that UCT is too ``optimistic- in some cases, leading to a regret OexpexpD where D is the depth of the tree. We propose alternative bandit algorithms for tree search. First, a modification of UCT using a confidence sequence that scales exponentially with the horizon depth is proven to have a regret O2^D \sqrt{n}, but does not adapt to possible smoothness in the tree. We then analyze Flat-UCB performed on the leaves and provide a finite regret bound with high probability. Then, we introduce a UCB-based Bandit Algorithm for Smooth Trees which takes into account actual smoothness of the rewards for performing efficient ``cuts- of sub-optimal branches with high confidence. Finally, we present an incremental tree search version which applies when the full tree is too big possibly infinite to be entirely represented and show that with high probability, essentially only the optimal branches is indefinitely developed. We illustrate these methods on a global optimization problem of a Lipschitz function, given noisy data.

Keywords : Bandit algorithms tree search exploration-exploitation tradeoff upper confidence bounds minimax game reinforcement learning

Author: Pierre-Arnaud Coquelin - Rémi Munos -



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