EFFICIENT AND ACCURATE PARALLEL GENETIC ALGORITHMS

Transcription

1 EFFICIENT AND ACCURATE PARALLEL GENETIC ALGORITHMS

2 GENETIC ALGORITHMS AND EVOLUTIONARY COMPUTATION

3 EFFICIENT AND ACCURATE PARALLEL GENETIC ALGORITHMS by Erick Cantti-Paz Lawrence Livermore National Lab, U.S.A. SPRINGER SCIENCE+BUSINESS MEDIA, LLC

4 Library of Congress Cataloging-in-Publication Data Cantü-Paz, Erick. Efficient and accurate parallel genetic algorithms / by Erick Cantü-Paz. p. cm. (Genetic algorithms and evolutionary computation ; 1) Includes bibliographical references and index. ISBN ISBN (ebook) DOI / Genetic algorithms-data processing. 2. Parallel processing (Electronic computers) I. Title. II. Series. QA402.5.C dc Copyright 2001 Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 2001 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, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC. Printed on acid-free paper.

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6 Series Foreword Genetic Algorithms and Evolutionary Computation David E. Goldberg, Consulting Editor University of Illinois at Urbana-Champaign Researchers and practitioners alike are increasingly turning to search, optimization, and machine-learning procedures based on natural selection and natural genetics to solve problems across the spectrum of human endeavor. These genetic algorithms and techniques of evolutionary computation are solving problems and inventing new hardware and software that rival human designs. Genetic Algorithms and Evolutionary Computation publishes research monographs, edited collections, and graduate-level texts in this rapidly growing field. Primary areas of coverage include the theory, implementation, and application of genetic algorithms (GAs), evolution strategies (ESs), evolutionary programming (EP), learning classifier systems (LCSs) and other variants of genetic and evolutionary computation (GEC). The series also publishes texts in related fields such as artificial life, adaptive behavior, artificial immune systems, agent-based systems, neural computing, fuzzy systems, and quantum computing as long as GEC techniques are part of or inspiration for the system being described. Erick CantU-Paz's monograph is the first book in this series, and in many ways it is an exemplar of the kind of book we hope to attract. The book is an elaboration of Erick's groundbreaking dissertation on the design of efficient parallel genetic algorithms. It combines facetwise and exact design theory, careful bounding empirical investigation, and a keen eye toward practice in a text that has been written in an accessible, logical, and thorough manner. In the end, it gives us critical scaling laws that accurately govern the particular parallel architectures that have been analyzed and that approximately bound many architectures that have not. I believe it is fair to say that prior to this work, the design of parallel GAs was something of an empirical black art, guided largely by ad hoc trial and error experimentation. Following this work, the solution quality, duration, connectivity, deme size, deme count, and migration rates succumb to the magic of Erick's careful analysis, experimentation, and exposition. This is not to say that parallel GAs are now completely understood. The very nature of complex systems design almost ensures that we can never get to the bottom of the subject. On the other hand, our understanding of parallel GAs is much enhanced by this work, and I urgently recommend that all readers interested in parallel genetic and evolutionary algorithms study this important book carefully and soon.

7 Contents Preface Acknowledgments 1 INTRODUCTION 1 1 An Introduction to Genetic Algorithms 2 2 A Classification of Parallel GAs 6 3 Summary 11 2 THE GAMBLER'S RUIN AND POPULATION SIZING 13 1 Background 14 2 Deciding Well Between two BBs 18 3 The Gambler's Ruin Model 20 4 Experimental Verification 23 5 Noise and Population Sizing 28 6 Effect of Selection Pressure 30 7 Summary 31 3 MASTER-SLAVE PARALLEL GAs 33 1 Background 34 2 Analysis 36 3 Experiments 42 4 Asynchronous Master-Slaves 43 5 A Distributed Panmictic Population 45 6 Summary 47 4 BOUNDING CASES OF GAs WITH MULTIPLE DEMES 49 1 Background 50 2 Parallel Speedups 54 3 Isolated Demes 55 4 Fully-Connected Demes 58 5 Summary 64 5 MARKOV CHAIN MODELS OF MULTIPLE DEMES 67 1 Fully-Connected Demes with Maximum Migration Rates 68 2 Arbitrary Migration Rates 73 3 Arbitrary Topologies 74 4 Experiments 77 5 Summary 80 6 MIGRATION RATES AND OPTIMAL TOPOLOGIES 81 1 Degree of Connectivity 82 2 Multiple Epochs and Choosing a Topology 87 xi xv

8 x PARALLEL GENETIC ALGORITHMS 3 Parallel Demes in the Long Run 94 4 Summary 96 7 MIGRATION AND SELECTION PRESSURE 97 1 Selection Pressure 98 2 Takeover Times Selection Intensity Experiments Superlinear Speedups Variance and the Higher Cumulants Summary FINE-GRAINED AND HIERARCHICAL PARALLEL GAs Fine-Grained Parallel GAs Hierarchical Parallel GAs Optimal Hierarchical Parallel GAs An Example of Optimal Design Summary SUMMARY, EXTENSIONS, AND CONCLUSIONS Summary Extensions Conclusions 141 References 145 Index 159

9 Preface As genetic algorithms (GAs) become increasingly popular, they are applied to difficult problems that may require considerable computations. In such cases, parallel implementations of GAs become necessary to reach high-quality solutions in reasonable times. But, even though their mechanics are simple, parallel GAs are complex non-linear algorithms that are controlled by many parameters, which are not well understood. This book is about the design of parallel GAs. It presents theoretical developments that improve our understanding of the effect of the algorithm's parameters on its search for quality and efficiency. These developments are used to formulate guidelines on how to choose the parameter values that minimize the execution time while consistently reaching solutions of high quality. The book can be read in several ways, depending on the readers' interests and their previous knowledge about these algorithms. Newcomers to the field will find the background material in each chapter useful to become acquainted with previous work, and to understand the problems that must be faced to design efficient and reliable algorithms. Potential users of parallel GAs that may have doubts about their practicality or reliability may be more confident after reading this book and understanding the algorithms better. Those who are ready to try a parallel GA on their applications may choose to skim through the background material, and use the results directly without following the derivations in detail. These readers will find that using the results can help them to choose the type of parallel GA that best suits their needs, without having to invest the time to implement and test various options. Once that is settled, even the most experienced users dread the long and frustrating experience of configuring their algorithms by trial and error. The guidelines contained herein will shorten dramatically the time spent tweaking the algorithm, although some experimentation may still be needed for fine-tuning. In addition to the results that practitioners will find applicable to the design of parallel GAs, those interested in the mathematical analysis of GAs may find the techniques used here useful for their own endeavors. Indeed, some of the

10 xii PARALLEL GENETIC ALGORITHMS analysis presented here is relevant not only to parallel GAs, but also to their serial counterparts. This book is based on my Ph.D. dissertation (Cantu-Paz, 1999a) The presentation of the material assumes that the reader has a good understanding of the basic concepts of GAs and statistics. However, some of the derivations use concepts that may not be familiar to many readers, and so I explain them briefly the first time that they appear. Organization The first chapter contains a brief introduction to GAs. In addition, it presents the four common types of parallel GAs, and it explains the design problems that users face when using these algorithms. Chapters 2 and 3 are about GAs with a single population. Chapter 2 deals with finding an appropriate population size for simple serial GAs, and introduces an abstract model of GAs that will be used in the remainder of the book to study parallel GAs. Chapter 3 shows how to parallelize GAs using a simple master-slave algorithm, and how to minimize its execution time. Since this is the simplest parallel algorithm that we examine, we consider this case to be the lower bound on the acceptable performance of parallel GAs. Other parallel GAs should perform better than the simple master-slave. Most of the book focuses on algorithms with multiple populations. These are the most popular parallel GAs and potentially the most efficient, because they require fewer communications than the master-slave. Unfortunately, they are also the most complex, as one must face difficult and interrelated design alternatives. Some of the main problems are to determine (1) the size and the number of demes (subpopulations), (2) the topology that interconnects the demes, and (3) the migration rate that controls how many individuals migrate between demes. It is difficult to study all these problems at once, so we consider first two cases that represent lower and upper bounds of the migration rate and the connectivity between demes. Practitioners do not use these extreme cases very often, but the analysis sheds some light on the intermediate configurations, and introduces the methodology that will be used in the following chapters. The study of bounding cases spans two chapters. First, Chapter 4 extends the population sizing theory of serial GAs to the parallel bounding cases, and

11 Preface xiii shows how to minimize the execution time while maintaining a constant target solution quality. Then, Chapter 5 uses Markov chains to analyze the long-term behavior of communicating demes. Chapter 5 also introduces the first model for intermediate topologies. This initial model is very accurate, but it requires considerable computations, and may not be practical to study large numbers of demes. Nevertheless, the Markov chains are used to verify the accuracy of the simpler models derived in the next chapter. Chapter 6 finds approximate equations that capture the effects of the topology and migration rate on the quality and cost of multideme GAs. These simple models are then used to minimize the execution time. Chapter 7 studies how the individuals that migrate between populations may affect the speed of convergence of the algorithm. The controversial issue of superlinear speedups in parallel GAs is explored in detail in this chapter. At this point, the book has investigated the benefits of master-slave and multiple-deme parallel GAs in isolation, but they may be combined into hierarchical algorithms that offer even greater advantages. Chapter 8 describes how to minimize the execution time by allocating the available processors to combinations of demes and slaves. This chapter also includes a discussion on fine-grained parallel GAs, and it contains a complete step-by-step example that illustrates how to determine the optimal configuration of a parallel GA for a particular problem and hardware environment. Finally, Chapter 9 contains a summary of the results, recommendations for further research, and the conclusions of this study.

12 Acknowledgments Undoubtedly, the person who most influenced this work is my Ph.D. advisor, David E. Goldberg. I wish to thank him for his guidance, support, and encouragement during my time at the University of Illinois. Besides of many technical and methodological issues, I learned from him much about the "human side" of engineering. I wish to thank Prof. Geneva Belford, Prof. Sylvian Ray, and Prof. Josep Torrellas for their comments on my work and for their patience when delays occurred. During my stay at Illinois, I had the opportunity to learn many different things as I worked with each of them. I was very fortunate to be surrounded by a very productive and cheerful research group in the Illinois Genetic Algorithms Lab. There are very few places in the world where one has access to so many resources to do research on genetic algorithms and have so much fun at the same time. I wish to thank all the students, visitors, and librarians at IlliGAL for their friendship and for all the technical discussions. Numerous anonymous reviewers made useful comments on parts of this book that have been submitted to conferences and journals. I am grateful for their careful reading of the papers, and for their suggestions that helped to improve the exposition of some of the ideas contained here. My explorations of parallel GAs began when I was an undergraduate student at the Instituto Tecno16gico Aut6nomo de Mexico. Back then, my advisor, Marcelo Mejia, encouraged me to work on a senior project on parallel GAs, a topic about which neither of us knew much. I was lucky to have the support of Marcelo and the help of other professors such as Francisco Cervantes, Jose Incera, and Carlos Zozaya, who also served as a model and inspiration. That experience with the senior project taught me about the fun of research, which continued in graduate school and beyond. My sisters and I were blessed with wonderful parents who gave us the perfect example of how to live a happy life. No words can express my gratitude for everything that they have done; I can only hope to be able to transmit their wisdom to my own children. Finally, I thank Javiera, my wife, for all her love and understanding, both in graduate school and during the preparation of this book. Her support kept me going during hard and sometimes frustrating times. My research was sponsored by a number of sources: a Fulbright-Garda Robles Fellowship, administered by USIA and by CONACyT; a teach-

13 xvi PARALLEL GENETIC ALGORITHMS ing assistantship from the Department of Computer Science, UIUC; and a research assistantship from the Beckman Institute, UIUC. The work was sponsored by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grants number F , F and F Research funding for this project was also provided by a grant from the US Army Research Laboratory Program, Cooperative Agreement DAALOl The US Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies and endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the US Government. Portions of this work were performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.