# Scaling Laws For Mixed Quantization

arXiv: https://arxiv.org/abs/2410.06722 · HTML: https://mlsanity.com/paper/2410.06722/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 53 references that are in the map.
- First posted on arXiv 9 Oct 2024.

## Builds on (most cited first, 29 of 29)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [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)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [AWQ: Activation-aware Weight Quantization for LLM Compression and Acceleration](https://mlsanity.com/paper/2306.00978/) (2023)
- [Mixed Precision Training](https://mlsanity.com/paper/1710.03740/) (2017)
- [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)
- [OmniQuant: Omnidirectionally Calibrated Quantization for Large Language Models](https://mlsanity.com/paper/2308.13137/) (2023)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [SpQR: A Sparse-Quantized Representation for Near-Lossless LLM Weight Compression](https://mlsanity.com/paper/2306.03078/) (2023)
- [Extreme Compression of Large Language Models via Additive Quantization](https://mlsanity.com/paper/2401.06118/) (2024)
- [Outlier Suppression: Pushing the Limit of Low-bit Transformer Language Models](https://mlsanity.com/paper/2209.13325/) (2022)
- [QTIP: Quantization with Trellises and Incoherence Processing](https://mlsanity.com/paper/2406.11235/) (2024)
- [VS-Quant: Per-vector Scaled Quantization for Accurate Low-Precision Neural Network Inference](https://mlsanity.com/paper/2102.04503/) (2021)
- [FlightLLM: Efficient Large Language Model Inference with a Complete Mapping Flow on FPGAs](https://mlsanity.com/paper/2401.03868/) (2024)
- [SliM-LLM: Salience-Driven Mixed-Precision Quantization for Large Language Models](https://mlsanity.com/paper/2405.14917/) (2024)
- [Any-Precision LLM: Low-Cost Deployment of Multiple, Different-Sized LLMs](https://mlsanity.com/paper/2402.10517/) (2024)
- [With Shared Microexponents, A Little Shifting Goes a Long Way](https://mlsanity.com/paper/2302.08007/) (2023)
- [LQER: Low-Rank Quantization Error Reconstruction for LLMs](https://mlsanity.com/paper/2402.02446/) (2024)
- [Transformer-VQ: Linear-Time Transformers via Vector Quantization](https://mlsanity.com/paper/2309.16354/) (2023)
- [Mixed Precision Training With 8-bit Floating Point](https://mlsanity.com/paper/1905.12334/) (2019)
- [Learning from Students: Applying t-Distributions to Explore Accurate and Efficient Formats for LLMs](https://mlsanity.com/paper/2405.03103/) (2024)
- [TernaryLLM: Ternarized Large Language Model](https://mlsanity.com/paper/2406.07177/) (2024)
- [Revisiting Block-based Quantisation: What is Important for Sub-8-bit LLM Inference?](https://mlsanity.com/paper/2310.05079/) (2023)

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

- [Scaling Laws for Predicting Downstream Performance in LLMs](https://mlsanity.com/paper/2410.08527/) (2024)
- [Mixed Precision Scaling Laws for Quantized Pretraining](https://mlsanity.com/paper/2610.17637/) (2026)

## Abstract (arXiv)

Post-training quantization of Large Language Models (LLMs) has proven effective in reducing the memory and computational requirements for inference. In this study, we focus on a straightforward question: When aiming for a target accuracy or perplexity with low-precision quantization, how much high-precision computation needs to be preserved, and how fine-grained this quantization would need to be as we scale LLMs to larger sizes? We first introduce two critical metrics, named the quantization ratio (Qr) and quantization block size (Qb). The former measures the number of parameters quantized to low-precision arithmetic normalized by the total parameter count, whereas the latter defines the number of values within a block that share a scaling factor, akin to the block size concept introduced in the FP4 format in NVIDIA's Blackwell architecture. Through extensive and carefully controlled experiments across different models and quantization methods, we propose a unified scaling law on post-training quantization (PTQ) that can predict loss degeneration for varying Qr and Qb. For Qr, our scaling law implies that parameter scaling and ratio scaling have a multiplicative relationship. Consequently, larger models are more amenable to a higher quantization ratio Qr, thus supporting an increase in the adoption of mixed quantization for inference. Regarding Qb, our findings indicate that a small block size, similar to that used in Blackwell, is not essential for large models. Employing a small Qb can instead unnecessarily complicate the design of the hardware circuit.