IEEE Computational Intelligence Magazine - November 2021 - 52

similarity to overcome the issue of imprecise estimation of
population distribution, especially in high-dimensional space
with only a small number of samples. Moreover, a mirror transformation
extended from opposition-based learning is adopted
to avoid premature convergence. Compared with the state-ofthe-art
algorithms on benchmark and real-world problems,
MFEA-GSMT is superior to or competitive with state-of-theart
algorithms on most MTO problems. The proposed algorithm
not only improves the convergence speed but also finds a
higher-quality solution or solution set.
The proposed algorithm has shown promising performance,
yet there remain some interesting potential future
directions. For example, compared with other MFEAs, the
proposed algorithm needs an extra ()
On K22
computation of
Eqs. (4) and (5), where n is the number of decision variables
and K is the number of component tasks in the multitasking
optimization. The additional computational cost in knowledge
transfer is tolerable for off-line optimization, however, improving
the computational efficiency of the knowledge transfer is
an important future direction. The effectiveness of mirror
TABLE IV Mean and standard deviation of the loss of the models in the training data during the evolution,
test data during the evolution, and test data using the EA's optimal solution as the initialization parameter.
MTO PROBLEM
MODEL 1 WITH 151 VARIABLES
(TASK 1)
MODEL 2 WITH 301 VARIABLES
(TASK 2)
MTO PROBLEM
MODEL 1 WITH 151 VARIABLES
(TASK 1)
MODEL 2 WITH 301 VARIABLES
(TASK 2)
SOEA
2036.29+
(296.00)
4021.26+
(955.15)
SOEA
939.22+
(124.22)
1885.58+
(435.46)
THE LOSS OF THE MODELS IN THE TRAINING DATA DURING THE EVOLUTION
MFEA/GHS
MFEA
2473.27+
(495.5)
5780.77+
(1191.03)
MFEA
MFEA-II
1493.56+
(177.53)
3539.90+
(1213.44)
1116.68+
(266.05)
2617.47+
(605.46)
MFEA-II
672.53+
(87.00)
1460.46+
(439.81)
2415.24+
(946.13)
4615.20+
(1977.94)
THE LOSS OF THE MODELS IN THE TEST DATA DURING THE EVOLUTION
MFEA/GHS
1061.5+
(432.92)
1987.18+
(840.55)
MFEA-GSMT
1144.30
(80.17)
1178.74
(71.52)
MFEA-GSMT
514.81
(41.40)
536.03
(30.49)
THE LOSS OF THE MODELS IN THE TEST DATA USING THE EA'S OPTIMAL SOLUTION AS THE
INITIALIZATION PARAMETER
MTO PROBLEM
MODEL 1 WITH 151 VARIABLES
(TASK 1)
MODEL 2 WITH 301 VARIABLES
(TASK 2)
SOEA
539.79+
(104.08)
552.41+
(152.37)
MFEA
497.77+
(71.71)
520.53+
(78.21)
MFEA-II
481.88+
(81.53)
515.23+
(76.67)
MFEA/GHS
469.73+
(41.71)
549.43+
(158.58)
MFEA-GSMT
425.21
(35.77)
423.78
(37.28)
Model 1 Train Loss
Model 2 Train Loss
9.5
8.5
9
7.5
8
7
0.1
0.2
0.3
0.4
Number of Fitness Evaluations ×104
(a)
0.5
10
7
8
9
0.1
0.2
0.3
0.4
Number of Fitness Evaluations ×104
(b)
SOEA MFEA MFEA-II MFEA-GHS MFEA-GSMT
FIGURE 5 The convergence trend of the model error on the training data during the evolution.
0.5
52 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2021
Loss (Log Scale)
Loss (Log Scale)
IEEE Computational Intelligence Magazine - November 2021

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