Stability and Generalization of the Decentralized Stochastic Gradient Descent
College of Computer, National University of Defense Technology（国防科技大学计算机学院）；Scientific Computing & Imaging Institute, University of Utah（犹他大学科学计算与成像研究所）
The stability and generalization of stochastic gradient-based methods provide valuable insights into understanding the algorithmic performance of machine learning models. As the main workhorse for deep learning, stochastic gradient descent has received a considerable amount of studies. Nevertheless, the community paid little attention to its decentralized variants. In this paper, we provide a novel formulation of the decentralized stochastic gradient descent. Leveraging this formulation together with (non)convex optimization theory, we establish the first stability and generalization guarantees for the decentralized stochastic gradient descent. Our theoretical results are built on top of a few common and mild assumptions and reveal that the decentralization deteriorates the stability of SGD for the first time. We verify our theoretical findings by using a variety of decentralized settings and benchmark machine learning models.
In this work, we introduce an asynchronous decentralized accelerated stochastic gradient descent type of method for decentralized stochastic optimization, considering communication and synchronization are the major bottlenecks. We establish $\mathcal{O}(1/ε)$ (resp., $\mathcal{O}(1/\sqrtε)$) communication complexity and $\mathcal{O}(1/ε^2)$ (resp., $\mathcal{O}(1/ε)$) sampling complexity for solving general convex (resp., strongly convex) problems.
Decentralized stochastic gradient method emerges as a promising solution for solving large-scale machine learning problems. This paper studies the decentralized Markov chain gradient descent (DMGD) algorithm - a variant of the decentralized stochastic gradient methods where the random samples are taken along the trajectory of a Markov chain. This setting is well-motivated when obtaining independent samples is costly or impossible, which excludes the use of the traditional stochastic gradient algorithms. Specifically, we consider the first- and zeroth-order versions of decentralized Markov chai…
University of Rochester（罗切斯特大学）；ETH Zurich（苏黎世联邦理工学院）；University of California, Davis（加利福尼亚大学戴维斯分校）；IBM T. J. Watson Research Center（IBM托马斯·J·沃森研究中心）；Tencent AI lab（腾讯人工智能实验室）
Most distributed machine learning systems nowadays, including TensorFlow and CNTK, are built in a centralized fashion. One bottleneck of centralized algorithms lies on high communication cost on the central node. Motivated by this, we ask, can decentralized algorithms be faster than its centralized counterpart? Although decentralized PSGD (D-PSGD) algorithms have been studied by the control community, existing analysis and theory do not show any advantage over centralized PSGD (C-PSGD) algorithms, simply assuming the application scenario where only the decentralized network is available. In th…
University of California, Santa Barbara（加州大学圣巴巴拉分校）；University of Texas at Austin（得克萨斯大学奥斯汀分校）；University of Pennsylvania（宾夕法尼亚大学）
We consider the problem of decentralized consensus optimization, where the sum of $n$ smooth and strongly convex functions are minimized over $n$ distributed agents that form a connected network. In particular, we consider the case that the communicated local decision variables among nodes are quantized in order to alleviate the communication bottleneck in distributed optimization. We propose the Quantized Decentralized Gradient Descent (QDGD) algorithm, in which nodes update their local decision variables by combining the quantized information received from their neighbors with their local in…
Distributed descent-based methods are an essential toolset to solving optimization problems in multi-agent system scenarios. Here the agents seek to optimize a global objective function through mutual cooperation. Oftentimes, cooperation is achieved over a wireless communication network that is prone to delays and errors. There are many scenarios wherein the objective function is either non-differentiable or merely observable. In this paper, we present a cross-entropy based distributed stochastic approximation algorithm (SA) that finds a minimum of the objective, using only samples. We call th…
Temple University（天普大学）；University of Pittsburgh（匹兹堡大学）
Decentralized training has been actively studied in recent years. Although a wide variety of methods have been proposed, yet the decentralized momentum SGD method is still underexplored. In this paper, we propose a novel periodic decentralized momentum SGD method, which employs the momentum schema and periodic communication for decentralized training. With these two strategies, as well as the topology of the decentralized training system, the theoretical convergence analysis of our proposed method is difficult. We address this challenging problem and provide the condition under which our propo…
Xi’an Jiaotong University（西安交通大学）；Fudan University（复旦大学）；The Hong Kong Polytechnic University（香港理工大学）；The Hong Kong University of Science and Technology（香港科技大学）
Electronic Information School, Wuhan University（武汉大学）；Department of Electrical and Computer Engineering, North Carolina State University（北卡罗来纳州立大学）
While machine learning has achieved remarkable results in a wide variety of domains, the training of models often requires large datasets that may need to be collected from different individuals. As sensitive information may be contained in the individual's dataset, sharing training data may lead to severe privacy concerns. Therefore, there is a compelling need to develop privacy-aware machine learning methods, for which one effective approach is to leverage the generic framework of differential privacy. Considering that stochastic gradient descent (SGD) is one of the most commonly adopted met…
We study infinite-horizon discounted two-player zero-sum Markov games, and develop a decentralized algorithm that provably converges to the set of Nash equilibria under self-play. Our algorithm is based on running an Optimistic Gradient Descent Ascent algorithm on each state to learn the policies, with a critic that slowly learns the value of each state. To the best of our knowledge, this is the first algorithm in this setting that is simultaneously rational (converging to the opponent's best response when it uses a stationary policy), convergent (converging to the set of Nash equilibria under…
University of Rochester（罗切斯特大学）；IBM T. J. Watson Research Center（IBM T. J. 沃森研究中心）；ETH Zurich（苏黎世联邦理工学院）；Tencent AI Lab（腾讯人工智能实验室）
Most commonly used distributed machine learning systems are either synchronous or centralized asynchronous. Synchronous algorithms like AllReduce-SGD perform poorly in a heterogeneous environment, while asynchronous algorithms using a parameter server suffer from 1) communication bottleneck at parameter servers when workers are many, and 2) significantly worse convergence when the traffic to parameter server is congested. Can we design an algorithm that is robust in a heterogeneous environment, while being communication efficient and maintaining the best-possible convergence rate? In this pape…
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…
Guanghua School of Management, Peking University（北京大学光华管理学院）；Center for Applied Statistics, Renmin University of China（中国人民大学应用统计中心）；School of Statistics, Renmin University of China（中国人民大学统计学院）
We study a fully decentralized federated learning algorithm, which is a novel gradient descent algorithm executed on a communication-based network. For convenience, we refer to it as a network gradient descent (NGD) method. In the NGD method, only statistics (e.g., parameter estimates) need to be communicated, minimizing the risk of privacy. Meanwhile, different clients communicate with each other directly according to a carefully designed network structure without a central master. This greatly enhances the reliability of the entire algorithm. Those nice properties inspire us to carefully stu…
Fudan University（复旦大学）；Tsinghua University（清华大学）；Beijing National Research Center for Information Science and Technology（北京信息科学与技术国家研究中心）；The Chinese University of Hong Kong（香港中文大学）
In decentralized learning, a network of nodes cooperate to minimize an overall objective function that is usually the finite-sum of their local objectives, and incorporates a non-smooth regularization term for the better generalization ability. Decentralized stochastic proximal gradient (DSPG) method is commonly used to train this type of learning models, while the convergence rate is retarded by the variance of stochastic gradients. In this paper, we propose a novel algorithm, namely DPSVRG, to accelerate the decentralized training by leveraging the variance reduction technique. The basic ide…
Machine learning has begun to play a central role in many applications. A multitude of these applications typically also involve datasets that are distributed across multiple computing devices/machines due to either design constraints (e.g., multiagent systems) or computational/privacy reasons (e.g., learning on smartphone data). Such applications often require the learning tasks to be carried out in a decentralized fashion, in which there is no central server that is directly connected to all nodes. In real-world decentralized settings, nodes are prone to undetected failures due to malfunctio…
The University of Queensland（昆士兰大学）；Griffith University（格里菲斯大学）；Monash University（莫纳什大学）
Actuated by the growing attention to personal healthcare and the pandemic, the popularity of E-health is proliferating. Nowadays, enhancement on medical diagnosis via machine learning models has been highly effective in many aspects of e-health analytics. Nevertheless, in the classic cloud-based/centralized e-health paradigms, all the data will be centrally stored on the server to facilitate model training, which inevitably incurs privacy concerns and high time delay. Distributed solutions like Decentralized Stochastic Gradient Descent (D-SGD) are proposed to provide safe and timely diagnostic…
Department of Computer Science, University of Rochester（罗切斯特大学计算机科学系）；Department of Computational Mathematics, Science and Engineering, Michigan State University（密歇根州立大学计算数学、科学与工程系）；Department of Computer Science, ETH Zurich（苏黎世联邦理工学院计算机科学系）；Tencent AI Lab（腾讯AI实验室）
While training a machine learning model using multiple workers, each of which collects data from their own data sources, it would be most useful when the data collected from different workers can be {\em unique} and {\em different}. Ironically, recent analysis of decentralized parallel stochastic gradient descent (D-PSGD) relies on the assumption that the data hosted on different workers are {\em not too different}. In this paper, we ask the question: {\em Can we design a decentralized parallel stochastic gradient descent algorithm that is less sensitive to the data variance across workers?} I…
Chongqing University（重庆大学）；Nanyang Technological University（南洋理工大学）；University of Alberta（阿尔伯塔大学）
Decentralized learning has gained great popularity to improve learning efficiency and preserve data privacy. Each computing node makes equal contribution to collaboratively learn a Deep Learning model. The elimination of centralized Parameter Servers (PS) can effectively address many issues such as privacy, performance bottleneck and single-point-failure. However, how to achieve Byzantine Fault Tolerance in decentralized learning systems is rarely explored, although this problem has been extensively studied in centralized systems. In this paper, we present an in-depth study towards the Byzanti…
Tokyo University of Science（东京理科大学）；Tsuda University（津田塾大学）
This paper proposes a communication strategy for decentralized learning on wireless systems. Our discussion is based on the decentralized parallel stochastic gradient descent (D-PSGD), which is one of the state-of-the-art algorithms for decentralized learning. The main contribution of this paper is to raise a novel open question for decentralized learning on wireless systems: there is a possibility that the density of a network topology significantly influences the runtime performance of D-PSGD. In general, it is difficult to guarantee delay-free communications without any communication deteri…
Harvard University（哈佛大学）；Massachusetts Institute of Technology（麻省理工学院）
This work addresses decentralized online optimization in non-stationary environments. A network of agents aim to track the minimizer of a global time-varying convex function. The minimizer evolves according to a known dynamics corrupted by an unknown, unstructured noise. At each time, the global function can be cast as a sum of a finite number of local functions, each of which is assigned to one agent in the network. Moreover, the local functions become available to agents sequentially, and agents do not have a prior knowledge of the future cost functions. Therefore, agents must communicate wi…
As deep learning models are usually massive and complex, distributed learning is essential for increasing training efficiency. Moreover, in many real-world application scenarios like healthcare, distributed learning can also keep the data local and protect privacy. A popular distributed learning strategy is federated learning, where there is a central server storing the global model and a set of local computing nodes updating the model parameters with their corresponding data. The updated model parameters will be processed and transmitted to the central server, which leads to heavy communicati…
IBM Research（IBM研究院）；George Mason University（乔治梅森大学）；Duke Unversity（杜克大学）
Distributed Deep Learning (DDL) is essential for large-scale Deep Learning (DL) training. Synchronous Stochastic Gradient Descent (SSGD) 1 is the de facto DDL optimization method. Using a sufficiently large batch size is critical to achieving DDL runtime speedup. In a large batch setting, the learning rate must be increased to compensate for the reduced number of parameter updates. However, a large learning rate may harm convergence in SSGD and training could easily diverge. Recently, Decentralized Parallel SGD (DPSGD) has been proposed to improve distributed training speed. In this paper, we …
University of California, Santa Barbara（加州大学圣巴巴拉分校）；University of British Columbia（不列颠哥伦比亚大学）
Decentralized learning offers privacy and communication efficiency when data are naturally distributed among agents communicating over an underlying graph. Motivated by overparameterized learning settings, in which models are trained to zero training loss, we study algorithmic and generalization properties of decentralized learning with gradient descent on separable data. Specifically, for decentralized gradient descent (DGD) and a variety of loss functions that asymptote to zero at infinity (including exponential and logistic losses), we derive novel finite-time generalization bounds. This co…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very recently, a line of work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs. However, existing generalization error bounds are unable to explain the good generalization performance of over-parameterized DNNs. The major limitation of most existing …
Princeton University（普林斯顿大学）；Beijing Institute of Big Data Research（北京大数据研究院）
A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are updated. General initialization schemes as well as general regimes for the network width and training data size are considered. In the over-parametrized regime, it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the labels. In addition, it is proved that throughout the training process the functions represented by the neural network model are unif…
We introduce a framework for designing primal methods under the decentralized optimization setting where local functions are smooth and strongly convex. Our approach consists of approximately solving a sequence of sub-problems induced by the accelerated augmented Lagrangian method, thereby providing a systematic way for deriving several well-known decentralized algorithms including EXTRA arXiv:1404.6264 and SSDA arXiv:1702.08704. When coupled with accelerated gradient descent, our framework yields a novel primal algorithm whose convergence rate is optimal and matched by recently derived lower …
Decentralized stochastic optimization is the basic building block of modern collaborative machine learning, distributed estimation and control, and large-scale sensing. Since involved data usually contain sensitive information like user locations, healthcare records and financial transactions, privacy protection has become an increasingly pressing need in the implementation of decentralized stochastic optimization algorithms. In this paper, we propose a decentralized stochastic gradient descent algorithm which is embedded with inherent privacy protection for every participating agent against o…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We study the training and generalization of deep neural networks (DNNs) in the over-parameterized regime, where the network width (i.e., number of hidden nodes per layer) is much larger than the number of training data points. We show that, the expected $0$-$1$ loss of a wide enough ReLU network trained with stochastic gradient descent (SGD) and random initialization can be bounded by the training loss of a random feature model induced by the network gradient at initialization, which we call a neural tangent random feature (NTRF) model. For data distributions that can be classified by NTRF mod…
Running Stochastic Gradient Descent (SGD) in a decentralized fashion has shown promising results. In this paper we propose Moniqua, a technique that allows decentralized SGD to use quantized communication. We prove in theory that Moniqua communicates a provably bounded number of bits per iteration, while converging at the same asymptotic rate as the original algorithm does with full-precision communication. Moniqua improves upon prior works in that it (1) requires zero additional memory, (2) works with 1-bit quantization, and (3) is applicable to a variety of decentralized algorithms. We demon…
Distributed training algorithms of deep neural networks show impressive convergence speedup properties on very large problems. However, they inherently suffer from communication related slowdowns and communication topology becomes a crucial design choice. Common approaches supported by most machine learning frameworks are: 1) Synchronous decentralized algorithms relying on a peer-to-peer All Reduce topology that is sensitive to stragglers and communication delays. 2) Asynchronous centralised algorithms with a server based topology that is prone to communication bottleneck. Researchers also sug…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Rapid advances in data collection and processing capabilities have allowed for the use of increasingly complex models that give rise to nonconvex optimization problems. These formulations, however, can be arbitrarily difficult to solve in general, in the sense that even simply verifying that a given point is a local minimum can be NP-hard [1]. Still, some relatively simple algorithms have been shown to lead to surprisingly good empirical results in many contexts of interest. Perhaps the most prominent example is the success of the backpropagation algorithm for training neural networks. Several…
Data parallelism has become the de facto standard for training Deep Neural Network on multiple processing units. In this work we propose DC-S3GD, a decentralized (without Parameter Server) stale-synchronous version of the Delay-Compensated Asynchronous Stochastic Gradient Descent (DC-ASGD) algorithm. In our approach, we allow for the overlap of computation and communication, and compensate the inherent error with a first-order correction of the gradients. We prove the effectiveness of our approach by training Convolutional Neural Network with large batches and achieving state-of-the-art result…
Most modern learning problems are highly overparameterized, meaning that there are many more parameters than the number of training data points, and as a result, the training loss may have infinitely many global minima (parameter vectors that perfectly interpolate the training data). Therefore, it is important to understand which interpolating solutions we converge to, how they depend on the initialization point and the learning algorithm, and whether they lead to different generalization performances. In this paper, we study these questions for the family of stochastic mirror descent (SMD) al…
While momentum-based methods, in conjunction with stochastic gradient descent (SGD), are widely used when training machine learning models, there is little theoretical understanding on the generalization error of such methods. In this work, we first show that there exists a convex loss function for which algorithmic stability fails to establish generalization guarantees when SGD with standard heavy-ball momentum (SGDM) is run for multiple epochs. Then, for smooth Lipschitz loss functions, we analyze a modified momentum-based update rule, i.e., SGD with early momentum (SGDEM), and show that it …
In this paper, we propose to solve a regularized distributionally robust learning problem in the decentralized setting, taking into account the data distribution shift. By adding a Kullback-Liebler regularization function to the robust min-max optimization problem, the learning problem can be reduced to a modified robust minimization problem and solved efficiently. Leveraging the newly formulated optimization problem, we propose a robust version of Decentralized Stochastic Gradient Descent (DSGD), coined Distributionally Robust Decentralized Stochastic Gradient Descent (DR-DSGD). Under some mi…
Michigan State University（密歇根州立大学）；Princeton University（普林斯顿大学）
This paper proposes a novel proximal-gradient algorithm for a decentralized optimization problem with a composite objective containing smooth and non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with by gradient and proximal updates, respectively. The proposed algorithm is closely related to a previous algorithm, PG-EXTRA \cite{shi2015proximal}, but has a few advantages. First of all, agents use uncoordinated step-sizes, and the stable upper bounds on step-sizes are independent of network topologies. The step-sizes depend on local objective functions, and they can be as…
A major challenge in understanding the generalization of deep learning is to explain why (stochastic) gradient descent can exploit the network architecture to find solutions that have good generalization performance when using high capacity models. We find simple but realistic examples showing that this phenomenon exists even when learning linear classifiers --- between two linear networks with the same capacity, the one with a convolutional layer can generalize better than the other when the data distribution has some underlying spatial structure. We argue that this difference results from a …
Universitat Pompeu Fabra（庞培法布拉大学）；Element AI（Element AI公司）；Mila（米拉实验室）；University of Toronto（多伦多大学）；Vector Institute（矢量研究院）
We study the generalization properties of the popular stochastic optimization method known as stochastic gradient descent (SGD) for optimizing general non-convex loss functions. Our main contribution is providing upper bounds on the generalization error that depend on local statistics of the stochastic gradients evaluated along the path of iterates calculated by SGD. The key factors our bounds depend on are the variance of the gradients (with respect to the data distribution) and the local smoothness of the objective function along the SGD path, and the sensitivity of the loss function to pert…
Generalization Error Analysis for Attack-Free and Byzantine-Resilient Decentralized Learning with Data Heterogeneity
School of Computer Science and Engineering, Sun Yat-Sen University（中山大学计算机科学与工程学院）；College of Computer, National University of Defense Technology（国防科技大学计算机学院）
LAYUP: Asynchronous decentralized gradient descent with LAYer-wise UPdates
Center for Cognitive Interaction Technology, Universität Bielefeld, Germany（认知交互技术中心，比勒菲尔德大学，德国）；Bielefeld University（比勒菲尔德大学）；Department of Computer Science, Royal Holloway, University of London, United Kingdom（计算机科学系，伦敦大学皇家霍洛威学院，英国）
Decentralized learning enables edge users to collaboratively train models by exchanging information via device-to-device communication, yet prior works have been limited to wireless networks with fixed topologies and reliable workers. In this work, we propose an asynchronous decentralized stochastic gradient descent (DSGD) algorithm, which is robust to the inherent computation and communication failures occurring at the wireless network edge. We theoretically analyze its performance and establish a non-asymptotic convergence guarantee. Experimental results corroborate our analysis, demonstrati…
It has long been argued that minibatch stochastic gradient descent can generalize better than large batch gradient descent in deep neural networks. However recent papers have questioned this claim, arguing that this effect is simply a consequence of suboptimal hyperparameter tuning or insufficient compute budgets when the batch size is large. In this paper, we perform carefully designed experiments and rigorous hyperparameter sweeps on a range of popular models, which verify that small or moderately large batch sizes can substantially outperform very large batches on the test set. This occurs …
Duke University（杜克大学）；Monta Vista High School（蒙塔维斯塔高中）；University of Nevada, Reno（内华达大学里诺分校）
Distributed learning systems have enabled training large-scale models over large amount of data in significantly shorter time. In this paper, we focus on decentralized distributed deep learning systems and aim to achieve differential privacy with good convergence rate and low communication cost. To achieve this goal, we propose a new learning algorithm LEASGD (Leader-Follower Elastic Averaging Stochastic Gradient Descent), which is driven by a novel Leader-Follower topology and a differential privacy model.We provide a theoretical analysis of the convergence rate and the trade-off between the …
