Reconsidering the Progress Rate Theory for Evolution 
              Strategies in Finite Dimensions

                A. Auger and N. Hansen


Abstract. This paper investigates the limits of the predictions based
  on the classical progress rate theory for Evolution Strategies.  We
  explain on the sphere function why positive progress rates give
  convergence in mean, negative progress rates divergence in mean and
  show that almost sure convergence can take place despite divergence
  in mean. Hence step-sizes associated to negative progress can
  actually lead to almost sure convergence. Based on these results we
  provide an alternative progress rate definition related to almost
  sure convergence.  We present Monte Carlo simulations to investigate
  the discrepancy between both progress rates and therefore both types
  of convergence. This discrepancy vanishes when dimension
  increases. The observation is supported by an asymptotic estimation
  of the new progress rate definition.


In: Proceedings of the Genetic and Evolutionary Computation Conference
(GECCO 2006), pp.445-452, ACM Press, 2006. 


ERRATUM: 
  Theorem 2, last equation should read 
     sigma^n = sigma^*_opt ||X_n|| / d    (instead sigma^*_opt / ||X_n|| / d) 

ADDENDUM: Also for Figure 3, we set lambda=5. 

BIBTEX: 
@conference{auger2006reconsidering,
  title={Reconsidering the progress rate theory for evolution strategies in finite dimensions},
  author={Auger, A. and Hansen, N.},
  booktitle={Proceedings of the 8th annual conference on genetic and evolutionary computation {GECCO}},
  pages={445--452},
  year={2006},
  organization={ACM}
}