Adaptive Genetic Algorithm Using Harmony Search Farhad Nadi, Ahamad Tajudin Khader, and Mohammed Azmi Al-Betar School of Computer Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia

[email protected], [email protected], [email protected]

ABSTRACT

2. METHODOLOGY

Evolutionary algorithm is one of the major classes of stochastic search methods. This algorithm searches the problem space by exploring and exploiting the search space. The balance between exploration and exploitation will change throughout the search process. Maintaining the right balance between the exploration and exploitation in the search process is crucial for the success of the search process. The parameter values of the algorithm play a crucial role in determining the nature of the search, whether explorative or exploitative. In this paper, we propose an adaptive parameter controlling approach using harmony search. During the search process, harmony search directs the search from the current state to a desired state by determining suitable parameter values such that the balance between exploration and exploitation is suitable for that state transition. The preliminary results of the proposed method is comparable with those from the literature.

An adaptive parameter control method is proposed. Current State (CS) of the algorithm in terms of convergence behavior will be expressed in the interval [0, 1] from the fully diverse to fully converged respectively. Exploration and exploitation are used to control the search behaviour towards a desired state. The parameter values could change the search towards exploration or exploitation, thus, affecting the diversity of the search space. A strategy, expressed as Desired State (DS), will be used to determine the future state of the search. Each parameter is given a weight that reflects the suitability of the parameter for achieving the DS. This measure expresses the current state of the algorithm during the run. DS value is in the interval of [0, 1] which reflect states from fully exploration to fully exploitation respectively. The value of CS would affect DS. A total weight of each parameter will result in a value called Proposed Set (P S). The value of P S shows how severe that parameter values could explore or exploit the search space. A P S value near to 0 means the parameter values are more suitable for exploration while values near to 1 are more suitable for exploitation of the search space. The parameters have been given a weight based on their orders. The weighting will be based on the effectiveness of the parameters on either exploration or exploitation. The less will be the given weight to the parameter values with more suitability for exploration. In turn, The weight will be more for the parameters that are more suitable for exploitation. A simple geometric series is used to model the weighting scheme of the parameters. The proposed method utilized Harmony Search Algorithm (HSA) for selecting best parameters’ values for a set of parameters. The fittness of HSA is defined as |P S −DS|, which is to be minimized. The more the difference of PS and DS are near to zero the better is the quality of the PS.

Categories and Subject Descriptors G.2.1 [DISCRETE MATHEMATICS]: Combinatorics—Combinatorial algorithms

General Terms Algorithms, Design, Experimentation, Performance, Theory

Keywords Parameter control, Exploration, Exploitation, Harmony Search Algorithm

1.

INTRODUCTION

Traditionally, parameter values will get tuned before the actual run and their values remain static during the run. In literature this is referred to as parameter tuning. In turn, the term parameter control covers all the methods where the values of the parameters will be adjusted during the run [5]. Parameter control methods mostly have been categorized into three different subcategories namely deterministic, adaptive, and self-adaptive methods[3, 5]. Normally adaptive methods will adjust the parameters’ values regarding to a feedback from the algorithm in run[5]. Feedbacks are mostly based on the quality of the solutions or speed of the algorithm [7].

3. EXPERIMENTAL SETUP The experiments are conducted based on the benchmark used in [2]. For the course of experiments we have used multi-modal problem generator (MPG) of Spears [6, 8]. The heights are linearly distributed for all of the peaks, and lowest height is 0.5. The number of peaks in the experiments is chosen as in [2]. Five parameters have been considered for parameter control method including crossover rate, mutation rate, selection pressure, survivor selection, and population size. A simple steady state GA is used for the course of the experiments. Uniform crossover and random resetting mutation are used for recombination operators. Tournament selection used for parent selection and delete oldest strategy is used for survival selection. The parameters of the HSA are tuned as follows, HM CR =

Copyright is held by the author/owner(s). GECCO’10, July 7–11, 2010, Portland, Oregon, USA. ACM 978-1-4503-0072-8/10/07.

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0.9, P AR = 0.1, HM S = 1, and N I = 1000. The parameter control algorithm (HSA) is called after every 5 generations. Following the steps in [1], we have run the algorithm on each benchmark 100 independent runs. For each of the experiments, the algorithm is stopped either after 10, 000 generations or if it reaches the optimum that is, 1. Table 1 shows Mean of Best Fitness (M BF ) of the taken results compared with [2] and [1]. The results of the very first 25 runs have been used to calculate the related measures. The proposed method here is compared with two different meta-algorithms and hand-tuned algorithm.

The HSA has shown a good performance in finding the optimized parameter sets. As the optimization process here is not a very complex problem, it could be replaced with other algorithms although it would not have effect on the performance.

4. CONCLUSIONS A new methodology is proposed that utilize harmony search for adaptive parameter controlling in GA. The proposed method directs the search based on a predefined strategy referred to as desired state. The algorithm behavior could change from explorative to exploitative and vice versa. The desired state of the algorithm and also the current state will determine the future state of the algorithm. Applying the proposed parameter values to the algorithm would direct the search towards the desired state. Parameters are given a weight based on how effective they are in changing the search behavior. A HSA is utilized to find suitable set of parameter values which could direct the search towards the desired state. A set of parameter values construct a candidate harmony. Fitness function of HSA, evaluate the candidate harmonies based on the desired state. The comparison of the preliminary results with some of the recent works in literature has shown reasonable efficiency of the proposed method.

Table 1: Comparing the proposed method with metaalgorithms methods [1] Peaks 1 2 5 10 25 50 100 250 500 1000

hand-tuned 1.0 1.0 1.0 0.9961 0.9885 0.9876 0.9853 0.9847 0.9865 0.9891

meta-GA 1.0 1.0 0.988 0.993 0.994 0.994 0.983 0.992 0.989 0.987

REVAC 1.0 1.0 1.0 0.996 0.991 0.995 0.989 0.966 0.970 0.985

Proposed Method 1.0 1.0 1.0 1.0 0.9958 0.9980 0.9933 0.9912 0.9888 0.9861

5. ACKNOWLEDGMENTS The first and third authors would like to thank the Institute of Postgraduate Studies Universiti Sains Malaysia for supporting this research under fellowship scheme.

The results of 100 runs have been used for comparing the proposed method with two parameter control methods as shown in Table 2.

6. REFERENCES [1] W. de Landgraaf, A. Eiben, and V. Nannen. Parameter Calibration using Meta-Algorithms. In IEEE Congress on Evolutionary Computation, pages 71–78. IEEE, 2007. [2] A. Eiben, M. Schut, and A. de Wilde. Is self-adaptation of selection pressure and population size possible? - a case study. In Parallel Problem Solving from Nature - PPSN IX, pages 900–909. 2006. [3] A. Eiben and J. Smith. Introduction to Evolutionary Computing. Natural Computing Series. Springer, 2 edition, 2007. [4] A. E. Eiben, E. Marchiori, and V. Valko. Evolutionary algorithms with on-the-fly population size adjustment. In Parallel Problem Solving from Nature - PPSN VIII, pages 41–50. 2004. [5] G. Eiben and M. C. Schut. New ways to calibrate evolutionary algorithms. In Advances in Metaheuristics for Hard Optimization, pages 153–177. 2008. [6] K. A. D. Jong and W. M. Spears. A formal analysis of the role of multi-point crossover in genetic algorithms. Ann. Math. Artif. Intell., 5(1):1–26, 1992. [7] S. K. Smit and A. E. Eiben. Comparing Parameter Tuning Methods for Evolutionary Algorithms. In IEEE Congress on Evolutionary Computation (CEC), pages 399–406, May 2009. [8] W. M. Spears. Evolutionary Algorithms: The Role of Mutation and Recombination (Natural Computing Series). Springer, June 2000.

Table 2: Comparing GAHSAT [5], APGA from [4], and the propose method. GAHSAT APGA Proposed Method Peaks SR(%) AES SR AES SR(%) AES 1 100 989 100 1100 100 2016 2 100 969 100 1129 100 1919 5 100 1007 100 1119 100 2024 10 89 1075 95 1104 100 1815 25 63 1134 54 1122 89 1681 50 45 1194 35 1153 66 1743 100 14 1263 22 1216 36 1171 250 12 1217 8 1040 13 2090 500 7 1541 6 1161 9 2159 1000 4 1503 1 910 7 2921

As preliminary results, the comparisons show the applicability of the proposed method. The overall M BF factor in all the cases shows a better performance comparing to other methods. According to the results shown in Table 1, it could be inferred that parameter control methods could be more effective than meta algorithms. In other words, controlling the parameters during the run will result in better outcomes. The proposed method has also shown a reasonable performance in front of (self-)adaptive methods according to comparison in Table 2. While the SR measure shows better results in the taken results, AES reveals that these results are taken in longer time comparing to other methods. This could be described by the slow rate in changing from explorative state towards exploitative state in the defined model for DS.

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[email protected], [email protected], [email protected]

ABSTRACT

2. METHODOLOGY

Evolutionary algorithm is one of the major classes of stochastic search methods. This algorithm searches the problem space by exploring and exploiting the search space. The balance between exploration and exploitation will change throughout the search process. Maintaining the right balance between the exploration and exploitation in the search process is crucial for the success of the search process. The parameter values of the algorithm play a crucial role in determining the nature of the search, whether explorative or exploitative. In this paper, we propose an adaptive parameter controlling approach using harmony search. During the search process, harmony search directs the search from the current state to a desired state by determining suitable parameter values such that the balance between exploration and exploitation is suitable for that state transition. The preliminary results of the proposed method is comparable with those from the literature.

An adaptive parameter control method is proposed. Current State (CS) of the algorithm in terms of convergence behavior will be expressed in the interval [0, 1] from the fully diverse to fully converged respectively. Exploration and exploitation are used to control the search behaviour towards a desired state. The parameter values could change the search towards exploration or exploitation, thus, affecting the diversity of the search space. A strategy, expressed as Desired State (DS), will be used to determine the future state of the search. Each parameter is given a weight that reflects the suitability of the parameter for achieving the DS. This measure expresses the current state of the algorithm during the run. DS value is in the interval of [0, 1] which reflect states from fully exploration to fully exploitation respectively. The value of CS would affect DS. A total weight of each parameter will result in a value called Proposed Set (P S). The value of P S shows how severe that parameter values could explore or exploit the search space. A P S value near to 0 means the parameter values are more suitable for exploration while values near to 1 are more suitable for exploitation of the search space. The parameters have been given a weight based on their orders. The weighting will be based on the effectiveness of the parameters on either exploration or exploitation. The less will be the given weight to the parameter values with more suitability for exploration. In turn, The weight will be more for the parameters that are more suitable for exploitation. A simple geometric series is used to model the weighting scheme of the parameters. The proposed method utilized Harmony Search Algorithm (HSA) for selecting best parameters’ values for a set of parameters. The fittness of HSA is defined as |P S −DS|, which is to be minimized. The more the difference of PS and DS are near to zero the better is the quality of the PS.

Categories and Subject Descriptors G.2.1 [DISCRETE MATHEMATICS]: Combinatorics—Combinatorial algorithms

General Terms Algorithms, Design, Experimentation, Performance, Theory

Keywords Parameter control, Exploration, Exploitation, Harmony Search Algorithm

1.

INTRODUCTION

Traditionally, parameter values will get tuned before the actual run and their values remain static during the run. In literature this is referred to as parameter tuning. In turn, the term parameter control covers all the methods where the values of the parameters will be adjusted during the run [5]. Parameter control methods mostly have been categorized into three different subcategories namely deterministic, adaptive, and self-adaptive methods[3, 5]. Normally adaptive methods will adjust the parameters’ values regarding to a feedback from the algorithm in run[5]. Feedbacks are mostly based on the quality of the solutions or speed of the algorithm [7].

3. EXPERIMENTAL SETUP The experiments are conducted based on the benchmark used in [2]. For the course of experiments we have used multi-modal problem generator (MPG) of Spears [6, 8]. The heights are linearly distributed for all of the peaks, and lowest height is 0.5. The number of peaks in the experiments is chosen as in [2]. Five parameters have been considered for parameter control method including crossover rate, mutation rate, selection pressure, survivor selection, and population size. A simple steady state GA is used for the course of the experiments. Uniform crossover and random resetting mutation are used for recombination operators. Tournament selection used for parent selection and delete oldest strategy is used for survival selection. The parameters of the HSA are tuned as follows, HM CR =

Copyright is held by the author/owner(s). GECCO’10, July 7–11, 2010, Portland, Oregon, USA. ACM 978-1-4503-0072-8/10/07.

819

0.9, P AR = 0.1, HM S = 1, and N I = 1000. The parameter control algorithm (HSA) is called after every 5 generations. Following the steps in [1], we have run the algorithm on each benchmark 100 independent runs. For each of the experiments, the algorithm is stopped either after 10, 000 generations or if it reaches the optimum that is, 1. Table 1 shows Mean of Best Fitness (M BF ) of the taken results compared with [2] and [1]. The results of the very first 25 runs have been used to calculate the related measures. The proposed method here is compared with two different meta-algorithms and hand-tuned algorithm.

The HSA has shown a good performance in finding the optimized parameter sets. As the optimization process here is not a very complex problem, it could be replaced with other algorithms although it would not have effect on the performance.

4. CONCLUSIONS A new methodology is proposed that utilize harmony search for adaptive parameter controlling in GA. The proposed method directs the search based on a predefined strategy referred to as desired state. The algorithm behavior could change from explorative to exploitative and vice versa. The desired state of the algorithm and also the current state will determine the future state of the algorithm. Applying the proposed parameter values to the algorithm would direct the search towards the desired state. Parameters are given a weight based on how effective they are in changing the search behavior. A HSA is utilized to find suitable set of parameter values which could direct the search towards the desired state. A set of parameter values construct a candidate harmony. Fitness function of HSA, evaluate the candidate harmonies based on the desired state. The comparison of the preliminary results with some of the recent works in literature has shown reasonable efficiency of the proposed method.

Table 1: Comparing the proposed method with metaalgorithms methods [1] Peaks 1 2 5 10 25 50 100 250 500 1000

hand-tuned 1.0 1.0 1.0 0.9961 0.9885 0.9876 0.9853 0.9847 0.9865 0.9891

meta-GA 1.0 1.0 0.988 0.993 0.994 0.994 0.983 0.992 0.989 0.987

REVAC 1.0 1.0 1.0 0.996 0.991 0.995 0.989 0.966 0.970 0.985

Proposed Method 1.0 1.0 1.0 1.0 0.9958 0.9980 0.9933 0.9912 0.9888 0.9861

5. ACKNOWLEDGMENTS The first and third authors would like to thank the Institute of Postgraduate Studies Universiti Sains Malaysia for supporting this research under fellowship scheme.

The results of 100 runs have been used for comparing the proposed method with two parameter control methods as shown in Table 2.

6. REFERENCES [1] W. de Landgraaf, A. Eiben, and V. Nannen. Parameter Calibration using Meta-Algorithms. In IEEE Congress on Evolutionary Computation, pages 71–78. IEEE, 2007. [2] A. Eiben, M. Schut, and A. de Wilde. Is self-adaptation of selection pressure and population size possible? - a case study. In Parallel Problem Solving from Nature - PPSN IX, pages 900–909. 2006. [3] A. Eiben and J. Smith. Introduction to Evolutionary Computing. Natural Computing Series. Springer, 2 edition, 2007. [4] A. E. Eiben, E. Marchiori, and V. Valko. Evolutionary algorithms with on-the-fly population size adjustment. In Parallel Problem Solving from Nature - PPSN VIII, pages 41–50. 2004. [5] G. Eiben and M. C. Schut. New ways to calibrate evolutionary algorithms. In Advances in Metaheuristics for Hard Optimization, pages 153–177. 2008. [6] K. A. D. Jong and W. M. Spears. A formal analysis of the role of multi-point crossover in genetic algorithms. Ann. Math. Artif. Intell., 5(1):1–26, 1992. [7] S. K. Smit and A. E. Eiben. Comparing Parameter Tuning Methods for Evolutionary Algorithms. In IEEE Congress on Evolutionary Computation (CEC), pages 399–406, May 2009. [8] W. M. Spears. Evolutionary Algorithms: The Role of Mutation and Recombination (Natural Computing Series). Springer, June 2000.

Table 2: Comparing GAHSAT [5], APGA from [4], and the propose method. GAHSAT APGA Proposed Method Peaks SR(%) AES SR AES SR(%) AES 1 100 989 100 1100 100 2016 2 100 969 100 1129 100 1919 5 100 1007 100 1119 100 2024 10 89 1075 95 1104 100 1815 25 63 1134 54 1122 89 1681 50 45 1194 35 1153 66 1743 100 14 1263 22 1216 36 1171 250 12 1217 8 1040 13 2090 500 7 1541 6 1161 9 2159 1000 4 1503 1 910 7 2921

As preliminary results, the comparisons show the applicability of the proposed method. The overall M BF factor in all the cases shows a better performance comparing to other methods. According to the results shown in Table 1, it could be inferred that parameter control methods could be more effective than meta algorithms. In other words, controlling the parameters during the run will result in better outcomes. The proposed method has also shown a reasonable performance in front of (self-)adaptive methods according to comparison in Table 2. While the SR measure shows better results in the taken results, AES reveals that these results are taken in longer time comparing to other methods. This could be described by the slow rate in changing from explorative state towards exploitative state in the defined model for DS.

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