🤖 AI Summary
This work addresses the high computational cost of traditional spline regression, which relies on grid search and cross-validation to select resolution hyperparameters. By leveraging approximation theory and ANOVA decomposition, the authors derive—for the first time—a closed-form analytical solution for the optimal resolution, thereby eliminating costly hyperparameter tuning. The proposed method, KORE, estimates bias and noise via two pilot fits and rapidly computes leave-one-out error using the PRESS identity. KORE reveals a Kolmogorov-optimal scaling law dependent solely on effective density and independent of input dimensionality, and it integrates multiple model selection criteria—including GCV, Cp, AIC, and BIC—for efficient modeling. On additive and sparse pairwise functions up to 80 dimensions, KORE achieves the accuracy of exhaustive cross-validation with only about one-eighth of the fitting effort; across 36 real-world tabular datasets, it attains the highest accuracy per unit computation among 21 competing methods.
📝 Abstract
Hyperparameter tuning almost always means search: fit the model at every value on a grid, score each by cross-validation, and keep the winner. For spline regression that search is unnecessary. The optimal resolution can be solved for in closed form, to the accuracy an exhaustive search reaches, at a fraction of the compute. Three ingredients make this possible: classical approximation theory pins the squared bias to a known power of the resolution G, exactly the Kolmogorov n-width of the smoothness class; the basis dimension is an explicit polynomial in G; and leave-one-out error follows from a single fit via the PRESS identity. Balancing the two known curves gives the minimizer analytically. We extend this calculus to many coordinates by replacing ambient input dimension with interaction order, the number of active low-order components in an ANOVA decomposition, yielding a scaling law in which the optimal resolution and error are power functions of the effective density (sample size per active component), with input dimension absent from the exponent. The law becomes an algorithm. KORE (Kolmogorov-optimal Order-aware Resolution Estimation) fits two pilot resolutions, solves a leverage-calibrated 2x2 system for the bias and noise scales, and evaluates the closed-form plug-in resolution with a tiny leave-one-out certificate: about a dozen fits instead of a full grid sweep, with a consistency guarantee as the sample grows. Across additive and sparse pairwise targets up to 80 input dimensions, KORE matches exhaustive 3-fold cross-validation and the full classical ladder (GCV, Mallows' Cp, AIC, BIC) while fitting roughly 8x fewer models; on 36 real tabular datasets it ranks first among 21 methods in accuracy per unit of compute, ahead of tuned boosters and kernel machines. When complexity lives in low interaction order, solving for the resolution beats searching for it.