Optimal exponential bounds for aggregation of density estimators

TL;DR

This paper introduces an optimal, fully adaptive aggregation method for density estimation that outperforms traditional empirical risk minimization and exponential weights approaches, achieving minimax optimal rates.

Contribution

It develops a new penalty-based aggregation estimator with sharp oracle inequalities and proves its optimality and adaptivity without relying on the unknown density's sup-norm.

Findings

The proposed estimator is fully adaptive and does not depend on the unknown density's sup-norm.

It achieves minimax optimal rates in density estimation.

Lower bounds confirm the sharpness of the deviation term in the oracle inequalities.

Abstract

We consider the problem of model selection type aggregation in the context of density estimation. We first show that empirical risk minimization is sub-optimal for this problem and it shares this property with the exponential weights aggregate, empirical risk minimization over the convex hull of the dictionary functions, and all selectors. Using a penalty inspired by recent works on the $Q$-aggregation procedure, we derive a sharp oracle inequality in deviation under a simple boundedness assumption and we show that the rate is optimal in a minimax sense. Unlike the procedures based on exponential weights, this estimator is fully adaptive under the uniform prior. In particular, its construction does not rely on the sup-norm of the unknown density. By providing lower bounds with exponential tails, we show that the deviation term appearing in the sharp oracle inequalities cannot be improved.