Blessings and curse of smoothness and phase transitions in nonparametric regressions: a nonasymptotic perspective

When the regression function belongs to the standard 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 well known that the minimax optimal rate of convergence in mean squared error (MSE) is $\left(\frac{\sigma^{2}}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ when $\gamma$ is finite and the sample size $n\rightarrow\infty$. From a nonasymptotic viewpoint that does not take $n$ to infinity, this paper shows that: for the standard H\"older and Sobolev classes, the minimax optimal rate is $\frac{\sigma^{2}\left(\gamma+1\right)}{n}$ ($\succsim\left(\frac{\sigma^{2}}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$) when $\frac{n}{\sigma^{2}}\precsim\left(\gamma+1\right)^{2\gamma+3}$ and $\left(\frac{\sigma^{2}}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ ($\succsim\frac{\sigma^{2}\left(\gamma+1\right)}{n}$) when $\frac{n}{\sigma^{2}}\succsim\left(\gamma+1\right)^{2\gamma+3}$. To establish these results, we derive upper and lower bounds on the covering and packing numbers for the generalized H\"older class where the absolute value of the $k$th ($k=0,...,\gamma$) derivative is bounded by a parameter $R_{k}$ and the $\gamma$th derivative is $R_{\gamma+1}-$Lipschitz (and also for the generalized ellipsoid class of smooth functions). Our bounds sharpen the classical metric entropy results for the standard classes, and give the general dependence on $\gamma$ and $R_{k}$. By deriving the minimax optimal MSE rates under various (well motivated) $R_{k}$s for the smooth classes with the help of our new entropy bounds, we show several interesting results that cannot be shown with the existing entropy bounds in the literature.