Monte Carlo *-Minimax Search
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Monte-Carlo Tree Search Solver
Monte-Carlo Tree Search Solver Mark H.M. Winands1, Yngvi BjÄornsson2, and Jahn-Takeshi Saito1 1 Games and AI Group, MICC, Universiteit Maastricht, Maastricht, The Netherlands fm.winands,[email protected] 2 School of Computer Science, Reykjav¶³kUniversity, Reykjav¶³k,Iceland [email protected] Abstract. Recently, Monte-Carlo Tree Search (MCTS) has advanced the ¯eld of computer Go substantially. In this article we investigate the application of MCTS for the game Lines of Action (LOA). A new MCTS variant, called MCTS-Solver, has been designed to play narrow tacti- cal lines better in sudden-death games such as LOA. The variant di®ers from the traditional MCTS in respect to backpropagation and selection strategy. It is able to prove the game-theoretical value of a position given su±cient time. Experiments show that a Monte-Carlo LOA program us- ing MCTS-Solver defeats a program using MCTS by a winning score of 65%. Moreover, MCTS-Solver performs much better than a program using MCTS against several di®erent versions of the world-class ®¯ pro- gram MIA. Thus, MCTS-Solver constitutes genuine progress in using simulation-based search approaches in sudden-death games, signi¯cantly improving upon MCTS-based programs. 1 Introduction For decades ®¯ search has been the standard approach used by programs for playing two-person zero-sum games such as chess and checkers (and many oth- ers). Over the years many search enhancements have been proposed for this framework. However, in some games where it is di±cult to construct an accurate positional evaluation function (e.g., Go) the ®¯ approach was hardly success- ful. -
Adversarial Search
Adversarial Search In which we examine the problems that arise when we try to plan ahead in a world where other agents are planning against us. Outline 1. Games 2. Optimal Decisions in Games 3. Alpha-Beta Pruning 4. Imperfect, Real-Time Decisions 5. Games that include an Element of Chance 6. State-of-the-Art Game Programs 7. Summary 2 Search Strategies for Games • Difference to general search problems deterministic random – Imperfect Information: opponent not deterministic perfect Checkers, Backgammon, – Time: approximate algorithms information Chess, Go Monopoly incomplete Bridge, Poker, ? information Scrabble • Early fundamental results – Algorithm for perfect game von Neumann (1944) • Our terminology: – Approximation through – deterministic, fully accessible evaluation information Zuse (1945), Shannon (1950) Games 3 Games as Search Problems • Justification: Games are • Games as playground for search problems with an serious research opponent • How can we determine the • Imperfection through actions best next step/action? of opponent: possible results... – Cutting branches („pruning“) • Games hard to solve; – Evaluation functions for exhaustive: approximation of utility – Average branching factor function chess: 35 – ≈ 50 steps per player ➞ 10154 nodes in search tree – But “Only” 1040 allowed positions Games 4 Search Problem • 2-player games • Search problem – Player MAX – Initial state – Player MIN • Board, positions, first player – MAX moves first; players – Successor function then take turns • Lists of (move,state)-pairs – Goal test -
Αβ-Based Play-Outs in Monte-Carlo Tree Search
αβ-based Play-outs in Monte-Carlo Tree Search Mark H.M. Winands Yngvi Bjornsson¨ Abstract— Monte-Carlo Tree Search (MCTS) is a recent [9]. In the context of game playing, Monte-Carlo simulations paradigm for game-tree search, which gradually builds a game- were first used as a mechanism for dynamically evaluating tree in a best-first fashion based on the results of randomized the merits of leaf nodes of a traditional αβ-based search [10], simulation play-outs. The performance of such an approach is highly dependent on both the total number of simulation play- [11], [12], but under the new paradigm MCTS has evolved outs and their quality. The two metrics are, however, typically into a full-fledged best-first search procedure that replaces inversely correlated — improving the quality of the play- traditional αβ-based search altogether. MCTS has in the past outs generally involves adding knowledge that requires extra couple of years substantially advanced the state-of-the-art computation, thus allowing fewer play-outs to be performed in several game domains where αβ-based search has had per time unit. The general practice in MCTS seems to be more towards using relatively knowledge-light play-out strategies for difficulties, in particular computer Go, but other domains the benefit of getting additional simulations done. include General Game Playing [13], Amazons [14] and Hex In this paper we show, for the game Lines of Action (LOA), [15]. that this is not necessarily the best strategy. The newest version The right tradeoff between search and knowledge equally of our simulation-based LOA program, MC-LOAαβ , uses a applies to MCTS. -
Artificial Intelligence Spring 2019 Homework 2: Adversarial Search
Artificial Intelligence Spring 2019 Homework 2: Adversarial Search PROGRAMMING In this assignment, you will create an adversarial search agent to play the 2048-puzzle game. A demo of the game is available here: gabrielecirulli.github.io/2048. I. 2048 As A Two-Player Game II. Choosing a Search Algorithm: Expectiminimax III. Using The Skeleton Code IV. What You Need To Submit V. Important Information VI. Before You Submit I. 2048 As A Two-Player Game 2048 is played on a 4×4 grid with numbered tiles which can slide up, down, left, or right. This game can be modeled as a two player game, in which the computer AI generates a 2- or 4-tile placed randomly on the board, and the player then selects a direction to move the tiles. Note that the tiles move until they either (1) collide with another tile, or (2) collide with the edge of the grid. If two tiles of the same number collide in a move, they merge into a single tile valued at the sum of the two originals. The resulting tile cannot merge with another tile again in the same move. Usually, each role in a two-player games has a similar set of moves to choose from, and similar objectives (e.g. chess). In 2048 however, the player roles are inherently asymmetric, as the Computer AI places tiles and the Player moves them. Adversarial search can still be applied! Using your previous experience with objects, states, nodes, functions, and implicit or explicit search trees, along with our skeleton code, focus on optimizing your player algorithm to solve 2048 as efficiently and consistently as possible. -
Smartk: Efficient, Scalable and Winning Parallel MCTS
SmartK: Efficient, Scalable and Winning Parallel MCTS Michael S. Davinroy, Shawn Pan, Bryce Wiedenbeck, and Tia Newhall Swarthmore College, Swarthmore, PA 19081 ACM Reference Format: processes. Each process constructs a distinct search tree in Michael S. Davinroy, Shawn Pan, Bryce Wiedenbeck, and Tia parallel, communicating only its best solutions at the end of Newhall. 2019. SmartK: Efficient, Scalable and Winning Parallel a number of rollouts to pick a next action. This approach has MCTS. In SC ’19, November 17–22, 2019, Denver, CO. ACM, low communication overhead, but significant duplication of New York, NY, USA, 3 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnneffort across processes. 1 INTRODUCTION Like root parallelization, our SmartK algorithm achieves low communication overhead by assigning independent MCTS Monte Carlo Tree Search (MCTS) is a popular algorithm tasks to processes. However, SmartK dramatically reduces for game playing that has also been applied to problems as search duplication by starting the parallel searches from many diverse as auction bidding [5], program analysis [9], neural different nodes of the search tree. Its key idea is it uses nodes’ network architecture search [16], and radiation therapy de- weights in the selection phase to determine the number of sign [6]. All of these domains exhibit combinatorial search processes (the amount of computation) to allocate to ex- spaces where even moderate-sized problems can easily ex- ploring a node, directing more computational effort to more haust computational resources, motivating the need for par- promising searches. allel approaches. SmartK is our novel parallel MCTS algo- Our experiments with an MPI implementation of SmartK rithm that achieves high performance on large clusters, by for playing Hex show that SmartK has a better win rate and adaptively partitioning the search space to efficiently utilize scales better than other parallelization approaches. -
Monte-Carlo Tree Search
Monte-Carlo Tree Search PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Maastricht, op gezag van de Rector Magnificus, Prof. mr. G.P.M.F. Mols, volgens het besluit van het College van Decanen, in het openbaar te verdedigen op donderdag 30 september 2010 om 14:00 uur door Guillaume Maurice Jean-Bernard Chaslot Promotor: Prof. dr. G. Weiss Copromotor: Dr. M.H.M. Winands Dr. B. Bouzy (Universit´e Paris Descartes) Dr. ir. J.W.H.M. Uiterwijk Leden van de beoordelingscommissie: Prof. dr. ir. R.L.M. Peeters (voorzitter) Prof. dr. M. Muller¨ (University of Alberta) Prof. dr. H.J.M. Peters Prof. dr. ir. J.A. La Poutr´e (Universiteit Utrecht) Prof. dr. ir. J.C. Scholtes The research has been funded by the Netherlands Organisation for Scientific Research (NWO), in the framework of the project Go for Go, grant number 612.066.409. Dissertation Series No. 2010-41 The research reported in this thesis has been carried out under the auspices of SIKS, the Dutch Research School for Information and Knowledge Systems. ISBN 978-90-8559-099-6 Printed by Optima Grafische Communicatie, Rotterdam. Cover design and layout finalization by Steven de Jong, Maastricht. c 2010 G.M.J-B. Chaslot. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronically, mechanically, photocopying, recording or otherwise, without prior permission of the author. Preface When I learnt the game of Go, I was intrigued by the contrast between the simplicity of its rules and the difficulty to build a strong Go program. -
CS234 Notes - Lecture 14 Model Based RL, Monte-Carlo Tree Search
CS234 Notes - Lecture 14 Model Based RL, Monte-Carlo Tree Search Anchit Gupta, Emma Brunskill June 14, 2018 1 Introduction In this lecture we will learn about model based RL and simulation based tree search methods. So far we have seen methods which attempt to learn either a value function or a policy from experience. In contrast model based approaches first learn a model of the world from experience and then use this for planning and acting. Model-based approaches have been shown to have better sample efficiency and faster convergence in certain settings. We will also have a look at MCTS and its variants which can be used for planning given a model. MCTS was one of the main ideas behind the success of AlphaGo. Figure 1: Relationships among learning, planning, and acting 2 Model Learning By model we mean a representation of an MDP < S; A; R; T; γ> parametrized by η. In the model learning regime we assume that the state and action space S; A are known and typically we also assume conditional independence between state transitions and rewards i.e P [st+1; rt+1jst; at] = P [st+1jst; at]P [rt+1jst; at] Hence learning a model consists of two main parts the reward function R(:js; a) and the transition distribution P (:js; a). k k k k K Given a set of real trajectories fSt ;At Rt ; :::; ST gk=1 model learning can be posed as a supervised learning problem. Learning the reward function R(s; a) is a regression problem whereas learning the transition function P (s0js; a) is a density estimation problem. -
Best-First and Depth-First Minimax Search in Practice
Best-First and Depth-First Minimax Search in Practice Aske Plaat, Erasmus University, [email protected] Jonathan Schaeffer, University of Alberta, [email protected] Wim Pijls, Erasmus University, [email protected] Arie de Bruin, Erasmus University, [email protected] Erasmus University, University of Alberta, Department of Computer Science, Department of Computing Science, Room H4-31, P.O. Box 1738, 615 General Services Building, 3000 DR Rotterdam, Edmonton, Alberta, The Netherlands Canada T6G 2H1 Abstract Most practitioners use a variant of the Alpha-Beta algorithm, a simple depth-®rst pro- cedure, for searching minimax trees. SSS*, with its best-®rst search strategy, reportedly offers the potential for more ef®cient search. However, the complex formulation of the al- gorithm and its alleged excessive memory requirements preclude its use in practice. For two decades, the search ef®ciency of ªsmartº best-®rst SSS* has cast doubt on the effectiveness of ªdumbº depth-®rst Alpha-Beta. This paper presents a simple framework for calling Alpha-Beta that allows us to create a variety of algorithms, including SSS* and DUAL*. In effect, we formulate a best-®rst algorithm using depth-®rst search. Expressed in this framework SSS* is just a special case of Alpha-Beta, solving all of the perceived drawbacks of the algorithm. In practice, Alpha-Beta variants typically evaluate less nodes than SSS*. A new instance of this framework, MTD(ƒ), out-performs SSS* and NegaScout, the Alpha-Beta variant of choice by practitioners. 1 Introduction Game playing is one of the classic problems of arti®cial intelligence. -
General Branch and Bound, and Its Relation to A* and AO*
ARTIFICIAL INTELLIGENCE 29 General Branch and Bound, and Its Relation to A* and AO* Dana S. Nau, Vipin Kumar and Laveen Kanal* Laboratory for Pattern Analysis, Computer Science Department, University of Maryland College Park, MD 20742, U.S.A. Recommended by Erik Sandewall ABSTRACT Branch and Bound (B&B) is a problem-solving technique which is widely used for various problems encountered in operations research and combinatorial mathematics. Various heuristic search pro- cedures used in artificial intelligence (AI) are considered to be related to B&B procedures. However, in the absence of any generally accepted terminology for B&B procedures, there have been widely differing opinions regarding the relationships between these procedures and B &B. This paper presents a formulation of B&B general enough to include previous formulations as special cases, and shows how two well-known AI search procedures (A* and AO*) are special cases o,f this general formulation. 1. Introduction A wide class of problems arising in operations research, decision making and artificial intelligence can be (abstractly) stated in the following form: Given a (possibly infinite) discrete set X and a real-valued objective function F whose domain is X, find an optimal element x* E X such that F(x*) = min{F(x) I x ~ X}) Unless there is enough problem-specific knowledge available to obtain the optimum element of the set in some straightforward manner, the only course available may be to enumerate some or all of the elements of X until an optimal element is found. However, the sets X and {F(x) [ x E X} are usually tThis work was supported by NSF Grant ENG-7822159 to the Laboratory for Pattern Analysis at the University of Maryland. -
Trees, Binary Search Trees, Heaps & Applications Dr. Chris Bourke
Trees Trees, Binary Search Trees, Heaps & Applications Dr. Chris Bourke Department of Computer Science & Engineering University of Nebraska|Lincoln Lincoln, NE 68588, USA [email protected] http://cse.unl.edu/~cbourke 2015/01/31 21:05:31 Abstract These are lecture notes used in CSCE 156 (Computer Science II), CSCE 235 (Dis- crete Structures) and CSCE 310 (Data Structures & Algorithms) at the University of Nebraska|Lincoln. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License 1 Contents I Trees4 1 Introduction4 2 Definitions & Terminology5 3 Tree Traversal7 3.1 Preorder Traversal................................7 3.2 Inorder Traversal.................................7 3.3 Postorder Traversal................................7 3.4 Breadth-First Search Traversal..........................8 3.5 Implementations & Data Structures.......................8 3.5.1 Preorder Implementations........................8 3.5.2 Inorder Implementation.........................9 3.5.3 Postorder Implementation........................ 10 3.5.4 BFS Implementation........................... 12 3.5.5 Tree Walk Implementations....................... 12 3.6 Operations..................................... 12 4 Binary Search Trees 14 4.1 Basic Operations................................. 15 5 Balanced Binary Search Trees 17 5.1 2-3 Trees...................................... 17 5.2 AVL Trees..................................... 17 5.3 Red-Black Trees.................................. 19 6 Optimal Binary Search Trees 19 7 Heaps 19 -
Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space
applied sciences Article Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space Dawid Połap 1,* ID , Karolina K˛esik 1, Marcin Wo´zniak 1 ID and Robertas Damaševiˇcius 2 ID 1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland; [email protected] (K.K.); [email protected] (M.W.) 2 Department of Software Engineering, Kaunas University of Technology, Studentu 50, LT-51368, Kaunas, Lithuania; [email protected] * Correspondence: [email protected] Received: 16 December 2017; Accepted: 13 February 2018 ; Published: 16 February 2018 Abstract: The increasing exploration of alternative methods for solving optimization problems causes that parallelization and modification of the existing algorithms are necessary. Obtaining the right solution using the meta-heuristic algorithm may require long operating time or a large number of iterations or individuals in a population. The higher the number, the longer the operation time. In order to minimize not only the time, but also the value of the parameters we suggest three proposition to increase the efficiency of classical methods. The first one is to use the method of searching through the neighborhood in order to minimize the solution space exploration. Moreover, task distribution between threads and CPU cores can affect the speed of the algorithm and therefore make it work more efficiently. The second proposition involves manipulating the solutions space to minimize the number of calculations. In addition, the third proposition is the combination of the previous two. All propositions has been described, tested and analyzed due to the use of various test functions. -
Backtracking Search (Csps) ■Chapter 5 5.3 Is About Local Search Which Is a Very Useful Idea but We Won’T Cover It in Class
CSC384: Intro to Artificial Intelligence Backtracking Search (CSPs) ■Chapter 5 5.3 is about local search which is a very useful idea but we won’t cover it in class. 1 Hojjat Ghaderi, University of Toronto Constraint Satisfaction Problems ● The search algorithms we discussed so far had no knowledge of the states representation (black box). ■ For each problem we had to design a new state representation (and embed in it the sub-routines we pass to the search algorithms). ● Instead we can have a general state representation that works well for many different problems. ● We can build then specialized search algorithms that operate efficiently on this general state representation. ● We call the class of problems that can be represented with this specialized representation CSPs---Constraint Satisfaction Problems. ● Techniques for solving CSPs find more practical applications in industry than most other areas of AI. 2 Hojjat Ghaderi, University of Toronto Constraint Satisfaction Problems ●The idea: represent states as a vector of feature values. We have ■ k-features (or variables) ■ Each feature takes a value. Domain of possible values for the variables: height = {short, average, tall}, weight = {light, average, heavy}. ●In CSPs, the problem is to search for a set of values for the features (variables) so that the values satisfy some conditions (constraints). ■ i.e., a goal state specified as conditions on the vector of feature values. 3 Hojjat Ghaderi, University of Toronto Constraint Satisfaction Problems ●Sudoku: ■ 81 variables, each representing the value of a cell. ■ Values: a fixed value for those cells that are already filled in, the values {1-9} for those cells that are empty.