# Theory-optimal Quantization Based on Flatness

arXiv: https://arxiv.org/abs/2605.18800 · HTML: https://mlsanity.com/paper/2605.18800/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 30 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 19 of 19)

- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [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)
- [Deep Compression: Compressing Deep Neural Networks with Pruning, Trained Quantization and Huffman Coding](https://mlsanity.com/paper/1510.00149/) (2015)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [AWQ: Activation-aware Weight Quantization for LLM Compression and Acceleration](https://mlsanity.com/paper/2306.00978/) (2023)
- [Sequence-Level Knowledge Distillation](https://mlsanity.com/paper/1606.07947/) (2016)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [SpinQuant: LLM quantization with learned rotations](https://mlsanity.com/paper/2405.16406/) (2024)
- [Early-Learning Regularization Prevents Memorization of Noisy Labels](https://mlsanity.com/paper/2007.00151/) (2020)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [Language model compression with weighted low-rank factorization](https://mlsanity.com/paper/2207.00112/) (2022)
- [FlatQuant: Flatness Matters for LLM Quantization](https://mlsanity.com/paper/2410.09426/) (2024)
- [OstQuant: Refining Large Language Model Quantization with Orthogonal and Scaling Transformations for Better Distribution Fitting](https://mlsanity.com/paper/2501.13987/) (2025)
- [A Systematic Classification of Knowledge, Reasoning, and Context within the ARC Dataset](https://mlsanity.com/paper/1806.00358/) (2018)
- [Knowledge Distillation Using Hierarchical Self-Supervision Augmented Distribution](https://mlsanity.com/paper/2109.03075/) (2021)

## Abstract (arXiv)

Post-training quantization has emerged as a widely adopted technique for compressing and accelerating the inference of Large Language Models (LLMs). The primary challenges in LLMs quantization stem from activation outliers, which significantly degrade model performance especially at lower bit precision. While recent approaches attempt to mitigate outliers through linear transformations across feature dimensions, our analysis reveals that the transformed weights and activations still exhibit persistent outlier patterns with concentrated magnitude distributions. In this paper, we first model the mathematical relationship between quantization error and outliers, and then introduce a new metric Flatness to quantify the distribution of outliers. Based on this, we derive the theoretical optimal solution with respect to Flatness. Building on these insights, we propose Bidirectional Diagonal Quantization (BDQ), a novel post-training quantization framework that effectively disperses outlier patterns through optimized matrix transformations. BDQ strategically distributes outlier magnitudes across matrix dimensions via learned diagonal operations. Extensive experiments demonstrate that BDQ establishes a new quantization benchmark. It achieves less than 1% accuracy drop in W4A4 quantization on the LLaMA-3-8B model. In the more challenging W2A4KV16 experiment, compared to state-of-the-art approaches, BDQ reduces the performance gap by 39.1% on the DeepSeek-R1-Distill-LLaMA-70B model.