Comparison of Different Surrogate Models for the JADE Algorithm

Konrad Krawczyk

and Jarosław Arabas

Institute of Computer Science, Warsaw University of Technology, Warsaw, Poland

Keywords:

Evolutionary Algorithms, Surrogate Models, Differential Evolution, Ranking Procedure, JADE, CEC.

Abstract:

We investigate the performance of various regression-based surrogate models integrated with a ranking proce-

dure in the Adaptive Differential Evolution with an Optional External Archive (JADE) method. We perform

regression of the ﬁtness function by the surrogate model to reduce the number of ﬁtness evaluations needed

to achieve the optimization progress. The surrogate model training process should be relatively cheap since

training is performed many times along with the optimization process. Therefore we investigate surrogate

models based on k Nearest Neighbors, Random Forests, and Support Vector Machines. We test the effective-

ness of JADE with and without the surrogate models using the CEC2013 benchmark set for single-criterion

continuous optimization. Experimental data conﬁrm the beneﬁts of using the surrogate models and indicate

the difference in efﬁciency improvement between the considered models.

1 INTRODUCTION

In this text, we aim to improve the efﬁciency of the

Adaptive Differential Evolution with an Optional Ex-

ternal Archive (JADE) algorithm (Zhang and Gao,

2013), which is a popular evolutionary algorithm and

has demonstrated remarkable performance in single-

criterion continuous optimization tasks. However, the

computational cost associated with the ﬁtness evalu-

ations required by the JADE algorithm can become

a bottleneck, especially when dealing with computa-

tionally expensive simulations or real-world systems.

This problem can be tackled by using surrogate mod-

els, also known as metamodels, to reduce the com-

puting time by providing an estimation of the ﬁt-

ness function. We introduce a methodology to han-

dle the surrogate model adaptation and investigate

several candidates for the surrogate models to be in-

tegrated with JADE. We evaluate the efﬁciency of

surrogate-assisted JADE using the CEC2013 bound-

constrained, single-criterion benchmark suite.

Surrogate models can either directly return the es-

timation of the ﬁtness function value of each individ-

ual or they can approximate the result of comparison

of ﬁtness values between two or more individuals. For

example, in the case of Differential Evolution (DE),

the selection process is based on the comparison be-

tween each offspring and its parent. One way of using

https://orcid.org/0000-0003-2012-0803

https://orcid.org/0000-0002-5699-947X

surrogate models would be to compute the surrogate

ﬁtness values and to perform selection on the basis

of their comparison. Note that the result of compar-

ison affects the selection rather than the ﬁtness val-

ues, therefore we may be less careful about the ap-

proximation accuracy of the ﬁtness function with the

surrogate model, provided that the proper relationship

of the individuals’ ﬁtness is preserved by the surro-

gate values. Hence we can distinguish between the

regression-based and the classiﬁcation-based surro-

gate models. Example applications of both concepts

for the DE can be found, e.g., in (Zhang and Sander-

son, 2007) and in (X. Lu and Yao, 2011).

When the surrogate model of the ﬁtness function

is used, several decisions need to be made:

• Individual vs. universal metamodel: The sur-

rogate function can be deﬁned for each individual

separately, or it can be universal for all population

members. If the variability pattern of the ﬁtness

landscape is relatively similar among individuals,

a single surrogate function may sufﬁce; otherwise,

assigning each individual its own surrogate func-

tion might lead to better approximations and ul-

timately improve the optimization process (Ong

et al., 2004).

• Frequency of re-training the surrogate func-

tion: Retraining the surrogate model after each

generation can provide a more accurate represen-

tation of the problem space, but it is quite expen-

sive. Updating the model after a few generations

186

Krawczyk, K. and Arabas, J.

Comparison of Different Surrogate Models for the JADE Algorithm.

DOI: 10.5220/0012166100003595

In Proceedings of the 15th International Joint Conference on Computational Intelligence (IJCCI 2023), pages 186-194

ISBN: 978-989-758-674-3; ISSN: 2184-3236

can reduce the computation cost but it can result

in a poorer quality of the model, so the frequency

of retraining reﬂects the desired trade-off between

computational time and solution quality.

• Handling the true ﬁtness function samples:

Surrogate models are built on the basis of the true

ﬁtness function samples. Initially, such a sample

can be created using a limited number of evalua-

tions when the true ﬁtness function is computed.

As the optimization continues, the sample can

be expanded by evaluating only a portion of the

points created by the optimization process. This

dynamic approach helps maintain a balance be-

tween exploration and exploitation and may result

in a more accurate surrogate model over time (He

et al., 2023).

• Metamodel adaptation style: The surrogate

model can be retrained either from scratch, or it

can be adaptively, incrementally changed. This

can involve updating the model’s parameters or

structure, which can help in reducing the compu-

tational cost while maintaining a reasonable level

of accuracy (Picheny et al., 2013).

• Surrogate model class: There are numerous

types of surrogate models that can be used in opti-

mization, including but not limited to kriging, lin-

ear regression, support vector machines, artiﬁcial

neural networks, Gaussian process regression, k-

nearest neighbors, and decision trees. The choice

of the surrogate model depends on the problem

domain, the available data, and the desired trade-

off between model complexity and computational

cost (Zaborski and Ma

ndziuk, 2022),(Zhang et al.,

2022).

We deﬁne a regression-based surrogate model to

be commonly used by all population individuals

(a global surrogate model). The surrogate model

is adaptively changed in each iteration of JADE.

The surrogate model adaptation process is performed

after the offspring population is completed. We

adopted the approximate ranking procedure proposed

in (S. Kern and Koumoutsakos, 2006) as a procedure

to control the process of computing true ﬁtness val-

ues, when the surrogate model approach is coupled

with the CMA-ES method. The main idea is to con-

trol the surrogate model quality without knowing the

correct ranking of the complete population.

In this study, we compare the performance of

four widely used regression-based surrogate models

(k Nearest Neighbors, Random Forest, XGBoost, and

Support Vector Regression) when integrated with the

JADE algorithm with the aforementioned ranking-

based model adaptation procedure for single-criterion

continuous optimization problems. We aim to iden-

tify the most suitable surrogate model for JADE

by considering their performance and efﬁciency im-

provement.

To achieve this, we use the CEC2013 bench-

mark set for single-criterion continuous optimization,

which covers a wide range of test functions with var-

ious optimization challenges. By analyzing the per-

formance of JADE with a surrogate model employed

in the ranking procedure, we aim to provide insights

into the effectiveness of different surrogate models

and their potential to enhance the JADE algorithm’s

performance.

2 RELATED WORK

2.1 JADE Algorithm

The Adaptive Differential Evolution with an Optional

External Archive (JADE) is an extended version of

the Differential Evolution algorithm, as presented

in (Zhang and Sanderson, 2009). A key aspect of

JADE is its self-regulating control of the mutation

and crossover parameters, F and CR, which are essen-

tial for achieving a balance between exploration and

exploitation during the optimization process. JADE

utilizes a learning strategy that updates F and CR

based on successful mutations, allowing the algorithm

to adjust to the problem’s characteristics throughout

the optimization process. This adaptive approach re-

moves the need for manual parameter tuning, making

JADE more robust and effective for solving a wide

range of optimization problems.

Another remarkable feature of JADE is the op-

tional utilization of an external archive, which stores

the solutions that are replaced during the evolution-

ary process. The external archive enhances the al-

gorithm’s ability to preserve population diversity and

prevent premature convergence. By integrating the

archived solutions into the mutation process, JADE

can efﬁciently explore the search space and identify

promising regions, resulting in improved optimization

performance.

In JADE, the DE/current-to-pbest/k mutation

scheme is used. It selects the p% best individuals

from the current population, where p% is a user-

deﬁned parameter, and combines them with the cur-

rent individual and a randomly selected individual

from either the current population or the external

archive. This mutation operation can be expressed as

follows:

= x

+ F · (x

− x

) + F · (x

− x

) (1)

Comparison of Different Surrogate Models for the JADE Algorithm

187

where m

is the mutant, x

is the target vector, x

a vector chosen from the fraction of p% best ﬁt vec-

tors from the current population, and x

and x

are

randomly selected individuals from either the current

population or the external archive. A more compre-

hensive pseudocode of JADE is provided in section 3,

along with the adaptations made to include the surro-

gate model adaptation.

2.2 Approximate Ranking Procedure

In (S. Kern and Koumoutsakos, 2006) the method

called approximate ranking procedure as assistance

to the surrogate model approach implemented in the

CMA-ES is proposed. Its main task is to help control

model quality without knowing the correct ranking of

the complete population. In that procedure, in the be-

ginning, a surrogate model is prepared based on the

training set S. With the use of the prepared surro-

gate model, a sorting order for µ best individuals is

prepared. Next, a fraction of offspring individuals is

evaluated with the original ﬁtness function and added

to S, the surrogate model is updated, and the sorting

order is established anew. If this results in a change of

the sorting order, the process of evaluating a fraction

of offspring, model update, and ranking is repeated

until the ranking stabilizes.

2.3 Considered Surrogate Models

K Nearest Neighbours. kNN is a non-parametric,

instance-based model (Dudani, 1976) which uses an

archive of points S. When given an input point x, the

model deﬁnes its output

f (x) as:

f (x) =

∑

i=1

· f (x

)

∑

i=1

(2)

where f (x

) is the approximated function value for the

i-th closest neighbor of the point x from the set S, w

is its weight, and the summation runs over the k near-

est neighbors. The weights are assigned to the neigh-

bors based on the inverse of their distance to the query

point:

||x

− x||

(3)

where || · || is the assumed norm (in our case, Eu-

clidean).

kNN is a non-parametric method, which means it

does not make any assumptions about the approxi-

mated function formula. This ﬂexibility allows it to

capture complex relationships between the input and

output variables, making it suitable for a wide range

of optimization problems (James et al., 2021). The

choice of k inﬂuences the model’s performance, with

smaller values providing a more ﬂexible ﬁt and larger

values offering a smoother approximation.

Random Forest Regression. RFR is an ensemble

learning method that combines multiple regression

trees to produce a more accurate and robust predic-

tion (Breiman, 2001).

Each regression tree is constructed independently

using a bootstrapped sample of the training set S. Dur-

ing the construction of each tree, a random subset

of features is selected at each node, which helps to

decorrelate the individual trees and improve the over-

all performance of the ensemble. The ﬁnal prediction

for an input point x is obtained by averaging the out-

puts of all the individual trees in the forest:

f (x) =

∑

i=1

(x) (4)

where

(x) is the output of the i-th decision tree, and

T is the total number of trees in the forest.

The number of trees (T ) and the size of the ran-

dom feature subset at each node are important hyper-

parameters that inﬂuence the performance of the RFR

model.

XGBoost. XGBoost builds on the principles of gra-

dient boosting, a method that sequentially combines

regression trees into a weighted average that can ac-

curately model complex relationships between input

and output variables (Chen and Guestrin, 2016). The

key idea behind XGBoost is to iteratively learn a re-

gression tree that can best correct the errors made by

the previously combined trees. The ﬁnal prediction

for an input point x is deﬁned as:

f (x) =

∑

i=1

(x) (5)

where T is the number of trees,

(x) is the predic-

tion of the i-th tree for the input point x, and w

is the

corresponding weight. Important hyperparameters in

XGBoost include the maximum tree depth, learning

rate (shrinkage), the number of trees, and regulariza-

tion parameters (L1 and L2).

Support Vector Regression. SVR (Drucker et al.,

1997) is a regression version of the Support Vector

Machine (Cortes and Vapnik, 1995). SVR yields its

prediction as a weighted sum of a form:

f (x) =

∑

∈S

k(x, x

) + b (6)

ECTA 2023 - 15th International Conference on Evolutionary Computation Theory and Applications

188

where w

are the weights, b is the bias term, and

k(x, x

) is the kernel function which takes as argu-

ments a point x

from the dataset S and the input point

x. The archive points whose corresponding weights

are nonzero are called the support vectors.

A critical aspect of SVR is the proper choice of

the kernel function which determines the transforma-

tion of the input data and the shape of the approxima-

tion hyperplane. Commonly used kernels include the

linear, polynomial, radial basis, and sigmoid. Other

important hyperparameters in SVR include the regu-

larization parameter C, which controls the trade-off

between model complexity and the degree of allowed

error, and the ε parameter, which deﬁnes the accept-

able error margin around the optimal hyperplane.

3 JADE WITH A SURROGATE

MODEL

The use of the fraction of p% best points in the

JADE mutation scheme is analogous to the selection

scheme used by CMA-ES. This motivated us to cou-

ple JADE with a ranking procedure originally formu-

P ← {x

,...,x

}

S ← (x

, f (x

)), k = 1,...,µ

= 0.5, µ

= 0.5

while !stop do

0, S

pbest

← f ind pbest(q(P))

for i := 1 to µ do

= randn

(µ

,0.1)

= randc

(µ

,0.1)

← sample(P)

pbest

← sample(P

pbest

)

← x

+ F

· (x

pbest

− x

+ x

− x

)

← crossover(x

)

end

f ,q(O),S ← ranking procedure(O,S)

q(P) ←

f (P)

for i := 1 to µ do

if q(o

) < q(x

) then

← o

← S

∪ {CR

}

← S

∪ {F

}

end

= (1 − c) · µ

+ c · mean

)

= (1 − c) · µ

+ c · mean

)

end

Algorithm 1: JADE with the surrogate model-based

ranking procedure.

lated for CMA-ES in (S. Kern and Koumoutsakos,

2006). Pseudocodes of the resulting JADE version

and of the approximate ranking procedure are pre-

sented as Algorithm 1 and Algorithm 2, respectively.

In the modiﬁed JADE, we distinguish between the

true ﬁtness function and the surrogate function, de-

noted with f and

f , respectively. The ﬁtness value

of a particular individual x

, maintained in a popula-

tion, is denoted by q(x

) and is assigned, depending on

the context, either by the true ﬁtness or the surrogate

function. Every point evaluated by the true ﬁtness

function is stored in the archive S which is organized

as a FIFO queue. This archive is used to train the

surrogate models and should be distinguished from

the optional archive of points, which is maintained by

JADE and is used to increase the number of difference

vectors.

The algorithm starts by generating the initial pop-

ulation P and evaluating all the individuals in the pop-

ulation using the original ﬁtness function f . As the

optimization loop begins, the population of p% best

individuals is identiﬁed on the basis of either the orig-

inal ﬁtness function (when available) or the surrogate

model. The offspring individuals are generated using

the mutation and crossover, and they are evaluated us-

ing the ranking procedure that updates the surrogate

model. After that, the parents’ ﬁtness values obtained

using the previous surrogate model are re-calculated

using the updated surrogate model. The selection of

f ← build

surrogate model(S)

q(P) ←

f (P)

(P) ← sort according to q(P)

S ← S ∪ {(x

, f (x

))}, i = 1, ..., n

init

for i := 1 to (µ − n

init

)/n

f ← build surrogate model(S)

q(P) ←

f (P)

(P) ← sort according to q(P)

if r

i−1

(P)[1 : pbest] = r

(P)[1 : pbest]

then

break

end

else

S ← S ∪ {(x

, f (x

))}, x = 1, ..., n

end

if i > 2 then

init

← min(n

init

+ n

, µ - n

)

end

else if i < 2 then

init

← max(n

, n

init

- n

)

end

Algorithm 2: Ranking procedure (P,S).

Comparison of Different Surrogate Models for the JADE Algorithm

189

the new parent population is based on the ﬁtness val-

ues obtained with the use of the same surrogate model

for both old parents and new offspring. Thus the new

parent population is completed.

4 EXPERIMENTAL STUDY

4.1 Experiment Conditions

CEC2013 Benchmark Suite. We tested the JADE

supported by the aforementioned regression-based

surrogate models using the CEC2013 benchmark

suite (Liang et al., 2013) which deﬁnes 28 challeng-

ing optimization functions that have a wide spectrum

of landscapes and difﬁculties. Thus it is particularly

well-suited for assessing the effectiveness of surro-

gate models.

The CEC2013 benchmark suite ﬁtness functions

are challenging for regression models, particularly in

terms of local estimation – many of them are highly

nonlinear and non-convex, making it difﬁcult for re-

gression models to accurately capture the underlying

relationships between input and output variables, par-

ticularly in local regions. Some of them present noisy

landscapes, and some are locally asymmetrical about

their local minima. In this study, we focus on prob-

lems with D = 10 dimensions.

Parameter Settings. To ensure statistically mean-

ingful results, we executed 51 independent runs for

each combination of JADE with a surrogate model

and for each optimization problem from the bench-

mark suite. In each run, we recorded the best achieved

true ﬁtness value and treated it as a result of that

run. In each independent run, the algorithm was

given a budget of 100,000 ﬁtness evaluations, as it

was suggested in the CEC2013 benchmark suite. The

JADE population contained µ = 20 individuals, and

the search range was deﬁned as [−100, 100]

. The

optimization process was terminated either after us-

ing the admissible budget or after reaching a point

whose ﬁtness differed no more than 10

−8

from the

global optimum which has been deﬁned in the bench-

mark suite. Bound constraints imposed by the bench-

mark suite were handled with the reﬂection method,

according to the guidelines from (Biedrzycki et al.,

2018).

For the kNN regression model, the value of k was

set to D + 2 after a preliminary tuning. In the case of

SVR, RFR, and XGBoost, we used the default param-

eters’ settings provided in the libraries that implement

these methods (Chen and Guestrin, 2016), (Pedregosa

et al., 2011).

4.2 Results

Efﬁciency of Surrogate Models and the Archive

Size. The ﬁrst series of experiments was aimed at

selecting the appropriate archive size for each surro-

gate model. We analyzed statistics of the results ob-

tained by surrogate model-assisted JADE for archive

size ranging from 2 · µ up to 50 · µ, where µ stands for

the population size. The results of the experiments

are provided in Fig. 2, 3 in the form of boxplots. The

median value is indicated by a horizontal line within

the box. The width of the box corresponds with the

interquartile range, and the whiskers correspond to

the distance between extreme values and the ﬁrst/third

quartiles. Outliers are represented with bullets.

Labels on axis X represent various surrogate

model settings: <model> <n> means that it is a

combination of JADE with a speciﬁc surrogate model

and the size of the archive of points (expressed as the

population size multiplier <n>).

In the case of kNN, RFR, and XGBoost, the

performance of the surrogate-assisted JADE usually

grows with the size of the archive used for creating

the surrogate model, but for some optimization prob-

lems, e.g. F15, F16, F23, it appears that an optimum

value of the archive size can be observed. Moreover,

the relationship between the archive size and the opti-

mization efﬁciency is similar between the three afore-

mentioned model types.

In the SVR case, quite often the quality of results

is worse than the results of JADE alone. Moreover,

the results get worse along with the model size. Clar-

iﬁcation of this effect needs much deeper investiga-

tion, we hypothesize that perhaps the kernel formula

and/or the kernel parameter are responsible for this

behavior.

Comparison Between Surrogate Model Efﬁciency.

For each surrogate model, we selected the best-

performing archive size using the Wilcoxon test in

pairs by comparing the results yielded by JADE with

and without the surrogate model across 28 different

CEC2013 functions. Then we aggregated the results

by counting the number of wins, draws, and losses of

a surrogate model-assisted JADE with JADE without

the model. The winning population size for the sur-

rogate model was equivalent to the value for each the

difference between the number of wins and the num-

ber of losses was greatest. The archive size of 20 gave

the best performance within each surrogate model.

Then we applied the same methodology to the

comparison between the results yielded by the sur-

rogate models with the optimal archive size. Ta-

ble 1 summarizes the results. Each row in the table

ECTA 2023 - 15th International Conference on Evolutionary Computation Theory and Applications

190

Table 1: Results of the Wilcoxon test for each surrogate

model and the standard JADE version for functions from

CEC2013 benchmark.

function kNN RFR XGBoost SVR

F1 * * - *

F2 + * * -

F3 - * * -

F4 + * * -

F5 * * - -

F6 * - - -

F7 + + + -

F8 * * * *

F9 + + + *

F10 * - - -

F11 * * - -

F12 + + + -

F13 + * + -

F14 - - - -

F15 + + + -

F16 - - * -

F17 * - * -

F18 + + * -

F19 * * * -

F20 + + + -

F21 + - - -

F22 - - - -

F23 + + + -

F24 + * + -

F25 + * + -

F26 * + * -

F27 + + + -

F28 * - - -

14/10/4 9/11/8 10/9/9 0/3/25

corresponds to a speciﬁc function, and the columns

represent each surrogate model with its best archive

size. We indicate the results with symbols “+”, “–

”, and “*”, when the surrogate-model assisted JADE

was performing better than JADE alone, worse than

JADE alone, or there was no statistically signiﬁ-

cant difference between the compared JADE versions.

The last row of the table summarizes the number of

wins/draws/losses of the surrogate model supported

JADE versus JADE alone.

Among the considered surrogate models, a clear

winner is kNN. JADE coupled with kNN yields re-

sults that are superior to JADE without a surrogate

model in the case of 14 optimization problems from

the benchmark suite, in 10 cases the results were com-

parable, and only in 4 cases, pure JADE was the win-

ner. Notably, it never happened that some other model

improved and kNN deteriorated the quality of JADE.

SVR consistently underperforms compared to other

methods, probably due to an improper choice of ker-

nel or its parameters.

Analysis of Convergence Curves. The Wilcoxon

test-based comparison relates to the best results af-

ter spending the admissible budget of ﬁtness evalu-

ations. Here we provide the convergence curves to

compare the intermediate results obtained for several

selected CEC2013 problems. The convergence curves

(Fig. 1) illustrate the evolution of the best-performing

point ﬁtness in each of 51 independent runs, averaged

over the independent runs. The results for surrogate

model-assisted JADE are obtained for the best archive

size.

In certain instances, JADE coupled with a surro-

gate model, clearly converges faster and yields bet-

ter results compared to the standard JADE. This un-

derlines the effectiveness of the JADE with a sur-

rogate model in swiftly navigating the search space

and reaching near-optimal solutions in a shorter time

frame. A good example is problem 27, where most

models show behavior like that, except the SVR

model. However, there are also scenarios where the

surrogate model assisted JADE initially converges

faster but plateaus at a certain level. This could be

indicative of getting stuck in local optima, which is a

well-known issue in evolutionary algorithms. These

observations provide insights into potential areas of

reﬁnement for our method, such as incorporating

mechanisms to escape from local optima and to reach

global ones. For problem 5 we can see this effect for

SVR and XGBoost models. In addition, there are also

situations where the standard JADE converges faster

initially but is eventually overtaken by its surrogate

model-assisted versions, which ﬁnally achieve better

results. This suggests that while the standard JADE

may be quicker to ﬁnd promising regions in the search

space for some of the functions, surrogate model as-

sistance could improve the results in the longer run.

For kNN and RFR, problem 20 is a good example of

such behavior.

5 CONCLUSION

We show the potential of using surrogate models

to improve the efﬁciency of the JADE algorithm.

Among tested methods, kNN with an archive size

equal to 20 achieved the best performance on the

CEC2013 benchmark suite. Note that kNN is a rela-

tively cheap model because it needs no training phase

to build the internal model structure.

Our ﬁndings highlight the important role of surro-

gate model selection. While certain benchmark func-

tions delivered comparable results regardless of the

surrogate model used, we have shown that most func-

tions reacted differently to varying models, leading to

signiﬁcant differences in the results. This points to the

necessity of understanding the speciﬁc characteristics

and requirements of each function when choosing the

most suitable model.

Despite the promising outcomes, we recognize

that there is always potential for improving our

method. Future research will thus concentrate on en-

hancing the performance of analyzed models on the

test functions where they currently lag.

Comparison of Different Surrogate Models for the JADE Algorithm

191

Figure 1: Convergence curves of problems 5, 20, and 27.

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Figure 2: Boxplot results for functions 1 - 14 from CEC2013 Benchmark.

Comparison of Different Surrogate Models for the JADE Algorithm

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Figure 3: Boxplot results for functions 15 - 28 from CEC2013 Benchmark.

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