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Andrew James Kelley

Publications and source records attributed to Andrew James Kelley.

7 recordsLinked to original sources
arXiv2026

All Constant Mutation Rates for the $(1+1)$ Evolutionary Algorithm

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.

cs.NE↗
arXiv2026

All Mutation Rates $c/n$ for the $(1+1)$ Evolutionary Algorithm

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.

cs.NE↗
arXiv2024

The Runtime of Random Local Search on the Generalized Needle Problem

In their recent work, C. Doerr and Krejca (Transactions on Evolutionary Computation, 2023) proved upper bounds on the expected runtime of the randomized local search heuristic on generalized Needle functions. Based on these upper bounds, they deduce in a not fully rigorous manner a drastic influence of the needle radius $k$ on the runtime. In this short article, we add the missing lower bound necessary to determine the influence of parameter $k$ on the runtime. To this aim, we derive an exact description of the expected runtime, which also significantly improves the upper bound given by C. Doerr and Krejca. We also describe asymptotic estimates of the expected runtime.

cs.NE↗
arXiv2023

Fourier Analysis Meets Runtime Analysis: Precise Runtimes on Plateaus

We propose a new method based on discrete Fourier analysis to analyze the time evolutionary algorithms spend on plateaus. This immediately gives a concise proof of the classic estimate of the expected runtime of the $(1+1)$ evolutionary algorithm on the Needle problem due to Garnier, Kallel, and Schoenauer (1999). We also use this method to analyze the runtime of the $(1+1)$ evolutionary algorithm on a new benchmark consisting of $n/\ell$ plateaus of effective size $2^\ell-1$ which have to be optimized sequentially in a LeadingOnes fashion. Using our new method, we determine the precise expected runtime both for static and fitness-dependent mutation rates. We also determine the asymptotically optimal static and fitness-dependent mutation rates. For $\ell = o(n)$, the optimal static mutation rate is approximately $1.59/n$. The optimal fitness dependent mutation rate, when the first $k$ fitness-relevant bits have been found, is asymptotically $1/(k+1)$. These results, so far only proven for the single-instance problem LeadingOnes, thus hold for a much broader class of problems. We expect similar extensions to be true for other important results on LeadingOnes. We are also optimistic that our Fourier analysis approach can be applied to other plateau problems as well.

cs.NE↗
arXiv2019

Maximal subgroup growth of a few polycyclic groups

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_k = \langle x_1, x_2, ..., x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i < j \rangle$. So $G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z} \rtimes ... \rtimes \mathbb{Z})$. Then for all $k \geq 2$, we calculate $m_n(G_k)$, the number of maximal subgroups of $G_k$ of index $n$, exactly. Also, for infinitely many groups $H_k$ of the form $\mathbb{Z}^2 \rtimes G_2$, we calculate $m_n(H_k)$ exactly.

math.GR↗
arXiv2018

Maximal subgroup growth of some metabelian groups

Let $m_n(G)$ denote the number of maximal subgroups of $G$ of index $n$. An upper bound is given for the degree of maximal subgroup growth of all polycyclic metabelian groups $G$ (i.e., for $\limsup \frac{\log m_n(G)}{\log n}$, the degree of polynomial growth of $m_n(G)$). A condition is given for when this upper bound is attained. For $G = \mathbb{Z}^k \rtimes \mathbb{Z}$, where $A \in GL(k,\mathbb{Z})$, it is shown that $m_n(G)$ grows like a polynomial of degree equal to the number of blocks in the rational canonical form of $A$. The leading term of this polynomial is the number of distinct roots (in $\mathbb{C}$) of the characteristic polynomial of the smallest block.

math.GR↗