Population Variance Based Empirical Analysis of. the Behavior of Differential Evolution Variants

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1 Applied Mathematical Sciences, Vol. 9, 2015, no. 66, HIKARI Ltd, Population Variance Based Empirical Analysis of the Behavior of Differential Evolution Variants S. Thangavelu, G. Jeyakumar and C. Shunmuga Velyautham Department of Computer Science and Engineering, Amrita School of Engineering,Amrita Vishwa Vidyapeetham Coimbatore, India Copyright 2014 S. Thangavelu, G. Jeyakumar and C. Shunmuga Velyautham. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Differential Evolution (DE) is a simple but efficient Evolutionary Algorithm (EA) for stochastic real parameter optimization. With various types of mutation and crossover applicable to DE, there exist many variants of DE. The empirical comparisons between the performances of these variants on chosen benchmarking problems are well reported in literature. However, attempts to analyze the reason for such identified behavior of the variants are scarce. As an attempt in this direction, this paper empirically analyzes the performance as well as the reason for such performance of 14 classical DE variants on 4 benchmarking functions with different modality and decomposability. The empirical analysis is carried out by measuring the mean objective function values (MOV), success rate (Sr), probability of convergence (Pc), quality measure (Qm) and empirical evolution of the variance of the population (Evar). The study also includes reporting evidences for the variants suffering with stagnation and/or premature convergence. Keywords: Differential Evolution, Population Variance, Stagnation and Premature Convergence 1 Introduction Differential Evolution (DE) [1] is a well known Evolutionary Algorithm (EA) for solving optimization problems in continuous spaces. The superiority of DE has been tested and proved on many benchmarking problems and real-world applications [2, 3]. Like other EAs, DE also employs mutation, recombination and

2 3250 S. Thangavelu et al. selection operations during evolution. However, DE has some unique characteristics which makes it different from other algorithms in the EA family. The mutation operation in DE, differential mutation, adds the weighted difference between a pair of parent vectors to a target vector to produce a mutant vector. Between the target vector and the mutant vector, a recombination operation is carried out to produce a trial vector. It is followed by a one-to-one greedy selection between the target vector and the trial vector. With various types of mutation and recombination there exist many strategies for generating trial vectors and consequently many DE variants in the literature. However, no variant has turned out to be superior in solving wide range of problems. Even though there are various studies [5, 6, 7, 8 and 9] reported in the literature on the performance of DE variants, the reasons for performance difference of the variants have not been properly addressed. There are few works reported in the literature to study the convergence nature and explorative power of the DE variants [10, 11]. This necessitates investigation on the inherent evolution mechanism of the variants, based on their mutation and recombination operations. This paper is an attempt to identify the reason for such performance of the variants, in light of the population diversity during the course of evolution. The remainder of the paper is organized as follow. Section 2 describes the algorithmic structure of DE, Section 3 briefs the exploration and exploitation process in DE and Section 4 presents the related works. In Section 5 design of experiment is presented. The results are presented and discussed in Section 6 and finally Section 7 concludes the paper. 2 Differential Evolution Population Initialization X(0) {x 1(0),...,x NP(0)} g 0 Compute { f(x 1(g)),...,f(x NP(g)) } while the stopping condition is false do for i = 1 to NP do MutantVector: y i generate mutant(x(g)) TrialVector: z i crossover(x i(g),y i) if f(z i) < f(x i(g)) then x i(g+1) z i else x i(g+1) x i(g) end if end for g g+1 Compute{ f(x 1(g)),...,f(x NP(g))} end while As depicted in Figure 1, the DE algorithm starts with initialization of a NP D-dimensional vectors, called initial population, denoted as X(0). After population initialization, an iterative process comprising mutation, recombination and selection is started. At each generation (iteration i) a new population X(i) is generated until the stopping criteria is satisfied. Figure1. Description of DE algorithm

3 Population variance based empirical analysis of the behavior 3251 We choose seven different mutation operations (rand/1, best/1, rand/2, best/2, current-to-rand/1, current-to-best/1 and rand-to-best/1) and two recombination operations (bin and exp), which resulted 14 different DE variants viz (DE/rand/1/bin, DE/rand/1/exp, DE/best/1/bin, DE/best/1/exp, DE/rand/2/bin, DE/rand/2/exp, DE/best/2/bin, DE/best/2/exp, DE/current-to-rand/1/bin, DE/current-to-rand/1/exp, DE/current-to-best/1/bin, DE/current-to-best/1/exp, DE/rand-to-best/1/bin and DE/rand-to-best/1/exp). The performance of these 14 fourteen variants on a test suite with 14 benchmark problems has been reported in [5 and 6]. This paper focuses on identifying the possible reasons for the reported performance of the above said DE variants, as an initial attempt. 3 Exploration and Exploitation of DE The search behavior of any EA (and hence DE) depends on its exploration and exploitation processes. The former explores the search space for finding the better candidate. On the other hand, the exploitation process focuses the search towards the desired local region. As stated by Beyer [13] and Feoktistov [14] the ability of an EA to find a global optimal solution depends on its ability to find the right relation between exploitation of the elements found so far and exploration of the search space. In DE, the variation operators viz. mutation and crossover operators are often attributed for the exploration process and the selection operator for the exploitation process. The variation operators often produce new candidate solutions and the selection operator selects suitable candidate(s) among the existing candidates. It is fairly understandable that the exploration process increases the population variability and the exploitation process decreases the population variability. Population Variance can be considered a good measure of population variability [15]. It serves as a measure to understand the diversity in the population, at every generation. Since each variation operator has a distinctive way of exploring the search space, the inherent evolutionary behavior of each DE variant is different. The empirical evolution of a DE variant, for the given random initial population, ends up in either one of the following three circumstances: (1) successful convergence (2) premature convergence and (3) stagnation. This paper is a preliminary attempt to identify the circumstances under which each of the DE variant descends in the search space (assuming a minimization problem) during its evolution and depict suitable empirical evidence for such cases. 4 Related Works With existence of many variants for mutation and crossover operators in DE, choosing an appropriate mutation/crossover operator for a given problem is crucial. Obviously, the performance of DE is largely affected by the strategy chosen for mutation and crossover. However, such choice is purely problem dependent. No DE variant, till now, has proven to be best suited for all kinds of

4 3252 S. Thangavelu et al. problems, which is quiet understandable from No Free Lunch Theorem [12]. Extensive empirical analysis on the performance of DE variants reported in the literature provides insight about the efficacy of different DE variants. Mezura-Montes et al in [7] reported performance of eight DE variants, and identified DE/rand/1/bin, DE/best/1/bin, DE/current-to-rand/1/bin and DE/rand/2/dir as the competitive variants. A comparison of ten different variants of DE to solve an optimal design problem was reported in [8]. This study concluded DE/best/*/* variants to be better than DE/rand/*/* variants. Mezura-Montes et al. in [9] presented an extensive analysis of different variants of DE on 24 benchmark problems. There have been few theoretical investigations too towards this direction. The relationship between the control parameters of DE and its evolution of population diversity is analyzed in [20], and based on the analysis, the control parameters are set up to avoid the premature convergence. A parameter adaptation technique based on the population variability is proposed in [21]. An idea to modify the selection operator to improve the balance between exploration and exploitation is proposed in [17]. Some mechanisms for diversity enhancement for DE is proposed and tested in [16, 22]. An empirical comparative study between DE/rand/1/bin and DE/best/1/bin by their population convergence measure and by their empirical evolution of population variance is presented in [10] and [11], respectively. Theoretical expressions to measure the population diversity of the DE variants also derived in the literature [19, 24]. All the above studies on the population diversity of DE have been carried out scarcely on one or two variants of DE. This necessitates extensive empirical analysis on the evolutionary behavior of different DE variants. This work is an attempt in this direction. 5 Experimental Design This paper investigates the performance of the above mentioned 14 classical DE variants on benchmark functions with 30 dimensions [7, 23] viz. 1. f1 Step Function(Unimodal Separble) 2. f2 Schwefel s Function 1.2(Unimodal Nonseparble) 3. f3 Generalized Restrigin s Function (Multimodal Separable) and 4. f4 - Generalized Griewank's Function(Multimodal NonSeparable). Classical DE has only three parameters to be set: NP (population size), F (mutation scale factor) and Cr (crossover rate). The NP has been set a moderate

5 Population variance based empirical analysis of the behavior 3253 value of 60. The F value is set as based on [7]. To decide the Cr values a bootstrap test was conducted for every variant-benchmark function with varying values of Cr as {0.0, 0.1, 0.2, 0.4, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}. The maximum number generation (MaxGen) is set as 3000 (and hence the maximum number of function evaluations (MaxnFE) is 180,000 (ie., 3000 * NP)). The stopping criterion has been fixed as a tolerance error of 1 x Since, DE is a stochastic algorithm, for each variant-benchmarking function 100 independent runs were performed with random initialization of the initial population for every run. At each run, DE variants will stop its evolution either upon reaching the stopping criteria or upon exhausting maximum allotted function evaluations. The performance analysis of the DE variants has been carried out by measuring their Mean Objective function Value (MOV), Success rate (Sr), Probability of Convergence (Pc) and Q-measure (Qm). The success rate is calculated as the percentage of number of successful runs out of total runs for each function,, where nc_f is total number of successful runs made by a variant for a function and nt_t is the total number of runs, in this current work nt_t = 100. A run is considered successful when the tolerance error is reached before the maximum number of generations. Pc% is measured as (nc total number of successful runs made by a variant for all the benchmarking functions, nt total number of runs (for our experiment nt = 4 functions * 100 runs = 400 runs)) [5, 6, 14]. The Quality measure is calculated as, where, with j =1,,nc are successful runs and FEj is the number of function evaluations in j th trial. The variants with good convergence rate and more probability of convergence will have lower values of quality measure. Subsequently, the empirical analysis of the DE variants has been carried out by observing their population variance (Evar). Analyzing the evolution pattern of the population variance, generation by generation, will provide insight about the exploration capability of the underlying variance operators of the given DE variant. In DE algorithm the variation operators perturb independently each component of the candidate in the population [11]. Hence, the population variance is measured independently for all D components of the candidates at all the 100 runs and the average of that is used for the study. 6 Experimental Results and Discussion Table 1 presents the MOV, Standard deviation, Sr, Pc% and Qm values obtained by all the 14 variants on solving the 4 benchmark functions. As can be seen from the results, no variant has turned out to be the best in solving all the functions. But,

6 3254 S. Thangavelu et al. DE/rand/2/bin and DE/rand-to-best/1/bin managed to solve three functions; DE/best/2/bin and DE/rand/1/bin have solved two functions. It is worth observing that the top four positions in terms of both higher probability of convergence and lower quality measure are secured by the same set of variants DE/rand-to-best/1/bin, DE/rand/1/bin, DE/best/2/bin and DE/rand/2/bin. The variants DE/current-to-rand/1/exp, DE/current-to-best/1/exp, DE/best/1/exp and DE/best/1/bin have not solved any of the 4 functions. All other remaining variants have solved a maximum of only one function each. In terms of Sr, between the extremes (0 and 100), there are many variants that provide both successful and unsuccessful runs in solving the given function like DE/rand/1/bin and DE/best/1/bin. The convergence analysis of different DE variants on different benchmarking functions, based on their performance analysis above, has been carried out. This involved observing the best objective function value as well as cumulative mean and standard deviation obtained by a DE variant in every 300 generations for a random run. Table 2 presents the convergence behavior of DE/rand/2/bin and DE/rand-to-best/1/bin on functions f1, f3 and f4 for a random run. The convergence behavior of DE/best/2/bin and DE/best/2/exp variants on function f2 is shown in Table 3. It is evident from Tables 2 and 3, that the objective function value converges faster (within lesser number of generations) to the global optimum. Table 1. The Pc(%), Qm, MOV, standard deviation and Sr measured for the 14 variants Sno Variant Pc(%) MOV Std Sr MOV Std Sr 1 DE/rand/1/bin E E E E DE/rand/1/exp E E E E DE/best/1/bin E E E E DE/best/1/exp E E E E DE/rand/2/bin E E E E DE/rand/2/exp E E E E DE/best/2/bin E E E E DE/best/2/exp E E E E DE/current-to-rand/1/bin E E E E DE/current-to-rand/1/exp E E E E DE/current-to-best/1/bin E E E E DE/current-to-best/1/exp E E E E DE/rand-to-best/1/bin E E E E DE/rand-to-best/1/exp E E E E Sno Variant Qm MOV Std Sr MOV Std Sr 1 DE/rand/1/bin E E E E DE/rand/1/exp E E E E DE/best/1/bin E E E E DE/best/1/exp E E E E DE/rand/2/bin E E E E DE/rand/2/exp E E E E DE/best/2/bin E E E E DE/best/2/exp E E E E DE/current-to-rand/1/bin E E E E DE/current-to-rand/1/exp E E E E DE/current-to-best/1/bin E E E E DE/current-to-best/1/exp E E E E DE/rand-to-best/1/bin E E E E DE/rand-to-best/1/exp E E E E f1 f3 f2 f4

7 Population variance based empirical analysis of the behavior 3255 Table 2. The evolution of best objective function value, its mean and standard deviation for the functions f1, f3 and f4 by the variants DE/rand/2/bin and DE/rand-to-best/1/bin DE/rand/2/bin f1 f3 f4 G ObjValue Mean Stddev G ObjValue Mean Stddev G ObjValue Mean Stddev DE/rand-to-best/1/bin f1 f3 f4 G ObjValue Mean Stddev G ObjValue Mean Stddev G ObjValue Mean Stddev The DE/rand/2/bin and DE/rand-to-best/1/bin variants have taken more number of generations to converge while solving f3, than f1 and f4. Thus the best performing variants are characterized by fast convergence to the global optimum of a given function within the maximum number of generations. The convergence behavior of worst performing variants i.e. DE/current- to-rand/ 1/exp and DE/current-to-best/1/exp, (with 0 success rates) on all functions f1 and f2 is depicted in Table 4. The results show that in all the function cases the variants have very slow convergence. This slow convergence of the worst performing variants often results in stagnation of evolution. In spite of enough diversity available in the population as can be seen in Table 4, these variants often stagnate and find it difficult to reach better solutions faster in successive generation. The evidence of these variants for not giving any successful run on any of the functions, due to stagnation, is apparent from the results. On the other hand, the variants DE/best/1/bin and DE/best/1/exp which too did not solve any of the functions displayed a different convergence behavior by virtue of their greedy mutation strategy. As shown in Table 5, for function f1, these variants converge very soon to a point in the search space with no subsequent convergence leading to premature convergence.

8 3256 S. Thangavelu et al. Table 3. The evolution of best objective function value, its mean and standard deviation for the function f2 by the variants DE/best/2/bin and DE/best/2/exp DE/best/2/bin DE/best/2/exp G ObjValue Mean Stddev G ObjValue Mean Stddev Table 4. The evolution of best objective function value, its mean and standard deviation by the variants DE/current-to-rand/1/exp and DE/current-to-best/1/exp f1 DE/current-to-rand/1/exp G ObjValue Mean Stddev G ObjValue Mean Stddev DE/current-to-best/1/exp f1 G ObjValue Mean Stddev G ObjValue Mean Stddev f2 f2 Table 6 shows the convergence of the objective function values for both sample successful and unsuccessful runs for the variants DE/rand/1/bin and DE/best/1/bin on a function each by way of an example. The results show that both DE/rand/1/bin and DE/best/1/bin suffer with premature convergence on solving the functions f1 and f4. It can be observed from the above results that the successful variants show slow/fast and steady convergence to the global optimum. The unsuccessful variants either fall in local optimum due to its fast and unsteady convergence or in

9 Population variance based empirical analysis of the behavior 3257 stagnation due to slow convergence. The partially successful variants some time show steady convergence and at other times show fast or slow convergence. Table 5. The evolution of best objective function value, its mean and standard deviation by the variants DE/best /1/bin and DE/best/1/exp DE/best/1/bin - f1 DE/best/1/exp f1 G ObjValue Mean Stddev G ObjValue Mean Stddev Table 6. The evolution of best objective function value, its mean and standard deviation during the successful and unsuccessful run of the variants DE/rand/1/bin and DE/best/1/bin DE/rand/1/bin - f2 DE/best/1/bin - f4 Successful Run UnSuccessful Run Successful Run UnSuccessful Run G ObjValue Mean Stddev ObjValue Mean Stddev ObjValue Mean Stddev ObjValue Mean Stddev The empirical evolution of population variance measured for all the 14 variants on all the four functions are presented in Tables 7, 8, 9 and 10. It is interesting to observe from the results that the variants which are reaching the global optimum have maintained slow and steady convergence of population variance. In f1, the most successful variants viz DE/rand/1/exp, DE/rand/2/exp, DE/current-to- rand/ 1/bin, DE/current-to-best/1/bin DE/rand-to-best/1/bin and DE/rand-to-best/1/exp show steady maintenance of population variance, till they reach the global optimum. This is also observed in the cases of the best performing DE variants in solving other functions, except f2. Interestingly in the case of f2, the population variance increased with generation. Even with increased variability in the population, DE/best/2/* variants could make Pc % = 100.

10 3258 S. Thangavelu et al. Tables 7. Comparison of the evolution of population variance recorded at each generation of the 100 runs, by all the variants for f1 G rand/1/bin best/1/bin rand/2/bin best/2/bin c-t-r/1/bin c-t-b/1/bin r-t-b/1/bin G rand/1/exp best/1/exp rand/2/exp best/2/exp c-t-r/1/exp c-t-b/1/exp r-t-b/1/exp Table 8. Comparison of the evolution of population variance recorded at each generation of the 100 runs, by all the variants for f2 G rand/1/bin best/1/bin rand/2/bin best/2/bin c-t-r/1/bin c-t-b/1/bin r-t-b/1/bin E E E E E E E E E E E E E E E E+19 G rand/1/exp best/1/exp rand/2/exp best/2/exp c-t-r/1/exp c-t-b/1/exp r-t-b/1/exp E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+26

11 Population variance based empirical analysis of the behavior 3259 Table 9. Comparison of the evolution of population variance recorded at each generation of the 100 runs, by all the variants for f3 G rand/1/bin best/1/bin rand/2/bin best/2/bin c-t-r/1/bin c-t-b/1/bin r-t-b/1/bin G rand/1/exp best/1/exp rand/2/exp best/2/exp c-t-r/1/exp c-t-b/1/exp r-t-b/1/exp Table 10. Comparison of the evolution of population variance recorded at each generation of the 100 runs all the variants for f4 G rand/1/bin best/1/bin rand/2/bin best/2/bin c-t-r/1/bin c-t-b/1/bin r-t-b/1/bin G rand/1/exp best/1/exp rand/2/exp best/2/exp c-t-r/1/exp c-t-b/1/exp r-t-b/1/exp For a difference, the DE/rand/1/bin and DE/rand-to-best/1/bin variants showed a sudden increase in the population variance. In the case of DE/best/1/exp, DE/current-to-rand/1/exp and DE/current-to-best/1/exp variants which experienced stagnation, the population remains diverse but with no search progress.

12 3260 S. Thangavelu et al. The variants with exponential crossover variants are more prone to such problem than binomial counterparts. By the virtue of the population spread in the non interesting region of the search space the trial vectors arising out of the exploration process are not capable of replacing the candidate in the current population leading to stagnation [18]. In case of DE/rand/1/bin, by virtue of its explorative nature, the population variance is high. On the other hand, the greedy DE/best/1/bin loses its population diversity soon. 7 Conclusion This paper is an attempt to empirically analyze the performance of classical DE variants. Fourteen different DE variants are benchmarked on 4 test functions with different features. Initially, the performance of the variants is identified by measuring MOV, Sr, Pc% and Qm. The empirical evolution of the population variance is then observed to identify the reason for such performance of the variants. The results are analyzed with the intention of identifying how the exploration and exploitation capabilities of the variants are balanced in those variants. It is found generally that the variants, failing to balance exploration and exploitation, suffer with premature convergence and stagnation problem. Suitable empirical evidence for such cases is reported. Early detection of the premature convergence and stagnation during evolution and redirecting the search in the desired vicinity of the search space to improvise the searching capability of the DE variants is a promising avenue for further research. References [1] R. Storn and K. Price Differential evolution a simple and efficient adaptive scheme for global optimization over continuous spaces, Technical, Report TR , ICSI, (1995). [2] K. Price, R.M. Storn and J.A.Lampinen, Differential Evolution : A practical Approach to Global Optimzation, Springer-Verlag, (2005). [3] J. Vesterstrom and R.Thomsen, A comparative study of differential evolution particle swarm optimization and evolutionary algorithm on numerical benchmark problems, Proceedings - IEEE Congress on Evolutionary Computation (CEC 2004), 3 (2004), [4] R. Storn and K. Price, Differential Evolution a simple and efficient heuristic strategy for global optimization and continuous spaces, Journal of Global Optimization, 11 (1997),

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15 Population variance based empirical analysis of the behavior 3263 [24] G. Jeyakumar and C. Shunmuga Velayutham, A Comparative Study on Theoretical and Empirical Evolution of the Population Variance of the Differential Evolution Variants, Lecture Notes in Computer Science (LNCS-6457), Springer-Verlag Berlin Heidelberg, (2010), Received: December 15, 2015; Published: April 20, 2015