  Recent theoretical studies have shown that self-adjusting mechanisms can
provably outperform the best static parameters in evolutionary algorithms on
discrete problems. However, the majority of these studies concerned elitist
algorithms and we do not have a clear answer on whether the same mechanisms can
be applied for non-elitist algorithms.
  We study one of the best-known parameter control mechanisms, the one-fifth
success rule, to control the offspring population size $\lambda$ in the
non-elitist ${(1 , \lambda)}$ EA. It is known that the ${(1 , \lambda)}$ EA has
a sharp threshold with respect to the choice of $\lambda$ where the runtime on
OneMax changes from polynomial to exponential time. Hence, it is not clear
whether parameter control mechanisms are able to find and maintain suitable
values of $\lambda$.
  We show that the answer crucially depends on the success rate $s$ (i.,e. a
one-$(s+1)$-th success rule). We prove that, if the success rate is
appropriately small, the self-adjusting ${(1 , \lambda)}$ EA optimises OneMax
in $O(n)$ expected generations and $O(n \log n)$ expected evaluations. A small
success rate is crucial: we also show that if the success rate is too large,
the algorithm has an exponential runtime on OneMax.
