Stochastic Zeroth Order Gradient and Hessian Estimators: Variance Reduction and Refined Bias Bounds

TL;DR

This paper introduces variance-reduced stochastic zeroth order gradient and Hessian estimators using random orthogonal directions, providing tighter bounds and improved bias analysis, with empirical validation.

Contribution

It proposes novel variance reduction techniques for zeroth order estimators using random orthogonal directions and refines bias bounds for smooth functions.

Findings

Variance of estimators decreases significantly with the number of directions.

When k=n, variances of estimators become negligibly small.

Empirical results support theoretical variance and bias bounds.

Abstract

We study stochastic zeroth order gradient and Hessian estimators for real-valued functions in $\mathbb{R}^n$. We show that, via taking finite difference along random orthogonal directions, the variance of the stochastic finite difference estimators can be significantly reduced. In particular, we design estimators for smooth functions such that, if one uses $ \Theta \left( k \right) $ random directions sampled from the Stiefel's manifold $ \text{St} (n,k) $ and finite-difference granularity $\delta$, the variance of the gradient estimator is bounded by $ \mathcal{O} \left( \left( \frac{n}{k} - 1 \right) + \left( \frac{n^2}{k} - n \right) \delta^2 + \frac{ n^2 \delta^4 }{ k } \right) $, and the variance of the Hessian estimator is bounded by $\mathcal{O} \left( \left( \frac{n^2}{k^2} - 1 \right) + \left( \frac{n^4}{k^2} - n^2 \right) \delta^2 + \frac{n^4 \delta^4 }{k^2} \right) $. When $k = n$, the variances become negligibly small. In addition, we provide improved bias bounds for the estimators. The bias of both gradient and Hessian estimators for smooth function $f$ is of order $\mathcal{O} \left( \delta^2 \Gamma \right)$, where $\delta$ is the finite-difference granularity, and $ \Gamma $ depends on high order derivatives of $f$. Our results are evidenced by empirical observations.