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Sharp Debiasing for Smooth Functional Estimation in Banach Spaces

Woonyoung Chang, Arun Kumar Kuchibhotla

Abstract

This paper studies the estimation of smooth functionals $f(θ)$ of a mean parameter $θ= \mathbb{E}_P[W]$ 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--Esséen 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.

Sharp Debiasing for Smooth Functional Estimation in Banach Spaces

Abstract

This paper studies the estimation of smooth functionals of a mean parameter for a distribution 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--Esséen bounds for both -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 without requiring any structural assumptions (e.g., sparsity). We discuss computational relaxations that enables polynomial-time implementation for a range of matrix functionals.

Paper Structure

This paper contains 42 sections, 34 theorems, 454 equations, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contributions
  4. Notation and Preliminaries
  5. Problem Setup and Methods
  6. Results for m-Smooth Functionals
  7. Moment Bounds
  8. Asymptotic Normality
  9. Results for Infinitely Differentiable Functionals
  10. Moment Bounds
  11. Asymptotic Normality
  12. Applications
  13. Precision Matrix Estimation
  14. Estimation of Linear Regression Functional
  15. Computational Relaxation under Product Structure of Derivatives
  16. ...and 27 more sections

Key Result

Proposition 1.1

For any $\tilde{\theta}\in \Theta$, the following deterministic identity holds: where Here, $\mathcal{J}$ is the Riemann-Liouville operator where $\mathcal{J}^0g=g(1)$ and $\mathcal{J}^m g = \int_0^1 g(t)(1-t)^{m-1}\,dt/\Gamma(m)$ for $m>0$, and $\Delta^{(s)}(t) = \mathcal{D}^s f(\tilde{\theta} + t(\theta - \tilde{\theta})) - \mathcal{D}^s f(\tilde{\theta})$, $t\in[0,1]$.$\blacktrian

Theorems & Definitions (56)

  • Proposition 1.1
  • Theorem 2.1
  • Proposition 2.2
  • Example 1: Hilbert Space
  • Example 2: Smooth Banach Space
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Corollary 4.2
  • ...and 46 more