optimization by model fitting chapter 9 luke, essentials of metaheuristics, 2011 byung-hyun ha r1

Optimization by Model Fitting

Chapter 9

Luke, Essentials of Metaheuristics, 2011

Byung-Hyun Ha

R1

2

Outline

Introduction

Model fitting by classification

Model fitting with distribution

Summary

3

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

4

Introduction

Example: T-problem with 5 jobs Training by sampling 15 solutions from population of 120 ones,

• and question: what is the quality of 2-5-3-1-0?

How?• By classification or using probability distribution

0-2-3-4-1(23) 4-1-0-3-2(15) 1-2-3-4-0(12) 0-3-1-2-4(19) 1-4-2-3-0(11)

1-2-4-3-0(11) 1-3-4-0-2(15) 2-1-4-3-0(10) 0-3-2-4-1(24) 1-3-2-0-4(16)

3-4-2-1-0(15) 0-1-3-2-4(19) 2-0-3-1-4(16) 2-4-3-0-1(16) 3-1-0-2-4(19)

4-3-2-1-0(15) 0-4-2-3-1(20) 3-4-0-1-2(15) 4-1-3-0-2(15) 3-1-4-2-0(12)

4-1-3-0-2(15) 1-0-2-3-4(18) 0-4-3-1-2(15) 1-0-2-4-3(17) 3-4-2-1-0(15)

4-2-0-1-3(16) 1-2-3-4-0(12) 4-1-2-3-0(11) 0-4-2-1-3(19) 1-2-4-3-0(11)

0-1-4-3-2(15) 3-2-4-1-0(15) 4-2-3-0-1(17) 0-4-3-2-1(21) 3-1-2-4-0(13)

1-4-2-3-0(11) 0-3-1-2-4(19) 4-1-3-0-2(15) 3-0-1-2-4(19) 2-4-3-1-0(13)

2-3-4-0-1(17) 0-3-4-2-1(21) 0-2-4-3-1(22) 0-4-1-2-3(14) 4-3-2-0-1(18)

3-4-2-0-1(18) 1-4-2-0-3(11) 4-0-2-1-3(19) 0-1-2-3-4(18) 4-3-2-1-0(15)

...

0-1-3-2-4(19) 2-4-3-0-1(16) 2-4-3-1-0(13)

1-0-2-3-4(18) 1-0-2-4-3(17) 3-4-1-0-2(15)

0-3-4-2-1(21) 0-4-1-2-3(14) 2-1-3-0-4(14)

4-1-0-3-2(15) 0-3-1-2-4(19) 2-0-4-1-3(18)

3-2-4-1-0(15) 0-4-3-2-1(21) 4-0-3-1-2(15)

solution space as populationsampling

a sample asrepresentatives of population

something we can do?

5

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

0-2-3-4-1(23) 4-1-0-3-2(15) 1-2-3-4-0(12)

1-2-4-3-0(11) 1-3-4-0-2(15) 2-1-4-3-0(10)

3-4-2-1-0(15) 0-1-3-2-4(19) 2-0-3-1-4(16)

4-3-2-1-0(15) 0-4-2-3-1(20) 3-4-0-1-2(15)

training set

a classificationmodel

induction

Is 2-5-3-1-0 a good solution?

query or test

Give me a good solution!

generation

6


Examples of classification algorithms Graduate students by naggings of professors ;-) Decision trees by C4.5 and ID3

• c.f., http://www-users.cs.umn.edu/~kumar/dmbook/ch4.pdf

k-nearest-neighbor (kNN) by kNN algorithm Neural networks by backpropagation algorithm

Tid Refund MaritalStatus

TaxableIncome Cheat

1 Yes Single 125K No

2 No Married 100K No

3 No Single 70K No

4 Yes Married 120K No

5 No Divorced 95K Yes

6 No Married 60K No

7 Yes Divorced 220K No

8 No Single 85K Yes

9 No Married 75K No

10 No Single 90K Yes10

Refund

MarSt

TaxInc

YESNO

NO

NO

Yes No

MarriedSingle, Divorced

< 80K > 80K

records (training set)

a decision tree for classification (or prediction)

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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 classificationmodel

Is 2-5-3-1-0 a good solution?

Give me a good solution!

discriminative model

generative model

rejection sampling

region-based sampling

8


Examples Inducing a decision tree Generating children from a decision tree

x

y

1.0

1.0

0.4

0.6

0.7

0.70.50.30.0

good bad

bad good

y

0.7

x

0.3

y

0.6

x

0.5

good bady

0.4

bad

x

0.7

bad good

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Example (cont’d) A model that specifies the probability

y

x x

x

y

y y

1.0

1.0

0.4

0.6

0.7

0.70.50.30.0

good bad

0.00 0.75

0.7

0.3

0.6

0.5

0.80 0.25

0.4

0.14

x

0.7

0.00 1.00

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Example (Talbi, 2009) Application of rule-based classifier into crossover operator Rules

• If X4 = 5 and X5 < 2, then class = best

• ...

Patterns matching the rules 5 1 • ...

Possible crossover?• 2 1 7 2 1 3 4 3 2 1 7 5 8 1 7 4• 3 2 5 7 8 0 7 4 3 2 5 5 1 1 4 3

11

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)

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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• an 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 n2

1,000 genes? 1,000,000 numbers

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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)?

14


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

15


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

16


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 Algorit

hm (BOA)

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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

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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

optimization by model fitting chapter 9 luke, essentials of metaheuristics, 2011 byung-hyun ha r1

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