Near-Oracle Robustification of Finite-Difference Stochastic Gradient Estimators via Cheap Pilot Calibration

2026-07-26

AI Summary

In black-box optimization where only noisy function evaluations are available, finite-difference gradient estimators suffer from unstable performance due to their reliance on perturbation step sizes that depend on unknown model constants. This work proposes a pilot calibration mechanism requiring minimal simulation budget to adaptively estimate these critical parameters and set the perturbation step size accordingly. For the first time, it is theoretically established that this strategy enables finite-difference estimators to achieve the same first-order mean squared error as if the optimal step size were known a priori. The approach is broadly applicable across diverse objective functions and difference schemes, and numerical experiments demonstrate its robustness and near-oracle optimality, significantly outperforming conventional methods that require manual tuning.

Abstract

We study stochastic gradient estimation in black-box environments where only noisy simulation observations of function values are available. Finite-difference (FD) methods are among the most widely used zeroth-order gradient estimators in such settings, by measuring the change in function values against a perturbation size. While the optimal order in choosing this perturbation size with respect to the simulation budget is well understood, the optimal constant factor relies on model characteristics that are typically unknown and viewed to be as difficult to estimate as the gradient itself. Consequently, FD estimators are often based on ad hoc tuning of the perturbation size, which may exhibit highly unstable performance across problem instances. In this paper, we challenge this conventional wisdom from both theoretical and practical perspectives. We show that, by pilot-estimating these model quantities using a negligible fraction of the simulation budget, substantial robustness is attained in the resulting FD estimators. Theoretically, we show that using a perturbation size governed by this pilot estimation can already achieve an MSE that is first-order identical to the ``oracle" MSE as if the optimal perturbation size is known in advance. Moreover, we show how such an approach is competitive against any choices of prescribed perturbation size, even if they are designed to be minimax-optimal over reasonable classes of target functions and FD schemes. Our proposed pilot estimation is practically easy to run, and a variety of numerical experiments demonstrate both the robustness and near-oracle optimality of our estimator relative to conventional FD schemes based on ad hoc tuning.

Problem

• stochastic gradient estimation
• finite-difference
• black-box optimization
• perturbation size
• zeroth-order methods

Innovation

• finite-difference
• stochastic gradient estimation
• pilot calibration
• zeroth-order optimization
• robustness