A new approach to locally adaptive polynomial regression
This addresses a fundamental challenge in locally adaptive nonparametric regression, though it appears to be an incremental improvement over existing methods like Lepski's method.
The paper tackles adaptive bandwidth selection in nonparametric regression by introducing a new procedure based on ℓ₂-norms of interval projections, obtaining non-asymptotic risk bounds that adapt near-optimally to the local Hölder exponent of the regression function simultaneously at all points.
Adaptive bandwidth selection is a fundamental challenge in nonparametric regression. This paper introduces a new bandwidth selection procedure inspired by the optimality criteria for $\ell_0$-penalized regression. Although similar in spirit to Lepski's method and its variants in selecting the largest interval satisfying an admissibility criterion, our approach stems from a distinct philosophy, utilizing criteria based on $\ell_2$-norms of interval projections rather than explicit point and variance estimates. We obtain non-asymptotic risk bounds for the local polynomial regression methods based on our bandwidth selection procedure which adapt (near-)optimally to the local Hölder exponent of the underlying regression function simultaneously at all points in its domain. Furthermore, we show that there is a single ideal choice of a global tuning parameter in each case under which the above-mentioned local adaptivity holds. The optimal risks of our methods derive from the properties of solutions to a new ``bandwidth selection equation'' which is of independent interest. We believe that the principles underlying our approach provide a new perspective to the classical yet ever relevant problem of locally adaptive nonparametric regression.