# Scale Sensitivity in Low-Bit Post-Training Quantization: Curvature of the Quantization Error Landscape

arXiv: https://arxiv.org/abs/2609.37416 · HTML: https://mlsanity.com/paper/2609.37416/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 32 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [DeepSeek-V3 Technical Report](https://mlsanity.com/paper/2412.19437/) (2024)
- [BRECQ: Pushing the Limit of Post-Training Quantization by Block Reconstruction](https://mlsanity.com/paper/2102.05426/) (2021)
- [A White Paper on Neural Network Quantization](https://mlsanity.com/paper/2106.08295/) (2021)
- [GPTAQ: Efficient Finetuning-Free Quantization for Asymmetric Calibration](https://mlsanity.com/paper/2504.02692/) (2025)
- [BitNet b1.58 2B4T Technical Report](https://mlsanity.com/paper/2504.12285/) (2025)
- [Rethinking Residual Errors in Compensation-based LLM Quantization](https://mlsanity.com/paper/2604.07955/) (2026)
- [Optimal Post-Training Quantization Scales and Where to Find Them](https://mlsanity.com/paper/2606.10890/) (2026)

## Abstract (arXiv)

Post-training quantization (PTQ) methods in the GPTQ family minimize a layer-wise reconstruction error on a uniform grid whose scale must be chosen; the common max-based choice degrades sharply at low bit-widths. We study how sensitive this objective is to the scale. For a layer with i.i.d. Gaussian weights and calibration activations of sufficiently large effective rank, we prove that, as the width grows, the normalized round-to-nearest loss converges with high probability, uniformly over all scales, to the mean-squared error of a uniform quantizer applied to a standard Gaussian; we verify the effective-rank condition for wide, randomly initialized MLPs with odd Lipschitz activations and isotropic Gaussian calibration data. The limiting objective has a unique nondegenerate minimizer, whose scale decreases strictly with the number of levels and whose curvature with respect to relative scale errors decays approximately exponentially with the bit-width. GPTQ experiments on five LLMs show the same trend: the scale rule changes perplexity substantially at 2--3 bits and negligibly from 6 bits on, and a local measure of GPTQ scale sensitivity decreases with bit-width in line with the Gaussian curvature. The Gaussian-optimal scale fails on raw weights; after Hadamard incoherence processing it matches the best searched rule at 3 bits and above without any search, but remains clearly worse at 2 bits.