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Ta b l e 2 . Comparison of the standard parameter set against the specific best parameter set;
denotes the optimization with 9900 iterations, i.e., 297000 function evaluations
Tests
Reference in [3]
Best parameter set
specificspecific standard
Function
gbest
lbest
( W, C 1 ,C 2 )
Ackley
2.58
2.62
18.34
17.6628
17.5891
(0.7893,0.3647, 2.3541)
Gen. Schwefel
2154
2155
3794
3508
3360
(0.7893,2.4098, 0.3647)
Griewank
0.0135 0.0135
0.0395
0.0308
0.0009
(0.6778, 2.1142, 1.3503)
Rastrigin
6.12
6.12
169.9 140.4876
144.8155
(0.7893,2.4098, 0.3647)
Rosenbrock
0.851
0.86
4.298
8.1579
12.6648
(0.7123,1.8782, 0.5902)
Schwefel
0
0
0
0
0.1259
more than one set
Sphere
0
0
0
0
0
more than one set
found the global optimum) on gbest for all functions with the selected parameter sets.
Table 2 shows our results for the specific parameters for the different functions (300000
evaluations + 30000 evaluations for feature computation; denoted as “specific”), the
same parameter set subtracting one percent evaluations for the feature computation (to
demonstrate if we used this one percent of computation time to extract the features, i.e.,
a total of 300000 evaluations; “specific ”) and the comparison to the standard parame-
ter set included in our code. Additionally, the comparison to the results of the original
paper of Bratton and Kennedy is included in the table.
The extensive search shows that the best specific parameter sets for the functions
Gen. Schwefel and Rastrigin is comparable. The same effect is also supported by the
features of both objective functions. This denotes that both functions are assigned the
same class in our classifier. All the specific parameter sets are the base of our classes
for each function. With the identified classes and the computed set of features for each
function we can train the classifier.
4.3
Learning and Classification
As classifier we use a C4.5 decision tree. In our implementation we use WEKA's J4.8
implementation [15]. As learning input we compute 300 independent instances for each
function. Each instance consists of 32 function features. The decision tree is created
based upon the training data and evaluated by stratified 10-fold cross-validation (re-
peated 100 times). Based on the results of the extensive search we merge the classes
for the objectives Gen. Schwefel and Rastrigin into one class. These functions share
the same specific parameter set, i.e., the same parameter configuration performs best
for both functions. Upon these six distinct classes we evaluate the model with cross
validation. The mean accuracy of the 100 repetitions is 84.32% with a standard devi-
ation of 0.29. Table 3 shows the confusion matrix of a sample classification. As it can
be seen, there are 1769 of 2100 instances classified correctly (this means 15.76 percent
of the instances are misclassified). The instances of the functions Ackley and Schwefel
are classified correctly with an accuracy of 99.7 percent, that is only one instance of
these classes is misclassified. The class for Gen. Schwefel and Rastrigin has an accu-
racy of 97.2 percent. The class Rosenbrock has a slightly lower accuracy, but still only
5.7 percent of its members are misclassified. The high number of incorrect instances
is essentially due to the inability of the model to separate the functions Sphere and
Griewank. The majority of the misclassified instances, 306 of 331, are instances of the
Griewank or Sphere class that are classified as the other class.
 
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