Following early work on Hessian-free methods for deep learning, we study a stochastic generalized Gauss-Newton method (SGN) for training DNNs. SGN is a second-order optimization method, with efficient iterations, that we demonstrate to often require substantially fewer iterations than standard SGD to converge. As the name suggests, SGN uses a Gauss-Newton approximation for the Hessian matrix, and, in order to compute an approximate search direction, relies on the conjugate gradient method combined with forward and reverse automatic differentiation. Despite the success of SGD and its first-order variants, and despite Hessian-free methods based on the Gauss-Newton Hessian approximation having been already theoretically proposed as practical methods for training DNNs, we believe that SGN has a lot of undiscovered and yet not fully displayed potential in big mini-batch scenarios. For this setting, we demonstrate that SGN does not only substantially improve over SGD in terms of the number of iterations, but also in terms of runtime. This is made possible by an efficient, easy-to-use and flexible implementation of SGN we propose in the Theano deep learning platform, which, unlike Tensorflow and Pytorch, supports forward automatic differentiation. This enables researchers to further study and improve this promising optimization technique and hopefully reconsider stochastic second-order methods as competitive optimization techniques for training DNNs; we also hope that the promise of SGN may lead to forward automatic differentiation being added to Tensorflow or Pytorch. Our results also show that in big mini-batch scenarios SGN is more robust than SGD with respect to its hyperparameters (we never had to tune its step-size for our benchmarks!), which eases the expensive process of hyperparameter tuning that is instead crucial for the performance of first-order methods.
First-order methods like stochastic gradient descent(SGD) are recently the popular optimization method to train deep neural networks (DNNs), but second-order methods are scarcely used because of the overpriced computing cost in getting the high-order information. In this paper, we propose the Damped Newton Stochastic Gradient Descent(DN-SGD) method and Stochastic Gradient Descent Damped Newton(SGD-DN) method to train DNNs for regression problems with Mean Square Error(MSE) and classification problems with Cross-Entropy Loss(CEL), which is inspired by a proved fact that the hessian matrix of la…
We present practical Levenberg-Marquardt variants of Gauss-Newton and natural gradient methods for solving non-convex optimization problems that arise in training deep neural networks involving enormous numbers of variables and huge data sets. Our methods use subsampled Gauss-Newton or Fisher information matrices and either subsampled gradient estimates (fully stochastic) or full gradients (semi-stochastic), which, in the latter case, we prove convergent to a stationary point. By using the Sherman-Morrison-Woodbury formula with automatic differentiation (backpropagation) we show how our method…
The University of Tokyo（东京大学）；The Institute of Statistical Mathematics（统计数学研究所）
We develop a minimax rate analysis to describe the reason that deep neural networks (DNNs) perform better than other standard methods. For nonparametric regression problems, it is well known that many standard methods attain the minimax optimal rate of estimation errors for smooth functions, and thus, it is not straightforward to identify the theoretical advantages of DNNs. This study tries to fill this gap by considering the estimation for a class of non-smooth functions that have singularities on hypersurfaces. Our findings are as follows: (i) We derive the generalization error of a DNN esti…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；SAS Institute（SAS研究所）
In this work we describe an Adaptive Regularization using Cubics (ARC) method for large-scale nonconvex unconstrained optimization using Limited-memory Quasi-Newton (LQN) matrices. ARC methods are a relatively new family of optimization strategies that utilize a cubic-regularization (CR) term in place of trust-regions and line-searches. LQN methods offer a large-scale alternative to using explicit second-order information by taking identical inputs to those used by popular first-order methods such as stochastic gradient descent (SGD). Solving the CR subproblem exactly requires Newton's method,…
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）
First-order methods such as stochastic gradient descent (SGD) are currently the standard algorithm for training deep neural networks. Second-order methods, despite their better convergence rate, are rarely used in practice due to the prohibitive computational cost in calculating the second-order information. In this paper, we propose a novel Gram-Gauss-Newton (GGN) algorithm to train deep neural networks for regression problems with square loss. Our method draws inspiration from the connection between neural network optimization and kernel regression of neural tangent kernel (NTK). Different f…
University of Maryland Baltimore County（马里兰大学巴尔的摩县分校）
This paper proposes a set of new error criteria and learning approaches, Adaptive Normalized Risk-Averting Training (ANRAT), to attack the non-convex optimization problem in training deep neural networks (DNNs). Theoretically, we demonstrate its effectiveness on global and local convexity lower-bounded by the standard $L_p$-norm error. By analyzing the gradient on the convexity index $λ$, we explain the reason why to learn $λ$ adaptively using gradient descent works. In practice, we show how this method improves training of deep neural networks to solve visual recognition tasks on the MNIST an…
LG Electronics Canada（LG电子加拿大公司）；LG Electronics USA（LG电子美国公司）
Batch Normalization (BN) is a popular technique for training Deep Neural Networks (DNNs). BN uses scaling and shifting to normalize activations of mini-batches to accelerate convergence and improve generalization. The recently proposed Iterative Normalization (IterNorm) method improves these properties by whitening the activations iteratively using Newton's method. However, since Newton's method initializes the whitening matrix independently at each training step, no information is shared between consecutive steps. In this work, instead of exact computation of whitening matrix at each time ste…
Incremental Gauss--Newton Methods with Superlinear Convergence Rates
School of Data Science, Fudan University（复旦大学数据科学学院）；National University of Singapore（新加坡国立大学）；Department of Computer Science and Engineering, The Chinese University of Hong Kong（香港中文大学计算机科学与工程学系）
University of California, Berkeley（加州大学伯克利分校）；Stanford University（斯坦福大学）；ICSI（国际计算机科学研究所）
In second-order optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration. Randomized sketching has emerged as a powerful technique for constructing estimates of the Hessian which can be used to perform approximate Newton steps. This involves multiplication by a random sketching matrix, which introduces a trade-off between the computational cost of sketching and the convergence rate of the optimization algorithm. A theoretically desirable but practically much too expensive choice is to use a dense Gaussian sketching matrix, which prod…
We introduce a new second order stochastic algorithm to estimate the entropically regularized optimal transport cost between two probability measures. The source measure can be arbitrary chosen, either absolutely continuous or discrete, while the target measure is assumed to be discrete. To solve the semi-dual formulation of such a regularized and semi-discrete optimal transportation problem, we propose to consider a stochastic Gauss-Newton algorithm that uses a sequence of data sampled from the source measure. This algorithm is shown to be adaptive to the geometry of the underlying convex opt…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；RIKEN Center for Advanced Intelligence Project（理化学研究所高级智能中心）；EPFL School of Computer and Communication Sciences（洛桑联邦理工学院计算机与通信科学学院）
In this thesis, we disentangle the generalized Gauss-Newton and approximate inference for Bayesian deep learning. The generalized Gauss-Newton method is an optimization method that is used in several popular Bayesian deep learning algorithms. Algorithms that combine the Gauss-Newton method with the Laplace and Gaussian variational approximation have recently led to state-of-the-art results in Bayesian deep learning. While the Laplace and Gaussian variational approximation have been studied extensively, their interplay with the Gauss-Newton method remains unclear. Recent criticism of priors and…
RIKEN Center for AI Project（日本理化学研究所人工智能项目中心）；EPFL（洛桑联邦理工学院）；Technical University of Denmark（丹麦技术大学）
Deep neural networks (DNN) and Gaussian processes (GP) are two powerful models with several theoretical connections relating them, but the relationship between their training methods is not well understood. In this paper, we show that certain Gaussian posterior approximations for Bayesian DNNs are equivalent to GP posteriors. This enables us to relate solutions and iterations of a deep-learning algorithm to GP inference. As a result, we can obtain a GP kernel and a nonlinear feature map while training a DNN. Surprisingly, the resulting kernel is the neural tangent kernel. We show kernels obtai…
Shell International Exploration & Production, Inc.（壳牌国际勘探与生产公司）
Training in supervised deep learning is computationally demanding, and the convergence behavior is usually not fully understood. We introduce and study a second-order stochastic quasi-Gauss-Newton (SQGN) optimization method that combines ideas from stochastic quasi-Newton methods, Gauss-Newton methods, and variance reduction to address this problem. SQGN provides excellent accuracy without the need for experimenting with many hyper-parameter configurations, which is often computationally prohibitive given the number of combinations and the cost of each training process. We discuss the implemen…
Shanghai Jiao Tong University（上海交通大学）；Tsinghua University（清华大学）；University of Electronic Science and Technology（电子科技大学）
Deep neural networks (DNNs) usually contain massive parameters, but there is redundancy such that it is guessed that the DNNs could be trained in low-dimensional subspaces. In this paper, we propose a Dynamic Linear Dimensionality Reduction (DLDR) based on low-dimensional properties of the training trajectory. The reduction is efficient, which is supported by comprehensive experiments: optimization in 40 dimensional spaces can achieve comparable performance as regular training over thousands or even millions of parameters. Since there are only a few optimization variables, we develop a quasi-N…
Approximate Newton methods are a standard optimization tool which aim to maintain the benefits of Newton's method, such as a fast rate of convergence, whilst alleviating its drawbacks, such as computationally expensive calculation or estimation of the inverse Hessian. In this work we investigate approximate Newton methods for policy optimization in Markov Decision Processes (MDPs). We first analyse the structure of the Hessian of the objective function for MDPs. We show that, like the gradient, the Hessian exhibits useful structure in the context of MDPs and we use this analysis to motivate tw…
University of Tübingen（蒂宾根大学）；MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）
Curvature in form of the Hessian or its generalized Gauss-Newton (GGN) approximation is valuable for algorithms that rely on a local model for the loss to train, compress, or explain deep networks. Existing methods based on implicit multiplication via automatic differentiation or Kronecker-factored block diagonal approximations do not consider noise in the mini-batch. We present ViViT, a curvature model that leverages the GGN's low-rank structure without further approximations. It allows for efficient computation of eigenvalues, eigenvectors, as well as per-sample first- and second-order direc…
Inception Institute of Artificial Intelligence（人工智能起源研究所）；University of Electronic Science and Technology of China（电子科技大学）；ByteDance AI Lab（字节跳动人工智能实验室）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Orthogonality is widely used for training deep neural networks (DNNs) due to its ability to maintain all singular values of the Jacobian close to 1 and reduce redundancy in representation. This paper proposes a computationally efficient and numerically stable orthogonalization method using Newton's iteration (ONI), to learn a layer-wise orthogonal weight matrix in DNNs. ONI works by iteratively stretching the singular values of a weight matrix towards 1. This property enables it to control the orthogonality of a weight matrix by its number of iterations. We show that our method improves the pe…
Domain decomposition architectures and Gauss-Newton training for physics-informed neural networks
Alexander Heinlein Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, Delft, 2628 CD, the Netherlands, Taniya Kapoor Artificial Intelligence Group, Wageningen University \& Research, Wageningen, The Netherlands
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学（KAUST））
Despite the impressive performance of deep neural networks (DNNs) on numerous vision tasks, they still exhibit yet-to-understand uncouth behaviours. One puzzling behaviour is the subtle sensitive reaction of DNNs to various noise attacks. Such a nuisance has strengthened the line of research around developing and training noise-robust networks. In this work, we propose a new training regularizer that aims to minimize the probabilistic expected training loss of a DNN subject to a generic Gaussian input. We provide an efficient and simple approach to approximate such a regularizer for arbitrary …
University of Oxford（牛津大学）；The Alan Turing Institute for Data Science（艾伦·图灵数据科学研究所）
We propose a Randomised Subspace Gauss-Newton (R-SGN) algorithm for solving nonlinear least-squares optimization problems, that uses a sketched Jacobian of the residual in the variable domain and solves a reduced linear least-squares on each iteration. A sublinear global rate of convergence result is presented for a trust-region variant of R-SGN, with high probability, which matches deterministic counterpart results in the order of the accuracy tolerance. Promising preliminary numerical results are presented for R-SGN on logistic regression and on nonlinear regression problems from the CUTEst …
Gauss-Newton Temporal Difference Learning with Nonlinear Function Approximation
Center for Data Science, Peking University（北京大学数据科学中心）；National University of Singapore（新加坡国立大学）；Beijing International Center for Mathematical Research, Peking University（北京国际数学研究中心（北京大学））
