IEEE Computational Intelligence Magazine - May 2018 - 21
optimizes the support vectors for each class at the same time
that it considers the other subpopulations for solving the multi-
class problem. Algorithm 1 describes MC2ESVM. Generally, it
follows these steps:
1) In line 1, for each class, a population is randomly created.
The number of variables for each population depends on
the number of samples in the training set for the given class.
2) In lines from 3 to 5, the fitness value is assigned for each indi-
vidual of each class (population). For doing so, an individual
from other classes is randomly selected to build the multiclass
classifier.This is part of the cooperative coevolution.
3) Lines from 7 to 16 are the evolutionary process, as follows:
a) Line 9 selects the best individual from other classes and
the evolutionary operators are applied to create an off-
spring, in line 11.
b) Line 12 computes the fitness value of the offspring
solution by concatenating it with the best solutions
from other classes.
c) In line 13, the best solutions among the parents and off-
spring are selected to be included in the next generation.
4) Once the evolutionary process is over, the final solutions,
i.e., the support vectors for the multiclass problem are
obtained from the best individuals of each class; this is
done in line 17.
The details of MC2ESVM are given in the remainder of this
section. First, in Section III-A, we explain the optimization
problem for finding the support vectors for a multiclass prob-
lem. Next, Section III-B presents the representation adopted in
the evolutionary optimization as well as the evolutionary oper-
ators. Section III-C describes the final steps to construct the
multiclass SVM and in Section III-D, we discuss the extension
to learn nonlinear functions.
Algorithm 1 MC2ESVM.
require: X, the set of samples,
Y, the set of classes labels,
c, the regularization term,
p, the population size,
E, maximum number of evaluations.
Ensure: The set of support vectors
1: Generate randomly an initial population, Py for each
class y ! Y
2: for each y ! Y do
3:
Select randomly an individual from each class y l ! Y y
4:
Construct full solutions by combining the selected individuals of each class
5:
Evaluate the full solutions using the fitness function
6: end for
7: while a stopping criterion is not met do
8:
for each y ! Y do
9:
Select the best individual from each population
yl ! Y y
10:
11:
12:
13:
for each individual in the current class do
Apply evolutionary operators to create an offspring
Evaluate the offspring with the fitness function
Add the offspring to the next generation if it
improves its parents
end for
end for
end while
Construct the final solution based on the best individuals
of each population
14:
15:
16:
17:
Setting the partial derivatives to zero gives:
A. Fitness Functions: optimization problem
n
r
2L
= 0 " w r = / a ri x i
2w r
i=1
2L
r
= 0 " b i = C - a ri .
2p ri
In MC2ESVM, each subproblem aims at learning the support
vector for a given class label. Therefore, the optimization prob-
lem for the rth class is formulated as follows:
minr 1 w r
wr, p 2
2
nr
+ C / p ri
i=1
subject to w r , x i $ 1 - p ri , p ri $ 0
(2)
where n r is the number of samples for the rth class.
For the sake of simplicity, we have omitted the bias term in
our formulation. Moreover, since a population is focused on
learning the support vectors for a single class, the class label is
not part of Equation 2. This leads to a simpler dual problem,
with no additional constraints. For the dual formulation, the
constraint is incorporated in the objective function by using the
Lagrange multipliers:
L ^w, p r , a ri , b ri h = 1 w r
2
2
nr
+ C / p ri
i=1
- / a ri 6 w r , x i - 1 + p ri @ - / b ri p ri
nr
nr
i=1
i=1
subject to a ri , b ri $ 0.
(4)
(5)
Substituting Equations 4 and 5 in Equation 3, gives the fol-
lowing dual optimization problem:
nr
nr
1 / ar ar x , x - / ar
min
r
2 i, j = 1 i j i j i = 1 i
ai
subject to 0 # a ri # C.
(6)
At this point, Equation 6 learns the support vectors for a
single class label. Since the goal of MC2ESVM is to gain bene-
fit from the cooperative evolution of each class label, an
additional term is added to Equation 6 that considers the infor-
mation from other classes. Thus, the optimization problem in
cooperative evolution is stated as:
nr
nr
nr
1 / a r a r x , x - / a r + 1 / P ^x h
min
i
r
2 i, j = 1 i j i j i = 1 i n r i = 1
ai
(3)
subject to 0 # a ri # C
may 2018 | IEEE ComputatIonal IntEllIgEnCE magazInE
(7)
21
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