I have used ES algorithms a couple of times and this question got me thinking a lot. I found a (partial) answer on the paper "Evolution strategies: A comprehensive introduction" by Beyer and Schwefel, Section 3.3.1.2: if the search space is real-valued and unconstrained, and we have no a-priori information, the Gaussian distribution stems from the maximum entropy principle, i.e., it is maximally unbiased. Two additional important factors are (i) the flexibility of Gaussian distributions, and (ii) the availability of some partial results in term of convergence speed. For the first point there is a discussion in Section 3 of the paper "Completely Derandomized Self-Adaptation in Evolution Strategies" by Hansen and Ostermeier, in terms of mapping between a problem and a solution: "using arbitrary normal mutation distributions

with zero mean [...] yields equivalence between an adaptive general linear encoding/decoding and the adaptation of all distribution parameters in the covariance matrix.".

I would be glad to learn more on the topic by someone more experienced than me.

The multivariate normal distribution has a couple of advantages:

(1) Maximum Entropy (as stated in the comment of Simone)

(2) Rotation Symmetry (there should be no preferred sampling along the coordinate axis)

(3) Unimodal, centered in zero (offspring that are far away to their parents should occur with less probability than offspring that are close to their parent. There should be no bias introduced by the mutation (this is the 'task' of the selection.)

(3) It can be rotated, scaled, adapted, and sampled easily (as used in the Covariance Matrix Adaptation, or ES with individual step-sizes).

(4) Infinite support (which makes it possible to prove global probabilistic convergence for infinite time. Practically the radius of gaussian distributions is however limited to ca. six times the standard deviation why the Cauchy distribution was suggested in some papers as an alternative.

(5) Nowadays, it is also true that most empirical and theoretical results are on ES with normal distributions, why for this variant the theoretical comprehension is relatively good. There are however also results for geometric, Binominal and mixed distributions (ES for integer and mixed integer optimization).

Thanks Michael, very comprehensive answer. You mention that there are good results for other distributions. I'm currently looking at uniform ring distributions, that allow me to specifically restrict an upper and lower bound for step sizes, as part of an ongoing research on convergence control. In my experiments so far, I've seen that it is hard to control a minimum step size for normal distributions, unless using rejection sampling, which I don't like a lot, for efficiency reasons. My goal is to restrict very small steps until later on, to disallow a fast convergence to local optima. This is why I looked at uniform rings. However, I loose some good properties of gaussians when switching to uniform, like the infinite support, but I don't know if that its an important or unimportant feature. My question is: do you have any knowledge regarding the performance of ES with uniform distributions?

Alejandro, the length of a multivariate standard normal distribution has mode sqrt(n-1), where n is the dimension, and standard deviation approximately sqrt(1/2). That is, even in moderate dimension we virtually never sample very small or very large steps. Normalizing all steps to a given length (typically sqrt(n)) is also not problematic. I don't understand why rejection sampling to prevent small steps to be realized could be inefficient in this context, because they are rare anyway.

Hi Nikolaus, thanks a lot for the feedback!

I was just speaking my mind, and didn't give too much thought to that statement of efficiency. In general I try not to do rejection sampling, specially in high dimensions (500 or more) because of efficiency, however, in this particular case I had not thought too much if it indeed was an issue. Thanks for pointing out my mistake.

I'm looking at uniform distributions because I'm trying to adapt a somewhat new technique called "thresheld convergence" (TC) to ES, which consists in disallowing steps below a given threshold to prevent early convergence. We already did it with DE, and it has also being done with PSO, SA, and a few other heuristics. I've seen some mechanisms in standard (1+1) ES to prevent early convergence, like adapting the sigma parameter. However, I want to try TC in the "purest" possible form first, and then move on to more bizarre formulations. And so far TC has been done mainly with ring distributions, because it is very easy to adjust a minimum and maximum step. That's why I asked if there are any known results regarding ES with uniform ring distributions.

Conference Paper Differential Evolution with Thresheld Convergence

Alejandro, in ESs, (premature) convergence is related to convergence of the step-size parameter sigma to zero. As far as I can see it is not related to whether or not small steps are sampled or accepted occasionally. You can prevent (premature) convergence only by not letting sigma go to zero.

Hi Nikolaus. Thanks for the answer.

Indeed, sigma dropping to zero is one of the things I'm trying to prevent. However, the "theory" behind TC states that sampling solutions too near in solution space can lead to a bias in the estimation of the "goodness" of different attraction basins. The whole idea behind TC is to sample at a bird's eye first, to (as unbiased as possible) identify good attraction basins, and then let the optimizer converge on the most promising basins.

If some samples are too close while other are too far apart, then we can end with many samples from a basin, and only a few from other basins, leading to an over- or under-estimation of the fitness of the basins. At least for DE, PSO and SA, this idea seems sound. I'm trying to apply it to ES and see if it works there. That's why, besides adjusting sigma, I want to reject solutions that are too close, and see if something improves.

I don't know if there are results for this type of modification to ES.

Gaussian or Cauchy distributions during mutation phase in case of Evolutionary Programming. Gaussian distribution implies a slower convergence rate, while Cauchy 

distribution implies a faster rate. User needs to select one suitable for the application. Definitely faster convergence rate have more chances to convergence to a local solution.

I have a question?

literature claims that if sample is less that 2000 Shapiro-Wilk test is best fit to check normality but if sample is more than 2000 which test is best fit and what would its value?

I need ans with reference or mail at [email protected]

Divergence to check various possible solutions according to normal distribution and most of problems follow in their best solutions the normal distribution