低选择压力和先验噪声下单变量边缘分布算法的性能分析
Runtime Analysis of the Univariate Marginal Distribution Algorithm under Low Selective Pressure and Prior Noise
摘要：
我们执行一个严格的运行时分析的单变量边际分布算法上的LeadingOnes功能，一个著名的基准功能，在理论界的演化计算与决策变量之间的高度相关性。对于大小为$n$的问题实例，当前已知的关于预期运行时间的上限是$\mathcal{O}(nλ\logλ+n^2)$（Dang and Lehre，GECCO 2015），而理解算法如何处理变量依赖关系所需的下限仍然缺失。出于这一动机，我们表明，该算法需要一个$e^{Ω(μ)}$运行时具有很高的概率，并在预期中，如果选择压力低，否则，我们得到一个下界的$Ω(\frac{nλ}{\log(λ-μ)})$的预期运行时间。此外，我们第一次考虑算法的功能下的先验噪声模型，并获得一个$\mathcal{O}(n^2)$预期运行时间的最佳参数设置。最后，我们的理论结果伴随着实证研究结果，不仅与严格的分析相匹配，而且还提供了新的见解算法的行为。
英文摘要：
We perform a rigorous runtime analysis for the Univariate Marginal Distribution Algorithm on the LeadingOnes function, a well-known benchmark function in the theory community of evolutionary computation with a high correlation between decision variables. For a problem instance of size $n$, the currently best known upper bound on the expected runtime is $\mathcal{O}(nλ\logλ+n^2)$ (Dang and Lehre, GECCO 2015), while a lower bound necessary to understand how the algorithm copes with variable dependencies is still missing. Motivated by this, we show that the algorithm requires a $e^{Ω(μ)}$ runtime with high probability and in expectation if the selective pressure is low; otherwise, we obtain a lower bound of $Ω(\frac{nλ}{\log(λ-μ)})$ on the expected runtime. Furthermore, we for the first time consider the algorithm on the function under a prior noise model and obtain an $\mathcal{O}(n^2)$ expected runtime for the optimal parameter settings. In the end, our theoretical results are accompanied by empirical findings, not only matching with rigorous analyses but also providing new insights into the behaviour of the algorithm.
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