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Sharp Bounds on the Runtime of the (1+1) EA via Drift Analysis and Analytic Combinatorial Tools

by   Hsien-Kuei Hwang, et al.

The expected running time of the classical (1+1) EA on the OneMax benchmark function has recently been determined by Hwang et al. (2018) up to additive errors of O(( n)/n). The same approach proposed there also leads to a full asymptotic expansion with errors of the form O(n^-K n) for any K>0. This precise result is obtained by matched asymptotics with rigorous error analysis (or by solving asymptotically the underlying recurrences via inductive approximation arguments), ideas radically different from well-established techniques for the running time analysis of evolutionary computation such as drift analysis. This paper revisits drift analysis for the (1+1) EA on OneMax and obtains that the expected running time E(T), starting from n/2 one-bits, is determined by the sum of inverse drifts up to logarithmic error terms, more precisely ∑_k=1^ n/21/Δ(k) - c_1 n < E(T) <∑_k=1^ n/21/Δ(k) - c_2 n, where Δ(k) is the drift (expected increase of the number of one-bits from the state of n-k ones) and c_1,c_2 >0 are explicitly computed constants. This improves the previous asymptotic error known for the sum of inverse drifts from Õ(n^2/3) to a logarithmic error and gives for the first time a non-asymptotic error bound. Using standard asymptotic techniques, the difference between E(T) and the sum of inverse drifts is found to be (e/2) n+O(1).


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