In stochastic optimization problems where only noisy zeroth-order (ZO) oracles are available, the Kiefer-Wolfowitz algorithm and its randomized counterparts are widely used as gradient estimators. Existing algorithms generate the random perturbations from certain distributions with a zero mean and an isotropic (either identity or scalar) covariance matrix. In contrast, this work considers the generalization where the perturbations may have an anisotropic covariance based on the ZO oracle history. We propose to feed the second-order approximation into the covariance matrix of the random perturbation, so it is dubbed as Hessian-aided random perturbation (HARP). HARP collects two or more (depending on the specific estimator form) ZO oracle calls per iteration to construct the gradient and the Hessian estimators. We prove HARP’s almost-surely convergence and derive its convergence rate under standard assumptions. We demonstrate, with theoretical guarantees and numerical experiments, that HARP is less sensitive to ill-conditioning and more query-efficient than other gradient approximation schemes whose random perturbations have an isotropic covariance.

中文翻译：

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使用嘈杂零阶预言的 Hessian 辅助随机扰动 (HARP)


在只有噪声零阶 (ZO) 预言可用的随机优化问题中，Kiefer-Wolfowitz 算法及其随机算法被广泛用作梯度估计器。现有算法从具有零均值和各向同性（单位或标量）协方差矩阵的某些分布生成随机扰动。相比之下，这项工作考虑了基于 ZO 预言历史的扰动可能具有各向异性协方差的概括。我们建议将二阶近似输入到随机扰动的协方差矩阵中，因此它被称为 Hessian 辅助随机扰动（HARP）。 HARP 在每次迭代中收集两个或多个（取决于具体估计器形式）ZO 预言机调用来构造梯度和 Hessian 估计器。我们证明了 HARP 几乎肯定收敛，并在标准假设下推导出其收敛率。我们通过理论保证和数值实验证明，与随机扰动具有各向同性协方差的其他梯度近似方案相比，HARP 对病态条件不太敏感，并且查询效率更高。