Adaptive gradient methods such as Adam have gained extreme popularity due to their success in training complex neural networks and less sensitivity to hyperparameter tuning compared to SGD. However, it has been recently shown that Adam can fail to converge and might cause poor generalization -- this lead to the design of new, sophisticated adaptive methods which attempt to generalize well while being theoretically reliable. In this technical report we focus on AdaBound, a promising, recently proposed optimizer. We present a stochastic convex problem for which AdaBound can provably take arbitrarily long to converge in terms of a factor which is not accounted for in the convergence rate guarantee of Luo et al. (2019). We present a new $O(\sqrt T)$ regret guarantee under different assumptions on the bound functions, and provide empirical results on CIFAR suggesting that a specific form of momentum SGD can match AdaBound's performance while having less hyperparameters and lower computational costs.
DiDi AI Labs（滴滴人工智能实验室）；Peking University（北京大学）；Zhejiang University（浙江大学）；University of Southern California（南加州大学）；Beijing Institute of Big Data Research（北京大数据研究院）
Adaptive optimization methods such as AdaGrad, RMSprop and Adam have been proposed to achieve a rapid training process with an element-wise scaling term on learning rates. Though prevailing, they are observed to generalize poorly compared with SGD or even fail to converge due to unstable and extreme learning rates. Recent work has put forward some algorithms such as AMSGrad to tackle this issue but they failed to achieve considerable improvement over existing methods. In our paper, we demonstrate that extreme learning rates can lead to poor performance. We provide new variants of Adam and AMSG…
Huazhong University of Science and Technology（华中科技大学）
Takagi-Sugeno-Kang (TSK) fuzzy systems are very useful machine learning models for regression problems. However, to our knowledge, there has not existed an efficient and effective training algorithm that ensures their generalization performance, and also enables them to deal with big data. Inspired by the connections between TSK fuzzy systems and neural networks, we extend three powerful neural network optimization techniques, i.e., mini-batch gradient descent, regularization, and AdaBound, to TSK fuzzy systems, and also propose three novel techniques (DropRule, DropMF, and DropMembership) spe…
Center for Data Science（数据科学中心）；Peking University（北京大学）；Key Lab. of Machine Perception (MOE)（机器感知教育部重点实验室）；School of EECS, Peking University（北京大学电子工程与计算机科学学院）；Peng Cheng Laboratory（鹏城实验室）
Adaptive gradient methods, especially Adam-type methods (such as Adam, AMSGrad, and AdaBound), have been proposed to speed up the training process with an element-wise scaling term on learning rates. However, they often generalize poorly compared with stochastic gradient descent (SGD) and its accelerated schemes such as SGD with momentum (SGDM). In this paper, we propose a new adaptive method called DecGD, which simultaneously achieves good generalization like SGDM and obtain rapid convergence like Adam-type methods. In particular, DecGD decomposes the current gradient into the product of two …
University of Maryland（马里兰大学）；Microsoft（微软）；Tsinghua University（清华大学）
Adaptive gradient methods such as RMSProp and Adam use exponential moving estimate of the squared gradient to compute adaptive step sizes, achieving better convergence than SGD in face of noisy objectives. However, Adam can have undesirable convergence behaviors due to unstable or extreme adaptive learning rates. Methods such as AMSGrad and AdaBound have been proposed to stabilize the adaptive learning rates of Adam in the later stage of training, but they do not outperform Adam in some practical tasks such as training Transformers \cite{transformer}. In this paper, we propose an adaptive lear…
The multiplicative structure of parameters and input data in the first layer of neural networks is explored to build connection between the landscape of the loss function with respect to parameters and the landscape of the model function with respect to input data. By this connection, it is shown that flat minima regularize the gradient of the model function, which explains the good generalization performance of flat minima. Then, we go beyond the flatness and consider high-order moments of the gradient noise, and show that Stochastic Gradient Descent (SGD) tends to impose constraints on these…
The University of Iowa（爱荷华大学）；Augusta University（奥古斯塔大学）
Stochastic Gradient Descent (SGD) is the key learning algorithm for many machine learning tasks. Because of its computational costs, there is a growing interest in accelerating SGD on HPC resources like GPU clusters. However, the performance of parallel SGD is still bottlenecked by the high communication costs even with a fast connection among the machines. A simple approach to alleviating this problem, used in many existing efforts, is to perform communication every few iterations, using a constant averaging period. In this paper, we show that the optimal averaging period in terms of converge…
The communication bottleneck has been a critical problem in large-scale distributed deep learning. In this work, we study distributed SGD with random block-wise sparsification as the gradient compressor, which is ring-allreduce compatible and highly computation-efficient but leads to inferior performance. To tackle this important issue, we improve the communication-efficient distributed SGD from a novel aspect, that is, the trade-off between the variance and second moment of the gradient. With this motivation, we propose a new detached error feedback (DEF) algorithm, which shows better converg…
University of Freiburg（弗莱堡大学）；University of North Carolina（北卡罗来纳大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）
First-order stochastic methods for solving large-scale non-convex optimization problems are widely used in many big-data applications, e.g. training deep neural networks as well as other complex and potentially non-convex machine learning models. Their inexpensive iterations generally come together with slow global convergence rate (mostly sublinear), leading to the necessity of carrying out a very high number of iterations before the iterates reach a neighborhood of a minimizer. In this work, we present a first-order stochastic algorithm based on a combination of homotopy methods and SGD, cal…
We empirically show that the test error of deep networks can be estimated by simply training the same architecture on the same training set but with a different run of Stochastic Gradient Descent (SGD), and measuring the disagreement rate between the two networks on unlabeled test data. This builds on -- and is a stronger version of -- the observation in Nakkiran & Bansal '20, which requires the second run to be on an altogether fresh training set. We further theoretically show that this peculiar phenomenon arises from the \emph{well-calibrated} nature of \emph{ensembles} of SGD-trained models…
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 …
SGD and Weight Decay Secretly Minimize the Rank of Your Neural Network
Texas A&M University（德克萨斯农工大学）；Princeton University（普林斯顿大学）；Massachusetts Institute of Technology（麻省理工学院）；Center for Brains, Mind, and Machines（脑、心智与机器中心）
First-order methods such as stochastic gradient descent (SGD) are currently the standard algorithm for training deep neural networks. Second-order methods, despite their better convergence rate, are rarely used in practice due to the prohibitive computational cost in calculating the second-order information. In this paper, we propose a novel Gram-Gauss-Newton (GGN) algorithm to train deep neural networks for regression problems with square loss. Our method draws inspiration from the connection between neural network optimization and kernel regression of neural tangent kernel (NTK). Different f…
Facebook AI Research（Facebook人工智能研究院）；New York University（纽约大学）
Momentum-based acceleration of stochastic gradient descent (SGD) is widely used in deep learning. We propose the quasi-hyperbolic momentum algorithm (QHM) as an extremely simple alteration of momentum SGD, averaging a plain SGD step with a momentum step. We describe numerous connections to and identities with other algorithms, and we characterize the set of two-state optimization algorithms that QHM can recover. Finally, we propose a QH variant of Adam called QHAdam, and we empirically demonstrate that our algorithms lead to significantly improved training in a variety of settings, including a…
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所（ISTA））；University of Surrey（萨里大学）；ETH Zurich（苏黎世联邦理工学院）
Decentralized optimization is emerging as a viable alternative for scalable distributed machine learning, but also introduces new challenges in terms of synchronization costs. To this end, several communication-reduction techniques, such as non-blocking communication, quantization, and local steps, have been explored in the decentralized setting. Due to the complexity of analyzing optimization in such a relaxed setting, this line of work often assumes \emph{global} communication rounds, which require additional synchronization. In this paper, we consider decentralized optimization in the simpl…
Google Research（谷歌研究院）；Princeton University（普林斯顿大学）；Cornell University（康奈尔大学）
Stochastic Gradient Descent (SGD) has been the method of choice for learning large-scale non-convex models. While a general analysis of when SGD works has been elusive, there has been a lot of recent progress in understanding the convergence of Gradient Flow (GF) on the population loss, partly due to the simplicity that a continuous-time analysis buys us. An overarching theme of our paper is providing general conditions under which SGD converges, assuming that GF on the population loss converges. Our main tool to establish this connection is a general converse Lyapunov like theorem, which impl…
We provide theoretical convergence guarantees on training Generative Adversarial Networks (GANs) via SGD. We consider learning a target distribution modeled by a 1-layer Generator network with a non-linear activation function $ϕ(\cdot)$ parametrized by a $d \times d$ weight matrix $\mathbf W_*$, i.e., $f_*(\mathbf x) = ϕ(\mathbf W_* \mathbf x)$. Our main result is that by training the Generator together with a Discriminator according to the Stochastic Gradient Descent-Ascent iteration proposed by Goodfellow et al. yields a Generator distribution that approaches the target distribution of $f_*$…
T-Brain AI Center（T-Brain人工智能中心）；SK telecom（SK电信）；Carnegie Mellon University（卡内基梅隆大学）；Superb AI（卓越人工智能）
When training a machine learning model with observational data, it is often encountered that some values are systemically missing. Learning from the incomplete data in which the missingness depends on some covariates may lead to biased estimation of parameters and even harm the fairness of decision outcome. This paper proposes how to adjust the causal effect of covariates on the missingness when training models using stochastic gradient descent (SGD). Inspired by the design of doubly robust estimator and its theoretical property of double robustness, we introduce stochastic doubly robust gradi…
Bias-Aware Minimisation: Understanding and Mitigating Estimator Bias in Private SGD
Institute for AI in Medicine, Technical University of Munich（慕尼黑工业大学医学人工智能研究所）；Konrad Zuse School for Excelllence in Reliable AI（康拉德·楚泽卓越可靠人工智能学院）；V7 Labs（V7实验室）；Imperial College London（帝国理工学院）
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）
Stochastic approximation (SA) and stochastic gradient descent (SGD) algorithms are work-horses for modern machine learning algorithms. Their constant stepsize variants are preferred in practice due to fast convergence behavior. However, constant step stochastic iterative algorithms do not converge asymptotically to the optimal solution, but instead have a stationary distribution, which in general cannot be analytically characterized. In this work, we study the asymptotic behavior of the appropriately scaled stationary distribution, in the limit when the constant stepsize goes to zero. Specific…
We study the problem of how to distribute the training of large-scale deep learning models in the parallel computing environment. We propose a new distributed stochastic optimization method called Elastic Averaging SGD (EASGD). We analyze the convergence rate of the EASGD method in the synchronous scenario and compare its stability condition with the existing ADMM method in the round-robin scheme. An asynchronous and momentum variant of the EASGD method is applied to train deep convolutional neural networks for image classification on the CIFAR and ImageNet datasets. Our approach accelerates t…
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…
Minimax-Optimal Generalization Bounds for Smooth Deep Neural Networks Trained by (Stochastic) Gradient Descent
Mathematical Institute for Machine Learning and Data Science, Catholic University of Eichstätt-Ingolstadt（机器学习与数据科学数学研究所，埃施特哈特-因戈尔施塔特天主教大学）；Department of Computer Science, RPTU Kaiserslautern-Landau（计算机科学系，凯斯莱特恩-兰道大学）；Department of Mathematics, The University of Hong Kong（数学系，香港大学）；School of Mathematics and Statistics, The University of Sydney（数学与统计学学院，悉尼大学）
Backpropagation (BP) is the most successful and widely used algorithm in deep learning. However, the computations required by BP are challenging to reconcile with known neurobiology. This difficulty has stimulated interest in more biologically plausible alternatives to BP. One such algorithm is the inference learning algorithm (IL). IL has close connections to neurobiological models of cortical function and has achieved equal performance to BP on supervised learning and auto-associative tasks. In contrast to BP, however, the mathematical foundations of IL are not well-understood. Here, we deve…
Center for Computational and Data-Intensive Science and Engineering（计算与数据密集型科学与工程中心）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）
We present a different view on stochastic optimization, which goes back to the splitting schemes for approximate solutions of ODE. In this work, we provide a connection between stochastic gradient descent approach and first-order splitting scheme for ODE. We consider the special case of splitting, which is inspired by machine learning applications and derive a new upper bound on the global splitting error for it. We present, that the Kaczmarz method is the limit case of the splitting scheme for the unit batch SGD for linear least squares problem. We support our findings with systematic empiric…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）；Microsoft Research（微软研究院）；University of Toronto（多伦多大学）
We propose a unified framework for neural net normalization, regularization and optimization, which includes Path-SGD and Batch-Normalization and interpolates between them across two different dimensions. Through this framework we investigate issue of invariance of the optimization, data dependence and the connection with natural gradients.
EPFL（洛桑联邦理工学院）；UC San Diego（加州大学圣地亚哥分校）；UC Berkeley（加州大学伯克利分校）；Columbia University（哥伦比亚大学）
Having similar behavior at training time and test time $-$ what we call a "What You See Is What You Get" (WYSIWYG) property $-$ is desirable in machine learning. Models trained with standard stochastic gradient descent (SGD), however, do not necessarily have this property, as their complex behaviors such as robustness or subgroup performance can differ drastically between training and test time. In contrast, we show that Differentially-Private (DP) training provably ensures the high-level WYSIWYG property, which we quantify using a notion of distributional generalization. Applying this connect…
Is Stochastic Gradient Descent Effective? A PDE Perspective on Machine Learning processes
Departamento de Matemáticas, Universidad Autónoma de Madrid, ICMAT - Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM（数学系，马德里自治大学，ICMAT数学科学研究所，CSIC-UAM-UC3M-UCM）
This paper revisits the so-called vanishing gradient phenomenon, which commonly occurs in deep randomly initialized neural networks. Leveraging an in-depth analysis of neural chains, we first show that vanishing gradients cannot be circumvented when the network width scales with less than O(depth), even when initialized with the popular Xavier and He initializations. Second, we extend the analysis to second-order derivatives and show that random i.i.d. initialization also gives rise to Hessian matrices with eigenspectra that vanish as networks grow in depth. Whenever this happens, optimizers a…
Meituan（美团）；University of Chinese Academy of Sciences（中国科学院大学）；University of Macau（澳门大学）
Over past few years afterward the birth of ResNet, skip connection has become the defacto standard for the design of modern architectures due to its widespread adoption, easy optimization and proven performance. Prior work has explained the effectiveness of the skip connection mechanism from different perspectives. In this work, we deep dive into the model's behaviors with skip connections which can be formulated as a learnable Markov chain. An efficient Markov chain is preferred as it always maps the input data to the target domain in a better way. However, while a model is explained as a Mar…
Improving Clinical Dataset Condensation with Mode Connectivity-based Trajectory Surrogates
Institute of Biomedical Engineering, University of Oxford, UK（生物医学工程研究所，牛津大学，英国）；GlaxoSmithKline, London, UK（葛兰素史克，伦敦，英国）；Nuffield Department of Primary Care Health Sciences, University of Oxford, UK（初级保健健康科学尼弗尔部门，牛津大学，英国）；Oxford-Suzhou Institute of Advanced Research (OSCAR), Suzhou, China（牛津-苏州先进研究所（OSCAR），苏州，中国）
Center for Brains, Minds and Machines, McGovern Institute（麦戈文研究所脑、心智与机器中心）；Massachusetts Institute of Technology（麻省理工学院）
Gradient backpropagation (BP) requires symmetric feedforward and feedback connections -- the same weights must be used for forward and backward passes. This "weight transport problem" (Grossberg 1987) is thought to be one of the main reasons to doubt BP's biologically plausibility. Using 15 different classification datasets, we systematically investigate to what extent BP really depends on weight symmetry. In a study that turned out to be surprisingly similar in spirit to Lillicrap et al.'s demonstration (Lillicrap et al. 2014) but orthogonal in its results, our experiments indicate that: (1) …
Bocconi University（博科尼大学）；Istituto Nazionale di Fisica Nucleare（意大利国家核物理研究所）；Politecnico di Torino（都灵理工大学）；International Centre for Theoretical Physics（国际理论物理中心）
Learning in Deep Neural Networks (DNN) takes place by minimizing a non-convex high-dimensional loss function, typically by a stochastic gradient descent (SGD) strategy. The learning process is observed to be able to find good minimizers without getting stuck in local critical points, and that such minimizers are often satisfactory at avoiding overfitting. How these two features can be kept under control in nonlinear devices composed of millions of tunable connections is a profound and far reaching open question. In this paper we study basic non-convex one- and two-layer neural network models w…
We study differentially private (DP) algorithms for stochastic non-convex optimization. In this problem, the goal is to minimize the population loss over a $p$-dimensional space given $n$ i.i.d. samples drawn from a distribution. We improve upon the population gradient bound of ${\sqrt{p}}/{\sqrt{n}}$ from prior work and obtain a sharper rate of $\sqrt[4]{p}/\sqrt{n}$. We obtain this rate by providing the first analyses on a collection of private gradient-based methods, including adaptive algorithms DP RMSProp and DP Adam. Our proof technique leverages the connection between differential priva…
In safety-critical but computationally resource-constrained applications, deep learning faces two key challenges: lack of robustness against adversarial attacks and large neural network size (often millions of parameters). While the research community has extensively explored the use of robust training and network pruning independently to address one of these challenges, only a few recent works have studied them jointly. However, these works inherit a heuristic pruning strategy that was developed for benign training, which performs poorly when integrated with robust training techniques, includ…
IST Austria（奥地利科学技术学院）；IIT Bombay（印度理工学院孟买分校）；NeuralMagic
We consider the problem of distributed mean estimation (DME), in which $n$ machines are each given a local $d$-dimensional vector $x_v \in \mathbb{R}^d$, and must cooperate to estimate the mean of their inputs $μ= \frac 1n\sum_{v = 1}^n x_v$, while minimizing total communication cost. DME is a fundamental construct in distributed machine learning, and there has been considerable work on variants of this problem, especially in the context of distributed variance reduction for stochastic gradients in parallel SGD. Previous work typically assumes an upper bound on the norm of the input vectors, a…
Using an extended and formalized version of the Q/C map analysis of Poole et al. (2016), along with Neural Tangent Kernel theory, we identify the main pathologies present in deep networks that prevent them from training fast and generalizing to unseen data, and show how these can be avoided by carefully controlling the "shape" of the network's initialization-time kernel function. We then develop a method called Deep Kernel Shaping (DKS), which accomplishes this using a combination of precise parameter initialization, activation function transformations, and small architectural tweaks, all of w…
Auburn University（奥本大学）；Sony AI（索尼人工智能）；Baidu Research（百度研究院）；University of Oregon（俄勒冈大学）
The federated learning (FL) framework enables edge clients to collaboratively learn a shared inference model while keeping privacy of training data on clients. Recently, many heuristics efforts have been made to generalize centralized adaptive optimization methods, such as SGDM, Adam, AdaGrad, etc., to federated settings for improving convergence and accuracy. However, there is still a paucity of theoretical principles on where to and how to design and utilize adaptive optimization methods in federated settings. This work aims to develop novel adaptive optimization methods for FL from the pers…
