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

Joshua Hill

cs.LGJul 14, 2026

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

Authors: Joshua Hill

Organizations: Baseten Labs, Inc. · David R. Cheriton School of Computer Science, University of Waterloo

Abstract

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 $f(S)=c\bigl(1-\prod_{i\in S}(1-a_i)\bigr)$ , 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.

Explore similar work

Jul 1, 2026cs.LG

Beyond Activation Alignment:The Alignment-Diversity Tradeoff in Task-Aware LLM Quantization

Mixed-precision quantization (MPQ) has become a key technique for deploying large language models under stringent memory and compute constraints. We first identify a phenomenon that we term the Perplexity Illusion: layers ranked as important by perplexity-based sensitivity show little rank correlation with those that are most influential for complex reasoning performance, with Kendall

τ\approx 0

in our analysis. We further reveal an Alignment-Diversity Tradeoff: using only target-task calibration data can degrade post-quantization performance, whereas incorporating general-domain data stabilizes sensitivity estimation and improves robustness across tasks. Based on these observations, we propose TASA (Task-Aware Sensitivity Analysis), a two-level framework that jointly optimizes calibration-data composition and mixed-precision bit allocation. Specifically, TASA searches for a calibration-data mixture using a training-free gradient-trace alignment criterion, and then aggregates perplexity and reasoning-oriented sensitivity signals to guide both inter-layer and intra-layer bit allocation. Experiments on LLaMA-3-8B and Qwen2.5-7B reveal a precision inversion: appropriately allocated 3.5-bit models can match or surpass less task-aware 4-bit baselines. At an average precision of 3.5 bits, TASA matches or outperforms several competitive 4-bit uniform baselines in aggregate accuracy, and improves over the strongest W3 baseline on GSM8K by more than 20 absolute points on LLaMA-3-8B. These results show that calibration-data composition substantially affects task-sensitive quantization, a factor underexplored in prior work.

Fei Wang, Chao Xue, Taoran Liu +3

Large Language Model Quantization Mixed-Precision Quantization Illusions Diversity

Ask PDF

Jul 25, 2026cs.LG

MixQuant: Adaptive Mixed-Precision Quantization for Large Language Models

Mixed-precision quantization improves the accuracy of post-training quantization by allocating higher bitwidths to sensitive layers, but existing methods solve the allocation for a single fixed memory budget. In practice the budget varies across deployments and is unknown at calibration time. Adaptive quantization addresses this with one offline calibration that serves any budget, yet current methods score layer sensitivity in a manner that does not consider its dependency on quantization levels of other layers. We show that a layer's sensitivity depends strongly on the bitwidths of its upstream layers and that this dependence shifts the resulting preferred bit allocation. We propose MixQuant, a technique-agnostic adaptive framework that wraps any base quantizer. MixQuant marginalizes each layer's distortion over random quantized upstream configurations to obtain budget-agnostic scores, calibrates the quantizer's parameters on plans the allocator itself produces, and penalizes allocations that leave layers at the lowest bitwidths. A single greedy pass then serves any budget at deployment. Across Llama-3.2-3B, Llama-2-7B, and Mistral-7B under AWQ and GPTQ, MixQuant outperforms adaptive and mixed-precision baselines in every setting, improving average accuracy by up to 8 points and reducing perplexity from 12.43 to 10.70 at the tightest budget, while matching an ILP solver at negligible deployment cost.

Ashitabh Misra, Madhav Agrawal, Arham Jain +1

Mixed-Precision Quantization Low-Bit Mix

Ask PDF

Sep 22, 2026cs.LG

Beyond Scalar Sensitivity: Activation-Aware Mixed-Precision LLM Quantization with Cross-Layer Refinement

Mixed-precision weight quantization is commonly formulated as a Multiple-Choice Knapsack Problem (MCKP), yet existing solvers rely on scalar sensitivity proxies that collapse each weight matrix's Hessian into a single number and treat every module independently. We prove that even the optimal scalar proxy incurs multiplicative distortion up to

\sqrt{κ(\mathbf{A})κ(\mathbf{B})}

relative to the full activation-aware quadratic, where

κ(\mathbf{A})

and

κ(\mathbf{B})

denote the condition numbers of the input- and output-side Hessian factors. This bound varies from

10^1

10^{13}

for typical LLM modules, making inter-module sensitivity ranking unreliable. To address these limitations, we propose Cross-layer Activation-aware Sensitivity Allocation (CASA), a two-phase method. In Stage 1, the scalar proxy is replaced by an activation-aware metric derived from the Kronecker-factored Hessian, reducing the MCKP to a form whose continuous relaxation admits a closed-form solution. In Stage 2, a cross-layer-aware local search evaluates bit-width updates using the end-to-end model loss. Experiments on multiple LLMs across different bit budgets show that CASA achieves lower perplexity than the latest scalar-proxy baselines, especially at ultra-low bit-widths (

<3

bits per weight). Moreover, the performance gain in zero-shot accuracy tracks the per-model average condition-number over modules, confirming the distortion bound as a practical indicator of scalar-proxy failure.

Akihiro Yoshida, Yuma Ichikawa

Mixed-Precision Quantization Model Weights

Ask PDF