Generalized Heavy-tailed Mutation for Evolutionary Algorithms

For every mutation rate $p \in (0, 1)$, and for all $\varepsilon > 0$, there is a fitness function $f : \{0,1\}^n \to \mathbb{R}$ with a unique maximum for which the optimal mutation rate for the $(1+1)$ evolutionary algorithm on $f$ is in $(p-\varepsilon, p+\varepsilon)$. In other words, the set of optimal mutation rates for the $(1+1)$ EA is dense in the interval $[0, 1]$. To show that, this paper introduces DistantSteppingStones, a fitness function which consists of large plateaus separated by large fitness valleys.

For every real number $c \geq 1$ and for all $\varepsilon > 0$, there is a fitness function $f : \{0,1\}^n \to \mathbb{R}$ for which the optimal mutation rate for the $(1+1)$ evolutionary algorithm on $f$, denoted $p_n$, satisfies $p_n \approx c/n$ in that $|np_n - c| < \varepsilon$. In other words, the set of all $c \geq 1$ for which the mutation rate $c/n$ is optimal for the $(1+1)$ EA is dense in the interval $[1, \infty)$. To show this, a fitness function is introduced which is called HillPathJump.

For the (1+1) evolutionary algorithm with standard bit mutation, we study the sensitivity of the expected hitting time $H_p=\mathbb{E}_x T$ to the mutation rate. We first point out an easily overlooked formalization pitfall: the improvement event is not monotone in the mutation mask, so the unsigned (total-influence) form of the Margulis-Russo formula does not apply; the correct object is the signed endpoint difference. Second, we give an exact three-dimensional separation: two fitness functions share the entire one-step success-rate curve, yet their expected hitting times are two different exact rational numbers; hence one-step success-rate quantities do not determine the expected hitting time. Building on the runtime derivative $H'_p=(I-Q_p)^{-1}Q'_p H_p$, we construct computable double-residual sign certificates, prove a per-initial-state sign theorem on OneMax (for every non-optimal initial state, $\partial_c H<0$ on $0<c<1$, where $p=c/n$; at $c=1$ only the distance-one state is stationary), and extend the framework to non-lumpable positive linear families: an explicit non-lumpability witness, a block-interval double-residual certificate that covers all states without enumerating them, a uniform sign bound $\partial_c\mathbb{E}T\le -9n/16$ over the whole interval $c\in[1/4,1/2]$ for an explicit family at all even scales $n\ge 8$, and a heterogeneous instance certificate $H'_x\le -1/6$ on 57 of 63 states across $c=1$. All finite verifications use exact rational arithmetic. A bounded systematic literature search did not uncover this exact combination, although the underlying tools are well established; we therefore make no novelty claim beyond the stated combination.