Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 74

4.7 Summary

1. Various mutation operators for ES exist, from the randomized standard
Gaussian mutation of ES to the derandomized CMA.
2. In order to minimize computational and implementation effort while offering
a greater flexibility than unbiased mutation, the BMO was developed. Unlike
the principle of unbiasedness of mutation the BMO was introduced as a self-
adaptive operator for biasing mutations into beneficial directions.
3. The operator makes use of a self-adaptive bias coecient vector ξ ,which
determines the direction of the bias. The absolute bias is computed by mul-
tiplication with the step sizes. A number of variants like the sBMO and
cBMO have been introduced.
4. Another variant is the DMO whose bias is not controlled self-adaptively,
but with the help of the descent direction estimated by two successive pop-
ulations' centers. This bias adaptation technique is reasonable as long as
the assumption of locality is true, i.e. the estimated direction to the global
optimum can be derived from the local information.
5. Under the assumption that the bias direction can be determined self-
adaptively or with the help of the DMO technique, the intuition that bi-
ased mutation is superior to unbiased mutation on monotone functions or
on monotone parts of functions could be confirmed theoretically.
6. Experimental analysis on typical test functions showed the superiority of
biased mutations on multimodal functions. On rosenbrock and rosenbrock
with noise the BMO variants and the DMO achieve considerable improve-
ments. The biased techniques also allow to speed up the evolution along the
ridge of the functions sharp and parabolic ridge.
7. On constrained functions the BMO and the DMO improve the convergence
to the optimum, while standard methods like DP suffer from premature step
size reduction.
8. In all experiments the population ratio of (15,100) shows a high stability of
the BMO and DMO results.
9. The mutation of the bias coecient ξ with settings around γ

≈ 0 . 1achieves

the best experimental results.
10. Statistical tests, i.e. the Wilcoxon rank-sum test on selected experimental
data, validated the relevance of the experimental results.