Optimal aggregation of affine estimators

Stochastics Seminar
Thursday, September 1, 2011 - 15:05
1 hour (actually 50 minutes)
Skiles 006
Electrical and Computer Engineering, Duke University
We consider the problem of combining a (possibly uncountably infinite) set of affine estimators in non-parametric regression model with heteroscedastic Gaussian noise. Focusing onthe exponentially weighted aggregate, we prove a PAC-Bayesian type inequality that leads tosharp 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 literatureon 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 rangeof tuning parameters nor splitting the set of observations. We also illustrate numerically thegood performance achieved by the exponentially weighted aggregate. (This is a joint work with Arnak Dalalyan.)