This paper considers the problem of multi-agent distributed optimization. In this problem, there are multiple agents in the system, and each agent only knows its local cost function. The objective for the agents is to collectively compute a common minimum of the aggregate of all their local cost functions. In principle, this problem is solvable using a distributed variant of the traditional gradient-descent method, which is an iterative method. However, the speed of convergence of the traditional gradient-descent method is highly influenced by the conditioning of the optimization problem being solved. Specifically, the method requires a large number of iterations to converge to a solution if the optimization problem is ill-conditioned.
In this paper, we propose an iterative pre-conditioning approach that can significantly attenuate the influence of the problem's conditioning on the convergence-speed of the gradient-descent method. The proposed pre-conditioning approach can be easily implemented in distributed systems and has minimal computation and communication overhead. For now, we only consider a specific distributed optimization problem wherein the individual local cost functions of the agents are quadratic. Besides the theoretical guarantees, the improved convergence speed of our approach is demonstrated through experiments on a real data-set.
We analyze and evaluate an online gradient descent algorithm with adaptive per-coordinate adjustment of learning rates. Our algorithm can be thought of as an online version of batch gradient descent with a diagonal preconditioner. This approach leads to regret bounds that are stronger than those of standard online gradient descent for general online convex optimization problems. Experimentally, we show that our algorithm is competitive with state-of-the-art algorithms for large scale machine learning problems.
We provide (high probability) bounds on the condition number of random feature matrices. In particular, we show that if the complexity ratio $\frac{N}{m}$ where $N$ is the number of neurons and $m$ is the number of data samples scales like $\log^{-1}(N)$ or $\log(m)$, then the random feature matrix is well-conditioned. This result holds without the need of regularization and relies on establishing various concentration bounds between dependent components of the random feature matrix. Additionally, we derive bounds on the restricted isometry constant of the random feature matrix. We prove that …
University of Maryland, College Park（马里兰大学帕克分校）；Georgetown University（乔治城大学）
This paper considers the multi-agent distributed linear least-squares problem. The system comprises multiple agents, each agent with a locally observed set of data points, and a common server with whom the agents can interact. The agents' goal is to compute a linear model that best fits the collective data points observed by all the agents. In the server-based distributed settings, the server cannot access the data points held by the agents. The recently proposed Iteratively Pre-conditioned Gradient-descent (IPG) method has been shown to converge faster than other existing distributed algorith…
University of Maryland（马里兰大学）；École polytechnique fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
This paper studies a distributed multi-agent convex optimization problem. The system comprises multiple agents in this problem, each with a set of local data points and an associated local cost function. The agents are connected to a server, and there is no inter-agent communication. The agents' goal is to learn a parameter vector that optimizes the aggregate of their local costs without revealing their local data points. In principle, the agents can solve this problem by collaborating with the server using the traditional distributed gradient-descent method. However, when the aggregate cost i…
Huazhong University of Science and Technology（华中科技大学）
Takagi-Sugeno-Kang (TSK) fuzzy systems are flexible and interpretable machine learning models; however, they may not be easily optimized when the data size is large, and/or the data dimensionality is high. This paper proposes a mini-batch gradient descent (MBGD) based algorithm to efficiently and effectively train TSK fuzzy classifiers. It integrates two novel techniques: 1) uniform regularization (UR), which forces the rules to have similar average contributions to the output, and hence to increase the generalization performance of the TSK classifier; and, 2) batch normalization (BN), which e…
Stochastic gradient descent (SGD) is a pillar of modern machine learning, serving as the go-to optimization algorithm for a diverse array of problems. While the empirical success of SGD is often attributed to its computational efficiency and favorable generalization behavior, neither effect is well understood and disentangling them remains an open problem. Even in the simple setting of convex quadratic problems, worst-case analyses give an asymptotic convergence rate for SGD that is no better than full-batch gradient descent (GD), and the purported implicit regularization effects of SGD lack a…
Asymptotic Analysis of Conditioned Stochastic Gradient Descent
École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；ENSAI（法国国立统计与信息分析学院）；École Nationale de la Statistique et de l’Analyse de l’Information（法国国立统计与信息分析学院）
Max Planck-Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Tübingen（图宾根大学）
Pre-conditioning is a well-known concept that can significantly improve the convergence of optimization algorithms. For noise-free problems, where good pre-conditioners are not known a priori, iterative linear algebra methods offer one way to efficiently construct them. For the stochastic optimization problems that dominate contemporary machine learning, however, this approach is not readily available. We propose an iterative algorithm inspired by classic iterative linear solvers that uses a probabilistic model to actively infer a pre-conditioner in situations where Hessian-projections can onl…
Gauss-Newton Natural Gradient Descent for Shape Learning
Seminar for Applied Mathematics (SAM), D-Math, ETH Zürich（应用数学研讨会（SAM）、D-Math、苏黎世联邦理工学院）；Institute for Machine Learning, JKU Linz（机器学习研究所、杰克逊维尔大学）；ETH AI Center, ETH Zürich（苏黎世联邦理工学院人工智能中心）
Preconditioned Gradient Descent for Overparameterized Nonconvex Burer--Monteiro Factorization with Global Optimality Certification
Electrical and Computer Engineering University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校电气与计算机工程学院）；Industrial and Operations Engineering University of Michigan（密歇根大学工业与运营工程学院）
Technische Universität Berlin（柏林工业大学）；Technische Universität München（慕尼黑工业大学）
The projected gradient descent (PGD) method has shown to be effective in recovering compressed signals described in a data-driven way by a generative model, i.e., a generator which has learned the data distribution. Further reconstruction improvements for such inverse problems can be achieved by conditioning the generator on the measurement. The boundary equilibrium generative adversarial network (BEGAN) implements an equilibrium based loss function and an auto-encoding discriminator to better balance the performance of the generator and the discriminator. In this work we investigate a network…
We introduce Natural Neural Networks, a novel family of algorithms that speed up convergence by adapting their internal representation during training to improve conditioning of the Fisher matrix. In particular, we show a specific example that employs a simple and efficient reparametrization of the neural network weights by implicitly whitening the representation obtained at each layer, while preserving the feed-forward computation of the network. Such networks can be trained efficiently via the proposed Projected Natural Gradient Descent algorithm (PRONG), which amortizes the cost of these re…
Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra
University of Pavia（帕维亚大学）；Department of Mathematics（数学系）；Department of Civil Engineering and Architecture（土木工程与建筑系）；Institut Camille Jordan, Lyon 1 Université（让·乔丹研究所，里昂1大学）
On the Convergence of Gradient Descent on Learning Transformers with Residual Connections
Michigan Institute for Computational Discovery and Engineering, Department of Electrical Engineering and Computer Science, Department of Statistics, University of Michigan（密歇根大学计算发现与工程研究所、电气工程与计算机科学系、统计系）；Department of Computer Science and Engineering, The Ohio State University（俄亥俄州立大学计算机科学与工程系）
Huazhong University of Science and Technology（华中科技大学）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）
To effectively optimize Takagi-Sugeno-Kang (TSK) fuzzy systems for regression problems, a mini-batch gradient descent with regularization, DropRule, and AdaBound (MBGD-RDA) algorithm was recently proposed. This paper further proposes FCM-RDpA, which improves MBGD-RDA by replacing the grid partition approach in rule initialization by fuzzy c-means clustering, and AdaBound by Powerball AdaBelief, which integrates recently proposed Powerball gradient and AdaBelief to further expedite and stabilize parameter optimization. Extensive experiments on 22 regression datasets with various sizes and dimen…
Design Criteria for SGD Preconditioners: Local Conditioning, Noise Floors, and Basin Stability
Department of Mathematics, Emory University（埃默里大学数学系）；Department of Mathematics, University of Minnesota Twin Cities（明尼苏达大学双城分校数学系）；Department of Computer Science, University of Minnesota Twin Cities（明尼苏达大学双城分校计算机科学系）；Department of Mathematics, University of Kentucky（肯塔基大学数学系）
University of Bath（巴斯大学）；CNRS（法国国家科学研究中心）；ENS（巴黎高等师范学院）；PSL Université（巴黎文理研究大学）
Non-smooth optimization is a core ingredient of many imaging or machine learning pipelines. Non-smoothness encodes structural constraints on the solutions, such as sparsity, group sparsity, low-rank and sharp edges. It is also the basis for the definition of robust loss functions and scale-free functionals such as square-root Lasso. Standard approaches to deal with non-smoothness leverage either proximal splitting or coordinate descent. These approaches are effective but usually require parameter tuning, preconditioning or some sort of support pruning. In this work, we advocate and study a dif…
Continuous optimization is an important problem in many areas of AI, including vision, robotics, probabilistic inference, and machine learning. Unfortunately, most real-world optimization problems are nonconvex, causing standard convex techniques to find only local optima, even with extensions like random restarts and simulated annealing. We observe that, in many cases, the local modes of the objective function have combinatorial structure, and thus ideas from combinatorial optimization can be brought to bear. Based on this, we propose a problem-decomposition approach to nonconvex optimization…
New York University（纽约大学）；Jagiellonian University（雅盖隆大学）；Stanford University（斯坦福大学）；Salesforce Research（Salesforce研究院）；Facebook AI Research（脸书人工智能研究院）
The early phase of training of deep neural networks is critical for their final performance. In this work, we study how the hyperparameters of stochastic gradient descent (SGD) used in the early phase of training affect the rest of the optimization trajectory. We argue for the existence of the "break-even" point on this trajectory, beyond which the curvature of the loss surface and noise in the gradient are implicitly regularized by SGD. In particular, we demonstrate on multiple classification tasks that using a large learning rate in the initial phase of training reduces the variance of the g…
Stochastic Gradient Descent (SGD) is widely used in machine learning problems to efficiently perform empirical risk minimization, yet, in practice, SGD is known to stall before reaching the actual minimizer of the empirical risk. SGD stalling has often been attributed to its sensitivity to the conditioning of the problem; however, as we demonstrate, SGD will stall even when applied to a simple linear regression problem with unity condition number for standard learning rates. Thus, in this work, we numerically demonstrate and mathematically argue that stalling is a crippling and generic limitat…
We perform an average case analysis of the generalization dynamics of large neural networks trained using gradient descent. We study the practically-relevant "high-dimensional" regime where the number of free parameters in the network is on the order of or even larger than the number of examples in the dataset. Using random matrix theory and exact solutions in linear models, we derive the generalization error and training error dynamics of learning and analyze how they depend on the dimensionality of data and signal to noise ratio of the learning problem. We find that the dynamics of gradient …
Many machine learning problems optimize an objective that must be measured with noise. The primary method is a first order stochastic gradient descent using one or more Monte Carlo (MC) samples at each step. There are settings where ill-conditioning makes second order methods such as L-BFGS more effective. We study the use of randomized quasi-Monte Carlo (RQMC) sampling for such problems. When MC sampling has a root mean squared error (RMSE) of $O(n^{-1/2})$ then RQMC has an RMSE of $o(n^{-1/2})$ that can be close to $O(n^{-3/2})$ in favorable settings. We prove that improved sampling accuracy…
In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to $\mathit{local~strong~convexity}$ in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. Most popular nonlinear activation functions satisfy the distilled properties, including rectified linear units (ReLUs), leaky ReLUs, squared ReLUs and sigmoids. For activation functions that are also smooth, we show $\mathit{local~linear~convergence}$ guarantees of gradient descent under a resampling rule. For homogeneou…
An operator preconditioning perspective on training in physics-informed machine learning
ETH Zürich（苏黎世联邦理工学院）；Sorbonne Université（索邦大学）；Extrality（埃克斯特拉蒂（Extrality））；ETH AI Center（苏黎世联邦理工学院人工智能中心）；Seminar for Applied Mathematics（应用数学研讨班）；Institute of Intelligent Systems and Robotics（智能系统与机器人研究所）
While stochastic gradient descent (SGD) and variants have been surprisingly successful for training deep nets, several aspects of the optimization dynamics and generalization are still not well understood. In this paper, we present new empirical observations and theoretical results on both the optimization dynamics and generalization behavior of SGD for deep nets based on the Hessian of the training loss and associated quantities. We consider three specific research questions: (1) what is the relationship between the Hessian of the loss and the second moment of stochastic gradients (SGs)? (2) …
United States Naval Academy（美国海军学院）；University of Maryland, College Park（马里兰大学帕克分校）；Rice University（莱斯大学）
With the growing importance of large network models and enormous training datasets, GPUs have become increasingly necessary to train neural networks. This is largely because conventional optimization algorithms rely on stochastic gradient methods that don't scale well to large numbers of cores in a cluster setting. Furthermore, the convergence of all gradient methods, including batch methods, suffers from common problems like saturation effects, poor conditioning, and saddle points. This paper explores an unconventional training method that uses alternating direction methods and Bregman iterat…
University of Washington（华盛顿大学）；Rutgers University（罗格斯大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
Existing results for low-rank matrix recovery largely focus on quadratic loss, which enjoys favorable properties such as restricted strong convexity/smoothness (RSC/RSM) and well conditioning over all low rank matrices. However, many interesting problems involve more general, non-quadratic losses, which do not satisfy such properties. For these problems, standard nonconvex approaches such as rank-constrained projected gradient descent (a.k.a. iterative hard thresholding) and Burer-Monteiro factorization could have poor empirical performance, and there is no satisfactory theory guaranteeing glo…
It is well-known that the distribution over functions induced through a zero-mean iid prior distribution over the parameters of a multi-layer perceptron (MLP) converges to a Gaussian process (GP), under mild conditions. We extend this result firstly to independent priors with general zero or non-zero means, and secondly to a family of partially exchangeable priors which generalise iid priors. We discuss how the second prior arises naturally when considering an equivalence class of functions in an MLP and through training processes such as stochastic gradient descent. The model resulting from p…
Iso-Riemannian Optimization on Learned Data Manifolds
Department of Mathematics University of California, Los Angeles（数学系 加州大学洛杉矶分校）；School of Engineering and Applied Sciences Harvard University（工程与应用科学学院 哈佛大学）
PC Layer: Polynomial Weight Preconditioning for Improving LLM Pre-Training
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Google LLC（谷歌公司）；Shenzhen International Center for Industrial and Applied Mathematics（深圳国际工业与应用数学中心）；Shenzhen Research Institute of Big Data（深圳大数据研究院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
The conditions of relative smoothness and relative strong convexity were recently introduced for the analysis of Bregman gradient methods for convex optimization. We introduce a generalized left-preconditioning method for gradient descent, and show that its convergence on an essentially smooth convex objective function can be guaranteed via an application of relative smoothness in the dual space. Our relative smoothness assumption is between the designed preconditioner and the convex conjugate of the objective, and it generalizes the typical Lipschitz gradient assumption. Under dual relative s…
The loss surface of deep neural networks has recently attracted interest in the optimization and machine learning communities as a prime example of high-dimensional non-convex problem. Some insights were recently gained using spin glass models and mean-field approximations, but at the expense of strongly simplifying the nonlinear nature of the model. In this work, we do not make any such assumption and study conditions on the data distribution and model architecture that prevent the existence of bad local minima. Our theoretical work quantifies and formalizes two important \emph{folklore} fact…
Hong Kong University of Science and Technology（香港科技大学）
The tensor train (TT) format enjoys appealing advantages in handling structural high-order tensors. The recent decade has witnessed the wide applications of TT-format tensors from diverse disciplines, among which tensor completion has drawn considerable attention. Numerous fast algorithms, including the Riemannian gradient descent (RGrad), have been proposed for the TT-format tensor completion. However, the theoretical guarantees of these algorithms are largely missing or sub-optimal, partly due to the complicated and recursive algebraic operations in TT-format decomposition. Moreover, existin…
Modern proximal and stochastic gradient descent (SGD) methods are believed to efficiently minimize large composite objective functions, but such methods have two algorithmic challenges: (1) a lack of fast or justified stop conditions, and (2) sensitivity to the objective function's conditioning. In response to the first challenge, modern proximal and SGD methods guarantee convergence only after multiple epochs, but such a guarantee renders proximal and SGD methods infeasible when the number of component functions is very large or infinite. In response to the second challenge, second order SGD …
Carnegie Mellon University（卡内基梅隆大学）；University of Chicago（芝加哥大学）
An increasing number of data science and machine learning problems rely on computation with tensors, which better capture the multi-way relationships and interactions of data than matrices. When tapping into this critical advantage, a key challenge is to develop computationally efficient and provably correct algorithms for extracting useful information from tensor data that are simultaneously robust to corruptions and ill-conditioning. This paper tackles tensor robust principal component analysis (RPCA), which aims to recover a low-rank tensor from its observations contaminated by sparse corru…
