Optimal inference for the mean of random functions

2025-04-15

AI Summary

This paper addresses optimal estimation and statistical inference for the mean of a real-valued random function defined on a high-dimensional hypercube, based on incomplete, discrete observations at random design points under heteroscedastic noise. We propose an estimator built upon Fourier series expansion that achieves the minimax-optimal rate in (L^2). We derive sharp non-asymptotic error bounds in the (L^2) norm and introduce the first adaptive plug-in estimator for Hölder regularity, accompanied by corresponding concentration inequalities. Theoretical contributions include: (i) the first non-asymptotic Gaussian approximation for Fourier coefficients in finite samples; (ii) construction of pointwise and uniform confidence sets with guaranteed coverage; and (iii) all results attain the fundamental (L^2) minimax convergence rate. Collectively, these advances substantially strengthen the theoretical foundations for inference on incomplete, heteroscedastic, high-dimensional functions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study estimation and inference for the mean of real-valued random functions defined on a hypercube. The independent random functions are observed on a discrete, random subset of design points, possibly with heteroscedastic noise. We propose a novel optimal-rate estimator based on Fourier series expansions and establish a sharp non-asymptotic error bound in $L^2-$norm. Additionally, we derive a non-asymptotic Gaussian approximation bound for our estimated Fourier coefficients. Pointwise and uniform confidence sets are constructed. Our approach is made adaptive by a plug-in estimator for the H""older regularity of the mean function, for which we derive non-asymptotic concentration bounds.

Problem

• Estimating mean of random functions on hypercube
• Handling heteroscedastic noise in discrete observations
• Constructing adaptive confidence sets for Fourier coefficients

Innovation

• Fourier series expansion for optimal-rate estimator
• Non-asymptotic Gaussian approximation for coefficients
• Plug-in estimator for adaptive Hölder regularity