Optimization by Model Fitting Chapter 9 Luke, Essentials of Metaheuristics, 2011 Byung-Hyun Ha R1

1 Outline  Introduction  Model fitting by classification  Model fitting with distribution  Summary

2 Introduction  Exploring and/or exploiting solution space  Construction or composition  Tweak or mutation  Recombination or crossover .. other ways?  In perspective of statistics  Population and sampling e.g., a set of all students, a sample of students for examining their height  Tweaking in search (metaheuristics) Sampling space of candidate solutions to select high-quality ones  An alternative to selecting and Tweaking by (statistical) model  Classification model Graduate students ;-), decision trees, neural networks, …  Probability distribution

3 Introduction  Example: T-problem with 5 jobs  Training by sampling 15 solutions from population of 120 ones, and question: what is the quality of ?  How? By classification or using probability distribution (23) (15) (12) (19) (11) (11) (15) (10) (24) (16) (15) (19) (16) (16) (19) (15) (20) (15) (15) (12) (15) (18) (15) (17) (15) (16) (12) (11) (19) (11) (15) (15) (17) (21) (13) (11) (19) (15) (19) (13) (17) (21) (22) (14) (18) (18) (11) (19) (18) (15) (19) (16) (13) (18) (17) (15) (21) (14) (14) (15) (19) (18) (15) (21) (15) solution space as population sampling a sample as representatives of population something we can do?

4 Model Fitting by Classification  Classification problem  Given a collection of records, to find a model for class attribute as a function of the values of other attributes  Fitting a model, or  model induction, machine learning (23) (15) (12) (11) (15) (10) (15) (19) (16) (15) (20) (15) training set a classification model induction Is a good solution? query or test Give me a good solution! generation

5 Model Fitting by Classification  Examples of classification algorithms  Graduate students by naggings of professors ;-)  Decision trees by C4.5 and ID3 c.f.,  k-nearest-neighbor (kNN) by kNN algorithm  Neural networks by backpropagation algorithm records (training set) a decision tree for classification (or prediction)

6 Model Fitting by Classification  Classification problem (revisited)  Given a collection of records, to find a model for class attribute as a function of the values of other attributes  Application of classification to search  Given a collection of solutions, to find a model for fitness as a function of the values of components of solutions  Generating children from the model  Rejection sampling with discriminative models Algorithm 115 and 117  Region-based sampling with generative models Algorithm 116  Learnable Evolution Model (LEM)  Algorithm 114 a classification model Is a good solution? Give me a good solution! discriminative model generative model rejection sampling region-based sampling

7 Model Fitting by Classification  Examples  Inducing a decision tree  Generating children from a decision tree x y goodbad good y 0.7 x 0.3 y 0.6 x 0.5 goodbad y 0.4 bad x 0.7 badgood

8 Model Fitting by Classification  Example (cont’d)  A model that specifies the probability y xx x y yy goodbad x

9 Model Fitting by Classification  Example (Talbi, 2009)  Application of rule-based classifier into crossover operator  Rules If X 4 = 5 and X 5 < 2, then class = best...  Patterns matching the rules    5  1  ...  Possible crossover?

10 Model Fitting with a Distribution  An alternative form of model  A distribution of an infinite-sized population A set of candidate solutions: a sample from population  Working with sample distribution  Estimation of Distribution Algorithm  Representing distribution of infinite population with a number of samples  Loop: sampling a set of individuals  assessing them  adjust the distribution to reflect the new fitness results  Algorithm 118: An Abstract Estimation of Distribution Algorithm (EDA)

11 Model Fitting with a Distribution  Representing distributions for genotype with n genes  Using n-dimensional histogram A fairly high-resolution grid to accurately represent the distribution c.f., kd-tree or quadtree A fairly high amount of grid points a n when distribution of each gene is discretized into a pieces  Using parametric distribution e.g., m number of gaussian curves How many gaussian curves? n-dimensional gaussian: mean vector of size n and a covariance matrix of size n 2 1,000 genes? 1,000,000 numbers

12 Model Fitting with a Distribution  Representing distributions (cont’d)  Using marginal distributions Projecting full distribution into a single dimension for each gene Representing single distribution, again 1-dimensional array as a histogram 1-dimensional gaussians as a parametric representation Size of representation? Problems (very big)?

13 Model Fitting with a Distribution  Univariate Estimation of Distribution Algorithms  Population-Based Incremental Learning (PBIL) Genes having finite discrete values n marginal distributions with n genes, initially uniform Representation? Truncation selection of good solutions sampled using distribution Gradual marginal distribution update Algorithm 119: Population-Based Incremental Learning  Univariate Marginal Distribution Algorithm (UMDA) A variation on PBIL Any selection procedure, allowed Entirely replacing distribution D each time around (  = 1) Large sample, required (why?)  Compact Genetic Algorithm (cGA) Genes having boolean values Updating each marginal distribution by pairwise comparison of individuals c.f., Modeling finite population instead of infinite one Algorithm 120: The Compact Genetic Algorithm

14 Model Fitting with a Distribution  Univariate Estimation of Distribution Algorithms (cont’d)  Real-valued representations By discretization of each marginal distribution Histogram approach Using PBIL directly By parametric approach e.g., using single gaussian Unbiased estimators of mean and variance for parameter estimation Updating each marginal distribution by linear combination Using multiple distributions c.f., Expectation Maximization (EM) algorithm

15 Model Fitting with a Distribution  Multivariate Estimation of Distribution Algorithms  Problems in univariate estimation (using marginal distributions) Assumption of no linkage between genes c.f., cooperative coevolution  An alternative Using bivariate distributions One distribution for every pair of genes Using triple genes per distribution, using quadruple …  A better way Multivariate distribution for strongly-linked genes, selectively e.g., Bayes Network c.f., not only about how good, but also about why it is good (Hierarchical) Bayesian Optimization Algorithm Algorithm 121: An Abstract Version of the Bayesian Optimization Algorithm (BOA)

16 Hybrid Metaheuristics (Talbi, 2009)  Combining with X  Mathematical programming approaches Enumeration algorithms Relaxation and decomposition methods Branch and cut and price algorithms  Constraint programming  Data mining techniques  Multiobjective optimization  Classical hybrid approaches  Low-level relay hybrids  Low-level teamwork hybrids  High-level relay hybrids  High-level teamwork hybrids

17 Summary  Exploring and/or exploiting solution space  In perspective of statistics  Model fitting by classification  Employing decision trees, kNN, neural networks  Generating children from the model  Model fitting with a distribution  Estimation of Distribution Algorithm  Representing distributions n-dimensional histogram, parametric distributions, marginal distributions  Univariate Estimation of Distribution Algorithms Problems  Multivariate Estimation of Distribution Algorithms Bayes Network