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Model Free Prediction with Uncertainty Assessment

Yuling Jiao, Lican Kang, Jin Liu, Heng Peng, Heng Zuo

TL;DR

This work tackles statistical inference in deep nonparametric regression by introducing a conditional diffusion framework that learns the conditional distribution $P_{Y|X}$ and converts deep estimation into conditional mean estimation. The authors establish an end-to-end convergence rate for the conditional diffusion model and prove asymptotic normality of the generated samples, enabling construction of confidence regions for $f_0(x)=\mathbb{E}(Y|X=x)$. The analysis combines score estimation, EM discretization, and drift-estimation error with rigorous TV-distance bounds, and is validated through simulation and real-data experiments (Wine Quality, Abalone, and Superconductivity datasets). This approach provides robust uncertainty quantification in high-dimensional, deep-regression contexts and lays groundwork for further developments in conditional generative inference and diffusion-based uncertainty quantification.

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.

Model Free Prediction with Uncertainty Assessment

TL;DR

This work tackles statistical inference in deep nonparametric regression by introducing a conditional diffusion framework that learns the conditional distribution and converts deep estimation into conditional mean estimation. The authors establish an end-to-end convergence rate for the conditional diffusion model and prove asymptotic normality of the generated samples, enabling construction of confidence regions for . The analysis combines score estimation, EM discretization, and drift-estimation error with rigorous TV-distance bounds, and is validated through simulation and real-data experiments (Wine Quality, Abalone, and Superconductivity datasets). This approach provides robust uncertainty quantification in high-dimensional, deep-regression contexts and lays groundwork for further developments in conditional generative inference and diffusion-based uncertainty quantification.

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.
Paper Structure (27 sections, 14 theorems, 195 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 27 sections, 14 theorems, 195 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Work
  4. Notations and Paper Organization
  5. Preliminaries
  6. Problem Setting
  7. Theoretical Analysis
  8. Proof Sketch
  9. Bound $\mathbb{E}_{\mathcal{D},\mathcal{T},\mathcal{Z}}[\mathrm{TV}(\check{p}_{T_0},\widetilde{p}_{T_0})]$
  10. Bound $\mathrm{TV}(p_{T_0},p_0)$
  11. Bound $\mathbb{E}_{\mathcal{D},\mathcal{T},\mathcal{Z}}[\mathrm{TV}(\widetilde{p}_{T_0}, p_{T_0})]$
  12. Numerical Experiments
  13. Simulation Studies
  14. Real Data Analysis
  15. The Wine Quality Dataset
  16. ...and 12 more sections

Key Result

Theorem 4.1

Suppose that Assumptions ass: bounded_support-ass: smoothness_y hold and the drift estimator $\widehat{\boldsymbol{s}}$ defined in eq: edrift is constructed as introduced in Theorem th: drift_estimation. Let $m > n^{\frac{d_{\mathcal{X}} + d_{\mathcal{Y}} + 5}{ d_{\mathcal{X}} + d_{\mathcal{Y}} + 3}

Figures (3)

  • Figure 1: (Left) Estimated density and actual density on the test set. (Right) The prediction intervals on the test set.
  • Figure 2: (Top) Estimated density and actual density on the test set. (Bottom) The prediction intervals on the test set.
  • Figure 3: (Left) Estimated density and actual density on the test set. (Right) The prediction intervals on the test set.

Theorems & Definitions (32)

  • Definition 2.1: $\mathrm{TV}$ Distance
  • Definition 2.2: ReLU DNNs
  • Definition 2.3: Covering number
  • Remark 4.1
  • Remark 4.2
  • Theorem 4.1: End-to-End Convergence Rate
  • Remark 4.3
  • Theorem 4.2: Asymptotic Normality
  • Remark 4.4
  • Lemma 5.1
  • ...and 22 more