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1 Root 08M
2 Root Tree Fuego Root Tree Root Root 2 Fuego Root CPU Root 64CPU Chaslot Root Root 1
3 RAVE Leaf Tree Root Root Fuego ( ) Root Root Tree
4 4.6 Root Tree Root Root Root
5 MiniMax Leaf Tree Root Root fuego fuego [5] Bonanza :Root ( 4 64 ) vs Fuego :Root ( 4 64 RAVE ) vs Fuego(RAVE ) :Root ( 4 64 ) vs Mogo :Root ( 4 64 ) vs Fuego :Root ( 4 64 ) vs Mogo :Root ( 64 )vstree (1 8 ) :Root ( 64 )vstree (1 8 ) F tree ( ) F tree ( ) :F80s : Fuego,Root ( ),Tree :F80s : Fuego,Root ( ),Tree :F80s : Root ( :F80s : Root ( ) (0 0.2) ( ) ( ) ( ) ( )
![2 fuego [5]...................... 27 3.3 Bonanza............ 29 3.4.............................. 31 3.5.................................. 32 4.1 9 :Root ( 4 64 ) vs Fuego....... 35 4.](/docs-images/42/23098801/images/page_5.jpg "2 9 :Root ( 4 64 RAVE ) vs Fuego(RAVE )...................................... 36 4.3 9 :Root ( 4 64 ) vs Mogo....... 36 4.4 19 :Root ( 4 64 ) vs Fuego...... 37 4.5 19 :Root ( 4 64 ) vs Mogo...... 38 4.")
6 (0 0.2) ( ) ( ) ( ) (0,8 1.0)
7 2.1 ( ) Single-Run Multiple-Runs GnuGo3.6 (T.Cazenave et al. 2006) Root Tree GnuGo (G.Chaslot et al. 2008) Root Tree GnuGo (G.Chaslot et al. 2008) (%)(8 8Root ) (%)(8 8Root ) Root 64 Root vs (%) Root 64 Root vs (%) ( 3.4% (187/5500 )) ( 2.3% (159/6759 )) ( 8.7% (2277/26064 )) ( 4.9% (1314/27032 )) ( 5 ) ( 5 )
8 [12] 1996 [17] 1997 IBM Deep Blue Kasparov [11] ,4 70 ( ) GnuGo ( ) 7
![Kasparov [11] 2009 4 19](/docs-images/42/23098801/images/page_8.jpg "3,4 70 ( ) 19 10 171")
9 1.2 CPU ( ) Tree [6, 7] CPU Root [1] CPU Tree Tree [7, 5] Chaslot Root Tree [2] [2] Root Root Root Tree Root CPU Chaslot Root [2] (2 ) Chaslot RAVE[22] Chaslot 16CPU CPU Root Root Tree Root 64 CPU Fuego Fuego RAVE 8
![3 3 Tree Root CPU Chaslot Root [2] (2 ) Chaslot](/docs-images/42/23098801/images/page_9.jpg "RAVE[22] Chaslot 16CPU CPU Root Root Tree Root 64 CPU")
10 Tree Chaslot Root [25] Root CPU Root Root Root Root : 3 : 4 : 5 : 9
11 2 2.1 UCT ( ) ( 3 3 ( 2.1)) ( ) 2.1:
12 : ( ) (9 ) (19 ) minimax minimax ( ) Max Min ( 2.2) 2.2 (Mini ) ( 2.2-1) (Max ) (Mini ) ( 2.2-2) (Mini ) (Max ) ( 2.2-3) Max ( ) (Mini ) ( 2.2-4) 55 A [21] 11
13 Minimax Minimax [4] C 65 D D 65 (Max ) 65 (Mini ) : MiniMax 4 [10] minimax GunGo[20].GnuGo GnuGo 5 2,3 2 / / 2 12
14 / 1. 2., minimax A B B A ( i s i X i 1 0 X i X i = X i (2.1) s i X i i ( )
15 2.3: 14
16 2.1.3 ( ) [23]. ( A ) ( B ) A B B ( ) UCT UCB1 [16] Auer 15
17 UCB1 [19] UCB1 ( ) UCB1 UCB(i) = X i + c 2log(n) n i (2.2) 2.2 UCB1 X j j n j j n ( )+( ) c c c = 2 UCB UCT(UCB applied to Trees) L.Koscis UCB1 UCT(UCB applied to Trees) [15] UCT UCT 2.4 UCT,UCB1 UCB1 UCB1 UCB UCB1 ( ) UCB1 (UCB1 ) 2. ( ) 16
![Trees) L.Koscis UCB1 UCT(UCB applied to Trees) [15] UCT UCT 2.](/docs-images/42/23098801/images/page_17.jpg "4 UCT,UCB1 UCB1 UCB1 UCB1 2.5 1. UCB1 ( ) UCB1 (UCB1 ) 2.")
18 UCB1 3 ( ) UCB1 1 4 ( ) 2.4: 17
19 2.5: 2.2 UCT MoGo CrazyStone[3, 9] UCT MoGo 3 3 [9] RAVE UCT-RAVE(Rapid Action Value Estimate)[8] RAVE i i 18
![[9] 3 3 2.](/docs-images/42/23098801/images/page_19.jpg "2.2 RAVE UCT-RAVE(Rapid Action")
20 RAVE X E UCT X uct UCT-RAVE X value X value = βx RAV E + (1 β)x uct (2.3) k β = (2.4) 3s + k s k 2.3 CPU ( ) 19
21 Leaf Leaf ( 2.6) Leaf [1, 2] ( ) UCT ( ) p1 p4 2.6: Leaf Leaf 20
22 Leaf Leaf [13] Tree Tree ( 2.7) Tree ( ) UCB [9] Tree CPU Tree [6, 7]. 2.7: Tree Tree UCB Chaslot [2] 21
23 4 Enzenberger Tree C++ volatile IA-32 Intel-64 CPU [6] UCB, 9 2,19 3, PC PC Gelly [7] Tree Chaslot (Virtual Loss) [2] Virtual Loss ( ) Chaslot Virtual Loss Root Root ( 2.8) Root [1] ( ) (p1 p4) Root Root. Cazenave Chaslot [1, 2], Cazaneve 22
24 2.8: Root Chaslot 3. UCT A,B,C 3 B Root Root Root 3 23
25 2.9: Root Cazenave Root [1] Root Zen Zen Root [14] Zen Cazenave [1] Single-Run(Root) At-the-leaves(Leaf) Multiple- Runs( ) At-the-leaves Multiple-Runs 16 Multiple-Runs Single-Run 16 ( 2.2) 2.2: Single-Run Multiple-Runs GnuGo3.6 (T.Cazenave et al. 2006) 1CPU 2CPUs 4CPUs 8CPUs 16CPUs Single-Run(Root) Multiple-Runs Chaslot 16CPU Leaf Root Tree 24
26 [2] 2.3: 13 1 Root Tree GnuGo (G.Chaslot et al. 2008) 1CPU 2CPUs 4CPUs 16CPUs Leaf Root Tree : 9 Root Tree GnuGo (G.Chaslot et al. 2008) (s) Root Tree Root Tree ( 2.3) 9 Root Tree ( 2.4) [2] 13 Root Tree 9 13 Leaf Root Tree 25
27 3 Root Root 3.1 Fuego 3.2 Root 3.3 Root Fuego ( ) fuego Fuego[5] 3.1: fuego Computer Olympiad [18] 1 Fuego GtpEngine GTP(Go Text Protocol) 1 C++ 26
28 3.2: fuego [5] 27
29 SmartGame Go Minimax GoUct UCT Fuego RAVE UCT MoGo [9] Fuego [6], MPI 2 Tree Virtual Loss 3.2 Root 2.9 B B [25] ( ). 2 28
30 3.3: Bonanza 29
31 1. 2. Bonanza[?] 1 [26] 2 Bonanza [24] Root ( 3.4). Root 1 ( ) B A A Root ( 3.5) Fuego0.3.2 MPI 1 (p1) (pe) MPI 2 ( ) p1 3 p1 4 p1 pe GoGui-twogtp 30
32 1. 5 pe p1 GoGui-twogtp 3.4: 2.9 B 4 A 2 B 1 C A Root 3.4 Root 2.9 B 2 31
33 3.5: 32
34 Fuego version PC 8 16GB CPU CPU Xeon E GHz 2 MPI MPICH2 1 C++ Mogo release , 19 ( ) Fuego Root MPI Fuego PC 64 Root Root Root Tree LockFree Fuego 16 [6]. PC 8 Tree F t ree Tree
35 Fuego 30 F80s Fuego Root Root Tree Root Tree Root 34
36 4.4 Root 4.1: 9 :Root ( 4 64 ) vs Fuego CPU Root ( ) Fuego Fuego Root Fuego Root 19 Root MoGo 4.3,4.5 CPU Fuego 9 MoGo CPU CPU Root Chaslot RAVE Root Chaslot Root ( ) 35
37 4.2: 9 :Root ( 4 64 RAVE ) vs Fuego(RAVE ) 4.3: 9 :Root ( 4 64 ) vs Mogo 36
38 4.2 Root Fuego RAVE 9 RAVE Chaslot Root RAVE Fuego 4.4: 19 :Root ( 4 64 ) vs Fuego 4.5 Root Tree Root Fuego Tree Root (64 ) Tree Virtual Loss 4.6, 4.7 9, 19 Tree Root Chaslot 64 Root 9 6 Tree Tree Enzenberger Fuego Tree [6] Martin Müller 16CPU Fuego 8 37
39 4.5: 19 :Root ( 4 64 ) vs Mogo Tree UCB1 (2 ) [6] 4.6 Root Tree Root Tree Tree Root 8 Tree Fuego F tree F tree 8 Root ( 8 8 Root ) 4.6,4.6 19,9 8 8 Root F tree MoGo F tree Fuego CPU 8 8 Root 67.0 ( 4.6 ) MoGo Root 77% Root Fuego Root
40 4.6: 9 :Root ( 64 )vstree (1 8 ) 4.7: 19 :Root ( 64 )vstree (1 8 ) 4.1: 9 (%)(8 8Root ) 8 8 Root vs F tree vs Mogo
41 4.2: 19 (%)(8 8Root ) 8 8 Root vs F tree vs Mogo Root 4.7, Root 64 Root Root Root 7 4.3: 9 8 8Root 64 Root vs (%) 8 8 Root Root : Root 64 Root vs (%) 8 8 Root Root
42 4.8 Root 9, 19 Fuego 64 Root (64 Root ) Fuego 200 Root 8 8 Root 8 8 Root F tree (4.6 ) Root 8 8 Root 97 ( 4.5,4.6 ) ( 4.7,4.8 ) (30 ) F tree F tree F tree F tree F tree F tree F tree (( )-1 (F tree 1) (4.1) F tree A Root A 1 Root A F tree Root 9 MoGo , Root Root F tree F tree
43 4.8 A7 A4 (F tree ) A7 A A4 CPU 64 9 A4 A A7 4.9 D1 E8 F tree D1 E E8 E8 CPU 16 D : ( 3.4% (187/5500 )) Root F tree : ( 2.3% (159/6759 )) Root F tree : ( 8.7% (2277/26064 )) Root F tree
44 4.8: ( 4.9% (1314/27032 )) Root F tree A B C D E F G H J A B C D E F G H J 4.8: F tree ( ) A B C D E F G H J A B C D E F G H J 4.9: F tree ( ) 43
45 4.9: 4.8 ( 5 ) 1 A4 29,416 2 H5 25,984 3 A7 25,917 4 H4 20,460 5 H6 2,618 1 A7 25, H4 20, H5 25, A4 29, H6 2, : 4.9 ( 5 ) 1 E8 24,637 2 B2 23,276 3 D1 20,910 4 E9 10,894 5 B1 7,535 1 D1 20, B2 23, E8 24, E9 10, B1 7, Root Root 80 Fuego ( F80s ) F80s 9, ( Fuego Mogo( 10 ) ) F80s ( ) A A = F80s A (4.2) F80s 1. Fuego( 1 64 ) 2. Root Fuego( ) 3. Root Fuego( ) 4.Tree Fuego( ) 44
46 4.10,4.11 Root Tree 4 64 Root Tree : 9 :F80s : Fuego,Root ( ),Tree 45
47 4.11: 19 :F80s : Fuego,Root ( ),Tree Root 19 Root 512 ( Root Root Root 19 Root F80s ( 4.13)
48 4.12: 19 :F80s : Root ( : 19 :F80s : Root ( ) 47
49 4.9.4 (2 ) Root = (4.3) F80s , , 19 (0.6 ) Root ( , ) 0.8 ( 8 ) Root 9, 19 ( 0.4) ( , ) :
50 4.14: 9 (0 0.2) 4.15: 9 ( ) 49
51 4.16: 9 ( ) 4.17: 9 ( ) 50
52 4.18: 9 ( ) 4.19: 19 (0 0.2) 51
53 4.20: 19 ( ) 4.21: 19 ( ) 52
54 4.22: 19 ( ) 4.23: 19 (0,8 1.0) 53
55 4.9.5 Root A n L (A) = [ x L ] (4.4) [30/100] = 0.3 9, , K K K Tree Root ( 1 ) 4.24: 9 54
56 4.25: 19 55
57 5 5.1 Fuego 3 Tree Root 2 Chaslot Tree 9 64 Root Tree Tree 4 64 Root 19 Tree 8 64 Root (4.9.4) Root Tree Root 10 1 Fuego Root Tree 64 8 Tree 1 8 Root ( 8 8 ) Root Tree 8 8 Root (4.6 ) Root 9 19 ( ) 64 Root 8 8 (4.6 ) 56
58 F tree F tree 8 Tree Fuego F tree 30 F tree F tree Root CPU Root Root Fuego Mogo Root 4 64 (4.4 ) Root (4.9.3 ) Root Root ( 4.19) Root Root 57
59 5.2 1 Fuego 10 Root Root 4.13 Root Root Root 58
60 59
61 5.1: 19 60
62 5.2: : 19 61
63 5.4: : 19 62
64 5.6: 9 5.7: 9 63
65 5.8: 9 5.9: 9 64
66 5.10: 9 65
67 [1] T. Cazenave and N. Jouandeau. On the parallelization of UCT. In Proceedings of the Computer Games Workshop, pp , [2] G. M. J-B. Chaslot, M. H.M Winands, and H. J. van den Herik. Parallel Monte- Carlo tree search. In Proceedings of the 6th International Conference on Computer and Games, Vol of Lecture Notes in Computer Science, pp , [3] R. Coulom. Computing Elo ratings of move patterns in the game of Go. ICGA Journal, Vol. 30, No. 4, pp , [4] D.E.Knuth and R.W.Moore. An analysis of alpha-beta pruning. In Artificial Intelligence, pp , [5] M. Enzenberger and M. Müller. Fuego - an open-source framework for board games and Go engine based on Monte-Carlo tree search. TR 09-08, University of Alberta, [6] M. Enzenberger and M. Müller. A lock-free multithreaded Monte-Carlo tree search algorithm. In Advances in Computer Games 12, [7] S. Gelly, J. B. Hoock, A. Rimmel, O. Teytaud, and Y. Kalemkarian. The parallelization of Monte-Carlo planning. In Proceedings of the 5th International Conference on Informatics in Control, Automation, and Robotics, pp , [8] S. Gelly and D. Silver. Combining online and offline knowledge in UCT. In Proceedings of the 24th International Conference on Machine Learning, pp , [9] S. Gelly, Y. Wang, Remi Munos, and O. Teytaud. Modification of UCT with patterns in Monte-Carlo Go. Technical Report 6062, INRIA, [10] GPSshogi [11] J.Schaeffer and A.Plaat. Kasparov versus deep blue: The re-match. In International Computer Chess Association Journal, pp , [12] J.Scheaffer. One jump ahead: Challenging human supremacy in chechkers. In Springer-Verlag, [13] H. Kato and I. Takeuchi. Parallel Monte-Carlo tree search with simulation servers. In Proceedings of the 13th Game Programming Workshop, pp ,
68 [14] H. Kato and I. Takeuchi. Zen. In Proceedings of the 14th Game Programming Workshop, pp , [15] Kocsis.L and Szepesvari.C. Bandit based monte-carlo planning. In 17th European Conference on Machine Learning(ECML 2006), pp , [16] Lai.T.L and Robbins.H. Asymptotically efficient adaptive allocation rules. In Advances in Applied Mathematics, pp. 4 22, [17] M.Buro. The othello match of the year: Takeshi murakami vs logistello. In International Computer Chess Association Journal, pp , [18] M. Müller. Fuego at the Computer Olympiad in Pamplona 2009: a tournament report. TR 09-09, University of Alberta, [19] P.Auer and N.Cesa-Bianchi.P.Fischer. Finite-time analysis of the multiarmed bandit problem machine learning. pp , [20] GNU Go GNU Project [21] Jonathan Schaeffer, Neil Burch, Yngvi Bj?rnsson, Akihiro Kishimoto, Martin M?ller, Robert Lake, Paul Lu, and Steve Sutphen. Checker is solved. In Science 14 September 2007, pp , [22] S.Gelly.Y.Wang, R.Mumos, and O.Teytaud. Modification of uct with patterns in monte-carlo go. In RR-6062-INRIA, pp. 1 19, [23] Yoshimoto.H, Yoshizoe.K, Kaneko.T, Kishimoto.A, and Taura.K. Monte carlo go has a way to go. In Twenty-First National Conference on Artificial Intelligence(AAAI-06), pp , [24],,,. :? In Proceedings of the 14th Game Programming Workshop, pp , [25],, , [26],,,,. -. In Proceedings of the 14th Game Programming Workshop, pp ,
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