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

TL;DR

This work extends Witt's tight runtime bound for the (1+1)-EA with standard bit mutation on linear functions to arbitrary static unary unbiased mutation operators described by a flip-distribution $\mathcal{D}= (p_0,\dots,p_n)$ with mean $\chi$. It proves an upper bound of $(1\pm o(1))\frac{n}{p_1}\ln n$ under $p_1=\Theta(1)$ and $\chi=O(1)$ for any linear function, and a matching lower bound under $p_1+p_{n-1}=n^{-o(1)}$ for functions with a unique global optimum; it also shows that stochastic domination fails for these operators and that OneMax is not always the easiest case. The results illuminate when heavy-tailed and other non-standard mutations preserve or alter the classic $\Theta(n\ln n)$ runtime, with implications for mutation-operator design and black-box complexity analyses.

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 $Θ(1)$. 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~$χ$. 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 $χ$ have qualitatively different trajectories close to the optimum.

Theorems & Definitions (44)

• Theorem 1
• Lemma 2
• proof
• Theorem 3: Multiplicative Drift Theorem doerr2012multiplicative
• Theorem 4: Variable Drift, lower bound
• Theorem 5: Additive Chernoff Bound
• Theorem 6: Multiplicative Chernoff Bound
• Theorem 7
• proof
• Theorem 8