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E-QRGMM: Efficient Generative Metamodeling for Covariate-Dependent Uncertainty Quantification

Zhiyang Liang, Qingkai Zhang

TL;DR

This work tackles covariate-dependent uncertainty quantification in simulation-based decision-making by introducing Efficient Quantile-Regression-Based Generative Metamodeling (E-QRGMM). The method accelerates QRGMM by replacing dense quantile grids with gradient-informed cubic Hermite interpolation on a sparse central region while preserving tails with the original approach, yielding a grid complexity of $O(n^{1/5})$ and the same convergence rate $O_p(n^{-1/2})$. Theoretical analysis decomposes errors into interpolation, quantile regression, and gradient estimation components and shows the total error remains optimal under balanced scaling; empirical results on synthetic and inventory datasets demonstrate favorable distributional fidelity and dramatically reduced training time, enabling practical bootstrap-based covariate-dependent confidence intervals for arbitrary estimands. Overall, E-QRGMM provides a scalable, distributionally faithful surrogate for simulators that supports real-time, covariate-conditioned uncertainty quantification and decision support.

Abstract

Covariate-dependent uncertainty quantification in simulation-based inference is crucial for high-stakes decision-making but remains challenging due to the limitations of existing methods such as conformal prediction and classical bootstrap, which struggle with covariate-specific conditioning. We propose Efficient Quantile-Regression-Based Generative Metamodeling (E-QRGMM), a novel framework that accelerates the quantile-regression-based generative metamodeling (QRGMM) approach by integrating cubic Hermite interpolation with gradient estimation. Theoretically, we show that E-QRGMM preserves the convergence rate of the original QRGMM while reducing grid complexity from $O(n^{1/2})$ to $O(n^{1/5})$ for the majority of quantile levels, thereby substantially improving computational efficiency. Empirically, E-QRGMM achieves a superior trade-off between distributional accuracy and training speed compared to both QRGMM and other advanced deep generative models on synthetic and practical datasets. Moreover, by enabling bootstrap-based construction of confidence intervals for arbitrary estimands of interest, E-QRGMM provides a practical solution for covariate-dependent uncertainty quantification.

E-QRGMM: Efficient Generative Metamodeling for Covariate-Dependent Uncertainty Quantification

TL;DR

This work tackles covariate-dependent uncertainty quantification in simulation-based decision-making by introducing Efficient Quantile-Regression-Based Generative Metamodeling (E-QRGMM). The method accelerates QRGMM by replacing dense quantile grids with gradient-informed cubic Hermite interpolation on a sparse central region while preserving tails with the original approach, yielding a grid complexity of and the same convergence rate . Theoretical analysis decomposes errors into interpolation, quantile regression, and gradient estimation components and shows the total error remains optimal under balanced scaling; empirical results on synthetic and inventory datasets demonstrate favorable distributional fidelity and dramatically reduced training time, enabling practical bootstrap-based covariate-dependent confidence intervals for arbitrary estimands. Overall, E-QRGMM provides a scalable, distributionally faithful surrogate for simulators that supports real-time, covariate-conditioned uncertainty quantification and decision support.

Abstract

Covariate-dependent uncertainty quantification in simulation-based inference is crucial for high-stakes decision-making but remains challenging due to the limitations of existing methods such as conformal prediction and classical bootstrap, which struggle with covariate-specific conditioning. We propose Efficient Quantile-Regression-Based Generative Metamodeling (E-QRGMM), a novel framework that accelerates the quantile-regression-based generative metamodeling (QRGMM) approach by integrating cubic Hermite interpolation with gradient estimation. Theoretically, we show that E-QRGMM preserves the convergence rate of the original QRGMM while reducing grid complexity from to for the majority of quantile levels, thereby substantially improving computational efficiency. Empirically, E-QRGMM achieves a superior trade-off between distributional accuracy and training speed compared to both QRGMM and other advanced deep generative models on synthetic and practical datasets. Moreover, by enabling bootstrap-based construction of confidence intervals for arbitrary estimands of interest, E-QRGMM provides a practical solution for covariate-dependent uncertainty quantification.
Paper Structure (39 sections, 6 theorems, 82 equations, 3 figures, 5 tables, 2 algorithms)

This paper contains 39 sections, 6 theorems, 82 equations, 3 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related literature
  3. Quantile-regression-based generative metamodeling
  4. Other generative models for approximating conditional distribution
  5. Problem formulation
  6. Efficient quantile-regression-based generative metamodeling
  7. Quantile-regression-based generative metamodeling
  8. Cubic Hermite interpolation
  9. Gradient estimation
  10. Grid point design
  11. Efficient covariate-dependent uncertainty quantification
  12. Theoretical analysis
  13. Numerical experiments
  14. Experimental Setup
  15. Estimands of interest
  16. ...and 24 more sections

Key Result

Proposition 1

Under Assumption assumption:linear, and Assumptions ass:pathwise and ass:residual_smoothness in app:assumptions, the gradient $D(\tau\mid \mathbf{x})$ satisfies where $\Lambda(\tau) = \lim_{\delta \to 0^+} \frac{1}{2\delta} \mathbb{E}\left[ \mathbf{x}\mathbf{x}^\top \cdot \mathbbm{1}\{-\delta < Y - \mathbf{x}^\top \boldsymbol{\beta}(\tau) < \delta\} \right].$

Figures (3)

  • Figure 1: Comparison of E-QRGMM and QRGMM on normal test distribution with $n=10^4$. Left: E-QRGMM achieves the same distributional accuracy with fewer grid points. Right: In E-QRGMM, quantile regression remains the dominant cost, and total time is much lower than in QRGMM.
  • Figure 2: Comparison of E-QRGMM and QRGMM under a normal test distribution with $n = 10^5$. Left: E-QRGMM achieves the same accuracy with fewer grid points. Right: Compared to the $n = 10^4$ case, gradient estimation takes up an even smaller portion.
  • Figure 3: Estimated versus true gradient curves of $\frac{d}{d\tau}\boldsymbol{\beta}(\tau)$ under the normal test distribution, . The blue lines represent the true gradients, while the yellow lines show the estimated gradients. Vertical red lines at $\tau=0.1$ and $\tau=0.9$ mark the tail regions. Notably, the estimated gradients closely match the ground truth in the central region, but exhibit significant numerical inaccuracies near the distributional tails.

Theorems & Definitions (6)

  • Proposition 1: Gradient estimation
  • Lemma 1: Cubic Hermite Interpolation Error
  • Lemma 2: Quantile Regression Estimation Error
  • Lemma 3: Gradient Estimation Error
  • Theorem 1: Convergence Rate of E-QRGMM
  • Lemma 4: hong2010pathwise