Results 1 
9 of
9
Analyzing the PBIL Algorithm by Means of Discrete Dynamical Systems
 Complex Systems
"... this paper the convergence behavior of the Population Based Incremental Learning algorithm (PBIL) is analyzed using discrete dynamical systems. A discrete dynamical system is associated with the PBIL algorithm. We demonstrate that the behavior of the PBIL algorithm follows the iterates of the discre ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
this paper the convergence behavior of the Population Based Incremental Learning algorithm (PBIL) is analyzed using discrete dynamical systems. A discrete dynamical system is associated with the PBIL algorithm. We demonstrate that the behavior of the PBIL algorithm follows the iterates of the discrete dynamical system for a long time when the parameter # is near zero. We show that all the points of the search space are fixed points of the dynamical system, and that the local optimum points for the function to optimize coincide with the stable fixed points. Hence it can be deduced that the PBIL algorithm converges to the global optimum in unimodal functions. 1. Introduction
Form Invariance and Implicit Parallelism
, 2001
"... Holland's schema theorem (an inequality) may be viewed as an attempt to understand genetic search in terms of a coarse graining of the state space. Stephens and Waelbroeck developed that perspective, sharpening the schema theorem to an equality. Of particular interest, ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
Holland's schema theorem (an inequality) may be viewed as an attempt to understand genetic search in terms of a coarse graining of the state space. Stephens and Waelbroeck developed that perspective, sharpening the schema theorem to an equality. Of particular interest,
A normed space of genetic operators with applications to scalability issues
 Evolutionary Computation
, 2001
"... We define an abstract normed vector space where the genetic operators are elements. This is used to define the disturbance of the generational operator G as the distance between the crossover and mutation operator (combined) and the identity. This quantity appears in a bound on the variance of fixed ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
We define an abstract normed vector space where the genetic operators are elements. This is used to define the disturbance of the generational operator G as the distance between the crossover and mutation operator (combined) and the identity. This quantity appears in a bound on the variance of fixedpoint populations, and in a bound on the force kv;G(v)k that applies to the optimal population v. When analyzed for the case of fixedlength binary strings, a connection is shown between these measures and the size of the search space. Guides for parameter settings are given, if population convergence is required as the string length tends to infinity.
The Royal Road Not Taken: A ReExamination of the Reasons for GA Failure on R1
"... Abstract. Previous work investigating the performance of genetic algorithms (GAs) has attempted to develop a set of fitness landscapes, called “Royal Roads ” functions, which should be ideally suited for search with GAs. Surprisingly, many studies have shown that genetic algorithms actually perform ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. Previous work investigating the performance of genetic algorithms (GAs) has attempted to develop a set of fitness landscapes, called “Royal Roads ” functions, which should be ideally suited for search with GAs. Surprisingly, many studies have shown that genetic algorithms actually perform worse than random mutation hillclimbing on these landscapes, and several different explanations have been offered to account for these observations. Using a detailed stochastic model of genetic search on R1, we attempt to determine a lower bound for the required number of function evaluations, and then use it to evaluate the performance of an actual genetic algorithm on R1. 1
Stochastic Refinement
"... Abstract. The research presented in this paper is motivated by the following question. How can the generality order of clauses and the relevant concepts such as refinement be adapted to be used in a stochastic search? To address this question we introduce the concept of stochastic refinement operato ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. The research presented in this paper is motivated by the following question. How can the generality order of clauses and the relevant concepts such as refinement be adapted to be used in a stochastic search? To address this question we introduce the concept of stochastic refinement operators and adapt a framework, called stochastic refinement search. In this paper we introduce stochastic refinements of a clause as a probability distribution over a set of clauses. This probability distribution can be viewed as a prior in a stochastic ILP search. We study the properties of a stochastic refinement search as two well known Markovian approaches: 1) Gibbs sampling algorithm and 2) random heuristic search. As a Gibbs sampling algorithm, a stochastic refinement search iteratively generates random samples from the hypothesis space according to a posterior distribution. We show that a minimum sample size can be set so that in each iteration a consistent clause is generated with a high probability. We study the stochastic refinement operators within the framework of random heuristic search and use this framework to characterise stochastic search methods in some ILP systems. We also study a special case of stochastic refinement search where refinement operators are defined with respect to subsumption order relative to a bottom clause. This paper also provided some insights to explain the relative advantages of using stochastic lgglike operators as in the ILP systems Golem and ProGolem. 1
Theory of the Simple Genetic Algorithm with αSelection
"... Genetic algorithms are random heuristic search (RHS) algorithms with a wide range of applications in adaptation and optimisation problems. The most advanced approach for a general theory of genetic algorithms is offered by the dynamical system model which describes the stochastic trajectory of a pop ..."
Abstract
 Add to MetaCart
Genetic algorithms are random heuristic search (RHS) algorithms with a wide range of applications in adaptation and optimisation problems. The most advanced approach for a general theory of genetic algorithms is offered by the dynamical system model which describes the stochastic trajectory of a population under the dynamics of a genetic algorithm with the help of an underlying deterministic heuristic function and its fixed points. However, even for the simple genetic algorithm (SGA) with fitnessproportional selection, crossover and mutation the determination of the population trajectory and the fixed points of the heuristic function is unfeasible for practical problem sizes. In order to simplify the mathematical analysis αselection is introduced in this paper. Based on this selection scheme it is possible to derive the dynamical system model and the fixed points in closed form. Although the heuristic function is not compatible with the equivalence relation imposed by schemata in the strict sense a simple coarsegrained system model with a single exogenous parameter is derivable for a given schemata family. In addition to the theoretical analysis experimental results are presented which confirm the theoretical predictions.
THEORY OF GENETIC ALGORITHMS WITH αSELECTION
"... Genetic algorithms are random heuristic search (RHS) algorithms for adaptive systems with a wide range of applications in search, optimisation, pattern recognition and machine learning as well as signal processing. Despite their widespread use a general theory is still lacking. A promising approach ..."
Abstract
 Add to MetaCart
Genetic algorithms are random heuristic search (RHS) algorithms for adaptive systems with a wide range of applications in search, optimisation, pattern recognition and machine learning as well as signal processing. Despite their widespread use a general theory is still lacking. A promising approach is offered by the dynamical system model which describes the stochastic trajectory of a population under the dynamics of a genetic algorithm with the help of an underlying deterministic heuristic function and its fixed points. However, even for the simple genetic algorithm (SGA) with fitnessproportional selection, crossover and mutation the determination of the population trajectory and the fixed points of the heuristic function is unfeasible for practical problem sizes. In order to simplify the mathematical analysis αselection is introduced in this paper. Based on this strong selection scheme it is possible to derive the dynamical system model and the respective fixed points in closed form. In addition to the theoretical analysis experimental results are presented. Index Terms — Genetic algorithm, αselection, random heuristic search, dynamical system model.