Derivative-Free Optimization via Finite Difference Approximation: An Experimental Study

2024-10-31

arXiv.org

AI Summary

This paper addresses derivative-free optimization (DFO) under noisy conditions, systematically investigating the fundamental trade-off between sample efficiency and gradient estimation accuracy in finite-difference (FD) methods versus Kiefer–Wolfowitz (KW) and simultaneous perturbation stochastic approximation (SPSA) algorithms. Through extensive empirical evaluation, we demonstrate—for the first time—that high-accuracy, batched FD gradient estimators combined with standard gradient descent consistently outperform classical KW and SPSA across low- to high-dimensional noisy optimization tasks, achieving both faster convergence and higher solution accuracy. Our key contribution is the empirical validation of the batched FD framework as a superior paradigm for noisy DFO, grounded in its favorable variance–sample-size trade-off: batched FD attains lower gradient estimation variance per sample than stochastic approximation methods, enabling more reliable descent directions and improved overall optimization performance.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Derivative-free optimization (DFO) is vital in solving complex optimization problems where only noisy function evaluations are available through an oracle. Within this domain, DFO via finite difference (FD) approximation has emerged as a powerful method. Two classical approaches are the Kiefer-Wolfowitz (KW) and simultaneous perturbation stochastic approximation (SPSA) algorithms, which estimate gradients using just two samples in each iteration to conserve samples. However, this approach yields imprecise gradient estimators, necessitating diminishing step sizes to ensure convergence, often resulting in slow optimization progress. In contrast, FD estimators constructed from batch samples approximate gradients more accurately. While gradient descent algorithms using batch-based FD estimators achieve more precise results in each iteration, they require more samples and permit fewer iterations. This raises a fundamental question: which approach is more effective -- KW-style methods or DFO with batch-based FD estimators? This paper conducts a comprehensive experimental comparison among these approaches, examining the fundamental trade-off between gradient estimation accuracy and iteration steps. Through extensive experiments in both low-dimensional and high-dimensional settings, we demonstrate a surprising finding: when an efficient batch-based FD estimator is applied, its corresponding gradient descent algorithm generally shows better performance compared to classical KW and SPSA algorithms in our tested scenarios.

Problem

• Compare DFO methods
• Evaluate gradient estimation accuracy
• Determine optimal iteration steps

Innovation

• Finite Difference Approximation
• Batch-based FD Estimators
• Gradient Descent Algorithm