# First-Order Error Matters: Accurate Compensation for Quantized Large Language Models

arXiv: https://arxiv.org/abs/2507.11017 · HTML: https://mlsanity.com/paper/2507.11017/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 30 references that are in the map.
- First posted on arXiv 15 Jul 2025.

## Builds on (most cited first, 24 of 24)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [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)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [Competition-Level Code Generation with AlphaCode](https://mlsanity.com/paper/2203.07814/) (2022)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [Up or Down? Adaptive Rounding for Post-Training Quantization](https://mlsanity.com/paper/2004.10568/) (2020)
- [BRECQ: Pushing the Limit of Post-Training Quantization by Block Reconstruction](https://mlsanity.com/paper/2102.05426/) (2021)
- [SpinQuant: LLM quantization with learned rotations](https://mlsanity.com/paper/2405.16406/) (2024)
- [OmniQuant: Omnidirectionally Calibrated Quantization for Large Language Models](https://mlsanity.com/paper/2308.13137/) (2023)
- [Optimal Brain Compression: A Framework for Accurate Post-Training Quantization and Pruning](https://mlsanity.com/paper/2208.11580/) (2022)
- [QDrop: Randomly Dropping Quantization for Extremely Low-bit Post-Training Quantization](https://mlsanity.com/paper/2203.05740/) (2022)
- [Embodied Intelligence via Learning and Evolution](https://mlsanity.com/paper/2102.02202/) (2021)
- [A Survey on Dialogue Systems: Recent Advances and New Frontiers](https://mlsanity.com/paper/1711.01731/) (2017)
- [RPTQ: Reorder-based Post-training Quantization for Large Language Models](https://mlsanity.com/paper/2304.01089/) (2023)
- [An empirical study of LLaMA3 quantization: from LLMs to MLLMs](https://mlsanity.com/paper/2404.14047/) (2024)
- [GPTAQ: Efficient Finetuning-Free Quantization for Asymmetric Calibration](https://mlsanity.com/paper/2504.02692/) (2025)
- [A Survey of Low-bit Large Language Models: Basics, Systems, and Algorithms](https://mlsanity.com/paper/2409.16694/) (2024)

## Built on by (most cited first, 8 of 8)

- [Fine-Grained Post-Training Quantization for Large Vision Language Models with Quantization-Aware Integrated Gradients](https://mlsanity.com/paper/2603.17809/) (2026)
- [CoreQ: Learning-Free Mismatch Correction and Successive Rounding for Quantization](https://mlsanity.com/paper/2602.05902/) (2026)
- [Multi-Resolution Flow Matching: Training-Free Diffusion Acceleration via Staged Sampling](https://mlsanity.com/paper/2607.01642/) (2026)
- [MGVQ: Synergizing Multi-dimensional Sensitivity-Aware and Gradient-Hessian Fusion for Vector Quantization](https://mlsanity.com/paper/2605.24019/) (2026)
- [MARR: Module-Adaptive Residual Reconstruction for Low-Bit Post-Training Quantization](https://mlsanity.com/paper/2605.17997/) (2026)
- [SchurQuant: Groupwise Discrete Optimization for Layer-Wise LLM Quantization](https://mlsanity.com/paper/2608.15567/) (2026)
- [SandwichQuant: Which Parameters Matter Before and After Quantization?](https://mlsanity.com/paper/2608.24173/) (2026)
- [G2PTQ: Improving LLM Post-Training Quantization with Generalized Gradient Compensation](https://mlsanity.com/paper/2609.31009/) (2026)

## Abstract (arXiv)

Post-training quantization (PTQ) offers an efficient approach to compressing large language models (LLMs), significantly reducing memory access and computational costs. Existing compensation-based weight calibration methods often rely on a second-order Taylor expansion to model quantization error, under the assumption that the first-order term is negligible in well-trained full-precision models. However, we reveal that the progressive compensation process introduces accumulated first-order deviations between latent weights and their full-precision counterparts, making this assumption fundamentally flawed. To address this, we propose FOEM, a novel PTQ method that explicitly incorporates first-order gradient terms to improve quantization error compensation. FOEM approximates gradients by performing a first-order Taylor expansion around the pre-quantization weights. This yields an approximation based on the difference between latent and full-precision weights as well as the Hessian matrix. When substituted into the theoretical solution, the formulation eliminates the need to explicitly compute the Hessian, thereby avoiding the high computational cost and limited generalization of backpropagation-based gradient methods. This design introduces only minimal additional computational overhead. Extensive experiments across a wide range of models and benchmarks demonstrate that FOEM consistently outperforms the classical GPTQ method. In 3-bit weight-only quantization, FOEM reduces the perplexity of Llama3-8B by 17.3% and increases the 5-shot MMLU accuracy from 53.8% achieved by GPTAQ to 56.1%. Moreover, FOEM can be seamlessly combined with advanced techniques such as SpinQuant, delivering additional gains under the challenging W4A4KV4 setting and further narrowing the performance gap with full-precision baselines, surpassing existing state-of-the-art methods.