Calibrated Derivative-Process Sensitivity for Gaussian-Process Variable Selection

arXiv:2609.33549v1 Announce Type: cross Abstract: Automatic relevance determination (ARD), the default tool for variable selection in Gaussian-process (GP) regression, ranks inputs by inverse lengthscales -- which measure how fast a function varies, not how much an input contributes to prediction -- and offers no calibrated rule for deciding which inputs to keep. The prediction-centred alternative, the derivative sensitivity $\nu_j = \mathbb{E}[(\partial f/\partial x_j)^2]$, is available in closed form from a fitted GP, but turning it into a selection rule is harder than it looks: at a null input the estimator is a degenerate quadratic form, so Wald and Bernstein-von Mises cutoffs are anti-conservative, and the natural residual bootstrap is mis-scaled. We show that a studentized multiplier bootstrap of the GP derivative process repairs both, prove its validity through an invariance principle for quadratic forms, and obtain asymptotic family-wise and false-discovery-rate control across inputs. Over 100 replications the rule controls FDR wherever inputs are truly null, while uncalibrated derivative rankings breach the target by up to 2x and a Bernstein-von Mises cutoff by 2.2x; at matched FDR it loses no power; it holds under a Mat\'ern kernel and input correlation up to 0.99; on real data with planted and authentic null inputs it admits 5-12x fewer spurious inputs; it costs 5-18% of the GP fit; and a block-averaged variant retains validity at cost linear in $n$.

Abstract:Automatic relevance determination (ARD), the default tool for variable selection in Gaussian-process (GP) regression, ranks inputs by inverse lengthscales -- which measure how fast a function varies, not how much an input contributes to prediction -- and offers no calibrated rule for deciding which inputs to keep. The prediction-centred alternative, the derivative sensitivity $\nu_j = \mathbb{E}[(\partial f/\partial x_j)^2]$, is available in closed form from a fitted GP, but turning it into a selection rule is harder than it looks: at a null input the estimator is a degenerate quadratic form, so Wald and Bernstein-von Mises cutoffs are anti-conservative, and the natural residual bootstrap is mis-scaled. We show that a studentized multiplier bootstrap of the GP derivative process repairs both, prove its validity through an invariance principle for quadratic forms, and obtain asymptotic family-wise and false-discovery-rate control across inputs. Over 100 replications the rule controls FDR wherever inputs are truly null, while uncalibrated derivative rankings breach the target by up to 2x and a Bernstein-von Mises cutoff by 2.2x; at matched FDR it loses no power; it holds under a Matérn kernel and input correlation up to 0.99; on real data with planted and authentic null inputs it admits 5-12x fewer spurious inputs; it costs 5-18% of the GP fit; and a block-averaged variant retains validity at cost linear in $n$.

Machine Learning (stat.ML); Machine Learning (cs.LG); Numerical Analysis (math.NA)

arXiv:2609.33549 [stat.ML]

(or arXiv:2609.33549v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.33549 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Calibrated Derivative-Process Sensitivity for Gaussian-Process Variable Selection

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