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Estimate Hitting Time by Hitting Probability for Elitist Evolutionary Algorithms

Jun He, Siang Yew Chong, Xin Yao

TL;DR

A novel drift analysis method is developed to estimate hitting probability, where paths are introduced to handle multimodal fitness landscapes and explicit expressions are constructed to compute hitting probability, significantly simplifying the estimation process.

Abstract

Drift analysis is a powerful tool for analyzing the time complexity of evolutionary algorithms. However, it requires manual construction of drift functions to bound hitting time for each specific algorithm and problem. To address this limitation, general linear drift functions were introduced for elitist evolutionary algorithms. But calculating linear bound coefficients effectively remains a problem. This paper proposes a new method called drift analysis of hitting probability to compute these coefficients. Each coefficient is interpreted as a bound on the hitting probability of a fitness level, transforming the task of estimating hitting time into estimating hitting probability. A novel drift analysis method is then developed to estimate hitting probability, where paths are introduced to handle multimodal fitness landscapes. Explicit expressions are constructed to compute hitting probability, significantly simplifying the estimation process. One advantage of the proposed method is its ability to estimate both the lower and upper bounds of hitting time and to compare the performance of two algorithms in terms of hitting time. To demonstrate this application, two algorithms for the knapsack problem, each incorporating feasibility rules and greedy repair respectively, are compared. The analysis indicates that neither constraint handling technique consistently outperforms the other.

Estimate Hitting Time by Hitting Probability for Elitist Evolutionary Algorithms

TL;DR

A novel drift analysis method is developed to estimate hitting probability, where paths are introduced to handle multimodal fitness landscapes and explicit expressions are constructed to compute hitting probability, significantly simplifying the estimation process.

Abstract

Drift analysis is a powerful tool for analyzing the time complexity of evolutionary algorithms. However, it requires manual construction of drift functions to bound hitting time for each specific algorithm and problem. To address this limitation, general linear drift functions were introduced for elitist evolutionary algorithms. But calculating linear bound coefficients effectively remains a problem. This paper proposes a new method called drift analysis of hitting probability to compute these coefficients. Each coefficient is interpreted as a bound on the hitting probability of a fitness level, transforming the task of estimating hitting time into estimating hitting probability. A novel drift analysis method is then developed to estimate hitting probability, where paths are introduced to handle multimodal fitness landscapes. Explicit expressions are constructed to compute hitting probability, significantly simplifying the estimation process. One advantage of the proposed method is its ability to estimate both the lower and upper bounds of hitting time and to compare the performance of two algorithms in terms of hitting time. To demonstrate this application, two algorithms for the knapsack problem, each incorporating feasibility rules and greedy repair respectively, are compared. The analysis indicates that neither constraint handling technique consistently outperforms the other.

Paper Structure

This paper contains 35 sections, 19 theorems, 112 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. The Markov Chain for the Elitist EA
  5. Probability of Transition between Fitness Levels
  6. Digraphs and Paths
  7. Hitting Time and Hitting Probability
  8. Lower and Upper Bounds on Hitting Time
  9. Previous Results of Drift Analysis Using Linear Drift Functions
  10. Previous Results in the Fitness Level Method
  11. Estimate Hitting time by Hitting Probability
  12. Exact Hitting Time
  13. Lower and Upper Bounds on Hitting Time
  14. Drift Analysis of Hitting Probability
  15. Drift Function
  16. ...and 20 more sections

Key Result

Proposition 1

Given a fitness level partition $(S_0, \ldots, S_K)$ and two levels $k,\ell: 0\le \ell<k \le K$, the hitting probability $h(X_k,S_\ell)$ satisfies

Figures (4)

  • Figure 1: The x-axis represents the state and the y-axis represents the fitness value. Each arc is a transition. Two paths from $S_{12}$ to $S_1$ are highlighted.
  • Figure 2: The digraph of the two (1+1) EAs on Instance KP1. Vertices represent fitness levels (feasible solution area). Arcs represent transitions. $n=12$.
  • Figure 3: The digraph of the two (1+1) EAs on Instance KP2. Vertices represent fitness levels (feasible solution area). Arcs represent transitions. $n=12$.
  • Figure 4: The digraph of the two (1+1) EAs on Instance KP3. Vertices represent fitness levels (feasible solution area). Arcs represent transitions. $n=12$.

Theorems & Definitions (23)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Definition 3
  • Theorem 3
  • Corollary 1
  • ...and 13 more