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 $O(d\ln(N)/\sqrt{N})$. When used with the default parameters, Adam doesn't converge, however, and just like constant step-size SGD, it moves away from the initialization point faster than Adagrad, which might explain its practical success. Finally, we obtain the tightest dependency on the heavy ball momentum decay rate $β_1$ among all previous convergence bounds for non-convex Adam and Adagrad, improving from $O((1-β_1)^{-3})$ to $O((1-β_1)^{-1})$.
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。
Clipping Improves Adam-Norm and AdaGrad-Norm when the Noise Is Heavy-Tailed
Moscow Institute of Physics and Technology, Russia（莫斯科物理技术学院）；Ivannikov Institute for System Programming RAS, Russia（伊万诺夫系统编程研究所）；Sber AI Lab, Russia（俄罗斯Sber人工智能实验室）；Innopolis University, Russia（因诺波利斯大学）；The Russian Presidential Academy of National Economy and Public Administration, Russia（俄罗斯总统经济与公共行政学院）；Skolkovo Institute of Science and Technology, Russia（斯克沃尔佐夫科学与技术研究所）；Mohamed bin Zayed University of Artificial Intelligence, UAE（穆罕默德·本·扎耶德人工智能大学）；Machine Learning and Optimization Laboratory (MLO), EPFL, Lausanne, Switzerland（机器学习与优化实验室（MLO），EPFL，拉瓦尔普勒斯，瑞士）
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Facebook AI Research（Facebook AI 研究院）
Adaptive gradient methods such as AdaGrad and its variants update the stepsize in stochastic gradient descent on the fly according to the gradients received along the way; such methods have gained widespread use in large-scale optimization for their ability to converge robustly, without the need to fine-tune the stepsize schedule. Yet, the theoretical guarantees to date for AdaGrad are for online and convex optimization. We bridge this gap by providing theoretical guarantees for the convergence of AdaGrad for smooth, nonconvex functions. We show that the norm version of AdaGrad (AdaGrad-Norm) …
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…
ENS Paris-Saclay（巴黎萨克雷高等师范学校）；University of Genova（热那亚大学）；Université Toulouse 3 Paul Sabatier（图卢兹第三大学）
We prove that the iterates produced by, either the scalar step size variant, or the coordinatewise variant of AdaGrad algorithm, are convergent sequences when applied to convex objective functions with Lipschitz gradient. The key insight is to remark that such AdaGrad sequences satisfy a variable metric quasi-Fejér monotonicity property, which allows to prove convergence.
Google Research（谷歌研究院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Google Research - Brain（谷歌研究院-大脑团队）
We revisit the problem of empirical risk minimziation (ERM) with differential privacy. We show that noisy AdaGrad, given appropriate knowledge and conditions on the subspace from which gradients can be drawn, achieves a regret comparable to traditional AdaGrad plus a well-controlled term due to noise. We show a convergence rate of $O(\text{Tr}(G_T)/T)$, where $G_T$ captures the geometry of the gradient subspace. Since $\text{Tr}(G_T)=O(\sqrt{T})$ we can obtain faster rates for convex and Lipschitz functions, compared to the $O(1/\sqrt{T})$ rate achieved by known versions of noisy (stochastic) …
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。
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…
A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad and Muon
Université de Toulouse, INP, IRIT, Toulouse, France（图卢兹大学，INP，IRIT，图卢兹，法国）；IA Artificial and Natural Intelligence Toulouse Institute (ANITI)（图卢兹3IA人工智能与自然智能研究所（ANITI））；University of Namur（纳慕尔大学）
Changchun University of Science and Technology（长春理工大学）；Northeast Normal University（东北师范大学）
Adaptive gradient algorithm (AdaGrad) and its variants, such as RMSProp, Adam, AMSGrad, etc, have been widely used in deep learning. Although these algorithms are faster in the early phase of training, their generalization performance is often not as good as stochastic gradient descent (SGD). Hence, a trade-off method of transforming Adam to SGD after a certain iteration to gain the merits of both algorithms is theoretically and practically significant. To that end, we propose a decreasing scaling transition scheme to achieve a smooth and stable transition from Adam to SGD, which is called DST…
Academy of Mathematics and Systems Science（数学与系统科学研究院）；Chinese Academy of Sciences（中国科学院）；University of Chinese Academy of Sciences（中国科学院大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）；University of Notre Dame（圣母大学）
As one of the most fundamental stochastic optimization algorithms, stochastic gradient descent (SGD) has been intensively developed and extensively applied in machine learning in the past decade. There have been some modified SGD-type algorithms, which outperform the SGD in many competitions and applications in terms of convergence rate and accuracy, such as momentum-based SGD (mSGD) and adaptive gradient algorithm (AdaGrad). Despite these empirical successes, the theoretical properties of these algorithms have not been well established due to technical difficulties. With this motivation, we f…
Accelerated gradient-based methods are being extensively used for solving non-convex machine learning problems, especially when the data points are abundant or the available data is distributed across several agents. Two of the prominent accelerated gradient algorithms are AdaGrad and Adam. AdaGrad is the simplest accelerated gradient method, which is particularly effective for sparse data. Adam has been shown to perform favorably in deep learning problems compared to other methods. In this paper, we propose a new fast optimizer, Generalized AdaGrad (G-AdaGrad), for accelerating the solution o…
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…
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…
Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions
University of Science and Technology of China（中国科学技术大学）；Microsoft Research Asia（微软亚洲研究院）；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）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）
Oden Institute, UT Austin（德克萨斯大学奥斯汀分校奥登研究所）；Department of Mathematics, UT Austin（德克萨斯大学奥斯汀分校数学系）
We prove that the norm version of the adaptive stochastic gradient method (AdaGrad-Norm) achieves a linear convergence rate for a subset of either strongly convex functions or non-convex functions that satisfy the Polyak Lojasiewicz (PL) inequality. The paper introduces the notion of Restricted Uniform Inequality of Gradients (RUIG)---which is a measure of the balanced-ness of the stochastic gradient norms---to depict the landscape of a function. RUIG plays a key role in proving the robustness of AdaGrad-Norm to its hyper-parameter tuning in the stochastic setting. On top of RUIG, we develop a…
Massachusetts Institute of Technology（麻省理工学院）；University of California, San Diego（加利福尼亚大学圣地亚哥分校）
The Polyak-Lojasiewicz (PL) inequality is a sufficient condition for establishing linear convergence of gradient descent, even in non-convex settings. While several recent works use a PL-based analysis to establish linear convergence of stochastic gradient descent methods, the question remains as to whether a similar analysis can be conducted for more general optimization methods. In this work, we present a PL-based analysis for linear convergence of generalized mirror descent (GMD), a generalization of mirror descent with a possibly time-dependent mirror. GMD subsumes popular first order opti…
In this paper, we propose a new, simplified high probability analysis of AdaGrad for smooth, non-convex problems. More specifically, we focus on a particular accelerated gradient (AGD) template (Lan, 2020), through which we recover the original AdaGrad and its variant with averaging, and prove a convergence rate of $\mathcal O (1/ \sqrt{T})$ with high probability without the knowledge of smoothness and variance. We use a particular version of Freedman's concentration bound for martingale difference sequences (Kakade & Tewari, 2008) which enables us to achieve the best-known dependence of $\log…
The success of deep learning can be attributed to various factors such as increase in computational power, large datasets, deep convolutional neural networks, optimizers etc. Particularly, the choice of optimizer affects the generalization, convergence rate, and training stability. Stochastic Gradient Descent (SGD) is a first order iterative optimizer that updates the gradient uniformly for all parameters. This uniform update may not be suitable across the entire training phase. A rudimentary solution for this is to employ a fine-tuned learning rate scheduler which decreases learning rate as a…
Although stochastic gradient descent (SGD) method and its variants (e.g., stochastic momentum methods, AdaGrad) are the choice of algorithms for solving non-convex problems (especially deep learning), there still remain big gaps between the theory and the practice with many questions unresolved. For example, there is still a lack of theories of convergence for SGD and its variants that use stagewise step size and return an averaged solution in practice. In addition, theoretical insights of why adaptive step size of AdaGrad could improve non-adaptive step size of {\sgd} is still missing for non…
Boston University College of Engineering（波士顿大学工程学院）；University of Science and Technology of China（中国科学技术大学）
Recently, Stochastic Gradient Descent (SGD) and its variants have become the dominant methods in the large-scale optimization of machine learning (ML) problems. A variety of strategies have been proposed for tuning the step sizes, ranging from adaptive step sizes to heuristic methods to change the step size in each iteration. Also, momentum has been widely employed in ML tasks to accelerate the training process. Yet, there is a gap in our theoretical understanding of them. In this work, we start to close this gap by providing formal guarantees to a few heuristic optimization methods and propos…
Adam is a widely used optimization method for training deep learning models. It computes individual adaptive learning rates for different parameters. In this paper, we propose a generalization of Adam, called Adambs, that allows us to also adapt to different training examples based on their importance in the model's convergence. To achieve this, we maintain a distribution over all examples, selecting a mini-batch in each iteration by sampling according to this distribution, which we update using a multi-armed bandit algorithm. This ensures that examples that are more beneficial to the model tr…
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…
How do simple rotations affect the implicit bias of Adam?
Committee on Computational and Applied Mathematics, University of Chicago（计算与应用数学委员会，芝加哥大学）；Department of Electrical & Computer Engineering, University of Washington（电气与计算机工程系，华盛顿大学）；NSF-Simons National Institute for Theory and Mathematics in Biology（NSF-西蒙斯国家理论与生物数学研究所）；Department of Statistics, Department of Computer Science, and Data Science Institute, University of Chicago（统计系、计算机科学系和数据科学研究所，芝加哥大学）
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 …
New York University（纽约大学）；Boston University（波士顿大学）；Northeastern University（东北大学）
In this work, we describe a generic approach to show convergence with high probability for both stochastic convex and non-convex optimization with sub-Gaussian noise. In previous works for convex optimization, either the convergence is only in expectation or the bound depends on the diameter of the domain. Instead, we show high probability convergence with bounds depending on the initial distance to the optimal solution. The algorithms use step sizes analogous to the standard settings and are universal to Lipschitz functions, smooth functions, and their linear combinations. This method can be …
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…
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-…
On the Convergence of Adaptive Gradient Methods for Nonconvex Optimization
Indiana University（印第安纳大学）；The Pennsylvania State University（宾夕法尼亚州立大学）；The University of Hong Kong（香港大学）；Rice University（莱斯大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Stochastic gradient descent is the method of choice for large scale optimization of machine learning objective functions. Yet, its performance is greatly variable and heavily depends on the choice of the stepsizes. This has motivated a large body of research on adaptive stepsizes. However, there is currently a gap in our theoretical understanding of these methods, especially in the non-convex setting. In this paper, we start closing this gap: we theoretically analyze in the convex and non-convex settings a generalized version of the AdaGrad stepsizes. We show sufficient conditions for these st…
Several recently proposed stochastic optimization methods that have been successfully used in training deep networks such as RMSProp, Adam, Adadelta, Nadam are based on using gradient updates scaled by square roots of exponential moving averages of squared past gradients. In many applications, e.g. learning with large output spaces, it has been empirically observed that these algorithms fail to converge to an optimal solution (or a critical point in nonconvex settings). We show that one cause for such failures is the exponential moving average used in the algorithms. We provide an explicit exa…
University of Toronto（多伦多大学）；Vector Institute（矢量研究所）
The vast majority of successful deep neural networks are trained using variants of stochastic gradient descent (SGD) algorithms. Recent attempts to improve SGD can be broadly categorized into two approaches: (1) adaptive learning rate schemes, such as AdaGrad and Adam, and (2) accelerated schemes, such as heavy-ball and Nesterov momentum. In this paper, we propose a new optimization algorithm, Lookahead, that is orthogonal to these previous approaches and iteratively updates two sets of weights. Intuitively, the algorithm chooses a search direction by looking ahead at the sequence of fast weig…
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…
Shizuoka University（静冈大学）；Shonan Institute of Technology（湘南工科大学）
Recent studies incorporate Nesterov's accelerated gradient method for the acceleration of gradient based training. The Nesterov's Accelerated Quasi-Newton (NAQ) method has shown to drastically improve the convergence speed compared to the conventional quasi-Newton method. This paper implements NAQ for non-convex optimization on Tensorflow. Two modifications have been proposed to the original NAQ algorithm to ensure global convergence and eliminate linesearch. The performance of the proposed algorithm - mNAQ is evaluated on standard non-convex function approximation benchmark problems and micro…
Data61, CSIRO（联邦科学与工业研究组织数据61研究所）；Australian National University（澳大利亚国立大学）；Google DeepMind（谷歌DeepMind）
Advanced optimization algorithms such as Newton method and AdaGrad benefit from second order derivative or second order statistics to achieve better descent directions and faster convergence rates. At their heart, such algorithms need to compute the inverse or inverse square root of a matrix whose size is quadratic of the dimensionality of the search space. For high dimensional search spaces, the matrix inversion or inversion of square root becomes overwhelming which in turn demands for approximate methods. In this work, we propose a new matrix approximation method which divides a matrix into …
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 ha…
Forecasting of time series in continuous systems becomes an increasingly relevant task due to recent developments in IoT and 5G. The popular forecasting model ARIMA is applied to a large variety of applications for decades. An online variant of ARIMA applies the Online Newton Step in order to learn the underlying process of the time series. This optimization method has pitfalls concerning the computational complexity and convergence. Thus, this work focuses on the computational less expensive Online Gradient Descent optimization method, which became popular for learning of neural networks in r…
On Convergence of Adam for Stochastic Optimization under Relaxed Assumptions
Center for Data Science and School of Mathematical Sciences, Zhejiang University（浙江大学数据科学中心与数学科学学院）；Center for Data Science, Zhejiang University（浙江大学数据科学中心）
University of Padua（帕多瓦大学）；Università di Bologna（博洛尼亚大学）
Stochastic gradient descent (SGD) is the main approach for training deep networks: it moves towards the optimum of the cost function by iteratively updating the parameters of a model in the direction of the gradient of the loss evaluated on a minibatch. Several variants of SGD have been proposed to make adaptive step sizes for each parameter (adaptive gradient) and take into account the previous updates (momentum). Among several alternative of SGD the most popular are AdaGrad, AdaDelta, RMSProp and Adam which scale coordinates of the gradient by square roots of some form of averaging of the sq…
University of Maryland（马里兰大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
RMSProp and ADAM continue to be extremely popular algorithms for training neural nets but their theoretical convergence properties have remained unclear. Further, recent work has seemed to suggest that these algorithms have worse generalization properties when compared to carefully tuned stochastic gradient descent or its momentum variants. In this work, we make progress towards a deeper understanding of ADAM and RMSProp in two ways. First, we provide proofs that these adaptive gradient algorithms are guaranteed to reach criticality for smooth non-convex objectives, and we give bounds on the r…
