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A First Runtime Analysis of NSGA-III on a Many-Objective Multimodal Problem: Provable Exponential Speedup via Stochastic Population Update

Andre Opris

TL;DR

A rigorous runtime analysis of NSGA-III on the many-objective OneJumpZeroJump benchmark provides runtime bounds where the number of objectives is constant, and shows that NSGA-III is faster than NSGA-II by a factor of N/n^(d/2) for N=omega(n^(d/2)) .

Abstract

The NSGA-III is a prominent algorithm in evolutionary many-objective optimization. It is well-suited for optimizing functions with more than three objectives, setting it apart from the classic NSGA-II. However, theoretical insights about NSGA-III of when and why it performs well are still in its early development. This paper addresses this point and conducts a rigorous runtime analysis of NSGA-III on the many-objective $\OJZJfull$ benchmark ($\OJZJ$ for short), providing runtime bounds where the number of objectives is constant. We show that NSGA-III finds the Pareto front of $\OJZJ$ in time $O(n^{k+d/2}+ μn \ln(n))$ where $n$ is the problem size, $d$ is the number of objectives, $k$ is the gap size, a problem specific parameter, if its population size $μ\in 2^{O(n)}$ is at least $(2n/d+1)^{d/2}$. Notably, NSGA-III is faster than NSGA-II by a factor of $μ/n^{d/2}$ for some $μ\in ω(n^{d/2})$. We also show that a stochastic population update, proposed by~\citet{UpBian}, provably guarantees a speedup of order $Θ((k/b)^{k-1})$ in the runtime where $b>0$ is a constant. Besides~\cite{DoerrNearTight}, this is the first rigorous runtime analysis of NSGA-III on \OJZJ. Proving these bounds requires a much deeper understanding of the population dynamics of NSGA-III than previous papers achieved.

A First Runtime Analysis of NSGA-III on a Many-Objective Multimodal Problem: Provable Exponential Speedup via Stochastic Population Update

TL;DR

A rigorous runtime analysis of NSGA-III on the many-objective OneJumpZeroJump benchmark provides runtime bounds where the number of objectives is constant, and shows that NSGA-III is faster than NSGA-II by a factor of N/n^(d/2) for N=omega(n^(d/2)) .

Abstract

The NSGA-III is a prominent algorithm in evolutionary many-objective optimization. It is well-suited for optimizing functions with more than three objectives, setting it apart from the classic NSGA-II. However, theoretical insights about NSGA-III of when and why it performs well are still in its early development. This paper addresses this point and conducts a rigorous runtime analysis of NSGA-III on the many-objective benchmark ( for short), providing runtime bounds where the number of objectives is constant. We show that NSGA-III finds the Pareto front of in time where is the problem size, is the number of objectives, is the gap size, a problem specific parameter, if its population size is at least . Notably, NSGA-III is faster than NSGA-II by a factor of for some . We also show that a stochastic population update, proposed by~\citet{UpBian}, provably guarantees a speedup of order in the runtime where is a constant. Besides~\cite{DoerrNearTight}, this is the first rigorous runtime analysis of NSGA-III on \OJZJ. Proving these bounds requires a much deeper understanding of the population dynamics of NSGA-III than previous papers achieved.
Paper Structure (9 sections, 9 theorems, 17 equations, 1 figure, 2 algorithms)

This paper contains 9 sections, 9 theorems, 17 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structural Results
  4. The Many-Objective Jump Function
  5. Population Dynamics of NSGA-III on OJZJ
  6. Upper Runtime Bounds
  7. Lower Runtime Bounds
  8. Conclusions
  9. Acknowledgements

Key Result

Lemma 1

Consider NSGA-III on a $d$-objective function $f$ with $\varepsilon_{\text{nad}} \geq f_{\max}$ and a set $\mathcal{R}_p$ of reference points for $p \in \mathbb{N}$ with $p \geq 2d^{3/2}f_{\max}$. Let $P_t$ be its current population. Assume $\mu \geq \lvert{I}\rvert$ if $a=0$ or $\mu \geq 2\lvert{I}

Figures (1)

  • Figure 1: Illustrating how search points with fitness vector $v$ are associated with reference points (dots on line through $(1,0)$ and $(0,1)$ connected by dashed lines through the origin) for $m=2$ objectives in the normalized objective space. The vector $v$ is associated to the nearest reference point to its right.

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 8 more