Information-Theoretic Lower Bounds for Zero-Order Stochastic Gradient Estimation
This work addresses the fundamental limits of gradient estimation in optimization and machine learning, highlighting a gap for developing more efficient methods.
The paper establishes an information-theoretic lower bound of Ω(√(d/T)) for zero-order stochastic gradient estimation and shows that the finite difference method achieves a rate of O(d^{4/3}/√T) under specific conditions, indicating it is not minimax optimal.
In this paper we analyze the necessary number of samples to estimate the gradient of any multidimensional smooth (possibly non-convex) function in a zero-order stochastic oracle model. In this model, an estimator has access to noisy values of the function, in order to produce the estimate of the gradient. We also provide an analysis on the sufficient number of samples for the finite difference method, a classical technique in numerical linear algebra. For $T$ samples and $d$ dimensions, our information-theoretic lower bound is $Ω(\sqrt{d/T})$. We show that the finite difference method for a bounded-variance oracle has rate $O(d^{4/3}/\sqrt{T})$ for functions with zero third and higher order derivatives. These rates are tight for Gaussian oracles. Thus, the finite difference method is not minimax optimal, and therefore there is space for the development of better gradient estimation methods.