SGD with momentum (SGDM) has been widely applied in many machine learning tasks, and it is often applied with dynamic stepsizes and momentum weights tuned in a stagewise manner. Despite of its empirical advantage over SGD, the role of momentum is still unclear in general since previous analyses on SGDM either provide worse convergence bounds than those of SGD, or assume Lipschitz or quadratic objectives, which fail to hold in practice. Furthermore, the role of dynamic parameters has not been addressed. In this work, we show that SGDM converges as fast as SGD for smooth objectives under both strongly convex and nonconvex settings. We also establish \textit{the first} convergence guarantee for the multistage setting, and show that the multistage strategy is beneficial for SGDM compared to using fixed parameters. Finally, we verify these theoretical claims by numerical experiments.
The starting assumptions to study the convergence and complexity of gradient-type methods may be the smoothness (also called Lipschitz continuity of gradient) and the strong convexity. In this note, we revisit these two basic properties from a new perspective that motivates their definitions and equivalent characterizations, along with an improved linear convergence of the gradient descent method.
In this paper, we deal with the Front Steepest Descent algorithm for multi-objective optimization. We point out that the algorithm from the literature is often incapable, by design, of spanning large portions of the Pareto front. We thus introduce some modifications within the algorithm aimed to overcome this significant limitation. We prove that the asymptotic convergence properties of the algorithm are preserved and numerically show that the proposed method significantly outperforms the original one.
The iteration complexity of the block-coordinate descent (BCD) type algorithm has been under extensive investigation. It was recently shown that for convex problems the classical cyclic BCGD (block coordinate gradient descent) achieves an $\mathcal{O}(1/r)$ complexity ($r$ is the number of passes of all blocks). However, such bounds are at least linearly depend on $K$ (the number of variable blocks), and are at least $K$ times worse than those of the gradient descent (GD) and proximal gradient (PG) methods. In this paper, we aim to close such theoretical performance gap between cyclic BCD and …
It is well known that the finite step-size ($h$) in Gradient Descent (GD) implicitly regularizes solutions to flatter minima. A natural question to ask is "Does the momentum parameter $β$ play a role in implicit regularization in Heavy-ball (H.B) momentum accelerated gradient descent (GD+M)?". To answer this question, first, we show that the discrete H.B momentum update (GD+M) follows a continuous trajectory induced by a modified loss, which consists of an original loss and an implicit regularizer. Then, we show that this implicit regularizer for (GD+M) is stronger than that of (GD) by factor …
IBM TJ Watson Research Center（IBM托马斯·J·沃森研究中心）；Georgia Institute of Technology（佐治亚理工学院）
Momentum method has been used extensively in optimizers for deep learning. Recent studies show that distributed training through K-step averaging has many nice properties. We propose a momentum method for such model averaging approaches. At each individual learner level traditional stochastic gradient is applied. At the meta-level (global learner level), one momentum term is applied and we call it block momentum. We analyze the convergence and scaling properties of such momentum methods. Our experimental results show that block momentum not only accelerates training, but also achieves better r…
Institute of Informatics, University of Warsaw（华沙大学信息学研究所）
We study the overparametrization bounds required for the global convergence of stochastic gradient descent algorithm for a class of one hidden layer feed-forward neural networks, considering most of the activation functions used in practice, including ReLU. We improve the existing state-of-the-art results in terms of the required hidden layer width. We introduce a new proof technique combining nonlinear analysis with properties of random initializations of the network. First, we establish the global convergence of continuous solutions of the differential inclusion being a nonsmooth analogue of…
University of Utah（犹他大学）；University of Kentucky（肯塔基大学）
Momentum plays a crucial role in stochastic gradient-based optimization algorithms for accelerating or improving training deep neural networks (DNNs). In deep learning practice, the momentum is usually weighted by a well-calibrated constant. However, tuning hyperparameters for momentum can be a significant computational burden. In this paper, we propose a novel \emph{adaptive momentum} for improving DNNs training; this adaptive momentum, with no momentum related hyperparameter required, is motivated by the nonlinear conjugate gradient (NCG) method. Stochastic gradient descent (SGD) with this n…
The University of Edinburgh（爱丁堡大学）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Moscow Institute of Physics and Technology (MIPT)（莫斯科物理技术学院）
In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these methods are studied. We choose to perform our analysis in a setting in which all of the above methods are equivalent. We prove global nonassymptotic linear convergence rates for all methods and various measures of success, including primal function values, primal iterates (in L2 sense…
State Key Laboratory of Scientific and Engineering Computing（科学与工程计算国家重点实验室）；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）
Following the same routine as [SSJ20], we continue to present the theoretical analysis for stochastic gradient descent with momentum (SGD with momentum) in this paper. Differently, for SGD with momentum, we demonstrate it is the two hyperparameters together, the learning rate and the momentum coefficient, that play the significant role for the linear rate of convergence in non-convex optimization. Our analysis is based on the use of a hyperparameters-dependent stochastic differential equation (hp-dependent SDE) that serves as a continuous surrogate for SGD with momentum. Similarly, we establis…
While momentum-based methods, in conjunction with stochastic gradient descent (SGD), are widely used when training machine learning models, there is little theoretical understanding on the generalization error of such methods. In this work, we first show that there exists a convex loss function for which algorithmic stability fails to establish generalization guarantees when SGD with standard heavy-ball momentum (SGDM) is run for multiple epochs. Then, for smooth Lipschitz loss functions, we analyze a modified momentum-based update rule, i.e., SGD with early momentum (SGDEM), and show that it …
The Ohio State University（俄亥俄州立大学）；Southern University of Science and Technology（南方科技大学）
Existing convergence analyses of Q-learning mostly focus on the vanilla stochastic gradient descent (SGD) type of updates. Despite the Adaptive Moment Estimation (Adam) has been commonly used for practical Q-learning algorithms, there has not been any convergence guarantee provided for Q-learning with such type of updates. In this paper, we first characterize the convergence rate for Q-AMSGrad, which is the Q-learning algorithm with AMSGrad update (a commonly adopted alternative of Adam for theoretical analysis). To further improve the performance, we propose to incorporate the momentum restar…
Maynooth University（梅努斯大学）；Max-Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）
We consider gradient descent with `momentum', a widely used method for loss function minimization in machine learning. This method is often used with `Nesterov acceleration', meaning that the gradient is evaluated not at the current position in parameter space, but at the estimated position after one step. In this work, we show that the algorithm can be improved by extending this `acceleration' --- by using the gradient at an estimated position several steps ahead rather than just one step ahead. How far one looks ahead in this `super-acceleration' algorithm is determined by a new hyperparamet…
University of Science & Technology of China（中国科学技术大学）；Microsoft Research Asia（微软亚洲研究院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
The momentum acceleration technique is widely adopted in many optimization algorithms. However, there is no theoretical answer on how the momentum affects the generalization performance of the optimization algorithms. This paper studies this problem by analyzing the implicit regularization of momentum-based optimization. We prove that on the linear classification problem with separable data and exponential-tailed loss, gradient descent with momentum (GDM) converges to the L2 max-margin solution, which is the same as vanilla gradient descent. That means gradient descent with momentum accelerati…
Stochastic dual coordinate descent with adaptive heavy ball momentum for linearly constrained convex optimization
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University（教育部数学线性代数重点实验室，北京航空航天大学数学科学学院）
Momentum-based gradient descent methods for Lie groups
Departament de Matemàtiques, Universitat de València（瓦伦西亚大学数学系）；Departamento de Matemática Aplicada, Ciencia e Ingeniería de los Materiales y Tecnología Electrónica, Universidad Rey Juan Carlos（雷乌恩卡洛斯大学应用数学系）；Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)（数学科学研究所）；Departamento de Matemáticas, Universidad de Córdoba（科尔多瓦大学数学系）
We present a coupled system of ODEs which, when discretized with a constant time step/learning rate, recovers Nesterov's accelerated gradient descent algorithm. The same ODEs, when discretized with a decreasing learning rate, leads to novel stochastic gradient descent (SGD) algorithms, one in the convex and a second in the strongly convex case. In the strongly convex case, we obtain an algorithm superficially similar to momentum SGD, but with additional terms. In the convex case, we obtain an algorithm with a novel order $k^{3/4}$ learning rate. We prove, extending the Lyapunov function approa…
Riemannian Momentum Tracking: Distributed Optimization with Momentum on Compact Submanifolds
Zhejiang Normal University（浙江师范大学）；China Mobile Research Institute（中国移动研究院）；Xi’an Jiaotong University（西安交通大学）；State Grid Corporation of China（国家电网有限公司）；Institute of Cyber-Systems and Control, Zhejiang University（浙江大学控制科学与工程学院）；Huzhou University（湖州师范学院）；Centre for Frontier Artificial Intelligence Research, A*STAR（新加坡国立研究理事会前沿人工智能研究中心）
We present a novel, practical, and provable approach for solving diagonally constrained semi-definite programming (SDP) problems at scale using accelerated non-convex programming. Our algorithm non-trivially combines acceleration motions from convex optimization with coordinate power iteration and matrix factorization techniques. The algorithm is extremely simple to implement, and adds only a single extra hyperparameter -- momentum. We prove that our method admits local linear convergence in the neighborhood of the optimum and always converges to a first-order critical point. Experimentally, w…
Asynchronous algorithms have attracted much attention recently due to the crucial demands on solving large-scale optimization problems. However, the accelerated versions of asynchronous algorithms are rarely studied. In this paper, we propose the "momentum compensation" technique to accelerate asynchronous algorithms for convex problems. Specifically, we first accelerate the plain Asynchronous Gradient Descent, which achieves a faster $O(1/\sqrtε)$ (v.s. $O(1/ε)$) convergence rate for non-strongly convex functions, and $O(\sqrtκ\log(1/ε))$ (v.s. $O(κ\log(1/ε))$) for strongly convex functions t…
In this paper we propose stochastic gradient-free methods and accelerated methods with momentum for solving stochastic optimization problems. All these methods rely on stochastic directions rather than stochastic gradients. We analyze the convergence behavior of these methods under the mean-variance framework, and also provide a theoretical analysis about the inclusion of momentum in stochastic settings which reveals that the momentum term we used adds a deviation of order $\mathcal{O}(1/k)$ but controls the variance at the order $\mathcal{O}(1/k)$ for the $k$th iteration. So it is shown that,…
We propose a new policy gradient method, named homotopic policy mirror descent (HPMD), for solving discounted, infinite horizon MDPs with finite state and action spaces. HPMD performs a mirror descent type policy update with an additional diminishing regularization term, and possesses several computational properties that seem to be new in the literature. We first establish the global linear convergence of HPMD instantiated with Kullback-Leibler divergence, for both the optimality gap, and a weighted distance to the set of optimal policies. Then local superlinear convergence is obtained for bo…
A new gradient-based optimization approach by automatically scheduling the learning rate has been proposed recently, which is called Binary Forward Exploration (BFE). The Adaptive version of BFE has also been discussed thereafter. In this paper, the improved algorithms based on them will be investigated, in order to optimize the efficiency and robustness of the new methodology. This improved approach provides a new perspective to scheduling the update of learning rate and will be compared with the stochastic gradient descent, aka SGD algorithm with momentum or Nesterov momentum and the most su…
We introduce Kalman Gradient Descent, a stochastic optimization algorithm that uses Kalman filtering to adaptively reduce gradient variance in stochastic gradient descent by filtering the gradient estimates. We present both a theoretical analysis of convergence in a non-convex setting and experimental results which demonstrate improved performance on a variety of machine learning areas including neural networks and black box variational inference. We also present a distributed version of our algorithm that enables large-dimensional optimization, and we extend our algorithm to SGD with momentum…
A theoretical, and potentially also practical, problem with stochastic gradient descent is that trajectories may escape to infinity. In this note, we investigate uniform boundedness properties of iterates and function values along the trajectories of the stochastic gradient descent algorithm and its important momentum variant. Under smoothness and $R$-dissipativity of the loss function, we show that broad families of step-sizes, including the widely used step-decay and cosine with (or without) restart step-sizes, result in uniformly bounded iterates and function values. Several important appli…
Alternating gradient-descent-ascent (AltGDA) is an optimization algorithm that has been widely used for model training in various machine learning applications, which aims to solve a nonconvex minimax optimization problem. However, the existing studies show that it suffers from a high computation complexity in nonconvex minimax optimization. In this paper, we develop a single-loop and fast AltGDA-type algorithm that leverages proximal gradient updates and momentum acceleration to solve regularized nonconvex minimax optimization problems. By leveraging the momentum acceleration technique, we pr…
