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

Abstract: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 $\sqrt{\kappa(\mathbf{A})\kappa(\mathbf{B})}$ relative to the full activation-aware quadratic, where $\kappa(\mathbf{A})$ and $\kappa(\mathbf{B})$ denote the condition numbers of the input- and output-side Hessian factors. This bound varies from $10^1$ to $10^{13}$ 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.

Machine Learning (cs.LG); Applications (stat.AP); Methodology (stat.ME); Machine Learning (stat.ML)

arXiv:2609.25916 [cs.LG]

(or arXiv:2609.25916v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.25916 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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