We analyze Local SGD (aka parallel or federated SGD) and Minibatch SGD in the heterogeneous distributed setting, where each machine has access to stochastic gradient estimates for a different, machine-specific, convex objective; the goal is to optimize w.r.t. the average objective; and machines can only communicate intermittently. We argue that, (i) Minibatch SGD (even without acceleration) dominates all existing analysis of Local SGD in this setting, (ii) accelerated Minibatch SGD is optimal when the heterogeneity is high, and (iii) present the first upper bound for Local SGD that improves over Minibatch SGD in a non-homogeneous regime.
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…
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…
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…
It is well known that, for most datasets, the use of large-size minibatches for Stochastic Gradient Descent (SGD) typically leads to slow convergence and poor generalization. On the other hand, large minibatches are of great practical interest as they allow for a better exploitation of modern GPUs. Previous literature on the subject concentrated on how to adjust the main SGD parameters (in particular, the learning rate) when using large minibatches. In this work we introduce an additional feature, that we call minibatch persistency, that consists in reusing the same minibatch for K consecutive…
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎综合理工学院电信学院LTCI研究所）；ENS Paris, CREST-ENSAE, CNRS（巴黎高等师范学院、CREST-ENSAE、法国国家科学研究中心）；Mila and DIRO, Université de Montréal（蒙特利尔大学Mila研究所与计算机科学与运筹学系）
Stochastic Gradient Descent (SGD) is being used routinely for optimizing non-convex functions. Yet, the standard convergence theory for SGD in the smooth non-convex setting gives a slow sublinear convergence to a stationary point. In this work, we provide several convergence theorems for SGD showing convergence to a global minimum for non-convex problems satisfying some extra structural assumptions. In particular, we focus on two large classes of structured non-convex functions: (i) Quasar (Strongly) Convex functions (a generalization of convex functions) and (ii) functions satisfying the Poly…
The noise in stochastic gradient descent (SGD), caused by minibatch sampling, is poorly understood despite its practical importance in deep learning. This work presents the first systematic study of the SGD noise and fluctuations close to a local minimum. We first analyze the SGD noise in linear regression in detail and then derive a general formula for approximating SGD noise in different types of minima. For application, our results (1) provide insight into the stability of training a neural network, (2) suggest that a large learning rate can help generalization by introducing an implicit re…
University of Utah（犹他大学）；Cornell University（康奈尔大学）
The minibatch stochastic gradient descent method (SGD) is widely applied in deep learning due to its efficiency and scalability that enable training deep networks with a large volume of data. Particularly in the distributed setting, SGD is usually applied with large batch size. However, as opposed to small-batch SGD, neural network models trained with large-batch SGD can hardly generalize well, i.e., the validation accuracy is low. In this work, we introduce a novel regularization technique, namely distinctive regularization (DReg), which replicates a certain layer of the deep network and enco…
Université de Lille（里尔大学）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）；National University of Singapore（新加坡国立大学）
Stochastic gradient descent (SGD) is a cornerstone of machine learning. When the number N of data items is large, SGD relies on constructing an unbiased estimator of the gradient of the empirical risk using a small subset of the original dataset, called a minibatch. Default minibatch construction involves uniformly sampling a subset of the desired size, but alternatives have been explored for variance reduction. In particular, experimental evidence suggests drawing minibatches from determinantal point processes (DPPs), distributions over minibatches that favour diversity among selected items. …
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…
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…
Deep learning thrives with large neural networks and large datasets. However, larger networks and larger datasets result in longer training times that impede research and development progress. Distributed synchronous SGD offers a potential solution to this problem by dividing SGD minibatches over a pool of parallel workers. Yet to make this scheme efficient, the per-worker workload must be large, which implies nontrivial growth in the SGD minibatch size. In this paper, we empirically show that on the ImageNet dataset large minibatches cause optimization difficulties, but when these are address…
KAIST（韩国科学技术院）；Center for Superintelligence, SNU（首尔大学超级智能中心）；New York University（纽约大学）；Facebook AI Research（Facebook人工智能研究院）
Although stochastic gradient descent (SGD) is a driving force behind the recent success of deep learning, our understanding of its dynamics in a high-dimensional parameter space is limited. In recent years, some researchers have used the stochasticity of minibatch gradients, or the signal-to-noise ratio, to better characterize the learning dynamics of SGD. Inspired from these work, we here analyze SGD from a geometrical perspective by inspecting the stochasticity of the norms and directions of minibatch gradients. We propose a model of the directional concentration for minibatch gradients thro…
We demonstrate that training ResNet-50 on ImageNet for 90 epochs can be achieved in 15 minutes with 1024 Tesla P100 GPUs. This was made possible by using a large minibatch size of 32k. To maintain accuracy with this large minibatch size, we employed several techniques such as RMSprop warm-up, batch normalization without moving averages, and a slow-start learning rate schedule. This paper also describes the details of the hardware and software of the system used to achieve the above performance.
Purdue University（普渡大学）；University of Minnesota（明尼苏达大学）
Past research has indicated that the covariance of the Stochastic Gradient Descent (SGD) error done via minibatching plays a critical role in determining its regularization and escape from low potential points. Motivated by some new research in this area, we prove universality results by showing that noise classes that have the same mean and covariance structure of SGD via minibatching have similar properties. We mainly consider the Multiplicative Stochastic Gradient Descent (M-SGD) algorithm as introduced in previous work, which has a much more general noise class than the SGD algorithm done …
We present an approach towards convex optimization that relies on a novel scheme which converts online adaptive algorithms into offline methods. In the offline optimization setting, our derived methods are shown to obtain favourable adaptive guarantees which depend on the harmonic sum of the queried gradients. We further show that our methods implicitly adapt to the objective's structure: in the smooth case fast convergence rates are ensured without any prior knowledge of the smoothness parameter, while still maintaining guarantees in the non-smooth setting. Our approach has a natural extensio…
KAUST（阿卜杜拉国王科技大学）；Cairo University（开罗大学）；Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；Facebook AI Research（脸书人工智能研究院）
We present a unified theorem for the convergence analysis of stochastic gradient algorithms for minimizing a smooth and convex loss plus a convex regularizer. We do this by extending the unified analysis of Gorbunov, Hanzely \& Richtárik (2020) and dropping the requirement that the loss function be strongly convex. Instead, we only rely on convexity of the loss function. Our unified analysis applies to a host of existing algorithms such as proximal SGD, variance reduced methods, quantization and some coordinate descent type methods. For the variance reduced methods, we recover the best known c…
Department of Computer and Information Sciences, University of Pennsylvania, Philadelphia, PA, U.S.A.（宾夕法尼亚大学计算机与信息科学系）；Hal c o g lu Data Science Institute, University of California San Diego, San Diego, CA, U.S.A.（加州大学圣地亚哥分校Hal c o g lu数据科学研究所）
For the past 5 years, the ILSVRC competition and the ImageNet dataset have attracted a lot of interest from the Computer Vision community, allowing for state-of-the-art accuracy to grow tremendously. This should be credited to the use of deep artificial neural network designs. As these became more complex, the storage, bandwidth, and compute requirements increased. This means that with a non-distributed approach, even when using the most high-density server available, the training process may take weeks, making it prohibitive. Furthermore, as datasets grow, the representation learning potentia…
University of California, Berkeley（加州大学伯克利分校）；Google Research（谷歌研究院）
We present a novel Metropolis-Hastings method for large datasets that uses small expected-size minibatches of data. Previous work on reducing the cost of Metropolis-Hastings tests yield variable data consumed per sample, with only constant factor reductions versus using the full dataset for each sample. Here we present a method that can be tuned to provide arbitrarily small batch sizes, by adjusting either proposal step size or temperature. Our test uses the noise-tolerant Barker acceptance test with a novel additive correction variable. The resulting test has similar cost to a normal SGD upda…
SGD with a Constant Large Learning Rate Can Converge to Local Maxima
The University of Tokyo（东京大学）；ENS（巴黎高等师范学院）；Université PSL（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Université de Paris（巴黎大学）；University of California, Berkeley（加州大学伯克利分校）
The stochastic proximal point (SPP) methods have gained recent attention for stochastic optimization, with strong convergence guarantees and superior robustness to the classic stochastic gradient descent (SGD) methods showcased at little to no cost of computational overhead added. In this article, we study a minibatch variant of SPP, namely M-SPP, for solving convex composite risk minimization problems. The core contribution is a set of novel excess risk bounds of M-SPP derived through the lens of algorithmic stability theory. Particularly under smoothness and quadratic growth conditions, we s…
Stochastic Gradient Descent (SGD) is a central tool in machine learning. We prove that SGD converges to zero loss, even with a fixed (non-vanishing) learning rate - in the special case of homogeneous linear classifiers with smooth monotone loss functions, optimized on linearly separable data. Previous works assumed either a vanishing learning rate, iterate averaging, or loss assumptions that do not hold for monotone loss functions used for classification, such as the logistic loss. We prove our result on a fixed dataset, both for sampling with or without replacement. Furthermore, for logistic …
University of Bucharest（布加勒斯特大学）；The Research Institute of the University of Bucharest (ICUB)（布加勒斯特大学研究所（ICUB））
Stochastic optimization lies at the core of most statistical learning models. The recent great development of stochastic algorithmic tools focused significantly onto proximal gradient iterations, in order to find an efficient approach for nonsmooth (composite) population risk functions. The complexity of finding optimal predictors by minimizing regularized risk is largely understood for simple regularizations such as $\ell_1/\ell_2$ norms. However, more complex properties desired for the predictor necessitates highly difficult regularizers as used in grouped lasso or graph trend filtering. In …
École polytechnique fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Google Research（谷歌研究院）；Hebrew University of Jerusalem (HUJI)（耶路撒冷希伯来大学）；Massachusetts Institute of Technology (MIT)（麻省理工学院）；Toyota Technological Institute at Chicago (TTIC)（芝加哥丰田理工学院）
We study the power of learning via mini-batch stochastic gradient descent (SGD) on the population loss, and batch Gradient Descent (GD) on the empirical loss, of a differentiable model or neural network, and ask what learning problems can be learnt using these paradigms. We show that SGD and GD can always simulate learning with statistical queries (SQ), but their ability to go beyond that depends on the precision $ρ$ of the gradient calculations relative to the minibatch size $b$ (for SGD) and sample size $m$ (for GD). With fine enough precision relative to minibatch size, namely when $b ρ$ is…
Aiming towards the minimizers: fast convergence of SGD for overparametrized problems
Halicioğlu Data Science Institute, University of California San Diego（加州大学圣地亚哥分校哈利乔格鲁数据科学研究所）；University of Washington（华盛顿大学）；Cornell University（康奈尔大学）
Univ. Lille（里尔大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）；Université de Bâle（巴塞尔大学）；EPFL（洛桑联邦理工学院）
One of the key challenges in decentralized and federated learning is to design algorithms that efficiently deal with highly heterogeneous data distributions across agents. In this paper, we revisit the analysis of the popular Decentralized Stochastic Gradient Descent algorithm (D-SGD) under data heterogeneity. We exhibit the key role played by a new quantity, called neighborhood heterogeneity, on the convergence rate of D-SGD. By coupling the communication topology and the heterogeneity, our analysis sheds light on the poorly understood interplay between these two concepts. We then argue that …
Large-scale nonconvex optimization problems are ubiquitous in modern machine learning, and among practitioners interested in solving them, Stochastic Gradient Descent (SGD) reigns supreme. We revisit the analysis of SGD in the nonconvex setting and propose a new variant of the recently introduced expected smoothness assumption which governs the behaviour of the second moment of the stochastic gradient. We show that our assumption is both more general and more reasonable than assumptions made in all prior work. Moreover, our results yield the optimal $\mathcal{O}(\varepsilon^{-4})$ rate for fin…
Why SGD is not Brownian Motion: A New Perspective on Stochastic Dynamics
Basic Research of Artificial Intelligence Laboratory (BRAIn Lab)（人工智能基础研究实验室（BRAIn Lab））；Innopolis University（因诺普利斯大学）；P.N. Lebedev Physical Institute of the Russian Academy of Sciences（俄罗斯科学院皮亚琴佐·列别杰夫物理研究所）
We propose Multi-Level Local SGD, a distributed gradient method for learning a smooth, non-convex objective in a heterogeneous multi-level network. Our network model consists of a set of disjoint sub-networks, with a single hub and multiple worker nodes; further, worker nodes may have different operating rates. The hubs exchange information with one another via a connected, but not necessarily complete communication network. In our algorithm, sub-networks execute a distributed SGD algorithm, using a hub-and-spoke paradigm, and the hubs periodically average their models with neighboring hubs. W…
RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）；The University of Tokyo（东京大学）；University of Tokyo（东京大学）
Recent studies have demonstrated that noise in stochastic gradient descent (SGD) is closely related to generalization: A larger SGD noise, if not too large, results in better generalization. Since the covariance of the SGD noise is proportional to $η^2/B$, where $η$ is the learning rate and $B$ is the minibatch size of SGD, the SGD noise has so far been controlled by changing $η$ and/or $B$. However, too large $η$ results in instability in the training dynamics and a small $B$ prevents scalable parallel computation. It is thus desirable to develop a method of controlling the SGD noise without …
We provide a new analysis of local SGD, removing unnecessary assumptions and elaborating on the difference between two data regimes: identical and heterogeneous. In both cases, we improve the existing theory and provide values of the optimal stepsize and optimal number of local iterations. Our bounds are based on a new notion of variance that is specific to local SGD methods with different data. The tightness of our results is guaranteed by recovering known statements when we plug $H=1$, where $H$ is the number of local steps. The empirical evidence further validates the severe impact of data …
Beyond Optimal Rates in Stochastic Optimization: Trajectory-Adaptive Stopping Rules
UC Berkeley（加州大学伯克利分校）；Inria（法国国家信息与自动化研究所）；École Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；Department of Statistics, UC Berkeley（加州大学伯克利分校统计系）
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…
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…
Communication-Efficient Adaptive Batch Size Strategies for Distributed Local Gradient Methods
Perelman School of Medicine, University of Pennsylvania（宾夕法尼亚大学佩雷尔曼医学院）；The University of Chicago Booth School of Business（芝加哥大学布斯商学院）；Northwestern University（西北大学）；University of Southern California Marshall School of Business（南加州大学马歇尔商学院）
In the vanishing learning rate regime, stochastic gradient descent (SGD) is now relatively well understood. In this work, we propose to study the basic properties of SGD and its variants in the non-vanishing learning rate regime. The focus is on deriving exactly solvable results and discussing their implications. The main contributions of this work are to derive the stationary distribution for discrete-time SGD in a quadratic loss function with and without momentum; in particular, one implication of our result is that the fluctuation caused by discrete-time dynamics takes a distorted shape and…
SGD for Variational Inference: Tackling Unbounded Variance via Preconditioning and Dynamic Batching
MaLGa；DIBRIS Università degli Studi di Genova（DIBRIS基因瓦大学）；Università degli Studi di Genova（基因瓦大学）；DIMA Università degli Studi di Genova（DIMA基因瓦大学）；IIT Università degli Studi di Genova（IIT基因瓦大学）
The Ohio State University（俄亥俄州立大学）；University of Minnesota（明尼苏达大学）；Syracuse University（雪城大学）；Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）
Federated Learning (FL) refers to the paradigm where multiple worker nodes (WNs) build a joint model by using local data. Despite extensive research, for a generic non-convex FL problem, it is not clear, how to choose the WNs' and the server's update directions, the minibatch sizes, and the local update frequency, so that the WNs use the minimum number of samples and communication rounds to achieve the desired solution. This work addresses the above question and considers a class of stochastic algorithms where the WNs perform a few local updates before communication. We show that when both the…
For infinitesimal learning rates, stochastic gradient descent (SGD) follows the path of gradient flow on the full batch loss function. However moderately large learning rates can achieve higher test accuracies, and this generalization benefit is not explained by convergence bounds, since the learning rate which maximizes test accuracy is often larger than the learning rate which minimizes training loss. To interpret this phenomenon we prove that for SGD with random shuffling, the mean SGD iterate also stays close to the path of gradient flow if the learning rate is small and finite, but on a m…
Lossy gradient compression, with either unbiased or biased compressors, has become a key tool to avoid the communication bottleneck in centrally coordinated distributed training of machine learning models. We analyze the performance of two standard and general types of methods: (i) distributed quantized SGD (D-QSGD) with arbitrary unbiased quantizers and (ii) distributed SGD with error-feedback and biased compressors (D-EF-SGD) in the heterogeneous (non-iid) data setting. Our results indicate that D-EF-SGD is much less affected than D-QSGD by non-iid data, but both methods can suffer a slowdow…
