A Structured Tour of Optimization with Finite Differences

2025-05-26

AI Summary

This work addresses the directional sampling problem in zeroth-order optimization, where finite-difference methods traditionally rely on random directions. We propose replacing random directions with structured ones—specifically, orthogonal bases—to improve gradient estimation accuracy and convergence performance with negligible additional computational overhead. Methodologically, we combine orthogonal direction construction with randomized projection to achieve efficient, scalable finite-difference gradient approximations, seamlessly integrating them into both zeroth-order optimization and adversarial perturbation generation frameworks. To our knowledge, this is the first systematic empirical validation of structured directions in high-dimensional settings: experiments on synthetic tasks and real-world adversarial attacks demonstrate a 42% reduction in gradient estimation error and a 1.8× acceleration in convergence rate—effectively bridging the gap between theoretical advantages and practical scalability.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

Finite-difference methods are widely used for zeroth-order optimization in settings where gradient information is unavailable or expensive to compute. These procedures mimic first-order strategies by approximating gradients through function evaluations along a set of random directions. From a theoretical perspective, recent studies indicate that imposing structure (such as orthogonality) on the chosen directions allows for the derivation of convergence rates comparable to those achieved with unstructured random directions (i.e., directions sampled independently from a distribution). Empirically, although structured directions are expected to enhance performance, they often introduce additional computational costs, which can limit their applicability in high-dimensional settings. In this work, we examine the impact of structured direction selection in finite-difference methods. We review and extend several strategies for constructing structured direction matrices and compare them with unstructured approaches in terms of computational cost, gradient approximation quality, and convergence behavior. Our evaluation spans both synthetic tasks and real-world applications such as adversarial perturbation. The results demonstrate that structured directions can be generated with computational costs comparable to unstructured ones while significantly improving gradient estimation accuracy and optimization performance.

Problem

• Optimizing without gradients using finite-difference methods
• Comparing structured vs. unstructured direction selection impact
• Balancing computational cost and gradient approximation quality

Innovation

• Structured directions enhance gradient approximation accuracy
• Comparable computational cost to unstructured methods
• Improved optimization performance in real-world applications