Enhanced Derivative-Free Optimization Using Adaptive Correlation-Induced Finite Difference Estimators

ArXiv ID: 2502.20819 “View on arXiv”

Authors: Unknown

Abstract

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.

Keywords: Derivative-free Optimization, Stochastic Approximation, Finite Difference, Gradient Estimation, Algorithm Design

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper features advanced mathematics including stochastic approximation theory, convergence proofs, and complex derivations for a new finite difference estimator, justifying a high math score. It includes comprehensive numerical experiments on benchmark DFO problems, but lacks the extensive backtesting on real-world financial data or implementation details typically seen in Street Traders, placing it in Holy Grail.

  flowchart TD
    subgraph A ["Research Goal"]
        Goal["“Enhance Derivative-Free Optimization<br>(DFO) efficiency”"]
    end

    subgraph B ["Key Methodology"]
        B1["“Correlation-Induced<br>Finite Difference Estimate”"]
        B2["“Adaptive<br>Sampling Strategy”"]
        B3["“Stochastic<br>Line Search”"]
    end

    subgraph C ["Computational Process"]
        C1["“Batch-based Gradient Estimation<br>(Dynamic Sample Size)”"]
        C2["“Adaptive Step Size Tuning”"]
    end

    subgraph D ["Outcomes"]
        D1["“Consistency & Convergence<br>(Same rate as KW/SPSA)”"]
        D2["“Superior Empirical Performance”"]
    end

    Goal --> B1
    Goal --> B2
    Goal --> B3
    
    B1 --> C1
    B2 --> C1
    B3 --> C2
    
    C1 --> D1
    C2 --> D2
    C1 --> D2