# Scaling Laws for Precision

arXiv: https://arxiv.org/abs/2411.04330 · HTML: https://mlsanity.com/paper/2411.04330/

## Key facts

- Cited by 95 papers in the map (data through 14 Oct 2026).
- Builds on 90 of its 99 references that are in the map.
- First posted on arXiv 7 Nov 2024.

## Builds on (most cited first, 30 of 90)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [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)
- [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)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Beyond the Imitation Game: Quantifying and extrapolating the capabilities of language models](https://mlsanity.com/paper/2206.04615/) (2022)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [BLOOM: A 176B-Parameter Open-Access Multilingual Language Model](https://mlsanity.com/paper/2211.05100/) (2022)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [Megatron-LM: Training Multi-Billion Parameter Language Models Using Model Parallelism](https://mlsanity.com/paper/1909.08053/) (2019)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference](https://mlsanity.com/paper/1712.05877/) (2017)
- [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)
- [Scaling Language Models: Methods, Analysis & Insights from Training Gopher](https://mlsanity.com/paper/2112.11446/) (2021)
- [Reproducible scaling laws for contrastive language-image learning](https://mlsanity.com/paper/2212.07143/) (2022)
- [Mixed Precision Training](https://mlsanity.com/paper/1710.03740/) (2017)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [StarCoder: may the source be with you!](https://mlsanity.com/paper/2305.06161/) (2023)

## Built on by (most cited first, 30 of 95)

- [ParetoQ: Improving Scaling Laws in Extremely Low-bit LLM Quantization](https://mlsanity.com/paper/2502.02631/) (2025)
- [How Do Large Language Monkeys Get Their Power (Laws)?](https://mlsanity.com/paper/2502.17578/) (2025)
- [Power Lines: Scaling Laws for Weight Decay and Batch Size in LLM Pre-training](https://mlsanity.com/paper/2505.13738/) (2025)
- [Joint MoE Scaling Laws: Mixture of Experts Can Be Memory Efficient](https://mlsanity.com/paper/2502.05172/) (2025)
- [Remote Sensing SpatioTemporal Vision-Language Models: A Comprehensive Survey](https://mlsanity.com/paper/2412.02573/) (2024)
- [QuEST: Stable Training of LLMs with 1-Bit Weights and Activations](https://mlsanity.com/paper/2502.05003/) (2025)
- [Low-Bit Quantization Favors Undertrained LLMs: Scaling Laws for Quantized LLMs with 100T Training Tokens](https://mlsanity.com/paper/2411.17691/) (2024)
- [INT v.s. FP: A Comprehensive Study of Fine-Grained Low-bit Quantization Formats](https://mlsanity.com/paper/2510.25602/) (2025)
- [Scaling Law for Quantization-Aware Training](https://mlsanity.com/paper/2505.14302/) (2025)
- [MixLLM: LLM Quantization with Global Mixed-precision between Output-features and Highly-efficient System Design](https://mlsanity.com/paper/2412.14590/) (2024)
- [Scaling Inference-Efficient Language Models](https://mlsanity.com/paper/2501.18107/) (2025)
- [Quamba2: A Robust and Scalable Post-training Quantization Framework for Selective State Space Models](https://mlsanity.com/paper/2503.22879/) (2025)
- [Training Dynamics Impact Post-Training Quantization Robustness](https://mlsanity.com/paper/2510.06213/) (2025)
- [MetaOpenFOAM 2.0: Large Language Model Driven Chain of Thought for Automating CFD Simulation and Post-Processing](https://mlsanity.com/paper/2502.00498/) (2025)
- [Benchmarking Post-Training Quantization in LLMs: Comprehensive Taxonomy, Unified Evaluation, and Comparative Analysis](https://mlsanity.com/paper/2502.13178/) (2025)
- [Kinetics: Rethinking Test-Time Scaling Laws](https://mlsanity.com/paper/2506.05333/) (2025)
- [Model Merging Scaling Laws in Large Language Models](https://mlsanity.com/paper/2509.24244/) (2025)
- [Prescriptive Scaling Laws for Data Constrained Training](https://mlsanity.com/paper/2605.01640/) (2026)
- [Compression Scaling Laws:Unifying Sparsity and Quantization](https://mlsanity.com/paper/2502.16440/) (2025)
- [Improved Scaling Laws in Linear Regression via Data Reuse](https://mlsanity.com/paper/2506.08415/) (2025)
- [Scaling LLM Test-Time Compute with Mobile NPU on Smartphones](https://mlsanity.com/paper/2509.23324/) (2025)
- [Does quantization affect models' performance on long-context tasks?](https://mlsanity.com/paper/2505.20276/) (2025)
- [Why Do Some Inputs Break Low-Bit LLM Quantization?](https://mlsanity.com/paper/2506.12044/) (2025)
- [Scaling with Collapse: Efficient and Predictable Training of LLM Families](https://mlsanity.com/paper/2509.25087/) (2025)
- [Scaling Laws for Floating Point Quantization Training](https://mlsanity.com/paper/2501.02423/) (2025)
- [Physics of Skill Learning](https://mlsanity.com/paper/2501.12391/) (2025)
- [Optimal Learning Rate Schedules under Functional Scaling Laws: Power Decay and Warmup-Stable-Decay](https://mlsanity.com/paper/2602.06797/) (2026)
- [Universal Properties of Activation Sparsity in Modern Large Language Models](https://mlsanity.com/paper/2509.00454/) (2025)
- [LaX: Boosting Low-Rank Training of Foundation Models via Latent Crossing](https://mlsanity.com/paper/2505.21732/) (2025)
- [Predictable Scale: Part II, Farseer: A Refined Scaling Law in Large Language Models](https://mlsanity.com/paper/2506.10972/) (2025)

## Abstract (arXiv)

Low precision training and inference affect both the quality and cost of language models, but current scaling laws do not account for this. In this work, we devise "precision-aware" scaling laws for both training and inference. We propose that training in lower precision reduces the model's "effective parameter count," allowing us to predict the additional loss incurred from training in low precision and post-train quantization. For inference, we find that the degradation introduced by post-training quantization increases as models are trained on more data, eventually making additional pretraining data actively harmful. For training, our scaling laws allow us to predict the loss of a model with different parts in different precisions, and suggest that training larger models in lower precision may be compute optimal. We unify the scaling laws for post and pretraining quantization to arrive at a single functional form that predicts degradation from training and inference in varied precisions. We fit on over 465 pretraining runs and validate our predictions on model sizes up to 1.7B parameters trained on up to 26B tokens.