Adaptive gradient methods (AGMs) have become popular in optimizing the nonconvex problems in deep learning area. We revisit AGMs and identify that the adaptive learning rate (A-LR) used by AGMs varies significantly across the dimensions of the problem over epochs (i.e., anisotropic scale), which may lead to issues in convergence and generalization. All existing modified AGMs actually represent efforts in revising the A-LR. Theoretically, we provide a new way to analyze the convergence of AGMs and prove that the convergence rate of \textsc{Adam} also depends on its hyper-parameter $ε$, which has been overlooked previously. Based on these two facts, we propose a new AGM by calibrating the A-LR with an activation ({\em softplus}) function, resulting in the \textsc{Sadam} and \textsc{SAMSGrad} methods \footnote{Code is available at https://github.com/neilliang90/Sadam.git.}. We further prove that these algorithms enjoy better convergence speed under nonconvex, non-strongly convex, and Polyak-Łojasiewicz conditions compared with \textsc{Adam}. Empirical studies support our observation of the anisotropic A-LR and show that the proposed methods outperform existing AGMs and generalize even better than S-Momentum in multiple deep learning tasks.
University of Minnesota（明尼苏达大学）；MIT-IBM Watson AI Lab, IBM Research（麻省理工学院-IBM沃森人工智能实验室，IBM研究院）；Northeastern University（东北大学）；Princeton University（普林斯顿大学）
The adaptive momentum method (AdaMM), which uses past gradients to update descent directions and learning rates simultaneously, has become one of the most popular first-order optimization methods for solving machine learning problems. However, AdaMM is not suited for solving black-box optimization problems, where explicit gradient forms are difficult or infeasible to obtain. In this paper, we propose a zeroth-order AdaMM (ZO-AdaMM) algorithm, that generalizes AdaMM to the gradient-free regime. We show that the convergence rate of ZO-AdaMM for both convex and nonconvex optimization is roughly a…
Huawei Technologies Research and Development（华为技术研发）；University College London（伦敦大学学院）
In this paper, we present C-ADAM, the first adaptive solver for compositional problems involving a non-linear functional nesting of expected values. We proof that C-ADAM converges to a stationary point in $\mathcal{O}(δ^{-2.25})$ with $δ$ being a precision parameter. Moreover, we demonstrate the importance of our results by bridging, for the first time, model-agnostic meta-learning (MAML) and compositional optimisation showing fastest known rates for deep network adaptation to-date. Finally, we validate our findings in a set of experiments from portfolio optimisation and meta-learning. Our res…
Boston University（波士顿大学）；IBM T. J. Watson Research Center（IBM T. J. 沃森研究中心）；University of Iowa（爱荷华大学）
Adam is a widely used stochastic optimization method for deep learning applications. While practitioners prefer Adam because it requires less parameter tuning, its use is problematic from a theoretical point of view since it may not converge. Variants of Adam have been proposed with provable convergence guarantee, but they tend not be competitive with Adam on the practical performance. In this paper, we propose a new method named Adam$^+$ (pronounced as Adam-plus). Adam$^+$ retains some of the key components of Adam but it also has several noticeable differences: (i) it does not maintain the m…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；University of California, Los Angeles（加州大学洛杉矶分校）
Stochastic gradient descent (SGD) has taken the stage as the primary workhorse for large-scale machine learning. It is often used with its adaptive variants such as AdaGrad, Adam, and AMSGrad. This paper proposes an adaptive stochastic gradient descent method for distributed machine learning, which can be viewed as the communication-adaptive counterpart of the celebrated Adam method - justifying its name CADA. The key components of CADA are a set of new rules tailored for adaptive stochastic gradients that can be implemented to save communication upload. The new algorithms adaptively reuse the…
Purdue University（普渡大学）；Sensetime Research（商汤科技研究院）；Chinese University of Hong Kong（香港中文大学）；University of Hong Kong（香港大学）
Although adaptive optimization algorithms such as Adam show fast convergence in many machine learning tasks, this paper identifies a problem of Adam by analyzing its performance in a simple non-convex synthetic problem, showing that Adam's fast convergence would possibly lead the algorithm to local minimums. To address this problem, we improve Adam by proposing a novel adaptive gradient descent algorithm named AdaX. Unlike Adam that ignores the past gradients, AdaX exponentially accumulates the long-term gradient information in the past during training, to adaptively tune the learning rate. We…
Peking University（北京大学）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Center for Data Science, Peking University（北京大学数据科学中心）；Beijing Institute of Big Data Research（北京大数据研究院）
First-order optimization algorithms have been proven prominent in deep learning. In particular, algorithms such as RMSProp and Adam are extremely popular. However, recent works have pointed out the lack of ``long-term memory" in Adam-like algorithms, which could hamper their performance and lead to divergence. In our study, we observe that there are benefits of weighting more of the past gradients when designing the adaptive learning rate. We therefore propose an algorithm called the Nostalgic Adam (NosAdam) with theoretically guaranteed convergence at the best known convergence rate. NosAdam …
University of Minnesota - Twin Cities（明尼苏达大学双城分校）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；IBM Research（IBM研究院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper studies a class of adaptive gradient based momentum algorithms that update the search directions and learning rates simultaneously using past gradients. This class, which we refer to as the "Adam-type", includes the popular algorithms such as the Adam, AMSGrad and AdaGrad. Despite their popularity in training deep neural networks, the convergence of these algorithms for solving nonconvex problems remains an open question. This paper provides a set of mild sufficient conditions that guarantee the convergence for the Adam-type methods. We prove that under our derived conditions, these…
A Control Theoretic Framework for Adaptive Gradient Optimizers in Machine Learning
Division of Data and Decision Sciences（数据与决策科学部）；Tata Consultancy Services Research（塔塔咨询服务公司研究院）；University of Maryland（马里兰大学）；Department of Mechanical Engineering（机械工程系）
本文基于状态空间建模与控制理论传递函数提出自适应梯度优化通用框架及 Adam 新变体 AdamSSM，证明其收敛性，并在图像分类和语言建模任务上验证其泛化与收敛优势。
We formulate the problem of neural network optimization as Bayesian filtering, where the observations are the backpropagated gradients. While neural network optimization has previously been studied using natural gradient methods which are closely related to Bayesian inference, they were unable to recover standard optimizers such as Adam and RMSprop with a root-mean-square gradient normalizer, instead getting a mean-square normalizer. To recover the root-mean-square normalizer, we find it necessary to account for the temporal dynamics of all the other parameters as they are geing optimized. The…
LTCI, Télécom Paris, Institut Polytechnique de Paris（LTCI，巴黎高等电信学院，巴黎理工学院）
Although ADAM is a very popular algorithm for optimizing the weights of neural networks, it has been recently shown that it can diverge even in simple convex optimization examples. Several variants of ADAM have been proposed to circumvent this convergence issue. In this work, we study the ADAM algorithm for smooth nonconvex optimization under a boundedness assumption on the adaptive learning rate. The bound on the adaptive step size depends on the Lipschitz constant of the gradient of the objective function and provides safe theoretical adaptive step sizes. Under this boundedness assumption, w…
Towards Simple and Provable Parameter-Free Adaptive Gradient Methods
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Department of Computer Science, University of California, Los Angeles（加州大学洛杉矶分校计算机科学系）；School of Computing and Data Science, the University of Hong Kong（香港大学计算科学与数据科学学院）；Bytedance Inc（字节跳动公司）
提出 AdaGrad++ 和 Adam++ 两种简单无参数自适应梯度方法，在无需预设学习率的情况下实现与 AdaGrad 和 Adam 相当的收敛保证。
University of Science and Technology of China（中国科学技术大学）；Zhejiang Cainiao Supply Chain Management Co. Ltd（浙江菜鸟供应链管理有限公司）；Advanced Digital Sciences Center（先进数字科学中心）
Adaptive learning rate methods have been successfully applied in many fields, especially in training deep neural networks. Recent results have shown that adaptive methods with exponential increasing weights on squared past gradients (i.e., ADAM, RMSPROP) may fail to converge to the optimal solution. Though many algorithms, such as AMSGRAD and ADAMNC, have been proposed to fix the non-convergence issues, achieving a data-dependent regret bound similar to or better than ADAGRAD is still a challenge to these methods. In this paper, we propose a novel adaptive method weighted adaptive algorithm (W…
Adaptive gradient methods have shown excellent performances for solving many machine learning problems. Although multiple adaptive gradient methods were recently studied, they mainly focus on either empirical or theoretical aspects and also only work for specific problems by using some specific adaptive learning rates. Thus, it is desired to design a universal framework for practical algorithms of adaptive gradients with theoretical guarantee to solve general problems. To fill this gap, we propose a faster and universal framework of adaptive gradients (i.e., SUPER-ADAM) by introducing a univer…
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…
National University of Defense Technology（国防科技大学）；Tsinghua University（清华大学）；University of Utah（犹他大学）
Heavy ball momentum is crucial in accelerating (stochastic) gradient-based optimization algorithms for machine learning. Existing heavy ball momentum is usually weighted by a uniform hyperparameter, which relies on excessive tuning. Moreover, the calibrated fixed hyperparameter may not lead to optimal performance. In this paper, to eliminate the effort for tuning the momentum-related hyperparameter, we propose a new adaptive momentum inspired by the optimal choice of the heavy ball momentum for quadratic optimization. Our proposed adaptive heavy ball momentum can improve stochastic gradient de…
We propose NovoGrad, an adaptive stochastic gradient descent method with layer-wise gradient normalization and decoupled weight decay. In our experiments on neural networks for image classification, speech recognition, machine translation, and language modeling, it performs on par or better than well tuned SGD with momentum and Adam or AdamW. Additionally, NovoGrad (1) is robust to the choice of learning rate and weight initialization, (2) works well in a large batch setting, and (3) has two times smaller memory footprint than Adam.
This paper proposes a conjugate-gradient-based Adam algorithm blending Adam with nonlinear conjugate gradient methods and shows its convergence analysis. Numerical experiments on text classification and image classification show that the proposed algorithm can train deep neural network models in fewer epochs than the existing adaptive stochastic optimization algorithms can.
The University of Tokyo（东京大学）；RIKEN Center for AIP（理化学研究所人工智能研究中心）；Microsoft Research Asia（微软亚洲研究院）
Adaptive Moment Estimation (Adam), which combines Adaptive Learning Rate and Momentum, would be the most popular stochastic optimizer for accelerating the training of deep neural networks. However, it is empirically known that Adam often generalizes worse than Stochastic Gradient Descent (SGD). The purpose of this paper is to unveil the mystery of this behavior in the diffusion theoretical framework. Specifically, we disentangle the effects of Adaptive Learning Rate and Momentum of the Adam dynamics on saddle-point escaping and flat minima selection. We prove that Adaptive Learning Rate can es…
College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics（南京航空航天大学计算机科学与技术学院）；MIIT Key Laboratory of Pattern Analysis and Machine Intelligence（工业和信息化部模式分析与机器智能重点实验室）
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 …
IFM Lab, Florida State University（佛罗里达州立大学IFM实验室（IFM Lab, Florida State University））
Optimization algorithms with momentum, e.g., (ADAM), have been widely used for building deep learning models due to the faster convergence rates compared with stochastic gradient descent (SGD). Momentum helps accelerate SGD in the relevant directions in parameter updating, which can minify the oscillations of parameters update route. However, there exist errors in some update steps in optimization algorithms with momentum like ADAM. The fixed momentum weight (e.g., β_1 in ADAM) will propagate errors in momentum computing. In this paper, we introduce a novel optimization algorithm, namely Discr…
A common way to train neural networks is the Backpropagation. This algorithm includes a gradient descent method, which needs an adaptive step size. In the area of neural networks, the ADAM-Optimizer is one of the most popular adaptive step size methods. It was invented in \cite{Kingma.2015} by Kingma and Ba. The $5865$ citations in only three years shows additionally the importance of the given paper. We discovered that the given convergence proof of the optimizer contains some mistakes, so that the proof will be wrong. In this paper we give an improvement to the convergence proof of the ADAM-…
Convergence and convergence rate analyses of adaptive methods, such as Adaptive Moment Estimation (Adam) and its variants, have been widely studied for nonconvex optimization. The analyses are based on assumptions that the expected or empirical average loss function is Lipschitz smooth (i.e., its gradient is Lipschitz continuous) and the learning rates depend on the Lipschitz constant of the Lipschitz continuous gradient. Meanwhile, numerical evaluations of Adam and its variants have clarified that using small constant learning rates without depending on the Lipschitz constant and hyperparamet…
Department of Mathematics, Stanford University（斯坦福大学数学系）；Program in Applied and Computational Mathematics, Princeton University（普林斯顿大学应用与计算数学系）；Department of Mathematics, Princeton University（普林斯顿大学数学系）
The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss curve: fast initial convergence, oscillations, and large spikes in the late phase. The sign gradient descent (signGD) flow, which is the limit of Adam when taking the learning rate to 0 while keeping the momentum parameters fixed, is used to explain the fast initial convergence. For the late phase of Adam, three different types of qualitative patterns are observed depending on th…
Despite superior training outcomes, adaptive optimization methods such as Adam, Adagrad or RMSprop have been found to generalize poorly compared to Stochastic gradient descent (SGD). These methods tend to perform well in the initial portion of training but are outperformed by SGD at later stages of training. We investigate a hybrid strategy that begins training with an adaptive method and switches to SGD when appropriate. Concretely, we propose SWATS, a simple strategy which switches from Adam to SGD when a triggering condition is satisfied. The condition we propose relates to the projection o…
University of California, Los Angeles（加州大学洛杉矶分校）；The University of Hong Kong（香港大学）；Carnegie Mellon University（卡内基梅隆大学）
Adaptive gradient methods such as Adam have gained increasing popularity in deep learning optimization. However, it has been observed that compared with (stochastic) gradient descent, Adam can converge to a different solution with a significantly worse test error in many deep learning applications such as image classification, even with a fine-tuned regularization. In this paper, we provide a theoretical explanation for this phenomenon: we show that in the nonconvex setting of learning over-parameterized two-layer convolutional neural networks starting from the same random initialization, for …
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…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Ohio State University（俄亥俄州立大学）；University of Virginia（弗吉尼亚大学）
Adaptive gradient methods, which adopt historical gradient information to automatically adjust the learning rate, despite the nice property of fast convergence, have been observed to generalize worse than stochastic gradient descent (SGD) with momentum in training deep neural networks. This leaves how to close the generalization gap of adaptive gradient methods an open problem. In this work, we show that adaptive gradient methods such as Adam, Amsgrad, are sometimes "over adapted". We design a new algorithm, called Partially adaptive momentum estimation method, which unifies the Adam/Amsgrad w…
University of Calgary（卡尔加里大学）；Wuhan University（武汉大学）；The University of Hong Kong（香港大学）
Adaptive optimization algorithms, such as Adam and RMSprop, have shown better optimization performance than stochastic gradient descent (SGD) in some scenarios. However, recent studies show that they often lead to worse generalization performance than SGD, especially for training deep neural networks (DNNs). In this work, we identify the reasons that Adam generalizes worse than SGD, and develop a variant of Adam to eliminate the generalization gap. The proposed method, normalized direction-preserving Adam (ND-Adam), enables more precise control of the direction and step size for updating weigh…
Department of Computer Science, The University of Iowa（爱荷华大学计算机科学系）；Machine Intelligence Technology, Alibaba Group（阿里巴巴集团机器智能技术）
Since its invention in 2014, the Adam optimizer has received tremendous attention. On one hand, it has been widely used in deep learning and many variants have been proposed, while on the other hand their theoretical convergence property remains to be a mystery. It is far from satisfactory in the sense that some studies require strong assumptions about the updates, which are not necessarily applicable in practice, while other studies still follow the original problematic convergence analysis of Adam, which was shown to be not sufficient to ensure convergence. Although rigorous convergence anal…
Non-convergence of Adam and other adaptive stochastic gradient descent optimization methods for non-vanishing learning rates
University of Münster（明斯特大学）；The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）
In the context of stochastic gradient descent(SGD) and adaptive moment estimation (Adam),researchers have recently proposed optimization techniques that transition from Adam to SGD with the goal of improving both convergence and generalization performance. However, precisely how each approach trades off early progress and generalization is not well understood; thus, it is unclear when or even if, one should transition from one approach to the other. In this work, by first studying the convex setting, we identify potential contributors to observed differences in performance between SGD and Adam…
Adaptive Moment Estimation (ADAM) is a very popular training algorithm for deep neural networks and belongs to the family of adaptive gradient descent optimizers. However to the best of the authors knowledge no complete convergence analysis exists for ADAM. The contribution of this paper is a method for the local convergence analysis in batch mode for a deterministic fixed training set, which gives necessary conditions for the hyperparameters of the ADAM algorithm. Due to the local nature of the arguments the objective function can be non-convex but must be at least twice continuously differen…
MADA: Meta-Adaptive Optimizers through hyper-gradient Descent
Amazon Web Services（亚马逊网络服务）；Technion – Israel Institute of Technology（以色列理工学院）；University of Minnesota（明尼苏达大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of California Los Angeles（加利福尼亚大学洛杉矶分校）
Yale University（耶鲁大学）；University of Central Florida（中佛罗里达大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We propose ACProp (Asynchronous-centering-Prop), an adaptive optimizer which combines centering of second momentum and asynchronous update (e.g. for $t$-th update, denominator uses information up to step $t-1$, while numerator uses gradient at $t$-th step). ACProp has both strong theoretical properties and empirical performance. With the example by Reddi et al. (2018), we show that asynchronous optimizers (e.g. AdaShift, ACProp) have weaker convergence condition than synchronous optimizers (e.g. Adam, RMSProp, AdaBelief); within asynchronous optimizers, we show that centering of second momentu…
University of California, Berkeley（加利福尼亚大学伯克利分校）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
Adaptive optimization methods, which perform local optimization with a metric constructed from the history of iterates, are becoming increasingly popular for training deep neural networks. Examples include AdaGrad, RMSProp, and Adam. We show that for simple overparameterized problems, adaptive methods often find drastically different solutions than gradient descent (GD) or stochastic gradient descent (SGD). We construct an illustrative binary classification problem where the data is linearly separable, GD and SGD achieve zero test error, and AdaGrad, Adam, and RMSProp attain test errors arbitr…
We provide a simple proof of convergence covering both the Adam and Adagrad adaptive optimization algorithms when applied to smooth (possibly non-convex) objective functions with bounded gradients. We show that in expectation, the squared norm of the objective gradient averaged over the trajectory has an upper-bound which is explicit in the constants of the problem, parameters of the optimizer, the dimension $d$, and the total number of iterations $N$. This bound can be made arbitrarily small, and with the right hyper-parameters, Adam can be shown to converge with the same rate of convergence …
Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analyses
School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)（数据科学学院和深圳大数据研究院，香港中文大学（深圳））
