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    This study introduces Hessian-aided random perturbation (HARP) for stochastic optimization with noisy zeroth-order oracles. HARP improves gradient estimation by using anisotropic covariance, enhancing efficiency and handling ill-conditioning.

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    Area of Science:

    • Optimization Theory
    • Machine Learning
    • Numerical Analysis

    Background:

    • Stochastic optimization problems often rely on noisy zeroth-order (ZO) oracles for gradient estimation.
    • Traditional methods like Kiefer-Wolfowitz use isotropic random perturbations, limiting their effectiveness in certain scenarios.

    Purpose of the Study:

    • To develop a novel gradient estimation technique for stochastic optimization using ZO oracles.
    • To enhance robustness against ill-conditioning and improve query efficiency compared to existing methods.

    Main Methods:

    • Proposing Hessian-aided random perturbation (HARP), which utilizes anisotropic covariance matrices informed by ZO oracle history.
    • Incorporating second-order information into the perturbation's covariance matrix for improved gradient and Hessian estimation.
    • Analyzing convergence properties and deriving convergence rates under standard assumptions.

    Main Results:

    • Demonstrated almost-sure convergence of the HARP algorithm.
    • Established theoretical convergence rates for the proposed method.
    • Numerical experiments confirm HARP's superior performance in handling ill-conditioned problems and its query efficiency.

    Conclusions:

    • HARP offers a significant advancement in gradient estimation for stochastic optimization with noisy ZO oracles.
    • The method's ability to leverage anisotropic perturbations provides theoretical and practical advantages over isotropic approaches.