# Hessian of Perplexity for Large Language Models by PyTorch autograd (Open Source)

arXiv: https://arxiv.org/abs/2504.04520 · HTML: https://mlsanity.com/paper/2504.04520/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 16 references that are in the map.
- First posted on arXiv 6 Apr 2025.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [An Improved Analysis of Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2007.07989/) (2020)
- [Pushing the Limits of Large Language Model Quantization via the Linearity Theorem](https://mlsanity.com/paper/2411.17525/) (2024)

## Built on by (most cited first, 2 of 2)

- [Learning Rate Matters: Vanilla LoRA May Suffice for LLM Fine-tuning](https://mlsanity.com/paper/2602.04998/) (2026)
- [Horizon-LM: A RAM-Centric Architecture for LLM Training](https://mlsanity.com/paper/2602.04816/) (2026)

## Abstract (arXiv)

Computing the full Hessian matrix -- the matrix of second-order derivatives for an entire Large Language Model (LLM) is infeasible due to its sheer size. In this technical report, we aim to provide a comprehensive guide on how to accurately compute at least a small portion of the Hessian for LLMs using PyTorch autograd library. We also demonstrate how to compute the full diagonal of the Hessian matrix using multiple samples of vector-Hessian Products (HVPs). We hope that both this guide and the accompanying GitHub code will be valuable resources for practitioners and researchers interested in better understanding the behavior and structure of the Hessian in LLMs.