IEEE Computational Intelligence Magazine - May 2022 - 73

Z min AUC w;Sh
+]
]
]
]
]]
]
]
]
]
]
\
w
T
T
[min
AUC ;
+
=+ w 2
2
/ / wx wxj
i =1 j=1
TT
g
w
#
#
sT sT=+
w 2
2
/ / wx wxj
i =1
j=1
sT T
g
#
2 +where
S' is a simple dataset sampled
from S with a sampling rate s randomly,
and the proportions of positive and negative
instances of S' and S are the same.
If the computational cost of AUCC is 1,
then the computational cost of AUCE
can be expressed as
co ts AUC 22E
s
==+
+
sT T
TT
-
-
1
(7)
In summary, by randomly sampling
the dataset, a cheap task is constructed,
which forms a multitasking AUC optimization
environment with the original
expensive task. Next, the knowledge
available between the constructing
cheap task and the expensive task is discussed
to explain the feasibility of the
constructed multitasking AUC optimization
environment.
In MTO, the available knowledge
across tasks can be measured by the overlap
among the optimal solutions and the
similarity among fitness landscapes [28].
For the constructed multitasking AUC
optimization environment, all tasks have
the same search space, that is, they all need
to search for an optimal classifier w. The
similarity can be quantified by Spearman's
rank correlation. Randomly generate
1,000,000 points in the search space w,
and then calculate their objective function
values on the constructed cheap and
expensive tasks to obtain the Spearman's
rank (or the ordinal) correlation coefficient,
which is considered as a representation
of the similarity among fitness
landscapes. The above steps are repeated 30
times. Each sampling process for constructing
the cheap task is independent.
The sample ratio s is set to 10%. Figure 3
shows the rank (ordinal) correlation
among fitness landscapes on 15 datasets
under 50 runs. The datasets used are introduced
in detail in Section VI.A. The averm
^
SS -h$$i
+
#
+C
^w S'h
(6)
m
^ SS -h$$i
+
E ^
age correlation coefficient is greater than
0.5 in most cases, which shows that the fitness
landscapes of the cheap and expensive
tasks are indeed similar. This phenomenon
further illustrates that the constructed
multitasking AUC optimization environment
is feasible. In addition, on the 15th
dataset, the variance of the rank (ordinal)
correlation among fitness landscapes is
extremely great, which shows that different
sampling samples under a fixed sampling
rate can greatly affect the similarity
among fitness landscapes of the constructed
cheap task and the expensive task. This
is because cheap tasks composed of different
sampling samples may contain different
preference knowledge, resulting in
different degrees of deviation between the
data distribution of these cheap tasks and
the expensive task.
B. Dynamic Adjustment
Strategy of AUCC
Compared with the optimization of the
AUCE task, the AUCC task's optimization
has a less difficult and computational cost,
but the knowledge carried by the simple
sampling dataset in AUCC is limited. For
example, as shown in Fig. 3, in problems
3, 4, 8, 10, and 15, the average value of
rank (ordinal) correlations among fitness
landscapes are not very high. Therefore,
this section proposes a dynamic adjustment
strategy of AUCC to make full use
of the knowledge in different sampling
datasets, as shown in Algorithm 1.
Algorithm 1 Dynamic Adjustment
Strategy of AUCC.
Input:
S: the whole dataset used in AUCE;
wE: the optimal classifier obtained on
the current AUCE task;
s: the sampling rate;
Output:
S': the adjusted dataset used in AUCC;
ff ! Q+4.
ff f
8. ff fi
9. end for
10.
= scorei T10!##
scor {, },
{, };
-
11. for i 1= to T+ do
12.
13.
14.
15.
i # -
+
!
end if
16. end for
17. end for
18. for i 1= to T− do
19.
20.
21.
22.
if (ff>i
-
end if
23. end for
24. end for
25. S'!Select the top sT+
and sTpositive
instances
negative instances from S
according to score+ and score−;
j
!
for j 1= to T− do
if () thenffj
scorescore 1ii
+++ ;
escore iT score10i
i
+ =
+ ## !
-
+
f wx ;--!
--! , -
i
T
E
i
f wx ;
++! , ;+
T
+ ! E
i
+
i
i
5. end for
6. for i 1= to T− do
7.
;
-
1. ,;
2. for i 1= to T+ do
3.
for j 1= to T+ do
+) then
scorescore 1ii
--+ ;
1
0.8
0.6
0.4
0.2
12345678 9
ID
10 11 12 13 14 15
FIGURE 3 Rank (ordinal) correlation among fitness landscapes on 15 data sets under 50 runs.
MAY 2022 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 73
Rank Correlation Among Fitness Landscapes
IEEE Computational Intelligence Magazine - May 2022

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