Decentralized Sparse Linear Regression via Gradient-Tracking: Linear Convergence and Statistical Guarantees
Texas A&M University（德克萨斯农工大学）；Purdue University（普渡大学）；The Pennsylvania State University（宾夕法尼亚州立大学）；The University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Marie Maros, Gesualdo Scutari, Ying Sun, Guang Cheng
备注：
The order of the first three authors is alphabetic. Final revised version
We study sparse linear regression over a network of agents, modeled as an undirected graph and no server node. The estimation of the $s$-sparse parameter is formulated as a constrained LASSO problem wherein each agent owns a subset of the $N$ total observations. We analyze the convergence rate and statistical guarantees of a distributed projected gradient tracking-based algorithm under high-dimensional scaling, allowing the ambient dimension $d$ to grow with (and possibly exceed) the sample size $N$. Our theory shows that, under standard notions of restricted strong convexity and smoothness of the loss functions, suitable conditions on the network connectivity and algorithm tuning, the distributed algorithm converges globally at a {\it linear} rate to an estimate that is within the centralized {\it statistical precision} of the model, $O(s\log d/N)$. When $s\log d/N=o(1)$, a condition necessary for statistical consistency, an $\varepsilon$-optimal solution is attained after $\mathcal{O}(κ\log (1/\varepsilon))$ gradient computations and $O (κ/(1-ρ) \log (1/\varepsilon))$ communication rounds, where $κ$ is the restricted condition number of the loss function and $ρ$ measures the network connectivity. The computation cost matches that of the centralized projected gradient algorithm despite having data distributed; whereas the communication rounds reduce as the network connectivity improves. Overall, our study reveals interesting connections between statistical efficiency, network connectivity \& topology, and convergence rate in high dimensions.
Decentralized stochastic optimization has recently benefited from gradient tracking methods \cite{DSGT_Pu,DSGT_Xin} providing efficient solutions for large-scale empirical risk minimization problems. In Part I \cite{GT_SAGA} of this work, we develop \textbf{\texttt{GT-SAGA}} that is based on a decentralized implementation of SAGA \cite{SAGA} using gradient tracking and discuss regimes of practical interest where \textbf{\texttt{GT-SAGA}} outperforms existing decentralized approaches in terms of the total number of local gradient computations. In this paper, we describe \textbf{\texttt{GT-SVRG}…
University of Toronto（多伦多大学）；Westlake University（西湖大学）；EPFL（洛桑联邦理工学院）；CISPA Helmholtz Center for Information Security（亥姆霍兹信息安全中心）
Gradient tracking (GT) is an algorithm designed for solving decentralized optimization problems over a network (such as training a machine learning model). A key feature of GT is a tracking mechanism that allows to overcome data heterogeneity between nodes. We develop a novel decentralized tracking mechanism, $K$-GT, that enables communication-efficient local updates in GT while inheriting the data-independence property of GT. We prove a convergence rate for $K$-GT on smooth non-convex functions and prove that it reduces the communication overhead asymptotically by a linear factor $K$, where $…
University of Minnesota Twin Cities（明尼苏达大学双城分校）；IBM Research AI（IBM研究院AI）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Many modern large-scale machine learning problems benefit from decentralized and stochastic optimization. Recent works have shown that utilizing both decentralized computing and local stochastic gradient estimates can outperform state-of-the-art centralized algorithms, in applications involving highly non-convex problems, such as training deep neural networks. In this work, we propose a decentralized stochastic algorithm to deal with certain smooth non-convex problems where there are $m$ nodes in the system, and each node has a large number of samples (denoted as $n$). Differently from the maj…
A Variance-Reduced Stochastic Gradient Tracking Algorithm for Decentralized Optimization with Orthogonality Constraints
State Key Laboratory of Scientific and Engineering Computing, 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科学与工程计算国家重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）
This paper studies a decentralized stochastic gradient tracking (DSGT) algorithm for non-convex empirical risk minimization problems over a peer-to-peer network of nodes, which is in sharp contrast to the existing DSGT only for convex problems. To ensure exact convergence and handle the variance among decentralized datasets, each node performs a stochastic gradient (SG) tracking step by using a mini-batch of samples, where the batch size is designed to be proportional to the size of the local dataset. We explicitly evaluate the convergence rate of DSGT with respect to the number of iterations …
CISPA Helmholtz Center for Information Security（亥姆霍兹信息安全研究中心）；EPFL（洛桑联邦理工学院）
We consider decentralized machine learning over a network where the training data is distributed across $n$ agents, each of which can compute stochastic model updates on their local data. The agent's common goal is to find a model that minimizes the average of all local loss functions. While gradient tracking (GT) algorithms can overcome a key challenge, namely accounting for differences between workers' local data distributions, the known convergence rates for GT algorithms are not optimal with respect to their dependence on the mixing parameter $p$ (related to the spectral gap of the connect…
Shanghai Jiao Tong University（上海交通大学）；Iowa State University（爱荷华州立大学）
Decentralized sparsity learning has attracted a significant amount of attention recently due to its rapidly growing applications. To obtain the robust and sparse estimators, a natural idea is to adopt the non-smooth median loss combined with a $\ell_1$ sparsity regularizer. However, most of the existing methods suffer from slow convergence performance caused by the {\em double} non-smooth objective. To accelerate the computation, in this paper, we proposed a decentralized surrogate median regression (deSMR) method for efficiently solving the decentralized sparsity learning problem. We show tha…
Compressed Gradient Tracking for Decentralized Optimization Over General Directed Networks
Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；Shenzhen Research Institute of Big Data（深圳大数据研究院）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）；Department of Mathematics, Michigan State University（密歇根州立大学数学系）
Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking
National Special Education Resource Center for Children with Autism, Zhejiang Normal University（浙江师范大学生自闭症教育资源中心）；School of Computer Science and Technology, Zhejiang Normal University（浙江师范大学计算机科学与技术学院）；China Mobile Research Institute（中国移动研究院）；Institute of Cyber-Systems and Control, Zhejiang University（浙江大学控制系统研究所）；SGIT AI Lab, State Grid Corporation of China（国网SGIT人工智能实验室）；Zhejiang Key Laboratory of Intelligent Education Technology and Application, Zhejiang Normal University（浙江智能教育技术与应用重点实验室）；School of Information Engineering, Huzhou University（湖州大学信息工程学院）；Centre for Frontier Artificial Intelligence Research, A*STAR（A*STAR前沿人工智能研究中心）
Sequential estimation of a vector of linear regression coefficients is considered under both centralized and decentralized setups. In sequential estimation, the number of observations used for estimation is determined by the observed samples, hence is random, as opposed to fixed-sample-size estimation. Specifically, after receiving a new sample, if a target accuracy level is reached, we stop and estimate using the samples collected so far; otherwise we continue to receive another sample. It is known that finding an optimum sequential estimator, which minimizes the average sample number for a g…
We present a framework for supervised subspace tracking, when there are two time series $x_t$ and $y_t$, one being the high-dimensional predictors and the other being the response variables and the subspace tracking needs to take into consideration of both sequences. It extends the classic online subspace tracking work which can be viewed as tracking of $x_t$ only. Our online sufficient dimensionality reduction (OSDR) is a meta-algorithm that can be applied to various cases including linear regression, logistic regression, multiple linear regression, multinomial logistic regression, support ve…
SPARKLE: A Unified Single-Loop Primal-Dual Framework for Decentralized Bilevel Optimization
Peking University（北京大学）；IBM Research（IBM研究院）；University of Pennsylvania（宾夕法尼亚大学）；National Engineering Labratory for Big Data Analytics and Applications（国家大数据分析与应用工程实验室）；AI for Science Institute（科学智能研究所）
Accelerated Gradient Tracking over Time-varying Graphs for Decentralized Optimization
Institute of Robotics and Automatic Information Systems, College of Artificial Intelligence, Nankai University（南开大学人工智能学院机器人与自动信息系统研究所）；State Key Lab of General AI, School of Intelligence Science and Technology, Peking University（北京大学智能科学与技术学院通用人工智能省部共建国家重点实验室）
U.S. Army Research Laboratory（美国陆军研究实验室）；University of Pennsylvania（宾夕法尼亚大学）；University of Nice Sophia-Antipolis（尼斯-索菲亚·安提波利斯大学）
We consider multi-agent stochastic optimization problems over reproducing kernel Hilbert spaces (RKHS). In this setting, a network of interconnected agents aims to learn decision functions, i.e., nonlinear statistical models, that are optimal in terms of a global convex functional that aggregates data across the network, with only access to locally and sequentially observed samples. We propose solving this problem by allowing each agent to learn a local regression function while enforcing consensus constraints. We use a penalized variant of functional stochastic gradient descent operating simu…
Shenzhen Research Institute of Big Data（深圳大数据研究院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Hong Kong University of Science and Technology（香港科技大学）
Due to the rapid growth of smart agents such as weakly connected computational nodes and sensors, developing decentralized algorithms that can perform computations on local agents becomes a major research direction. This paper considers the problem of decentralized Principal components analysis (PCA), which is a statistical method widely used for data analysis. We introduce a technique called subspace tracking to reduce the communication cost, and apply it to power iterations. This leads to a decentralized PCA algorithm called \texttt{DeEPCA}, which has a convergence rate similar to that of th…
Distributed Optimization with Streaming Data: A Temporal Weighting Perspective
School of Electrical, Computer and Energy Engineering, Arizona State University（亚利桑那州立大学电气、计算机与能源工程学院）；Department of Electrical Engineering (ISY), Linköping University（林雪平大学电气工程系）
Texas A&M University（德克萨斯农工大学）；University of Minnesota（明尼苏达大学）
We consider a distributed non-convex optimization where a network of agents aims at minimizing a global function over the Stiefel manifold. The global function is represented as a finite sum of smooth local functions, where each local function is associated with one agent and agents communicate with each other over an undirected connected graph. The problem is non-convex as local functions are possibly non-convex (but smooth) and the Steifel manifold is a non-convex set. We present a decentralized Riemannian stochastic gradient method (DRSGD) with the convergence rate of $\mathcal{O}(1/\sqrt{K…
Decentralized solutions to finite-sum minimization are of significant importance in many signal processing, control, and machine learning applications. In such settings, the data is distributed over a network of arbitrarily-connected nodes and raw data sharing is prohibitive often due to communication or privacy constraints. In this article, we review decentralized stochastic first-order optimization methods and illustrate some recent improvements based on gradient tracking and variance reduction, focusing particularly on smooth and strongly-convex objective functions. We provide intuitive ill…
Decentralized methods to solve finite-sum minimization problems are important in many signal processing and machine learning tasks where the data is distributed over a network of nodes and raw data sharing is not permitted due to privacy and/or resource constraints. In this article, we review decentralized stochastic first-order methods and provide a unified algorithmic framework that combines variance-reduction with gradient tracking to achieve both robust performance and fast convergence. We provide explicit theoretical guarantees of the corresponding methods when the objective functions are…
This paper presents a differentially private algorithm for linear regression learning in a decentralized fashion. Under this algorithm, privacy budget is theoretically derived, in addition to that the solution error is shown to be bounded by $O(t)$ for $O(1/t)$ descent step size and $O(\exp(t^{1-e}))$ for $O(t^{-e})$ descent step size.
University of Cambridge（剑桥大学）；University of Pennsylvania（宾夕法尼亚大学）；University of Maryland（马里兰大学）
The problem of decentralized multi-robot target tracking asks for jointly selecting actions, e.g., motion primitives, for the robots to maximize target tracking performance with local communications. One major challenge for practical implementations is to make target tracking approaches scalable for large-scale problem instances. In this work, we propose a general-purpose learning architecture toward collaborative target tracking at scale, with decentralized communications. Particularly, our learning architecture leverages a graph neural network (GNN) to capture local interactions of the robot…
Decentralized Nonconvex Composite Federated Learning with Gradient Tracking and Momentum
School of Cyber Science & Engineering, Southeast University（东南大学网络空间科学与工程学院）；Institute of Artificial Intelligence (TeleAI) of China Telecom（中国电信人工智能研究院）；School of Mathematics, Southeast University（东南大学数学学院）；Purple Mountain Laboratories（紫金山实验室）
Sporadic Gradient Tracking over Directed Graphs: A Theoretical Perspective on Decentralized Federated Learning
School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA（电气与计算机工程学院，普渡大学，西拉法基，印第安纳州，美国）；Department of Computer Science and Engineering, Yonsei University, Seoul, South Korea（计算机科学与工程系，延世大学，首尔，韩国）；Department of Electrical Engineering, University at Buffalo -- SUNY, Buffalo, NY, USA（电气工程系，布法罗大学——SUNY，布法罗，纽约州，美国）
Johnson Controls Inc.（江森自控有限公司）；Iowa State University（爱荷华州立大学）；New York University（纽约大学）
We propose a novel policy gradient method for multi-agent reinforcement learning, which leverages two different variance-reduction techniques and does not require large batches over iterations. Specifically, we propose a momentum-based decentralized policy gradient tracking (MDPGT) where a new momentum-based variance reduction technique is used to approximate the local policy gradient surrogate with importance sampling, and an intermediate parameter is adopted to track two consecutive policy gradient surrogates. Moreover, MDPGT provably achieves the best available sample complexity of $\mathca…
Decentralized Federated Learning with Gradient Tracking over Time-Varying Directed Networks
School of Electrical, Computer and Energy Engineering, Arizona State University（亚利桑那州立大学电气、计算机与能源工程学院）；Department of Electrical and Computer Engineering, Princeton University（普林斯顿大学电气与计算机工程系）
In this paper, we present Neural-Swarm, a nonlinear decentralized stable controller for close-proximity flight of multirotor swarms. Close-proximity control is challenging due to the complex aerodynamic interaction effects between multirotors, such as downwash from higher vehicles to lower ones. Conventional methods often fail to properly capture these interaction effects, resulting in controllers that must maintain large safety distances between vehicles, and thus are not capable of close-proximity flight. Our approach combines a nominal dynamics model with a regularized permutation-invariant…
Rutgers Business School（罗格斯大学商学院）；The Chinese University of Hong Kong（香港中文大学）；Florida State University（佛罗里达州立大学）
Stochastic gradient Langevin dynamics (SGLD) and stochastic gradient Hamiltonian Monte Carlo (SGHMC) are two popular Markov Chain Monte Carlo (MCMC) algorithms for Bayesian inference that can scale to large datasets, allowing to sample from the posterior distribution of the parameters of a statistical model given the input data and the prior distribution over the model parameters. However, these algorithms do not apply to the decentralized learning setting, when a network of agents are working collaboratively to learn the parameters of a statistical model without sharing their individual data …
Visual tracking is fundamentally the problem of regressing the state of the target in each video frame. While significant progress has been achieved, trackers are still prone to failures and inaccuracies. It is therefore crucial to represent the uncertainty in the target estimation. Although current prominent paradigms rely on estimating a state-dependent confidence score, this value lacks a clear probabilistic interpretation, complicating its use. In this work, we therefore propose a probabilistic regression formulation and apply it to tracking. Our network predicts the conditional probabilit…
We study the decentralized optimization problem where a network of $n$ agents seeks to minimize the average of a set of heterogeneous non-convex cost functions distributedly. State-of-the-art decentralized algorithms like Exact Diffusion~(ED) and Gradient Tracking~(GT) involve communicating every iteration. However, communication is expensive, resource intensive, and slow. In this work, we analyze a locally updated GT method (LU-GT), where agents perform local recursions before interacting with their neighbors. While local updates have been shown to reduce communication overhead in practice, t…
There is growing interest in large-scale machine learning and optimization over decentralized networks, e.g. in the context of multi-agent learning and federated learning. Due to the imminent need to alleviate the communication burden, the investigation of communication-efficient distributed optimization algorithms - particularly for empirical risk minimization - has flourished in recent years. A large fraction of these algorithms have been developed for the master/slave setting, relying on a central parameter server that can communicate with all agents. This paper focuses on distributed optim…
Decentralized machine learning is a promising emerging paradigm in view of global challenges of data ownership and privacy. We consider learning of linear classification and regression models, in the setting where the training data is decentralized over many user devices, and the learning algorithm must run on-device, on an arbitrary communication network, without a central coordinator. We propose COLA, a new decentralized training algorithm with strong theoretical guarantees and superior practical performance. Our framework overcomes many limitations of existing methods, and achieves communic…
Regression-based Inverter Control for Decentralized Optimal Power Flow and Voltage Regulation
2 Department of Electrical Engineering \& Computer Sciences, UC Berkeley, Berkeley, USA；3 Department of Mechanical Engineering, UC Berkeley, Berkeley, USA；4 Delft Center for Systems；Control, Delft University of Technology, Delft, The Netherlands
Laboratoire Jean-Perrin UMR 8237, Sorbonne Université（索邦大学让-佩兰实验室）；SABILAb；CMAP, Ecole Polytechnique, Université Paris-Saclay（巴黎萨克雷大学综合理工学院应用数学中心）；Heuritech
The mother machine is a popular microfluidic device that allows long-term time-lapse imaging of thousands of cells in parallel by microscopy. It has become a valuable tool for single-cell level quantitative analysis and characterization of many cellular processes such as gene expression and regulation, mutagenesis or response to antibiotics. The automated and quantitative analysis of the massive amount of data generated by such experiments is now the limiting step. In particular the segmentation and tracking of bacteria cells imaged in phase-contrast microscopy---with error rates compatible wi…
Decentralized Online Regularized Learning Over Random Time-Varying Graphs
East China Normal University（华东师范大学）；No.2 High School of East China Normal University（华东师范大学第二附属中学）；Institute of Systems Science, 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系统科学研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Ningbo University（宁波大学）
Nanjing University of Science and Technology（南京理工大学）
Vehicle tracking has become one of the key applications of wireless sensor networks (WSNs) in the fields of rescue, surveillance, traffic monitoring, etc. However, the increased tracking accuracy requires more energy consumption. In this letter, a decentralized vehicle tracking strategy is conceived for improving both tracking accuracy and energy saving, which is based on adjusting the intersection area between the fixed sensing area and the dynamic activation area. Then, two deep reinforcement learning (DRL) aided solutions are proposed relying on the dynamic selection of the activation area …
Distributed algorithms have been playing an increasingly important role in many applications such as machine learning, signal processing, and control. Significant research efforts have been devoted to developing and analyzing new algorithms for various applications. In this work, we provide a fresh perspective to understand, analyze, and design distributed optimization algorithms. Through the lens of multi-rate feedback control, we show that a wide class of distributed algorithms, including popular decentralized/federated schemes, can be viewed as discretizing a certain continuous-time feedbac…
