Sharp Oracle Inequalities for Aggregation of Affine Estimators

TL;DR

This paper develops sharp oracle inequalities for aggregating affine estimators in non-parametric regression with heteroscedastic Gaussian noise, providing adaptive, minimax optimal estimators without discretization or data splitting.

Contribution

It introduces a PAC-Bayesian framework for exponential aggregation of affine estimators, achieving sharp oracle inequalities in both discrete and continuous settings.

Findings

The aggregate adapts optimally in the minimax sense.

The method applies to various estimators like kernel ridge regression and shrinking estimators.

Numerical experiments show strong performance of the proposed aggregation.

Abstract

We consider the problem of combining a (possibly uncountably infinite) set of affine estimators in non-parametric regression model with heteroscedastic Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a PAC-Bayesian type inequality that leads to sharp oracle inequalities in discrete but also in continuous settings. The framework is general enough to cover the combinations of various procedures such as least square regression, kernel ridge regression, shrinking estimators and many other estimators used in the literature on statistical inverse problems. As a consequence, we show that the proposed aggregate provides an adaptive estimator in the exact minimax sense without neither discretizing the range of tuning parameters nor splitting the set of observations. We also illustrate numerically the good performance achieved by the exponentially weighted aggregate.