IEEE Computational Intelligence Magazine - November 2022 - 58

TABLE I Formal notations used in this paper.
NOTATION
m
BCiðTiÞ
Pi
pij
N
z
SIS
ISTi
BTISTi
SISTi
maxgen
tD
aD
ATi
pATi1; ...;pATi10
SWorse
ISTi
DESCRIPTION
the number of classes in original multiclass classification problem MC
the i-th decomposed binary classification
sub-problem (task) for MC
the population that evolves for BCiðTiÞ
thej-th individual in the population Pi
the population size of each sub-problem (task)
the threshold that used to determine the " ill-solved " tasks
the set of " ill-solved " tasks
an " ill-solved " task in SIS
the balanced task that constructed for an
imbalanced task ISTi in SIS
the final selected assisting set for task ISTi
the maximum number of generations
the number of generations for independent evolution
the number of generations for assisting evolution
an assisting task in SISTi
10 individuals with the highest accuracy for the taskATi
an subset of SISTi
whose offspring are worse than
those of the original task ISTi
EMT-MC. To be specific, in the
proposed method, an individual pij in
population Pi denotes a weight w oflinear
perceptron (in this paper, parameter
b is assumed as 0 for simplicity). Thus,
pij ¼ w ¼ðw1; ...wdÞ, where d is the
dimensions of data, and wt 2 R
ðt 2f1; ...dgÞ.
At the beginning ofEMT-MC, random
initialization is used for each population,
Pi, which is a widely used
initialization scheme in many EAs.
During the evolution, for each population
the evolutionary operators and
environmental selection that suggested
in NSGA-II [47] are adopted to generate
the offspring. Specifically, a mating
selection strategy based on a binary
tournament selection mechanism is
adopted for mating, and the N better
individuals are generated as parents,
denoted by P0
i . Meanwhile the variation
operators, including simulated binary
crossover (SBX) and polynomial mutation,
are performed on the population
P0
independently, our proposed algorithm
(EMT-MC) views each binary classification
sub-problem as a task, and the " wellsolved "
tasks (sub-problems) assist the " illsolved "
tasks (sub-problems) by using the
evolutionary multitasking learning, which
can achieve the final multiclass classifiers
with better quality. Thedetails oftheproposed
EMT-MC are presented in the following
section.
III. The Proposed Method
Before giving the details of the proposed
method, Table I firstly lists the
formal notations used in this paper
which include those for the multiclass
classification problem and our method.
A. Framework ofthe Proposed
Method
Theproposedmethod(EMT-MC) utilizes
the idea of multitaskinglearningtosolve
the multiclass classification problem. To be
specific, the OVO decomposition strategy
is firstly adopted to transform the original
multiclass classification problem MC
into mðm 1Þ=2binary classification
sub-problems fBC1; .. .; BCmðm1Þ=2g.
Then, each sub-problem BCi is viewed as
atask Ti, i 2f1; .. .; mðm 1Þ=2g.
For each task Ti, a population Pi is evolved
to achieve its corresponding binary classifiers.
In population Pi,the j-th individual
pij denotes a binary classifier obtained for
the binary classification sub-problemBCi.
In this work, the linear perceptron is
used as the binary classifier, since it is a
light classifier and can be obtained with
low computational cost. Specifically,
the linear perceptron modelfðxÞ is formulated
as:
fðxÞ¼ signðwT x þ bÞ
where
TPR ¼
signðxÞ¼
1; x < 0
þ1; x50
w 2 Rd is the learned linear weight. x 2
Rd is the data sample to be classified. b 2
R is a threshold. The basic mechanism
of perceptron can be described as follows.
In a binary classification problem,
the label of a sample vector x is determined
according to the output state of
wT x þ b.If wT x þ b < 0, the label
of x is classified as negative (-1). Otherwise,
it is labeled as positive (+1).
According to the descriptions above,
the real encoding scheme is adopted in
58 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2022
TP þFN
FP
FPR ¼
FP þTN
(2)
i to generate N offspring. After that,
the environmental selection (non-dominated
sorting and crowding distance) is
performed on the combination of parent
and offspring populations. Finally,
N better individuals are selected as the
population Pi for the next generation.
During the evolution ofEMT-MC,
each individual pij in the population Pi
is evaluated by two objectives TPR
(true positive rate) and FPR (false positive
rate) which are widely used metrics
for the binary classification [48]. Their
formulations are depicted as follows.
TP
(3Þ
(4)
where TP (true positive) denotes the
number of the positive samples that are
classified as positive. Similarly, TN (true
negative) denotes the number of the
negative samples that are classified as
negative. FP (false positive) denotes the
number of negative samples that are
classified as positive. FN (false negative)
denotes the number of positive samples
that are classified as negative.
The basic idea ofEMT-MC is to use
the " well-solved " binary classification
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