Drift Theory in Continuous Search Spaces: Expected Hitting Time of the (1+1)-ES with 1/5 Success Rule
This provides the first non-asymptotic analysis for this algorithm and applies drift analysis to continuous domains, which is foundational for theoretical evolutionary computation research.
The paper tackled the problem of analyzing runtime bounds for continuous optimization algorithms by applying drift analysis to the (1+1) Evolution Strategy with the one-fifth success rule on the sphere function, proving bounds akin to linear convergence with a convergence rate dependency of Θ(1/d).
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.