  Recently it has been proved that a simple algorithm configurator called
ParamRLS can efficiently identify the optimal neighbourhood size to be used by
stochastic local search to optimise two standard benchmark problem classes. In
this paper we analyse the performance of algorithm configurators for tuning the
more sophisticated global mutation operator used in standard evolutionary
algorithms, which flips each of the $n$ bits independently with probability
$\chi/n$ and the best value for $\chi$ has to be identified. We compare the
performance of configurators when the best-found fitness values within the
cutoff time $\kappa$ are used to compare configurations against the actual
optimisation time for two standard benchmark problem classes, Ridge and
LeadingOnes. We rigorously prove that all algorithm configurators that use
optimisation time as performance metric require cutoff times that are at least
as large as the expected optimisation time to identify the optimal
configuration. Matters are considerably different if the fitness metric is
used. To show this we prove that the simple ParamRLS-F configurator can
identify the optimal mutation rates even when using cutoff times that are
considerably smaller than the expected optimisation time of the best parameter
value for both problem classes.
