Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 62

BMO sBMO cBMO DMO

ES 1.421E-09 0.1253 1.421E-09 1.603E-09

BMO 1.421E-09 0.01301 1.421E-08

sBMO 1.421E-08 1.421E-09

cBMO 2.059E-08

We assume a significance level of p l =0 . 05.Theabovetableshowsthe p w -values,
which indicate the probability that the null hypothesis, i.e. the two data sets show
the same distribution, in particular the same median, holds. The comparison
of the ES to the biased mutation variants reveals a significant superiority of
the BMO (1 . 4 E

− 9 << 0 . 05), the cBMO (1 . 4 E − 9 << 0 . 05) and the DMO

(1 . 6 E 9 << 0 . 05). This is not the case for the sBMO. It exhibits the same

distribution with probability p

−

0 . 1253, which is not significant. Furthermore,
the test shows that the BMO is significantly better than the sBMO, the DMO
and even the cBMO (0 . 01301 << 0 . 05). The variants cBMO and DMO are
significantly better than sBMO. BMO and cBMO cannot be distinguished with
a significant probability. Consequently, we can assert the order 1. BMO and
cBMO,2.DMO,3.sBMO,4.ES.
The new strategy parameters have to be controlled adequately, otherwise the
search will get stuck. The sBMO with an N-dimensional bias, but only one
step size is not able to control the bias properly. We analyze population ratios
and the mutation parameter γ in the next paragraph. The comparison on the
function rastrigin, see table 4.4, upper part, does not reveal any superiority of
the biased mutation variants. The standard ES as well as the DMO only show
average results during the approximation of the optimum. We observe a slight
deterioration of the BMO and the DMO in comparison to the other variants. A
similar behavior can be observed on the function griewank, see table 4.4, lower
part. Neither an improvement nor a deterioration is obtained by the usage of
the self-adaptive bias.

≤

Table 4.4. Experimental results of the BMO variants on the function rastrigin (upper
part) and the function griewank (lower part). No significant superiority of any variant
can be observed.

ES BMO cBMO DMO

rastrigin
best 0.99 3.97 0.99 1.98

median 2.98 11.93 2.98 5.96

worst 6.96 45.76 6.96 11.93

mean 3.22 16.55 3.58 6.76

dev 1.80 10.20 1.64 2.79

griewank
best 0 0 0 0

median 0 0.009 0 0.014

worst 0.022 0.036 0.017 0.145

mean 0.004 0.013 0.003 0.023

dev 0.006 0.013 0.006 0.029