Clustering is a fundamental problem in many areas, which aims to partition a given data set into groups based on some distance measure, such that the data points in the same group are similar while that in different groups are dissimilar. Due to its importance and NP-hardness, a lot of methods have been proposed, among which evolutionary algorithms are a class of popular ones. Evolutionary clustering has found many successful applications, but all the results are empirical, lacking theoretical support. This paper fills this gap by proving that the approximation performance of the GSEMO (a simple multi-objective evolutionary algorithm) for solving the three popular formulations of clustering, i.e., $k$-center, $k$-median and $k$-means, can be theoretically guaranteed. Furthermore, we prove that evolutionary clustering can have theoretical guarantees even when considering fairness, which tries to avoid algorithmic bias, and has recently been an important research topic in machine learning.

Evolutionary algorithms (EAs) are a kind of nature-inspired general-purpose optimization algorithm, and have shown empirically good performance in solving various real-word optimization problems. During the past two decades, promising results on the running time analysis (one essential theoretical aspect) of EAs have been obtained, while most of them focused on isolated combinatorial optimization problems, which do not reflect the general-purpose nature of EAs. To provide a general theoretical explanation of the behavior of EAs, it is desirable to study their performance on general classes of combinatorial optimization problems. To the best of our knowledge, the only result towards this direction is the provably good approximation guarantees of EAs for the problem class of maximizing monotone submodular functions with matroid constraints. The aim of this work is to contribute to this line of research. Considering that many combinatorial optimization problems involve non-monotone or non-submodular objective functions, we study the general problem classes, maximizing submodular functions with/without a size constraint and maximizing monotone approximately submodular functions with a size constraint. We prove that a simple multi-objective EA called GSEMO-C can generally achieve good approximation guarantees in polynomial expected running time.

Evolutionary algorithms (EAs) are general-purpose optimization algorithms, inspired by natural evolution. Recent theoretical studies have shown that EAs can achieve good approximation guarantees for solving the problem classes of submodular optimization, which have a wide range of applications, such as maximum coverage, sparse regression, influence maximization, document summarization and sensor placement, just to name a few. Though they have provided some theoretical explanation for the general-purpose nature of EAs, the considered submodular objective functions are defined only over sets or multisets. To complement this line of research, this paper studies the problem class of maximizing monotone submodular functions over sequences, where the objective function depends on the order of items. We prove that for each kind of previously studied monotone submodular objective functions over sequences, i.e., prefix monotone submodular functions, weakly monotone and strongly submodular functions, and DAG monotone submodular functions, a simple multi-objective EA, i.e., GSEMO, can always reach or improve the best known approximation guarantee after running polynomial time in expectation. Note that these best-known approximation guarantees can be obtained only by different greedy-style algorithms before. Empirical studies on various applications, e.g., accomplishing tasks, maximizing information gain, search-and-tracking and recommender systems, show the excellent performance of the GSEMO.

Evolutionary algorithms (EAs) have found many successful real-world applications, where the optimization problems are often subject to a wide range of uncertainties. To understand the practical behaviors of EAs theoretically, there are a series of efforts devoted to analyzing the running time of EAs for optimization under uncertainties. Existing studies mainly focus on noisy and dynamic optimization, while another common type of uncertain optimization, i.e., robust optimization, has been rarely touched. In this paper, we analyze the expected running time of the (1+1)-EA solving robust linear optimization problems (i.e., linear problems under robust scenarios) with a cardinality constraint $k$. Two common robust scenarios, i.e., deletion-robust and worst-case, are considered. Particularly, we derive tight ranges of the robust parameter $d$ or budget $k$ allowing the (1+1)-EA to find an optimal solution in polynomial running time, which disclose the potential of EAs for robust optimization.

机会受到限制的优化问题允许建模问题，其中涉及随机组件的约束仅应以较小的概率侵犯。进化算法已应用于这种情况，并证明可以实现高质量的结果。在本文中，我们有助于对进化算法的理论理解，以进行偶然的优化。我们研究独立且正态分布的随机组件的场景。考虑到简单的单对象（1+1）〜EA，我们表明，施加额外的统一约束已经导致局部最佳选择，对于非常有限的场景和指数优化时间。因此，我们引入了问题的多目标公式，该公式可以摆脱预期成本及其差异。我们表明，在使用此公式时，多目标进化算法是非常有效的，并获得一组解决方案，该解决方案包含最佳解决方案，以适用于施加在约束上的任何可能的置信度。此外，我们证明这种方法还可以用于计算一组最佳解决方案，以限制最小跨越树问题。为了在多目标配方中呈指数指数的折衷，我们提出并分析了改进的凸多目标方法。关于NP-固定随机最小重量占主导地位问题的实例的实验研究证实了多目标和改进的凸多目标方法的益处。

非主导的分类遗传算法II（NSGA-II）是现实应用中最强烈使用的多目标进化算法（MOEA）。然而，与几个通过数学手段分析的几个简单的MOES相反，到目前为止，NSGA-II也不存在这种研究。在这项工作中，我们表明，数学运行时分析也可用于NSGA-II。结果，我们证明，由于持续因素大于帕累托前方大小的人口大小，具有两个经典突变算子的NSGA-II和三种不同的选择父母的方式满足与Semo和GSEMO相同的渐近运行时保证基本ineminmax和Lotz基准函数的算法。但是，如果人口大小仅等于帕累托前面的大小，那么NSGA-II就无法有效地计算完整的帕累托前部（对于指数迭代，人口总是错过帕累托前部的恒定分数） 。我们的实验证实了上述研究结果。

本文展示了如何适应$ k $ -MEANS问题的几种简单和经典的基于采样的算法，以使用离群值设置。最近，Bhaskara等人。 （Neurips 2019）展示了如何将古典$ K $ -MEANS ++算法适应与异常值的设置。但是，他们的算法需要输出$ o（\ log（k）\ cdot z）$ outiers，其中$ z $是true Outliers的数量，以匹配$ o（\ log k）$ - 近似值的$ k的近似保证$ -Means ++。在本文中，我们以他们的想法为基础，并展示了如何适应几个顺序和分布式的$ k $ - 均值算法，但使用离群值来设置，但具有更强的理论保证：我们的算法输出$（1+ \ VAREPSILON）z $ OUTLIERS Z $ OUTLIERS在实现$ o（1 / \ varepsilon）$ - 近似目标函数的同时。在顺序世界中，我们通过改编Lattanzi和Sohler的最新算法来实现这一目标（ICML 2019）。在分布式设置中，我们适应了Guha等人的简单算法。 （IEEE Trans。知道和数据工程2003）以及Bahmani等人的流行$ K $ -Means $ \ | $。 （PVLDB 2012）。我们技术的理论应用是一种具有运行时间$ \ tilde {o}（nk^2/z）$的算法，假设$ k \ ll z \ ll n $。这与Omacle模型中此问题的$ \ Omega（NK^2/z）$的匹配下限相互补。

我们研究社会上公平$（\ ell_p，k）$的近似算法 - $ m $组的聚类问题，其特殊案例包括社会公平的$ k $ -Median（$ p = 1 $）和社会公平的$ k $ - 均值（$ p = 2 $）问题。我们提出（1）一个多项式时间$（5+2 \ sqrt {6}）^p $ - approximation，最多$ k+m $中心（2）a $（5+2 \ sqrt {6}+\ \ \ \ \ \ \ \ \ \ \ \ \ \\ epsilon）^p $ - approximation with $ k $中心$ n^{2^{o（p）} \ cdot m^2} $，和（3）a $（15+6 \ sqrt {6}） ^p $ k $中心的时间$ k^{m} \ cdot \ text {poly}（n）$。第一个结果是通过使用一系列线性程序的迭代圆形方法的细化来获得的。后两个结果是通过将最多$ K+M $中心的解决方案转换为使用（2）的稀疏方法的$ K $中心的解决方案，并通过详尽的搜索（3）。我们还将算法的性能与现有的双色算法以及基准数据集中的$ K $中心近似算法的恰好比较，并发现我们的算法在实践中也优于现有方法。

In noisy evolutionary optimization, sampling is a common strategy to deal with noise. By the sampling strategy, the fitness of a solution is evaluated multiple times (called \emph{sample size}) independently, and its true fitness is then approximated by the average of these evaluations. Most previous studies on sampling are empirical, and the few theoretical studies mainly showed the effectiveness of sampling with a sufficiently large sample size. In this paper, we theoretically examine what strategies can work when sampling with any fixed sample size fails. By constructing a family of artificial noisy examples, we prove that sampling is always ineffective, while using parent or offspring populations can be helpful on some examples. We also construct an artificial noisy example to show that when using neither sampling nor populations is effective, a tailored adaptive sampling (i.e., sampling with an adaptive sample size) strategy can work. These findings may enhance our understanding of sampling to some extent, but future work is required to validate them in natural situations.

The NSGA-II is one of the most prominent algorithms to solve multi-objective optimization problems. Despite numerous successful applications, several studies have shown that the NSGA-II is less effective for larger numbers of objectives. In this work, we use mathematical runtime analyses to rigorously demonstrate and quantify this phenomenon. We show that even on the simple OneMinMax benchmark, where every solution is Pareto optimal, the NSGA-II also with large population sizes cannot compute the full Pareto front (objective vectors of all Pareto optima) in sub-exponential time when the number of objectives is at least three. Our proofs suggest that the reason for this unexpected behavior lies in the fact that in the computation of the crowding distance, the different objectives are regarded independently. This is not a problem for two objectives, where any sorting of a pair-wise incomparable set of solutions according to one objective is also such a sorting according to the other objective (in the inverse order).

最近，已经进行了多目标进化优化器NSGA-II的第一个数学运行时分析（AAAI 2022，GECCO 2022（出现），ARXIV 2022）。我们通过对由两个多模式目标组成的基准问题进行该算法的第一个运行时分析继续进行这一研究。我们证明，如果人口尺寸$ n $至少是帕累托阵线的四倍，那么NSGA-II具有四种不同方法的NSGA-II选择父母，并且位于Bit Wise突变将优化OnejumpzeroJump基准，其跳高尺寸〜$ 2 \ le lek \ le n/4 $ in Time $ o（n n^k）$。当使用快速突变（最近提出的重型突变操作员）时，此保证将提高$ k^{\ omega（k）} $。总体而言，这项工作表明，NSGA-II至少与全球SEMO算法有关OnejumpZeroJump问题的局部优势。

In many real-world optimization problems, the objective function evaluation is subject to noise, and we cannot obtain the exact objective value. Evolutionary algorithms (EAs), a type of general-purpose randomized optimization algorithm, have been shown to be able to solve noisy optimization problems well. However, previous theoretical analyses of EAs mainly focused on noise-free optimization, which makes the theoretical understanding largely insufficient for the noisy case. Meanwhile, the few existing theoretical studies under noise often considered the one-bit noise model, which flips a randomly chosen bit of a solution before evaluation; while in many realistic applications, several bits of a solution can be changed simultaneously. In this paper, we study a natural extension of one-bit noise, the bit-wise noise model, which independently flips each bit of a solution with some probability. We analyze the running time of the (1+1)-EA solving OneMax and LeadingOnes under bit-wise noise for the first time, and derive the ranges of the noise level for polynomial and super-polynomial running time bounds. The analysis on LeadingOnes under bit-wise noise can be easily transferred to one-bit noise, and improves the previously known results. Since our analysis discloses that the (1+1)-EA can be efficient only under low noise levels, we also study whether the sampling strategy can bring robustness to noise. We prove that using sampling can significantly increase the largest noise level allowing a polynomial running time, that is, sampling is robust to noise.

多样性最大化是数据汇总，Web搜索和推荐系统中广泛应用的基本问题。给定$ n $元素的$ x $元素，它要求选择一个$ k \ ll n $元素的子集$ s $，具有最大\ emph {多样性}，这是由$ s $中元素之间的差异量化的。在本文中，我们关注流媒体环境中公平限制的多样性最大化问题。具体而言，我们考虑了最大值的多样性目标，该目标选择了一个子集$ s $，该子集$ s $最大化了其中任何一对不同元素之间的最小距离（不同）。假设集合$ x $通过某些敏感属性（例如性别或种族）将$ m $ discoint组分为$ m $ discoint组，确保\ emph {fairness}要求所选的子集$ s $包含每个组$ i的$ k_i $ e元素\在[1，m] $中。流算法应在一个通过中顺序处理$ x $，并返回具有最大\ emph {多样性}的子集，同时保证公平约束。尽管对多样性的最大化进行了广泛的研究，但唯一可以与最大值多样性目标和公平性约束的唯一已知算法对数据流非常低效。由于多样性最大化通常是NP-HARD，因此我们提出了两个在数据流中最大化的公平多样性的近似算法，其中第一个是$ \ frac {1- \ varepsilon} {4} {4} $ - 近似于$ m = 2 $，其中$ \ varepsilon \ in（0,1）$，第二个实现了$ \ frac {1- \ varepsilon} {3m+2} $ - 任意$ m $的近似值。现实世界和合成数据集的实验结果表明，两种算法都提供了与最新算法相当的质量解决方案，同时在流式设置中运行多个数量级。

我们提供了一个新的双标准$ \ tilde {o}（\ log ^ 2 k）$竞争算法，可解释$ k $ -means群集。最近解释了$ k $ -means最近由Dasgupta，Frost，Moshkovitz和Rashtchian（ICML 2020）引入。它由易于解释和理解（阈值）决策树或图表描述。可解释的$ k $ -means集群的成本等于其集群成本的总和;每个群集的成本等于从群集中点到该群集的中心的平方距离之和。我们的随机双标准算法构造了一个阈值决策树，将数据设置为$（1+ \ delta）k $群集（其中$ \ delta \ In（0,1）$是算法的参数）。此群集的成本是大多数$ \ tilde {o}（1 / \ delta \ cdot \ log ^ 2 k）$乘以最佳不受约束$ k $ -means群集的成本。我们表明这一界限几乎是最佳的。

The k-means method is a widely used clustering technique that seeks to minimize the average squared distance between points in the same cluster. Although it offers no accuracy guarantees, its simplicity and speed are very appealing in practice. By augmenting k-means with a simple, randomized seeding technique, we obtain an algorithm that is O(log k)-competitive with the optimal clustering. Experiments show our augmentation improves both the speed and the accuracy of k-means, often quite dramatically.

K-means++ is an important algorithm to choose initial cluster centers for the k-means clustering algorithm. In this work, we present a new algorithm that can solve the $k$-means++ problem with near optimal running time. Given $n$ data points in $\mathbb{R}^d$, the current state-of-the-art algorithm runs in $\widetilde{O}(k )$ iterations, and each iteration takes $\widetilde{O}(nd k)$ time. The overall running time is thus $\widetilde{O}(n d k^2)$. We propose a new algorithm \textsc{FastKmeans++} that only takes in $\widetilde{O}(nd + nk^2)$ time, in total.

基于中心的聚类（例如，$ k $ -means，$ k $ -Medians）和使用线性子空间的聚类是两种最受欢迎的技术，可以将真实数据分配到较小的群集中。但是，当数据由敏感人群组组成时，不同敏感组的每点的聚集成本显着不同，可能会导致与公平相关的危害（例如，服务质量不同）。社会公平聚类的目的是最大程度地降低所有组中每点聚类的最大成本。在这项工作中，我们提出了一个统一的框架，以解决社会公平的基于中心的聚类和线性子空间聚类，并为这些问题提供实用，高效的近似算法。我们进行了广泛的实验，以表明在多个基准数据集上，我们的算法要么紧密匹配或超越最先进的基线。

在设计聚类算法时，初始中心的选择对于学习簇的质量至关重要。在本文中，我们基于数据的构建，我们开发了一种新的初始化方案，称为$ k $ -Median问题（例如图形引起的离散空间），基于数据的构造。从树中，我们提出了一种新颖有效的搜索算法，用于良好的初始中心，随后可用于本地搜索算法。我们提出的HST初始化可以产生与另一种流行初始化方法$ K $ -Median ++的初始中心，具有可比的效率。 HST初始化也可以扩展到差异隐私（DP）的设置，以生成私人初始中心。我们表明，应用DP本地搜索后，我们的私有HST初始化会改善对近似错误的先前结果，并在小因素内接近下限。实验证明了理论的合理性，并证明了我们提出的方法的有效性。我们的方法也可以扩展到$ k $ -MEANS问题。

在聚类问题中，中央决策者通过顶点给出完整的公制图，并且必须提供最小化某些目标函数的顶点的聚类。在公平的聚类问题中，顶点以颜色（例如，组中的成员身份）赋予，并且有效群集的功能也可能包括该群集中的颜色的表示。在公平集群中的事先工作假设完全了解集团成员资格。在本文中，我们通过假设通过概率分配不完美了解集团成员资格的知识。我们在此具有近似率保证的更常规设置中呈现聚类算法。我们还解决了“公制成员资格”的问题，其中不同的群体的概念和距离。使用我们所提出的算法以及基线进行实验，以验证我们的方法，并且当组成员资格不确定时，验证我们的方法以及表面细微的问题。

我们重新审视了Chierichetti等人首先引入的公平聚类问题，该问题要求每个受保护的属性在每个集群中具有近似平等的表示。即，余额财产。现有的公平聚类解决方案要么是不可扩展的，要么无法在聚类目标和公平之间实现最佳权衡。在本文中，我们提出了一种新的公平概念，我们称之为$ tau $ $ $ - fair公平，严格概括了余额财产，并实现了良好的效率与公平折衷。此外，我们表明，简单的基于贪婪的圆形算法有效地实现了这一权衡。在更一般的多价受保护属性的设置下，我们严格地分析了算法的理论特性。我们的实验结果表明，所提出的解决方案的表现优于所有最新算法，即使对于大量簇，也可以很好地工作。