Phase transitions in nonparametric regressions

TL;DR

This paper investigates phase transitions in the minimax optimal rates of mean integrated squared error for nonparametric regression, revealing how these rates change with sample size and smoothness, and introduces new metric entropy bounds for smooth function classes.

Contribution

It develops new metric entropy bounds for smooth functions, enabling the analysis of phase transitions in minimax rates in nonparametric regression.

Findings

Identifies phase transitions in minimax MISE rates based on sample size and smoothness.

Provides new metric entropy bounds that improve and generalize existing results.

Shows how optimal smoothness exploitation varies with sample size.

Abstract

When the unknown regression function of a single variable is known to have derivatives up to the $(\gamma+1)$th order bounded in absolute values by a common constant everywhere or a.e. (i.e., $(\gamma+1)$th degree of smoothness), the minimax optimal rate of the mean integrated squared error (MISE) is stated as $\left(\frac{1}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ in the literature. This paper shows that: (i) if $n\leq\left(\gamma+1\right)^{2\gamma+3}$, the minimax optimal MISE rate is $\frac{\log n}{n\log(\log n)}$ and the optimal degree of smoothness to exploit is roughly $\max\left\{ \left\lfloor \frac{\log n}{2\log\left(\log n\right)}\right\rfloor ,\,1\right\} $; (ii) if $n>\left(\gamma+1\right)^{2\gamma+3}$, the minimax optimal MISE rate is $\left(\frac{1}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ and the optimal degree of smoothness to exploit is $\gamma+1$. The fundamental contribution of this paper is a set of metric entropy bounds we develop for smooth function classes. Some of our bounds are original, and some of them improve and/or generalize the ones in the literature (e.g., Kolmogorov and Tikhomirov, 1959). Our metric entropy bounds allow us to show phase transitions in the minimax optimal MISE rates associated with some commonly seen smoothness classes as well as non-standard smoothness classes, and can also be of independent interest outside the nonparametric regression problems.