Interpolating Local and Global Search by Controlling the Variance of Standard Bit Mutation

A key property underlying the success of evolutionary algorithms (EAs) is their global search behavior, which allows the algorithms to `jump' from a current state to other parts of the search space, thereby avoiding to get stuck in local optima. This property is obtained through a random choice of the radius at which offspring are sampled from previously evaluated solutions. It is well known that, thanks to this global search behavior, the probability that an EA using standard bit mutation finds a global optimum of an arbitrary function $f:\{0,1\}^n \to \mathbb{R}$ tends to one as the number of function evaluations grows. This advantage over heuristics using a fixed search radius, however, comes at the cost of using non-optimal step sizes also in those regimes in which the optimal rate is stable for a long time. This downside results in significant performance losses for many standard benchmark problems. We introduce in this work a simple way to interpolate between the random global search of EAs and their deterministic counterparts which sample from a fixed radius only. To this end, we introduce \emph{normalized standard bit mutation}, in which the binomial choice of the search radius is replaced by a normal distribution. Normalized standard bit mutation allows a straightforward way to control its variance, and hence the degree of randomness involved. We experiment with a self-adjusting choice of this variance, and demonstrate its effectiveness for the two classic benchmark problems LeadingOnes and OneMax. Our work thereby also touches a largely ignored question in discrete evolutionary computation: multi-dimensional parameter control.