On the Pinsker bound of inner product kernel regression in large dimensions

TL;DR

This paper derives the exact minimax risk and Pinsker constant for inner product kernel regression in high dimensions, revealing precise theoretical limits of prediction accuracy as the dimension grows.

Contribution

It provides the first precise calculation of the Pinsker constant for inner product kernel regression in large dimensions, extending understanding of minimax risks in high-dimensional kernel methods.

Findings

Exact minimax risk determined for high-dimensional inner product kernel regression.

Identification of the Pinsker constant associated with the excess risk.

Analysis applicable when sample size scales as a power of the dimension.

Abstract

Building on recent studies of large-dimensional kernel regression, particularly those involving inner product kernels on the sphere $\mathbb{S}^{d}$, we investigate the Pinsker bound for inner product kernel regression in such settings. Specifically, we address the scenario where the sample size $n$ is given by $\alpha d^{\gamma}(1+o_{d}(1))$ for some $\alpha, \gamma>0$. We have determined the exact minimax risk for kernel regression in this setting, not only identifying the minimax rate but also the exact constant, known as the Pinsker constant, associated with the excess risk.