While momentum-based methods, in conjunction with stochastic gradient descent (SGD), are widely used when training machine learning models, there is little theoretical understanding on the generalization error of such methods. In this work, we first show that there exists a convex loss function for which algorithmic stability fails to establish generalization guarantees when SGD with standard heavy-ball momentum (SGDM) is run for multiple epochs. Then, for smooth Lipschitz loss functions, we analyze a modified momentum-based update rule, i.e., SGD with early momentum (SGDEM), and show that it admits an upper-bound on the generalization error. Thus, our results show that machine learning models can be trained for multiple epochs of SGDEM with a guarantee for generalization. Finally, for the special case of strongly convex loss functions, we find a range of momentum such that multiple epochs of standard SGDM, as a special form of SGDEM, also generalizes. Extending our results on generalization, we also develop an upper-bound on the expected true risk, in terms of the number of training steps, the size of the training set, and the momentum parameter. Experimental evaluations verify the consistency between the numerical results and our theoretical bounds and the effectiveness of SGDEM for smooth Lipschitz loss functions.
College of Computer, National University of Defense Technology（国防科技大学计算机学院）；Scientific Computing & Imaging Institute, University of Utah（犹他大学科学计算与成像研究所）
The stability and generalization of stochastic gradient-based methods provide valuable insights into understanding the algorithmic performance of machine learning models. As the main workhorse for deep learning, stochastic gradient descent has received a considerable amount of studies. Nevertheless, the community paid little attention to its decentralized variants. In this paper, we provide a novel formulation of the decentralized stochastic gradient descent. Leveraging this formulation together with (non)convex optimization theory, we establish the first stability and generalization guarantee…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very recently, a line of work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs. However, existing generalization error bounds are unable to explain the good generalization performance of over-parameterized DNNs. The major limitation of most existing …
Princeton University（普林斯顿大学）；Beijing Institute of Big Data Research（北京大数据研究院）
A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are updated. General initialization schemes as well as general regimes for the network width and training data size are considered. In the over-parametrized regime, it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the labels. In addition, it is proved that throughout the training process the functions represented by the neural network model are unif…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We study the training and generalization of deep neural networks (DNNs) in the over-parameterized regime, where the network width (i.e., number of hidden nodes per layer) is much larger than the number of training data points. We show that, the expected $0$-$1$ loss of a wide enough ReLU network trained with stochastic gradient descent (SGD) and random initialization can be bounded by the training loss of a random feature model induced by the network gradient at initialization, which we call a neural tangent random feature (NTRF) model. For data distributions that can be classified by NTRF mod…
It is well known that the finite step-size ($h$) in Gradient Descent (GD) implicitly regularizes solutions to flatter minima. A natural question to ask is "Does the momentum parameter $β$ play a role in implicit regularization in Heavy-ball (H.B) momentum accelerated gradient descent (GD+M)?". To answer this question, first, we show that the discrete H.B momentum update (GD+M) follows a continuous trajectory induced by a modified loss, which consists of an original loss and an implicit regularizer. Then, we show that this implicit regularizer for (GD+M) is stronger than that of (GD) by factor …
Most modern learning problems are highly overparameterized, meaning that there are many more parameters than the number of training data points, and as a result, the training loss may have infinitely many global minima (parameter vectors that perfectly interpolate the training data). Therefore, it is important to understand which interpolating solutions we converge to, how they depend on the initialization point and the learning algorithm, and whether they lead to different generalization performances. In this paper, we study these questions for the family of stochastic mirror descent (SMD) al…
Maynooth University（梅努斯大学）；Max-Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）
We consider gradient descent with `momentum', a widely used method for loss function minimization in machine learning. This method is often used with `Nesterov acceleration', meaning that the gradient is evaluated not at the current position in parameter space, but at the estimated position after one step. In this work, we show that the algorithm can be improved by extending this `acceleration' --- by using the gradient at an estimated position several steps ahead rather than just one step ahead. How far one looks ahead in this `super-acceleration' algorithm is determined by a new hyperparamet…
University of Science and Technology of China（中国科学技术大学）；University of Warwick（华威大学）
Federated learning (FL) provides a communication-efficient approach to solve machine learning problems concerning distributed data, without sending raw data to a central server. However, existing works on FL only utilize first-order gradient descent (GD) and do not consider the preceding iterations to gradient update which can potentially accelerate convergence. In this paper, we consider momentum term which relates to the last iteration. The proposed momentum federated learning (MFL) uses momentum gradient descent (MGD) in the local update step of FL system. We establish global convergence pr…
Changchun University of Science and Technology（长春理工大学）；Northeast Normal University（东北师范大学）
The plain stochastic gradient descent and momentum stochastic gradient descent have extremely wide applications in deep learning due to their simple settings and low computational complexity. The momentum stochastic gradient descent uses the accumulated gradient as the updated direction of the current parameters, which has a faster training speed. Because the direction of the plain stochastic gradient descent has not been corrected by the accumulated gradient. For the parameters that currently need to be updated, it is the optimal direction, and its update is more accurate. We combine the adva…
A major challenge in understanding the generalization of deep learning is to explain why (stochastic) gradient descent can exploit the network architecture to find solutions that have good generalization performance when using high capacity models. We find simple but realistic examples showing that this phenomenon exists even when learning linear classifiers --- between two linear networks with the same capacity, the one with a convolutional layer can generalize better than the other when the data distribution has some underlying spatial structure. We argue that this difference results from a …
Northeast Normal University（东北师范大学）；Beihua University（北华大学）
The study on the implicit regularization induced by gradient-based optimization is a longstanding pursuit. In the present paper, we characterize the implicit regularization of momentum gradient descent (MGD) with early stopping by comparing with the explicit $\ell_2$-regularization (ridge). In details, we study MGD in the continuous-time view, so-called momentum gradient flow (MGF), and show that its tendency is closer to ridge than the gradient descent (GD) [Ali et al., 2019] for least squares regression. Moreover, we prove that, under the calibration $t=\sqrt{2/λ}$, where $t$ is the time par…
State Key Laboratory of Scientific and Engineering Computing（科学与工程计算国家重点实验室）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Following the same routine as [SSJ20], we continue to present the theoretical analysis for stochastic gradient descent with momentum (SGD with momentum) in this paper. Differently, for SGD with momentum, we demonstrate it is the two hyperparameters together, the learning rate and the momentum coefficient, that play the significant role for the linear rate of convergence in non-convex optimization. Our analysis is based on the use of a hyperparameters-dependent stochastic differential equation (hp-dependent SDE) that serves as a continuous surrogate for SGD with momentum. Similarly, we establis…
Universitat Pompeu Fabra（庞培法布拉大学）；Element AI（Element AI公司）；Mila（米拉实验室）；University of Toronto（多伦多大学）；Vector Institute（矢量研究院）
We study the generalization properties of the popular stochastic optimization method known as stochastic gradient descent (SGD) for optimizing general non-convex loss functions. Our main contribution is providing upper bounds on the generalization error that depend on local statistics of the stochastic gradients evaluated along the path of iterates calculated by SGD. The key factors our bounds depend on are the variance of the gradients (with respect to the data distribution) and the local smoothness of the objective function along the SGD path, and the sensitivity of the loss function to pert…
Convergence detection of iterative stochastic optimization methods is of great practical interest. This paper considers stochastic gradient descent (SGD) with a constant learning rate and momentum. We show that there exists a transient phase in which iterates move towards a region of interest, and a stationary phase in which iterates remain bounded in that region around a minimum point. We construct a statistical diagnostic test for convergence to the stationary phase using the inner product between successive gradients and demonstrate that the proposed diagnostic works well. We theoretically …
The University of Edinburgh（爱丁堡大学）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Moscow Institute of Physics and Technology (MIPT)（莫斯科物理技术学院）
In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these methods are studied. We choose to perform our analysis in a setting in which all of the above methods are equivalent. We prove global nonassymptotic linear convergence rates for all methods and various measures of success, including primal function values, primal iterates (in L2 sense…
It has long been argued that minibatch stochastic gradient descent can generalize better than large batch gradient descent in deep neural networks. However recent papers have questioned this claim, arguing that this effect is simply a consequence of suboptimal hyperparameter tuning or insufficient compute budgets when the batch size is large. In this paper, we perform carefully designed experiments and rigorous hyperparameter sweeps on a range of popular models, which verify that small or moderately large batch sizes can substantially outperform very large batches on the test set. This occurs …
Hong Kong University of Science and Technology（香港科技大学）
Stochastic methods with coordinate-wise adaptive stepsize (such as RMSprop and Adam) have been widely used in training deep neural networks. Despite their fast convergence, they can generalize worse than stochastic gradient descent. In this paper, by revisiting the design of Adagrad, we propose to split the network parameters into blocks, and use a blockwise adaptive stepsize. Intuitively, blockwise adaptivity is less aggressive than adaptivity to individual coordinates, and can have a better balance between adaptivity and generalization. We show theoretically that the proposed blockwise adapt…
University of Utah（犹他大学）；University of Kentucky（肯塔基大学）
Momentum plays a crucial role in stochastic gradient-based optimization algorithms for accelerating or improving training deep neural networks (DNNs). In deep learning practice, the momentum is usually weighted by a well-calibrated constant. However, tuning hyperparameters for momentum can be a significant computational burden. In this paper, we propose a novel \emph{adaptive momentum} for improving DNNs training; this adaptive momentum, with no momentum related hyperparameter required, is motivated by the nonlinear conjugate gradient (NCG) method. Stochastic gradient descent (SGD) with this n…
For strongly convex objectives that are smooth, the classical theory of gradient descent ensures linear convergence relative to the number of gradient evaluations. An analogous nonsmooth theory is challenging. Even when the objective is smooth at every iterate, the corresponding local models are unstable and the number of cutting planes invoked by traditional remedies is difficult to bound, leading to convergences guarantees that are sublinear relative to the cumulative number of gradient evaluations. We instead propose a multipoint generalization of the gradient descent iteration for local op…
Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
Two major momentum-based techniques that have achieved tremendous success in optimization are Polyak's heavy ball method and Nesterov's accelerated gradient. A crucial step in all momentum-based methods is the choice of the momentum parameter $m$ which is always suggested to be set to less than $1$. Although the choice of $m < 1$ is justified only under very strong theoretical assumptions, it works well in practice even when the assumptions do not necessarily hold. In this paper, we propose a new momentum based method $\textit{ADINE}$, which relaxes the constraint of $m < 1$ and allows the lea…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We study to what extent may stochastic gradient descent (SGD) be understood as a "conventional" learning rule that achieves generalization performance by obtaining a good fit to training data. We consider the fundamental stochastic convex optimization framework, where (one pass, without-replacement) SGD is classically known to minimize the population risk at rate $O(1/\sqrt n)$, and prove that, surprisingly, there exist problem instances where the SGD solution exhibits both empirical risk and generalization gap of $Ω(1)$. Consequently, it turns out that SGD is not algorithmically stable in any…
We derive a simple and model-independent formula for the change in the generalization gap due to a gradient descent update. We then compare the change in the test error for stochastic gradient descent to the change in test error from an equivalent number of gradient descent updates and show explicitly that stochastic gradient descent acts to regularize generalization error by decorrelating nearby updates. These calculations depends on the details of the model only through the mean and covariance of the gradient distribution, which may be readily measured for particular models of interest. We d…
Recently there are a considerable amount of work devoted to the study of the algorithmic stability and generalization for stochastic gradient descent (SGD). However, the existing stability analysis requires to impose restrictive assumptions on the boundedness of gradients, strong smoothness and convexity of loss functions. In this paper, we provide a fine-grained analysis of stability and generalization for SGD by substantially relaxing these assumptions. Firstly, we establish stability and generalization for SGD by removing the existing bounded gradient assumptions. The key idea is the introd…
Optimal Rates for Generalization of Gradient Descent Methods with Deep Neural Networks
Mathematical Institute for Machine Learning and Data Science, KU Eichstätt-Ingolstadt（机器学习与数据科学数学研究所，埃施特哈特-英戈尔施塔特大学）；Department of Computer Science, RPTU Kaiserslautern-Landau（计算机科学系，凯撒斯劳滕-兰道大学）；Department of Mathematics, University of Hong Kong（数学系，香港大学）；School of Mathematics and Statistics, University of Sydney（数学与统计学学院，悉尼大学）
Stochastic Gradient Descent (SGD) methods are prominent for training machine learning and deep learning models. The performance of these techniques depends on their hyperparameter tuning over time and varies for different models and problems. Manual adjustment of hyperparameters is very costly and time-consuming, and even if done correctly, it lacks theoretical justification which inevitably leads to "rule of thumb" settings. In this paper, we propose a generic approach that utilizes the statistics of an unbiased gradient estimator to automatically and simultaneously adjust two paramount hyper…
Modern deep learning models employ considerably more parameters than required to fit the training data. Whereas conventional statistical wisdom suggests such models should drastically overfit, in practice these models generalize remarkably well. An emerging paradigm for describing this unexpected behavior is in terms of a \emph{double descent} curve, in which increasing a model's capacity causes its test error to first decrease, then increase to a maximum near the interpolation threshold, and then decrease again in the overparameterized regime. Recent efforts to explain this phenomenon theoret…
Northwest University（西北大学）；Southwestern University of Finance and Economics（西南财经大学）；University of Birmingham（伯明翰大学）
Randomized coordinate descent (RCD) is a popular optimization algorithm with wide applications in solving various machine learning problems, which motivates a lot of theoretical analysis on its convergence behavior. As a comparison, there is no work studying how the models trained by RCD would generalize to test examples. In this paper, we initialize the generalization analysis of RCD by leveraging the powerful tool of algorithmic stability. We establish argument stability bounds of RCD for both convex and strongly convex objectives, from which we develop optimal generalization bounds by showi…
IBM Research AI（IBM研究院人工智能中心）；Colorado State University-Pueblo（科罗拉多州立大学普韦布洛分校）
Distributionally robust optimization (DRO) problems are increasingly seen as a viable method to train machine learning models for improved model generalization. These min-max formulations, however, are more difficult to solve. We therefore provide a new stochastic gradient descent algorithm to efficiently solve this DRO formulation. Our approach applies gradient descent to the outer minimization formulation and estimates the gradient of the inner maximization based on a sample average approximation. The latter uses a subset of the data in each iteration, progressively increasing the subset siz…
An open question in the Deep Learning community is why neural networks trained with Gradient Descent generalize well on real datasets even though they are capable of fitting random data. We propose an approach to answering this question based on a hypothesis about the dynamics of gradient descent that we call Coherent Gradients: Gradients from similar examples are similar and so the overall gradient is stronger in certain directions where these reinforce each other. Thus changes to the network parameters during training are biased towards those that (locally) simultaneously benefit many exampl…
The Cartesian reverse derivative is a categorical generalization of reverse-mode automatic differentiation. We use this operator to generalize several optimization algorithms, including a straightforward generalization of gradient descent and a novel generalization of Newton's method. We then explore which properties of these algorithms are preserved in this generalized setting. First, we show that the transformation invariances of these algorithms are preserved: while generalized Newton's method is invariant to all invertible linear transformations, generalized gradient descent is invariant o…
We study the linear subspace fitting problem in the overparameterized setting, where the estimated subspace can perfectly interpolate the training examples. Our scope includes the least-squares solutions to subspace fitting tasks with varying levels of supervision in the training data (i.e., the proportion of input-output examples of the desired low-dimensional mapping) and orthonormality of the vectors defining the learned operator. This flexible family of problems connects standard, unsupervised subspace fitting that enforces strict orthonormality with a corresponding regression task that is…
University of Utah（犹他大学）；Rice University（莱斯大学）；National University of Defense Technology（国防科技大学）；University of California, Los Angeles（加州大学洛杉矶分校）
Stochastic gradient descent (SGD) with constant momentum and its variants such as Adam are the optimization algorithms of choice for training deep neural networks (DNNs). Since DNN training is incredibly computationally expensive, there is great interest in speeding up the convergence. Nesterov accelerated gradient (NAG) improves the convergence rate of gradient descent (GD) for convex optimization using a specially designed momentum; however, it accumulates error when an inexact gradient is used (such as in SGD), slowing convergence at best and diverging at worst. In this paper, we propose Sc…
The skip-connections used in residual networks have become a standard architecture choice in deep learning due to the increased training stability and generalization performance with this architecture, although there has been limited theoretical understanding for this improvement. In this work, we analyze overparameterized deep residual networks trained by gradient descent following random initialization, and demonstrate that (i) the class of networks learned by gradient descent constitutes a small subset of the entire neural network function class, and (ii) this subclass of networks is suffic…
Temple University（天普大学）；University of Pittsburgh（匹兹堡大学）
Decentralized training has been actively studied in recent years. Although a wide variety of methods have been proposed, yet the decentralized momentum SGD method is still underexplored. In this paper, we propose a novel periodic decentralized momentum SGD method, which employs the momentum schema and periodic communication for decentralized training. With these two strategies, as well as the topology of the decentralized training system, the theoretical convergence analysis of our proposed method is difficult. We address this challenging problem and provide the condition under which our propo…
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）
Despite being highly over-parametrized, and having the ability to fully interpolate the training data, deep networks are known to generalize well to unseen data. It is now understood that part of the reason for this is that the training algorithms used have certain implicit regularization properties that ensure interpolating solutions with "good" properties are found. This is best understood in linear over-parametrized models where it has been shown that the celebrated stochastic gradient descent (SGD) algorithm finds an interpolating solution that is closest in Euclidean distance to the initi…
Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Recent evidence has shown the existence of a so-called double-descent and even triple-descent behavior for the generalization error of deep-learning models. This important phenomenon commonly appears in implemented neural network architectures, and also seems to emerge in epoch-wise curves during the training process. A recent line of research has highlighted that random matrix tools can be used to obtain precise analytical asymptotics of the generalization (and training) errors of the random feature model. In this contribution, we analyze the whole temporal behavior of the generalization and …
We introduce Kalman Gradient Descent, a stochastic optimization algorithm that uses Kalman filtering to adaptively reduce gradient variance in stochastic gradient descent by filtering the gradient estimates. We present both a theoretical analysis of convergence in a non-convex setting and experimental results which demonstrate improved performance on a variety of machine learning areas including neural networks and black box variational inference. We also present a distributed version of our algorithm that enables large-dimensional optimization, and we extend our algorithm to SGD with momentum…
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 Ohio State University（俄亥俄州立大学）；Southern University of Science and Technology（南方科技大学）
Existing convergence analyses of Q-learning mostly focus on the vanilla stochastic gradient descent (SGD) type of updates. Despite the Adaptive Moment Estimation (Adam) has been commonly used for practical Q-learning algorithms, there has not been any convergence guarantee provided for Q-learning with such type of updates. In this paper, we first characterize the convergence rate for Q-AMSGrad, which is the Q-learning algorithm with AMSGrad update (a commonly adopted alternative of Adam for theoretical analysis). To further improve the performance, we propose to incorporate the momentum restar…
We provide (high probability) bounds on the condition number of random feature matrices. In particular, we show that if the complexity ratio $\frac{N}{m}$ where $N$ is the number of neurons and $m$ is the number of data samples scales like $\log^{-1}(N)$ or $\log(m)$, then the random feature matrix is well-conditioned. This result holds without the need of regularization and relies on establishing various concentration bounds between dependent components of the random feature matrix. Additionally, we derive bounds on the restricted isometry constant of the random feature matrix. We prove that …
Homophily modulates double descent generalization in graph convolution networks
Universität Basel（巴塞尔大学）；University of Science and Technology of China（中国科学技术大学）；Nanjing Normal University（南京师范大学）；University of Pennsylvania（宾夕法尼亚大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
NEC Corporation（日本电气股份有限公司）；Biometrics Research Laboratories, NEC Corporation（日本电气股份有限公司生物识别研究实验室）
In this paper, we propose a novel uniform generalization bound on the time and inverse temperature for stochastic gradient Langevin dynamics (SGLD) in a non-convex setting. While previous works derive their generalization bounds by uniform stability, we use Rademacher complexity to make our generalization bound independent of the time and inverse temperature. Using Rademacher complexity, we can reduce the problem to derive a generalization bound on the whole space to that on a bounded region and therefore can remove the effect of the time and inverse temperature from our generalization bound. …
