# Saturation Makes Quantization Error Additive: A Coverage Model with a Certificate

arXiv: https://arxiv.org/abs/2607.12266 · HTML: https://mlsanity.com/paper/2607.12266/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 57 references that are in the map.
- First posted on arXiv 14 Jul 2026.

## Builds on (most cited first, 28 of 28)

- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [AWQ: Activation-aware Weight Quantization for LLM Compression and Acceleration](https://mlsanity.com/paper/2306.00978/) (2023)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [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)
- [HAWQ: Hessian AWare Quantization of Neural Networks with Mixed-Precision](https://mlsanity.com/paper/1905.03696/) (2019)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [HAWQ-V2: Hessian Aware trace-Weighted Quantization of Neural Networks](https://mlsanity.com/paper/1911.03852/) (2019)
- [The Hydra Effect: Emergent Self-repair in Language Model Computations](https://mlsanity.com/paper/2307.15771/) (2023)
- [Improving KernelSHAP: Practical Shapley Value Estimation via Linear Regression](https://mlsanity.com/paper/2012.01536/) (2020)
- [Bayesian Optimization of Combinatorial Structures](https://mlsanity.com/paper/1806.08838/) (2018)
- [The Combinatorial Brain Surgeon: Pruning Weights That Cancel One Another in Neural Networks](https://mlsanity.com/paper/2203.04466/) (2022)
- [Mixture Compressor for Mixture-of-Experts LLMs Gains More](https://mlsanity.com/paper/2410.06270/) (2024)
- [Quantization Error Propagation: Revisiting Layer-Wise Post-Training Quantization](https://mlsanity.com/paper/2504.09629/) (2025)
- [Defining and Quantifying the Emergence of Sparse Concepts in DNNs](https://mlsanity.com/paper/2111.06206/) (2021)
- [GuidedQuant: Large Language Model Quantization via Exploiting End Loss Guidance](https://mlsanity.com/paper/2505.07004/) (2025)
- [AMQ: Enabling AutoML for Mixed-precision Weight-Only Quantization of Large Language Models](https://mlsanity.com/paper/2509.12019/) (2025)
- [SPEX: Scaling Feature Interaction Explanations for LLMs](https://mlsanity.com/paper/2502.13870/) (2025)
- [Identifying Sensitive Weights via Post-quantization Integral](https://mlsanity.com/paper/2503.01901/) (2025)
- [Shapley Value as Principled Metric for Structured Network Pruning](https://mlsanity.com/paper/2006.01795/) (2020)
- [FGMP: Fine-Grained Mixed-Precision Weight and Activation Quantization for Hardware-Accelerated LLM Inference](https://mlsanity.com/paper/2504.14152/) (2025)
- [Fine-Grained Post-Training Quantization for Large Vision Language Models with Quantization-Aware Integrated Gradients](https://mlsanity.com/paper/2603.17809/) (2026)
- [Mixed-Precision Quantization for Deep Vision Models with Integer Quadratic Programming](https://mlsanity.com/paper/2307.05657/) (2023)
- [AutoQRA: Joint Optimization of Mixed-Precision Quantization and Low-rank Adapters for Efficient LLM Fine-Tuning](https://mlsanity.com/paper/2602.22268/) (2026)
- [Pruning as a Cooperative Game: Surrogate-Assisted Layer Contribution Estimation for Large Language Models](https://mlsanity.com/paper/2602.07804/) (2026)
- [ScaleBITS: Scalable Bitwidth Search for Hardware-Aligned Mixed-Precision LLMs](https://mlsanity.com/paper/2602.17698/) (2026)
- [MiCo: End-to-End Mixed Precision Neural Network Co-Exploration Framework for Edge AI](https://mlsanity.com/paper/2508.09500/) (2025)

## Abstract (arXiv)

Mixed-precision quantization must decide which parts of a model to keep at higher precision. A common premise, shared by sensitivity-based methods such as HAWQ and CoopQ, is that the loss from quantizing a set of layers can be reconstructed from per-layer or pairwise sensitivities measured in isolation. We test this premise at the 4-bit weight-and-activation precisions now being deployed, treating the change in loss f(S) from quantizing a layer set S as a set function on the Boolean cube and analyzing it through two classical changes of basis. This analysis yields two findings. First, across configurations drawn from the deployment distribution, 85--93% of the variance of f is explained by per-layer effects alone. Second, a monotone transform of a sum of per-layer terms reproduces f's ranking of configurations, misordering at most 2% of pairs. We propose the coverage model f(S)=cbigl(1-prodi∈ S(1-ai)bigr), which reproduces the measured variance profile of f to within a few percent from its L fitted break-rates. This structure supports two predictors of a configuration's loss, each with L+1 parameters. The additive model is the optimal first-order predictor. By Parseval's identity its mean-squared error equals the variance of f left unexplained by per-layer effects, which we measure on full lattices, estimate out of sample at full-network scale, and report with every result as a certificate of how well any additive model can do. The coverage model itself is the second predictor. As allocators at matched memory, they attain the lowest KL divergence among the compared allocators on models from 30B to 355B parameters. Below four bits, the resulting allocations continue to solve code and reasoning tasks at budgets where allocations from gradient sensitivities no longer produce terminating generations.