Learning Probability Distributions in Continuous Evolutionary
Algorithms -- A Comparative Review

Stefan Kern, Sibylle D. Mueller, Nikolaus Hansen, Dirk Bueche, Jiri
Ocenasek, and Petros Koumoutsakos

We present a comparative review of Evolutionary Algorithms that
generate new population members by sampling a probability distribution
constructed during the optimization process.  We present a unifying
formulation for five such algorithms that enables us to characterize
them based on the parametrization of the probability distribution, the
learning methodology, and the use of historical information.  The
algorithms are evaluated on a number of test functions in order to
assess their relative strengths and weaknesses. This comparative
review helps to identify areas of applicability for the algorithms and
to guide future algorithmic developments.


ERRATA, that were corrected in the draft (but not in the final version): 

  Equation (7): "d = max(...) + 1/c" must be "d = max(...) + c"

  Equation (14): "d = max(...) + 1/c_sigma" must be "d = max(...) + c_sigma"

  The same two corrections hold for Table 2.

Many thanks to Christian Igel for pointing to these errors. 

Recently better parameter settings for c_sigma and d were found, 
improving in particular with large population size:

  c_sigma = (mu + 2)/(n + mu + 3) and 
  d = 1 + 2 * max(0, sqrt((mu-1)/(n+1)) - 1) + c_sigma

in Hansen, N, and S. Kern (2004). Evaluating the CMA Evolution Strategy
on Multimodal Test Functions. In Eighth International Conference on
Parallel Problem Solving from Nature PPSN VIII, Proceedings,
pp. 282-291, Berlin: Springer.