Probabilistic Analysis of the (1+1)-Evolutionary Algorithm

Abstract

We give a detailed analysis of the optimization time of the [Formula: see text]-Evolutionary Algorithm under two simple fitness functions (OneMax and LeadingOnes). The problem has been approached in the evolutionary algorithm literature in various ways and with different degrees of rigor. Our asymptotic approximations for the mean and the variance represent the strongest of their kind. The approach we develop is based on an asymptotic resolution of the underlying recurrences and can also be extended to characterize the corresponding limiting distributions. While most of our approximations can be derived by simple heuristic calculations based on the idea of matched asymptotics, the rigorous justifications are challenging and require a delicate error analysis.

Sharp bounds on the runtime of the (1+1) EA via drift analysis and analytic combinatorial tools

“…for some sufficiently large constant c > 0. We shall define, as in [15], a kind of normalized drift that is easier to handle. Here it becomes relevant to manipulate the number n, so that we write more formally…”

“…The purpose of this section is to analyze more precisely how far the sum of inverse drifts n/2 k=1 1/∆(k) differs from the expected optimization time E (T | X 0 = n/2 ) = en log n − C 1 n + (e/2) log n + O(1) derived in [15]. We know from the preceding analysis that the sum of inverse drifts overestimates E (T | X 0 = n/2 ) by a Θ(log n)-term.…”

“…The main tool to derive these results relies on increasingly refined drift theorems, most notably on variable drift analysis. At roughly the same time, Hwang et al [14] presented a drastically refined analysis, which determines the expected runtime of the (1+1) EA on OneMax up to terms of order O((log n)/n): the exact expression given is en log n − C 1 n + (e/2) log n + C 2 + O((log n)/n),…”

“…. are explicitly computable constants; see also [15] for the journal version and http://140.109.74.92/hk/?p=840 for the web version with a full asymptotic expansion. To obtain these precise re-sults, techniques fundamentally different from drift analysis and other established methods for the runtime analysis were used, namely matched asymptotics with rigorous error analysis.…”

“…where K is chosen large enough depending on the required tolerance error. In particular, by refining the analysis in [15], one has d 0 = 1 2 , d 1 = 9 8 and d 2 = 31…”