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

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((log n)/n). The same approach proposed there also leads to a full asymptotic expansion with errors of the form O(n-Klog 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 lceil n/2rceil one-bits, is determined by the sum of inverse drifts up to logarithmic error terms, more precisely sumk=1lfloor n/2rfloorfrac1Delta(k) - c1log n ≤ E(T) ≤ sumk=1lfloor n/2rfloorfrac1Delta(k) - c2log n, where Delta(k) is the drift (expected increase of the number of one-bits from the state of n-k ones) and c1,c2 >0 are explicitly computed constants. This improves the previous asymptotic error known for the sum of inverse drifts from tildeO(n2/3) 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)log n+O(1).