Multi-task statistical learning.

Summary The purpose of this thesis is to construct, calibrate and study multi-task estimators in a non-parametric and non-asymptotic frequentist framework. We place ourselves in the framework of kernel ridge regression and extend existing methods of multi-task regression. The key issue is the calibration of a matrix regularization parameter, which encodes the similarity between tasks. We propose a calibration method for this parameter, based on the estimation of the noise covariance matrix between tasks. We then give optimality guarantees for the obtained estimator, via an oracle inequality, and verify its behavior on simulated examples. We also obtain a precise framework for the risks of multi-task and single-task oracle estimators in some cases. This allows us to identify several interesting situations, where the multi-task oracle is more efficient than the single-task oracle, or vice versa. It also allows us to ensure that the oracle inequality forces the multi-task estimator to have a lower risk than the single-task estimator in the studied cases. The behavior of the multi-task and single-task oracles is verified on simulated examples.