1. Evaluation-Function Based Monte-Carlo LOA Mark H.M. Winands and Yngvi Björnsson

2. Department of Knowledge Engineering – Maastricht University Outline Background MC-LOA Evaluators and Simulations Experiments Conclusions

3. Department of Knowledge Engineering – Maastricht University Introduction Monte-Carlo Tree Search (Kocsis et al., 2006, Coulom 2007) ‏ Revolution in Go Applied in Amazons, Clobber, Havannah, Hex, GGP etc Here we focus on the sudden-death game Lines of Action (LOA) ‏

4. Department of Knowledge Engineering – Maastricht University Test Domain: Lines of Action (LOA) ‏ Two-person zero-sum chess-like connection game A move takes place in a straight line, exactly as many squares as there are pieces of either colour anywhere along the line of movement Goal is to group your pieces into one connected unit

5. Department of Knowledge Engineering – Maastricht University Computer LOA 1 st LOA program, Stanford AI laboratory 1975 Since 2000, it is also played at the Computer Olympiad Several programs: YL (Björnnson), Mona (Billings), Bing (Helmstetter), T-T (JAIST Lab), Lola (Coulom) ‏ All programs use αβ search + evaluation function αβ search: null-move, multi-cut, RPS Evaluation functions are quite advanced

6. Department of Knowledge Engineering – Maastricht University Monte-Carlo Tree Search (MCTS) ‏ A best-first search guided by the results of Monte-Carlo simulations We will discuss the steps for MC-LOA Play-out

7. Department of Knowledge Engineering – Maastricht University Selection Step Traversal from the root until we reach a position which is not part of the tree Selection Strategy: Controls exploitation and exploration balance UCT (Kocsis, 2006) and Progressive BIAS (PB) (Chaslot et al. 2008) Selects the child k of the node p according: UCT: v i is the value of the node i, n i is the visit count of i, and n p is the visit count of p. C is a coefficient PB: P mc is the transition probability of a move category mc, W is a constant UCT PB

8. Department of Knowledge Engineering – Maastricht University Progressive Bias: Transition Probabilities Transition Probabilities originally used in Realization Probability Search (Tsuruoka, 2002) ‏ Transition Probability: Probability that a move belonging to a category mc will be played Statistics are retrieved from self-play

9. Department of Knowledge Engineering – Maastricht University Progressive Bias: Move Categories Move categories in LOA: –Captures or non-captures –Origin and destination of the move’s from and to squares –Number of squares traveled from- or towards to the centre-of-mass 277 move categories

10. Department of Knowledge Engineering – Maastricht University Simulation / Play-out If the node has been visited fewer times than a threshold T, the next move is selected pseudo-randomly according to the simulation strategy Their move category weights At a leaf node: –All moves are generated and checked for a mate in one Play-out step: outside the tree, self-play

11. Department of Knowledge Engineering – Maastricht University Backpropagation Step The result of a simulated game k is back- propagated from the leaf node L, through the previously traversed path (+1 win, 0 draw, -1 loss) ‏ The value v of a node is computed by taking the average of the results of all simulated games made through this node

12. Department of Knowledge Engineering – Maastricht University MCTS – Solver: Backpropagation MCTS-Solver (Winands et al. 2008) For terminal positions assign the values ∞ (Win) or -∞ (Loss) ‏ The internal nodes are assigned in a minimax way

13. Department of Knowledge Engineering – Maastricht University Simulation Strategies MCTS does not require a positional evaluation function, but is based on the results of simulations Not good enough for LOA ‏ How to use a positional evaluation function? Four different strategies –Evaluation Cut-off –Corrective –Greedy –Mixed

14. Department of Knowledge Engineering – Maastricht University Evaluation function: (MIA 2006) ‏ Concentration Centralisation Centre-of-mass position Quads Mobility Walls Connectedness Uniformity Variance

15. Department of Knowledge Engineering – Maastricht University Evaluation Cut-Off Stops a simulated game before a terminal state is reached if, according to a heuristic knowledge, the game is judged to be effectively over Starting at second ply in the play-out, every three ply the evaluation function is called –If the value exceeds a certain threshold, a win is scored (700) ‏ –If the value is below a certain threshold, a loss is scored (-700) ‏ Function can return a quite trustworthy score, more so than even elaborate simulation strategies The game can thus be safely terminated both earlier and with a more accurate score than if continuing the simulation

16. Department of Knowledge Engineering – Maastricht University Corrective Minimizing the risk of choosing an obviously bad move First, we evaluate the position for which we are choosing a move Next, we generate the moves and scan them to get their weights If the move leads to a successor which has a lower evaluation score than its parent, we set the weight of a move to a preset minimum value (close to zero) ‏ If a move leads to a win, it will be immediately played

17. Department of Knowledge Engineering – Maastricht University Greedy The move leading to the position with the highest evaluation score is selected We evaluate only moves that have a good potential for being the best The k-best moves according to their move category weights are fully evaluated (k = 7) ‏ When a move leads to a position with an evaluation over a preset threshold, the play-out is stopped and scored as a win The remaining moves — which are not heuristically evaluated — are checked for a mate Evaluation function has small a random factor

18. Department of Knowledge Engineering – Maastricht University Mixed Greedy strategy could be too deterministic The Mixed strategy combines the Corrective strategy and the Greedy strategy The Corrective strategy is used in the selection step, i.e., at tree nodes where a simulation strategy is needed (i.e., n < T ) as well as in the first position entered in the play-out step For the remainder of the play-out the Greedy strategy is applied

19. Department of Knowledge Engineering – Maastricht University Experiments: MIA Test the strength against a non-MC program: MIA Won the 8 th, 9 th and 11 th Computer Olympiad αβ depth-first iterative-deepening search in the PVS frame work Enhancements: Transposition Table, ETC, killer moves, relative history heuristic etc. Variable depth: Multi-cut, (adaptive) null move, Enhanced realization probability search, quiescence search

20. Department of Knowledge Engineering – Maastricht University Parameter Tuning: Evaluation Cut-off Tune the bounds of Evaluation Cut-off Strategy 3 programs: No-Bound, MIA III and MIA 4.5 1000 game results, 1 sec per move

21. Department of Knowledge Engineering – Maastricht University Round-Robin Tournament Results 1000 Games 1 sec per move

22. Department of Knowledge Engineering – Maastricht University MC-LOA vs MIA Mixed Strategy 5 sec a move Root-Parallelization

23. Department of Knowledge Engineering – Maastricht University MC-LOA vs MIA No parallel version of MIA Two-threaded MC-LOA program competed against MIA, where MIA was given 50% more time Simulating a search-efficiency increase of 50% if MIA were to be given two processors A 1,000 game match resulted in a 52% winning percentage for MC-LOA

24. Department of Knowledge Engineering – Maastricht University Conclusions (1) ‏ Applying an evaluation function to stop simulations, increases the playing strength Mixed strategy of playing greedily in the play-out phase, but exploring more in the earlier selection phase, works the best

25. Department of Knowledge Engineering – Maastricht University Conclusions (2) ‏ MCTS using simple root-parallelization outperforms MIA MCTS programs are competitive with αβ-based programs in the game LOA