Sharp Debiasing for Smooth Functional Estimation in Banach Spaces

TL;DR

This paper introduces a cross-fitted estimator for smooth functional estimation in Banach spaces, providing non-asymptotic bounds and applications to high-dimensional problems.

Contribution

It develops a novel estimator with theoretical guarantees for smooth functionals in Banach spaces, applicable to high-dimensional statistical inference.

Findings

Achieves asymptotic normality under mild dimension conditions.

Provides non-asymptotic moment and Berry--Eséen bounds.

Enables polynomial-time computation for matrix functionals.

Abstract

This paper studies the estimation of smooth functionals $f(\theta)$ of a mean parameter $\theta = \mathbb{E}_P[W]$ for a distribution $P$ on a general Banach space. We propose a cross-fitted estimator based on a single sample splitting and establish non-asymptotic moment bounds and Berry--Ess\'een bounds for both $m$-smooth and infinitely smooth functionals under the finite moment assumptions. Our framework is applied to precision matrix estimation and the inference of projection parameters in high-dimensional regression. In these Euclidean settings, the proposed estimators achieve asymptotic normality under the dimension regime $d \log^2(en) = o(n)$ without requiring any structural assumptions (e.g., sparsity). We discuss computational relaxations that enables polynomial-time implementation for a range of matrix functionals.