Model Free Prediction with Uncertainty Assessment

TL;DR

This paper introduces a new framework using conditional diffusion models for deep nonparametric regression, enabling rigorous statistical inference through asymptotic properties and confidence regions.

Contribution

It develops an end-to-end convergence rate and establishes asymptotic normality for the proposed deep estimation method, bridging a gap in statistical inference for neural networks.

Findings

Theoretical convergence rate for the conditional diffusion model.

Asymptotic normality of generated samples.

Numerical experiments validate the methodology's effectiveness.

Abstract

Deep nonparametric regression, characterized by the utilization of deep neural networks to learn target functions, has emerged as a focus of research attention in recent years. Despite considerable progress in understanding convergence rates, the absence of asymptotic properties hinders rigorous statistical inference. To address this gap, we propose a novel framework that transforms the deep estimation paradigm into a platform conducive to conditional mean estimation, leveraging the conditional diffusion model. Theoretically, we develop an end-to-end convergence rate for the conditional diffusion model and establish the asymptotic normality of the generated samples. Consequently, we are equipped to construct confidence regions, facilitating robust statistical inference. Furthermore, through numerical experiments, we empirically validate the efficacy of our proposed methodology.