An analysis of mutative sigma-self-adaptation on linear fitness functions

This paper investigates sigma-self-adaptation for real valued evolutionary algorithms on linear fitness functions. We identify the step-size logarithm log sigma as a key quantity to understand strategy behavior. Knowing the bias of mutation, recombination, and selection on log sigma is sufficient to explain sigma-dynamics and strategy behavior in many cases, even from previously reported results on non-linear and/or noisy fitness functions. On a linear fitness function, if intermediate multi-recombination is applied on the object parameters, the i-th best and the i-th worst individual have the same sigma-distribution. Consequently, the correlation between fitness and step-size sigma is zero. Assuming additionally that sigma-changes due to mutation and recombination are unbiased, then sigma-self-adaptation enlarges sigma if and only if mu < lambda/2, given (mu, lambda)-truncation selection. Experiments show the relevance of the given assumptions.