IEEE Computational Intelligence Magazine - November 2021 - 42

Based on the similarity of the gene distributions,
which is also called gene similarity for short in this
article, MFEA-GSMT can achieve more fine-grained
and accurate knowledge transfer than methods based
on the population distribution.
on dynamic opposition-based learning is applied to the offspring
solutions to avoid premature convergence and explore
more promising search areas (lines 12-14). If the selective
crossover does not occur, then the polynomial mutation
(PM) [44] is used to generate offspring (lines 15-18). Each
generated offspring solution is randomly assigned the skill
factor from one of its parents (line 19) and evaluated on only
the skill factor task (line 22). Such selective evaluation can
significantly reduce the consumption of computing resources.
The details of the key components of MFEA-GSMT are
provided in the following subsections.
A. Knowledge Transfer Based on Inter-Task Gene Similarity
Many studies [13]-[18] have shown that MFEAs with positive
inter-task knowledge transfer can outperform their counterpart
single-task EAs. The inter-task correlation and the transferred
knowledge are critical to the success of MFEAs.
Algorithm 1 Framework of the proposed MFEA-GSMT
Require: N: the population siz e. K: number of optimization tasks.
G: the generation interval to perform the mirror transformation.
Ensure: The optimal solution or non-dominated set of each task.
1: Randomly generate a population P of size N.
2: Evaluate each solution of P on all optimization tasks.
3: Identify the skill factor of each solution in P based on Eq. (3).
4: g 1! .
5: While the termination criterion not satisfied do
6:
KLD ! the inter-task gene similarity matrix based on KLD
(see Algorithm 2).
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
21:
22:
23:
24:
For :/=12
iN
Randomly select x1
If xx
xx
oo
12
and x2
//N: the population size
from P for mating.
Identify the skill factors x1
== or rand < rmp
If mod(g, G)= =0
o2 ! Mirror transformation of o1
End if
Else
o1 ! Polynomial Mutation of x .1
o2 ! Polynomial Mutation of x .2
End if
20: OO oo
End for
gg .1! +
25: End while
Assign o1 and o2
randomly.
,
! {, }.12
Evaluate offspring population O.
P ! Perform elite selection on PO.,
where N ,ki1
(N ,kj2
the skill factor from one of their parents
using Eqs. (6-8).
x and x2
x of x1 and x .2
{, }12 ! SelectiveCrossover xx
(see Algorithm 3)
(, ,12 KL ).D
1) Inter-Task Gene Similarity Measurement
Before measuring the inter-task gene similarity, it is necessary
to characterize the distribution of each gene/variable on each
task. Given a subpopulation Pk
individuals, i.e., {, ,}Px xN
consisting of Nk
= f k
1
the univariate normal distribution (, )nv is used to estiki
ki
ki
2
mate the distribution of the i-th gene with respect to task k:
,,...,
(),,...,
nki
2
ki,
,,
vnki
Nk - 1 l=1
==
=
1 / li
Nk l =1
Nk
1 / xi n
2
xi 1
Nk
li-=1
,,
n
(4)
Based on the above definition of gene distributions, the
inter-task gene similarity can be measured with the KLD [30],
which is an estimation of how one probability distribution
diverges from another. KLD is a popular similarity measure of
two distributions and has achieved wide success in various fields.
The KLD of two gene distributions can be calculated by
2
KLD NN log2^h=+ki kj
12
,,
v
v
ki
kj
1
2
,
,
vn n
2vkj
ki+-1 21
, ,,
2
()
2,
kj ki
-
2
2
1
(5)
) indicates the distribution of the i-th
(j-th) gene on the sub-population associated with task
42 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2021
n-dimensional
where (, ,) and
x xx= f1ll,,ln
lN1= ,..., ,k with the skill factor of each individual being k,
N,, ,
For complex or black-box problems, the
decision-maker usually has little or no prior
knowledge of the inter-task correlation. To
avoid the notorious negative transfer, suitable
sources should be determined before knowledge
transfer. Thanks to the support of statistics
theories, online-density-estimation-based
knowledge transfer has recently become an
important candidate solution in this direction
[25]-[29]. However, the density estimation applied in
[25]-[29] is targeted at the individual distribution, which
tends to be imprecise especially in high-dimensional space
with only a small number of samples [2], [8]. To address this
issue, in this study the knowledge transfer sources are selected
based on the distributions of each gene/variable, rather than
on the individual distribution considering all genes/variables.
Specifically, we use the current population (i.e., N individuals)
to estimate the probability density function of each gene/
variable to overcome the imprecise estimation issue in highdimensional
space.
Based on the similarity of the gene distributions, which is
also called gene similarity for short in this article, MFEAGSMT
can achieve more fine-grained and accurate knowledge
transfer than methods based on the population distribution.
Particularly, the gene similarity estimated on different sub-populations
associated with different tasks is referred to as intertask
gene similarity. The inter-task gene similarity measurement
is detailed as follows.
IEEE Computational Intelligence Magazine - November 2021

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - November 2021
