# A new approach to locally adaptive polynomial regression

arXiv: https://arxiv.org/abs/2412.19802 · HTML: https://mlsanity.com/paper/2412.19802/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 47 references that are in the map.
- First posted on arXiv 27 Dec 2024.

## Builds on (most cited first, 5 of 5)

- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [Adaptive Estimation of Multivariate Piecewise Polynomials and Bounded Variation Functions by Optimal Decision Trees](https://mlsanity.com/paper/1911.11562/) (2019)
- [Prediction bounds for higher order total variation regularized least squares](https://mlsanity.com/paper/1904.10871/) (2019)
- [Divided Differences, Falling Factorials, and Discrete Splines: Another Look at Trend Filtering and Related Problems](https://mlsanity.com/paper/2003.03886/) (2020)
- [Minmax Trend Filtering: Generalizations of Total Variation Denoising via a Local Minmax/Maxmin Formula](https://mlsanity.com/paper/2410.03041/) (2024)

## Abstract (arXiv)

Adaptive bandwidth selection is a fundamental challenge in nonparametric regression. This paper introduces a new bandwidth selection procedure inspired by the optimality criteria for ℓ0-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 ℓ2-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 H"older 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.