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Information about Population sizing for entropy-based model buliding In genetic algorithms

Published on July 14, 2007

Author: kknsastry

Source: slideshare.net

This paper presents a population-sizing model for the entropy-based model building in genetic algorithms. Specifically, the population size required for building an accurate model is investigated. The effect of the selection pressure on population sizing is also incorporated. The proposed model indicates that the population size

required for building an accurate model scales as Θ(m log m), where m is the number of substructures and proportional to the problem size. Experiments are conducted to verify the derivations, and the results agree with the proposed model.

required for building an accurate model scales as Θ(m log m), where m is the number of substructures and proportional to the problem size. Experiments are conducted to verify the derivations, and the results agree with the proposed model.

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Motivation • Facetwise population sizing in GEC – Initial supply [Goldberg et al. 2001] – Decision-making [Goldberg et al. 1992] – Gambler’s ruin [Harik et al. 1997] • EDA—Model building is essential. • Population sizing for model building [Pelikan et al. 2003] • Better explanation and modeling are needed.

Roadmap • Entropy-based model building • Mutual information • The effect of selection • Distribution of mutual information under limited sampling • Building an accurate model • The effect of selection pressure • Conclusion

Entropy-based model building & Mutual information • Entropy: measurement of uncertainty. • Loss of entropy Gain in certainty Mutual information • Bivariate: MIMIC, BMDA • Multivariate: eCGA, BOA, EBNA, DSMGA • Most multivariate model building start from bivariate dependency detection.

Mutual information • Definition • Some facts: – –

Base: Bipolar Royal Road • Additively separable bipolar Royal road u 0 k • Given the minimal signal , the most difficult for model building. • Analytical simplicity, no gene-wise bias.

The effect of selection • 00******** and 11******** increase: • 10******** and 01******** decrease: • Define – – •

Growth of schemata and M.I. • • • Growth in mutual information

Limited sampling • In GAs, finite population limited sampling • Define two random variables: – :Signal of mutual information between two independent genes under n random samples. – :Signal of mutual information between two dependent genes under n random samples. • Ideally:

Distribution of mutual information [Hutter and Zaffalon, 2004] • •

Empirical verification

Building an accurate model • Define • Decision error • Building an accurate model • Finally

Verification of O(22k) DSMGA, m=10

Verification of O(mlogm) eCGA DSMGA

Effect of selection pressure • Quantitative, order statistics • Qualitative, consider truncation selection • Higher s – More growth of Hopt – Fewer number of effective samples

Empirical results on selection pressure Future work: Empirically, larger k larger s*

Summary and Conclusions • Refine the required population sizing for model building – From – To • Correct to • Preliminarily incorporate selection pressure into population-sizing model. – Qualitatively show the existence of s*

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