## Efficient selectivity and backup operators in Monte-Carlo tree search (2006)

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Venue: | In: Proceedings Computers and Games 2006 |

Citations: | 135 - 2 self |

### BibTeX

@INPROCEEDINGS{Coulom06efficientselectivity,

author = {Rémi Coulom},

title = {Efficient selectivity and backup operators in Monte-Carlo tree search},

booktitle = {In: Proceedings Computers and Games 2006},

year = {2006},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

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### Abstract

Abstract. Monte-Carlo evaluation consists in estimating a position by averaging the outcome of several random continuations, and can serve as an evaluation function at the leaves of a min-max tree. This paper presents a new framework to combine tree search with Monte-Carlo evaluation, that does not separate between a min-max phase and a Monte-Carlo phase. Instead of backing-up the min-max value close to the root, and the average value at some depth, a more general backup operator is defined that progressively changes from averaging to min-max as the number of simulations grows. This approach provides a fine-grained control of the tree growth, at the level of individual simulations, and allows efficient selectivity methods. This algorithm was implemented in a 9 × 9 Go-playing program, Crazy Stone, that won the 10th KGS computer-Go tournament. 1

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Citation Context ... go to zero. Ideas for this kind of algorithm can be found in two fields of research: n-armed bandit problems, and discrete stochastic optimization. n-armed bandit techniques (Sutton and Barto’s book =-=[25]-=- provides an introduction) are the basis for the MonteCarlo tree search algorithm of Chang, Fu and Marcus [12]. Optimal schemes for the allocation of simulations in discrete stochastic optimization [1... |

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Citation Context ...so refute other moves of a node. Since formal methods seem difficult to apply, the backup operator of Crazy Stone was determined empirically, by an algorithm similar to the temporal difference method =-=[24]-=-. In the beginning, the backup method for internal nodes was the external-node method. 1,500 positions were sampled at random from ,s6 R. Coulom self-play games. For each of these 1,500 positions, the... |

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Citation Context ...tion inaccuracies. Other algorithms with better asymptotic properties (given enough time and memory, they will find an optimal action) have been proposed in the formalism of Markov decision processes =-=[12, 19, 22]-=-. This paper presents a new algorithm for combining Monte-Carlo evaluation with tree search. Its basic structure is described in Section 2. Its selectivity and backup operators are presented in the fo... |

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Citation Context ...ns is Monte-Carlo evaluation. Monte-Carlo evaluation consists in averagings2 R. Coulom the outcome of several continuations. It is an usual technique in games with randomness or partial observability =-=[5, 23, 26, 14, 17]-=-, but can also be applied to deterministic games, by choosing actions at random until a terminal state is reached [1, 9, 10]. The accuracy of Monte-Carlo evaluation can be improved with tree search. J... |

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Citation Context ...er boards. For 19x19, an approach based on a global tree search does not seem reasonable. Generalizing the tree search with high-level tactical objectives such as Cazenave and Helmstetter’s algorithm =-=[11]-=- might be an interesting solution. Acknowledgements I thank Bruno Bouzy and Guillaume Chaslot, for introducing me to Monte-Carlo Go. A lot of the inspiration for the research presented in this paper c... |

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Citation Context ...tion inaccuracies. Other algorithms with better asymptotic properties (given enough time and memory, they will find an optimal action) have been proposed in the formalism of Markov decision processes =-=[12, 19, 22]-=-. This paper presents a new algorithm for combining Monte-Carlo evaluation with tree search. Its basic structure is described in Section 2. Its selectivity and backup operators are presented in the fo... |

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Citation Context ...compare the expected values of many random variables, this theorem allows to compute a probability that the expected value of one variable is larger than the expected value of another variable. Bouzy =-=[9, 7]-=- used this principle to propose progressive pruning. Progressive pruning cuts off moves whose probability of being best according to the distribution of the central-limit theorem falls below some thre... |

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Citation Context ...ccurate static position evaluator [15, 8]. When played on a 9 × 9 grid, the complexity of the game of Go, in terms of the number of legal positions, is inferior to the complexity of the game of chess =-=[2, 27]-=-, and the number of legal moves per position is similar. Nevertheless, chess-programming techniques fail to produce a player stronger than experienced humans. One reason is that tree search cannot be ... |

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Citation Context ...5] provides an introduction) are the basis for the MonteCarlo tree search algorithm of Chang, Fu and Marcus [12]. Optimal schemes for the allocation of simulations in discrete stochastic optimization =-=[13, 16, 3]-=-, could also be applied to Monte-Carlo tree search. Although they provide interesting sources of inspiration, the theoretical frameworks of n-armed bandit problems and discrete stochastic optimization... |

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Citation Context ...5] provides an introduction) are the basis for the MonteCarlo tree search algorithm of Chang, Fu and Marcus [12]. Optimal schemes for the allocation of simulations in discrete stochastic optimization =-=[13, 16, 3]-=-, could also be applied to Monte-Carlo tree search. Although they provide interesting sources of inspiration, the theoretical frameworks of n-armed bandit problems and discrete stochastic optimization... |

1 |
Computer Go tournaments on KGS. http://www.weddslist.com/ kgs/, 2005. A Random Simulations in Crazy Stone The most basic method to perform random simulations in computer-Go consists in selecting legal moves uniformly at random, with the exception of eye-f
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Citation Context ...ssible to find better methods. 5 Game Results As indicated in the abstract, Crazy Stone won the 10th KGS computer-Go tournament, ahead of 8 participants, including GNU Go, Neuro Go, Viking 5, and Aya =-=[28]-=-. This is a spectacular result, but this was only a 6-round tournament, and luck was probably one of the main factor in this victory. In order to test the strength of Crazy Stone more accurately, 100-... |