Aggregation of Affine Estimators
This work provides theoretical guarantees for high-probability performance in statistical estimator aggregation, addressing a gap in existing results for researchers in machine learning and statistics.
The paper tackles the problem of aggregating affine estimators for fixed design regression, proving that a generalized Q-aggregation scheme achieves sharp oracle inequalities with high probability and simultaneously attains the best known bounds for various aggregation tasks.
We consider the problem of aggregating a general collection of affine estimators for fixed design regression. Relevant examples include some commonly used statistical estimators such as least squares, ridge and robust least squares estimators. Dalalyan and Salmon (2012) have established that, for this problem, exponentially weighted (EW) model selection aggregation leads to sharp oracle inequalities in expectation, but similar bounds in deviation were not previously known. While results indicate that the same aggregation scheme may not satisfy sharp oracle inequalities with high probability, we prove that a weaker notion of oracle inequality for EW that holds with high probability. Moreover, using a generalization of the newly introduced $Q$-aggregation scheme we also prove sharp oracle inequalities that hold with high probability. Finally, we apply our results to universal aggregation and show that our proposed estimator leads simultaneously to all the best known bounds for aggregation, including $\ell_q$-aggregation, $q \in (0,1)$, with high probability.