Renewal Theory Model of First Hitting Time Analysis for Continuous Evolutionary Algorithms
ZHOU Zhensheng1,2, WANG Lin2, FENG Fujian1,2, TAN Mian1,2, HE Xing2, ZHANG Zaijun3
1. School of Data Sciences and Information Engineering, Guizhou Minzu University, Guiyang 550025; 2. Key Laboratory of Pattern Recognition and Intelligent Systems of Guizhou Province, Guizhou Minzu University, Guiyang 550025; 3. School of Mathematics and Statistics, Qiannan Normal University for Nationalities, Duyun 558000
Abstract In the research on upper bound of the first hitting time for continuous evolutionary algorithms,strong assumptions are required and less attention is given to its lower bound . In this paper, martingale theory and renewal process are introduced and combined with Wald's inequality and renewal theorem. A renewal theory model based on progress rate is proposed to estimate the upper and lower bounds of the expected first hitting time of evolution strategies. The renewal theory model relies on the initial population and the probability density function of the progress rate, providing an estimation advantage for the analysis of the first hitting time of evolutionary strategies. To verify the validity of the proposed renewal theory model, experiments are conducted to estimate the expected first hitting time. Firstly, the expected first hitting time of (1,λ)evolution strategies with uniform mutation on a two-dimensional inclined plane problem is calculated. The closed-form expression for the relationship between(1,λ)evolution strategies population size and the time upper and lower bounds is obtained. It is proved that the expected first hitting time is not negatively correlated with the population size. Next, the expected first hitting time of evolution strategies with uniform mutation on a five-dimensional hyperplane problem is calculated, and the closed-form expression for theoretical upper and lower bounds is derived. Numerical experiments show that the theoretically calculated upper and lower bounds are consistent with the actual expected first hitting time, which provides a theoretical tool for analyzing the first hitting time of evolution strategies.
Fund:National Natural Science Foundation of China(No.62241206), Science and Technology Program of Guizhou Province(No.QKH-Basic-ZK[2022]195,QKH-Basic-ZK[2023]143,QKH-Basic-ZK[2022]550), Natural Science Research Project of Department of Education of Guizhou Province(No.QJJ[2023]061,QJJ[2023]012,QJJ[2022]015), Open Project of Guizhou Key Laboratory of Pattern Recognition and Intelligent System(No.GZMUKL[2022]KF01,GZMUKL[2022]KF05)
Corresponding Authors:
WANG Lin, Ph.D., professor. His research interests include image processing, pattern recognition and intelligent control.
About author:: ZHOU Zhensheng, master student. His research interests include theoretical analysis of evolutionary computation. FENG Fujian, Ph.D., professor. His research interests include intelligent computing and microcomputation.TAN Mian, master, associate professor. Her research interests include theoretical ana-lysis of evolutionary computation and intelligent computation.HE Xing, Ph.D. candidate. His research interests include intelligent computing and ma-chine learning.ZHANG Zaijun, Ph.D., associate profe-ssor. His research interests include algorithm design and analysis and deep learning.
ZHOU Zhensheng,WANG Lin,FENG Fujian等. Renewal Theory Model of First Hitting Time Analysis for Continuous Evolutionary Algorithms[J]. Pattern Recognition and Artificial Intelligence, 2023, 36(10): 918-930.
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2023, Vol. 36 
