Solve for the Hyperparameter, Skip the Search: Kolmogorov-Optimal Scaling Laws for Spline Regression

Researchers propose KORE (Kolmogorov-optimal Order-aware Resolution Estimation), a method that solves for optimal hyperparameters in spline regression analytically rather than through expensive grid search. The approach reduces computational cost by ~8x while matching exhaustive cross-validation performance across high-dimensional datasets.

KORE addresses a fundamental inefficiency in machine learning: the brute-force hyperparameter tuning that dominates model selection pipelines. Traditional approaches require fitting dozens or hundreds of candidate models, evaluating each through cross-validation—a computationally wasteful process that scales poorly with dataset complexity. This research leverages classical approximation theory to derive closed-form solutions for optimal resolution in spline regression, replacing search with a mathematically principled calculation.

The breakthrough combines three theoretical insights: the bias-variance tradeoff can be characterized precisely using Kolmogorov n-widths from approximation theory, basis dimension scales polynomially with resolution, and leave-one-out cross-validation error can be computed efficiently via the PRESS identity. For high-dimensional problems, the authors replace input dimension with interaction order—counting only active low-order components in ANOVA decompositions—yielding scaling laws where the exponent depends on effective sample density rather than ambient dimensionality.

Empirical validation demonstrates KORE's practical value. On 36 real tabular datasets, it ranked first among 21 competing methods when evaluated on accuracy-per-computation, outperforming tuned gradient boosting and kernel methods. The algorithm requires only ~12 model fits versus full grid sweeps, with theoretical consistency guarantees as sample size increases. This efficiency gain matters for practitioners working with limited computational budgets and large datasets.

The advancement signals broader momentum in automated machine learning toward replacing expensive search procedures with theoretical approximations. While spline regression occupies a specific niche, the meta-insight—that domain knowledge plus mathematical structure can eliminate hyperparameter search entirely—suggests similar approaches may apply to other model families.

• →KORE solves for optimal spline regression hyperparameters analytically, reducing computational cost by ~8x compared to grid search
• →The method balances known bias and noise curves derived from approximation theory to find the optimal resolution in closed form
• →Interaction order replaces input dimension in scaling laws for high-dimensional problems, making the approach dimension-agnostic
• →Empirical testing across 36 real datasets ranks KORE first among 21 methods in accuracy-per-unit-compute
• →The algorithm requires only ~12 model fits with theoretical consistency guarantees, eliminating traditional cross-validation grid sweeps