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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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
17 Multiple video streams control for the synchronous delivery and playback 1085404 2006 3 10 Web IP 1 1 1 3,,, i Abstract Multiple video streams control for the synchronous delivery and playback Yoshiyuki
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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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14 A Method of Article Retrieval Utilizing Characteristics in Newspaper Articles 1055104 2003 1 31 1 1 tf-idf tf-idf i Abstract A Method of Article Retrieval Utilizing Characteristics in Newspaper Articles
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〈論文〉興行データベースから「古典芸能」の定義を考える
Abstract The long performance database of rakugo and kabuki was totaled, and it is found that few programs are repeated in both genres both have the frequency differential of performance. It is a question
A Contrastive Study of Japanese and Korean by Analyzing Mistranslation from Japanese into Korean Yukitoshi YUTANI Japanese, Korean, contrastive study, mistranslation, Japanese-Korean dictionary It is already
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Vol. 52 No. 1 257 268 (Jan. 2011) 1 2, 1 1 measurement. In this paper, a dynamic road map making system is proposed. The proposition system uses probe-cars which has an in-vehicle camera and a GPS receiver.
A Study on Traffic Characteristics in Multi-hop Wireless Networks 2010 3 Yoichi Yamasaki ( ) 21 Local Area Network (LAN) LAN LAN LAN (AP, Access Point) LAN AP LAN AP AP AP (MWN, Multi-hop Wireless Network)
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GPGPU 2013 1008 2015 1 23 Abstract In recent years, with the advance of microscope technology, the alive cells have been able to observe. On the other hand, from the standpoint of image processing, the
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1,a) 2,b) 3 Modeling of Agitation Method in Automatic Mahjong Table using Multi-Agent Simulation Hiroyasu Ide 1,a) Takashi Okuda 2,b) Abstract: Automatic mahjong table refers to mahjong table which automatically
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2011 Vol.60 No.2 p.138 147 Performance of the Japanese long-term care benefit: An International comparison based on OECD health data Mie MORIKAWA[1] Takako TSUTSUI[2] [1]National Institute of Public Health,
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25 D Effects of viewpoints of head mounted wearable 3D display on human task performance 1140322 2014 2 28 D HMD HMD HMD HMD 3D HMD HMD HMD HMD i Abstract Effects of viewpoints of head mounted wearable
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23 FPGA CUDA Performance Comparison of FPGA Array with CUDA on Poisson Equation ([email protected]), ([email protected]), ([email protected]), ([email protected]),
Table 1. Reluctance equalization design. Fig. 2. Voltage vector of LSynRM. Fig. 4. Analytical model. Table 2. Specifications of analytical models. Fig
Mover Design and Performance Analysis of Linear Synchronous Reluctance Motor with Multi-flux Barrier Masayuki Sanada, Member, Mitsutoshi Asano, Student Member, Shigeo Morimoto, Member, Yoji Takeda, Member
IPSJ SIG Technical Report Vol.2014-EIP-63 No /2/21 1,a) Wi-Fi Probe Request MAC MAC Probe Request MAC A dynamic ads control based on tra
1,a) 1 1 2 1 Wi-Fi Probe Request MAC MAC Probe Request MAC A dynamic ads control based on traffic Abstract: The equipment with Wi-Fi communication function such as a smart phone which are send on a regular
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21 Stock price forecast using text mining 1100323 2010 3 1 Q-Learning Support-Vector-Machine NIKKEI NET Infoseek MSN 10 1 12 22 170 121 10 9 15 12 22 85 2 85 10 i Abstract Stock price forecast using text
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MEMOIRS OF SHONAN INSTITUTE OF TECHNOLOGY Vol. 41, No. 1, 2007 * * 2 * 3 * 4 * 5 * 6 * 7 * 8 Service Learning for Engineering Students Satsuki TASAKA*, Mitsutoshi ISHIMURA* 2, Hikaru MIZUTANI* 3, Naoyuki
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The Transformation of Educational Competition in Japan Conceptual Framework and Problems OWAKI Yasuhirc School of Education, Osaka Kyoiku University, Kashiwara, Osaka 582-8582, JAPAN I intend to compose
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