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In this work, we develop first-order (Hessian-free) and zero-order (derivative-free) implementations of the Cubically regularized Newton method for solving general non-convex optimization problems. For that, we employ finite difference…

Optimization and Control · Mathematics 2023-09-06 Nikita Doikov , Geovani Nunes Grapiglia

While the superior performance of second-order optimization methods such as Newton's method is well known, they are hardly used in practice for deep learning because neither assembling the Hessian matrix nor calculating its inverse is…

Machine Learning · Computer Science 2020-09-16 Siyuan Shen , Tianjia Shao , Kun Zhou , Chenfanfu Jiang , Feng Luo , Yin Yang

We define a second-order neural network stochastic gradient training algorithm whose block-diagonal structure effectively amounts to normalizing the unit activations. Investigating why this algorithm lacks in robustness then reveals two…

Machine Learning · Computer Science 2017-05-29 Jean Lafond , Nicolas Vasilache , Léon Bottou

In this paper we present a novel quasi-Newton algorithm for use in stochastic optimisation. Quasi-Newton methods have had an enormous impact on deterministic optimisation problems because they afford rapid convergence and computationally…

Systems and Control · Electrical Eng. & Systems 2019-09-04 Adrian Wills , Thomas Schön

Progress in deep learning is slowed by the days or weeks it takes to train large models. The natural solution of using more hardware is limited by diminishing returns, and leads to inefficient use of additional resources. In this paper, we…

Machine Learning · Computer Science 2017-12-12 Shankar Krishnan , Ying Xiao , Rif A. Saurous

For training fully-connected neural networks (FCNNs), we propose a practical approximate second-order method including: 1) an approximation of the Hessian matrix and 2) a conjugate gradient (CG) based method. Our proposed approximate…

Machine Learning · Computer Science 2018-12-07 Sheng-Wei Chen , Chun-Nan Chou , Edward Y. Chang

Due to the rapid growth of data and computational resources, distributed optimization has become an active research area in recent years. While first-order methods seem to dominate the field, second-order methods are nevertheless attractive…

Machine Learning · Computer Science 2018-06-21 Celestine Dünner , Aurelien Lucchi , Matilde Gargiani , An Bian , Thomas Hofmann , Martin Jaggi

Due to the effectiveness of second-order algorithms in solving classical optimization problems, designing second-order optimizers to train deep neural networks (DNNs) has attracted much research interest in recent years. However, because of…

Machine Learning · Computer Science 2024-03-06 Ying Sun , Hongwei Yong , Lei Zhang

Diagonal preconditioners are computationally feasible approximate to second-order optimizers, which have shown significant promise in accelerating training of deep learning models. Two predominant approaches are based on Adam and…

Machine Learning · Computer Science 2025-10-16 Bingbin Liu , Rachit Bansal , Depen Morwani , Nikhil Vyas , David Alvarez-Melis , Sham M. Kakade

Optimization in Deep Learning is mainly dominated by first-order methods which are built around the central concept of backpropagation. Second-order optimization methods, which take into account the second-order derivatives are far less…

Machine Learning · Computer Science 2021-04-09 Fares B. Mehouachi , Chaouki Kasmi

Using quasi-Newton methods in stochastic optimization is not a trivial task given the difficulty of extracting curvature information from the noisy gradients. Moreover, pre-conditioning noisy gradient observations tend to amplify the noise.…

Optimization and Control · Mathematics 2024-04-02 Andre Carlon , Luis Espath , Raul Tempone

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…

Machine Learning · Computer Science 2020-07-02 Christopher Thiele , Mauricio Araya-Polo , Detlef Hohl

We present Exact Gauss-Newton (EGN), a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the…

Machine Learning · Computer Science 2025-10-16 Mikalai Korbit , Adeyemi D. Adeoye , Alberto Bemporad , Mario Zanon

While first-order methods are popular for solving optimization problems that arise in large-scale deep learning problems, they come with some acute deficiencies. To diminish such shortcomings, there has been recent interest in applying…

Machine Learning · Computer Science 2023-10-05 Mahsa Yousefi , Angeles Martinez

In this paper, we propose a generic and simple strategy for utilizing stochastic gradient information in optimization. The technique essentially contains two consecutive steps in each iteration: 1) computing and normalizing each block…

Machine Learning · Computer Science 2018-04-24 Adams Wei Yu , Lei Huang , Qihang Lin , Ruslan Salakhutdinov , Jaime Carbonell

We introduce ADAHESSIAN, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the HESSIAN. Second order algorithms are among the most powerful…

Machine Learning · Computer Science 2021-04-30 Zhewei Yao , Amir Gholami , Sheng Shen , Mustafa Mustafa , Kurt Keutzer , Michael W. Mahoney

We propose a second-order (Hessian or Hessian-free) based optimization method for variational inference inspired by Gaussian backpropagation, and argue that quasi-Newton optimization can be developed as well. This is accomplished by…

Machine Learning · Statistics 2017-03-30 Kai Fan , Ziteng Wang , Jeff Beck , James Kwok , Katherine Heller

This paper presents a novel natural gradient and Hessian-free (NGHF) optimisation framework for neural network training that can operate efficiently in a distributed manner. It relies on the linear conjugate gradient (CG) algorithm to…

Machine Learning · Computer Science 2021-03-16 Adnan Haider , Chao Zhang , Florian L. Kreyssig , Philip C. Woodland

Accelerating the convergence of second-order optimization, particularly Newton-type methods, remains a pivotal challenge in algorithmic research. In this paper, we extend previous work on the \textbf{Quadratic Gradient (QG)} and rigorously…

Optimization and Control · Mathematics 2026-04-01 John Chiang

In this paper, we consider stochastic second-order methods for minimizing a finite summation of nonconvex functions. One important key is to find an ingenious but cheap scheme to incorporate local curvature information. Since the true…

Optimization and Control · Mathematics 2021-03-26 Minghan Yang , Dong Xu , Hongyu Chen , Zaiwen Wen , Mengyun Chen