Generalized Random Direction Newton Algorithms for Stochastic Optimization

We present a finite-time analysis of two smoothed functional stochastic approximation algorithms for simulation-based optimization. The first is a two time-scale gradient-based method, while the second is a three time-scale Newton-based algorithm that estimates both the gradient and the Hessian of the objective function $J$. Both algorithms involve zeroth order estimates for the gradient/Hessian. Although the asymptotic convergence of these algorithms has been established in prior work, finite-time guarantees of two-timescale stochastic optimization algorithms in zeroth order settings have not been provided previously. For our Newton algorithm, we derive mean-squared error bounds for the Hessian estimator and establish a finite-time bound on $\min\limits_{0 \le m \le T} \mathbb{E}\| \nabla J(θ(m)) \|^2$, showing convergence to first-order stationary points. The analysis explicitly characterizes the interaction between multiple time-scales and the propagation of estimation errors. We further identify step-size choices that balance dominant error terms and achieve near-optimal convergence rates. We also provide corresponding finite-time guarantees for the gradient algorithm under the same framework. The theoretical results are further validated through experiments on the Continuous Mountain Car environment.

We study online inference and asymptotic covariance estimation for the stochastic gradient descent (SGD) algorithm. While classical methods (such as plug-in and batch-means estimators) are available, they either require inaccessible second-order (Hessian) information or suffer from slow convergence. To address these challenges, we propose a novel, fully online de-biased covariance estimator that eliminates the need for second-order derivatives while significantly improving estimation accuracy. Our method employs a bias-reduction technique to achieve a convergence rate of $n^{(α-1)/2} \sqrt{\log n}$, outperforming existing Hessian-free alternatives.

Accurate Zeroth-Order (ZO) Hessian estimation is a cornerstone of derivative-free methods, essential for tasks such as bilevel optimization, Bayesian inference, and uncertainty quantification. However, obtaining a complete suite of low-variance estimators for the Hessian and its inverse in high-dimensional settings remains a significant challenge. To address this, we propose a unified framework that reinterprets ZO Hessian approximation through the lens of single-step Policy Optimization (PO). This perspective establishes a theoretical equivalence between general ZO Hessian estimators and the Hessian of a smoothed PO objective, unifying distinct classical randomized estimators as specific instances of baseline selection. Building on this foundation, we introduce ZoVH, a comprehensive suite of variance-reduced estimators for the full Hessian matrix, its regularized inverse, and the bias-corrected inverse Hessian-gradient product. ZoVH leverages two key techniques: (1) a unique optimal baseline derived to provably minimize variance, and (2) a query reuse strategy that incorporates historical function queries to enhance sample efficiency without inflating costs. Our rigorous theoretical analysis confirms the unbiasedness of the Hessian estimator, validates the variance optimality of our baseline, provides error bounds for the entire ZoVH suite, and establishes convergence guarantees for the resulting curvature-aware ZO algorithm. Extensive empirical results validate our theoretical findings, demonstrating that ZoVH achieves superior estimation accuracy and convergence performance in real-world applications. Code is available at https://github.com/Qjbtiger/ZoVH