We study distributed computing of the truncated singular value decomposition problem. We develop an algorithm that we call \texttt{LocalPower} for improving communication efficiency. Specifically, we uniformly partition the dataset among $m$ nodes and alternate between multiple (precisely $p$) local power iterations and one global aggregation. In the aggregation, we propose to weight each local eigenvector matrix with orthogonal Procrustes transformation (OPT). As a practical surrogate of OPT, sign-fixing, which uses a diagonal matrix with $\pm 1$ entries as weights, has better computation complexity and stability in experiments. We theoretically show that under certain assumptions \texttt{LocalPower} lowers the required number of communications by a factor of $p$ to reach a constant accuracy. We also show that the strategy of periodically decaying $p$ helps obtain high-precision solutions. We conduct experiments to demonstrate the effectiveness of \texttt{LocalPower}.
Singular Value Decomposition (SVD) is a well studied research topic in many fields and applications from data mining to image processing. Data arising from these applications can be represented as a matrix where it is large and sparse. Most existing algorithms are used to calculate singular values, left and right singular vectors of a large-dense matrix but not large and sparse matrix. Even if they can find SVD of a large matrix, calculation of large-dense matrix has high time complexity due to sequential algorithms. Distributed approaches are proposed for computing SVD of large matrices. Howe…
Technion - Israel Institute of Technology（以色列理工学院）
Convex optimization over the spectrahedron, i.e., the set of all real $n\times n$ positive semidefinite matrices with unit trace, has important applications in machine learning, signal processing and statistics, mainly as a convex relaxation for optimization problems with low-rank matrices. It is also one of the most prominent examples in the theory of first-order methods for convex optimization in which non-Euclidean methods can be significantly preferable to their Euclidean counterparts. In particular, the desirable choice is the Matrix Exponentiated Gradient (MEG) method which is based on t…
University of Hyderabad（海得拉巴大学）；Central University of Rajasthan（拉贾斯坦中央大学）
With the abundance of data in recent years, interesting challenges are posed in the area of recommender systems. Producing high quality recommendations with scalability and performance is the need of the hour. Singular Value Decomposition(SVD) based recommendation algorithms have been leveraged to produce better results. In this paper, we extend the SVD technique further for scalability and performance in the context of 1) multi-threading 2) multiple computational units (with the use of Graphical Processing Units) and 3) distributed computation. We propose block based matrix factorization (BMF…
LocalKMeans: Convergence of Lloyd's Algorithm with Distributed Local Iterations
Computer Science and Engineering, University of California, San Diego（计算机科学与工程系，加州大学圣地亚哥分校）；Electrical and Computer Engineering, University of California, San Diego（电气与计算机工程系，加州大学圣地亚哥分校）；Computer Science and Engineering, Indian Institute of Technology, Bombay（计算机科学与工程系，印度理工学院孟买分校）；Halıcıoğlu Data Science Institute, University of California, San Diego（Halıcıoğlu数据科学研究所，加州大学圣地亚哥分校）
Shanghai Jiao Tong University（上海交通大学）；University of Southampton（南安普顿大学）
Sparse canonical correlation analysis (CCA) is a useful statistical tool to detect latent information with sparse structures. However, sparse CCA works only for two datasets, i.e., there are only two views or two distinct objects. To overcome this limitation, in this paper, we propose a sparse generalized canonical correlation analysis (GCCA), which could detect the latent relations of multiview data with sparse structures. Moreover, the introduced sparsity could be considered as Laplace prior on the canonical variates. Specifically, we convert the GCCA into a linear system of equations and im…
Purdue Univ.（普渡大学）；Univ. of Queensland（昆士兰大学）；ICSI（国际计算机科学研究所）；Univ. of California at Berkeley（加州大学伯克利分校）
First-order optimization methods, such as stochastic gradient descent (SGD) and its variants, are widely used in machine learning applications due to their simplicity and low per-iteration costs. However, they often require larger numbers of iterations, with associated communication costs in distributed environments. In contrast, Newton-type methods, while having higher per-iteration costs, typically require a significantly smaller number of iterations, which directly translates to reduced communication costs. In this paper, we present a novel distributed optimizer for classification problems,…
NS-RGS: Newton-Schulz based Riemannian gradient method for orthogonal group synchronization
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong（香港中文大学系统工程与工程管理学系）
In this paper, we present a scalable distributed implementation of the Sampled Limited-memory Symmetric Rank-1 (S-LSR1) algorithm. First, we show that a naive distributed implementation of S-LSR1 requires multiple rounds of expensive communications at every iteration and thus is inefficient. We then propose DS-LSR1, a communication-efficient variant that: (i) drastically reduces the amount of data communicated at every iteration, (ii) has favorable work-load balancing across nodes, and (iii) is matrix-free and inverse-free. The proposed method scales well in terms of both the dimension of the …
To address the communication bottleneck problem in distributed optimization within a master-worker framework, we propose LocalNewton, a distributed second-order algorithm with local averaging. In LocalNewton, the worker machines update their model in every iteration by finding a suitable second-order descent direction using only the data and model stored in their own local memory. We let the workers run multiple such iterations locally and communicate the models to the master node only once every few (say L) iterations. LocalNewton is highly practical since it requires only one hyperparameter,…
Imperial College London（伦敦帝国学院）；University of Maryland, College Park（马里兰大学帕克分校）
Distributed gradient descent (DGD) is an efficient way of implementing gradient descent (GD), especially for large data sets, by dividing the computation tasks into smaller subtasks and assigning to different computing servers (CSs) to be executed in parallel. In standard parallel execution, per-iteration waiting time is limited by the execution time of the straggling servers. Coded DGD techniques have been introduced recently, which can tolerate straggling servers via assigning redundant computation tasks to the CSs. In most of the existing DGD schemes, either with coded computation or coded …
