Stochastic zeroth-order gradient and Hessian estimators: variance reduction and refined bias bounds

Author:

Feng Yasong1,Wang Tianyu12

Affiliation:

1. Shanghai Center for Mathematical Sciences , Fudan University, Shanghai 200433, China

2. Shanghai Artificial Intelligence Laboratory , Shanghai 200232, China

Abstract

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 $ \varTheta \left ( k \right ) $ random directions sampled from the Stiefel manifold $ \text{St} (n,k) $ and finite-difference granularity $\delta $, the variance of the gradient estimator is bounded by $ \mathscr{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 $\mathscr{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 $\mathscr{O} \big( \delta ^2 \varGamma \big )$, where $\delta $ is the finite-difference granularity, and $ \varGamma $ depends on high-order derivatives of $f$. Our results are evidenced by empirical observations.

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Theory and Mathematics,Numerical Analysis,Statistics and Probability,Analysis

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