1+1 Basics

 

We consider a simple evolution strategy (ES), the so-called (1+1)-ES, see Procedure 1.

The $1/5$th rule states that $\sigma$ should be modified according to the rule

$$\sigma(t+1) := \left\{ \begin{array}{ll} \sigma(t) a,& \text... ...\\ \sigma(t) ,& \textrm{ if } P_s = 1/5\\ \end{array}\right.$$ (1)

where the factor $a$ is usually between $1.1$ and $1.5$ and $P_s$ denotes the success rate (Beyer, 2001). The factor $a$ depends particularly on the measurement period $g$, which is used to estimate the success rate $P_s$. During the measurement period, $g$ remains constant. For $g=n$, where $n$ denotes the problem dimension, Schwefel (1995) calculated $1/a \approx 0.817$. Beyer (2001) states that the ``choice of $a$ is relatively uncritical'' and that the $1/5$th rule has a ``remarkable validity domain.'' He also mentions limits of this rule.

Based on these theoretical results, we can derive certain scientific hypotheses. One might be formulated as follows: Given a spherical fitness landscape, the (1+1)-ES performs optimally, if the step-sizes $\sigma$ is modified according to the $1/5$th rule as stated in Eq. 1. This statement is related to the primary model.

In the experimental model, we relate primary questions or statements to questions about a particular type of experiment. At this level, we define an objective function, a starting point, a quality measure, and parameters used by the algorithm. These parameters are summarized in Table 1.

Table 1: $(1+1)$-ES parameters. The first three parameters belong to the algorithm design, whereas the remaining parameters are from the problem design 

Note, the quality measure is defined in the CONF file.