Optimal Estimation of High Dimensional Smooth Additive Function Based on Noisy Observations

Fan Zhou, Ping Li
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:12813-12823, 2021.

Abstract

Given $\bx_j = \btheta + \bepsilon_j$, $j=1,...,n$ where $\btheta \in \RR^d$ is an unknown parameter and $\bepsilon_j$ are i.i.d. Gaussian noise vectors, we study the estimation of $f(\btheta)$ for a given smooth function $f:\RR^d \rightarrow \RR$ equipped with an additive structure. We inherit the idea from a recent work which introduced an effective bias reduction technique through iterative bootstrap and derive a bias-reducing estimator. By establishing its normal approximation results, we show that the proposed estimator can achieve asymptotic normality with a looser constraint on smoothness compared with general smooth function due to the additive structure. Such results further imply that the proposed estimator is asymptotically efficient. Both upper and lower bounds on mean squared error are proved which shows the proposed estimator is minimax optimal for the smooth class considered. Numerical simulation results are presented to validate our analysis and show its superior performance of the proposed estimator over the plug-in approach in terms of bias reduction and building confidence intervals.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-zhou21c, title = {Optimal Estimation of High Dimensional Smooth Additive Function Based on Noisy Observations}, author = {Zhou, Fan and Li, Ping}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {12813--12823}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/zhou21c/zhou21c.pdf}, url = {https://proceedings.mlr.press/v139/zhou21c.html}, abstract = {Given $\bx_j = \btheta + \bepsilon_j$, $j=1,...,n$ where $\btheta \in \RR^d$ is an unknown parameter and $\bepsilon_j$ are i.i.d. Gaussian noise vectors, we study the estimation of $f(\btheta)$ for a given smooth function $f:\RR^d \rightarrow \RR$ equipped with an additive structure. We inherit the idea from a recent work which introduced an effective bias reduction technique through iterative bootstrap and derive a bias-reducing estimator. By establishing its normal approximation results, we show that the proposed estimator can achieve asymptotic normality with a looser constraint on smoothness compared with general smooth function due to the additive structure. Such results further imply that the proposed estimator is asymptotically efficient. Both upper and lower bounds on mean squared error are proved which shows the proposed estimator is minimax optimal for the smooth class considered. Numerical simulation results are presented to validate our analysis and show its superior performance of the proposed estimator over the plug-in approach in terms of bias reduction and building confidence intervals.} }
Endnote
%0 Conference Paper %T Optimal Estimation of High Dimensional Smooth Additive Function Based on Noisy Observations %A Fan Zhou %A Ping Li %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-zhou21c %I PMLR %P 12813--12823 %U https://proceedings.mlr.press/v139/zhou21c.html %V 139 %X Given $\bx_j = \btheta + \bepsilon_j$, $j=1,...,n$ where $\btheta \in \RR^d$ is an unknown parameter and $\bepsilon_j$ are i.i.d. Gaussian noise vectors, we study the estimation of $f(\btheta)$ for a given smooth function $f:\RR^d \rightarrow \RR$ equipped with an additive structure. We inherit the idea from a recent work which introduced an effective bias reduction technique through iterative bootstrap and derive a bias-reducing estimator. By establishing its normal approximation results, we show that the proposed estimator can achieve asymptotic normality with a looser constraint on smoothness compared with general smooth function due to the additive structure. Such results further imply that the proposed estimator is asymptotically efficient. Both upper and lower bounds on mean squared error are proved which shows the proposed estimator is minimax optimal for the smooth class considered. Numerical simulation results are presented to validate our analysis and show its superior performance of the proposed estimator over the plug-in approach in terms of bias reduction and building confidence intervals.
APA
Zhou, F. & Li, P.. (2021). Optimal Estimation of High Dimensional Smooth Additive Function Based on Noisy Observations. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:12813-12823 Available from https://proceedings.mlr.press/v139/zhou21c.html.

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