Stochastic gradient descent (SGD) has been widely studied in the literature from different angles, and is commonly employed for solving many big data machine learning problems. However, the averaging technique, which combines all iterative solutions into a single solution, is still under-explored. While some increasingly weighted averaging schemes have been considered in the literature, existing works are mostly restricted to strongly convex objective functions and the convergence of optimization error. It remains unclear how these averaging schemes affect the convergence of {\it both optimization error and generalization error} (two equally important components of testing error) for {\bf non-strongly convex objectives, including non-convex problems}. In this paper, we {\it fill the gap} by comprehensively analyzing the increasingly weighted averaging on convex, strongly convex and non-convex objective functions in terms of both optimization error and generalization error. In particular, we analyze a family of increasingly weighted averaging, where the weight for the solution at iteration $t$ is proportional to $t^α$ ($α> 0$). We show how $α$ affects the optimization error and the generalization error, and exhibit the trade-off caused by $α$. Experiments have demonstrated this trade-off and the effectiveness of polynomially increased weighted averaging compared with other averaging schemes for a wide range of problems including deep learning.
Consider a number of workers running SGD independently on the same pool of data and averaging the models every once in a while -- a common but not well understood practice. We study model averaging as a variance-reducing mechanism and describe two ways in which the frequency of averaging affects convergence. For convex objectives, we show the benefit of frequent averaging depends on the gradient variance envelope. For non-convex objectives, we illustrate that this benefit depends on the presence of multiple globally optimal points. We complement our findings with multicore experiments on both …
University of Stuttgart（斯图加特大学）；Universitat Pompeu Fabra（庞培法布拉大学）；Massachusetts Institute of Technology（麻省理工学院）；Istituto Italiano di Tecnologia（意大利理工学院）；Universita’ degli Studi di Genova（热那亚大学）
While stochastic gradient descent (SGD) is one of the major workhorses in machine learning, the learning properties of many practically used variants are poorly understood. In this paper, we consider least squares learning in a nonparametric setting and contribute to filling this gap by focusing on the effect and interplay of multiple passes, mini-batching and averaging, and in particular tail averaging. Our results show how these different variants of SGD can be combined to achieve optimal learning errors, hence providing practical insights. In particular, we show for the first time in the li…
Acceleration of stochastic gradient descent with momentum by averaging: finite-sample rates and asymptotic normality
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Daniels School of Business, Purdue University（普渡大学丹尼尔斯商学院）；Stern School of Business, New York University（纽约大学斯特恩商学院）
Momentum methods are now used pervasively within the machine learning community for training non-convex models such as deep neural networks. Empirically, they out perform traditional stochastic gradient descent (SGD) approaches. In this work we develop a Lyapunov analysis of SGD with momentum (SGD+M), by utilizing a equivalent rewriting of the method known as the stochastic primal averaging (SPA) form. This analysis is much tighter than previous theory in the non-convex case, and due to this we are able to give precise insights into when SGD+M may out-perform SGD, and what hyper-parameter sche…
University of Southern California（南加州大学）；Amazon Alexa AI（亚马逊 Alexa AI）
Local Stochastic Gradient Descent (SGD) with periodic model averaging (FedAvg) is a foundational algorithm in Federated Learning. The algorithm independently runs SGD on multiple workers and periodically averages the model across all the workers. When local SGD runs with many workers, however, the periodic averaging causes a significant model discrepancy across the workers making the global loss converge slowly. While recent advanced optimization methods tackle the issue focused on non-IID settings, there still exists the model discrepancy issue due to the underlying periodic model averaging. …
Microsoft Research（微软研究院）；University of Washington（华盛顿大学）；Stanford University（斯坦福大学）
This work characterizes the benefits of averaging schemes widely used in conjunction with stochastic gradient descent (SGD). In particular, this work provides a sharp analysis of: (1) mini-batching, a method of averaging many samples of a stochastic gradient to both reduce the variance of the stochastic gradient estimate and for parallelizing SGD and (2) tail-averaging, a method involving averaging the final few iterates of SGD to decrease the variance in SGD's final iterate. This work presents non-asymptotic excess risk bounds for these schemes for the stochastic approximation problem of leas…
Revisiting Gradient Normalization and Clipping for Nonconvex SGD under Heavy-Tailed Noise: Necessity, Sufficiency, and Acceleration
College of Computer Science and Technology National University of Defense Technology（计算机科学与技术学院 国防科技大学）；Center for Machine Learning Research (CMLR) Peking University（机器学习研究中心（CMLR） 北京大学）
Large neural network models present a hefty communication challenge to distributed Stochastic Gradient Descent (SGD), with a communication complexity of O(n) per worker for a model of n parameters. Many sparsification and quantization techniques have been proposed to compress the gradients, some reducing the communication complexity to O(k), where k << n. In this paper, we introduce a strategy called two-level gradient averaging (A2SGD) to consolidate all gradients down to merely two local averages per worker before the computation of two global averages for an updated model. A2SGD also retain…
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…
Facebook AI Research（脸书人工智能研究院）；Princeton University（普林斯顿大学）；Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）
First-order stochastic optimization methods are currently the most widely used class of methods for training deep neural networks. However, the choice of the optimizer has become an ad-hoc rule that can significantly affect the performance. For instance, SGD with momentum (SGD+M) is typically used in computer vision (CV) and Adam is used for training transformer models for Natural Language Processing (NLP). Using the wrong method can lead to significant performance degradation. Inspired by the dual averaging algorithm, we propose Modernized Dual Averaging (MDA), an optimizer that is able to pe…
Distributed training of deep learning models on large-scale training data is typically conducted with asynchronous stochastic optimization to maximize the rate of updates, at the cost of additional noise introduced from asynchrony. In contrast, the synchronous approach is often thought to be impractical due to idle time wasted on waiting for straggling workers. We revisit these conventional beliefs in this paper, and examine the weaknesses of both approaches. We demonstrate that a third approach, synchronous optimization with backup workers, can avoid asynchronous noise while mitigating for th…
Distributed training of deep learning models on large-scale training data is typically conducted with asynchronous stochastic optimization to maximize the rate of updates, at the cost of additional noise introduced from asynchrony. In contrast, the synchronous approach is often thought to be impractical due to idle time wasted on waiting for straggling workers. We revisit these conventional beliefs in this paper, and examine the weaknesses of both approaches. We demonstrate that a third approach, synchronous optimization with backup workers, can avoid asynchronous noise while mitigating for th…
Communication-efficient SGD algorithms, which allow nodes to perform local updates and periodically synchronize local models, are highly effective in improving the speed and scalability of distributed SGD. However, a rigorous convergence analysis and comparative study of different communication-reduction strategies remains a largely open problem. This paper presents a unified framework called Cooperative SGD that subsumes existing communication-efficient SGD algorithms such as periodic-averaging, elastic-averaging and decentralized SGD. By analyzing Cooperative SGD, we provide novel convergenc…
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…
University of California, Los Angeles（加州大学洛杉矶分校）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of Washington（华盛顿大学）；Microsoft Research（微软研究院）
There is an increasing realization that algorithmic inductive biases are central in preventing overfitting; empirically, we often see a benign overfitting phenomenon in overparameterized settings for natural learning algorithms, such as stochastic gradient descent (SGD), where little to no explicit regularization has been employed. This work considers this issue in arguably the most basic setting: constant-stepsize SGD (with iterate averaging or tail averaging) for linear regression in the overparameterized regime. Our main result provides a sharp excess risk bound, stated in terms of the full…
Communication overhead is one of the key challenges that hinders the scalability of distributed optimization algorithms. In this paper, we study local distributed SGD, where data is partitioned among computation nodes, and the computation nodes perform local updates with periodically exchanging the model among the workers to perform averaging. While local SGD is empirically shown to provide promising results, a theoretical understanding of its performance remains open. We strengthen convergence analysis for local SGD, and show that local SGD can be far less expensive and applied far more gener…
Microsoft Research（微软研究院）；Rutgers University（罗格斯大学）
Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth objective functions such as support vector machines. In this paper, we investigate the performance of SGD without such smoothness assumptions, as well as a running average scheme to convert the SGD iterates to a solution with optimal optimization accuracy. In this framework, we prove t…
The state-of-the-art deep learning algorithms rely on distributed training systems to tackle the increasing sizes of models and training data sets. Minibatch stochastic gradient descent (SGD) algorithm requires workers to halt forward/back propagations, to wait for gradients aggregated from all workers, and to receive weight updates before the next batch of tasks. This synchronous execution model exposes the overheads of gradient/weight communication among a large number of workers in a distributed training system. We propose a new SGD algorithm, DaSGD (Local SGD with Delayed Averaging), which…
Johns Hopkins University（约翰霍普金斯大学）；University of California, Los Angeles（加州大学洛杉矶分校）
Regularization for optimization is a crucial technique to avoid overfitting in machine learning. In order to obtain the best performance, we usually train a model by tuning the regularization parameters. It becomes costly, however, when a single round of training takes significant amount of time. Very recently, Neu and Rosasco show that if we run stochastic gradient descent (SGD) on linear regression problems, then by averaging the SGD iterates properly, we obtain a regularized solution. It left open whether the same phenomenon can be achieved for other optimization problems and algorithms. In…
Low precision operations can provide scalability, memory savings, portability, and energy efficiency. This paper proposes SWALP, an approach to low precision training that averages low-precision SGD iterates with a modified learning rate schedule. SWALP is easy to implement and can match the performance of full-precision SGD even with all numbers quantized down to 8 bits, including the gradient accumulators. Additionally, we show that SWALP converges arbitrarily close to the optimal solution for quadratic objectives, and to a noise ball asymptotically smaller than low precision SGD in strongly…
In distributed training of deep neural networks, parallel mini-batch SGD is widely used to speed up the training process by using multiple workers. It uses multiple workers to sample local stochastic gradient in parallel, aggregates all gradients in a single server to obtain the average, and update each worker's local model using a SGD update with the averaged gradient. Ideally, parallel mini-batch SGD can achieve a linear speed-up of the training time (with respect to the number of workers) compared with SGD over a single worker. However, such linear scalability in practice is significantly l…
Georgia Institute of Technology（佐治亚理工学院）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Reducing communication in training large-scale machine learning applications on distributed platform is still a big challenge. To address this issue, we propose a distributed hierarchical averaging stochastic gradient descent (Hier-AVG) algorithm with infrequent global reduction by introducing local reduction. As a general type of parallel SGD, Hier-AVG can reproduce several popular synchronous parallel SGD variants by adjusting its parameters. We show that Hier-AVG with infrequent global reduction can still achieve standard convergence rate for non-convex optimization problems. In addition, w…
Cornell University（康奈尔大学）；Higher School of Economics（高等经济学院）；Samsung-HSE Laboratory（三星-高等经济学院实验室）；Samsung AI Center in Moscow（莫斯科三星人工智能中心）；Lomonosov Moscow State University（莫斯科国立大学（罗蒙诺索夫莫斯科国立大学））
Deep neural networks are typically trained by optimizing a loss function with an SGD variant, in conjunction with a decaying learning rate, until convergence. We show that simple averaging of multiple points along the trajectory of SGD, with a cyclical or constant learning rate, leads to better generalization than conventional training. We also show that this Stochastic Weight Averaging (SWA) procedure finds much flatter solutions than SGD, and approximates the recent Fast Geometric Ensembling (FGE) approach with a single model. Using SWA we achieve notable improvement in test accuracy over co…
We consider speeding up stochastic gradient descent (SGD) by parallelizing it across multiple workers. We assume the same data set is shared among $N$ workers, who can take SGD steps and coordinate with a central server. While it is possible to obtain a linear reduction in the variance by averaging all the stochastic gradients at every step, this requires a lot of communication between the workers and the server, which can dramatically reduce the gains from parallelism. The Local SGD method, proposed and analyzed in the earlier literature, suggests machines should make many local steps between…
Federated Averaging (FedAvg), also known as Local SGD, is one of the most popular algorithms in Federated Learning (FL). Despite its simplicity and popularity, the convergence rate of FedAvg has thus far been undetermined. Even under the simplest assumptions (convex, smooth, homogeneous, and bounded covariance), the best-known upper and lower bounds do not match, and it is not clear whether the existing analysis captures the capacity of the algorithm. In this work, we first resolve this question by providing a lower bound for FedAvg that matches the existing upper bound, which shows the existi…
We revisit and extend model stitching (Lenc & Vedaldi 2015) as a methodology to study the internal representations of neural networks. Given two trained and frozen models $A$ and $B$, we consider a "stitched model'' formed by connecting the bottom-layers of $A$ to the top-layers of $B$, with a simple trainable layer between them. We argue that model stitching is a powerful and perhaps under-appreciated tool, which reveals aspects of representations that measures such as centered kernel alignment (CKA) cannot. Through extensive experiments, we use model stitching to obtain quantitative verifica…
Courant Institute, NYU（纽约大学柯朗研究所）；Center for Data Science, NYU（纽约大学数据科学中心）；Facebook AI Research（脸书人工智能研究院）
We study the problem of stochastic optimization for deep learning in the parallel computing environment under communication constraints. A new algorithm is proposed in this setting where the communication and coordination of work among concurrent processes (local workers), is based on an elastic force which links the parameters they compute with a center variable stored by the parameter server (master). The algorithm enables the local workers to perform more exploration, i.e. the algorithm allows the local variables to fluctuate further from the center variable by reducing the amount of commun…
Synchronous mini-batch SGD is state-of-the-art for large-scale distributed machine learning. However, in practice, its convergence is bottlenecked by slow communication rounds between worker nodes. A natural solution to reduce communication is to use the \emph{`local-SGD'} model in which the workers train their model independently and synchronize every once in a while. This algorithm improves the computation-communication trade-off but its convergence is not understood very well. We propose a non-asymptotic error analysis, which enables comparison to \emph{one-shot averaging} i.e., a single co…
The generalization of machine learning models has a complex dependence on the data, model and learning algorithm. We study train and test performance, as well as the generalization gap given by the mean of their difference over different data set samples to understand their ``typical" behavior. We derive an expression for the gap as a function of the covariance between the model parameter distribution and the train loss, and another expression for the average test performance, showing test generalization only depends on data-averaged parameter distribution and the data-averaged loss. We show t…
Large-scale machine learning training, in particular distributed stochastic gradient descent, needs to be robust to inherent system variability such as node straggling and random communication delays. This work considers a distributed training framework where each worker node is allowed to perform local model updates and the resulting models are averaged periodically. We analyze the true speed of error convergence with respect to wall-clock time (instead of the number of iterations), and analyze how it is affected by the frequency of averaging. The main contribution is the design of AdaComm, a…
Toyota Technological Institute at Chicago（芝加哥丰田工业大学）；EPFL（洛桑联邦理工学院）；University of Chicago（芝加哥大学）；Google（谷歌）；Weizmann Institute of Science（魏茨曼科学研究所）
We study local SGD (also known as parallel SGD and federated averaging), a natural and frequently used stochastic distributed optimization method. Its theoretical foundations are currently lacking and we highlight how all existing error guarantees in the convex setting are dominated by a simple baseline, minibatch SGD. (1) For quadratic objectives we prove that local SGD strictly dominates minibatch SGD and that accelerated local SGD is minimax optimal for quadratics; (2) For general convex objectives we provide the first guarantee that at least sometimes improves over minibatch SGD; (3) We sh…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Google Inc.（谷歌公司）；University of California, Santa Barbara（加利福尼亚大学圣巴巴拉分校）；DAMO Academy, Alibaba Group（阿里巴巴集团达摩院）
Decentralized SGD is an emerging training method for deep learning known for its much less (thus faster) communication per iteration, which relaxes the averaging step in parallel SGD to inexact averaging. The less exact the averaging is, however, the more the total iterations the training needs to take. Therefore, the key to making decentralized SGD efficient is to realize nearly-exact averaging using little communication. This requires a skillful choice of communication topology, which is an under-studied topic in decentralized optimization. In this paper, we study so-called exponential graph…
Research Center for Applied Mathematics and Machine Intelligence（应用数学与机器智能研究中心）；Zhejiang Lab（之江实验室）
Averaging neural network weights sampled by a backbone stochastic gradient descent (SGD) is a simple yet effective approach to assist the backbone SGD in finding better optima, in terms of generalization. From a statistical perspective, weight averaging (WA) contributes to variance reduction. Recently, a well-established stochastic weight averaging (SWA) method is proposed, which is featured by the application of a cyclical or high constant (CHC) learning rate schedule (LRS) in generating weight samples for WA. Then a new insight on WA appears, which states that WA helps to discover wider opti…
Effective regularisation during training can mean the difference between success and failure for deep neural networks. Recently, dither has been suggested as alternative to dropout for regularisation during batch-averaged stochastic gradient descent (SGD). In this article, we show that these methods fail without batch averaging and we introduce a new, parallel regularisation method that may be used without batch averaging. Our results for parallel-regularised non-batch-SGD are substantially better than what is possible with batch-SGD. Furthermore, our results demonstrate that dither and dropou…
Iterative procedures for parameter estimation based on stochastic gradient descent allow the estimation to scale to massive data sets. However, in both theory and practice, they suffer from numerical instability. Moreover, they are statistically inefficient as estimators of the true parameter value. To address these two issues, we propose a new iterative procedure termed averaged implicit SGD (AI-SGD). For statistical efficiency, AI-SGD employs averaging of the iterates, which achieves the optimal Cramér-Rao bound under strong convexity, i.e., it is an optimal unbiased estimator of the true pa…
International Computer Science Institute（国际计算机科学研究所）；University of California, Berkeley（加州大学伯克利分校）；Nanyang Technological University（南洋理工大学）
In this work we apply model averaging to parallel training of deep neural network (DNN). Parallelization is done in a model averaging manner. Data is partitioned and distributed to different nodes for local model updates, and model averaging across nodes is done every few minibatches. We use multiple GPUs for data parallelization, and Message Passing Interface (MPI) for communication between nodes, which allows us to perform model averaging frequently without losing much time on communication. We investigate the effectiveness of Natural Gradient Stochastic Gradient Descent (NG-SGD) and Restric…
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；School of Computer and Data Science, Nanyang Technological University（南洋理工大学计算机与数据科学学院）；School of Cyber Science & Technology, Shenzhen Campus of Sun Yat-sen University（中山大学深圳校区网络空间科学与技术学院）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Faculty of Engineering, the University of Sydney（悉尼大学工程学院）
We describe the neural-network training framework used in the Kaldi speech recognition toolkit, which is geared towards training DNNs with large amounts of training data using multiple GPU-equipped or multi-core machines. In order to be as hardware-agnostic as possible, we needed a way to use multiple machines without generating excessive network traffic. Our method is to average the neural network parameters periodically (typically every minute or two), and redistribute the averaged parameters to the machines for further training. Each machine sees different data. By itself, this method does …
The communication bottleneck has been a critical problem in large-scale distributed deep learning. In this work, we study distributed SGD with random block-wise sparsification as the gradient compressor, which is ring-allreduce compatible and highly computation-efficient but leads to inferior performance. To tackle this important issue, we improve the communication-efficient distributed SGD from a novel aspect, that is, the trade-off between the variance and second moment of the gradient. With this motivation, we propose a new detached error feedback (DEF) algorithm, which shows better converg…
Hierarchical Weight Averaging for Deep Neural Networks
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Pengcheng Laboratory（鹏城实验室）；Institute of High Performance Computing, Agency for Science, Technology and Research（新加坡科技研究局高性能计算研究所）
Yahoo Research（雅虎研究院）；The University of Iowa（爱荷华大学）；Nanjing University（南京大学）
We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent~(SGD) methods. To reduce the computation costs pertaining to the projections, we propose an Epoch-Projection Stochastic Gradient Descent~(Epro-SGD) method. The proposed Epro-SGD method consists of a sequence of epochs; it applies SGD to an augmented objective function at each iteration within the epoch, and then performs a projection at the end of each epoch. Gi…
University of Oxford（牛津大学）；University of Bristol（布里斯托大学）；University of Cambridge（剑桥大学）
We analyse and explain the increased generalisation performance of iterate averaging using a Gaussian process perturbation model between the true and batch risk surface on the high dimensional quadratic. We derive three phenomena \latestEdits{from our theoretical results:} (1) The importance of combining iterate averaging (IA) with large learning rates and regularisation for improved regularisation. (2) Justification for less frequent averaging. (3) That we expect adaptive gradient methods to work equally well, or better, with iterate averaging than their non-adaptive counterparts. Inspired by…
In distributed learning, local SGD (also known as federated averaging) and its simple baseline minibatch SGD are widely studied optimization methods. Most existing analyses of these methods assume independent and unbiased gradient estimates obtained via with-replacement sampling. In contrast, we study shuffling-based variants: minibatch and local Random Reshuffling, which draw stochastic gradients without replacement and are thus closer to practice. For smooth functions satisfying the Polyak-Łojasiewicz condition, we obtain convergence bounds (in the large epoch regime) which show that these s…
The stochastic gradient noise (SGN) is a significant factor in the success of stochastic gradient descent (SGD). Following the central limit theorem, SGN was initially modeled as Gaussian, and lately, it has been suggested that stochastic gradient noise is better characterized using $SαS$ Lévy distribution. This claim was allegedly refuted and rebounded to the previously suggested Gaussian noise model. This paper presents solid, detailed empirical evidence that SGN is heavy-tailed and better depicted by the $SαS$ distribution. Furthermore, we argue that different parameters in a deep neural ne…
