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 ar X iv :1 80 2. 03 20 9v 1 [ cs .N E ] 9 F eb 2 01 8