IEEE Computational Intelligence Magazine - May 2021 - 26

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2021

Loss

10-2
10-4
10-6

10-2

10-4

10-6

10-9

10-7

10-5

10-3

10-1

101

103

105

g = 1.6
a0 = 15

SGD (ADAM)
xNES

g = 1.6
a0 = 45

g = 3.7
a0 = 60

10-6

10-4

10-2

100

102

104

(b)

g = 3.7
a0 = 75

(a)

tNES

10-8
200 k
0

SGD (ADAM)

50 k 100 k 150 k
Number of Evaluation

g = 3.7, a0 = 75

tNES Transfer From Other a0
tNES Transfer From Other g

10-8
50 k 100 k 150 k 200 k
0
Number of Evaluation

100

100

0

102

102

10-8

104

104

g = 1.6, a0 = 45

10-6

10-4

10-2

100

102

104

g = 9.8
a0 = 45

g = 9.8
a0 = 60

g = 9.8
a0 = 75

50 k 100 k 150 k
Number of Evaluation

g = 9.8, a0 = 60

tNES Transfer From No Drag

10-8
200 k
0

SGD (ADAM)
xNES

g = 9.8
a0 = 15

xNES

50 k 100 k 150 k
Number of Evaluation

g = 9.8, a0 = 15

200 k

FIGURE 7 (a) The convergence trends from 20 independent SGD and xNES optimization runs, and 40 tNES optimization runs for selected scenarios are displayed. The bold lines indicate the mean
convergence path and the shaded areas indicate their 10th-90th interpercentile ranges. (b) The distribution of optimized loss on a test set given by tNES, xNES and SGD for all scenarios are compared. Some scenarios (e.g., g = 1.6, a 0 = 45 & g = 3.7, a 0 = 60) are challenging to SGD but not to the neuroevolution (xNES & tNES). For both g = 1.6, and g = 3.7, the results (in green blocks)
given source problem from the same g (i.e., from a 0 = 15 to a 0 = 30, from a 0 = 60 to a 0 = 75, and vice versa), and the results (in red blocks) given source problem from the same a 0 (i.e., from
g = 3.7 to g = 1.6, and from g = 9.8 to g = 3.7), are compared. For g = 9.8, the source problem came from the same scenario without considering the drag effect. Comparing their optimized loss
on a test set, the Friedman tests conclude a difference in performance between three optimization algorithms in all scenarios (at 5% significance level). In particular, xNES performs better than SGD
in all scenarios except for g = 3.7, a 0 = 75, whereas tNES performs significantly better than both SGD and xNES (as per the Mann-Whitney rank test at 5% significance level).

Optimized Loss

26



IEEE Computational Intelligence Magazine - May 2021

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