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Generalizing the Generalized Likelihood Ratio Method Through a Push-Out Leibniz Integration Approach

Xingyu Ren, Michael C. Fu

TL;DR

This work addresses gradient estimation of an expected performance ${\mathbb E}[\psi(X,\theta)]$ when the natural change of variables yields a parameter-dependent sample space or discontinuous $\psi$. It introduces a push-out GLR framework based on the Leibniz integral rule, which differentiates both the integrand and the moving domain, producing a decomposition into a density-driven term and a surface-term arising from the boundary, with a local-invertibility extension. The proposed estimator generalizes existing GLR methods, applies to a broader class of discontinuous performances, and can reduce variance at the cost of additional simulations, as illustrated by a toy simulation where the surface term reduces to a simple constant in a simple case. The approach promises simpler regularity requirements, links to perturbation-analysis concepts like SPA, and potential applicability to stochastic optimization problems with parameter-dependent domains.

Abstract

We generalize the generalized likelihood ratio (GLR) method through a novel push-out Leibniz integration approach. Extending the conventional push-out likelihood ratio (LR) method, our approach allows the sample space to be parameter-dependent after the change of variables. Specifically, leveraging the Leibniz integral rule enables differentiation of the parameter-dependent sample space, resulting in a surface integral in addition to the usual LR estimator, which may necessitate additional simulation. Furthermore, our approach extends to cases where the change of variables only locally exists. Notably, the derived estimator includes existing GLR estimators as special cases and is applicable to a broader class of discontinuous sample performances. Moreover, the derivation is streamlined and more straightforward, and the requisite regularity conditions are easier to understand and verify.

Generalizing the Generalized Likelihood Ratio Method Through a Push-Out Leibniz Integration Approach

TL;DR

This work addresses gradient estimation of an expected performance when the natural change of variables yields a parameter-dependent sample space or discontinuous . It introduces a push-out GLR framework based on the Leibniz integral rule, which differentiates both the integrand and the moving domain, producing a decomposition into a density-driven term and a surface-term arising from the boundary, with a local-invertibility extension. The proposed estimator generalizes existing GLR methods, applies to a broader class of discontinuous performances, and can reduce variance at the cost of additional simulations, as illustrated by a toy simulation where the surface term reduces to a simple constant in a simple case. The approach promises simpler regularity requirements, links to perturbation-analysis concepts like SPA, and potential applicability to stochastic optimization problems with parameter-dependent domains.

Abstract

We generalize the generalized likelihood ratio (GLR) method through a novel push-out Leibniz integration approach. Extending the conventional push-out likelihood ratio (LR) method, our approach allows the sample space to be parameter-dependent after the change of variables. Specifically, leveraging the Leibniz integral rule enables differentiation of the parameter-dependent sample space, resulting in a surface integral in addition to the usual LR estimator, which may necessitate additional simulation. Furthermore, our approach extends to cases where the change of variables only locally exists. Notably, the derived estimator includes existing GLR estimators as special cases and is applicable to a broader class of discontinuous sample performances. Moreover, the derivation is streamlined and more straightforward, and the requisite regularity conditions are easier to understand and verify.
Paper Structure (5 sections, 5 theorems, 53 equations, 2 figures)

This paper contains 5 sections, 5 theorems, 53 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Integrating the Leibniz integral rule with the push-out LR method
  3. Local change of variables via the inverse function theorem
  4. Simulation example
  5. Conclusion

Key Result

Theorem 1

Let $D_\theta\subset{\mathbb{R}}^n$ be a compact set. Suppose that there exists a function $\phi(\cdot,\cdot):U\times\Theta\mapsto{\mathbb{R}}^n$, where $U\subset {\mathbb{R}}^n$ is a fixed domain, such that $D_\theta=\phi(U,\theta)$. Suppose $\phi(\cdot,\cdot):{\mathbb{R}}^n\times\Theta\mapsto{\mat where $\text{div}$ is the divergence operator, i.e., $\text{div}(F)=\sum_{i=1}^n \partial_{x_i}F_i,

Figures (2)

  • Figure 1: The original domain $U_\theta$ and the perturbed domain $U_{\theta+\Delta\theta}$.
  • Figure 2: Simulation results: Point estimates and standard errors for $\frac{d}{d\theta}{\mathbb{E}}({\mathbf{1}}\{X<\theta\})$.

Theorems & Definitions (6)

  • Theorem 1
  • Proposition 1
  • Definition 1
  • Lemma 1
  • Proposition 2
  • Theorem 2