Spherical-Design Oracles for Zeroth-Order Optimisation: Deterministic, Variance-Free Gradient Estimation
Abstract
In this work, we develop a zeroth-order (ZO) method that estimates the gradient with minimal bias and variance. Instead of the random directions often used in the ZO literature, we consider central differences with directions given by a fixed spherical t-design, a finite point set that integrates every polynomial of degree at most t exactly. The oracle is deterministic and thus does not induce any variance.We also propose a general finite difference method in the higher-order-smoothness regime and show the bias of both finite difference methods. We then show the convergence of the proposed ZO method via the input-to-state stability concept. Numerical experiments show the superiority of our proposed method over the existing random ZO methods.
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BibTeX
@misc{farzin2026sphericaldesign,
title = {{Spherical-Design Oracles for Zeroth-Order Optimisation: Deterministic, Variance-Free Gradient Estimation}},
author = {Amir Ali Farzin and Yuen-Man Pun and Philipp Braun and Iman Shames},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24373},
eprint = {2610.24373},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24373},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spherical-Design Oracles for Zeroth-Order Optimisation: Deterministic, Variance-Free Gradient Estimation
AU - Amir Ali Farzin
AU - Yuen-Man Pun
AU - Philipp Braun
AU - Iman Shames
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24373
UR - https://arxiv.org/abs/2610.24373
AB - In this work, we develop a zeroth-order (ZO) method that estimates the gradient with minimal bias and variance. Instead of the random directions often used in the ZO literature, we consider central differences with directions given by a fixed spherical t-design, a finite point set that integrates every polynomial of degree at most t exactly. The oracle is deterministic and thus does not induce any variance.We also propose a general finite difference method in the higher-order-smoothness regime and show the bias of both finite difference methods. We then show the convergence of the proposed ZO method via the input-to-state stability concept. Numerical experiments show the superiority of our proposed method over the existing random ZO methods.
ER -