On Non-Elitist Evolutionary Algorithms Optimizing Fitness Functions with a Plateau
Anton V. Eremeev
备注：
14 pages, accepted for proceedings of Mathematical Optimization Theory and Operations Research (MOTOR 2020). arXiv admin note: text overlap with arXiv:1908.08686
发表信息：
Proc. of MOTOR 2020, LNCS, vol. 12095, Springer, Cham. 2020. pp.329-342
We consider the expected runtime of non-elitist evolutionary algorithms (EAs), when they are applied to a family of fitness functions with a plateau of second-best fitness in a Hamming ball of radius r around a unique global optimum. On one hand, using the level-based theorems, we obtain polynomial upper bounds on the expected runtime for some modes of non-elitist EA based on unbiased mutation and the bitwise mutation in particular. On the other hand, we show that the EA with fitness proportionate selection is inefficient if the bitwise mutation is used with the standard settings of mutation probability.
This paper aims to study how the population size affects the computation time of evolutionary algorithms in a rigorous way. The computation time of an evolutionary algorithm can be measured by either the expected number of generations (hitting time) or the expected number of fitness evaluations (running time) to find an optimal solution. Population scalability is the ratio of the expected hitting time between a benchmark algorithm and an algorithm using a larger population size. Average drift analysis is presented for comparing the expected hitting time of two algorithms and estimating lower a…
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The hardness of fitness functions is an important research topic in the field of evolutionary computation. In theory, the study can help understanding the ability of evolutionary algorithms. In practice, the study may provide a guideline to the design of benchmarks. The aim of this paper is to answer the following research questions: Given a fitness function class, which functions are the easiest with respect to an evolutionary algorithm? Which are the hardest? How are these functions constructed? The paper provides theoretical answers to these questions. The easiest and hardest fitness functi…
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This paper extends the runtime analysis of non-elitist evolutionary algorithms (EAs) with fitness-proportionate selection from the simple OneMax function to the linear functions. Not only does our analysis cover a larger class of fitness functions, it also holds for a wider range of mutation rates. We show that with overwhelmingly high probability, no linear function can be optimised in less than exponential time, assuming bitwise mutation rate $Θ(1/n)$ and population size $λ=n^k$ for any constant $k>2$. In contrast to this negative result, we also show that for any linear function with polyno…
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In this article a tool for the analysis of population-based EAs is used to derive asymptotic upper bounds on the optimization time of the algorithm solving Royal Roads problem, a test function with plateaus of fitness. In addition to this, limiting distribution of a certain subset of the population is approximated.
Volkswagen Group of America（美国大众汽车集团）；Volkswagen Group（大众汽车集团）；Terra Quantum（泰拉量子公司）；Leiden University（莱顿大学）
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The interplay between mutation and selection plays a fundamental role in the behaviour of evolutionary algorithms (EAs). However, this interplay is still not completely understood. This paper presents a rigorous runtime analysis of a non-elitist population-based EA that uses the linear ranking selection mechanism. The analysis focuses on how the balance between parameter $η$, controlling the selection pressure in linear ranking, and parameter $χ$ controlling the bit-wise mutation rate, impacts the runtime of the algorithm. The results point out situations where a correct balance between select…
It may seem very intuitive that for the maximization of the OneMax problem $\OM(x):=\sum_{i=1}^n{x_i}$ the best that an elitist unary unbiased search algorithm can do is to store a best so far solution, and to modify it with the operator that yields the best possible expected progress in function value. This assumption has been implicitly used in several empirical works. In [Doerr, Doerr, Yang: Optimal parameter choices via precise black-box analysis, TCS, 2020] it was formally proven that this approach is indeed almost optimal. In this work we prove that drift maximization is not optimal. Mor…
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