Nonparametric Regression on Fixed-Cardinality Subsets:Minimax Risk and Random-Design Effects

RESUMO

We study regression with subsets as covariates. The response is an unknown function of the input subset, and observations consist of noisy evaluations at uniformly sampled subsets, each containing exactly \(k\) items from a ground set of size \(d\). This problem arises in combination screening, bundle preference modeling, and other settings in which outcomes depend on collections of prescribed size. The fixed-cardinality constraint couples the membership coordinates, so standard product-domain notions of interaction and smoothness cannot be imported unchanged. We use the harmonic decomposition of the Johnson graph to define an intrinsic Johnson--Sobolev scale and determine the exact dimensions of low-order interaction spaces. We derive finite-sample risk bounds for harmonic projection and establish a nonasymptotic minimax characterization when the Johnson--Sobolev energy budget is controlled relative to the noise variance. We also show that stable least squares removes the signal-dependent fluctuation of empirical projection when the empirical Gram matrix is sufficiently well conditioned. For arbitrary signal-to-noise ratios, a rank-deficiency analysis quantifies the components left unidentified by the random design. Together, the rank-deficiency analysis and a centered completion estimator yield minimax bounds that match up to constants whenever \(\min(k,d-k)\) and the smoothness order are fixed. Numerical experiments reported in the Supplementary Material illustrate estimation under different interaction profiles and distinguish the effects of observation noise, random-design fluctuation, and incomplete coverage.