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Modelling Evolutionary Algorithms with Stochastic Differential Equations.

Jorge Pérez Heredia1

  • 1Department of Computer Science, University of Sheffield, Sheffield S1 4DP, United Kingdom jperezheredia1@sheffield.ac.uk.

Evolutionary Computation
|November 21, 2017
PubMed
Summary
This summary is machine-generated.

Stochastic differential equations (SDEs) offer a novel mathematical approach to analyze evolutionary algorithms (EAs). This method captures stochasticity, unlike ODEs, and simplifies analysis compared to Markov chains, especially for fixed budget scenarios.

Keywords:
(1+1) EASSWMTheorydriftevolutionary algorithmsfixed budgetlocal searchmetropolismodelling.non-elitismrandom walkstochastic differential equations

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Area of Science:

  • * Computational intelligence and machine learning.
  • * Mathematical analysis of algorithms.

Background:

  • * Traditional mathematical models for evolutionary algorithms (EAs), like ordinary differential equations (ODEs) and Markov chains, offer analytical advantages but often discard process information.
  • * There is a need for modeling techniques that retain stochasticity while facilitating analysis.

Purpose of the Study:

  • * To introduce and explore the utility of stochastic differential equations (SDEs) for analyzing the behavior of evolutionary algorithms (EAs).
  • * To develop analytical tools for fixed budget scenarios and provide new theoretical results for EA runtime analysis.

Main Methods:

  • * Application of stochastic differential equations (SDEs) to model the dynamics of evolutionary algorithms.
  • * Derivation of analogues to additive and multiplicative drift theorems.
  • * Development of a generalized multiplicative drift theorem applicable to non-elitist EAs.

Main Results:

  • * SDEs provide a balance between analytical tractability and information retention regarding process stochasticity.
  • * New drift theorems are presented, extending runtime analysis for EAs, including non-elitist variants.
  • * The framework is demonstrated on several heuristics: Random Walk (RW), Random Local Search (RLS), (1+1) EA, Metropolis Algorithm (MA), and Strong Selection Weak Mutation (SSWM).

Conclusions:

  • * Stochastic differential equations (SDEs) are a powerful and suitable tool for the mathematical analysis of evolutionary algorithms.
  • * The derived theorems offer valuable insights into EA performance, even in challenging optimization scenarios.
  • * The SDE approach enhances the understanding of various evolutionary computation methods.