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1 TD(λ) Ms. Pac-Man AI 1,a) Ms. Pac-Man AI Ms. Pac-Man UCT (Upper Confidence Bounds applied to Trees) TD(λ) UCT UCT Progressive bias Progressive bias UCT UCT Ms. Pac-Man UCT Progressive bias TD(λ) Improvement of Monte Carlo Tree Search AI using TD(λ) in Ms. Pac-Man Hiuchi Akihiko 1,a) Miwa Makoto 2 Tsuruoka Yoshimasa 3 Chikayama Takashi 3 Abstract: Due to its simplicity of the rules and high degree of difficulty, Ms. Pac-Man has become an attractive subject of research on digital game AI. In Ms. Mac-Man, Monte Carlo Tree Search-based, especially UCT-based players tend to give better performance than rule-based systems. In this study, we propose the improvement of Monte-Carlo tree search using the evaluation function that have been learned in TD (λ). At the use of the evaluation function in Monte Carlo tree search, we adopt the Progressive bias. Progressive bias makes the selection follow the evaluate function in opening phase of Monte-Carlo Tree Search and increase the search of favorable states. Experimental results obtained on a simulator show that the proposed method gives better scores than an existing UCT-based method. Keywords: Ms. Pac-Man, Monte-Carlo Tree Search, UCT, Progressive bias, TD(λ) 1. AI 1 Engineering Department, The University of Tokyo 2 School of Computer Science, University of Manchester 3 Graduate School of Engineering, The University of Tokyo a) hiuchi@logos.t.u-tokyo.ac.jp AI AI Ms. Pac-Man Ms. Pac-Man AI Ms. Pac-Man Pac-Man AI 1

2 [1] AI [2] Pac-Man Ms. Pac-Man Ms. Pac-Man Pac-Man Ms. Pac-Man IEEE AI Ms. Pac-Man AI [3] AI 2011 UCT[4] [5] 58,990 Ms. Pac-Man AI Ms. Pac-Man 921,360 AI Ms. Pac- Man AI TD(λ) Progressive bias Ms. Pac-Man 2. Ms. Pac-Man Pac-Man 1980 ( ) Pac-Man Ms. Pac- Man Ms. Pac-Man 1981 Pac-Man Pac-Man Pac-Man = 1 Ms. Pac-Man Ms. Pac-Man Pac-Man 1 Ms. Pac-Man Pac-Man Ms. Pac-Man 2.1 Ms. Pac-Man ( Blinky Pinky Inky Sue) A A A B B C C D D A A 3 10, ( 1 ) ( 2 ) 2

3 , ( 3 ) 1 7 (100 ) (200 ) (500 ) (700 ) (1,000 ) (2,000 ) (5,000 ) Ms. Pac-Man 3. Ms. Pac-Man Ms. Pac-Man Ms. Pac-Man 40,000 Ms. Pac-Man 15,000 [6] AI 3.1 UCT UCT UCT 2 UCT 2006 Kocsis [4] UCT UCT 2 ( 1 ) (1) Upper Confidence Bounds (UCB) ln T X i + C (1) T i X i i T i i T C X i ( 2 ) ( 3 ) Ms. Pac-man ( 4 ) UCT 3

4 3.1.2 Progressive bias Chaslot [7] UCT Progressive Strategies Progressive bias UCT UCB (2) ln T X i + C + f(t i ) (2) T i Guilllaume MANGO H i f(t i ) = Hi T i+1 Progressive bias UCT UCT Ms. Pac-Man Robles Lucas [6] AI Ms. Pac-Man 15,640 Ikehata Ito [5] Ms. Pac-Man UCT UCT Ikehata Ms. Pac-Man 58, V (s) 1 : s e(s) = 0 s : a s π a r s δ r + γv (s ) V (s) e(s) e(s) + 1 s : V (s) V (s) + αδe(s) e(s) γλe(s) s s s 3 TD(λ) ϵ-greedy ϵgreedy ϵ TD(λ) TD(λ) t s e t (s) (3) { γλet 1 (s) (s s t ) e t (s) = γλe t 1 (s) + 1 (s = s t ) (3) γ (0 γ 1) γ λ (0 λ 1) (4) 1 TD δ t = r t+1 + γv t (s t+1 ) V t (s t ) (4) V t (s t ) t s (3) (4) (5) V t (s) = αδ t e t (s) (5) α(0 α 1) 3 TD(λ) π

5 3.2.4 s V (s) s (6) V (s) = θ ϕ(s) (6) s ϕ(s) θ Ms. Pac-Man (7) V (s) = e aθ ϕ(s) (7) a 0 V (s) TD TD(λ) 4 θ V (s) (8) θ V (s) = ϕ(s) (8) θ V (s) (9) θ V (s) = V (s)(1 V (s))ϕ(s) (9) Ms. Pac-Man Ms. Pac-Man θ 1 : e = 0 s : a s π a r s δ r + γv (s ) V (s) e γλe + θ V (s) θ θ + αδe e(s) γλe(s) s s s 4 TD(λ) Lucas[8] (µ,λ)- Lucas 13 Ms. Pac-Man Ms. Pac-Man Pac-Man AI Koza [9] Pac-Man 5000 Pac-Man AI Ms. Pac-Man Szita Lorincz [10] Ms. Pac-Man AI [11] AI 8186 Matsumoto [12] ( ) Matsumoto 2009 Ms. Pac-Man CIG 30, Ms. Pac-Man 5

6 1 1-1/ Blinky 2-1/ Inky 3-1/ Sue 4-1/ Pinky 5 1/ Blinky 6 1/ Inky 7 1/ Sue 8 1/ Pinky 9 5 TD(λ) Progressive bias Ms. Pac-Man 4.1 TD(λ) 4 TD(λ) 1F () 1F 1F ϵ-greedy ϵ-greedy / Lucus [8] 4.2 UCT F Progressive bias Progressive bias TD(λ) (10) a, b e a(θ ϕ(si) b) (10) Progressive bias (11) ln T 1 D X i + C + (11) T i 1 + e a(θ ϕ(si) b) T i + 1 X i, T, T i, C θ, ϕ(s i ) s i D C UCT Ms. Pac-Man AI C# 6

7 6 7 CPU Core i5 2.5Hz 8GB 5.2 Ms. Pac-Man Flensbank Yannakakis [13] 5.3 TD(λ) α (11) α = α 0 N N 0 + Games (12) Geramifard [14] α 0 N 0 α 0 = 0.1 N 0 = 10, 000 Games γ 1 λ 0.9 ϵ 0.03 Progressive bias C = 2 D = a, b a = 0.2 b = 14.6 UCT TD(λ) TD(λ) Sue Sue 1 1,00 1, , Ms. Pac- Man Progressive bias TD(λ) UCT 100 UCT 7 2 UCT 7

8 8 2 UCT Max Min Mean UCT Progressive bias TD(λ) Progressive bias Ms. Pac-Man Ms. Pac-Man Progressive bias TD(λ) TD(λ) UCT TD(λ) Progressive bias Ms. Pac-Man [1] S. Karakovskiy and J. J. Togelius, The mario ai benchmark and competitions, IEEE Transactions on Computational Intelligence and AI in Games, No. 99, pp. 1 1, [2] S. Bojarski and C. B. Congdon, Realm: A rule-based evolutionary computation agent that learns to play mario, 2010 IEEE Symposium on Computational Intelligence and Games, pp , [3] Ms. Pac-man Competition(screen-capture version), PacManContest.html. [4] L. Kocsis and C. Szepesvári, Bandit based monte-carlo planning, Machine Learning: ECML 2006, pp , [5] N. Ikehata and T. Ito, Monte-carlo tree search in ms. pac-man, Computational Intelligence and Games, 2011 IEEE Conference, pp , [6] D. Robles and S. M. Lucas. A simple tree search method for playing ms. pac-man, Computational Intelligence and Games, 2009 IEEE Symposium, pp , IEEE, [7] G. Chaslot, M. Winands, H. Hherik, J. Uiterwijk and B. Bouzy Progressive strategies for Monte-Carlo tree search, New Mathematics and Natural Computation, Vol. 4, No. 3, pp. 343, [8] S. Lucas, Evolving a neural network location evaluator to play ms. pac-man, IEEE Symposium on Computational Intelligence and Games pp , 2005, [9] J. R. Koza, Genetic programming: On the programming of computers by means of natural selection (complex adaptive systems), A Bradford book, Vol. 1, [10] I. Szita and A. Lõrincz. Learning to play using lowcomplexity rule-based policies: Illustrations through ms. pac-man, Journal of Artificial Intelligence Research, Vol. 30, No. 1, pp , [11] R. Rubinstein. The cross-entropy method for combinatorial and continuous optimization, Methodology and computing in applied probability, Vol. 1, No. 2, pp , [12] H. Matsumoto, T. Ashida, Y. Ozasa, T. Maruyama, and R. Thawonmas. Ice pambush 3, Controller description paper, Vol. 203, [13] Ms. Pacman AI NET, com. [14] A. Geramifard, M. Bowling, M. Zinkevich, and R. Sutton, ilstd: Eligibility traces and convergence analysis, Advances in Neural Information Processing Systems 19: Proceedings of the 2006 Conference, pp MIT Press, Ms. Pac-Man 8

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