Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Evolutionary Algorithms Are Significantly More Robust to Noise When They Ignore It

Denis Antipov, Benjamin Doerr

TL;DR

The paper addresses optimization under noisy objective evaluations using no re-evaluations, focusing on the Leading-Ones benchmark. It provides rigorous runtime analyses for the no-re-evaluation $(1+1)$ EA under the one-bit and bitwise noise models, yielding $O(n^2)$-time guarantees for constant noise levels and standard mutation settings. The findings contrast with prior analyses that assume re-evaluations, showing that re-evaluations can be detrimental and are not strictly necessary. Complementary experiments support the theory, demonstrating robustness close to the noiseless case even at substantial noise. The results suggest broader applicability to single-objective optimization and motivate rethinking evaluation strategies in noisy black-box settings.

Abstract

Randomized search heuristics (RSHs) are known to have a certain robustness to noise. Mathematical analyses trying to quantify rigorously how robust RSHs are to a noisy access to the objective function typically assume that each solution is re-evaluated whenever it is compared to others. This aims at preventing that a single noisy evaluation has a lasting negative effect, but is computationally expensive and requires the user to foresee that noise is present (as in a noise-free setting, one would never re-evaluate solutions). In this work, we conduct the first mathematical runtime analysis of an evolutionary algorithm solving a single-objective noisy problem without re-evaluations. We prove that the $(1+1)$ evolutionary algorithm without re-evaluations can optimize the classic LeadingOnes benchmark with up to constant noise rates, in sharp contrast to the version with re-evaluations, where only noise with rates $O(n^{-2} \log n)$ can be tolerated. This result suggests that re-evaluations are much less needed than what was previously thought, and that they actually can be highly detrimental. The insights from our mathematical proofs indicate that this similar results are plausible for other classic benchmarks.

Evolutionary Algorithms Are Significantly More Robust to Noise When They Ignore It

TL;DR

The paper addresses optimization under noisy objective evaluations using no re-evaluations, focusing on the Leading-Ones benchmark. It provides rigorous runtime analyses for the no-re-evaluation EA under the one-bit and bitwise noise models, yielding -time guarantees for constant noise levels and standard mutation settings. The findings contrast with prior analyses that assume re-evaluations, showing that re-evaluations can be detrimental and are not strictly necessary. Complementary experiments support the theory, demonstrating robustness close to the noiseless case even at substantial noise. The results suggest broader applicability to single-objective optimization and motivate rethinking evaluation strategies in noisy black-box settings.

Abstract

Randomized search heuristics (RSHs) are known to have a certain robustness to noise. Mathematical analyses trying to quantify rigorously how robust RSHs are to a noisy access to the objective function typically assume that each solution is re-evaluated whenever it is compared to others. This aims at preventing that a single noisy evaluation has a lasting negative effect, but is computationally expensive and requires the user to foresee that noise is present (as in a noise-free setting, one would never re-evaluate solutions). In this work, we conduct the first mathematical runtime analysis of an evolutionary algorithm solving a single-objective noisy problem without re-evaluations. We prove that the evolutionary algorithm without re-evaluations can optimize the classic LeadingOnes benchmark with up to constant noise rates, in sharp contrast to the version with re-evaluations, where only noise with rates can be tolerated. This result suggests that re-evaluations are much less needed than what was previously thought, and that they actually can be highly detrimental. The insights from our mathematical proofs indicate that this similar results are plausible for other classic benchmarks.
Paper Structure (12 sections, 11 theorems, 38 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 11 theorems, 38 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Leading-Ones and Prior Noise
  4. The $(1 + 1)$ EA
  5. Auxiliary Tools
  6. Runtime Analysis
  7. One-bit Noise and One-bit Mutation
  8. Bitwise Noise and Bitwise Mutation
  9. Discussion of the Theoretical Results
  10. Experiments
  11. Statistical Tests
  12. Conclusion

Key Result

Theorem 1

Let $(X_t)_{t \ge 0}$ be a sequence of non-negative random variables with a finite state space $S \subseteq {\mathbb R}_0^+$ such that $0 \in S$. Let $T \coloneqq \inf\{t \ge 0 \mid X_t = 0\}$. If there exists $\delta > 0$ such that for all $s \in S \setminus \{0\}$ and for all $t \ge 0$ we have $E[

Figures (3)

  • Figure 1: The Markov Chain used in the proof of Lemma \ref{['lem:phase-one-bit']} and the possible transitions between the states.
  • Figure 2: The normalized mean best true fitnesses and their standard deviation of the $(1 + 1)$ EA with and without re-evaluations for different noise rates depending on the problem size. The results of the $(1 + 1)$ EA without re-evaluations are shown with solid lines and filled markers and the runtimes of the $(1 + 1)$ EA with re-evaluations are shown in dashed lines and unfilled markers.
  • Figure 3: The normalized mean runtimes and their standard deviation of the $(1 + 1)$ EA with and without re-evaluations for different noise rates depending on the problem size. The runtimes of the $(1 + 1)$ EA without re-evaluations are shown with solid lines and filled markers and the runtimes of the $(1 + 1)$ EA with re-evaluations are shown in dashed lines and unfilled markers.

Theorems & Definitions (19)

  • Theorem 1: Additive Drift Theorem HeY04, upper bound
  • Lemma 2: Lemma 7 in DoerrK15
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • Theorem 7
  • ...and 9 more