Distance metric learning (DML) is an important task that has found applications in many domains. The high computational cost of DML arises from the large number of variables to be determined and the constraint that a distance metric has to be a positive semi-definite (PSD) matrix. Although stochastic gradient descent (SGD) has been successfully applied to improve the efficiency of DML, it can still be computationally expensive because in order to ensure that the solution is a PSD matrix, it has to, at every iteration, project the updated distance metric onto the PSD cone, an expensive operation. We address this challenge by developing two strategies within SGD, i.e. mini-batch and adaptive sampling, to effectively reduce the number of updates (i.e., projections onto the PSD cone) in SGD. We also develop hybrid approaches that combine the strength of adaptive sampling with that of mini-batch online learning techniques to further improve the computational efficiency of SGD for DML. We prove the theoretical guarantees for both adaptive sampling and mini-batch based approaches for DML. We also conduct an extensive empirical study to verify the effectiveness of the proposed algorithms for DML.
UC San Diego（加州大学圣地亚哥分校）；Princeton University（普林斯顿大学）；Carnegie Mellon University（卡内基梅隆大学）
A number of competing hypotheses have been proposed to explain why small-batch Stochastic Gradient Descent (SGD)leads to improved generalization over the full-batch regime, with recent work crediting the implicit regularization of various quantities throughout training. However, to date, empirical evidence assessing the explanatory power of these hypotheses is lacking. In this paper, we conduct an extensive empirical evaluation, focusing on the ability of various theorized mechanisms to close the small-to-large batch generalization gap. Additionally, we characterize how the quantities that SGD…
Existing research shows that the batch size can seriously affect the performance of stochastic gradient descent~(SGD) based learning, including training speed and generalization ability. A larger batch size typically results in less parameter updates. In distributed training, a larger batch size also results in less frequent communication. However, a larger batch size can make a generalization gap more easily. Hence, how to set a proper batch size for SGD has recently attracted much attention. Although some methods about setting batch size have been proposed, the batch size problem has still n…
Mini-batch stochastic gradient methods (SGD) are state of the art for distributed training of deep neural networks. Drastic increases in the mini-batch sizes have lead to key efficiency and scalability gains in recent years. However, progress faces a major roadblock, as models trained with large batches often do not generalize well, i.e. they do not show good accuracy on new data. As a remedy, we propose a \emph{post-local} SGD and show that it significantly improves the generalization performance compared to large-batch training on standard benchmarks while enjoying the same efficiency (time-…
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…
Disney Research（迪士尼研究院）；KTH Royal Institute of Technology（瑞典皇家理工学院）
We study a mini-batch diversification scheme for stochastic gradient descent (SGD). While classical SGD relies on uniformly sampling data points to form a mini-batch, we propose a non-uniform sampling scheme based on the Determinantal Point Process (DPP). The DPP relies on a similarity measure between data points and gives low probabilities to mini-batches which contain redundant data, and higher probabilities to mini-batches with more diverse data. This simultaneously balances the data and leads to stochastic gradients with lower variance. We term this approach Diversified Mini-Batch SGD (DM-…
Stochastic gradient descent (SGD) is almost ubiquitously used for training non-convex optimization tasks. Recently, a hypothesis proposed by Keskar et al. [2017] that large batch methods tend to converge to sharp minimizers has received increasing attention. We theoretically justify this hypothesis by providing new properties of SGD in both finite-time and asymptotic regimes. In particular, we give an explicit escaping time of SGD from a local minimum in the finite-time regime and prove that SGD tends to converge to flatter minima in the asymptotic regime (although may take exponential time to…
Computer Science, Wayne State University（韦恩州立大学计算机科学（系））
Deep neural networks (DNNs) for supervised learning can be viewed as a pipeline of a feature extractor (i.e. last hidden layer) and a linear classifier (i.e. output layer) that is trained jointly with stochastic gradient descent (SGD). In each iteration of SGD, a mini-batch from the training data is sampled and the true gradient of the loss function is estimated as the noisy gradient calculated on this mini-batch. From the feature learning perspective, the feature extractor should be updated to learn meaningful features with respect to the entire data, and reduce the accommodation to noise in …
For SGD based distributed stochastic optimization, computation complexity, measured by the convergence rate in terms of the number of stochastic gradient calls, and communication complexity, measured by the number of inter-node communication rounds, are two most important performance metrics. The classical data-parallel implementation of SGD over $N$ workers can achieve linear speedup of its convergence rate but incurs an inter-node communication round at each batch. We study the benefit of using dynamically increasing batch sizes in parallel SGD for stochastic non-convex optimization by chara…
In this paper we aim to formally explain the phenomenon of fast convergence of SGD observed in modern machine learning. The key observation is that most modern learning architectures are over-parametrized and are trained to interpolate the data by driving the empirical loss (classification and regression) close to zero. While it is still unclear why these interpolated solutions perform well on test data, we show that these regimes allow for fast convergence of SGD, comparable in number of iterations to full gradient descent. For convex loss functions we obtain an exponential convergence bound …
Toshiba Corporation（东芝公司）；The University of Tokyo（东京大学）；RIKEN（理化学研究所）
Large-batch stochastic gradient descent (SGD) is widely used for training in distributed deep learning because of its training-time efficiency, however, extremely large-batch SGD leads to poor generalization and easily converges to sharp minima, which prevents naive large-scale data-parallel SGD (DP-SGD) from converging to good minima. To overcome this difficulty, we propose gradient noise convolution (GNC), which effectively smooths sharper minima of the loss function. For DP-SGD, GNC utilizes so-called gradient noise, which is induced by stochastic gradient variation and convolved to the los…
Recent advances in the theoretical understanding of SGD led to a formula for the optimal batch size minimizing the number of effective data passes, i.e., the number of iterations times the batch size. However, this formula is of no practical value as it depends on the knowledge of the variance of the stochastic gradients evaluated at the optimum. In this paper we design a practical SGD method capable of learning the optimal batch size adaptively throughout its iterations for strongly convex and smooth functions. Our method does this provably, and in our experiments with synthetic and real data…
Alibaba Group（阿里巴巴集团）；University of Pennsylvania（宾夕法尼亚大学）
The scale of deep learning nowadays calls for efficient distributed training algorithms. Decentralized momentum SGD (DmSGD), in which each node averages only with its neighbors, is more communication efficient than vanilla Parallel momentum SGD that incurs global average across all computing nodes. On the other hand, the large-batch training has been demonstrated critical to achieve runtime speedup. This motivates us to investigate how DmSGD performs in the large-batch scenario. In this work, we find the momentum term can amplify the inconsistency bias in DmSGD. Such bias becomes more evident …
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…
Multi-epoch, small-batch, Stochastic Gradient Descent (SGD) has been the method of choice for learning with large over-parameterized models. A popular theory for explaining why SGD works well in practice is that the algorithm has an implicit regularization that biases its output towards a good solution. Perhaps the theoretically most well understood learning setting for SGD is that of Stochastic Convex Optimization (SCO), where it is well known that SGD learns at a rate of $O(1/\sqrt{n})$, where $n$ is the number of samples. In this paper, we consider the problem of SCO and explore the role of…
Optimization in Deep Learning is mainly guided by vague intuitions and strong assumptions, with a limited understanding how and why these work in practice. To shed more light on this, our work provides some deeper understandings of how SGD behaves by empirically analyzing the trajectory taken by SGD from a line search perspective. Specifically, a costly quantitative analysis of the full-batch loss along SGD trajectories from common used models trained on a subset of CIFAR-10 is performed. Our core results include that the full-batch loss along lines in update step direction is highly parabolic…
We present a generic framework for trading off fidelity and cost in computing stochastic gradients when the costs of acquiring stochastic gradients of different quality are not known a priori. We consider a mini-batch oracle that distributes a limited query budget over a number of stochastic gradients and aggregates them to estimate the true gradient. Since the optimal mini-batch size depends on the unknown cost-fidelity function, we propose an algorithm, {\it EE-Grad}, that sequentially explores the performance of mini-batch oracles and exploits the accumulated knowledge to estimate the one a…
Mini-batch SGD with momentum is a fundamental algorithm for learning large predictive models. In this paper we develop a new analytic framework to analyze noise-averaged properties of mini-batch SGD for linear models at constant learning rates, momenta and sizes of batches. Our key idea is to consider the dynamics of the second moments of model parameters for a special family of "Spectrally Expressible" approximations. This allows to obtain an explicit expression for the generating function of the sequence of loss values. By analyzing this generating function, we find, in particular, that 1) t…
It has been experimentally observed that distributed implementations of mini-batch stochastic gradient descent (SGD) algorithms exhibit speedup saturation and decaying generalization ability beyond a particular batch-size. In this work, we present an analysis hinting that high similarity between concurrently processed gradients may be a cause of this performance degradation. We introduce the notion of gradient diversity that measures the dissimilarity between concurrent gradient updates, and show its key role in the performance of mini-batch SGD. We prove that on problems with high gradient di…
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…
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…
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…
Ajou University（亚洲大学）；Surromind；ETRI（韩国电子通信研究院）；Korea Institute of Science and Technology（韩国科学技术研究院）
Distributed deep learning is an effective way to reduce the training time of deep learning for large datasets as well as complex models. However, the limited scalability caused by network overheads makes it difficult to synchronize the parameters of all workers. To resolve this problem, gossip-based methods that demonstrates stable scalability regardless of the number of workers have been proposed. However, to use gossip-based methods in general cases, the validation accuracy for a large mini-batch needs to be verified. To verify this, we first empirically study the characteristics of gossip m…
In stochastic optimization, using large batch sizes during training can leverage parallel resources to produce faster wall-clock training times per training epoch. However, for both training loss and testing error, recent results analyzing large batch Stochastic Gradient Descent (SGD) have found sharp diminishing returns, beyond a certain critical batch size. In the hopes of addressing this, it has been suggested that the Kronecker-Factored Approximate Curvature (\mbox{K-FAC}) method allows for greater scalability to large batch sizes, for non-convex machine learning problems such as neural ne…
When using large-batch training to speed up stochastic gradient descent, learning rates must adapt to new batch sizes in order to maximize speed-ups and preserve model quality. Re-tuning learning rates is resource intensive, while fixed scaling rules often degrade model quality. We propose AdaScale SGD, an algorithm that reliably adapts learning rates to large-batch training. By continually adapting to the gradient's variance, AdaScale automatically achieves speed-ups for a wide range of batch sizes. We formally describe this quality with AdaScale's convergence bound, which maintains final obj…
When and Why Momentum Accelerates SGD:An Empirical Study
University of Science and Technology of China（中国科学技术大学）；Microsoft Research（微软研究院）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）
We study the statistical properties of the dynamic trajectory of stochastic gradient descent (SGD). We approximate the mini-batch SGD and the momentum SGD as stochastic differential equations (SDEs). We exploit the continuous formulation of SDE and the theory of Fokker-Planck equations to develop new results on the escaping phenomenon and the relationship with large batch and sharp minima. In particular, we find that the stochastic process solution tends to converge to flatter minima regardless of the batch size in the asymptotic regime. However, the convergence rate is rigorously proven to de…
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 Luxembourg（卢森堡大学）；University of Basel（巴塞尔大学）；MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）；ELLIS Tübingen（ELLIS蒂宾根）；Tübingen AI Center（蒂宾根人工智能中心）
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…
Mini-batch stochastic gradient descent (SGD) is state of the art in large scale distributed training. The scheme can reach a linear speedup with respect to the number of workers, but this is rarely seen in practice as the scheme often suffers from large network delays and bandwidth limits. To overcome this communication bottleneck recent works propose to reduce the communication frequency. An algorithm of this type is local SGD that runs SGD independently in parallel on different workers and averages the sequences only once in a while. This scheme shows promising results in practice, but elude…
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）
Jagiellonian University（雅盖隆大学）；Mila / University of Montreal（Mila / 蒙特利尔大学）；Facebook AI Research（Facebook人工智能研究院）；Ruhr University Bochum（波鸿鲁尔大学）；University of Edinburgh（爱丁堡大学）
Stochastic Gradient Descent (SGD) based training of neural networks with a large learning rate or a small batch-size typically ends in well-generalizing, flat regions of the weight space, as indicated by small eigenvalues of the Hessian of the training loss. However, the curvature along the SGD trajectory is poorly understood. An empirical investigation shows that initially SGD visits increasingly sharp regions, reaching a maximum sharpness determined by both the learning rate and the batch-size of SGD. When studying the SGD dynamics in relation to the sharpest directions in this initial phase…
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）
Institute of Science and Technology Austria（奥地利科学与技术研究所）；Sung Kyun Kwan University（顺天妇女大学）；New York University and Flatiron Institute（纽约大学和Flatiron研究所）；University of Southern California（南加州大学）
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…
Ruhr-University Bochum（波鸿鲁尔大学）；School of Informatics, University of Edinburgh（爱丁堡大学信息学院）；Jagiellonian University（雅盖隆大学）；MILA, Université de Montréal（蒙特利尔大学蒙特利尔学习算法研究所）；Facebook AI Research（脸书人工智能研究院）
We investigate the dynamical and convergent properties of stochastic gradient descent (SGD) applied to Deep Neural Networks (DNNs). Characterizing the relation between learning rate, batch size and the properties of the final minima, such as width or generalization, remains an open question. In order to tackle this problem we investigate the previously proposed approximation of SGD by a stochastic differential equation (SDE). We theoretically argue that three factors - learning rate, batch size and gradient covariance - influence the minima found by SGD. In particular we find that the ratio of…
Scaling the distributed deep learning to a massive GPU cluster level is challenging due to the instability of the large mini-batch training and the overhead of the gradient synchronization. We address the instability of the large mini-batch training with batch-size control and label smoothing. We address the overhead of the gradient synchronization with 2D-Torus all-reduce. Specifically, 2D-Torus all-reduce arranges GPUs in a logical 2D grid and performs a series of collective operation in different orientations. These two techniques are implemented with Neural Network Libraries (NNL). We have…
