Where Compression Hurts: Function-Space Sensitivity for Mixed-Precision Neural Network Quantization

Selecting layer-wise precision is a central combinatorial problem in mixed-precision post-training quantization and requires a reliable surrogate for the damage induced by each compression action. Prior work has studied this problem from several perspectives, focusing on how compression-induced perturbations affect model parameters, optimization objectives, or model outputs. We study a complementary question: how should a quantization direction be measured in output space, and what additional information is preserved by retaining its first-order tangent response rather than reducing it to a scalar sensitivity score? For a compression action inducing $\Delta W_{l,a}$, we analyze the first-order logit perturbation $J_l\Delta W_{l,a}$ and the associated $J_l^{\top }{J}_l$ quadratic form. The resulting sensitivity is specific to the chosen compression direction, is label-independent once the calibration inputs are fixed, and can be reused across different first-order analyses. Across ResNet-18, MobileNetV3-Small, and MobileNetV2, this function-space sensitivity provides an informative ranking of realized compression damage. Its local linear response closely tracks directly measured output distortion under isolated compression actions, while severity-controlled analyses show that the ranking cannot be explained solely by compression magnitude. We further find that the benefit of modeling tangent interference is network-dependent rather than universal. Under budgeted precision allocation, improved prediction of function-space distortion can also lead to better allocation decisions when compression actions are sufficiently distinguishable. These results identify Euclidean logit geometry as a reusable function-space complement to parameter-based, loss-based, and directly measured output-space sensitivity.