Analyzing the Population Based Incremental Learning Algorithm by Means of Discrete Dynamical Systems

In 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 Α 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]  Pedro Larrañaga,et al.  Combinatonal Optimization by Learning and Simulation of Bayesian Networks , 2000, UAI.

[2]  J. A. Lozano,et al.  Analyzing the PBIL Algorithm by Means of Discrete Dynamical Systems , 2000 .

[3]  J. A. Lozano,et al.  Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation , 2001 .

[4]  Markus Höhfeld,et al.  Improving the Generalization Performance of Multi-Layer-Perceptrons with Population-Based Incremental Learning , 1996, PPSN.

[5]  David E. Goldberg,et al.  A Survey of Optimization by Building and Using Probabilistic Models , 2002, Comput. Optim. Appl..

[6]  Heinz Mühlenbein The Equation for Response to Selection and Its Use for Prediction , 1997, Evolutionary Computation.

[7]  Martin Pelikan,et al.  Hill Climbing with Learning (An Abstraction of Genetic Algorithm) , 1995 .

[8]  Marcus Gallagher Multi-layer Perceptron Error Surfaces: Visualization, Structure and Modelling , 2000 .

[9]  Michèle Sebag,et al.  Extending Population-Based Incremental Learning to Continuous Search Spaces , 1998, PPSN.

[10]  Rahul Sukthankar,et al.  Evolving an intelligent vehicle for tactical reasoning in traffic , 1997, Proceedings of International Conference on Robotics and Automation.

[11]  Colin Fyfe Structured population-based incremental learning , 1998, Soft Comput..

[12]  Arnaud Berny Selection and Reinforcement Learning for Combinatorial Optimization , 2000, PPSN.

[13]  Pedro Larrañaga,et al.  Feature subset selection by genetic algorithms and estimation of distribution algorithms - A case study in the survival of cirrhotic patients treated with TIPS , 2001, Artif. Intell. Medicine.

[14]  A. Berny An adaptive scheme for real function optimization acting as a selection operator , 2000, 2000 IEEE Symposium on Combinations of Evolutionary Computation and Neural Networks. Proceedings of the First IEEE Symposium on Combinations of Evolutionary Computation and Neural Networks (Cat. No.00.

[15]  M. Servais,et al.  Function Optimisation Using Multiple-base Population Based Incremental Learning , 1997 .

[16]  Michael D. Vose The simple genetic algorithm - foundations and theory , 1999, Complex adaptive systems.

[17]  Edward R. Scheinerman Invitation to Dynamical Systems , 1995 .

[18]  Pedro Larrañaga,et al.  Optimization in Continuous Domains by Learning and Simulation of Gaussian Networks , 2000 .

[19]  H. Mühlenbein,et al.  From Recombination of Genes to the Estimation of Distributions I. Binary Parameters , 1996, PPSN.

[20]  Mike Mannion,et al.  Complex systems , 1997, Proceedings International Conference and Workshop on Engineering of Computer-Based Systems.

[21]  Heinz Mühlenbein,et al.  Evolutionary computation and Wright's equation , 2002, Theor. Comput. Sci..

[22]  Nicolas Monmarché,et al.  On the similarities between AS, BSC and PBIL: toward the birth of a new meta-heuristic , 1999 .

[23]  Michael D. Vose Random Heuristic Search , 1999, Theor. Comput. Sci..

[24]  M. Schmidt,et al.  Adding genetics to the standard PBIL algorithm , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[25]  Shumeet Baluja,et al.  Genetic Algorithms and Explicit Search Statistics , 1996, NIPS.

[26]  EngineeringSwarthmore CollegeSwarthmore Training Hidden Markov Models using Population-Based Learning , 1999 .

[27]  Shumeet Baluja,et al.  A Method for Integrating Genetic Search Based Function Optimization and Competitive Learning , 1994 .

[28]  Pedro Larrañaga,et al.  The Convergence Behavior of the PBIL Algorithm: A Preliminary Approach , 2001 .

[29]  Rich Caruana,et al.  Removing the Genetics from the Standard Genetic Algorithm , 1995, ICML.