Non-asymptotic Optimal Prediction Error for Growing-dimensional Partially Functional Linear Models

TL;DR

This paper derives non-asymptotic, rate-optimal bounds for prediction error in growing-dimensional partially functional linear models using RKHS, revealing a trade-off between predictor complexity and effective dimension.

Contribution

It provides the first non-asymptotic bounds for prediction error in PFLMs with diverging parameters, including upper and lower bounds, and establishes conditions for prediction consistency.

Findings

Optimal prediction error bounds derived

Prediction consistency shown with increasing multivariate predictors

Trade-off between non-functional predictors and kernel dimension

Abstract

Under the reproducing kernel Hilbert spaces (RKHS), we consider the penalized least-squares of the partially functional linear models (PFLM), whose predictor contains both functional and traditional multivariate parts, and the multivariate part allows a divergent number of parameters. From the non-asymptotic point of view, we focus on the rate-optimal upper and lower bounds of the prediction error. An exact upper bound for the excess prediction risk is shown in a non-asymptotic form under a more general assumption known as the effective dimension to the model, by which we also show the prediction consistency when the number of multivariate covariates $p$ slightly increases with the sample size $n$. Our new finding implies a trade-off between the number of non-functional predictors and the effective dimension of the kernel principal components to ensure prediction consistency in the increasing-dimensional setting. The analysis in our proof hinges on the spectral condition of the sandwich operator of the covariance operator and the reproducing kernel, and on sub-Gaussian and Berstein concentration inequalities for the random elements in Hilbert space. Finally, we derive the non-asymptotic minimax lower bound under the regularity assumption of the Kullback-Leibler divergence of the models.