Tight Runtime Bounds for Static Unary Unbiased Evolutionary Algorithms on Linear Functions

Abstract

In a seminal paper in 2013, Witt showed that the (1+1) Evolutionary Algorithm with standard bit mutation needs time \((1+o(1))n \ln n/p_1\) to find the optimum of any linear function, as long as the probability \(p_1\) to flip exactly one bit is \(\Theta (1)\). In this paper we investigate how this result generalizes if standard bit mutation is replaced by an arbitrary unbiased mutation operator. This situation is notably different, since the stochastic domination argument used for the lower bound by Witt no longer holds. In particular, starting closer to the optimum is not necessarily an advantage, and OneMax is no longer the easiest function for arbitrary starting positions. Nevertheless, we show that Witt’s result carries over if \(p_1\) is not too small, with different constraints for upper and lower bounds, and if the number of flipped bits has bounded expectation \(\chi \). Notably, this includes some of the heavy-tail mutation operators used in fast genetic algorithms, but not all of them. We also give examples showing that algorithms with unbounded \(\chi \) have qualitatively different trajectories close to the optimum.