TD 2048 TD 1 N N 2048 N TD N N N N N N 2048 N 2048 TD 2048 TD TD TD 2048 TD 2048 minimax 2048, 2048, TD, N i
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1 TD Computer Players Based on TD Learning for Game 2048 and Its Two-player Variant
2 TD 2048 TD 1 N N 2048 N TD N N N N N N 2048 N 2048 TD 2048 TD TD TD 2048 TD 2048 minimax 2048, 2048, TD, N i
3 Abstract Computer Players Based on TD Learning for Game 2048 and Its Two-player Variant Kazuto Oka One of the factors that affects the strength of a computer player based on TD learning in game 2048 is N-tuple networks used for the player. But it is unclear whether the N-tuple networks used so far are adequate. In 2048, if good N-tuple networks can be used, the strength of the computer player based on TD learning using this can be improved. However, it is difficult to investigate all the performance of N- tuple networks which can be enormous. Therefore, I propose systematic selection of N- tuple networks from ordering of N-tuples using own usefulness calculated by exhaustive analysis. In addition, I consider the ordering of the obtained N-tuple and discuss the characteristics of N-tuple with high usefulness and the property of I investigate the performance of the player using N-tuple networks selected by the proposed method. Also, by applying the player based on TD learning at 2048, I make a computer player based on TD learning in two-player I improve the player s strength through TD learning in self-game-play. As the baseline for TD learning, I use a player based on TD learning at I make a stronger player by combining the player of Battle type 2048 with the minimax method and compare the strength by playing against the player using the existing method. I combine the two-player 2048 player with the minimax method. I compare the strength of the two-player 2048 player with the strength of the player using the existing technique by buttle. ii
4 key words 2048, two-player 2048, TD learning, N-tuple networks iii
5 TD N TD TD TD N N N N N N iv
6 A N 42 A A v
7 k Szubert Jaśkowski [12] Wu 4 6 [4] N [9]6 (1) N (3 ) minimax 1 minimax/expectimax vi
8 5.3 N ( m = 8) 5 (2 ) minimax 1 minimax/expectimax (3 ) minimax 1 minimax/expectimax N m = 4 5 (2 ) minimax 1 minimax/expectimax vii
9 3.1 N N bit N TD (1): (2): (3): : 5 (2 ) minimax viii
10 1 1.1 αβ A [11] G. Cirulli 1 [2] GPCC 1 [14] GPCC Games and Puzzles Competitions on Computers URL= 1
11 [1] TD [12] TD 2048 TD TD 2
12 Szubert TD [12] N N N Wu TD [4] N Szubert N N TD TD N N TD N N 3
13 TD 2048 TD 2048 TD 2048 TD 5 N N N , ,660 expectimax 12 88,000 / N TD 2048 TD TD TD TD 3 N N 4 4
14 1.4 N N N TD 6 7 5
15 % 2 10%
16 2.2 TD TD Szubert 2048 TD TD s 8 f i (s) 1 i N Szubert GPCC
17 2.2 TD n N 8 [12] m N s N W j (1 j m) V m 8 V (s) = W j (f i (s)) j=1 i= N 8
18 2.2 TD (1) (2) (3) (1) (2) (3) k TD TD s A(s) {N, E, S, W} a R(s, a) N(s, a) arg max (R(s, a) + V (N(s, a))) (2.1) a A(s) TD 2048 TD 2048 TD
19 2.2 TD 2.3 Szubert Jaśkowski [12] 2.4 Wu 4 6 [4] TD A(s) arg min (V (N(s, a))) (2.2) a A(s) TD N 2.3, 2.4 Szubert Jaśkowski 2.3 N Wu 2.4 N [4] 2.3, 2.4 N TD 1 TD TD TD
20 2.2 TD TD [3] [13] N TD TD t s t 2.1 a r = R(s t, a) N(s t, a) 3 s t 1 TD d d = V (s t) + r V (s t 1) (2.3) W j (s t 1) W j(s t 1 ) = W j(s t 1 ) + α d 8m (2.4) α m N TD 8m t s t N(s t, a) s t t 1 t r 2.3,
21 3 N 3.1 N Szubert 2.3 Wu 2.4 N N N N N 3.1 N N N N 1 N N N 1 N N 9 N Szubert Wu N 4 [12, 4] N N N N 12
22 3.2 N 3.1 N N N N TD N 3.2 N 13
23 3.2 N N N N F = 18 F = i 18 i 16 i F N F = 18 Wu F = 16 2 F = 16 N N N F = 18 N = 16 N F = = 131, 072 F = = 32,
24 3.2 N N MB 67 MB 1.1 GB 17 GB bit N 1 N N N N 1 32bit N N = 6, 7 15
25 4 N 4.1 N N N TD N N 6 1 C 6 = C 7 = 119 N p i i i s i = 1, s i = 0 P P = C i=1 s ip i. E = j (P j c i=1 s jip i ) 2 10 TD 1,000,000 10, , , %, 261% 69%, 61% 1,000,000 16
26 4.1 N partial score rank of 6-tuples partial score rank of 7-tuples N = 6, N
27 4.1 N N ,161 24,530 9,644 9, ,207 20,576 9,563 9, ,900 23,504 8,543 8, ,483 23,338 7,918 7, Wu N N N 18
28 4.1 N ,190 11, ,320 10,573 10,988 11, ,930 23,304 10, N , , 7-87, N 1 N + 1 N
29 4.2 N average score (x1000) m=40 m=20 m=16 m=10 m= 8 m= 4 m= 2 m= number of game learned (x1,000,000) average score (x1000) m=10 m= 8 m= 6 m= 4 m= 2 m= number of game learned (x1,000,000) N N m TD ?? MB 7 20
30 4.2 N max score (x1000) m=40 m=20 m=16 m=10 m= 8 m= 4 m= 2 m= number of game learned (x1,000,000) max score (x1000) m=10 m= 8 m= 6 m= 4 m= 2 m= number of game learned (x1,000,000) GB N = 6 m = GB N = 7 m = GB 2 Intel Xeon E5645 CPU GHz 12 GB PC CentOS e15 g
31 4.2 N 250 average score (x1000) N=7, m=10 N=6, m=40 N=6, m=10 N=7, m= 4 N=6, m= number of game learned (x1,000,000) max score (x1000) N=7, m=10 N=6, m=40 N=6, m=10 N=7, m= 4 N=6, m= number of game learned (x1,000,000) N 6,000,000 TD 10,000 10,000 10,000 10,000 1 N 22
32 4.2 N average score (x1000) time for a move (µs) , 4.4, , 4.6, and 4.8 N N = 6 m = 20 m = 10 N = 6 m = 40 1,000,000 N = 6 N = 7 N = 6 N N = 7 6,000,0000 N = 7 m = 10 N = N m m N = 7 m = 10 N = 6 m = 40 23
33 4.2 N 4.9 N = 6 N = 7 m = ,625 1 N 4.3 N = 6 m = 40 N = 7 m = % N = 6 m = 40 N = 7 m = ,384 N = 7 m = 10 32, GB 32 GB PC 24
34 4.2 N 4.3 N = 6 m = 1 m = 2 m = 4 m = 6 m = 8 m = 10 m = N = 6 m = 16 m = 20 m = 25 m = 30 m = 35 m = 40 m = N = 7 m = 1 m = 2 m = 4 m = 6 m = 8 m =
35 TD 2048 TD N [9] [12, 4] TD N TD N [9] 6 ( 5.1) m = Wu N N 5,000, TD 2048 m = 5 26
36 5.1 (1) (2) (3) (4) (5) (6) (7) (8) 5.1 [9]6 (1) N TD m = 1 51,506 m = 2 113,313 m = 3 138,365 m = 4 169,649 m = 5 214,526 m = 6 212,384 m = 7 222,639 m = 8 230,513 m = 4 (Wu ) 178, N N 27
37 5.1 TD m = (9 ) m = 1 8 (11 ) 0% 25% 50% 75% (4 ) N , 5.3, m =
38 (1): 6 m = m = m = m = m = m = m = m = m = 4 (Wu ) % N 50% (2 90% 4 10% ) 29
39 (2): m = 4 (Wu ) m = m = m = m = m = m = m = m = m = 1,..., m = 8 50% (1 ) m = (9 ) 0%, 25%, 50%, 75% (4 ) N 0 5,000, % N 50% 30
40 (3): 00% % % % : 5 (2 ) minimax 1 m = 8 m = 4 00% % % % minimax expectimax 31
41 ¹ l q± l w N n m=2 m=4 m=8 minimax expectimax 7ÁÝûÛfçï NMey[QV}Š= (3 ) minimax 1 minimax/expectimax ¹ l q± l w N n m=2 m=4 m=8 minimax expectimax 7ÁÝûÛfçï NMey[QV}Š= 5.3 N ( m = 8) 5 (2 ) minimax 1 minimax/expectimax ,000 32
42 % minimax expectimax d 1, 3, 5, 7 0 1, 2, 3 expectimax expectimax d 1, 3, 5, 7 0 1, 2, 3 N N d = 1, 3, 5 (0, 1, 2 ) minimax m 2, 4, (3 ) minimax m = 8 N 5 (2 ) minimax 5.3 N minimax expectimax minimax N m minimax minimax expectimax minimax ( ) 33
43 ,000 minimax minimax d 1, 3, 5, 7 0 1, 2, 3 expectimax expectimax d 1, 3, 5, 7 0 1, 2, 3 1 N N d = 1, 3, 5 (0, 1, 2 ) minimax m 2, 4, 8 3 N 5.2 N d = 1, 3, 5 (0, 1, 2 ) minimax m 2, 4, (3 ) minimax m = 4 N 5 (2 ) minimax 5.3 N minimax 1 (m = 8) 34
44 ¹ l q± l w m=2 m=4 m=8 m=2 m=4 m=8 N n minimax expectimax 1Ç@ŠnÝûÛfçï NMey[QV}Š= PSÝûÌÛfçï NMey[QV}Š= (3 ) minimax 1 minimax/expectimax ¹ l q± l w m=2 m=4 m=8 m=2 m=4 m=8 N n minimax expectimax 1Ç@ŠnÝûÛfçï NMey[QV}Š= PSÝûÌÛfçï NMey[QV}Š= 5.5 N m = 4 5 (2 ) minimax 1 minimax/expectimax 35
45 [1, 15, 4, 12, 8] expectimax TD 2048 TD Szubert Jaśkowski [12] ,000,000 TD 100, Wu ,727 Wu expectimax 328,946 TD expectimax = 5 expectimax [10, 16] 1 N = 7 m = GB
46 7 [8, 6, 5, 7] N TD N N N N N N N N N N N N , ,660 N N 37
47 N TD 2048 N 2048 N [9] N 1 3 N TD N minimax minimax N minimax N minimax [8] 38
48 2 IACP 39
49 [1] Ahmad Zaky, Minimax and Expectimax Algorithm to Solve 2048, http: //informatika.stei.itb.ac.id/~rinaldi.munir/stmik/ genap/ Makalah2014/MakalahIF pdf, [2] Gabriele Cirulli, 2048, [3] Gerald Tesauro, TD-Gammon, a self-teaching backgammon program, achieves master-level play, Neural computation, Vol. 6, No. 2, pp , [4] I-Chen Wu, Kun-Hao Yeh, Chao-Chin Liang, Chia-Chuan Chang and Han Chiang, Multi-Stage Temporal Difference Learning for 2048, Technologies and Applications of Artificial Intelligence, Lecture Notes in Computer Science, Vol. 8916, pp , [5] Kazuto Oka, Kiminori Matsuzaki, Systematic Selection of N-Tuple Networks for 2048, CG 2016, pp , [6],, , pp. 9 18, [7],, N-tuple networks 2048, 58, [8], 2048 Vol. 12, No. 1, pp , [9] Kiminori Matsuzaki, Systematic Selection of N-tuple Networks with Consideration of Interinfluence for Game 2048, Proceedings of the 2016 Conference on Technologies and Applications of Artificial Intelligence (TAAI 2016), [10] Kun-Hao Yeh, Chao-Chin Liang, Kun-Hao Yeh and I-Chen Wu, 2048-bot tournament in Taiwan, 40
50 04/2048-bot-tournament-report-1104.pdf, [11],, bit,, [12] Marcin Szubert and Wojciech Jaśkowski, Temporal Difference Learning of N-Tuple Networks for the Game 2048, 2014 IEEE Conference on Computational Intelligence and Games, pp. 1 8, [13] Michiel van der Ree and Marco Wiering, Reinforcement learning in the game of Othello: Learning against a fixed opponent and learning from self-play, IEEE Symposium on Adaptive Dynamic Programming And Reinforcement Learning (AD- PRL), pp , [14], 2048, [2015], pp , [15] Philip Rodgers and John Levine, An Investigation into 2048 AI Strategies, 2014 IEEE Conference on Computational Intelligence and Games, pp. 1 2, [16] Wojciech Jaśkowski and Marcin Szubert, Game 2048 AI controller competition GECCO 2015, GECCO-2048-Competition/GECCO Competition-Results.pdf,
51 A N
52 A A No. Tuple Score 1 [ 0, 1, 2, 3, 4, 5] 31,161 2 [ 0, 1, 2, 3, 4, 6] 24,530 3 [ 0, 1, 2, 4, 6, 7] 22,207 4 [ 0, 1, 2, 4, 5, 6] 20,576 5 [ 0, 1, 4, 5, 6, 7] 20,512 6 [ 0, 1, 5, 8, 9, 13] 20,283 7 [ 0, 1, 2, 5, 6, 7] 19,970 8 [ 0, 1, 2, 3, 4, 8] 18,990 9 [ 0, 1, 3, 5, 6, 7] 18, [ 0, 1, 2, 6, 7, 10] 16, [ 0, 1, 2, 3, 5, 6] 16, [ 1, 2, 4, 5, 6, 7] 16, [ 0, 1, 2, 3, 4, 7] 15, [ 0, 1, 2, 6, 7, 11] 15, [ 0, 1, 2, 6, 10, 14] 15, [ 0, 1, 5, 9, 13, 14] 15,718 No. Tuple Score 68 [ 1, 2, 5, 6, 8, 9] 9, [ 1, 2, 5, 9, 13, 14] 9, [ 1, 4, 5, 6, 7, 10] 9, [ 1, 5, 6, 7, 9, 10] 9, [ 1, 4, 5, 6, 10, 11] 9, [ 1, 5, 6, 9, 10, 14] 9, [ 1, 5, 6, 9, 10, 11] 10, [ 1, 5, 6, 7, 8, 9] 10, [ 1, 2, 5, 9, 10, 13] 10, [ 1, 2, 5, 9, 10, 11] 10, [ 1, 5, 6, 9, 10, 13] 10, [ 1, 4, 5, 6, 9, 10] 10, [ 0, 1, 5, 6, 9, 10] 10, [ 0, 1, 2, 5, 6, 9] 10, [ 0, 1, 2, 5, 9, 10] 10, [ 0, 1, 5, 6, 8, 9] 10,936 43
53 A A No. Tuple Score 1 [ 0, 1, 2, 3, 4, 5, 6] 32,900 2 [ 0, 1, 2, 3, 4, 5, 9] 23,504 3 [ 0, 1, 2, 4, 5, 6, 7] 23,483 4 [ 0, 1, 3, 4, 5, 6, 7] 23,338 5 [ 0, 1, 2, 3, 4, 5, 7] 23,304 6 [ 0, 1, 2, 3, 4, 6, 10] 21,954 7 [ 0, 1, 2, 3, 4, 8, 9] 21,766 8 [ 0, 1, 2, 4, 5, 6, 9] 20,591 9 [ 0, 1, 2, 4, 5, 6, 10] 20, [ 0, 1, 3, 5, 6, 7, 9] 17, [ 0, 1, 5, 6, 8, 9, 13] 17, [ 0, 1, 2, 3, 4, 5, 8] 17, [ 0, 1, 2, 5, 6, 7, 9] 17, [ 0, 1, 2, 4, 6, 7, 10] 17, [ 0, 1, 5, 8, 9, 13, 14] 16, [ 0, 1, 4, 5, 6, 9, 10] 16,590 No. Tuple Score 119 [ 0, 1, 2, 6, 10, 11, 15] 7, [ 0, 1, 2, 5, 9, 13, 14] 7, [ 1, 2, 4, 5, 9, 13, 14] 8, [ 1, 4, 5, 6, 9, 10, 11] 8, [ 1, 4, 5, 6, 10, 11, 14] 8, [ 0, 1, 5, 7, 9, 10, 11] 9, [ 1, 2, 5, 9, 10, 11, 14] 9, [ 1, 2, 5, 6, 8, 9, 13] 9, [ 1, 4, 5, 6, 7, 9, 13] 9, [ 0, 1, 5, 6, 10, 11, 14] 9, [ 1, 5, 6, 7, 9, 10, 13] 9, [ 1, 4, 5, 6, 7, 10, 14] 9, [ 0, 1, 5, 6, 7, 9, 11] 9, [ 0, 1, 5, 9, 10, 11, 14] 9, [ 1, 2, 5, 8, 9, 10, 11] 9, [ 0, 1, 2, 6, 10, 11, 14] 9,986 44
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22 Web Research of a Web search support system based on individual sensitivity 1135117 2011 2 14 Web Web Web Web Web, i Abstract Research of a Web search support system based on individual sensitivity
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Vol.2, No.x, April 2015, pp.xx-xx ISSN xxxx-xxxx 2015 4 30 2015 5 25 253-8550 1100 Tel 0467-53-2111( ) Fax 0467-54-3734 http://www.bunkyo.ac.jp/faculty/business/ 1 [1, 2, 3, 4, 5, 8, 9, 10, 12, 15] The
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Kochi University of Technology Aca Title jjencode で難読化された JavaScript の検知 Author(s) 中村, 弘亮 Citation Date of 2018-03 issue URL http://hdl.handle.net/10173/1975 Rights Text version author Kochi, JAPAN http://kutarr.lib.kochi-tech.ac.jp/dspa
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21 Fault-Toleranted Authentication Data Distribution Protocol for Autonomous Distributed Networks 1125153 2010 3 2 i Abstract Fault-Toleranted Authentication Data Distribution Protocol for Autonomous Distributed
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1,a) 1,b) 1/f β Generation Method of Animation from Pictures with Natural Flicker Abstract: Some methods to create animation automatically from one picture have been proposed. There is a method that gives
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CHLAC 1 2 3 3,. (CHLAC), 1).,.,, CHLAC,.,. Suspicious Behavior Detection based on CHLAC Method Hideaki Imanishi, 1 Toyohiro Hayashi, 2 Shuichi Enokida 3 and Toshiaki Ejima 3 We have proposed a method for
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14 Application of Automatic Text Summarization for Question Answering System 1030260 2003 2 12 Prassie Posum Prassie Prassie i Abstract Application of Automatic Text Summarization for Question Answering
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28 Confined Decoding System for Medical Data Distributed by Secret Sharing Scheme and Its Security Evaluation 1195046 2017 3 6 DMAT i Abstract Confined Decoding System for Medical Data Distributed by Secret
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19 Estimation of Sound Source Zone using The Arrival Time Interval 1080351 2008 3 7 S/N 2 2 2 i Abstract Estimation of Sound Source Zone using The Arrival Time Interval Koichiro Kanai The microphone array
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