Selecting GPTQ Damping from the Calibration Set by Nested-Ridge Leave-One-Out

GPTQ-style post-training quantization adds λI to the calibration Hessian before inverting it, with λ a fixed fraction of the mean diagonal (one percent in the GPTQ authors' code, five percent in a maintained implementation) that ignores how many calibration rows there are. Small tabular, time-series and text-embedding foundation models can have fewer calibration rows than their layers are wide, yet no fixed fraction fits them: on the layers we measure, the best fixed multiplier moves from 0.0064 at aspect ratio d/n ∈ [0.5, 4) to 0.77 at d/n ≥ 4. Building on the known damping–ridge identity, we read the Cholesky factor GPTQ already builds, column by column, as the d nested ridge regressions that λ regularises. This reading decomposes the layer's population error exactly, under an idealized rounding model, into an oracle term and regression excess risks; under that model it bounds, from the population activation spectrum alone, what GPTQ can gain over rounding to nearest; and it makes λ estimable. We set each layer's penalty by minimizing the leave-one-out error summed over its regressions (LOOCV), whose whole path the same factorization gives with no extra quantization pass. On the 175 cells with n ≤ 512 from 25 layers of four open-weight foundation models, LOOCV, selected among six cross-validation criteria, has median excess cost (extra error over the best evaluated error, as a fraction of its gain over round-to-nearest) 0.0048 against 0.0843 for the one-percent convention and 0.0401 for the five-percent rule, lower than every fixed, searched or hindsight rule we compare. With the rule fixed in advance, a confirmation on re-split calibration pools and a model absent from the selection sweep again finds a lower excess cost than the one-percent convention at d/n ∈ [0.5, 4), by 0.0494 [0.0065, 0.1928], and than per-bucket hindsight multipliers.