We study Nesterov's accelerated gradient method with constant step-size and momentum parameters in the stochastic approximation setting (unbiased gradients with bounded variance) and the finite-sum setting (where randomness is due to sampling mini-batches). To build better insight into the behavior of Nesterov's method in stochastic settings, we focus throughout on objectives that are smooth, strongly-convex, and twice continuously differentiable. In the stochastic approximation setting, Nesterov's method converges to a neighborhood of the optimal point at the same accelerated rate as in the deterministic setting. Perhaps surprisingly, in the finite-sum setting, we prove that Nesterov's method may diverge with the usual choice of step-size and momentum, unless additional conditions on the problem related to conditioning and data coherence are satisfied. Our results shed light as to why Nesterov's method may fail to converge or achieve acceleration in the finite-sum setting.
In this paper, we propose Nesterov Accelerated Shuffling Gradient (NASG), a new algorithm for the convex finite-sum minimization problems. Our method integrates the traditional Nesterov's acceleration momentum with different shuffling sampling schemes. We show that our algorithm has an improved rate of $\mathcal{O}(1/T)$ using unified shuffling schemes, where $T$ is the number of epochs. This rate is better than that of any other shuffling gradient methods in convex regime. Our convergence analysis does not require an assumption on bounded domain or a bounded gradient condition. For randomized…
Shizuoka University（静冈大学）；Shonan Institute of Technology（湘南工科大学）；Hiroshima City University（广岛市立大学）
The Nesterov's accelerated quasi-Newton (L)NAQ method has shown to accelerate the conventional (L)BFGS quasi-Newton method using the Nesterov's accelerated gradient in several neural network (NN) applications. However, the calculation of two gradients per iteration increases the computational cost. The Momentum accelerated Quasi-Newton (MoQ) method showed that the Nesterov's accelerated gradient can be approximated as a linear combination of past gradients. This abstract extends the MoQ approximation to limited memory NAQ and evaluates the performance on a function approximation problem.
We present a unifying framework for adapting the update direction in gradient-based iterative optimization methods. As natural special cases we re-derive classical momentum and Nesterov's accelerated gradient method, lending a new intuitive interpretation to the latter algorithm. We show that a new algorithm, which we term Regularised Gradient Descent, can converge more quickly than either Nesterov's algorithm or the classical momentum algorithm.
Shonan Institute of Technology（湘南工科大学）；Shizuoka University（静冈大学）
Recently algorithms incorporating second order curvature information have become popular in training neural networks. The Nesterov's Accelerated Quasi-Newton (NAQ) method has shown to effectively accelerate the BFGS quasi-Newton method by incorporating the momentum term and Nesterov's accelerated gradient vector. A stochastic version of NAQ method was proposed for training of large-scale problems. However, this method incurs high stochastic variance noise. This paper proposes a stochastic variance reduced Nesterov's Accelerated Quasi-Newton method in full (SVR-NAQ) and limited (SVRLNAQ) memory…
Shizuoka University（静冈大学）；Shonan Institute of Technology（湘南工业大学）
A common problem in training neural networks is the vanishing and/or exploding gradient problem which is more prominently seen in training of Recurrent Neural Networks (RNNs). Thus several algorithms have been proposed for training RNNs. This paper proposes a novel adaptive stochastic Nesterov accelerated quasiNewton (aSNAQ) method for training RNNs. The proposed method aSNAQ is an accelerated method that uses the Nesterov's gradient term along with second order curvature information. The performance of the proposed method is evaluated in Tensorflow on benchmark sequence modeling problems. The…
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…
Shizuoka University（静冈大学）；Shonan Institute of Technology（湘南工科大学）
Incorporating second order curvature information in gradient based methods have shown to improve convergence drastically despite its computational intensity. In this paper, we propose a stochastic (online) quasi-Newton method with Nesterov's accelerated gradient in both its full and limited memory forms for solving large scale non-convex optimization problems in neural networks. The performance of the proposed algorithm is evaluated in Tensorflow on benchmark classification and regression problems. The results show improved performance compared to the classical second order oBFGS and oLBFGS me…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We study the algorithmic stability of Nesterov's accelerated gradient method. For convex quadratic objectives, Chen et al. (2018) proved that the uniform stability of the method grows quadratically with the number of optimization steps, and conjectured that the same is true for the general convex and smooth case. We disprove this conjecture and show, for two notions of algorithmic stability (including uniform stability), that the stability of Nesterov's accelerated method in fact deteriorates exponentially fast with the number of gradient steps. This stands in sharp contrast to the bounds in t…
We present two classes of differentially private optimization algorithms derived from the well-known accelerated first-order methods. The first algorithm is inspired by Polyak's heavy ball method and employs a smoothing approach to decrease the accumulated noise on the gradient steps required for differential privacy. The second class of algorithms are based on Nesterov's accelerated gradient method and its recent multi-stage variant. We propose a noise dividing mechanism for the iterations of Nesterov's method in order to improve the error behavior of the algorithm. The convergence rate analy…
Shizuoka University（静冈大学）；Shonan Institute of Technology（湘南工业大学）；Hiroshima City University（广岛市立大学）
Deep Q-learning method is one of the most popularly used deep reinforcement learning algorithms which uses deep neural networks to approximate the estimation of the action-value function. Training of the deep Q-network (DQN) is usually restricted to first order gradient based methods. This paper attempts to accelerate the training of deep Q-networks by introducing a second order Nesterov's accelerated quasi-Newton method. We evaluate the performance of the proposed method on deep reinforcement learning using double DQNs for global routing. The results show that the proposed method can obtain b…
Federated learning (FL) is a fast-developing technique that allows multiple workers to train a global model based on a distributed dataset. Conventional FL (FedAvg) employs gradient descent algorithm, which may not be efficient enough. Momentum is able to improve the situation by adding an additional momentum step to accelerate the convergence and has demonstrated its benefits in both centralized and FL environments. It is well-known that Nesterov Accelerated Gradient (NAG) is a more advantageous form of momentum, but it is not clear how to quantify the benefits of NAG in FL so far. This motiv…
Huazhong University of Science and Technology（华中科技大学）；Peking University（北京大学）；Cornell University（康奈尔大学）
Deep learning models are vulnerable to adversarial examples crafted by applying human-imperceptible perturbations on benign inputs. However, under the black-box setting, most existing adversaries often have a poor transferability to attack other defense models. In this work, from the perspective of regarding the adversarial example generation as an optimization process, we propose two new methods to improve the transferability of adversarial examples, namely Nesterov Iterative Fast Gradient Sign Method (NI-FGSM) and Scale-Invariant attack Method (SIM). NI-FGSM aims to adapt Nesterov accelerate…
Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. While many generalizations and extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the \emph{Bregman Lagrangian} which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-E…
Nesterov SGD is widely used for training modern neural networks and other machine learning models. Yet, its advantages over SGD have not been theoretically clarified. Indeed, as we show in our paper, both theoretically and empirically, Nesterov SGD with any parameter selection does not in general provide acceleration over ordinary SGD. Furthermore, Nesterov SGD may diverge for step sizes that ensure convergence of ordinary SGD. This is in contrast to the classical results in the deterministic scenario, where the same step size ensures accelerated convergence of the Nesterov's method over optim…
University of Alberta（阿尔伯塔大学）；University of Chicago（芝加哥大学）；Purdue University（普渡大学）
Nesterov's accelerated gradient methods (AGM) have been successfully applied in many machine learning areas. However, their empirical performance on training max-margin models has been inferior to existing specialized solvers. In this paper, we first extend AGM to strongly convex and composite objective functions with Bregman style prox-functions. Our unifying framework covers both the $\infty$-memory and 1-memory styles of AGM, tunes the Lipschiz constant adaptively, and bounds the duality gap. Then we demonstrate various ways to apply this framework of methods to a wide range of machine lear…
Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted much attention due to their low computational cost in each iteration. However, these algorithms might perform poorly especially if it is hard to approximate the Hessian well and efficiently. As far as we know, there is no effective way to handle this problem. In this paper, we resort to Nesterov's acceleration technique to improve the convergence performance of a class of second-order methods called approximate Newton. We give a theoretical analysis that Nesterov's acceleration technique…
Policy gradient methods are widely used in reinforcement learning algorithms to search for better policies in the parameterized policy space. They do gradient search in the policy space and are known to converge very slowly. Nesterov developed an accelerated gradient search algorithm for convex optimization problems. This has been recently extended for non-convex and also stochastic optimization. We use Nesterov's acceleration for policy gradient search in the well-known actor-critic algorithm and show the convergence using ODE method. We tested this algorithm on a scheduling problem. Here an …
The proximal point method (PPM) is a fundamental method in optimization that is often used as a building block for designing optimization algorithms. In this work, we use the PPM method to provide conceptually simple derivations along with convergence analyses of different versions of Nesterov's accelerated gradient method (AGM). The key observation is that AGM is a simple approximation of PPM, which results in an elementary derivation of the update equations and stepsizes of AGM. This view also leads to a transparent and conceptually simple analysis of AGM's convergence by using the analysis …
In this paper, we propose a simple acceleration scheme for Riemannian gradient methods by extrapolating iterates on manifolds. We show when the iterates are generated from Riemannian gradient descent method, the accelerated scheme achieves the optimal convergence rate asymptotically and is computationally more favorable than the recently proposed Riemannian Nesterov accelerated gradient methods. Our experiments verify the practical benefit of the novel acceleration strategy.
We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback-Leibler (KL) divergence as the objective functional. We show that an underdamped form of the Langevin algorithm performs accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm.
Center for Strategic Assessment and Consulting, Academy of Military Science（军事科学院战略评估咨询中心）
Momentum methods, including heavy-ball~(HB) and Nesterov's accelerated gradient~(NAG), are widely used in training neural networks for their fast convergence. However, there is a lack of theoretical guarantees for their convergence and acceleration since the optimization landscape of the neural network is non-convex. Nowadays, some works make progress towards understanding the convergence of momentum methods in an over-parameterized regime, where the number of the parameters exceeds that of the training instances. Nonetheless, current results mainly focus on the two-layer neural network, which…
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Information Transmission Problems RAS（俄罗斯科学院信息传输问题研究所）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机研究所）
Alternating minimization (AM) procedures are practically efficient in many applications for solving convex and non-convex optimization problems. On the other hand, Nesterov's accelerated gradient is theoretically optimal first-order method for convex optimization. In this paper we combine AM and Nesterov's acceleration to propose an accelerated alternating minimization algorithm. We prove $1/k^2$ convergence rate in terms of the objective for convex problems and $1/k$ in terms of the squared gradient norm for non-convex problems, where $k$ is the iteration counter. Our method does not require …
Monash University（莫纳什大学）；University of Queensland（昆士兰大学）；University of Waterloo（滑铁卢大学）
We present Nesterov-type acceleration techniques for Alternating Least Squares (ALS) methods applied to canonical tensor decomposition. While Nesterov acceleration turns gradient descent into an optimal first-order method for convex problems by adding a momentum term with a specific weight sequence, a direct application of this method and weight sequence to ALS results in erratic convergence behaviour. This is so because the tensor decomposition problem is non-convex and ALS is accelerated instead of gradient descent. Instead, we consider various restart mechanisms and suitable choices of mome…
Command and Control Engineering College, Army Engineering University of PLA（中国人民解放军陆军工程大学指挥与控制工程学院）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；Army Academy Of Artillery and Air Defense（陆军防空兵学院）
The extrapolation strategy raised by Nesterov, which can accelerate the convergence rate of gradient descent methods by orders of magnitude when dealing with smooth convex objective, has led to tremendous success in training machine learning tasks. In this article, the convergence of individual iterates of projected subgradient (PSG) methods for nonsmooth convex optimization problems is theoretically studied based on Nesterov's extrapolation, which we name individual convergence. We prove that Nesterov's extrapolation has the strength to make the individual convergence of PSG optimal for nonsm…
In this work we propose a differential geometric motivation for Nesterov's accelerated gradient method (AGM) for strongly-convex problems. By considering the optimization procedure as occurring on a Riemannian manifold with a natural structure, The AGM method can be seen as the proximal point method applied in this curved space. This viewpoint can also be extended to the continuous time case, where the accelerated gradient method arises from the natural block-implicit Euler discretization of an ODE on the manifold. We provide an analysis of the convergence rate of this ODE for quadratic object…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
We propose ADOM - an accelerated method for smooth and strongly convex decentralized optimization over time-varying networks. ADOM uses a dual oracle, i.e., we assume access to the gradient of the Fenchel conjugate of the individual loss functions. Up to a constant factor, which depends on the network structure only, its communication complexity is the same as that of accelerated Nesterov gradient method (Nesterov, 2003). To the best of our knowledge, only the algorithm of Rogozin et al. (2019) has a convergence rate with similar properties. However, their algorithm converges under the very re…
Center for Strategic Assessment and Consulting, Academy of Military Science（军事科学院战略评估咨询中心）
Momentum methods, such as heavy ball method~(HB) and Nesterov's accelerated gradient method~(NAG), have been widely used in training neural networks by incorporating the history of gradients into the current updating process. In practice, they often provide improved performance over (stochastic) gradient descent~(GD) with faster convergence. Despite these empirical successes, theoretical understandings of their accelerated convergence rates are still lacking. Recently, some attempts have been made by analyzing the trajectories of gradient-based methods in an over-parameterized regime, where th…
Georgia Institute of Technology（佐治亚理工学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Institute of Artificial Intelligence and Robotics for Society（深圳人工智能与机器人研究院）
Nesterov's Accelerated Gradient (NAG) for optimization has better performance than its continuous time limit (noiseless kinetic Langevin) when a finite step-size is employed \citep{shi2021understanding}. This work explores the sampling counterpart of this phenonemon and proposes a diffusion process, whose discretizations can yield accelerated gradient-based MCMC methods. More precisely, we reformulate the optimizer of NAG for strongly convex functions (NAG-SC) as a Hessian-Free High-Resolution ODE, change its high-resolution coefficient to a hyperparameter, inject appropriate noise, and discre…
University of Pennsylvania（宾夕法尼亚大学）；Stanford University（斯坦福大学）
Acceleration and momentum are the de facto standard in modern applications of machine learning and optimization, yet the bulk of the work on implicit regularization focuses instead on unaccelerated methods. In this paper, we study the statistical risk of the iterates generated by Nesterov's accelerated gradient method and Polyak's heavy ball method, when applied to least squares regression, drawing several connections to explicit penalization. We carry out our analyses in continuous-time, allowing us to make sharper statements than in prior work, and revealing complex interactions between earl…
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…
Pennsylvania State University（宾夕法尼亚州立大学）；Massachusetts Institute of Technology（麻省理工学院）；Rutgers University（罗格斯大学）
We study the problem of minimizing a strongly convex, smooth function when we have noisy estimates of its gradient. We propose a novel multistage accelerated algorithm that is universally optimal in the sense that it achieves the optimal rate both in the deterministic and stochastic case and operates without knowledge of noise characteristics. The algorithm consists of stages that use a stochastic version of Nesterov's method with a specific restart and parameters selected to achieve the fastest reduction in the bias-variance terms in the convergence rate bounds.
Inria（法国国家信息与自动化研究所）；PSL Research University（巴黎文理研究大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔理工学院）；MSR-Inria Joint Centre（微软研究院-法国国家信息与自动化研究所联合中心）
We introduce the continuized Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter. The two variables continuously mix following a linear ordinary differential equation and take gradient steps at random times. This continuized variant benefits from the best of the continuous and the discrete frameworks: as a continuous process, one can use differential calculus to analyze convergence and obtain analytical expressions for the parameters; and a discretization of the continuized process can be computed exactly with convergence r…
Stochastic dual coordinate ascent (SDCA) is an effective technique for solving regularized loss minimization problems in machine learning. This paper considers an extension of SDCA under the mini-batch setting that is often used in practice. Our main contribution is to introduce an accelerated mini-batch version of SDCA and prove a fast convergence rate for this method. We discuss an implementation of our method over a parallel computing system, and compare the results to both the vanilla stochastic dual coordinate ascent and to the accelerated deterministic gradient descent method of \cite{ne…
We consider nonlinear convergence acceleration methods for fixed-point iteration $x_{k+1}=q(x_k)$, including Anderson acceleration (AA), nonlinear GMRES (NGMRES), and Nesterov-type acceleration (corresponding to AA with window size one). We focus on fixed-point methods that converge asymptotically linearly with convergence factor $ρ<1$ and that solve an underlying fully smooth and non-convex optimization problem. It is often observed that AA and NGMRES substantially improve the asymptotic convergence behavior of the fixed-point iteration, but this improvement has not been quantified theoretica…
