Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 53

4.2.1 BMO Concept

Unlike directed mutation, the BMO does not change the skewness, but biases
the mean of the Gaussian distribution to lead the search into a more beneficial
direction. This is reflected in the success rate of reproducing superior offspring.
For the BMO we introduce a bias coe cient vector ξ which indicates the level
of bias relative to the standard deviation σ

ξ =( ξ 1 ,...,ξ N )with − 1 ≤ ξ i ≤ 1 . (4.30)

For every i ∈ 1 ,...,N the bias vector b =( b 1 ,...,b N ) is defined by:

b i = ξ i · σ i (4.31)

Since the absolute value of bias coecient ξ i is less than or equal to 1, the bias
will be bound to the step sizes σ i . This restriction prevents the search from being
biased too far away from the parent. Figure 4.2 illustrates the BMO. The BMO
follows the standard way of mutation

x := x + z . (4.32)

The mutation of the BMO works as follows:

z := ( σ 1 N 1 (0 , 1) + b 1 ,...,σ N N N (0 , 1) + b N ) (4.33)

=( σ 1 N 1 (0 , 1) + ξ 1 σ 1 ,...,σ N N N (0 , 1) + ξ N σ N ) (4.34)

=( σ 1 N 1 ( ξ 1 , 1) ,...,σ N N N ( ξ N , 1)) (4.35)

In terms of modifying the mutation strength, the aforementioned log-normal
rule is applied. Furthermore, in the BMO the bias coecients are mutated in
the following meta-EP way:

ξ i = ξ i + γ ·N

(0 , 1)

i =1 ,...,N.

(4.36)

The parameter γ is a new parameter introduced for the BMO to determine the
mutation strength on the bias. In section 4.5 recommendations will be proposed
to tune this parameter.
The BMO biases the mean of mutation and enables the ES to reproduce off-
spring outside the standard mutation ellipsoid. To direct the search, the BMO
enables the mutation ellipsoid to move within the bounds of the regular step sizes
σ . The bias moves the center of the Gaussian distribution within the bounds
of the step sizes. This is advantageous on ridge functions, on some multimodal
functions and at the edge of feasibility. Without the BMO the success rate to
reproduce better offspring is relatively low because many mutations lie beyond
the feasible search space or have got a worse fitness, as described in section 7.3.