Stochastic gradient descent (SGD) is central to simulation optimization, stochastic programming, and online M-estimation, where sampling effort is a decision variable. We study the mini-batch gradient noise as a sampling-design object. Under exchangeable fresh-sampling mini-batches, the conditional covariance given the de Finetti directing measure mu is b^{-1} G_mu(theta), and under identifiability the projected population object is b^{-1} G*(theta) -- projected Fisher information for correctly specified likelihoods, the sandwich partner of the Hessian otherwise. This identification fixes the noise matrix entering the diffusion analysis of constant-step SGD: the raw iterate path has a deterministic fluid limit, and the sqrt(b/eta)-scaled fluctuations satisfy a functional CLT with noise covariance G*; near a nondegenerate optimum the limit is Ornstein-Uhlenbeck, and its Lyapunov covariance scaled by eta/b matches the linearized discrete recursion at leading order. Under a curvature-noise compatibility condition mu_F > 0, we prove 1/N mean-square upper bounds and an i.i.d. parametric Fisher van Trees lower bound of the same rate order, with oracle-complexity guarantees depending on an effective dimension d_eff and condition number kappa_F. Numerical experiments verify the identification and confirm the Lyapunov predictions in direct SGD.
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
We introduce biased gradient oracles to capture a setting where the function measurements have an estimation error that can be controlled through a batch size parameter. Our proposed oracles are appealing in several practical contexts, for instance, risk measure estimation from a batch of independent and identically distributed (i.i.d.) samples, or simulation optimization, where the function measurements are `biased' due to computational constraints. In either case, increasing the batch size reduces the estimation error. We highlight the applicability of our biased gradient oracles in a risk-s…
Numerical evaluations have definitively shown that, for deep learning optimizers such as stochastic gradient descent, momentum, and adaptive methods, the number of steps needed to train a deep neural network halves for each doubling of the batch size and that there is a region of diminishing returns beyond the critical batch size. In this paper, we determine the actual critical batch size by using the global minimizer of the stochastic first-order oracle (SFO) complexity of the optimizer. To prove the existence of the actual critical batch size, we set the lower and upper bounds of the SFO com…
Practical results have shown that deep learning optimizers using small constant learning rates, hyperparameters close to one, and large batch sizes can find the model parameters of deep neural networks that minimize the loss functions. We first show theoretical evidence that the momentum method (Momentum) and adaptive moment estimation (Adam) perform well in the sense that the upper bound of the theoretical performance measure is small with a small constant learning rate, hyperparameters close to one, and a large batch size. Next, we show that there exists a batch size called the critical batc…
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…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
With the large rising of complex data, the nonconvex models such as nonconvex loss function and nonconvex regularizer are widely used in machine learning and pattern recognition. In this paper, we propose a class of mini-batch stochastic ADMMs (alternating direction method of multipliers) for solving large-scale nonconvex nonsmooth problems. We prove that, given an appropriate mini-batch size, the mini-batch stochastic ADMM without variance reduction (VR) technique is convergent and reaches a convergence rate of $O(1/T)$ to obtain a stationary point of the nonconvex optimization, where $T$ den…
We propose a projected semi-stochastic gradient descent method with mini-batch for improving both the theoretical complexity and practical performance of the general stochastic gradient descent method (SGD). We are able to prove linear convergence under weak strong convexity assumption. This requires no strong convexity assumption for minimizing the sum of smooth convex functions subject to a compact polyhedral set, which remains popular across machine learning community. Our PS2GD preserves the low-cost per iteration and high optimization accuracy via stochastic gradient variance-reduced tech…
NTT DATA Mathematical Systems Inc.（NTT DATA 数理系统公司）；The University of Tokyo（东京大学）
In this paper, we develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings. The use of mini-batches is becoming a golden standard in the machine learning community, because mini-batch settings stabilize the gradient estimate and can easily make good use of parallel computing. The core of our proposed method is the incorporation of our new "double acceleration" technique and variance reduction technique. We theoretically analyze our proposed method and show that our method much improves the min…
We study the problem of model selection in batch policy optimization: given a fixed, partial-feedback dataset and $M$ model classes, learn a policy with performance that is competitive with the policy derived from the best model class. We formalize the problem in the contextual bandit setting with linear model classes by identifying three sources of error that any model selection algorithm should optimally trade-off in order to be competitive: (1) approximation error, (2) statistical complexity, and (3) coverage. The first two sources are common in model selection for supervised learning, wher…
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 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…
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
Determining the appropriate batch size for mini-batch gradient descent is always time consuming as it often relies on grid search. This paper considers a resizable mini-batch gradient descent (RMGD) algorithm based on a multi-armed bandit for achieving best performance in grid search by selecting an appropriate batch size at each epoch with a probability defined as a function of its previous success/failure. This probability encourages exploration of different batch size and then later exploitation of batch size with history of success. At each epoch, the RMGD samples a batch size from its pro…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；VinAI Research（VinAI研究院）
Mini-batch optimal transport (m-OT) has been widely used recently to deal with the memory issue of OT in large-scale applications. Despite their practicality, m-OT suffers from misspecified mappings, namely, mappings that are optimal on the mini-batch level but are partially wrong in the comparison with the optimal transportation plan between the original measures. Motivated by the misspecified mappings issue, we propose a novel mini-batch method by using partial optimal transport (POT) between mini-batch empirical measures, which we refer to as mini-batch partial optimal transport (m-POT). Le…
In this work we develop a new algorithm for regularized empirical risk minimization. Our method extends recent techniques of Shalev-Shwartz [02/2015], which enable a dual-free analysis of SDCA, to arbitrary mini-batching schemes. Moreover, our method is able to better utilize the information in the data defining the ERM problem. For convex loss functions, our complexity results match those of QUARTZ, which is a primal-dual method also allowing for arbitrary mini-batching schemes. The advantage of a dual-free analysis comes from the fact that it guarantees convergence even for non-convex loss f…
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…
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-…
University of Edinburgh（爱丁堡大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We address the issue of using mini-batches in stochastic optimization of SVMs. We show that the same quantity, the spectral norm of the data, controls the parallelization speedup obtained for both primal stochastic subgradient descent (SGD) and stochastic dual coordinate ascent (SCDA) methods and use it to derive novel variants of mini-batched SDCA. Our guarantees for both methods are expressed in terms of the original nonsmooth primal problem based on the hinge-loss.
Stochastic dual coordinate ascent (SDCA) is an effective technique for solving regularized loss minimization problems in machine learning. This paper considers an extension of SDCA under the mini-batch setting that is often used in practice. Our main contribution is to introduce an accelerated mini-batch version of SDCA and prove a fast convergence rate for this method. We discuss an implementation of our method over a parallel computing system, and compare the results to both the vanilla stochastic dual coordinate ascent and to the accelerated deterministic gradient descent method of \cite{ne…
University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校）
The Iterative Hard Thresholding (IHT) algorithm has been considered extensively as an effective deterministic algorithm for solving sparse optimizations. The IHT algorithm benefits from the information of the batch (full) gradient at each point and this information is a crucial key for the convergence analysis of the generated sequence. However, this strength becomes a weakness when it comes to machine learning and high dimensional statistical applications because calculating the batch gradient at each iteration is computationally expensive or impractical. Fortunately, in these applications th…
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-…
The stochastic gradient descent (SGD) method and its variants are algorithms of choice for many Deep Learning tasks. These methods operate in a small-batch regime wherein a fraction of the training data, say $32$-$512$ data points, is sampled to compute an approximation to the gradient. It has been observed in practice that when using a larger batch there is a degradation in the quality of the model, as measured by its ability to generalize. We investigate the cause for this generalization drop in the large-batch regime and present numerical evidence that supports the view that large-batch met…
Huazhong University of Science and Technology（华中科技大学）
Takagi-Sugeno-Kang (TSK) fuzzy systems are flexible and interpretable machine learning models; however, they may not be easily optimized when the data size is large, and/or the data dimensionality is high. This paper proposes a mini-batch gradient descent (MBGD) based algorithm to efficiently and effectively train TSK fuzzy classifiers. It integrates two novel techniques: 1) uniform regularization (UR), which forces the rules to have similar average contributions to the output, and hence to increase the generalization performance of the TSK classifier; and, 2) batch normalization (BN), which e…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；VinAI Research（VinAI研究院）；Monash University（莫纳什大学）
Mini-batch optimal transport (m-OT) has been successfully used in practical applications that involve probability measures with a very high number of supports. The m-OT solves several smaller optimal transport problems and then returns the average of their costs and transportation plans. Despite its scalability advantage, the m-OT does not consider the relationship between mini-batches which leads to undesirable estimation. Moreover, the m-OT does not approximate a proper metric between probability measures since the identity property is not satisfied. To address these problems, we propose a n…
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 …
University of Edinburgh（爱丁堡大学）；Lehigh University（理海大学）
We propose a mini-batching scheme for improving the theoretical complexity and practical performance of semi-stochastic gradient descent applied to the problem of minimizing a strongly convex composite function represented as the sum of an average of a large number of smooth convex functions, and simple nonsmooth convex function. Our method first performs a deterministic step (computation of the gradient of the objective function at the starting point), followed by a large number of stochastic steps. The process is repeated a few times with the last iterate becoming the new starting point. The…
University of Edinburgh（爱丁堡大学）；Lehigh University（利哈伊大学）
We propose mS2GD: a method incorporating a mini-batching scheme for improving the theoretical complexity and practical performance of semi-stochastic gradient descent (S2GD). We consider the problem of minimizing a strongly convex function represented as the sum of an average of a large number of smooth convex functions, and a simple nonsmooth convex regularizer. Our method first performs a deterministic step (computation of the gradient of the objective function at the starting point), followed by a large number of stochastic steps. The process is repeated a few times with the last iterate be…
Microsoft Research AI（微软研究院AI）；Princeton University（普林斯顿大学）
We propose a reduction for non-convex optimization that can (1) turn an stationary-point finding algorithm into an local-minimum finding one, and (2) replace the Hessian-vector product computations with only gradient computations. It works both in the stochastic and the deterministic settings, without hurting the algorithm's performance. As applications, our reduction turns Natasha2 into a first-order method without hurting its performance. It also converts SGD, GD, SCSG, and SVRG into algorithms finding approximate local minima, outperforming some best known results.
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…
The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient
School of Statistics, Renmin University of China（中国人民大学统计学院）；School of Computing and Data Science, The University of Hong Kong（香港大学计算与数据科学学院）；School of Computing and Data Science & Institute of Data Science, The University of Hong Kong（香港大学计算与数据科学学院及数据科学研究所）
The mini-batch stochastic gradient descent (SGD) algorithm is widely used in training machine learning models, in particular deep learning models. We study SGD dynamics under linear regression and two-layer linear networks, with an easy extension to deeper linear networks, by focusing on the variance of the gradients, which is the first study of this nature. In the linear regression case, we show that in each iteration the norm of the gradient is a decreasing function of the mini-batch size $b$ and thus the variance of the stochastic gradient estimator is a decreasing function of $b$. For deep…
IBM T.J. Watson Research Center（IBM托马斯·J·沃森研究中心）；Amazon Alexa（亚马逊Alexa）；Morgan Stanley（摩根士丹利）；Rice University（莱斯大学）
This paper presents a methodology for selecting the mini-batch size that minimizes Stochastic Gradient Descent (SGD) learning time for single and multiple learner problems. By decoupling algorithmic analysis issues from hardware and software implementation details, we reveal a robust empirical inverse law between mini-batch size and the average number of SGD updates required to converge to a specified error threshold. Combining this empirical inverse law with measured system performance, we create an accurate, closed-form model of average training time and show how this model can be used to id…
Microsoft Research（微软研究院）；Disney Research（迪士尼研究院）；KTH Royal Institute of Technology（瑞典皇家理工学院）
The convergence speed of stochastic gradient descent (SGD) can be improved by actively selecting mini-batches. We explore sampling schemes where similar data points are less likely to be selected in the same mini-batch. In particular, we prove that such repulsive sampling schemes lowers the variance of the gradient estimator. This generalizes recent work on using Determinantal Point Processes (DPPs) for mini-batch diversification (Zhang et al., 2017) to the broader class of repulsive point processes. We first show that the phenomenon of variance reduction by diversified sampling generalizes in…
In high dimensions, most machine learning method perform fragile even there are a little outliers. To address this, we hope to introduce a new method with the base learner, such as Bayesian regression or stochastic gradient descent to solve the problem of the vulnerability in the model. Because the mini-batch gradient descent allows for a more robust convergence than the batch gradient descent, we work a method with the mini-batch gradient descent, called Mini-Batch Gradient Descent with Trimming (MBGDT). Our method show state-of-art performance and have greater robustness than several baselin…
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…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣地亚哥分校）；Google（谷歌）
Data compression is a popular technique for improving the efficiency of data processing workloads such as SQL queries and more recently, machine learning (ML) with classical batch gradient methods. But the efficacy of such ideas for mini-batch stochastic gradient descent (MGD), arguably the workhorse algorithm of modern ML, is an open question. MGD's unique data access pattern renders prior art, including those designed for batch gradient methods, less effective. We fill this crucial research gap by proposing a new lossless compression scheme we call tuple-oriented compression (TOC) that is in…
Mini-batch algorithms have been proposed as a way to speed-up stochastic convex optimization problems. We study how such algorithms can be improved using accelerated gradient methods. We provide a novel analysis, which shows how standard gradient methods may sometimes be insufficient to obtain a significant speed-up and propose a novel accelerated gradient algorithm, which deals with this deficiency, enjoys a uniformly superior guarantee and works well in practice.
Stochastic gradient descent plays a fundamental role in nearly all applications of deep learning. However its ability to converge to a global minimum remains shrouded in mystery. In this paper we propose to study the behavior of the loss function on fixed mini-batches along SGD trajectories. We show that the loss function on a fixed batch appears to be remarkably convex-like. In particular for ResNet the loss for any fixed mini-batch can be accurately modeled by a quadratic function and a very low loss value can be reached in just one step of gradient descent with sufficiently large learning r…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
We propose a metric for evaluating the generalization ability of deep neural networks trained with mini-batch gradient descent. Our metric, called gradient disparity, is the $\ell_2$ norm distance between the gradient vectors of two mini-batches drawn from the training set. It is derived from a probabilistic upper bound on the difference between the classification errors over a given mini-batch, when the network is trained on this mini-batch and when the network is trained on another mini-batch of points sampled from the same dataset. We empirically show that gradient disparity is a very promi…
Optimal Transport (OT) is a fundamental tool for comparing probability distributions, but its exact computation remains prohibitive for large datasets. In this work, we introduce novel families of upper and lower bounds for the OT problem constructed by aggregating solutions of mini-batch OT problems. The upper bound family contains traditional mini-batch averaging at one extreme and a tight bound found by optimal coupling of mini-batches at the other. In between these extremes, we propose various methods to construct bounds based on a fixed computational budget. Through various experiments, w…
Continuum-armed Bandit Optimization with Batch Pairwise Comparison Oracles
Department of Information Systems and Intelligence Business, Xi’an Jiaotong University, Xi’an 710049, China（西安交通大学信息系统与智能商业系）；Stern School of Business, New York University, New York, NY 10012, USA（纽约大学斯特恩商学院）；Naveen Jindal School of Management, University of Texas at Dallas, Richardson, TX 75080, USA（德克萨斯大学达拉斯分校奈维恩·金达管理学院）
University of Southern California（南加州大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）
We study the problem of training machine learning models incrementally with batches of samples annotated with noisy oracles. We select each batch of samples that are important and also diverse via clustering and importance sampling. More importantly, we incorporate model uncertainty into the sampling probability to compensate for poor estimation of the importance scores when the training data is too small to build a meaningful model. Experiments on benchmark image classification datasets (MNIST, SVHN, CIFAR10, and EMNIST) show improvement over existing active learning strategies. We introduce …
A new algorithm is proposed which accelerates the mini-batch k-means algorithm of Sculley (2010) by using the distance bounding approach of Elkan (2003). We argue that, when incorporating distance bounds into a mini-batch algorithm, already used data should preferentially be reused. To this end we propose using nested mini-batches, whereby data in a mini-batch at iteration t is automatically reused at iteration t+1. Using nested mini-batches presents two difficulties. The first is that unbalanced use of data can bias estimates, which we resolve by ensuring that each data sample contributes exa…
We study the problem of reducing the amount of labeled training data required to train supervised classification models. We approach it by leveraging Active Learning, through sequential selection of examples which benefit the model most. Selecting examples one by one is not practical for the amount of training examples required by the modern Deep Learning models. We consider the mini-batch Active Learning setting, where several examples are selected at once. We present an approach which takes into account both informativeness of the examples for the model, as well as the diversity of the examp…
Mini-batch sub-sampling in neural network training is unavoidable, due to growing data demands, memory-limited computational resources such as graphical processing units (GPUs), and the dynamics of on-line learning. In this study we specifically distinguish between static mini-batch sub-sampled loss functions, where mini-batches are intermittently fixed during training, resulting in smooth but biased loss functions; and the dynamic sub-sampling equivalent, where new mini-batches are sampled at every loss evaluation, trading bias for variance in sampling induced discontinuities. These render au…
Technical University of Munich（慕尼黑工业大学）；University of Hamburg（汉堡大学）；Friedrich-Alexander University Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希-亚历山大大学）
Federated learning has faced performance and network communication challenges, especially in the environments where the data is not independent and identically distributed (IID) across the clients. To address the former challenge, we introduce the federated-centralized concordance property and show that the federated single-mini-batch training approach can achieve comparable performance as the corresponding centralized training in the Non-IID environments. To deal with the latter, we present the federated multi-mini-batch approach and illustrate that it can establish a trade-off between the pe…
Modern deep neural network training is typically based on mini-batch stochastic gradient optimization. While the use of large mini-batches increases the available computational parallelism, small batch training has been shown to provide improved generalization performance and allows a significantly smaller memory footprint, which might also be exploited to improve machine throughput. In this paper, we review common assumptions on learning rate scaling and training duration, as a basis for an experimental comparison of test performance for different mini-batch sizes. We adopt a learning rate th…
Area under the receiver operating characteristics curve (AUC) is an important metric for a wide range of signal processing and machine learning problems, and scalable methods for optimizing AUC have recently been proposed. However, handling very large datasets remains an open challenge for this problem. This paper proposes a novel approach to AUC maximization, based on sampling mini-batches of positive/negative instance pairs and computing U-statistics to approximate a global risk minimization problem. The resulting algorithm is simple, fast, and learning-rate free. We show that the number of …
Batch Normalization (BN) uses mini-batch statistics to normalize the activations during training, introducing dependence between mini-batch elements. This dependency can hurt the performance if the mini-batch size is too small, or if the elements are correlated. Several alternatives, such as Batch Renormalization and Group Normalization (GN), have been proposed to address this issue. However, they either do not match the performance of BN for large batches, or still exhibit degradation in performance for smaller batches, or introduce artificial constraints on the model architecture. In this pa…
