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Learning to Simulate: Generative Metamodeling via Quantile Regression

L. Jeff Hong, Yanxi Hou, Qingkai Zhang, Xiaowei Zhang

TL;DR

This work introduces generative metamodeling to bridge the gap between slow stochastic simulators and the need for real-time decision support. It proposes QRGMM, which learns the conditional quantile function $F_Y^{-1}(\tau|\mathbf{x})$ offline on a grid and rapidly generates samples online via interpolation of $\hat{Q}(\tau|\mathbf{x})$, yielding approximate samples from $Y(\mathbf{x})$ while preserving distributional fidelity. A rigorous convergence analysis leverages a novel tail-partition scheme to establish uniform convergence of $\hat{Y}(\mathbf{x})$ to $Y(\mathbf{x})$ as $n,m\to\infty$; practical guidance shows $m=O(\sqrt{n})$ balances estimation and interpolation errors and mitigates quantile crossing through rearrangement. Extensions to multidimensional outputs and nonlinear quantile models broaden applicability, and numerical experiments—including artificial benchmarks and an esophageal cancer treatment simulator—demonstrate QRGMM’s superior distributional accuracy and dramatic online generation speed relative to competing generative models. Collectively, the framework enables real-time computation of arbitrary statistics from the simulator’s conditional distribution, with potential impact across operations research, healthcare, and finance.

Abstract

Stochastic simulation models effectively capture complex system dynamics but are often too slow for real-time decision-making. Traditional metamodeling techniques learn relationships between simulator inputs and a single output summary statistic, such as the mean or median. These techniques enable real-time predictions without additional simulations. However, they require prior selection of one appropriate output summary statistic, limiting their flexibility in practical applications. We propose a new concept: generative metamodeling. It aims to construct a "fast simulator of the simulator," generating random outputs significantly faster than the original simulator while preserving approximately equal conditional distributions. Generative metamodels enable rapid generation of numerous random outputs upon input specification, facilitating immediate computation of any summary statistic for real-time decision-making. We introduce a new algorithm, quantile-regression-based generative metamodeling (QRGMM), and establish its distributional convergence and convergence rate. Extensive numerical experiments demonstrate QRGMM's efficacy compared to other state-of-the-art generative algorithms in practical real-time decision-making scenarios.

Learning to Simulate: Generative Metamodeling via Quantile Regression

TL;DR

This work introduces generative metamodeling to bridge the gap between slow stochastic simulators and the need for real-time decision support. It proposes QRGMM, which learns the conditional quantile function offline on a grid and rapidly generates samples online via interpolation of , yielding approximate samples from while preserving distributional fidelity. A rigorous convergence analysis leverages a novel tail-partition scheme to establish uniform convergence of to as ; practical guidance shows balances estimation and interpolation errors and mitigates quantile crossing through rearrangement. Extensions to multidimensional outputs and nonlinear quantile models broaden applicability, and numerical experiments—including artificial benchmarks and an esophageal cancer treatment simulator—demonstrate QRGMM’s superior distributional accuracy and dramatic online generation speed relative to competing generative models. Collectively, the framework enables real-time computation of arbitrary statistics from the simulator’s conditional distribution, with potential impact across operations research, healthcare, and finance.

Abstract

Stochastic simulation models effectively capture complex system dynamics but are often too slow for real-time decision-making. Traditional metamodeling techniques learn relationships between simulator inputs and a single output summary statistic, such as the mean or median. These techniques enable real-time predictions without additional simulations. However, they require prior selection of one appropriate output summary statistic, limiting their flexibility in practical applications. We propose a new concept: generative metamodeling. It aims to construct a "fast simulator of the simulator," generating random outputs significantly faster than the original simulator while preserving approximately equal conditional distributions. Generative metamodels enable rapid generation of numerous random outputs upon input specification, facilitating immediate computation of any summary statistic for real-time decision-making. We introduce a new algorithm, quantile-regression-based generative metamodeling (QRGMM), and establish its distributional convergence and convergence rate. Extensive numerical experiments demonstrate QRGMM's efficacy compared to other state-of-the-art generative algorithms in practical real-time decision-making scenarios.
Paper Structure (40 sections, 14 theorems, 48 equations, 10 figures, 7 tables, 2 algorithms)

This paper contains 40 sections, 14 theorems, 48 equations, 10 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Quantile-Regression-Based Generative Metamodeling
  4. Framework
  5. Linear Quantile Regression Model
  6. Convergence in Distribution
  7. Preliminaries
  8. Partition Scheme
  9. Establishing the Convergence in Distribution
  10. Discussion on the Choice of $m$
  11. Rate of Convergence in the Middle Interval
  12. Quantile Crossing
  13. Frequency of Quantile Crossing.
  14. Post-processing with Rearrangement.
  15. Extensions
  16. ...and 25 more sections

Key Result

Lemma 1

For any $\tau$, let $\tilde{\boldsymbol{\beta}}(\tau)$ denote the linear quantile regression estimator of $\boldsymbol{\beta}(\tau)$ by solving problem eq:optqr. Fix any interval $[\tau_{\mathsf{L}},\tau_{\mathsf{U}}]\subset(0,1)$ with $0<\tau_{\mathsf{L}}<\tau_{\mathsf{U}}<1$. Under Assumptions ass as $n\to\infty$, uniformly for $\tau\in[\tau_{\mathsf{L}},\tau_{\mathsf{U}}]$. Here, as $n\to\inft

Figures (10)

  • Figure 1: Cumulative Distribution Function and Quantile Function of $Y(\mathbf{x})$.
  • Figure 2: Quantile Crossing Frequency with $m=O(\sqrt{n})$.
  • Figure 3: Generated Samples Approximating Conditional Distribution $Y$ Given $\mathbf{x}^*$.
  • Figure 4: Generated Samples Approximating $Y$ Distribution: Wasserstein Distance (WD) and KS Statistic (KS).
  • Figure 5: Convergence of KS Statistic as $n$ Increases.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1: Bahadur Representation
  • Proposition 1
  • Lemma 2: Portmanteau Theorem
  • Lemma 3
  • Proposition 2
  • Lemma 4
  • Theorem 1
  • ...and 7 more