Already Moderate Population Sizes Provably Yield Strong Robustness to Noise

TL;DR

This work proves that both the $(1+\lambda)$ EA and the $(1,\lambda)$ EA with population sizes $\lambda \ge C\ln(n)$ tolerate constant bit-wise noise per iteration on OneMax, without increasing the asymptotic runtime. A key methodological contribution is viewing the noiseless offspring as a biased uniform crossover between the parent and the noisy offspring, enabling precise drift-based analysis. The main result shows an overall fitness-evaluation runtime of $O\left(n\log\left(\frac{n}{\lambda}\right) + n\frac{\lambda\log\log\lambda}{\log\lambda}\right)$, which becomes $O(n\log n)$ when $\lambda=\Theta(\log n)$, matching the noiseless case and improving prior requirements on population size. Experimental results corroborate the theory, indicating little difference between the two EAs under noise and demonstrating the practical robustness of population-based EAs to bit-wise disturbances. The findings suggest a general principle: moderate population sizes can yield strong noise robustness in standard EAs, with potential applicability to broader algorithms and benchmarks.

Abstract

Experience shows that typical evolutionary algorithms can cope well with stochastic disturbances such as noisy function evaluations. In this first mathematical runtime analysis of the $(1+λ)$ and $(1,λ)$ evolutionary algorithms in the presence of prior bit-wise noise, we show that both algorithms can tolerate constant noise probabilities without increasing the asymptotic runtime on the OneMax benchmark. For this, a population size $λ$ suffices that is at least logarithmic in the problem size $n$. The only previous result in this direction regarded the less realistic one-bit noise model, required a population size super-linear in the problem size, and proved a runtime guarantee roughly cubic in the noiseless runtime for the OneMax benchmark. Our significantly stronger results are based on the novel proof argument that the noiseless offspring can be seen as a biased uniform crossover between the parent and the noisy offspring. We are optimistic that the technical lemmas resulting from this insight will find applications also in future mathematical runtime analyses of evolutionary algorithms.

Figures (3)

• Figure 1: Illustration of the four groups of bits in the proof of Lemma \ref{['lem:expected-non-noisy']}.
• Figure 2: Mean runtimes (number of fitness evaluations, normalized by $n \ln n$) and their standard deviations over $100$ runs of the $(1+\lambda)$ EA and the $(1,\lambda)$ EA on OneMax with noise rates $q = 1$ with varying problem size $n$.
• Figure 3: Mean runtimes (number of fitness evaluations) and their standard deviations over $100$ runs of the $(1+\lambda)$ EA and the $(1 + 1)$ EA on OneMax with noise rates $q = 0.01$ with varying problem size $n$.

Theorems & Definitions (20)

• Theorem 1: Theorem 6 in DoerrDY20
• Lemma 2: Lemma 1.4.9 in Doerr20bookchapter
• Lemma 3: Inequality 3.6.2 in vasic2012analytic
• Lemma 4: Lemma 1 in AntipovDK22
• Lemma 5
• proof
• Lemma 6
• proof
• Theorem 7
• Lemma 8