The Crowding Algorithm

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    The Crowding Approach to Niching in Genetic

    Algorithms

    Ole J. Mengshoel [email protected], NASA Ames Research Center, Mail Stop 269-3, Moffett Field, CA 94035

    David E. Goldberg [email protected] Genetic Algorithms Laboratory, Department of General Engineering, Univer-sity of Illinois at Urbana-Champaign, Urbana, IL 61801

    Abstract

    A wide range of niching techniques have been investigated in evolutionary and ge-netic algorithms. In this article, we focus on niching using crowding techniques inthe context of what we call local tournament algorithms. In addition to determinis-tic and probabilistic crowding, the family of local tournament algorithms includes theMetropolis algorithm, simulated annealing, restricted tournament selection, and par-allel recombinative simulated annealing. We describe an algorithmic and analyticalframework which is applicable to a wide range of crowding algorithms. As an ex-ample of utilizing this framework, we present and analyze the probabilistic crowdingniching algorithm. Like the closely related deterministic crowding approach, proba-bilistic crowding is fast, simple, and requires no parameters beyond those of classicalgenetic algorithms. In probabilistic crowding, sub-populations are maintained reliably,and we show that it is possible to analyze and predict how this maintenance takesplace. We also provide novel results for deterministic crowding, show how different

    crowding replacement rules can be combined in portfolios, and discuss population siz-ing. Our analysis is backed up by experiments that further increase the understandingof probabilistic crowding.

    KeywordsGenetic algorithms, niching, crowding, deterministic crowding, probabilistic crowd-ing, local tournaments, population sizing, portfolios.

    1 Introduction

    Niching algorithms and techniques constitute an important research area in geneticand evolutionary computation. The two main objectives of niching algorithms are (i)to converge to multiple, highly fit, and significantly different solutions, and (ii) to slowdown convergence in cases where only one solution is required. A wide range of nich-

    ing approaches have been investigated, including sharing (Goldberg and Richardson,1987; Goldberg et al., 1992; Darwen and Yao, 1996; Ptrowski, 1996; Mengshoel andWilkins, 1998), crowding (DeJong, 1975; Mahfoud, 1995; Harik, 1995; Mengshoel andGoldberg, 1999; Ando et al., 2005b), clustering (Yin, 1993; Hocaoglu and Sanderson,1997; Ando et al., 2005a), and other approaches (Goldberg and Wang, 1998). Our mainfocus here is on crowding, and in particular we take as starting point the crowding ap-proach known as deterministic crowding (Mahfoud, 1995). Strengths of deterministiccrowding are that it is simple, fast, and requires no parameters in addition to those of a

    c200X by the Massachusetts Institute of Technology Evolutionary Computation x(x): xxx-xxx

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    O. J. Mengshoel and D. E. Goldberg

    classical GA. Deterministic crowding has also been found to work well on test functionsas well as in applications.

    In this article, we present an algorithmic framework that supports different crowd-

    ing algorithms, including different replacement rules and the use of multiple replace-ment rules in portfolios. While our main emphasis is on the probabilistic crowding al-gorithm (Mengshoel and Goldberg, 1999; Mengshoel, 1999), we also investigate otherapproaches including deterministic crowding within. As the name suggests, proba-

    bilistic crowding is closely related to deterministic crowding, and as such shares manyof deterministic crowdings strong characteristics. The main difference is the use of aprobabilistic rather than a deterministic replacement rule (or acceptance function). Inprobabilistic crowding, stronger individuals do not always win over weaker individu-als, they win proportionally according to their fitness. Using a probabilistic acceptancefunction is shown to give stable, predictable convergence that approximates the nichingrule, a gold standard for niching algorithms.

    We also present here a framework for analyzing crowding algorithms. We con-sider performance at equilibrium and during convergence to equilibrium. Further, we

    introduce a novel portfolio mechanism and discuss the benefit of integrating differentreplacement rules by means of this mechanism. In particular, we show the advan-tage of integrating deterministic and probabilistic crowding when selection pressureunder probabilistic crowding only is low. Our analysis, which includes population siz-ing results, is backed up by experiments that confirm our analytical results and furtherincrease the understanding of how crowding and in particular probabilistic crowdingoperates.

    A final contribution of this article is to identify a class of algorithms to which bothdeterministic and probabilistic crowding belongs, local tournament algorithms. Othermembers of this class include the Metropolis algorithm (Metropolis et al., 1953), simu-lated annealing (Kirkpatrick et al., 1983), restricted tournament selection (Harik, 1995),elitist recombination (Thierens and Goldberg, 1994), and parallel recombinative simu-lated annealing (Mahfoud and Goldberg, 1995). Common to these algorithms is that

    competition is localized in that it occurs between genotypically similar individuals. Itturns out that slight variations in how tournaments are set up and take place are crucialto whether one obtains a niching algorithm or not. This class of algorithms is interesting

    because it is very efficient and can easily be applied in different settings, for exampleby changing or combining the replacement rules.

    We believe this work is significant for several reasons. As already mentioned, nich-ing algorithms reduce the effect of premature convergence or improve search for mul-tiple optima. Finding multiple optima is useful, for example, in situations where thereis uncertainty about the fitness function and robustness with respect to inaccuraciesin the fitness function is desired. Niching and crowding algorithms also play a fun-damental role in multi-objective optimization algorithms (Fonseca and Fleming, 1993;Deb, 2001) as well as in estimation of distribution algorithms (Pelikan and Goldberg,2001; Sastry et al., 2005). We enable further progress in these areas by explicitly stat-ing new and existing algorithms in an overarching framework, thereby improving theunderstanding of the crowding approach to niching. There are several informative ex-periments that compare different niching and crowding GAs (Ando et al., 2005b; Singhand Deb, 2006). However, less effort has been devoted to increasing the understand-ing of crowding from an analytical point of view, as we do here. Analytically, we alsomake a contribution with our portfolio framework, which enables easy combination ofdifferent replacement rules. Finally, while our focus is here on discrete multimodal

    2 Evolutionary Computation Volume x, Number x

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    Crowding in Genetic Algorithms

    optimization, a variant of probabilistic crowding has successfully been applied to hardmultimodal optimization problems in high-dimensional continuous spaces (Ballesterand Carter, 2003, 2004, 2006). We hope the present work will act as a catalyst to further

    progress also in this area.The rest of this article is organized as follows. Section 2 presents fundamental con-

    cepts. Section 3 discusses local tournament algorithms. Section 4 discusses our crowd-ing algorithms and replacement rules, including the probabilistic and deterministiccrowding replacement rules. In Section 5, we analyze several variants of probabilisticand deterministic crowding. In Section 6, we introduce and analyze our approach tointegrating different crowding replacement rules in a portfolio. Section 7 discusseshow our analysis compares to previous analysis, using Markov chains, of stochasticsearch algorithms including genetic algorithms. Section 8 contains experiments thatshed further light on probabilistic crowding, suggesting that it works well and in linewith our analysis. Section 9 concludes and points out directions for future research.

    2 Preliminaries

    To simplify the exposition we focus on GAs using binary strings, or bitstrings x 2f0; 1gm, of length m . Distance is measured using Hamming distance D ISTANCE(x; y)

    between two bitstrings, x;y 2 f0; 1gm. More formally, we have the following defini-tion.

    Definition 1 (Distance) Let x, y be bitstrings of length m and let, for xi 2 x and yi 2 ywhere 1 i m, d(xi; yi) = 0 ifxi = yi, d(xi; yi) = 1 otherwise. Now, the distance functionDISTANCE(x;y) is defined as follows:

    DISTANCE(x; y) =mXi=1

    d(xi; yi):

    Our DISTANCE definition is often called genotypic distance; when we discuss dis-tance in this article the above definition is generally assumed.

    A natural way to analyze a stochastic search algorithms operation on a probleminstance is to use discrete time Markov chains with discrete state spaces.

    Definition 2 (Markov chain) A (discrete time, discrete state space) Markov chain M is de-fined as a 3-tuple M = (S, V, P) where S= fs1, : : :, skg defines the set of k states whileV= (1, . . . ,k), a k-dimensional vector, defines the initial probability distribution. The con-ditional state transition probabilities Pcan be characterized by means of a k k matrix.

    Only time-homogenous Markov chains, where Pis constant, will be considered inthis article. The performance of stochastic search algorithms, both evolutionary algo-rithms and stochastic local search algorithms, can be formalized using Markov chains(Goldberg and Segrest, 1987; Nix and Vose, 1992; Harik et al., 1997; De Jo

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