Phase transitions in nonparametric regressions: a curse of exploiting higher degree smoothness assumptions in finite samples

When the regression function belongs to the smooth classes consisting of univariate functions with derivatives up to the $(\gamma+1)$th order bounded in absolute values by a common constant everywhere or a.e., it is generally viewed that exploiting higher degree smoothness assumption helps reduce the estimation error. This paper shows that the minimax optimal mean integrated squared error (MISE) rate increases in $\gamma$ when the sample size $n$ is small relative to $\left(\gamma+1\right)^{2\gamma+3}$ (e.g., $\left(\gamma+1\right)^{2\gamma+3}=262144$ when $\gamma=3$), and decreases in $\gamma$ when $n$ is large relative to $\left(\gamma+1\right)^{2\gamma+3}$. In particular, this phase transition property is shown to be achieved by common nonparametric procedures. Consider $\gamma_{1}$ and $\gamma_{2}$ such that $\gamma_{1}<\gamma_{2}$, where the $(\gamma_{2}+1)$th degree smoothness class is a subset of the $(\gamma_{1}+1)$th degree class. What is interesting about our results is that they imply, if $n$ is small relative to $\left(\gamma_{1}+1\right)^{2\gamma_{1}+3}$, the optimal rate achieved by the estimator constrained to be in the smoother class is larger. In data sets with fewer than hundreds-of-thousands observations, our results suggest that one should not exploit beyond the third degree of smoothness. To some extent, our results provide a theoretical basis for the widely adopted practical recommendation given by Gelman and Imbens (2019). The building blocks of our minimax optimality results are a set of metric entropy bounds we develop in this paper for smooth function classes. Some of our bounds are original, and some of them refine and/or generalize the ones in the literature.