Rutgers Business School（罗格斯商学院）；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）；Florida State University（佛罗里达州立大学）
Mert Gurbuzbalaban, Umut Şimşekli, Lingjiong Zhu
发表信息：
Published as a conference paper at International Conference on Machine Learning (ICML) 2021
In recent years, various notions of capacity and complexity have been proposed for characterizing the generalization properties of stochastic gradient descent (SGD) in deep learning. Some of the popular notions that correlate well with the performance on unseen data are (i) the `flatness' of the local minimum found by SGD, which is related to the eigenvalues of the Hessian, (ii) the ratio of the stepsize $η$ to the batch-size $b$, which essentially controls the magnitude of the stochastic gradient noise, and (iii) the `tail-index', which measures the heaviness of the tails of the network weights at convergence. In this paper, we argue that these three seemingly unrelated perspectives for generalization are deeply linked to each other. We claim that depending on the structure of the Hessian of the loss at the minimum, and the choices of the algorithm parameters $η$ and $b$, the SGD iterates will converge to a \emph{heavy-tailed} stationary distribution. We rigorously prove this claim in the setting of quadratic optimization: we show that even in a simple linear regression problem with independent and identically distributed data whose distribution has finite moments of all order, the iterates can be heavy-tailed with infinite variance. We further characterize the behavior of the tails with respect to algorithm parameters, the dimension, and the curvature. We then translate our results into insights about the behavior of SGD in deep learning. We support our theory with experiments conducted on synthetic data, fully connected, and convolutional neural networks.
Rutgers Business School（罗格斯商学院）；CNRS（法国国家科学研究中心）；INRIA（法国国家信息与自动化研究所）；DAMO Academy, Alibaba (US) Group（阿里巴巴（美国）集团达摩院）；Florida State University（佛罗里达州立大学）
Northwestern University（西北大学）；University of Washington（华盛顿大学）
The empirical success of deep learning is often attributed to SGD's mysterious ability to avoid sharp local minima in the loss landscape, as sharp minima are known to lead to poor generalization. Recently, empirical evidence of heavy-tailed gradient noise was reported in many deep learning tasks, and it was shown in Şimşekli (2019a,b) that SGD can escape sharp local minima under the presence of such heavy-tailed gradient noise, providing a partial solution to the mystery. In this work, we analyze a popular variant of SGD where gradients are truncated above a fixed threshold. We show that it ac…
Privacy of SGD under Gaussian or Heavy-Tailed Noise: Guarantees without Gradient Clipping
Inria, CNRS, Ecole Normale Supérieure PSL Research University（法国国家科学研究中心、巴黎高等师范学院PSL研究大学）；Department of Management Science and Information Systems（管理科学与信息系统系）；Faculty of Engineering and Natural Sciences（工程与自然科学学院）；Department of Mathematics（数学系）
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…
Cyclic and Randomized Stepsizes Invoke Heavier Tails in SGD than Constant Stepsize
Rutgers Business School（罗格斯商学院）；Princeton University（普林斯顿大学）；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）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；Florida State University（佛罗里达州立大学）
Revisiting Gradient Normalization and Clipping for Nonconvex SGD under Heavy-Tailed Noise: Necessity, Sufficiency, and Acceleration
College of Computer Science and Technology National University of Defense Technology（计算机科学与技术学院 国防科技大学）；Center for Machine Learning Research (CMLR) Peking University（机器学习研究中心（CMLR） 北京大学）
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；Florida State University（佛罗里达州立大学）；Rutgers University（罗格斯大学）；Princeton University（普林斯顿大学）；CNRS（法国国家科学研究中心）
Heavy-tail phenomena in stochastic gradient descent (SGD) have been reported in several empirical studies. Experimental evidence in previous works suggests a strong interplay between the heaviness of the tails and generalization behavior of SGD. To address this empirical phenomena theoretically, several works have made strong topological and statistical assumptions to link the generalization error to heavy tails. Very recently, new generalization bounds have been proven, indicating a non-monotonic relationship between the generalization error and heavy tails, which is more pertinent to the rep…
Why Adam Can Beat SGD: Second-Moment Normalization Yields Sharper Tails
Department of Electrical and Computer Engineering, The Ohio State University（俄亥俄州立大学电气与计算机工程系）；School of Electrical, Computer and Energy Engineering, Arizona State University（亚利桑那州立大学电气、计算机与能源工程学院）
While stochastic gradient descent (SGD) is still the \emph{de facto} algorithm in deep learning, adaptive methods like Clipped SGD/Adam have been observed to outperform SGD across important tasks, such as attention models. The settings under which SGD performs poorly in comparison to adaptive methods are not well understood yet. In this paper, we provide empirical and theoretical evidence that a heavy-tailed distribution of the noise in stochastic gradients is one cause of SGD's poor performance. We provide the first tight upper and lower convergence bounds for adaptive gradient methods under …
Emergence of heavy tails in homogenized stochastic gradient descent
School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；TU Dresden（德累斯顿工业大学）；ScaDS.ai Center for scalable data analytics and artificial intelligence（ScaDS.ai可扩展数据分析与人工智能中心）
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎理工学院巴黎电信学院LTCI实验室）；Department of Statistics, University of Oxford（牛津大学统计系）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）；Department of Management Science and Information Systems, Rutgers Business School（罗格斯大学商学院管理科学与信息系统系）
Stochastic gradient descent with momentum (SGDm) is one of the most popular optimization algorithms in deep learning. While there is a rich theory of SGDm for convex problems, the theory is considerably less developed in the context of deep learning where the problem is non-convex and the gradient noise might exhibit a heavy-tailed behavior, as empirically observed in recent studies. In this study, we consider a \emph{continuous-time} variant of SGDm, known as the underdamped Langevin dynamics (ULD), and investigate its asymptotic properties under heavy-tailed perturbations. Supported by recen…
Adynamical systems view of training generativemodels and the memorization phenomenon
International Institute for Theoretical Sciences（理论科学国际研究所）；Department of Computer Science and Automation（计算机科学与自动化系）；Indian Institute of Science（印度科学研究所）；Department of Electrical Engineering（电气工程系）；Indian Institute of Technology Bombay（博亚理工大学）
Approximate Heavy Tails in Offline (Multi-Pass) Stochastic Gradient Descent
Inria Paris（巴黎国立信息与自动化研究所）；CNRS（法国国家科学研究中心）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；CMAP（应用数学中心）；Ecole Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）
Faculty of Sciences（理学院）；University of Novi Sad（诺维萨德大学）；Faculty of Technical Sciences（技术科学学院）；Amazon Alexa AI（亚马逊Alexa人工智能部门）；Carnegie Mellon University（卡内基梅隆大学）
We introduce a general framework for nonlinear stochastic gradient descent (SGD) for the scenarios when gradient noise exhibits heavy tails. The proposed framework subsumes several popular nonlinearity choices, like clipped, normalized, signed or quantized gradient, but we also consider novel nonlinearity choices. We establish for the considered class of methods strong convergence guarantees assuming a strongly convex cost function with Lipschitz continuous gradients under very general assumptions on the gradient noise. Most notably, we show that, for a nonlinearity with bounded outputs and fo…
Salesforce Research（Salesforce研究院）；National University of Singapore（新加坡国立大学）；Princeton University（普林斯顿大学）
It is not clear yet why ADAM-alike adaptive gradient algorithms suffer from worse generalization performance than SGD despite their faster training speed. This work aims to provide understandings on this generalization gap by analyzing their local convergence behaviors. Specifically, we observe the heavy tails of gradient noise in these algorithms. This motivates us to analyze these algorithms through their Levy-driven stochastic differential equations (SDEs) because of the similar convergence behaviors of an algorithm and its SDE. Then we establish the escaping time of these SDEs from a local…
Boğaziçi University（海峡大学）；Télécom Paris（巴黎电信学院）；Institut Polytechnique de Paris（巴黎理工学院）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）；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）
Neural network compression techniques have become increasingly popular as they can drastically reduce the storage and computation requirements for very large networks. Recent empirical studies have illustrated that even simple pruning strategies can be surprisingly effective, and several theoretical studies have shown that compressible networks (in specific senses) should achieve a low generalization error. Yet, a theoretical characterization of the underlying cause that makes the networks amenable to such simple compression schemes is still missing. In this study, we address this fundamental …
Why is Normalization Preferred? A Worst-Case Complexity Theory for Stochastically Preconditioned SGD under Heavy-Tailed Noise
Department of Mathematics, University of California, Berkeley, CA, USA（加州大学伯克利分校数学系）；Department of EECS, University of California, Berkeley, CA, USA（加州大学伯克利分校电子工程与计算机科学系）；Department of IEOR, University of California, Berkeley, CA, USA（加州大学伯克利分校工业工程与运营研究系）
Peking University（北京大学）；University of Pennsylvania（宾夕法尼亚大学）
The phenomenon that stochastic gradient descent (SGD) favors flat minima has played a critical role in understanding the implicit regularization of SGD. In this paper, we provide an explanation of this striking phenomenon by relating the particular noise structure of SGD to its \emph{linear stability} (Wu et al., 2018). Specifically, we consider training over-parameterized models with square loss. We prove that if a global minimum $θ^*$ is linearly stable for SGD, then it must satisfy $\|H(θ^*)\|_F\leq O(\sqrt{B}/η)$, where $\|H(θ^*)\|_F, B,η$ denote the Frobenius norm of Hessian at $θ^*$, bat…
Implicit Compressibility of Overparametrized Neural Networks Trained with Heavy-Tailed SGD
Paris Research Center, Huawei Technologies France（华为技术法国巴黎研发中心）；Boğaziçi University（博加齐大学）；Université Gustave Eiffel（古斯塔夫·埃菲尔大学）；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）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
High-Probability PL-SGD with Markovian Noise: Optimal Mixing and Tail Dependence
Indian Institute of Technology Kharagpur（印度理工学院卡哈拉格普尔分校）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）；Purdue University（普渡大学）
In this paper, we propose a new accelerated stochastic first-order method called clipped-SSTM for smooth convex stochastic optimization with heavy-tailed distributed noise in stochastic gradients and derive the first high-probability complexity bounds for this method closing the gap in the theory of stochastic optimization with heavy-tailed noise. Our method is based on a special variant of accelerated Stochastic Gradient Descent (SGD) and clipping of stochastic gradients. We extend our method to the strongly convex case and prove new complexity bounds that outperform state-of-the-art results …
University of California, Los Angeles（加州大学洛杉矶分校）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of Washington（华盛顿大学）；Microsoft Research（微软研究院）
There is an increasing realization that algorithmic inductive biases are central in preventing overfitting; empirically, we often see a benign overfitting phenomenon in overparameterized settings for natural learning algorithms, such as stochastic gradient descent (SGD), where little to no explicit regularization has been employed. This work considers this issue in arguably the most basic setting: constant-stepsize SGD (with iterate averaging or tail averaging) for linear regression in the overparameterized regime. Our main result provides a sharp excess risk bound, stated in terms of the full…
Télécom Paris, Institut Polytechnique de Paris（巴黎理工学院巴黎电信学院）；University of Oxford（牛津大学）；Rutgers Business School（罗格斯商学院）；Facebook AI Research（脸书人工智能研究院）
The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the \emph{classical} central limit theorem (CLT) kicks in. This assumption is often made for mathematical convenience, since it enables SGD to be analyzed as a stochastic differential equation (SDE) driven by a Brownian motion. We argue that the Gaussianity assumption might fail to hold in deep learning settings and hence render the Brownian motion-based analyses inappropriate. Inspired by non-Gaussian natural phenomena, we consider the GN in a…
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…
LTCI, Télécom ParisTech, Université Paris-Saclay（巴黎萨克雷大学巴黎高等电信学院LTCI实验室）；Institute of Physics, École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院物理研究所）；Rutgers Business School（罗格斯商学院）
The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the classical central limit theorem (CLT) kicks in. This assumption is often made for mathematical convenience, since it enables SGD to be analyzed as a stochastic differential equation (SDE) driven by a Brownian motion. We argue that the Gaussianity assumption might fail to hold in deep learning settings and hence render the Brownian motion-based analyses inappropriate. Inspired by non-Gaussian natural phenomena, we consider the GN in a more g…
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…
LTCI Télécom Paris（巴黎电信）；Institut Polytechnique de Paris（巴黎综合理工学院）；University of Oxford（牛津大学）；Intel Labs（英特尔研究院）；The Alan Turing Institute（艾伦·图灵研究所）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）
Despite its success in a wide range of applications, characterizing the generalization properties of stochastic gradient descent (SGD) in non-convex deep learning problems is still an important challenge. While modeling the trajectories of SGD via stochastic differential equations (SDE) under heavy-tailed gradient noise has recently shed light over several peculiar characteristics of SGD, a rigorous treatment of the generalization properties of such SDEs in a learning theoretical framework is still missing. Aiming to bridge this gap, in this paper, we prove generalization bounds for SGD under …
Stochastic first-order methods such as Stochastic Extragradient (SEG) or Stochastic Gradient Descent-Ascent (SGDA) for solving smooth minimax problems and, more generally, variational inequality problems (VIP) have been gaining a lot of attention in recent years due to the growing popularity of adversarial formulations in machine learning. However, while high-probability convergence bounds are known to reflect the actual behavior of stochastic methods more accurately, most convergence results are provided in expectation. Moreover, the only known high-probability complexity results have been de…
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎理工学院巴黎电信学院 LTCI）；Rutgers University（罗格斯大学）
Stochastic gradient descent (SGD) has been widely used in machine learning due to its computational efficiency and favorable generalization properties. Recently, it has been empirically demonstrated that the gradient noise in several deep learning settings admits a non-Gaussian, heavy-tailed behavior. This suggests that the gradient noise can be modeled by using $α$-stable distributions, a family of heavy-tailed distributions that appear in the generalized central limit theorem. In this context, SGD can be viewed as a discretization of a stochastic differential equation (SDE) driven by a Lévy …
Johns Hopkins University（约翰斯·霍普金斯大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Harvard University（哈佛大学）
Stochastic gradient descent (SGD) has been shown to generalize well in many deep learning applications. In practice, one often runs SGD with a geometrically decaying stepsize, i.e., a constant initial stepsize followed by multiple geometric stepsize decay, and uses the last iterate as the output. This kind of SGD is known to be nearly minimax optimal for classical finite-dimensional linear regression problems (Ge et al., 2019). However, a sharp analysis for the last iterate of SGD in the overparameterized setting is still open. In this paper, we provide a problem-dependent analysis on the last…
The University of Iowa（爱荷华大学）；Alibaba Group（阿里巴巴集团）
Learning with a {\it convex loss} function has been a dominating paradigm for many years. It remains an interesting question how non-convex loss functions help improve the generalization of learning with broad applicability. In this paper, we study a family of objective functions formed by truncating traditional loss functions, which is applicable to both shallow learning and deep learning. Truncating loss functions has potential to be less vulnerable and more robust to large noise in observations that could be adversarial. More importantly, it is a generic technique without assuming the knowl…
Alan Turing Institute（艾伦·图灵研究所）；University of Oxford（牛津大学）；Florida State University（佛罗里达州立大学）；Rutgers Business School（罗格斯商学院）；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）
Gaussian noise injections (GNIs) are a family of simple and widely-used regularisation methods for training neural networks, where one injects additive or multiplicative Gaussian noise to the network activations at every iteration of the optimisation algorithm, which is typically chosen as stochastic gradient descent (SGD). In this paper we focus on the so-called `implicit effect' of GNIs, which is the effect of the injected noise on the dynamics of SGD. We show that this effect induces an asymmetric heavy-tailed noise on SGD gradient updates. In order to model this modified dynamics, we first…
Most convergence guarantees for stochastic gradient descent with momentum (SGDm) rely on iid sampling. Yet, SGDm is often used outside this regime, in settings with temporally correlated input samples such as continual learning and reinforcement learning. Existing work has shown that SGDm with a decaying step-size can converge under Markovian temporal correlation. In this work, we show that SGDm under covariate shift with a fixed step-size can be unstable and diverge. In particular, we show SGDm under covariate shift is a parametric oscillator, and so can suffer from a phenomenon known as reso…
