# HESTIA: A Hessian-Guided Differentiable Quantization-Aware Training Framework for Extremely Low-Bit LLMs

arXiv: https://arxiv.org/abs/2601.20745 · HTML: https://mlsanity.com/paper/2601.20745/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 37 references that are in the map.
- First posted on arXiv 28 Jan 2026.

## Builds on (most cited first, 30 of 32)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [Learned Step Size Quantization](https://mlsanity.com/paper/1902.08153/) (2019)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [HAWQ: Hessian AWare Quantization of Neural Networks with Mixed-Precision](https://mlsanity.com/paper/1905.03696/) (2019)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [LLM-QAT: Data-Free Quantization Aware Training for Large Language Models](https://mlsanity.com/paper/2305.17888/) (2023)
- [The Era of 1-bit LLMs: All Large Language Models are in 1.58 Bits](https://mlsanity.com/paper/2402.17764/) (2024)
- [Differentiable Soft Quantization: Bridging Full-Precision and Low-Bit Neural Networks](https://mlsanity.com/paper/1908.05033/) (2019)
- [Ternary Weight Networks](https://mlsanity.com/paper/1605.04711/) (2016)
- [Hutch++: Optimal Stochastic Trace Estimation](https://mlsanity.com/paper/2010.09649/) (2020)
- [PV-Tuning: Beyond Straight-Through Estimation for Extreme LLM Compression](https://mlsanity.com/paper/2405.14852/) (2024)
- [Optimizing Large Language Model Training Using FP4 Quantization](https://mlsanity.com/paper/2501.17116/) (2025)
- [ParetoQ: Improving Scaling Laws in Extremely Low-bit LLM Quantization](https://mlsanity.com/paper/2502.02631/) (2025)
- [Ultra-FineWeb: Efficient Data Filtering and Verification for High-Quality LLM Training Data](https://mlsanity.com/paper/2505.05427/) (2025)
- [Spectra: Surprising Effectiveness of Pretraining Ternary Language Models at Scale](https://mlsanity.com/paper/2407.12327/) (2024)
- [Distance-aware Quantization](https://mlsanity.com/paper/2108.06983/) (2021)
- [TernaryLLM: Ternarized Large Language Model](https://mlsanity.com/paper/2406.07177/) (2024)
- [Tequila: Trapping-free Ternary Quantization for Large Language Models](https://mlsanity.com/paper/2509.23809/) (2025)
- [Direct Quantized Training of Language Models with Stochastic Rounding](https://mlsanity.com/paper/2412.04787/) (2024)
- [LOTION: Smoothing the Optimization Landscape for Quantized Training](https://mlsanity.com/paper/2510.08757/) (2025)
- [iFairy: the First 2-bit Complex LLM with All Parameters in ±1, ± i](https://mlsanity.com/paper/2508.05571/) (2025)
- [Compute-Optimal Quantization-Aware Training](https://mlsanity.com/paper/2509.22935/) (2025)

## Abstract (arXiv)

As large language models (LLMs) continue to scale, deployment is increasingly bottlenecked by the memory wall, motivating a shift toward extremely low-bit quantization. However, most quantization-aware training (QAT) methods apply hard rounding and the straight-through estimator (STE) from the beginning of the training, which prematurely discretizes the optimization landscape and induces persistent gradient mismatch between latent weights and quantized weights, hindering effective optimization of quantized models. To address this, we propose Hestia, a Hessian-guided differentiable QAT framework for extremely low-bit LLMs, which replaces the rigid step function with a temperature-controlled softmax relaxation to maintain gradient flow early in training while progressively hardening quantization. Furthermore, Hestia leverages a tensor-wise Hessian trace metric as a lightweight curvature signal to drive fine-grained temperature annealing, enabling sensitivity-aware discretization across the model. Evaluations on Llama-3.2 show that Hestia consistently outperforms existing ternary QAT baselines, yielding average zero-shot improvements of 5.39% and 4.34% for the 1B and 3B models. These results indicate that Hessian-guided relaxation effectively recovers representational capacity, establishing a more robust training path for 1.58-bit LLMs. The code is available at https://github.com/hestia2026/Hestia.