Estimation of Distribution Algorithms with Kikuchi Approximations

The question of finding feasible ways for estimating probability distributions is one of the main challenges for Estimation of Distribution Algorithms (EDAs). To estimate the distribution of the selected solutions, EDAs use factorizations constructed according to graphical models. The class of factorizations that can be obtained from these probability models is highly constrained. Expanding the class of factorizations that could be employed for probability approximation is a necessary step for the conception of more robust EDAs. In this paper we introduce a method for learning a more general class of probability factorizations. The method combines a reformulation of a probability approximation procedure known in statistical physics as the Kikuchi approximation of energy, with a novel approach for finding graph decompositions. We present the Markov Network Estimation of Distribution Algorithm (MN-EDA), an EDA that uses Kikuchi approximations to estimate the distribution, and Gibbs Sampling (GS) to generate new points. A systematic empirical evaluation of MN-EDA is done in comparison with different Bayesian network based EDAs. From our experiments we conclude that the algorithm can outperform other EDAs that use traditional methods of probability approximation in the optimization of functions with strong interactions among their variables.

[1]  J. McCall,et al.  Incorporating a Metropolis method in a distribution estimation using Markov random field algorithm , 2005, 2005 IEEE Congress on Evolutionary Computation.

[2]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[3]  C. Reeves,et al.  Properties of fitness functions and search landscapes , 2001 .

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

[5]  P. Spirtes,et al.  Causation, prediction, and search , 1993 .

[6]  Heinz Mühlenbein,et al.  Schemata, Distributions and Graphical Models in Evolutionary Optimization , 1999, J. Heuristics.

[7]  Pedro Larrañaga,et al.  Estimation of Distribution Algorithms , 2002, Genetic Algorithms and Evolutionary Computation.

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

[9]  Payam Pakzad,et al.  Belief Propagation and Statistical Physics , 2002 .

[10]  Vasant Honavar,et al.  Evolutionary Synthesis of Bayesian Networks for Optimization , 2001 .

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

[12]  Steffen L. Lauritzen,et al.  Graphical models in R , 1996 .

[13]  R. Santana,et al.  The mixture of trees Factorized Distribution Algorithm , 2001 .

[14]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[15]  C. Bron,et al.  Algorithm 457: finding all cliques of an undirected graph , 1973 .

[16]  Wainwrighty Tommi Jaakkola,et al.  Tree-based reparameterization framework for analysis of beliefpropagation and related algorithmsMartin , 2001 .

[17]  Roberto Santana,et al.  The Factorized Distribution Algorithm and The Junction Tree: A Learning Perspective , 2005 .

[18]  R. Kikuchi.  A Theory of Cooperative Phenomena , 1951 .

[19]  Heinz Mühlenbein,et al.  Evolutionary Algorithms and the Boltzmann Distribution , 2002, FOGA.

[20]  David Maxwell Chickering,et al.  Dependency Networks for Inference, Collaborative Filtering, and Data Visualization , 2000, J. Mach. Learn. Res..

[21]  Roberto Santana.  A Markov Network Based Factorized Distribution Algorithm for Optimization , 2003, ECML.

[22]  Jordan B. Pollack,et al.  Modeling Building-Block Interdependency , 1998, PPSN.

[23]  G. Harik.  Linkage Learning via Probabilistic Modeling in the ECGA , 1999 .

[24]  T. Morita.  Formal Structure of the Cluster Variation Method , 1994 .

[25]  J. Jeffry Howbert,et al.  The Maximum Clique Problem , 2007 .

[26]  D. Goldberg,et al.  BOA: the Bayesian optimization algorithm , 1999 .

[27]  Heinz Mühlenbein,et al.  Blocked stochastic sampling versus Estimation of Distribution Algorithms , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).

[28]  William T. Freeman,et al.  Understanding belief propagation and its generalizations , 2003 .

[29]  William T. Freeman,et al.  Constructing free-energy approximations and generalized belief propagation algorithms , 2005, IEEE Transactions on Information Theory.