# Block-diagonal Hessian-free Optimization for Training Neural Networks

arXiv: https://arxiv.org/abs/1712.07296 · HTML: https://mlsanity.com/paper/1712.07296/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Dec 2017.

## Built on by (most cited first, 10 of 10)

- [ViViT: Curvature access through the generalized Gauss-Newton's low-rank structure](https://mlsanity.com/paper/2106.02624/) (2021)
- [Towards Quantifying the Hessian Structure of Neural Networks](https://mlsanity.com/paper/2505.02809/) (2025)
- [Batch Normalization Preconditioning for Neural Network Training](https://mlsanity.com/paper/2108.01110/) (2021)
- [Better Hessians Matter: Studying the Impact of Curvature Approximations in Influence Functions](https://mlsanity.com/paper/2509.23437/) (2025)
- [Debiasing Mini-Batch Quadratics for Applications in Deep Learning](https://mlsanity.com/paper/2410.14325/) (2024)
- [Train Large, Deploy Compact: Structured Compression for Compact Low-Rank Adaptation](https://mlsanity.com/paper/2510.00192/) (2025)
- [Fuzzy hyperparameters update in a second order optimization](https://mlsanity.com/paper/2403.15416/) (2024)
- [Recursive Deep Inverse Reinforcement Learning](https://mlsanity.com/paper/2504.13241/) (2025)
- [Importance-Guided Basis Selection for Low-Rank Decomposition of Large Language Models](https://mlsanity.com/paper/2605.01627/) (2026)
- [Scalable Kronecker-Fisher Approximation: Efficient Hessian Analysis for Billion-Parameter Language Models Compression](https://mlsanity.com/paper/2609.02451/) (2026)

## Abstract (arXiv)

Second-order methods for neural network optimization have several advantages over methods based on first-order gradient descent, including better scaling to large mini-batch sizes and fewer updates needed for convergence. But they are rarely applied to deep learning in practice because of high computational cost and the need for model-dependent algorithmic variations. We introduce a variant of the Hessian-free method that leverages a block-diagonal approximation of the generalized Gauss-Newton matrix. Our method computes the curvature approximation matrix only for pairs of parameters from the same layer or block of the neural network and performs conjugate gradient updates independently for each block. Experiments on deep autoencoders, deep convolutional networks, and multilayer LSTMs demonstrate better convergence and generalization compared to the original Hessian-free approach and the Adam method.