A matrix free and a low rank approximation preconditioner are proposed to accelerate the convergence of stochastic gradient descent (SGD) by exploiting curvature information sampled from Hessian-vector products or finite differences of parameters and gradients similar to the BFGS algorithm. Both preconditioners are fitted with an online updating manner minimizing a criterion that is free of line search and robust to stochastic gradient noise, and further constrained to be on certain connected Lie groups to preserve their corresponding symmetry or invariance, e.g., orientation of coordinates by the connected general linear group with positive determinants. The Lie group's equivariance property facilitates preconditioner fitting, and its invariance property saves any need of damping, which is common in second-order optimizers, but difficult to tune. The learning rate for parameter updating and step size for preconditioner fitting are naturally normalized, and their default values work well in most situations.
GMEMS Technologies and Spectimbre（GMEMS技术和Spectimbre（公司））；Fairview Way, Milpitas, CA 95035（366 Fairview Way, Milpitas, CA 95035（地址，非机构））
We study two types of preconditioners and preconditioned stochastic gradient descent (SGD) methods in a unified framework. We call the first one the Newton type due to its close relationship to the Newton method, and the second one the Fisher type as its preconditioner is closely related to the inverse of Fisher information matrix. Both preconditioners can be derived from one framework, and efficiently estimated on any matrix Lie groups designated by the user using natural or relative gradient descent minimizing certain preconditioner estimation criteria. Many existing preconditioners and meth…
Stochastic gradient descent (SGD) still is the workhorse for many practical problems. However, it converges slow, and can be difficult to tune. It is possible to precondition SGD to accelerate its convergence remarkably. But many attempts in this direction either aim at solving specialized problems, or result in significantly more complicated methods than SGD. This paper proposes a new method to estimate a preconditioner such that the amplitudes of perturbations of preconditioned stochastic gradient match that of the perturbations of parameters to be optimized in a way comparable to Newton met…
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）
MIT EECS（麻省理工学院电子工程与计算机科学系）；Google Research, New York（谷歌研究院纽约分部）
Adaptive methods such as Adam and RMSProp are widely used in deep learning but are not well understood. In this paper, we seek a crisp, clean and precise characterization of their behavior in nonconvex settings. To this end, we first provide a novel view of adaptive methods as preconditioned SGD, where the preconditioner is estimated in an online manner. By studying the preconditioner on its own, we elucidate its purpose: it rescales the stochastic gradient noise to be isotropic near stationary points, which helps escape saddle points. Furthermore, we show that adaptive methods can efficiently…
High-dimensional limit theorems for SGD: Momentum and Adaptive Step-sizes
Department of Statistics and Actuarial Science, University of Waterloo, Canada（统计与精算科学系，滑铁卢大学，加拿大）；Cheriton School of Computer Science, University of Waterloo, Canada（切尔顿计算机科学学院，滑铁卢大学，加拿大）
This paper proposes a family of online second order methods for possibly non-convex stochastic optimizations based on the theory of preconditioned stochastic gradient descent (PSGD), which can be regarded as an enhance stochastic Newton method with the ability to handle gradient noise and non-convexity simultaneously. We have improved the implementations of the original PSGD in several ways, e.g., new forms of preconditioners, more accurate Hessian vector product calculations, and better numerical stability with vanishing or ill-conditioned Hessian, etc.. We also have unrevealed the relationsh…
Optimizing the Preconditioner: A Black-box Online-to-Nonconvex Conversion with Static Regret Minimization Oracles
Center for Computational Science and Engineering, MIT（麻省理工学院计算科学与工程中心）；Department of Civil and Environmental Engineering, MIT（麻省理工学院土木与环境工程系）；Institute for Data, Systems, and Society, MIT（麻省理工学院数据、系统与社会研究所）
We provide the first generalization error analysis for black-box learning through derivative-free optimization. Under the assumption of a Lipschitz and smooth unknown loss, we consider the Zeroth-order Stochastic Search (ZoSS) algorithm, that updates a $d$-dimensional model by replacing stochastic gradient directions with stochastic differences of $K+1$ perturbed loss evaluations per dataset (example) query. For both unbounded and bounded possibly nonconvex losses, we present the first generalization bounds for the ZoSS algorithm. These bounds coincide with those for SGD, and rather surprising…
Stochastic convex optimization algorithms are the most popular way to train machine learning models on large-scale data. Scaling up the training process of these models is crucial, but the most popular algorithm, Stochastic Gradient Descent (SGD), is a serial method that is surprisingly hard to parallelize. In this paper, we propose an efficient distributed stochastic optimization method by combining adaptivity with variance reduction techniques. Our analysis yields a linear speedup in the number of machines, constant memory footprint, and only a logarithmic number of communication rounds. Cri…
This paper studies the performance of a recently proposed preconditioned stochastic gradient descent (PSGD) algorithm on recurrent neural network (RNN) training. PSGD adaptively estimates a preconditioner to accelerate gradient descent, and is designed to be simple, general and easy to use, as stochastic gradient descent (SGD). RNNs, especially the ones requiring extremely long term memories, are difficult to train. We have tested PSGD on a set of synthetic pathological RNN learning problems and the real world MNIST handwritten digit recognition task. Experimental results suggest that PSGD is …
Why is Normalization Preferred? A Worst-Case Complexity Theory for Stochastically Preconditioned SGD under Heavy-Tailed Noise
Department of Mathematics, University of California, Berkeley, CA, USA（加州大学伯克利分校数学系）；Department of EECS, University of California, Berkeley, CA, USA（加州大学伯克利分校电子工程与计算机科学系）；Department of IEOR, University of California, Berkeley, CA, USA（加州大学伯克利分校工业工程与运营研究系）
University of California, San Diego（加利福尼亚大学圣地亚哥分校）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Google（谷歌）
While significant progress has been made separately on analytics systems for scalable stochastic gradient descent (SGD) and private SGD, none of the major scalable analytics frameworks have incorporated differentially private SGD. There are two inter-related issues for this disconnect between research and practice: (1) low model accuracy due to added noise to guarantee privacy, and (2) high development and runtime overhead of the private algorithms. This paper takes a first step to remedy this disconnect and proposes a private SGD algorithm to address \emph{both} issues in an integrated manner…
We establish novel generalization bounds for learning algorithms that converge to global minima. We do so by deriving black-box stability results that only depend on the convergence of a learning algorithm and the geometry around the minimizers of the loss function. The results are shown for nonconvex loss functions satisfying the Polyak-Łojasiewicz (PL) and the quadratic growth (QG) conditions. We further show that these conditions arise for some neural networks with linear activations. We use our black-box results to establish the stability of optimization algorithms such as stochastic gradi…
SGD for Variational Inference: Tackling Unbounded Variance via Preconditioning and Dynamic Batching
MaLGa；DIBRIS Università degli Studi di Genova（DIBRIS基因瓦大学）；Università degli Studi di Genova（基因瓦大学）；DIMA Università degli Studi di Genova（DIMA基因瓦大学）；IIT Università degli Studi di Genova（IIT基因瓦大学）
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…
We study the problem of directly optimizing arbitrary non-differentiable task evaluation metrics such as misclassification rate and recall. Our method, named MetricOpt, operates in a black-box setting where the computational details of the target metric are unknown. We achieve this by learning a differentiable value function, which maps compact task-specific model parameters to metric observations. The learned value function is easily pluggable into existing optimizers like SGD and Adam, and is effective for rapidly finetuning a pre-trained model. This leads to consistent improvements since th…
Texas Advanced Computing Center（德州高级计算中心）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of Chicago（芝加哥大学）；Argonne National Laboratory（阿贡国家实验室）
Training neural networks with many processors can reduce time-to-solution; however, it is challenging to maintain convergence and efficiency at large scales. The Kronecker-factored Approximate Curvature (K-FAC) was recently proposed as an approximation of the Fisher Information Matrix that can be used in natural gradient optimizers. We investigate here a scalable K-FAC design and its applicability in convolutional neural network (CNN) training at scale. We study optimization techniques such as layer-wise distribution strategies, inverse-free second-order gradient evaluation, and dynamic K-FAC …
ÉTS Montréal（蒙特利尔高等工程技术学院）；Shenzhen Institutes of Advanced Technology（深圳先进技术研究院）
We propose a novel weakly supervised learning segmentation based on several global constraints derived from box annotations. Particularly, we leverage a classical tightness prior to a deep learning setting via imposing a set of constraints on the network outputs. Such a powerful topological prior prevents solutions from excessive shrinking by enforcing any horizontal or vertical line within the bounding box to contain, at least, one pixel of the foreground region. Furthermore, we integrate our deep tightness prior with a global background emptiness constraint, guiding training with information…
Salzburg University of Applied Sciences（萨尔茨堡应用科学大学）
In this work we present an "out-of-the-box" application of Machine Learning (ML) optimizers for an industrial optimization problem. We introduce a piecewise polynomial model (spline) for fitting of $\mathcal{C}^k$-continuos functions, which can be deployed in a cam approximation setting. We then use the gradient descent optimization context provided by the machine learning framework TensorFlow to optimize the model parameters with respect to approximation quality and $\mathcal{C}^k$-continuity and evaluate available optimizers. Our experiments show that the problem solution is feasible using T…
Accelerated zero-order SGD under high-order smoothness and overparameterized regime
Lomonosov MSU（罗蒙诺索夫莫斯科大学）；ISP RAS（俄罗斯科学院信息问题研究所）；HSE University（高等经济大学）；MIPT（莫斯科物理技术学院）；MSU Institute for Artificial Intelligence（莫斯科大学人工智能研究所）；Innopolis University（因诺波利斯大学）；IITP RAS（俄罗斯科学院信息传输问题研究所）；Skoltech（斯科尔科沃科学技术研究院）
Edmond and Lilly Safra Center for Brain Sciences（埃德蒙与莉莉·萨夫拉脑科学中心）；The Hebrew University of Jerusalem（耶路撒冷希伯来大学）
Despite their great success, there is still no comprehensive theoretical understanding of learning with Deep Neural Networks (DNNs) or their inner organization. Previous work proposed to analyze DNNs in the \textit{Information Plane}; i.e., the plane of the Mutual Information values that each layer preserves on the input and output variables. They suggested that the goal of the network is to optimize the Information Bottleneck (IB) tradeoff between compression and prediction, successively, for each layer. In this work we follow up on this idea and demonstrate the effectiveness of the Informati…
We present CYCLADES, a general framework for parallelizing stochastic optimization algorithms in a shared memory setting. CYCLADES is asynchronous during shared model updates, and requires no memory locking mechanisms, similar to HOGWILD!-type algorithms. Unlike HOGWILD!, CYCLADES introduces no conflicts during the parallel execution, and offers a black-box analysis for provable speedups across a large family of algorithms. Due to its inherent conflict-free nature and cache locality, our multi-core implementation of CYCLADES consistently outperforms HOGWILD!-type algorithms on sufficiently spa…
Large Deviation Upper Bounds and Improved MSE Rates of Nonlinear SGD: Heavy-tailed Noise and Power of Symmetry
Carnegie Mellon University（卡内基梅隆大学）；Faculty of Technical Sciences, University of Novi Sad（诺维萨德大学技术科学学院）；Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院）
Accurate estimates of network parameters are essential for modeling, monitoring, and control in power distribution systems. In this paper, we develop a physics-informed graphical learning algorithm to estimate network parameters of three-phase power distribution systems. Our proposed algorithm uses only readily available smart meter data to estimate the three-phase series resistance and reactance of the primary distribution line segments. We first develop a parametric physics-based model to replace the black-box deep neural networks in the conventional graphical neural network (GNN). Then we d…
Nonlinear Stochastic Gradient Descent and Heavy-tailed Noise: A Unified Framework and High-probability Guarantees
Carnegie Mellon University（卡内基梅隆大学）；Faculty of Technical Sciences, University of Novi Sad（诺威萨大学技术科学学院）；Faculty of Sciences, University of Novi Sad（诺威萨大学理学院）
Effective training of deep neural networks suffers from two main issues. The first is that the parameter spaces of these models exhibit pathological curvature. Recent methods address this problem by using adaptive preconditioning for Stochastic Gradient Descent (SGD). These methods improve convergence by adapting to the local geometry of parameter space. A second issue is overfitting, which is typically addressed by early stopping. However, recent work has demonstrated that Bayesian model averaging mitigates this problem. The posterior can be sampled by using Stochastic Gradient Langevin Dynam…
