数学代写|优化算法作业代写optimisation algorithms代考| Deceptiveness

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|优化算法作业代写optimisation algorithms代考|Deceptiveness

Especially annoying fitness landscapes show deceptiveness (or deceptivity). The gradient of deceptive objective functions leads the optimizer away from the optima, as illustrated in Fig. 1.e.

The term deceptiveness is mainly used in the Genetic Algorithm community in the context of the Schema Theorem. Schemas describe certain areas (hyperplanes) in the search space. If an optimization algorithm has discovered an area with a better average fitness compared to other regions, it will focus on exploring this region based on the assumption that highly fit areas are likely to contain the true optimum. Objective functions where this is not the case are called deceptive $[20,84,127]$. Examples for deceptiveness are the ND fitness landscapes $[17]$, trap functions $[1,59,112]$ like the one illustrated in Fig. 4 , and the fully deceptive problems given by Goldberg et al $[86,60]$.

数学代写|优化算法作业代写optimisation algorithms代考|The Problem: Neutrality

We consider the outcome of the application of a search operation to an element of the search space as neutral if it yields no change in the objective values $[15,172]$. It is challenging for optimization algorithms if the best solution candidate currently known is situated on a plane of the fitness landscape, i.e., all adjacent solution candidates have the same objective values. As illustrated in Fig. 1.f, an optimizer then cannot find any gradient information and thus, no direction in which to proceed in a systematic manner. From its point of view, each search operation will yield identical individuals. Furthermore, optimization algorithms usually maintain a list of the best individuals found, which will then overflow eventually or require pruning.

The degree of neutrality $\nu$ is defined as the fraction of neutral results among all possible products of the search operations $O p$ applied to a specific genotype $[15]$. We can generalize this measure to areas $G$ in the search space $G$ by averaging over all their elements. Regions where $\nu$ is close to one are considered as neutral.
\begin{aligned} &\forall g_{1} \in \mathbb{G} \Rightarrow \nu\left(g_{1}\right)=\frac{\left|\left{g_{2} \mid P\left(g_{2}=O p\left(g_{1}\right)\right)>0 \wedge \mathbf{f}\left(\operatorname{gpm}\left(g_{2}\right)\right)=\mathbf{f}\left(\operatorname{gpm}\left(g_{1}\right)\right)\right}\right|}{\left|\left{g_{2} \mid P\left(g_{2}=O p\left(g_{1}\right)\right)>0\right}\right|} \ &\forall G \subseteq \mathbb{G} \Rightarrow \nu(G)=\frac{1}{|G|} \sum_{g \in G} \nu(g) \end{aligned}

数学代写|优化算法作业代写optimisation algorithms代考|Problematic and Beneficial

The link between evolvability and neutrality has been discussed by many researchers. The evolvability of neutral parts of a fitness landscape depends on the optimization algorithm used. It is especially low for Hill Climbing and similar approaches, since the search operations cannot directly provide improvements or even changes. The optimization process then degenerates to a random walk, as illustrated in Fig. 1.f. The work of Beaudoin et al [17] on the ND fitness landscapes shows that neutrality may “destroy” useful information such as correlation.

Researchers in molecular evolution, on the other hand, found indications that the majority of mutations have no selective influence $[77,106]$ and that the transformation from genotypes to phenotypes is a many-to-one mapping. Wagner [226] states that neutrality in natural genomes is beneficial if it concerns only a subset of the properties peculiar to the offspring of a solution candidate while allowing meaningful modifications of the others. Toussaint and Igel [214] even go as far as declaring it a necessity for self-adaptation.
The theory of punctuated equilibria in biology introduced by Eldredge and Gould $[67,68]$ states that species experience long periods of evolutionary inactivity which are interrupted by sudden, localized, and rapid phenotypic evolutions $[47,134,12]$. It is assumed that the populations explore neutral layers during the time of stasis until, suddenly, a relevant change in a genotype leads to a better adapted phenotype [224] which then reproduces quickly.
The key to differentiating between “good” and “bad” neutrality is its degree $\nu$ in relation to the number of possible solutions maintained by the optimization algorithms. Smith et al [204] have used illustrative examples similar to Fig. 5 showing that a certain amount of neutral reproductions can foster the progress of optimization. In Fig. 5.a, basically the same scenario of premature convergence as in Fig. 3.a is depicted. The optimizer is drawn to a local optimum from which it cannot escape anymore. Fig. 5.b shows that a little shot of neutrality could form a bridge to the global optimum. The optimizer now has a chance to escape the smaller peak if it is able to find and follow that bridge, i.e., the evolvability of the system has increased. If this bridge gets wider, as sketched in Fig. 5.c, the chance of finding the global optimum increases as well. Of course, if the bridge gets too wide, the optimization process may end up in a scenario like in Fig. 1.f where it cannot find any direction. Furthermore, in this scenario we expect the neutral bridge to lead to somewhere useful, which is not necessarily the case in reality.

数学代写|优化算法作业代写optimisation algorithms代考|The Problem: Neutrality

\begin{aligned} &\forall g_{1} \in \mathbb{G} \Rightarrow \nu\left(g_{1}\right)=\frac{\left|\left{g_{2} \mid P \left(g_{2}=O p\left(g_{1}\right)\right)>0 \wedge \mathbf{f}\left(\operatorname{gpm}\left(g_{2}\right) \right)=\mathbf{f}\left(\operatorname{gpm}\left(g_{1}\right)\right)\right}\right|}{\left|\left{g_{2} \mid P\left(g_{2}=O p\left(g_{1}\right)\right)>0\right}\right|} \ &\forall G \subseteq \mathbb{G} \Rightarrow \nu( G)=\frac{1}{|G|} \sum_{g \in G} \nu(g) \end{aligned}\begin{aligned} &\forall g_{1} \in \mathbb{G} \Rightarrow \nu\left(g_{1}\right)=\frac{\left|\left{g_{2} \mid P \left(g_{2}=O p\left(g_{1}\right)\right)>0 \wedge \mathbf{f}\left(\operatorname{gpm}\left(g_{2}\right) \right)=\mathbf{f}\left(\operatorname{gpm}\left(g_{1}\right)\right)\right}\right|}{\left|\left{g_{2} \mid P\left(g_{2}=O p\left(g_{1}\right)\right)>0\right}\right|} \ &\forall G \subseteq \mathbb{G} \Rightarrow \nu( G)=\frac{1}{|G|} \sum_{g \in G} \nu(g) \end{aligned}

数学代写|优化算法作业代写optimisation algorithms代考|Problematic and Beneficial

Eldredge 和 Gould 提出的生物学中的间断平衡理论[67,68]指出物种经历了长时间的进化不活动，这些不活动被突然的、局部的和快速的表型进化打断[47,134,12]. 假设种群在停滞期间探索中性层，直到突然，基因型的相关变化导致更好的适应表型[224]，然后快速繁殖。

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