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Multi-Agent Machine Learning The Reinforcement Approach
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This book introduces some machine learning approaches about multi-agent learning. There are a number of algorithms that are typically used for system identification, adaptive control, adaptive signal processing, and machine learning.These algorithms all have particular similarities and differences
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Multi-Agent Machine
Learning
Multi-Agent Machine
Learning
A Reinforcement Approach
Howard M. Schwartz
Department of Systems and Computer Engineering
Carleton University
Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or
by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as
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Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in
preparing this book, they make no representations or warranties with respect to the accuracy or
completeness of the contents of this book and specically disclaim any implied warranties of
merchantability or tness for a particular purpose. No warranty may be created or extended by sales
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Library of Congress Cataloging-in-Publication Data:
Schwartz, Howard M., editor.
Multi-agent machine learning : a reinforcement approach / Howard M. Schwartz.
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-36208-2 (hardback)
1. Reinforcement learning. 2. Differential games. 3. Swarm intelligence. 4. Machine learning. I. Title.
Q325.6.S39 2014
519.3–dc23
2014016950
Printed in the United States of America
10987654321
Contents
Preface .................................................... ix
Chapter 1 A Brief Review of Supervised Learning ......... 1
1.1 Least Squares Estimates ........................... 1
1.2 Recursive Least Squares........................... 5
1.3 Least Mean Squares ............................... 6
1.4 Stochastic Approximation ......................... 10
References ........................................ 11
Chapter 2 Single-Agent Reinforcement Learning .......... 12
2.1 Introduction ....................................... 12
2.2 n-Armed Bandit Problem ........................... 13
2.3 The Learning Structure ............................ 15
2.4 The Value Function ................................ 17
2.5 The Optimal Value Functions ....................... 18
2.5.1 The Grid World Example ..................... 20
2.6 Markov Decision Processes ........................ 23
2.7 Learning Value Functions .......................... 25
2.8 Policy Iteration ..................................... 26
2.9 Temporal Difference Learning ...................... 28
2.10 TD Learning of the State-Action Function .......... 30
v
vi Contents
2.11 Q-Learning ........................................ 32
2.12 Eligibility Traces ................................... 33
References ........................................ 37
Chapter 3 Learning in Two-Player Matrix Games .......... 38
3.1 Matrix Games ...................................... 38
3.2 Nash Equilibria in Two-Player Matrix Games ....... 42
3.3 Linear Programming in Two-Player Zero-Sum Matrix
Games ............................................. 43
3.4 The Learning Algorithms ........................... 47
3.5 Gradient Ascent Algorithm ......................... 47
3.6 WoLF-IGA Algorithm ............................... 51
3.7 Policy Hill Climbing (PHC) ......................... 52
3.8 WoLF-PHC Algorithm .............................. 54
3.9 Decentralized Learning in Matrix Games ........... 57
3.10 Learning Automata ................................ 59
3.11 Linear Reward–Inaction Algorithm ................. 59
3.12 Linear Reward–Penalty Algorithm .................. 60
3.13 The Lagging Anchor Algorithm ..................... 60
3.14 L
R−I
Lagging Anchor Algorithm .................... 62
3.14.1 Simulation .................................. 68
References ........................................ 70
Chapter 4 Learning in Multiplayer Stochastic Games ...... 73
4.1 Introduction ....................................... 73
4.2 Multiplayer Stochastic Games ...................... 75
4.3 Minimax-Q Algorithm .............................. 79
4.3.1 2 × 2GridGame............................. 80
4.4 Nash Q-Learning ................................... 87
4.4.1 The Learning Process ......................... 95
4.5 The Simplex Algorithm ............................. 96
4.6 The Lemke–Howson Algorithm .................... 100
4.7 Nash-Q Implementation ............................ 107
4.8 Friend-or-Foe Q-Learning .......................... 111
4.9 Infinite Gradient Ascent ............................ 112
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