Drift theory in continuous search spaces: expected hitting time of the (1 + 1)-ES with 1/5 success rule

This paper explores the use of the standard approach for proving runtime bounds in discrete domains---often referred to as drift analysis---in the context of optimization on a continuous domain. Using this framework we analyze the (1+1) Evolution Strategy with one-fifth success rule on the sphere function. To deal with potential functions that are not lower-bounded, we formulate novel drift theorems. We then use the theorems to prove bounds on the expected hitting time to reach a certain target fitness in finite dimension d. The bounds are akin to linear convergence. We then study the dependency of the different terms on d proving a convergence rate dependency of Θ(1/d). Our results constitute the first non-asymptotic analysis for the algorithm considered as well as the first explicit application of drift analysis to a randomized search heuristic with continuous domain.

[1]  Jens Jägersküpper,et al.  Algorithmic analysis of a basic evolutionary algorithm for continuous optimization , 2007, Theor. Comput. Sci..

[2]  Angelika Steger,et al.  Drift Analysis and Evolutionary Algorithms Revisited , 2016, Combinatorics, Probability and Computing.

[3]  Jens Jägersküpper,et al.  Analysis of a Simple Evolutionary Algorithm for Minimization in Euclidean Spaces , 2003, ICALP.

[4]  Anne Auger,et al.  Analyzing the impact of mirrored sampling and sequential selection in elitist evolution strategies , 2011, FOGA '11.

[5]  J. Snyder.  Coupling , 1998, Critical Inquiry.

[6]  Anne Auger,et al.  Linear Convergence on Positively Homogeneous Functions of a Comparison Based Step-Size Adaptive Randomized Search: the (1+1) ES with Generalized One-fifth Success Rule , 2013, ArXiv.

[7]  Per Kristian Lehre,et al.  General Drift Analysis with Tail Bounds , 2013, ArXiv.

[8]  Jens Jagersk.  Analysis of a Simple Evolutionary Algorithm for Minimization in Euclidean Spaces , 2003 .

[9]  Benjamin Doerr,et al.  Multiplicative Drift Analysis , 2010, GECCO '10.

[10]  Ingo Rechenberg,et al.  Evolutionsstrategie : Optimierung technischer Systeme nach Prinzipien der biologischen Evolution , 1973 .

[11]  Petros Koumoutsakos,et al.  Learning probability distributions in continuous evolutionary algorithms – a comparative review , 2004, Natural Computing.

[12]  Elizabeth F. Wanner,et al.  Lyapunov Design of a Simple Step-Size Adaptation Strategy Based on Success , 2016, PPSN.

[13]  Jens Jägersküpper,et al.  How the (1+1) ES using isotropic mutations minimizes positive definite quadratic forms , 2006, Theor. Comput. Sci..

[14]  Per Kristian Lehre,et al.  Drift analysis , 2012, GECCO '12.

[15]  W. Vent,et al.  Rechenberg, Ingo, Evolutionsstrategie — Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. 170 S. mit 36 Abb. Frommann‐Holzboog‐Verlag. Stuttgart 1973. Broschiert , 1975 .

[16]  H. Thorisson.  Coupling, stationarity, and regeneration , 2000 .

[17]  Nikolaus Hansen,et al.  Completely Derandomized Self-Adaptation in Evolution Strategies , 2001, Evolutionary Computation.

[18]  Thomas Jansen,et al.  On the analysis of the (1+1) evolutionary algorithm , 2002, Theor. Comput. Sci..