Log-Linear Convergence and Divergence of the Scale-Invariant (1+1)-ES in Noisy Environments

Jebalia, Mohamed; Auger, Anne; Hansen, Nikolaus

doi:10.1007/s00453-010-9403-3

Cited by 31 publications

(31 citation statements)

References 28 publications

(64 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also restrict our work to noisy settings in which noise does not decrease to 0 around the optimum. This constrain makes our work different from [4]. In [5,6] we can find noise models related to ours but the results presented here are not covered by their analysis.…”

Section: Local Noisy Optimizationmentioning

confidence: 60%

“…1. Remarks: (i) Informally speaking, our theorem shows that if a scale invariant algorithm converges in the noise-free case, then it also converges in the noisy case with the exponential resampling rule, at least if parameters are large enough (a similar effect of constants was pointed out in [4] in a different setting).…”

Section: Theorem 1 Consider the Fitness Functionmentioning

confidence: 72%

“…Let us now consider p (4) n the probability that at least two points i a and i b at iteration n verify…”

Section: Theorem 1 Consider the Fitness Functionmentioning

confidence: 99%

“…We now consider n≥1 p (4) n , which is an upper bound on the probability that in at least one iteration there is a misranking of two individuals. If γ ′ and γ ′′ are large enough, n≥1 p…”

Section: Theorem 1 Consider the Fitness Functionmentioning

confidence: 99%

See 3 more Smart Citations

Log-log Convergence for Noisy Optimization

Astete-Morales

Liu

Teytaud

2014

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We consider noisy optimization problems, without the assumption of variance vanishing in the neighborhood of the optimum. We show mathematically that simple rules with exponential number of resamplings lead to a log-log convergence rate. In particular, in this case the log of the distance to the optimum is linear on the log of the number of resamplings. As well as with number of resamplings polynomial in the inverse step-size. We show empirically that this convergence rate is obtained also with polynomial number of resamplings. In this polynomial resampling setting, using classical evolution strategies and an ad hoc choice of the number of resamplings, we seemingly get the same rate as those obtained with specific Estimation of Distribution Algorithms designed for noisy setting. We also experiment non-adaptive polynomial resamplings. Compared to the state of the art, our results provide (i) proofs of log-log convergence for evolution strategies (which were not covered by existing results) in the case of objective functions with quadratic expectations and constant noise, (ii) log-log rates also for objective functions with expectation E[f (x)] = ||x − x * || p , where x * represents the optimum (iii) experiments with different parametrizations than those considered in the proof. These results propose some simple revaluation schemes. This paper extends [1].

show abstract

Section: Local Noisy Optimizationmentioning

confidence: 60%

Section: Theorem 1 Consider the Fitness Functionmentioning

confidence: 72%

“…Let us now consider p (4) n the probability that at least two points i a and i b at iteration n verify…”

Section: Theorem 1 Consider the Fitness Functionmentioning

confidence: 99%

Section: Theorem 1 Consider the Fitness Functionmentioning

confidence: 99%

See 2 more Smart Citations

Log-log Convergence for Noisy Optimization

Astete-Morales

Liu

Teytaud

2014

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…It has been shown that they are naturally robust in front of actuator 1 noise (see [14,6]). While other studies refer to noisy fitness values, with noise models such as additive or multiplicative noise, as in [13]. In the work presented here, we focus on the study of noisy functions such that the fitness values are perturbed by additive noise with constant variance all over the domain.…”

Section: State Of the Artmentioning

confidence: 99%

Analysis of runtime of optimization algorithms for noisy functions over discrete codomains

Akimoto

Astete-Morales

Teytaud

2015

Theoretical Computer Science

View full text Add to dashboard Cite

We consider in this work the application of optimization algorithms to problems over discrete codomains corrupted by additive unbiased noise.We propose a modification of the algorithms by repeating the fitness evaluation of the noisy function su ciently so that, with a fix probability, the function evaluation on the noisy case is identical to the true value.If the runtime of the algorithms on the noise-free case is known, the number of resampling is chosen accordingly. If not, the number of resampling is chosen regarding to the number of fitness evaluations, in an anytime manner.We conclude that if the additive noise is Gaussian, then the runtime on the noisy case, for an adapted algorithm using resamplings, is similar to the runtime on the noise-free case: we incur only an extra logarithmic factor. If the noise is non-Gaussian but with finite variance, then the total runtime of the noisy case is quadratic in function of the runtime on the noise-free case.

show abstract

Global convergence results for single parent evolution strategies

Jebalia

Khlaifi

Belgacem

2016

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

In this research article, we present theoretical techniques that can be used to investigate and comprehend the convergence behavior patterns of single parent evolution strategies. In the process, we determine instances of divergence or prove log‐linear convergence and estimate the related speed, for a single parent evolution strategies class. The tools and results provided herein entertain a wide range of evolution strategies variants of interest and can be readily adapted for global convergence studies of a multitude of continuous optimization methods, evolving a single solution. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

Log-Linear Convergence and Divergence of the Scale-Invariant (1+1)-ES in Noisy Environments

Cited by 31 publications

References 28 publications

Log-log Convergence for Noisy Optimization

Log-log Convergence for Noisy Optimization

Analysis of runtime of optimization algorithms for noisy functions over discrete codomains

Global convergence results for single parent evolution strategies

Contact Info

Product

Resources

About