Decision Making in Multiplayer Environments Application in Backgammon Variants

Transcription

1 Decision Making in Multiplayer Environments Application in Backgammon Variants PhD Thesis by Nikolaos Papahristou AI researcher Department of Applied Informatics Thessaloniki, Greece

2 Contributions Expert playing agents for the first time in Tavli games New training method for self-play learning Statistics for tavli games Palamedes: A free program where anyone can play against the trained agents 1 st place and gold medal in 2011 and 2015 Backgammon Computer Olympiad Nikolaos Papahristou PhD Thesis AI in backgammon Variants 2

3 AI history in Games 1993, TD-Gammon: Surpasses world-champion in backgammon 1994, Chinook: Beat world champion in Checkers 1997, Logistello: Beat world champion in Othello 1997, Deep Blue: Beat world champion in Chess 1998, Maven: Beat world champion in Scrabble 2007, Chinook: Weakly solving of checkers 2015, Cepheus: Effectively solving Heads up Limit Texas Holdem Poker 2016, AlphaGo: Beat legendary player at Go Nikolaos Papahristou PhD Thesis AI in backgammon Variants 3

4 Tavli Games - Motivation In Greece there are three popular variants Portes (similar to standard backgammon) Plakoto Fevga No previous research in these variants Can we make AI agents that play these games expertly? Nikolaos Papahristou PhD Thesis AI in backgammon Variants 4

5 Outline 1. Background 2. Learning to Play Tavli Games 3. Statistics and Match Play 4. Palamedes 5. Conclusion Nikolaos Papahristou PhD Thesis AI in backgammon Variants 5

6 Backgammon Games Portes Plakoto Fevga Tavli Match Nikolaos Papahristou PhD Thesis AI in backgammon Variants 6

7 Modes of Play Money games The goal of the player is to maximize his/her points from a single game Money games can be viewed as games played in a match of infinite length Backgammon Matches Players accumulate points until one player reaches a predefined number of points Typical match length is 5 or 7 points. Goal of the player is to win the match. Matches can also be comprised by different variants (e.g. Tavli match). In this thesis we deal with matches of one variant only. Nikolaos Papahristou PhD Thesis AI in backgammon Variants 7

8 Machine Learning Categories Supervised learning Examples of inputs and their desired outputs (labels) are given. Find a model that learns to map new inputs. Unsupervised learning No labels given. Find structure in data. Reinforcement learning No inputs/labels. Find best behavior by interacting a (dynamic) environment. Nikolaos Papahristou PhD Thesis AI in backgammon Variants 8

9 Reinforcement Learning (RL) MDP μ = (S, Α, P, R, I) S (s 1, s 2, s T ) is the state space A is the action space: Α(s) P is the transition model: P(s t+1 s t, α) R is the reward function: R(s t,, α, s t+1 ) Ι is the initial state Markov property Transitions and rewards are independent of history Nikolaos Papahristou PhD Thesis AI in backgammon Variants 9

10 RL - Value Function Methods A value function V: S R, maps a state to a real number value when following a policy π: V π s = E π R t s t = s π is better than π, if and only if V π (s) V π (s) for every state. All optimal policies share the same optimal value function: V s = max π Vπ s Nikolaos Papahristou PhD Thesis AI in backgammon Variants 10

11 Temporal Difference Learning TD(0) δ t = r t+1 + γv s t+1 V s t V s t V s t + aδ t δ t is called the temporal difference error V s t+1 is the target of the update α [0,1] : is the learning rate TD(0) uses bootstrapping is a stochastic approximation algorithm uses one-step backups Nikolaos Papahristou PhD Thesis AI in backgammon Variants 11

12 Temporal Difference Learning TD(λ) base the backup on more than one future reward multi-step backup V x t V x t + a γλ k k=0 λ [0,1] For λ=0 TD(λ) equivalent to TD(0) δ t+k For λ=1 TD(λ) resembles Monte Carlo sampling λ (0,1): offers a way of future states affecting the present. A carefully selected value of λ can speed up learning In practice λ is determined by trial and error Nikolaos Papahristou PhD Thesis AI in backgammon Variants 12

13 Function Approximation Previous algorithms work on small state spaces Real-world applications have large state spaces Computing and storing the values of all states impractical Solution: generalize from a limited subsets of states = function approximation Typical function approximation methods Neural Networks Decision trees Tile coding Radial Basis Functions (RBF) Nikolaos Papahristou PhD Thesis AI in backgammon Variants 13

14 Outline 1. Background 2. Learning to Play Tavli Games 3. Statistics and Match Play 4. Palamedes 5. Conclusion Nikolaos Papahristou PhD Thesis AI in backgammon Variants 14

15 Neural Network Architecture Outputs: W: Win game WD: Win Double game LD: Lose Double game OUTPUTS W WD LD OUTPUT LAYER ʃ ʃ ʃ Linear combination of outputs produces position score ʃ ʃ ʃ HIDDEN LAYER..... INPUT LAYER Backgammon position encoded at input layer Nikolaos Papahristou PhD Thesis AI in backgammon Variants 15

16 Evaluation Evaluation procedures: against Tavli3D (open source program) against stored weights of the same NN against previously trained agent games per benchmark Evaluation value: estimated points per game (ppg) Nikolaos Papahristou PhD Thesis AI in backgammon Variants 16

17 Training procedure Generate sample game Self-play: Neural Network used as evaluation function At every time step all legal afterstates are scored and the best one is played Update weights Apply TD(λ) update for every move of the game using the back-propagation procedure of the NN Repeat procedure until no more improvement is observed Nikolaos Papahristou PhD Thesis AI in backgammon Variants 17

18 Game sequence creation for learning Database of games already available Observe (or play against) expert(s) Self-play 1. Learning online (each update is done immediately after a move is played). 2. Learning offline (updates are done incrementally after the game ends) a) Forward offline: Updates are done starting from the first position of the game and ending at the terminal position. b) Reverse offline: Updates are done starting from the terminal position of the game and ending at the first. c) Reverse offline recalc: As previous, but recalculate target value after each update. Nikolaos Papahristou PhD Thesis AI in backgammon Variants 18

19 Comparison of sequence creation methods ppg vs pubeval ppg vs Fevga-1 ppg vs Plakoto1 0 Backgammon 0.5 Plakoto Fevga Reverse Offline Recalc Online Reverse Offline Forward Offline Each line average of 10 training runs -λ = 0, α = 0.1 -ΝΝ has 10 Hidden Units Nikolaos Papahristou PhD Thesis AI in backgammon Variants 19

20 Determining the target of the update a b c a: Update the values without flipping the board. b: Updates are split in two. c: Updates are done on the inverted value of the next player. Nikolaos Papahristou PhD Thesis AI in backgammon Variants 20

21 Plakoto features Plakoto-1 (raw encoding) 4 binary inputs for every point per player 1 input for the checkers off board per player 1 binary input for every point per player for pins Plakoto-2 (raw + smart features) Replaced the player pin units with the probability of the opponent pinning the points Nikolaos Papahristou PhD Thesis AI in backgammon Variants 21

22 Fevga features Fevga-1 (raw encoding) 4 binary inputs for every point per player 1 input for the checkers off board per player Fevga-2 (raw + smart features) existence of primes (the most powerful strategy in fevga) pipcount, existence of a race situation Fevga-3 (raw + smart + intermediate reward) Primes are treated as winning positions Strategy learned is based on the creation of primes This strategy is considered one of the most powerful by expert players Nikolaos Papahristou PhD Thesis AI in backgammon Variants 22

23 points per game (ppg) vs Tavli3D points per game (ppg) vs Tavli3D Training progress of all agents examined Plakoto-1 Plakoto-2 Plakoto Fevga2 Fevga3 Fevga4 Fevga5 0 Games trained (millions) Games trained (millions) Summary of techniques used by the various agents Plakoto Agent Updating method Sequence creation and update direction Fevga agent Updating method Sequence creation and update direction Intermediate reward Plakoto-1 b Forward offline Fevga-2 b Forward offline No Plakoto-2 b Forward offline Fevga-3 b Forward offline Yes Plakoto-3 c Reverse offline recalc Fevga-4 c Reverse offline recalc No Fevga-5 c Reverse offline recalc Yes Nikolaos Papahristou PhD Thesis AI in backgammon Variants 23

24 Final Training Setup NNs as game evaluation function of states Training examples by self-play Temporal Difference Learning (λ) for weight update Offline updates Updates start from the terminal position and work back to the starting position Incremental updating of weights Gradual decrease of α, λ parameters Nikolaos Papahristou PhD Thesis AI in backgammon Variants 24

25 Selected values of α and λ parameters. Games Trained Portes Plakoto Fevga λ=0.7 α=1 λ=0 α=0.3 λ=0.7 α= λ=0.7 α=0.3 λ=0 α=0.3 λ=0.7 α= λ=0.7 α=0.1 λ=0 α=0.1 λ=0.7 α= λ=0 α=0.3 λ=0 α=0.1 λ=0 α= λ=0 α=0.1 λ=0 α=0.1 λ=0 α= λ=0 α=0.1 λ=0 α=0.01 λ=0 α= λ=0 α= Nikolaos Papahristou PhD Thesis AI in backgammon Variants 25

26 New features added Plakoto-5 Race, PipDiff, PipBearoff, PipBearoff, ChFrontOfPin, Esc_Prob Fevga-6 Probability of making a prime instead of binary feature Race, PipDiff, PipBearoff, PipBearoff Nikolaos Papahristou PhD Thesis AI in backgammon Variants 26

27 Performance of the new bots New Bot Opponent ppg Portes-1(1-ply) Pubeval (1-ply) Plakoto-5(1-ply) Plakoto-4(1-ply) Plakoto-5(2-ply) Plakoto-4(1-ply) Fevga-6(1-ply) Fevga-4(1-ply) Fevga-6(2-ply) Fevga-4(1-ply) The number of games played are for 1-ply and for 2-ply. In order to speed up the testing time of 2-ply, the expansion of depth-2 was performed only for the best 15 candidate moves (forward pruning). Nikolaos Papahristou PhD Thesis AI in backgammon Variants 27

28 Outline 1. Background 2. Learning to Play Tavli Games 3. Statistics and Match Play 4. Palamedes 5. Conclusion Nikolaos Papahristou PhD Thesis AI in backgammon Variants 28

29 Motivation - Method Investigate first player advantage in Tavli games (Portes, Plakoto, Fevga) using simulation and the Palamedes bot. Extract useful statistics (e.g. % games won as double wins) for each game Construct effective match strategies Method: Self-Play simulations for every roll and every starting move Nikolaos Papahristou PhD Thesis AI in backgammon Variants 29

30 First player estimated equity of all opening rolls Nikolaos Papahristou PhD Thesis AI in backgammon Variants 30

31 Advantage of the first player SINGLE ROLLS Portes Plakoto Fevga DOUBLE ROLLS ALL ROLLS Notable points: First player has big advantage on Fevga Double rolls significantly increase first player chances (except on Fevga) Fairness ranking: 1)Backgammon 2)Portes 3)Plakoto 4)Fevga Nikolaos Papahristou PhD Thesis AI in backgammon Variants 31

32 Expected outcome (%) of the first player Fevga All Rolls 47.31% 10.22% 38.42% 4.05% Fevga Double Rolls 47.39% 12.89% 36.06% 3.66% Fevga Single Rolls 47.30% 9.69% 38.89% 4.12% Plakoto All Rolls 29.90% 22.77% 29.62% 17.71% Plakoto Double Rolls 27.85% 28.91% 27.30% 15.95% Plakoto Single Rolls 30.31% 21.55% 30.09% 18.06% Portes All Rolls 38.18% 14.62% 34.87% 12.31% Portes Double Rolls 41.84% 17.57% 30.68% 9.73% Portes Single Rolls 37.45% 14.03% 35.70% 12.82% 0% 100% Single Wins (WS) Double Wins (WD) Single Losses (LS) Double Losses (LD) Table 1. Gammon rates of Tavli variants Variant Gammon Rate Portes 26.85% Plakoto 40.48% Fevga 14.27% Nikolaos Papahristou PhD Thesis AI in backgammon Variants 32

33 Example of a MWC table (Fevga) A MATCH WINNING CHANCE (MWC) away B away Move selection Money Strategy: E = WS LS + 2 * (WD LD) Match Strategy: MWC = WS * mwc(a-1, B) + WD * mwc(a-2, B) + LS * mwc(a, B-1) + LD * mwc(a, B-2) Nikolaos Papahristou PhD Thesis AI in backgammon Variants 33

34 Match strategy vs Money Strategy Variant Match Wins Diff. moves Games WS Games WD Games LS Games LD Total game points Portes 5144 ± % Plakoto 5103 ± % Fevga 5067 ± % point matches All results from the point of the match-strategy player WS: Single wins WD: Double Wins LS: Single Losses LD: Double Losses Diff. moves: % of match strategy moves that is different from the ones that the money strategy would have made in its place. Nikolaos Papahristou PhD Thesis AI in backgammon Variants 34

35 Outline 1. Background 2. Learning to Play Tavli Games 3. Statistics and Match Play 4. Palamedes 5. Conclusion Nikolaos Papahristou PhD Thesis AI in backgammon Variants 35

36 Palamedes Free Software to play against all agents Windows and Android versions Developed in C++ using Qt Framework and Eigen library Nikolaos Papahristou PhD Thesis AI in backgammon Variants 36

37 Palamedes Features Several variants supported Human vs AI Look-ahead search (2-ply) Endgame databases supported Money-game and match modes Player Statistics Analysis of played moves (Windows only) Nikolaos Papahristou PhD Thesis AI in backgammon Variants 37

38 Palamedes Analytics installs 250 active users / day 1500 games / day 18 / session User results per game: Portes : ppg Plakoto: ppg Fevga: ppg Nikolaos Papahristou PhD Thesis AI in backgammon Variants 38

39 Palamedes in Computer Olympiads Participated two times in backgammon computer Olympiads (2011, 2015) 1 st place both times Game type: standard backgammon Opponents GNUBG: open source BGBlitz: commercial Nikolaos Papahristou PhD Thesis AI in backgammon Variants 39

40 Outline 1. Background 2. Learning to Play Tavli Games 3. Statistics and Match Play 4. Palamedes 5. Conclusion Nikolaos Papahristou PhD Thesis AI in backgammon Variants 40

41 Contributions Expert playing agents for the first time in Tavli games New training method for self-play learning Statistics for tavli games Palamedes: A free program where anyone can play against the trained agents Nikolaos Papahristou PhD Thesis AI in backgammon Variants 41

42 Future work Apply training algorithm to other games/env Make training algorithm multi-threaded Graying the NN black box Extend MWC tables for tavli matches Change Fevga rules to reduce first player adv. Add more endgame databases Palamedes: add deeper search (3-ply, 4-ply) Palamedes: tutor mode Nikolaos Papahristou PhD Thesis AI in backgammon Variants 42

43 Thank you! Nikos Papahristou