The power method is a classical algorithm with broad applications in machine learning tasks, including streaming PCA, spectral clustering, and low-rank matrix approximation. The distilled purpose of the vanilla power method is to determine the largest eigenvalue (in absolute modulus) and its eigenvector of a matrix. A momentum-based scheme can be used to accelerate the power method, but achieving an optimal convergence rate with existing algorithms critically relies on additional spectral information that is unavailable at run-time, and sub-optimal initializations can result in divergence. In this paper, we provide a pair of novel momentum-based power methods, which we call the delayed momentum power method (DMPower) and a streaming variant, the delayed momentum streaming method (DMStream). Our methods leverage inexact deflation and are capable of achieving near-optimal convergence with far less restrictive hyperparameter requirements. We provide convergence analyses for both algorithms through the lens of perturbation theory. Further, we experimentally demonstrate that DMPower routinely outperforms the vanilla power method and that both algorithms match the convergence speed of an oracle running existing accelerated methods with perfect spectral knowledge.
The problem of minimizing sum-of-nonconvex functions (i.e., convex functions that are average of non-convex ones) is becoming increasingly important in machine learning, and is the core machinery for PCA, SVD, regularized Newton's method, accelerated non-convex optimization, and more. We show how to provably obtain an accelerated stochastic algorithm for minimizing sum-of-nonconvex functions, by $\textit{adding one additional line}$ to the well-known SVRG method. This line corresponds to momentum, and shows how to directly apply momentum to the finite-sum stochastic minimization of sum-of-nonc…
Stochastic gradient descent (SGD) with stochastic momentum is popular in nonconvex stochastic optimization and particularly for the training of deep neural networks. In standard SGD, parameters are updated by improving along the path of the gradient at the current iterate on a batch of examples, where the addition of a ``momentum'' term biases the update in the direction of the previous change in parameters. In non-stochastic convex optimization one can show that a momentum adjustment provably reduces convergence time in many settings, yet such results have been elusive in the stochastic and n…
Convex-composite optimization, which minimizes an objective function represented by the sum of a differentiable function and a convex one, is widely used in machine learning and signal/image processing. Fast Iterative Shrinkage Thresholding Algorithm (FISTA) is a typical method for solving this problem and has a global convergence rate of $O(1 / k^2)$. Recently, this has been extended to multi-objective optimization, together with the proof of the $O(1 / k^2)$ global convergence rate. However, its momentum factor is classical, and the convergence of its iterates has not been proven. In this wo…
Recently, research on accelerated stochastic gradient descent methods (e.g., SVRG) has made exciting progress (e.g., linear convergence for strongly convex problems). However, the best-known methods (e.g., Katyusha) requires at least two auxiliary variables and two momentum parameters. In this paper, we propose a fast stochastic variance reduction gradient (FSVRG) method, in which we design a novel update rule with the Nesterov's momentum and incorporate the technique of growing epoch size. FSVRG has only one auxiliary variable and one momentum weight, and thus it is much simpler and has much …
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
Federated Learning is a popular distributed learning paradigm in machine learning. Meanwhile, composition optimization is an effective hierarchical learning model, which appears in many machine learning applications such as meta learning and robust learning. More recently, although a few federated composition optimization algorithms have been proposed, they still suffer from high sample and communication complexities. In the paper, thus, we propose a class of faster federated compositional optimization algorithms (i.e., MFCGD and AdaMFCGD) to solve the nonconvex distributed composition problem…
Changchun University of Science and Technology（长春理工大学）；Northeast Normal University（东北师范大学）
The plain stochastic gradient descent and momentum stochastic gradient descent have extremely wide applications in deep learning due to their simple settings and low computational complexity. The momentum stochastic gradient descent uses the accumulated gradient as the updated direction of the current parameters, which has a faster training speed. Because the direction of the plain stochastic gradient descent has not been corrected by the accumulated gradient. For the parameters that currently need to be updated, it is the optimal direction, and its update is more accurate. We combine the adva…
University of Pittsburgh（匹兹堡大学）；Duke University（杜克大学）
Mobile crowdsensing has gained significant attention in recent years and has become a critical paradigm for emerging Internet of Things applications. The sensing devices continuously generate a significant quantity of data, which provide tremendous opportunities to develop innovative intelligent applications. To utilize these data to train machine learning models while not compromising user privacy, federated learning has become a promising solution. However, there is little understanding of whether federated learning algorithms are guaranteed to converge. We reconsider model averaging in fede…
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…
Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
Two major momentum-based techniques that have achieved tremendous success in optimization are Polyak's heavy ball method and Nesterov's accelerated gradient. A crucial step in all momentum-based methods is the choice of the momentum parameter $m$ which is always suggested to be set to less than $1$. Although the choice of $m < 1$ is justified only under very strong theoretical assumptions, it works well in practice even when the assumptions do not necessarily hold. In this paper, we propose a new momentum based method $\textit{ADINE}$, which relaxes the constraint of $m < 1$ and allows the lea…
Optimal transport (OT)-based methods have a wide range of applications and have attracted a tremendous amount of attention in recent years. However, most of the computational approaches of OT do not learn the underlying transport map. Although some algorithms have been proposed to learn this map, they rely on kernel-based methods, which makes them prohibitively slow when the number of samples increases. Here, we propose a way to learn an approximate transport map and a parametric approximation of the Wasserstein barycenter. We build an approximated transport mapping by leveraging the closed-fo…
We propose a stochastic recursive momentum method for Riemannian non-convex optimization that achieves a near-optimal complexity of $\tilde{\mathcal{O}}(ε^{-3})$ to find $ε$-approximate solution with one sample. That is, our method requires $\mathcal{O}(1)$ gradient evaluations per iteration and does not require restarting with a large batch gradient, which is commonly used to obtain the faster rate. Extensive experiment results demonstrate the superiority of our proposed algorithm.
The Heavy Ball Method, proposed by Polyak over five decades ago, is a first-order method for optimizing continuous functions. While its stochastic counterpart has proven extremely popular in training deep networks, there are almost no known functions where deterministic Heavy Ball is provably faster than the simple and classical gradient descent algorithm in non-convex optimization. The success of Heavy Ball has thus far eluded theoretical understanding. Our goal is to address this gap, and in the present work we identify two non-convex problems where we provably show that the Heavy Ball momen…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Toyota Technical Institute of Chicago（芝加哥丰田技术研究所）
We present and analyze a momentum-based gradient method for training linear classifiers with an exponentially-tailed loss (e.g., the exponential or logistic loss), which maximizes the classification margin on separable data at a rate of $\widetilde{\mathcal{O}}(1/t^2)$. This contrasts with a rate of $\mathcal{O}(1/\log(t))$ for standard gradient descent, and $\mathcal{O}(1/t)$ for normalized gradient descent. This momentum-based method is derived via the convex dual of the maximum-margin problem, and specifically by applying Nesterov acceleration to this dual, which manages to result in a simp…
On the fast convergence of minibatch heavy ball momentum
Operations Research and Industrial Engineering（运筹学与工业工程）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Mathematics（数学）；New York University（纽约大学）；Computer Science and Engineering（计算机科学与工程）
Correlation clustering is a widely studied framework for clustering based on pairwise similarity and dissimilarity scores, but its best approximation algorithms rely on impractical linear programming relaxations. We present faster approximation algorithms that avoid these relaxations, for two well-studied special cases: cluster editing and cluster deletion. We accomplish this by drawing new connections to edge labeling problems related to the principle of strong triadic closure. This leads to faster and more practical linear programming algorithms, as well as extremely scalable combinatorial t…
Oxford-Man Institute of Quantitative Finance, University of Oxford（牛津大学牛津-曼量化金融研究所）
Momentum strategies are an important part of alternative investments and are at the heart of commodity trading advisors (CTAs). These strategies have, however, been found to have difficulties adjusting to rapid changes in market conditions, such as during the 2020 market crash. In particular, immediately after momentum turning points, where a trend reverses from an uptrend (downtrend) to a downtrend (uptrend), time-series momentum (TSMOM) strategies are prone to making bad bets. To improve the response to regime change, we introduce a novel approach, where we insert an online changepoint detec…
The Ohio State University（俄亥俄州立大学）；University of Michigan（密歇根大学）
Bilevel optimization has been widely applied in many important machine learning applications such as hyperparameter optimization and meta-learning. Recently, several momentum-based algorithms have been proposed to solve bilevel optimization problems faster. However, those momentum-based algorithms do not achieve provably better computational complexity than $\mathcal{\widetilde O}(ε^{-2})$ of the SGD-based algorithm. In this paper, we propose two new algorithms for bilevel optimization, where the first algorithm adopts momentum-based recursive iterations, and the second algorithm adopts recurs…
The Ohio State University（俄亥俄州立大学）；The University of Utah（犹他大学）；Duke University（杜克大学）
SARAH and SPIDER are two recently developed stochastic variance-reduced algorithms, and SPIDER has been shown to achieve a near-optimal first-order oracle complexity in smooth nonconvex optimization. However, SPIDER uses an accuracy-dependent stepsize that slows down the convergence in practice, and cannot handle objective functions that involve nonsmooth regularizers. In this paper, we propose SpiderBoost as an improved scheme, which allows to use a much larger constant-level stepsize while maintaining the same near-optimal oracle complexity, and can be extended with proximal mapping to handl…
SloMo-Fast: Slow-Momentum and Fast-Adaptive Teachers for Source-Free Continual Test-Time Adaptation
Center for Computational & Data Sciences, Independent University, Bangladesh（计算与数据科学中心，独立大学，孟加拉国）；University of Central Florida, USA（佛罗里达中央大学，美国）；University of Virginia, USA（弗吉尼亚大学，美国）；Amazon GenAI, USA（亚马逊生成人工智能，美国）
Double Momentum and Error Feedback for Clipping with Fast Rates and Differential Privacy
University of Basel（巴塞尔大学）；Mohamed bin Zayed University of Artificial Intelligence (MBZUAI)（穆罕默德·本·扎耶德人工智能大学（MBZUAI））；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学（KAUST））
A Fast and Flat Federated Learning Method via Weighted Momentum and Sharpness-Aware Minimization
The Hong Kong University of Science and Technology (Guangzhou)（香港科学与技术大学（广州））；National University of Singapore（新加坡国立大学）；Nanyang Technological University, Singapore（南洋理工大学，新加坡）；Shenzhen University（深圳大学）
Complexity of normalized stochastic first-order methods with momentum under heavy-tailed noise
Department of Mathematics, Linköping University（利乌普斯大学数学系）；Department of Industrial and Systems Engineering, University of Minnesota（明尼苏达大学工业与系统工程系）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）；Academy for Advanced Interdisciplinary Studies, Peking University（北京大学先进跨学科研究学院）
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；NVIDIA（英伟达）
We generalize gradient descent with momentum for optimization in differentiable games to have complex-valued momentum. We give theoretical motivation for our method by proving convergence on bilinear zero-sum games for simultaneous and alternating updates. Our method gives real-valued parameter updates, making it a drop-in replacement for standard optimizers. We empirically demonstrate that complex-valued momentum can improve convergence in realistic adversarial games - like generative adversarial networks - by showing we can find better solutions with an almost identical computational cost. W…
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…
Topological Data Analysis is a recent and fast growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. This paper is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of \tda\ for non experts.
