Enhanced Derivative-Free Optimization Using Adaptive Correlation-Induced Finite Difference Estimators
This work addresses efficiency and convergence issues in derivative-free optimization, which is incremental as it builds on existing methods like Kiefer-Wolfowitz and SPSA.
The paper tackles the problem of imprecise gradient estimates in derivative-free optimization by proposing an adaptive correlation-induced finite difference estimator and a stochastic line search technique, achieving the same convergence rate as traditional methods while demonstrating superior empirical performance in numerical experiments.
Gradient-based methods are well-suited for derivative-free optimization (DFO), where finite-difference (FD) estimates are commonly used as gradient surrogates. Traditional stochastic approximation methods, such as Kiefer-Wolfowitz (KW) and simultaneous perturbation stochastic approximation (SPSA), typically utilize only two samples per iteration, resulting in imprecise gradient estimates and necessitating diminishing step sizes for convergence. In this paper, we first explore an efficient FD estimate, referred to as correlation-induced FD estimate, which is a batch-based estimate. Then, we propose an adaptive sampling strategy that dynamically determines the batch size at each iteration. By combining these two components, we develop an algorithm designed to enhance DFO in terms of both gradient estimation efficiency and sample efficiency. Furthermore, we establish the consistency of our proposed algorithm and demonstrate that, despite using a batch of samples per iteration, it achieves the same convergence rate as the KW and SPSA methods. Additionally, we propose a novel stochastic line search technique to adaptively tune the step size in practice. Finally, comprehensive numerical experiments confirm the superior empirical performance of the proposed algorithm.