A new approach to locally adaptive polynomial regression
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909617291788288 |
|---|---|
| author | Chatterjee, Sabyasachi Goswami, Subhajit Mukherjee, Soumendu Sundar |
| author_facet | Chatterjee, Sabyasachi Goswami, Subhajit Mukherjee, Soumendu Sundar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19802 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new approach to locally adaptive polynomial regression Chatterjee, Sabyasachi Goswami, Subhajit Mukherjee, Soumendu Sundar Machine Learning Probability Statistics Theory Methodology 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. |
| title | A new approach to locally adaptive polynomial regression |
| topic | Machine Learning Probability Statistics Theory Methodology |
| url | https://arxiv.org/abs/2412.19802 |