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A Correlation-induced Finite Difference Estimator

Guo Liang, Guangwu Liu, Kun Zhang

TL;DR

This work tackles gradient estimation for stochastic black-box objectives using finite-difference methods. It introduces a bootstrap-based, sample-driven approach to estimate the unknown constants that govern the optimal perturbation, and couples this with a correlation-induced CFD estimator (Cor-CFD) that reuses pilot samples to create correlated diffs, achieving variance reduction and potential bias reduction. The authors provide consistency and asymptotic results, propose a practical implementation algorithm, and demonstrate effectiveness in derivative-free optimization, including high-dimensional problems. The combination of perturbation-constant estimation and correlated sample reuse offers a sample-efficient FD framework with strong theoretical guarantees and practical impact for DFO and simulation optimization.

Abstract

Finite difference (FD) approximation is a classic approach to stochastic gradient estimation when only noisy function realizations are available. In this paper, we first provide a sample-driven method via the bootstrap technique to estimate the optimal perturbation, and then propose an efficient FD estimator based on correlated samples at the estimated optimal perturbation. Furthermore, theoretical analyses of both the perturbation estimator and the FD estimator reveal that, {\it surprisingly}, the correlation enables the proposed FD estimator to achieve a reduction in variance and, in some cases, a decrease in bias compared to the traditional optimal FD estimator. Numerical results confirm the efficiency of our estimators and align well with the theory presented, especially in scenarios with small sample sizes. Finally, we apply the estimator to solve derivative-free optimization (DFO) problems, and numerical studies show that DFO problems with 100 dimensions can be effectively solved.

A Correlation-induced Finite Difference Estimator

TL;DR

This work tackles gradient estimation for stochastic black-box objectives using finite-difference methods. It introduces a bootstrap-based, sample-driven approach to estimate the unknown constants that govern the optimal perturbation, and couples this with a correlation-induced CFD estimator (Cor-CFD) that reuses pilot samples to create correlated diffs, achieving variance reduction and potential bias reduction. The authors provide consistency and asymptotic results, propose a practical implementation algorithm, and demonstrate effectiveness in derivative-free optimization, including high-dimensional problems. The combination of perturbation-constant estimation and correlated sample reuse offers a sample-efficient FD framework with strong theoretical guarantees and practical impact for DFO and simulation optimization.

Abstract

Finite difference (FD) approximation is a classic approach to stochastic gradient estimation when only noisy function realizations are available. In this paper, we first provide a sample-driven method via the bootstrap technique to estimate the optimal perturbation, and then propose an efficient FD estimator based on correlated samples at the estimated optimal perturbation. Furthermore, theoretical analyses of both the perturbation estimator and the FD estimator reveal that, {\it surprisingly}, the correlation enables the proposed FD estimator to achieve a reduction in variance and, in some cases, a decrease in bias compared to the traditional optimal FD estimator. Numerical results confirm the efficiency of our estimators and align well with the theory presented, especially in scenarios with small sample sizes. Finally, we apply the estimator to solve derivative-free optimization (DFO) problems, and numerical studies show that DFO problems with 100 dimensions can be effectively solved.
Paper Structure (39 sections, 9 theorems, 145 equations, 7 figures, 4 tables)

This paper contains 39 sections, 9 theorems, 145 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Literature Review
  4. Organization
  5. Background
  6. A Sample-driven Method for Constant Estimation
  7. Bootstrap Sample Mean and Variance
  8. Regression for Perturbation Selection
  9. Consistency of $\widehat{B}$
  10. Consistency of $\widehat{\beta}$$_v$
  11. The Optimal Perturbation for Pilot Samples
  12. The Correlation-induced FD Estimator
  13. Constructing the Estimator by Inducing Correlated Samples
  14. Asymptotic Analyses of the Cor-CFD Estimator
  15. Implementation
  16. ...and 24 more sections

Key Result

Theorem 1

Denote $\nu_4(h) = \mathbb{E}\left[Z_1(h)\right]^4$ and $\nu_4 = \underset{h\to 0}{\lim}\mathbb{E}\left[hZ_1(h)\right]^4$. If $\nu_4 < \infty$, then under Assumptions ass:continuity and ass:derivative5, as $n_b\to\infty$, where $D = \alpha^{(5)}(\theta_0)/120$ and $\phi(h)$ is a zero-mean error with

Figures (7)

  • Figure 1: Comparison of the estimates of the unknown constants and perturbation for case 1 in Example \ref{['exa:ksinx']}.
  • Figure 2: Comparison of the estimates of the unknown constants and perturbation for case 2 in Example \ref{['exa:ksinx']}.
  • Figure 3: Comparison of the MSE among different estimators for cases 1 and 2 in Example \ref{['exa:exa1']}.
  • Figure 4: Performances of the optimality gaps by different algorithms for Rosenbrock function w.r.t. different sample sizes.
  • Figure 5: Illustration of the MSE of different methods w.r.t. $r$ for cases 1 and 2 in Example \ref{['exa:ksinx']}.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Example 1
  • Example 2
  • Example 3: Queuing system
  • Corollary 6
  • Lemma 1
  • ...and 5 more