A Unified Framework for Center-based Clustering of Distributed Data
Carnegie Mellon University（卡内基梅隆大学）；Faculty of Technical Sciences, University of Novi Sad（诺维萨德大学技术科学学院）；Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院）
Aleksandar Armacki, Dragana Bajović, Dušan Jakovetić, Soummya Kar
We develop a family of distributed center-based clustering algorithms that work over networks of users. In the proposed scenario, users contain a local dataset and communicate only with their immediate neighbours, with the aim of finding a clustering of the full, joint data. The proposed family, termed Distributed Gradient Clustering (DGC-$\mathcal{F}_ρ$), is parametrized by $ρ\geq 1$, controling the proximity of users' center estimates, with $\mathcal{F}$ determining the clustering loss. Our framework allows for a broad class of smooth convex loss functions, including popular clustering losses like $K$-means and Huber loss. Specialized to popular clustering losses like $K$-means and Huber loss, DGC-$\mathcal{F}_ρ$ gives rise to novel distributed clustering algorithms DGC-KM$_ρ$ and DGC-HL$_ρ$, while novel clustering losses based on Logistic and Fair functions lead to DGC-LL$_ρ$ and DGC-FL$_ρ$. We provide a unified analysis and establish several strong results, under mild assumptions. First, we show that the sequence of centers generated by the methods converges to a well-defined notion of fixed point, under any center initialization and value of $ρ$. Second, we prove that, as $ρ$ increases, the family of fixed points produced by DGC-$\mathcal{F}_ρ$ converges to a notion of consensus fixed points. We show that consensus fixed points of DGC-$\mathcal{F}_ρ$ are equivalent to fixed points of gradient clustering over the full data, guaranteeing a clustering of the full data is produced. For the special case of Bregman losses, we show that our fixed points converge to the set of Lloyd points. Extensive numerical experiments on synthetic and real data confirm our theoretical findings, show strong performance of our methods and demonstrate the usefulness and wide range of potential applications of our general framework, such as outlier detection.
Insight Centre for Data Analytics School of Computer Science University College Dublin（都柏林大学学院计算机科学学院数据洞察中心）；School of Computer Science University College Dublin（都柏林大学学院计算机科学学院）
In this paper, we present a new approach of distributed clustering for spatial datasets, based on an innovative and efficient aggregation technique. This distributed approach consists of two phases: 1) local clustering phase, where each node performs a clustering on its local data, 2) aggregation phase, where the local clusters are aggregated to produce global clusters. This approach is characterised by the fact that the local clusters are represented in a simple and efficient way. And The aggregation phase is designed in such a way that the final clusters are compact and accurate while the ov…
Amasya University（阿马西亚大学）；Bilkent University（比尔肯特大学）；National Magnetic Resonance Research Center, Bilkent University（比尔肯特大学国家磁共振研究中心）；Sabuncu Brain Research Center, Bilkent University（比尔肯特大学萨本朱脑研究中心）
Melanoma is a fatal skin cancer that is curable and has dramatically increasing survival rate when diagnosed at early stages. Learning-based methods hold significant promise for the detection of melanoma from dermoscopic images. However, since melanoma is a rare disease, existing databases of skin lesions predominantly contain highly imbalanced numbers of benign versus malignant samples. In turn, this imbalance introduces substantial bias in classification models due to the statistical dominance of the majority class. To address this issue, we introduce a deep clustering approach based on the …
Low-Rank Representation (LRR) highly suffers from discarding the locality information of data points in subspace clustering, as it may not incorporate the data structure nonlinearity and the non-uniform distribution of observations over the ambient space. Thus, the information of the observational density is lost by the state-of-art LRR models, as they take a constant number of adjacent neighbors into account. This, as a result, degrades the subspace clustering accuracy in such situations. To cope with deficiency, in this paper, we propose to consider a hypergraph model to facilitate having a …
This paper provides new algorithms for distributed clustering for two popular center-based objectives, k-median and k-means. These algorithms have provable guarantees and improve communication complexity over existing approaches. Following a classic approach in clustering by \cite{har2004coresets}, we reduce the problem of finding a clustering with low cost to the problem of finding a coreset of small size. We provide a distributed method for constructing a global coreset which improves over the previous methods by reducing the communication complexity, and which works over general communicati…
College of Information and Computer Sciences, University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校信息与计算机科学学院）；Google Research New York（谷歌纽约研究院）
In this paper, we present distributed generalized clustering algorithms that can handle large scale data across multiple machines in spite of straggling or unreliable machines. We propose a novel data assignment scheme that enables us to obtain global information about the entire data even when some machines fail to respond with the results of the assigned local computations. The assignment scheme leads to distributed algorithms with good approximation guarantees for a variety of clustering and dimensionality reduction problems.
Graph clustering is a fundamental computational problem with a number of applications in algorithm design, machine learning, data mining, and analysis of social networks. Over the past decades, researchers have proposed a number of algorithmic design methods for graph clustering. However, most of these methods are based on complicated spectral techniques or convex optimisation, and cannot be applied directly for clustering many networks that occur in practice, whose information is often collected on different sites. Designing a simple and distributed clustering algorithm is of great interest, …
Distributed Clustering based on Distributional Kernel
National Key Laboratory for Novel Software Technology, Nanjing University（南京大学计算机软件新技术国家重点实验室）；School of Artificial Intelligence, Nanjing University（南京大学人工智能学院）；School of Information Technology, Deakin University（迪肯大学信息技术学院）
University of Massachusetts, Dartmouth（马萨诸塞大学达特茅斯分校）；Lovelace Respiratory Research Institute（洛夫莱斯呼吸研究所）
The last decades have seen a surge of interests in distributed computing thanks to advances in clustered computing and big data technology. Existing distributed algorithms typically assume {\it all the data are already in one place}, and divide the data and conquer on multiple machines. However, it is increasingly often that the data are located at a number of distributed sites, and one wishes to compute over all the data with low communication overhead. For spectral clustering, we propose a novel framework that enables its computation over such distributed data, with "minimal" communications …
In this work, we study the $k$-median and $k$-means clustering problems when the data is distributed across many servers and can contain outliers. While there has been a lot of work on these problems for worst-case instances, we focus on gaining a finer understanding through the lens of beyond worst-case analysis. Our main motivation is the following: for many applications such as clustering proteins by function or clustering communities in a social network, there is some unknown target clustering, and the hope is that running a $k$-median or $k$-means algorithm will produce clusterings which …
Indian Institute of Science（印度科学学院）；University of Maryland, College Park（马里兰大学帕克分校）；Microsoft Research（微软研究院）
Center-based clustering (e.g., $k$-means, $k$-medians) and clustering using linear subspaces are two most popular techniques to partition real-world data into smaller clusters. However, when the data consists of sensitive demographic groups, significantly different clustering cost per point for different sensitive groups can lead to fairness-related harms (e.g., different quality-of-service). The goal of socially fair clustering is to minimize the maximum cost of clustering per point over all groups. In this work, we propose a unified framework to solve socially fair center-based clustering an…
In this paper we target the class of modal clustering methods where clusters are defined in terms of the local modes of the probability density function which generates the data. The most well-known modal clustering method is the k-means clustering. Mean Shift clustering is a generalization of the k-means clustering which computes arbitrarily shaped clusters as defined as the basins of attraction to the local modes created by the density gradient ascent paths. Despite its potential, the Mean Shift approach is a computationally expensive method for unsupervised learning. Thus, we introduce two …
Harbin Institute of Technology（哈尔滨工业大学）；Beihang University（北京航空航天大学）；Jilin University（吉林大学）
The purpose of this paper is to improve the traditional K-means algorithm. In the traditional K mean clustering algorithm, the initial clustering centers are generated randomly in the data set. It is easy to fall into the local minimum solution when the initial cluster centers are randomly generated. The initial clustering center selected by K-means clustering algorithm which based on density is more representative. The experimental results show that the improved K clustering algorithm can eliminate the dependence on the initial cluster, and the accuracy of clustering is improved.
University of Science and Technology of China（中国科学技术大学）
We study the problem of $k$-center clustering with outliers in arbitrary metrics and Euclidean space. Though a number of methods have been developed in the past decades, it is still quite challenging to design quality guaranteed algorithm with low complexity for this problem. Our idea is inspired by the greedy method, Gonzalez's algorithm, for solving the problem of ordinary $k$-center clustering. Based on some novel observations, we show that this greedy strategy actually can handle $k$-center clustering with outliers efficiently, in terms of clustering quality and time complexity. We further…
Dual-Center Graph Clustering with Neighbor Distribution
Shandong University（山东大学）；Aerospace Information Research Institute, Chinese Academy of Sciences（中国科学院 aerospace information research institute）；Harbin Institute of Technology (Shenzhen)（哈尔滨工业大学（深圳））
In this paper, we consider the $k$-center/median/means clustering with outliers problems (or the $(k, z)$-center/median/means problems) in the distributed setting. Most previous distributed algorithms have their communication costs linearly depending on $z$, the number of outliers. Recently Guha et al. overcame this dependence issue by considering bi-criteria approximation algorithms that output solutions with $2z$ outliers. For the case where $z$ is large, the extra $z$ outliers discarded by the algorithms might be too large, considering that the data gathering process might be costly. In thi…
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Center for Excellence in Animal Evolution and Genetics, Chinese Academy of Sciences（中国科学院动物进化与遗传前沿交叉卓越创新中心）；Yunnan University（云南大学）
Matrix decomposition is one of the fundamental tools to discover knowledge from big data generated by modern applications. However, it is still inefficient or infeasible to process very big data using such a method in a single machine. Moreover, big data are often distributedly collected and stored on different machines. Thus, such data generally bear strong heterogeneous noise. It is essential and useful to develop distributed matrix decomposition for big data analytics. Such a method should scale up well, model the heterogeneous noise, and address the communication issue in a distributed sys…
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；University of Colorado Boulder（科罗拉多大学博尔德分校）
The unsupervised learning of community structure, in particular the partitioning vertices into clusters or communities, is a canonical and well-studied problem in exploratory graph analysis. However, like most graph analyses the introduction of immense scale presents challenges to traditional methods. Spectral clustering in distributed memory, for example, requires hundreds of expensive bulk-synchronous communication rounds to compute an embedding of vertices to a few eigenvectors of a graph associated matrix. Furthermore, the whole computation may need to be repeated if the underlying graph c…
We consider $K$-means clustering in networked environments (e.g., internet of things (IoT) and sensor networks) where data is inherently distributed across nodes and processing power at each node may be limited. We consider a clustering algorithm referred to as networked $K$-means, or $NK$-means, which relies only on local neighborhood information exchange. Information exchange is limited to low-dimensional statistics and not raw data at the agents. The proposed approach develops a parametric family of multi-agent clustering objectives (parameterized by $ρ$) and associated distributed $NK$-mea…
Georgia Institute of Technology（佐治亚理工学院）；Duke University（杜克大学）
One key use of k-means clustering is to identify cluster prototypes which can serve as representative points for a dataset. However, a drawback of using k-means cluster centers as representative points is that such points distort the distribution of the underlying data. This can be highly disadvantageous in problems where the representative points are subsequently used to gain insights on the data distribution, as these points do not mimic the distribution of the data. To this end, we propose a new clustering method called "distributional clustering", which ensures cluster centers capture the …
Microsoft Research（微软研究院）；Indian Institute of Science（印度科学学院）
$k$-means clustering is NP-hard in the worst case but previous work has shown efficient algorithms assuming the optimal $k$-means clusters are \emph{stable} under additive or multiplicative perturbation of data. This has two caveats. First, we do not know how to efficiently verify this property of optimal solutions that are NP-hard to compute in the first place. Second, the stability assumptions required for polynomial time $k$-means algorithms are often unreasonable when compared to the ground-truth clusters in real-world data. A consequence of multiplicative perturbation resilience is \emph{…
Distributed Gradient Clustering: Convergence and the Effect of Initialization
Carnegie Mellon University（卡内基梅隆大学）；Indian Institute of Technology Kharagpur（印度理工学院克勒格布尔分校）；Faculty of Technical Sciences, University of Novi Sad（诺维萨德大学技术科学学院）；Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院）
Near-Optimal Algorithms for Constrained k-Center Clustering with Instance-level Background Knowledge
School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学学院）；School of Accounting, Information Systems and Supply Chain, RMIT University（皇家墨尔本理工大学会计、信息系统与供应链学院）；School of Information Technology, Deakin University（德肯大学信息科技学院）；Western Sydney University（西悉尼大学）；CSIRO’s Data61（澳大利亚联邦科学与工业研究组织Data61）
Hong Kong University of Science and Technology（香港科技大学）
In this paper, we study the problem of fair clustering on the $k-$center objective. In fair clustering, the input is $N$ points, each belonging to at least one of $l$ protected groups, e.g. male, female, Asian, Hispanic. The objective is to cluster the $N$ points into $k$ clusters to minimize a classical clustering objective function. However, there is an additional constraint that each cluster needs to be fair, under some notion of fairness. This ensures that no group is either "over-represented" or "under-represented" in any cluster. Our work builds on the work of Chierichetti et al. (NIPS 2…
University of Maryland（马里兰大学）；Imperial College London（伦敦帝国学院）
Distributed implementations are crucial in speeding up large scale machine learning applications. Distributed gradient descent (GD) is widely employed to parallelize the learning task by distributing the dataset across multiple workers. A significant performance bottleneck for the per-iteration completion time in distributed synchronous GD is $straggling$ workers. Coded distributed computation techniques have been introduced recently to mitigate stragglers and to speed up GD iterations by assigning redundant computations to workers. In this paper, we consider gradient coding (GC), and propose …
University of Science and Technology of China（中国科学技术大学）
Clustering has many important applications in computer science, but real-world datasets often contain outliers. Moreover, the presence of outliers can make the clustering problems to be much more challenging. To reduce the complexities, various sampling methods have been proposed in past years. Namely, we take a small sample (uniformly or non-uniformly) from input and run an existing approximation algorithm on the sample. Comparing with existing non-uniform sampling methods, the uniform sampling approach has several significant benefits. For example, it only needs to read the data in one-pass …
Language-Assisted Image Clustering Guided by Discriminative Relational Signals and Adaptive Semantic Centers
School of Computer Science and Engineering, Southeast University, Nanjing, China（东南大学计算机科学与工程学院）；School of Computing Information Sciences, Saint Francis University, Hong Kong, China（圣弗朗西斯大学计算信息科学学院）；Department of Computer Science, City University of Hong Kong, Hong Kong, China（香港城市大学计算机科学系）
In data summarization we want to choose $k$ prototypes in order to summarize a data set. We study a setting where the data set comprises several demographic groups and we are restricted to choose $k_i$ prototypes belonging to group $i$. A common approach to the problem without the fairness constraint is to optimize a centroid-based clustering objective such as $k$-center. A natural extension then is to incorporate the fairness constraint into the clustering problem. Existing algorithms for doing so run in time super-quadratic in the size of the data set, which is in contrast to the standard $k…
In this work, we consider to improve the model estimation efficiency by aggregating the neighbors' information as well as identify the subgroup membership for each node in the network. A tree-based $l_1$ penalty is proposed to save the computation and communication cost. We design a decentralized generalized alternating direction method of multiplier algorithm for solving the objective function in parallel. The theoretical properties are derived to guarantee both the model consistency and the algorithm convergence. Thorough numerical experiments are also conducted to back up our theory, which …
Coordinated Sciences Laboratory（协同科学实验室）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Federated learning systems that jointly preserve Byzantine robustness and privacy have remained an open problem. Robust aggregation, the standard defense for Byzantine attacks, generally requires server access to individual updates or nonlinear computation -- thus is incompatible with privacy-preserving methods such as secure aggregation via multiparty computation. To this end, we propose SHARE (Secure Hierarchical Robust Aggregation), a distributed learning framework designed to cryptographically preserve client update privacy and robustness to Byzantine adversaries simultaneously. The key id…
University of Science and Technology of China（中国科学技术大学）
In this paper, we study the problem of {\em $k$-center clustering with outliers}. The problem has many important applications in real world, but the presence of outliers can significantly increase the computational complexity. Though a number of methods have been developed in the past decades, it is still quite challenging to design quality guaranteed algorithm with low complexity for this problem. Our idea is inspired by the greedy method, Gonzalez's algorithm, that was developed for solving the ordinary $k$-center clustering problem. Based on some novel observations, we show that a simple ra…
Random restart of a given algorithm produces many partitions to yield a consensus clustering. Ensemble methods such as consensus clustering have been recognized as more robust approaches for data clustering than single clustering algorithms. We propose the use of determinantal point processes or DPP for the random restart of clustering algorithms based on initial sets of center points, such as k-medoids or k-means. The relation between DPP and kernel-based methods makes DPPs suitable to describe and quantify similarity between objects. DPPs favor diversity of the center points within subsets. …
The $k$-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances. Therefore to improve on these ratios, one must go beyond the worst case. In this work, we take this approach and provide strong positive results both for the asymmetric and symmetric $k$-center problems under a natural input stability (promise) condition called $α$-perturbation resilience [Bilu and Linia 2012], which states that the optimal solutio…
Clustering in graphs has been a well-known research problem, particularly because most Internet and social network data is in the form of graphs. Organizations widely use spectral clustering algorithms to find clustering in graph datasets. However, applying spectral clustering to a large dataset is challenging due to computational overhead. While the distributed spectral clustering algorithm exists, they face the problem of data privacy and increased communication costs between the clients. Thus, in this paper, we propose a spectral clustering algorithm using federated learning (FL) to overcom…
University of Maryland（马里兰大学）；Wellesley College（韦尔斯利学院）；Carnegie Mellon University（卡内基梅隆大学）；Northwestern University（西北大学）
Clustering is a foundational problem in machine learning with numerous applications. As machine learning increases in ubiquity as a backend for automated systems, concerns about fairness arise. Much of the current literature on fairness deals with discrimination against protected classes in supervised learning (group fairness). We define a different notion of fair clustering wherein the probability that two points (or a community of points) become separated is bounded by an increasing function of their pairwise distance (or community diameter). We capture the situation where data points repres…
University of Sydney（悉尼大学）；TU Eindhoven（埃因霍温理工大学）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）
Due to the massively increasing amount of available geospatial data and the need to present it in an understandable way, clustering this data is more important than ever. As clusters might contain a large number of objects, having a representative for each cluster significantly facilitates understanding a clustering. Clustering methods relying on such representatives are called center-based. In this work we consider the problem of center-based clustering of trajectories. In this setting, the representative of a cluster is again a trajectory. To obtain a compact representation of the clusters a…
Federated Learning (FL) under distributed concept drift is a largely unexplored area. Although concept drift is itself a well-studied phenomenon, it poses particular challenges for FL, because drifts arise staggered in time and space (across clients). To the best of our knowledge, this work is the first to explicitly study data heterogeneity in both dimensions. We first demonstrate that prior solutions to drift adaptation that use a single global model are ill-suited to staggered drifts, necessitating multiple-model solutions. We identify the problem of drift adaptation as a time-varying clust…
The most widely used internal measure for clustering evaluation is the silhouette coefficient, whose naive computation requires a quadratic number of distance calculations, which is clearly unfeasible for massive datasets. Surprisingly, there are no known general methods to efficiently approximate the silhouette coefficient of a clustering with rigorously provable high accuracy. In this paper, we present the first scalable algorithm to compute such a rigorous approximation for the evaluation of clusterings based on any metric distances. Our algorithm hinges on a Probability Proportional to Siz…
Metric $k$-center clustering is a fundamental unsupervised learning primitive. Although widely used, this primitive is heavily affected by noise in the data, so that a more sensible variant seeks for the best solution that disregards a given number $z$ of points of the dataset, called outliers. We provide efficient algorithms for this important variant in the streaming model under the sliding window setting, where, at each time step, the dataset to be clustered is the window $W$ of the most recent data items. Our algorithms achieve $O(1)$ approximation and, remarkably, require a working memory…
Singapore University of Technology and Design（新加坡科技设计大学）
We study dynamic clustering problems from the perspective of online learning. We consider an online learning problem, called \textit{Dynamic $k$-Clustering}, in which $k$ centers are maintained in a metric space over time (centers may change positions) such as a dynamically changing set of $r$ clients is served in the best possible way. The connection cost at round $t$ is given by the \textit{$p$-norm} of the vector consisting of the distance of each client to its closest center at round $t$, for some $p\geq 1$ or $p = \infty$. We present a \textit{$Θ\left( \min(k,r) \right)$-regret} polynomia…
Approximation Algorithm for Constrained $k$-Center Clustering: A Local Search Approach
School of Accounting, Information Systems and Supply Chain, RMIT University（会计、信息系统与供应链学院，皇家墨尔本理工大学）；School of Mathematics and Statistics, Fuzhou University（数学与统计学院，福州大学）；School of Integrated Circuits, Guangdong University of Technology（集成电路学院，广东省技术大学）；School of Information Technology, Deakin University（信息技术学院，德金大学）；Western Sydney University（西澳大学）；CSIRO’s Data61 and Responsible AI Research (RAIR) Centre, Adelaide University（CSIRO的数据61和负责任的人工智能研究（RAIR）中心，阿德莱德大学）
We consider online $k$-means clustering where each new point is assigned to the nearest cluster center, after which the algorithm may update its centers. The loss incurred is the sum of squared distances from new points to their assigned cluster centers. The goal over a data stream $X$ is to achieve loss that is a constant factor of $L(X, OPT_k)$, the best possible loss using $k$ fixed points in hindsight. We propose a data parameter, $Λ(X)$, such that for any algorithm maintaining $O(k\text{poly}(\log n))$ centers at time $n$, there exists a data stream $X$ for which a loss of $Ω(Λ(X))$ is in…
Google Research（谷歌研究院）；New York University（纽约大学）；University of Michigan（密歇根大学）
The k-means objective is arguably the most widely-used cost function for modeling clustering tasks in a metric space. In practice and historically, k-means is thought of in a continuous setting, namely where the centers can be located anywhere in the metric space. For example, the popular Lloyd's heuristic locates a center at the mean of each cluster. Despite persistent efforts on understanding the approximability of k-means, and other classic clustering problems such as k-median and k-minsum, our knowledge of the hardness of approximation factors of these problems remains quite poor. In this …
Qualcomm Institute, University of California, San Diego（加州大学圣地亚哥分校高通研究所）；Ben Gurion University（本·古里安大学）
We study k-median clustering under the sequential no-substitution setting. In this setting, a data stream is sequentially observed, and some of the points are selected by the algorithm as cluster centers. However, a point can be selected as a center only immediately after it is observed, before observing the next point. In addition, a selected center cannot be substituted later. We give the first algorithm for this setting that obtains a constant approximation factor on the optimal risk under a random arrival order, an exponential improvement over previous work. This is also the first constant…
