# Exact Stochastic Second Order Deep Learning

arXiv: https://arxiv.org/abs/2104.03804 · HTML: https://mlsanity.com/paper/2104.03804/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 21 references that are in the map.
- First posted on arXiv 8 Apr 2021.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [A Progressive Batching L-BFGS Method for Machine Learning](https://mlsanity.com/paper/1802.05374/) (2018)

## Built on by (most cited first, 1 of 1)

- [Fractional is Better: Learnable Derivative Orders in Neural Operator Learning](https://mlsanity.com/paper/2610.20363/) (2026)

## Abstract (arXiv)

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 used despite superior theoretical properties. This inadequacy of second-order methods stems from its exorbitant computational cost, poor performance, and the ineluctable non-convex nature of Deep Learning. Several attempts were made to resolve the inadequacy of second-order optimization without reaching a cost-effective solution, much less an exact solution. In this work, we show that this long-standing problem in Deep Learning could be solved in the stochastic case, given a suitable regularization of the neural network. Interestingly, we provide an expression of the stochastic Hessian and its exact eigenvalues. We provide a closed-form formula for the exact stochastic second-order Newton direction, we solve the non-convexity issue and adjust our exact solution to favor flat minima through regularization and spectral adjustment. We test our exact stochastic second-order method on popular datasets and reveal its adequacy for Deep Learning.