Convergence properties of incremental Bayesian evolutionary algorithms with single Markov chains

Bayesian evolutionary algorithms (BEAs) are a probabilistic model of evolutionary computation for learning and optimization. Starting from a population of individuals drawn from a prior distribution, a Bayesian evolutionary algorithm iteratively generates a new population by estimating the posterior fitness distribution of parent individuals and then sampling from the distribution offspring individuals by variation and selection operators. Due to the non-homogeneity of their Markov chains, the convergence properties of the full BEAs are difficult to analyze. However, recent developments in Markov chain analysis for dynamic Monte Carlo methods provide a useful tool for studying asymptotic behaviors of adaptive Markov chain Monte Carlo methods including evolutionary algorithms. We apply these results to Investigate the convergence properties of Bayesian evolutionary algorithms with incremental data growth. We study the case of BEAs that generate single chains or have populations of size one. It is shown that under regularity conditions the incremental BEA asymptotically converges to a maximum a posteriori (MAP) estimate which is concentrated around the maximum likelihood estimate. This result relies on the observation that increasing the number of data items has an equivalent effect of reducing the temperature in simulated annealing.

[1]  Bruce E. Hajek,et al.  Cooling Schedules for Optimal Annealing , 1988, Math. Oper. Res..

[2]  H. Haario,et al.  Simulated annealing process in general state space , 1991, Advances in Applied Probability.

[3]  Günter Rudolph,et al.  Convergence analysis of canonical genetic algorithms , 1994, IEEE Trans. Neural Networks.

[4]  M. Schervish.  Theory of Statistics , 1995 .

[5]  Gerhard Winkler,et al.  Image analysis, random fields and dynamic Monte Carlo methods: a mathematical introduction , 1995, Applications of mathematics.

[6]  David B. Fogel,et al.  Using fitness distributions to design more efficient evolutionary computations , 1996, Proceedings of IEEE International Conference on Evolutionary Computation.

[7]  R. Tweedie,et al.  Rates of convergence of the Hastings and Metropolis algorithms , 1996 .

[8]  R. Tweedie,et al.  Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms , 1996 .

[9]  Stephen S. Wilson,et al.  Random iterative models , 1996 .

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

[11]  David H. Wolpert,et al.  No free lunch theorems for optimization , 1997, IEEE Trans. Evol. Comput..

[12]  David B. Fogel,et al.  Evolutionary Computation: The Fossil Record , 1998 .

[13]  Olivier François,et al.  An evolutionary strategy for global minimization and its Markov chain analysis , 1998, IEEE Trans. Evol. Comput..

[14]  D. Fogel,et al.  Fitness distributions in evolutionary computation: motivation and examples in the continuous domain. , 1999, Bio Systems.

[15]  Byoung-Tak Zhang.  A Bayesian framework for evolutionary computation , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[16]  H. Muhlenbein,et al.  The Factorized Distribution Algorithm for additively decomposed functions , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[17]  Byoung-Tak Zhang.  Bayesian Evolutionary Algorithms for Learning and Optimization , 2000 .