Accelerating Distributed SGD for Linear Regression using Iterative Pre-Conditioning
University of Maryland, College Park（马里兰大学帕克分校）；Georgetown University（乔治城大学）
Kushal Chakrabarti, Nirupam Gupta, Nikhil Chopra
备注：
Changes in the replacement: Application to distributed state estimation problem has been added in Appendix B. Related articles: arXiv:2003.07180v2 [math.OC] and arXiv:2008.02856v1 [math.OC]
This paper considers the multi-agent distributed linear least-squares problem. The system comprises multiple agents, each agent with a locally observed set of data points, and a common server with whom the agents can interact. The agents' goal is to compute a linear model that best fits the collective data points observed by all the agents. In the server-based distributed settings, the server cannot access the data points held by the agents. The recently proposed Iteratively Pre-conditioned Gradient-descent (IPG) method has been shown to converge faster than other existing distributed algorithms that solve this problem. In the IPG algorithm, the server and the agents perform numerous iterative computations. Each of these iterations relies on the entire batch of data points observed by the agents for updating the current estimate of the solution. Here, we extend the idea of iterative pre-conditioning to the stochastic settings, where the server updates the estimate and the iterative pre-conditioning matrix based on a single randomly selected data point at every iteration. We show that our proposed Iteratively Pre-conditioned Stochastic Gradient-descent (IPSG) method converges linearly in expectation to a proximity of the solution. Importantly, we empirically show that the proposed IPSG method's convergence rate compares favorably to prominent stochastic algorithms for solving the linear least-squares problem in server-based networks.
Stochastic gradient descent (SGD) is a pillar of modern machine learning, serving as the go-to optimization algorithm for a diverse array of problems. While the empirical success of SGD is often attributed to its computational efficiency and favorable generalization behavior, neither effect is well understood and disentangling them remains an open problem. Even in the simple setting of convex quadratic problems, worst-case analyses give an asymptotic convergence rate for SGD that is no better than full-batch gradient descent (GD), and the purported implicit regularization effects of SGD lack a…
Tsinghua University（清华大学）；Chinese Academy of Sciences（中国科学院）
Machine learning, especially deep neural networks, has been rapidly developed in fields including computer vision, speech recognition and reinforcement learning. Although Mini-batch SGD is one of the most popular stochastic optimization methods in training deep networks, it shows a slow convergence rate due to the large noise in gradient approximation. In this paper, we attempt to remedy this problem by building more efficient batch selection method based on typicality sampling, which reduces the error of gradient estimation in conventional Minibatch SGD. We analyze the convergence rate of the…
Nesterov SGD is widely used for training modern neural networks and other machine learning models. Yet, its advantages over SGD have not been theoretically clarified. Indeed, as we show in our paper, both theoretically and empirically, Nesterov SGD with any parameter selection does not in general provide acceleration over ordinary SGD. Furthermore, Nesterov SGD may diverge for step sizes that ensure convergence of ordinary SGD. This is in contrast to the classical results in the deterministic scenario, where the same step size ensures accelerated convergence of the Nesterov's method over optim…
University of Illinois at Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校）
The matrix completion problem seeks to recover a $d\times d$ ground truth matrix of low rank $r\ll d$ from observations of its individual elements. Real-world matrix completion is often a huge-scale optimization problem, with $d$ so large that even the simplest full-dimension vector operations with $O(d)$ time complexity become prohibitively expensive. Stochastic gradient descent (SGD) is one of the few algorithms capable of solving matrix completion on a huge scale, and can also naturally handle streaming data over an evolving ground truth. Unfortunately, SGD experiences a dramatic slow-down …
College of Computing（计算学院）；Georgia Institute of Technology（佐治亚理工学院）
In this work we consider the stochastic minimization of nonsmooth convex loss functions, a central problem in machine learning. We propose a novel algorithm called Accelerated Nonsmooth Stochastic Gradient Descent (ANSGD), which exploits the structure of common nonsmooth loss functions to achieve optimal convergence rates for a class of problems including SVMs. It is the first stochastic algorithm that can achieve the optimal O(1/t) rate for minimizing nonsmooth loss functions (with strong convexity). The fast rates are confirmed by empirical comparisons, in which ANSGD significantly outperfor…
We introduce a novel method to compute a rank $m$ approximation of the inverse of the Hessian matrix in the distributed regime. By leveraging the differences in gradients and parameters of multiple Workers, we are able to efficiently implement a distributed approximation of the Newton-Raphson method. We also present preliminary results which underline advantages and challenges of second-order methods for large stochastic optimization problems. In particular, our work suggests that novel strategies for combining gradients provide further information on the loss surface.
Communication overhead hinders the scalability of large-scale distributed training. Gossip SGD, where each node averages only with its neighbors, is more communication-efficient than the prevalent parallel SGD. However, its convergence rate is reversely proportional to quantity $1-β$ which measures the network connectivity. On large and sparse networks where $1-β\to 0$, Gossip SGD requires more iterations to converge, which offsets against its communication benefit. This paper introduces Gossip-PGA, which adds Periodic Global Averaging into Gossip SGD. Its transient stage, i.e., the iterations…
Modern proximal and stochastic gradient descent (SGD) methods are believed to efficiently minimize large composite objective functions, but such methods have two algorithmic challenges: (1) a lack of fast or justified stop conditions, and (2) sensitivity to the objective function's conditioning. In response to the first challenge, modern proximal and SGD methods guarantee convergence only after multiple epochs, but such a guarantee renders proximal and SGD methods infeasible when the number of component functions is very large or infinite. In response to the second challenge, second order SGD …
Stochastic Gradient Descent (SGD) is a popular optimization method which has been applied to many important machine learning tasks such as Support Vector Machines and Deep Neural Networks. In order to parallelize SGD, minibatch training is often employed. The standard approach is to uniformly sample a minibatch at each step, which often leads to high variance. In this paper we propose a stratified sampling strategy, which divides the whole dataset into clusters with low within-cluster variance; we then take examples from these clusters using a stratified sampling technique. It is shown that th…
Design Criteria for SGD Preconditioners: Local Conditioning, Noise Floors, and Basin Stability
Department of Mathematics, Emory University（埃默里大学数学系）；Department of Mathematics, University of Minnesota Twin Cities（明尼苏达大学双城分校数学系）；Department of Computer Science, University of Minnesota Twin Cities（明尼苏达大学双城分校计算机科学系）；Department of Mathematics, University of Kentucky（肯塔基大学数学系）
SOTA decentralized SGD algorithms can overcome the bandwidth bottleneck at the parameter server by using communication collectives like Ring All-Reduce for synchronization. While the parameter updates in distributed SGD may happen asynchronously there is still a synchronization barrier to make sure that the local training epoch at every learner is complete before the learners can advance to the next epoch. The delays in waiting for the slowest learners(stragglers) remain to be a problem in the synchronization steps of these state-of-the-art decentralized frameworks. In this paper, we propose t…
Stochastic Gradient Descent (SGD) is widely used in machine learning problems to efficiently perform empirical risk minimization, yet, in practice, SGD is known to stall before reaching the actual minimizer of the empirical risk. SGD stalling has often been attributed to its sensitivity to the conditioning of the problem; however, as we demonstrate, SGD will stall even when applied to a simple linear regression problem with unity condition number for standard learning rates. Thus, in this work, we numerically demonstrate and mathematically argue that stalling is a crippling and generic limitat…
We consider stochastic approximation for the least squares regression problem in the non-strongly convex setting. We present the first practical algorithm that achieves the optimal prediction error rates in terms of dependence on the noise of the problem, as $O(d/t)$ while accelerating the forgetting of the initial conditions to $O(d/t^2)$. Our new algorithm is based on a simple modification of the accelerated gradient descent. We provide convergence results for both the averaged and the last iterate of the algorithm. In order to describe the tightness of these new bounds, we present a matchin…
University of Maryland（马里兰大学）；École polytechnique fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
This paper studies a distributed multi-agent convex optimization problem. The system comprises multiple agents in this problem, each with a set of local data points and an associated local cost function. The agents are connected to a server, and there is no inter-agent communication. The agents' goal is to learn a parameter vector that optimizes the aggregate of their local costs without revealing their local data points. In principle, the agents can solve this problem by collaborating with the server using the traditional distributed gradient-descent method. However, when the aggregate cost i…
We analyze the dynamics of large batch stochastic gradient descent with momentum (SGD+M) on the least squares problem when both the number of samples and dimensions are large. In this setting, we show that the dynamics of SGD+M converge to a deterministic discrete Volterra equation as dimension increases, which we analyze. We identify a stability measurement, the implicit conditioning ratio (ICR), which regulates the ability of SGD+M to accelerate the algorithm. When the batch size exceeds this ICR, SGD+M converges linearly at a rate of $\mathcal{O}(1/\sqrtκ)$, matching optimal full-batch mome…
While stochastic gradient descent (SGD) and variants have been surprisingly successful for training deep nets, several aspects of the optimization dynamics and generalization are still not well understood. In this paper, we present new empirical observations and theoretical results on both the optimization dynamics and generalization behavior of SGD for deep nets based on the Hessian of the training loss and associated quantities. We consider three specific research questions: (1) what is the relationship between the Hessian of the loss and the second moment of stochastic gradients (SGs)? (2) …
Georgia Institute of Technology（佐治亚理工学院）；NEC Laboratories, USA（NEC实验室）
SGD is the widely adopted method to train CNN. Conceptually it approximates the population with a randomly sampled batch; then it evenly trains batches by conducting a gradient update on every batch in an epoch. In this paper, we demonstrate Sampling Bias, Intrinsic Image Difference and Fixed Cycle Pseudo Random Sampling differentiate batches in training, which then affect learning speeds on them. Because of this, the unbiased treatment of batches involved in SGD creates improper load balancing. To address this issue, we present Inconsistent Stochastic Gradient Descent (ISGD) to dynamically va…
Accelerating the inference of a trained DNN is a well studied subject. In this paper we switch the focus to the training of DNNs. The training phase is compute intensive, demands complicated data communication, and contains multiple levels of data dependencies and parallelism. This paper presents an algorithm/architecture space exploration of efficient accelerators to achieve better network convergence rates and higher energy efficiency for training DNNs. We further demonstrate that an architecture with hierarchical support for collective communication semantics provides flexibility in trainin…
IBM T. J. Watson Research Center（IBM托马斯·J·沃森研究中心）；University of Rochester（罗切斯特大学）
Deep neural networks have been shown to achieve state-of-the-art performance in several machine learning tasks. Stochastic Gradient Descent (SGD) is the preferred optimization algorithm for training these networks and asynchronous SGD (ASGD) has been widely adopted for accelerating the training of large-scale deep networks in a distributed computing environment. However, in practice it is quite challenging to tune the training hyperparameters (such as learning rate) when using ASGD so as achieve convergence and linear speedup, since the stability of the optimization algorithm is strongly influ…
New York University（纽约大学）；Jagiellonian University（雅盖隆大学）；Stanford University（斯坦福大学）；Salesforce Research（Salesforce研究院）；Facebook AI Research（脸书人工智能研究院）
The early phase of training of deep neural networks is critical for their final performance. In this work, we study how the hyperparameters of stochastic gradient descent (SGD) used in the early phase of training affect the rest of the optimization trajectory. We argue for the existence of the "break-even" point on this trajectory, beyond which the curvature of the loss surface and noise in the gradient are implicitly regularized by SGD. In particular, we demonstrate on multiple classification tasks that using a large learning rate in the initial phase of training reduces the variance of the g…
National University of Defense Technology（国防科技大学）；Tsinghua University（清华大学）；University of Utah（犹他大学）
Heavy ball momentum is crucial in accelerating (stochastic) gradient-based optimization algorithms for machine learning. Existing heavy ball momentum is usually weighted by a uniform hyperparameter, which relies on excessive tuning. Moreover, the calibrated fixed hyperparameter may not lead to optimal performance. In this paper, to eliminate the effort for tuning the momentum-related hyperparameter, we propose a new adaptive momentum inspired by the optimal choice of the heavy ball momentum for quadratic optimization. Our proposed adaptive heavy ball momentum can improve stochastic gradient de…
Asymptotic Analysis of Conditioned Stochastic Gradient Descent
École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；ENSAI（法国国立统计与信息分析学院）；École Nationale de la Statistique et de l’Analyse de l’Information（法国国立统计与信息分析学院）
Reinforcement Learning (RL) algorithms allow artificial agents to improve their action selections so as to increase rewarding experiences in their environments. Deep Reinforcement Learning algorithms require solving a nonconvex and nonlinear unconstrained optimization problem. Methods for solving the optimization problems in deep RL are restricted to the class of first-order algorithms, such as stochastic gradient descent (SGD). The major drawback of the SGD methods is that they have the undesirable effect of not escaping saddle points and their performance can be seriously obstructed by ill-c…
The University of Tokyo（东京大学）；RIKEN Center for AIP（理化学研究所人工智能研究中心）；Microsoft Research Asia（微软亚洲研究院）
Adaptive Moment Estimation (Adam), which combines Adaptive Learning Rate and Momentum, would be the most popular stochastic optimizer for accelerating the training of deep neural networks. However, it is empirically known that Adam often generalizes worse than Stochastic Gradient Descent (SGD). The purpose of this paper is to unveil the mystery of this behavior in the diffusion theoretical framework. Specifically, we disentangle the effects of Adaptive Learning Rate and Momentum of the Adam dynamics on saddle-point escaping and flat minima selection. We prove that Adaptive Learning Rate can es…
The University of Iowa（爱荷华大学）；Augusta University（奥古斯塔大学）
Stochastic Gradient Descent (SGD) is the key learning algorithm for many machine learning tasks. Because of its computational costs, there is a growing interest in accelerating SGD on HPC resources like GPU clusters. However, the performance of parallel SGD is still bottlenecked by the high communication costs even with a fast connection among the machines. A simple approach to alleviating this problem, used in many existing efforts, is to perform communication every few iterations, using a constant averaging period. In this paper, we show that the optimal averaging period in terms of converge…
This paper introduces Selective-Backprop, a technique that accelerates the training of deep neural networks (DNNs) by prioritizing examples with high loss at each iteration. Selective-Backprop uses the output of a training example's forward pass to decide whether to use that example to compute gradients and update parameters, or to skip immediately to the next example. By reducing the number of computationally-expensive backpropagation steps performed, Selective-Backprop accelerates training. Evaluation on CIFAR10, CIFAR100, and SVHN, across a variety of modern image models, shows that Selecti…
University of Utah（犹他大学）；University of Kentucky（肯塔基大学）
Momentum plays a crucial role in stochastic gradient-based optimization algorithms for accelerating or improving training deep neural networks (DNNs). In deep learning practice, the momentum is usually weighted by a well-calibrated constant. However, tuning hyperparameters for momentum can be a significant computational burden. In this paper, we propose a novel \emph{adaptive momentum} for improving DNNs training; this adaptive momentum, with no momentum related hyperparameter required, is motivated by the nonlinear conjugate gradient (NCG) method. Stochastic gradient descent (SGD) with this n…
This paper aims to accelerate the test-time computation of convolutional neural networks (CNNs), especially very deep CNNs that have substantially impacted the computer vision community. Unlike previous methods that are designed for approximating linear filters or linear responses, our method takes the nonlinear units into account. We develop an effective solution to the resulting nonlinear optimization problem without the need of stochastic gradient descent (SGD). More importantly, while previous methods mainly focus on optimizing one or two layers, our nonlinear method enables an asymmetric …
Tsinghua University（清华大学）；National Supercomputing Center in Wuxi（国家超级计算无锡中心）；Beihang University（北京航空航天大学）
This paper reports our efforts on swCaffe, a highly efficient parallel framework for accelerating deep neural networks (DNNs) training on Sunway TaihuLight, the current fastest supercomputer in the world that adopts a unique many-core heterogeneous architecture, with 40,960 SW26010 processors connected through a customized communication network. First, we point out some insightful principles to fully exploit the performance of the innovative many-core architecture. Second, we propose a set of optimization strategies for redesigning a variety of neural network layers based on Caffe. Third, we p…
This paper investigates the intriguing question of whether we can create learning algorithms that automatically generate training data, learning environments, and curricula in order to help AI agents rapidly learn. We show that such algorithms are possible via Generative Teaching Networks (GTNs), a general approach that is, in theory, applicable to supervised, unsupervised, and reinforcement learning, although our experiments only focus on the supervised case. GTNs are deep neural networks that generate data and/or training environments that a learner (e.g. a freshly initialized neural network…
Q-Newton: Hybrid Quantum-Classical Scheduling for Accelerating Neural Network Training with Newton's Gradient Descent
The University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；The University of Pittsburgh（匹兹堡大学）；The University of Chicago（芝加哥大学）；University of California Los Angeles（加州大学洛杉矶分校）
Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra
University of Pavia（帕维亚大学）；Department of Mathematics（数学系）；Department of Civil Engineering and Architecture（土木工程与建筑系）；Institut Camille Jordan, Lyon 1 Université（让·乔丹研究所，里昂1大学）
We study distributed optimization algorithms for minimizing the average of \emph{heterogeneous} functions distributed across several machines with a focus on communication efficiency. In such settings, naively using the classical stochastic gradient descent (SGD) or its variants (e.g., SVRG) with a uniform sampling of machines typically yields poor performance. It often leads to the dependence of convergence rate on maximum Lipschitz constant of gradients across the devices. In this paper, we propose a novel \emph{adaptive} sampling of machines specially catered to these settings. Our method r…
The information bottleneck principle (Shwartz-Ziv & Tishby, 2017) suggests that SGD-based training of deep neural networks results in optimally compressed hidden layers, from an information theoretic perspective. However, this claim was established on toy data. The goal of the work we present here is to test whether the information bottleneck principle is applicable to a realistic setting using a larger and deeper convolutional architecture, a ResNet model. We trained PixelCNN++ models as inverse representation decoders to measure the mutual information between hidden layers of a ResNet and in…
Seoul National University（首尔大学）；Yonsei University（延世大学）；Stanford University（斯坦福大学）
Near-data processing (NDP) refers to augmenting memory or storage with processing power. Despite its potential for acceleration computing and reducing power requirements, only limited progress has been made in popularizing NDP for various reasons. Recently, two major changes have occurred that have ignited renewed interest and caused a resurgence of NDP. The first is the success of machine learning (ML), which often demands a great deal of computation for training, requiring frequent transfers of big data. The second is the popularity of NAND flash-based solid-state drives (SSDs) containing mu…
This paper investigates accelerating the convergence of distributed optimization algorithms on non-convex problems. We propose a distributed primal-dual stochastic gradient descent~(SGD) equipped with "powerball" method to accelerate. We show that the proposed algorithm achieves the linear speedup convergence rate $\mathcal{O}(1/\sqrt{nT})$ for general smooth (possibly non-convex) cost functions. We demonstrate the efficiency of the algorithm through numerical experiments by training two-layer fully connected neural networks and convolutional neural networks on the MNIST dataset to compare wit…
Online Parallel Multi-Task Relationship Learning via Alternating Direction Method of Multipliers
School of Computer Science and Technology, Xidian University（西安电子科技大学计算机科学与技术学院）；Tencent AI Lab, Tencent Inc（腾讯公司腾讯AI实验室）；MBZUAI（穆罕默德·本·扎耶德人工智能大学）；School of Intelligent Engineering, Henan Institute of Technology（河南工程学院智能工程学院）
Carnegie Mellon University（卡内基梅隆大学）；University of Chicago（芝加哥大学）；Air Force Research Laboratory（美国空军研究实验室）
Tensors, which provide a powerful and flexible model for representing multi-attribute data and multi-way interactions, play an indispensable role in modern data science across various fields in science and engineering. A fundamental task is to faithfully recover the tensor from highly incomplete measurements in a statistically and computationally efficient manner. Harnessing the low-rank structure of tensors in the Tucker decomposition, this paper develops a scaled gradient descent (ScaledGD) algorithm to directly recover the tensor factors with tailored spectral initializations, and shows tha…
Stochastic variance reduced gradient (SVRG) is a popular variance reduction technique for accelerating stochastic gradient descent (SGD). We provide a first analysis of the method for solving a class of linear inverse problems in the lens of the classical regularization theory. We prove that for a suitable constant step size schedule, the method can achieve an optimal convergence rate in terms of the noise level (under suitable regularity condition) and the variance of the SVRG iterate error is smaller than that by SGD. These theoretical findings are corroborated by a set of numerical experime…
