Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 63

Table 4.5. Parameter study of γ for the BMO on the rosenbrock function. Settings
around γ =0 . 1 achieve the best approximation qualities.

0.001 0.01 0.05 0.1 0.15 0.5 1.0

best 1.54E-02 5.24E-03 3.95E-10 7.97E-10 6.53E-09 8.38E-05 8.80E-03

median 2.73E-02 1.87E-02 1.83E-08 1.22E-08 1.11E-07 1.59E-03 1.58E-02

worst 9.16E-01 6.43E-01 5.69E-07 4.72E-08 2.89E-07 8.45E-03 8.41E-01

mean 6.52E-02 4.59E-02 5.81E-08 1.47E-08 1.23E-07 1.89E-03 7.48E-02

dev 1.77E-01 1.26E-01 1.19E-07 1.14E-08 7.65E-08 1.49E-03 2.04E-01

Parameter Study of γ

Now we conduct a parameter study of the biased mutation parameter γ .Mu-
tation parameters of strategy variables often have a strong impact on self-
adaptation capabilities. Whether this is also the case for the BMO, table 4.5
answers. The parameter γ controls the mutation strength of the bias coecient
vector ξ .For γ values between 0.001 and 1.0 have been tested. There is a strong
influence of γ on the quality of the results. In fact, only the values γ 0 . 1, i.e.

γ =0 . 05, 0 . 1and0 . 15 let the BMO obtain a successful approximation of the
optimum. For higher values the mutation is too strong and does not adapt the
bias properly. For lower values the mutation is too weak and achieves no signif-
icant influence. As further experiments on other test functions reveal a similar
optimal behavior for γ , we recommend settings around γ

≈

0 . 1 being aware that
according to no free lunch other settings might be more appropriate for other
problems.

≈

Selection Pressure and Population Ratios

Selection pressure and population ratios influence the population's diversity
withing the strategy parameter search space. We vary the selection pressure
and the population ratio in order to explore the BMO's self-adaptation capabili-
ties. Table 4.6 shows the outcome of these experiments. As we can see, we obtain
no improvement of the mean results with an increase of the selection pressure
[(5,100);(15,300);(15,500)]. Among these three settings the (15,100) ratio obtains
the results with the highest stability and lowest standard deviation. Only the
(5,100)-ES achieves a better best and median . But as the distribution of the
results is relatively high, it pays with a slightly higher standard deviation. An
increase of offspring individuals is supposed to increase the population's diver-
sity. But this has no positive impact on the achieved results. No improvements
can be obtained with a high number of parents and a weak selection pressure.
We explain these results with a weakened connection between strategy variables
and fitness: a weak selection pressure allows solutions with comparatively bad
strategy parameters to survive. Hence, an evolution of strategy parameters is
almost impossible.