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Runtime Analysis for the NSGA-II: Provable Speed-Ups From Crossover

Benjamin Doerr, Zhongdi Qu

TL;DR

It is proved that the NSGA-II optimizes the OneJumpZeroJump benchmark asymptotically faster when crossover is employed, the first time such an advantage of crossover is proven for the NS GA-II.

Abstract

Very recently, the first mathematical runtime analyses for the NSGA-II, the most common multi-objective evolutionary algorithm, have been conducted. Continuing this research direction, we prove that the NSGA-II optimizes the OneJumpZeroJump benchmark asymptotically faster when crossover is employed. Together with a parallel independent work by Dang, Opris, Salehi, and Sudholt, this is the first time such an advantage of crossover is proven for the NSGA-II. Our arguments can be transferred to single-objective optimization. They then prove that crossover can speed up the $(μ+1)$ genetic algorithm in a different way and more pronounced than known before. Our experiments confirm the added value of crossover and show that the observed advantages are even larger than what our proofs can guarantee.

Runtime Analysis for the NSGA-II: Provable Speed-Ups From Crossover

TL;DR

It is proved that the NSGA-II optimizes the OneJumpZeroJump benchmark asymptotically faster when crossover is employed, the first time such an advantage of crossover is proven for the NS GA-II.

Abstract

Very recently, the first mathematical runtime analyses for the NSGA-II, the most common multi-objective evolutionary algorithm, have been conducted. Continuing this research direction, we prove that the NSGA-II optimizes the OneJumpZeroJump benchmark asymptotically faster when crossover is employed. Together with a parallel independent work by Dang, Opris, Salehi, and Sudholt, this is the first time such an advantage of crossover is proven for the NSGA-II. Our arguments can be transferred to single-objective optimization. They then prove that crossover can speed up the genetic algorithm in a different way and more pronounced than known before. Our experiments confirm the added value of crossover and show that the observed advantages are even larger than what our proofs can guarantee.
Paper Structure (17 sections, 8 theorems, 3 equations, 2 figures, 2 tables)

This paper contains 17 sections, 8 theorems, 3 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Previous Works
  3. Preliminaries
  4. The NSGA-II Algorithm
  5. The ($\mu+1$) Genetic Algorithm
  6. Benchmark Problems
  7. Runtime Analysis of the NSGA-II with Crossover
  8. Runtime Analysis of the $(\mu + 1)$ GA
  9. Experiments
  10. The NSGA-II Optimizing OneJumpZeroJump
  11. Settings
  12. Results
  13. The ($\mu+1$) GA Optimizing Jump
  14. Settings
  15. Runtime
  16. ...and 2 more sections

Key Result

Lemma 1

Consider an iteration $t$ of the NSGA-II algorithm optimizing the $\textsc{OneJumpZeroJump$_{n,k}$}\xspace$ benchmark with population size $N = c(n-2k+3)$ for some $c > 4$. Suppose $x\in P_t$ belongs to rank $1$. Then with the fair selection method, the uniform selection method, $N$ independent bina

Figures (2)

  • Figure 1: Average number of fitness evaluations needed for the ($\mu+1$) GA to optimize Jump$_{n,k}$ for $n=100$ and $k=4$ using uniform crossover with probability $0.9$ and bit-wise mutation.
  • Figure 2: Average number of fitness evaluations needed for the ($\mu+1$) GA to optimize Jump$_{n,k}$ for $n=1000$ and $k=4$ using uniform crossover with probability $0.9$ and bit-wise mutation.

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Corollary 2
  • proof
  • Lemma 3
  • proof
  • Corollary 4
  • proof
  • Lemma 5
  • proof
  • ...and 6 more