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**Monte Carlo Tree Search A Tutorial**

Supported by the Engineering and Physical Sciences Research Council (EPSRC) Peter Cowling University of York, UK Simon Lucas University of Essex, UK

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Further Reading Browne C., Powley E., Whitehouse D., Lucas S., Cowling P.I., Rohlfshagen P., Tavener S., Perez D. and Samothrakis S., Colton S. (2012): "A Survey of Monte Carlo Tree Search Methods" IEEE Transactions on Computational Intelligence and AI in Games, IEEE, 4 (1): 1-43.

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**Monte Carlo Tree Search (MCTS) Why know more?**

Go: Strong/World Champion MCTS players in: General Game Playing, Hex, Amazons, Arimaa, Khet, Shogi, Mancala, Morpion, Blokus, Focus, Chinese Checkers, Yavalath, Connect Four, Gomoku Morpion solitaire best ever Over 250 papers since about one per week on average. Promise for planning, real-time games, nondeterministic games, single-player puzzles, optimisation, simulation, PCG, …

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**Tutorial Structure Monte Carlo search Monte Carlo Tree Search (MCTS)**

The multi-armed bandit problem Monte Carlo Tree Search (MCTS) Selection, expansion, simulation, backpropagation Enhancements to the basic algorithm Handling uncertainty in MCTS MCTS for real-time games Conclusions, future research directions

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**Monte Carlo Search for Decision Making in Games**

No trees yet …

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**The Multi-Armed Bandit Problem**

At each step pull one arm Noisy/random reward signal In order to: Find the best arm Minimise regret Maximise expected return Show first part of Excel simulation

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**Which Arm to Pull? Mean so far Upper bound on variance**

UCB1 (Auer et al (2002)). Choose arm j so as to maximise: Flat Monte Carlo Share trials uniformly between arms ε-Greedy P(1- ε) – Best arm so far P(ε) – Random arm Show Flat MC, epsilon-greedy and UCB1 part of Excel demo Mean so far Upper bound on variance

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**Simulation Result (+1 = win, 0 = loss)**

Game Decisions Current position Multi-Armed Bandit Arm A Arm C Arm B Move C Move A Move B Position after move A Position after move B Position after move C Show Connect 4 with Flat, UCB1 players Emphasise simulation result = random game = arm pull Simulation Result (+1 = win, 0 = loss)

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**Discussion Anytime – stop whenever you like**

UCB1 formula minimises regret Grows like log(n) Needs only game rules: Move generation Terminal state evaluation Surprisingly effective, but… … doesn’t look ahead GIB here, Snails example here if time Move generation and terminal state evaluation allow random simulated games to be run

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MCTS / UCT Basics

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**Attractive Features Anytime Scalable No need for heuristic function**

Tackle complex games better than before May be logarithmically better with increased CPU No need for heuristic function Though usually better with one Next we’ll look at: General MCTS UCT in particular Note: this slide may be unnecessary – might be covered earlier

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MCTS: the main idea Tree policy: choose which node to expand (not necessarily leaf of tree) Default (simulation) policy: random playout until end of game

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**MCTS Algorithm Decompose into 6 parts: MCTS main algorithm**

Tree policy Expand Best Child (UCT Formula) Default Policy Back-propagate We’ll run through these then show demos

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MCTS Main Algorithm Check this point about BestChild BestChild simply picks best child node of root according to some criteria: e.g. best mean value In our pseudo-code BestChild is called from TreePolicy and from MctsSearch, but different versions can be used E.g. final selection can be the max value child or the most frequently visited one

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TreePolicy Note that node selected for expansion does not need to be a leaf of the tree The nonterminal test refers to the game state

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Expand Note: need to choose an appropriate data structure for the children (array, ArrayList, Map)

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**Best Child (UCT) This is the standard UCT equation**

Used in the tree Higher values of c lead to more exploration Other terms can be added, and usually are More on this later

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DefaultPolicy Each time a new node is added to the tree, the default policy randomly rolls out from the current state until a terminal state of the game is reached The standard is to do this uniformly randomly But better performance may be obtained by biasing with knowledge

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Backup Note that v is the new node added to the tree by the tree policy Back up the values from the added node up the tree to the root

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**MCTS Builds Asymmetric Trees (demo)**

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Connect 4 Demo

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**Enhancements to the Basic MCTS Algorithm**

Tree Policy Default Policy Domain: independent or specific

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**Selection/Expansion Enhancements (just a sample …)**

AMAF / RAVE First Play Urgency Learning E.g. Temporal Difference Learning to bias tree policy or default policy Bandit enhancements E.g. Bayesian selection Parameter Tuning

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**All Moves As First (AMAF), Rapid Value Action Estimates (RAVE)**

Additional term in UCT equation: Treat actions / moves the same independently of where they occur in the move sequence

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**Simulation Enhancements**

Nudge the simulation to be more realistic Without introducing bias At very low CPU cost Examples: Move-Average Sampling Technique (MAST) Last Good Reply (LGR) Patterns e.g. Bridge Completion: Heuristic value functions Paradoxically better simulation policy sometimes yields weaker play MAST – Finsson and Bjornsson – CADIAPLAYER for GGP - Maintain a table for each action independent of state. Bias action selection towards effective action using Gibbs sampling. Also PAST/FAST where predicates/features related to the state are checked and good predicates are preferentially chosen during simulation. LGR – Drake – Go – For each move remember the last reply to that move that resulted in a winning simulation. Drake and Baier added forgetting (LGRF) if a reply is subsequently shown to lead to a loss. Patterns – Go – Use local configurations to choose or ignore moves. Heuristic value functions – Winands + Bjornsson – Lines of Action – bias towards high value moves later in search. Paradox – Gelly and Silver – RLGO – presumably bias is the problem.

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Hex Demo Cameron’s Hex demo: show MCTS variants

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**Parallelisation Figure from Chaslot, Winands, van den Herik.**

Leaf parallelisation – many simulations per new leaf – problem of waiting for simulations to finish Root parallelisation – combine trees (at level 1) found independently (also known as ensemble MCTS). Majority voting beats averaging (Soejima, Kishimoto, Watanabe). Tree parallelisation – use a single tree accessed by many processes. Lock whole tree if Sims much slower than updates, else local locks. Also lock-free parallelisation (Enzenberger and Muller). Use “virtual loss” to diversify search away from simulations under investigation. Root parallelisation wins. We also find this with determinisation in games with hidden info and uncertainty

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**Handling Uncertainty and Incomplete Information in MCTS**

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**Probabilistic (stochastic) nodes**

CHANCE MAX MIN MAX Probabilistic (stochastic) planning E,g, dynamic scheduling Probabilistic planning – tends to have low branching factor at probabilty nodes – but card shuffling gives a gigantic branching factor Decision-making/optimisation under uncertainty

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**{ , , , ...} Hidden Information Information set: Actual state:**

Observation: , ...} E.g. Poker, Bridge, Security

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**Effects of Uncertainty and Hidden Information on the Game Tree**

choose subset 4 possible plays by me 50 possible random card draws 40C3 = 9880 different opponent plays My opponent has to play 3 out of 4 cards ...

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**... Reduced Branching Through Determinization**

4 possible plays by me 1 possible random card draws 4C3 = 4 different opponent plays FF-replan, GIB, Maven, …, determinisation idea. ... Use average results over many determinizations [Yoon, Fern & Givan 2007]

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**Determinization Sample states from the information set**

Analyse the individual perfect information games Combine the results at the end Successes Bridge (Ginsberg), Scrabble (Sheppard) Klondike solitaire (Bjarnason, Fern and Tadepalli) Probabilistic planning (Yoon, Fern and Givan) Problems Never tries to gather or hide information Suffers from strategy fusion and non-locality (Frank and Basin) GIB, Maven, more recent Scrabble work by Childs et al on opponent modelling/computing a good prior. Klondike solitaire – using UCT and other approache. Win rate about 37% - much higher than thought possible. FF Replan, uses FF and determinization. Strategy fusion – assumes different decisions possible based on unknown information. “If my opponent holds an Ace, I’ll do that, otherwise I’ll do this” – but whether or not the oppnent holds an ace is unknown. Non-locality – The opponent is leading us into situations unfavourable for us – so some parts of the state space become very unlikely. M.L. Ginsberg, “GIB: Imperfect information in a computationally challenging game,” Journal of Artificial Intelligence Research, vol. 14, 2002, p. 313–368. R. Bjarnason, A. Fern, and P. Tadepalli, “Lower bounding Klondike solitaire with Monte-Carlo planning,” Proc. ICAPS-2009, p. 26–33. S. Yoon, A. Fern, and R. Givan, “FF-Replan: A Baseline for Probabilistic Planning,” 1Proc. ICAPS-2007, p. 352–359. I. Frank and D. Basin, “Search in games with incomplete information: a case study using Bridge card play,” Artificial Intelligence, vol. 100, 1998, pp

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Cheating Easiest approach to AI for games with imperfect information: cheat and look at the hidden information and outcomes of future chance events This gives a deterministic game of perfect information, which can be searched with standard techniques (minimax, UCT, …) Obviously not a reasonable approach in practice, but gives a useful bound on performance Much used in commercial games Has: Perfect Inference Clairvoyance No Strategy Fusion or non-locality

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**Information Set Monte Carlo Tree Search (ISMCTS)**

Maintain a single tree Use each determinization only once Determinizations restrict search to a subtree of the information set tree. Single observer and multi=observer versions to handle partially observable moves Collect information about the value of a decision in the corresponding information set.

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**Lord of the Rings: The Confrontation**

Board game designed by Reiner Knizia Hidden information: can’t see the identities of opponent’s pieces Simultaneous moves: combat is resolved by both players simultaneously choosing a card Also Phantom 4,4,4, Dou Di Zhu and several other card games…

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**MCTS for Real-Time Games**

Limited roll-out budget Heuristic knowledge becomes important Action space may be too fine-grained Take Macro-actions Otherwise planning will be very short-term May be no terminal node in sight Again, use heuristic Tune simulation depth Next-state function may be expensive Consider making a simpler abstraction

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PTSP Demo

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**Summary MCTS: exciting area of research**

Many impressive achievements already With many more to come Some interesting directions: Applications beyond games Comparisons with other search methods Heuristic knowledge is important – more work needed on learning it Efficient parameter tuning POMDPs Uncertainty Macro-actions Hybridisation with other CI methods e.g. Evolution

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Questions? Browne C., Powley E., Whitehouse D., Lucas S., Cowling P.I., Rohlfshagen P., Tavener S., Perez D. and Samothrakis S., Colton S. (2012): "A Survey of Monte Carlo Tree Search Methods" IEEE Transactions on Computational Intelligence and AI in Games, IEEE, 4 (1): 1-43.

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**Advertising Break Sep 28: Essex workshop on AI and Games**

Strong MCTS Theme Organised by Essex Game Intelligence Group Next week: AIGAMEDEV Vienna AI summit Find out what games industry people think Resistance competition and MCTS tutorial (!)

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