Do additional target points speed up evolutionary algorithms?

doi:10.1016/j.tcs.2023.113757

Theoretical Computer Science

Volume 950, 16 March 2023, 113757

https://doi.org/10.1016/j.tcs.2023.113757 Get rights and content

Highlights

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We study the placement and size of additional target points on the expected hitting time of evolutionary algorithms.
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We show that adding exponentially many targets uniformly at random has a negligible effect on the expected optimisation.
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Considering all points within a Hamming-ball of size r around all optima, improvements depend on the considered function.
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On functions where search typically traverses through a single search point, adding one target can yield drastic speedups.

Abstract

Most runtime analyses of randomised search heuristics focus on the expected number of function evaluations to find a unique global optimum. We ask a fundamental question: if additional search points are declared optimal, or declared as desirable target points, do these additional optima speed up evolutionary algorithms? More formally, we analyse the expected hitting time of a target set $OPT \cup S$ where S is a set of non-optimal search points and OPT is the set of optima and compare it to the expected hitting time of OPT.

We show that the answer to our question depends on the number and placement of search points in S. For all black-box algorithms and all fitness functions with polynomial expected optimisation times we show that, if additional optima are placed randomly, even an exponential number of optima has a negligible effect on the expected optimisation time. Considering Hamming balls around all global optima gives an easier target for some algorithms and functions and can shift the phase transition with respect to offspring population sizes in the (1,λ) EA on OneMax. However, for the one-dimensional Ising model the time to reach Hamming balls of radius $(1 / 2 - ε) n$ around optima does not reduce the asymptotic expected optimisation time in the worst case. Finally, on functions where search trajectories typically join in a single search point, turning one search point into an optimum drastically reduces the expected optimisation time.

Section snippets

Motivation

Runtime analysis has emerged as a fruitful research area that is helping to develop and consolidate our understanding of the performance of evolutionary algorithms and many other randomised search heuristics [1], [2], [3], [4].

Many results have been obtained for problems from combinatorial optimisation [1] and for pseudo-Boolean functions $f : {0, 1}^{n} \to R$ . The latter includes frequently used benchmark functions like OneMax, linear functions, LeadingOnes, Ridge, Jump, Cliff, Plateau (see Section 2 for

Preliminaries

We consider the very general class of all black-box algorithms shown in Algorithm 1 that comprises meta-heuristics such as evolutionary algorithms, ant colony optimisation, particle swarm optimisation, estimation-of-distribution algorithms, artificial immune systems, tabu search, simulated annealing, and many more. Here ${OPT}_{f}$ denotes the set of optimal solutions for the function f and S is a set of additional target points. Note that in the classical setting $S = \emptyset$ .

We also show more specific

Random and worst-case target points

We first consider the simple scenario where additional target points are added uniformly at random as an average case scenario for the placement of additional optima. It is not surprising that these additional targets do not help much if they only make up a tiny fraction of the whole search space.

Adding targets around optima

We now consider scenarios where additional optima or target points are placed “close” to global optima. There are several possible notions of “close”. We could aim to reach a solution of a specified minimum fitness. This scenario is highly relevant for practice and has been investigated implicitly in several works (e.g. [43], [44]). Recently, it was studied explicitly under the term fixed target runtime analysis [18].

We cite a fixed-target result for LeadingOnes for illustration. It is notable

Best-case placement of additional optima

Finally, we consider a best possible placement of additional target points, and how much the expected running time can be decreased by carefully choosing additional targets.

A rather obvious example of a huge benefit through added optima is the function Trap, a deceptive function on which the (1+1) EA requires time $Θ (n^{n})$ [11]. Turning the trap into a target point turns the function into OneMax with an additional optimum at $0^{n}$ .

Theorem 5.1

The expected time for the (1+1) EA operating on Trap to find ${0^{n}, 1^{n}}$

Conclusions

Runtime analysis of randomised search heuristics concentrates on the expected number of fitness function evaluations until a certain target point (often a unique global optimum) is hit for the first time. We studied how the expected optimisation time changes if, in addition to global optima, additional target points are considered.

Our results points out that it depends on the size and placement of the target points as well as characteristics of the fitness function. We considered a worst-case

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Cited by (1)

Self-adjusting offspring population sizes outperform fixed parameters on the cliff function
2024, Artificial Intelligence
In the discrete domain, self-adjusting parameters of evolutionary algorithms (EAs) have emerged as a fruitful research area with many runtime analyses showing that self-adjusting parameters can outperform the best fixed parameters. Most existing runtime analyses focus on elitist EAs on simple problems, for which moderate performance gains were shown. Here we consider a much more challenging scenario: the multimodal function Cliff, defined as an example where a $(1, λ)$ EA is effective, and for which the best known upper runtime bound for standard EAs is $O (n^{25})$ .
We prove that a $(1, λ)$ EA self-adjusting the offspring population size λ using success-based rules optimises Cliff in $O (n)$ expected generations and $O (n \log n)$ expected evaluations. Along the way, we prove tight upper and lower bounds on the runtime for fixed λ (up to a logarithmic factor) and identify the runtime for the best fixed λ as $n^{η}$ for $η \approx 3.97677$ (up to sub-polynomial factors). Hence, the self-adjusting $(1, λ)$ EA outperforms the best fixed parameter by a factor of at least $n^{2.9767}$ .

^☆: This article belongs to Section C: Theory of natural computing, Edited by Lila Kari.

View full text

Do additional target points speed up evolutionary algorithms?☆

Highlights

Abstract

Section snippets

Motivation

Preliminaries

Random and worst-case target points

Adding targets around optima

Best-case placement of additional optima

Conclusions

Declaration of Competing Interest

Discrete Appl. Math.

Theor. Comput. Sci.

Theor. Comput. Sci.

Theor. Comput. Sci.

Theor. Comput. Sci.

Theor. Comput. Sci.

Theor. Comput. Sci.

Theor. Comput. Sci.

Theor. Comput. Sci.

Bioinspired Computation in Combinatorial Optimization – Algorithms and Their Computational Complexity

Analyzing Evolutionary Algorithms – The Computer Science Perspective

On the choice of the offspring population size in evolutionary algorithms

Evol. Comput.

Drift analysis and evolutionary algorithms revisited

Comb. Probab. Comput.

Mutation rate matters even when optimizing monotonic functions

Evol. Comput.

A general dichotomy of evolutionary algorithms on monotone functions

IEEE Trans. Evol. Comput.

Exponential slowdown for larger populations: the (μ+1)-EA on monotone functions

A new method for lower bounds on the running time of evolutionary algorithms

IEEE Trans. Evol. Comput.

Lower bounds from fitness levels made easy

Upper and lower bounds for randomized search heuristics in black-box optimization

Theory Comput. Syst.

Black-box search by unbiased variation

Algorithmica

Complexity theory for discrete black-box optimization heuristics

Fixed-target runtime analysis

Fixed-target runtime analysis

Algorithmica

Runtime analysis for maximizing population diversity in single-objective optimization

Exponential slowdown for larger populations: the ( $μ + 1$ )-EA on monotone functions