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https://github.com/cor3bit/awesome-soms

A curated list of resources for second-order stochastic optimization
https://github.com/cor3bit/awesome-soms

List: awesome-soms

awesome jax machine-learning optimization second-order-optimization

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A curated list of resources for second-order stochastic optimization

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# Awesome Second-Order Methods [![Awesome](https://awesome.re/badge.svg)](https://awesome.re)



A curated list of resources for second-order 

stochastic optimization 

methods in machine learning.



## Table of Contents

- [Books and Lecture Notes](#books-and-lecture-notes)

- [Papers](#papers)

- [Implementation in JAX](#implementation-in-jax)



## Books and Lecture Notes



- [Numerical Optimization](https://link.springer.com/book/10.1007/978-0-387-40065-5) by Jorge Nocedal and Stephen J. Wright, 2006 :dollar:

- [Introduction to Optimization and Data Fitting](https://www2.compute.dtu.dk/pubdb/pubs/5938-full.html) by H. B. Nielsen and K. Madsen, 2010

- [Optimization for Machine Learning](https://arxiv.org/abs/1909.03550) by Elad Hazan, 2019

- [Topics in Machine Learning: Neural Net Training Dynamics (Winter 2022)](https://www.cs.toronto.edu/~rgrosse/courses/csc2541_2022/) by Roger Grosse, University of Toronto, 2022



## Papers



### Overview



- [Optimization Methods for Large-Scale Machine Learning](https://arxiv.org/abs/1606.04838) 

by Léon Bottou, Frank E. Curtis, Jorge Nocedal, 2016.



- [Exact and inexact subsampled Newton methods for optimization](https://academic.oup.com/imajna/article/39/2/545/4959058) 

by Raghu Bollapragada, Richard H Byrd, Jorge Nocedal, 2018.



### Analysis of the Hessian



- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://arxiv.org/abs/1706.04454)

by Levent Sagun, Utku Evci, V. Ugur Guney, Yann Dauphin, Leon Bottou, 2017.



- [The Full Spectrum of Deepnet Hessians at Scale: Dynamics with SGD Training and Sample Size](https://arxiv.org/abs/1811.07062)

by Vardan Papyan, 2018.



- [PyHessian: Neural Networks Through the Lens of the Hessian](https://arxiv.org/abs/1912.07145)

by Zhewei Yao, Amir Gholami, Kurt Keutzer, Michael W. Mahoney, 2019.



- [A Deeper Look at the Hessian Eigenspectrum of Deep Neural Networks and its Applications to Regularization](https://arxiv.org/abs/2012.03801)

by Adepu Ravi Sankar, Yash Khasbage, Rahul Vigneswaran, Vineeth N Balasubramanian, 2020.



### Diagonal Scaling



- [AdaHessian: An Adaptive Second Order Optimizer for Machine Learning](https://arxiv.org/abs/2006.00719) 

by Zhewei Yao, Amir Gholami, Sheng Shen, Mustafa Mustafa, Kurt Keutzer, Michael W. Mahoney, 2020. Algorithm: AdaHessian



- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://arxiv.org/abs/2305.14342) 

by Hong Liu, Zhiyuan Li, David Hall, Percy Liang, Tengyu Ma, 2023. Algorithm: Sophia



### Hessian-free Optimization



- [Learning Recurrent Neural Networks with Hessian-Free Optimization](https://www.cs.toronto.edu/~jmartens/docs/RNN_HF.pdf) 

by James Martens, Ilya Sutskever, 2011.



- [Training Neural Networks with Stochastic Hessian-Free Optimization](https://arxiv.org/abs/1301.3641) 

by Ryan Kiros, 2013. Algorithm: SHF



### Quasi-Newton



- [A Stochastic Quasi-Newton Method for Large-Scale Optimization](https://arxiv.org/abs/1401.7020) 

by R.H. Byrd, S.L. Hansen, J. Nocedal, Y. Singer, 2014. 



- [A Multi-Batch L-BFGS Method for Machine Learning](https://arxiv.org/abs/1605.06049) 

by Albert S. Berahas, Jorge Nocedal, Martin Takáč, 2016. 



- [Stochastic Quasi-Newton with Line-Search Regularization](https://arxiv.org/abs/1909.01238) by Adrian Wills, Thomas Schön, 2019. Algorithm: SQN



- [Practical Quasi-Newton Methods for Training Deep Neural Networks](https://arxiv.org/abs/2006.08877) 

by Donald Goldfarb, Yi Ren, Achraf Bahamou, 2020.



### Gauss-Newton



- [Efficient Subsampled Gauss-Newton and Natural Gradient Methods for Training Neural Networks](https://arxiv.org/abs/1906.02353) 

by Yi Ren and Donald Goldfarb, 2019. Algorithm: SWM-GN, SWM-NG



- [On the Promise of the Stochastic Generalized Gauss-Newton Method for Training DNNs](https://arxiv.org/abs/2006.02409) 

by Matilde Gargiani et al., 2020. Algorithm: SGN



- [Stochastic Gauss-Newton Algorithms for Nonconvex Compositional Optimization](https://arxiv.org/abs/2002.07290) 

by Quoc Tran-Dinh et al., 2020. Algorithm: SGN with SARAH estimators



- [Nonlinear Least Squares for Large-Scale Machine Learning using Stochastic Jacobian Estimates](https://arxiv.org/abs/2107.05598) 

by Johannes J. Brust, 2021. Discusses using stochastic Jacobian estimates in nonlinear least squares for scalable machine learning.

Algorithm: NLLS1, NLLSL



- [Improving Levenberg-Marquardt Algorithm for Neural Networks](https://arxiv.org/abs/2212.08769) 

by Omead Pooladzandi and Yiming Zhou, 2022. Algorithm: LM



- [Rethinking Gauss-Newton for learning over-parameterized models](https://arxiv.org/abs/2302.02904) 

by Michael Arbel et al., 2023.



- [Exact Gauss-Newton Optimization for Training Deep Neural Networks](https://arxiv.org/abs/2405.14402) by Mikalai Korbit, Adeyemi D. Adeoye, Alberto Bemporad, Mario Zanon, 2024. Algorithm: EGN



### Fisher Information



- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://arxiv.org/abs/1503.05671) 

by James Martens and Roger Grosse, 2015. Algorithm: K-FAC



### Other



- [Second-order optimization with lazy Hessians](https://arxiv.org/abs/2212.00781) 

by Nikita Doikov, El Mahdi Chayti, Martin Jaggi, 2022.



## Implementation in JAX



- [Optax](https://github.com/google-deepmind/optax) - mostly first-order accelerated methods



- [Somax](https://github.com/cor3bit/somax) - second-order stochastic solvers



- [JAXopt](https://github.com/google/jaxopt) - deterministic second-order methods (e.g., Gauss-Newton, Levenberg Marquardt), stochastic first-order methods PolyakSGD, ArmijoSGD



- [KFAC-JAX](https://github.com/google-deepmind/kfac-jax) - implementation of KFAC from the DeepMind team



- [AdaHessianJax](https://github.com/nestordemeure/AdaHessianJax) - implementation of the AdaHessian optimizer by Nestor Demeure