Small Errors in Random Zeroth-Order Optimization Are Imaginary

SIAM Journal on Optimization, Volume 34, Issue 3, Page 2638-2670, September 2024.
Abstract. Most zeroth-order optimization algorithms mimic a first-order algorithm but replace the gradient of the objective function with some gradient estimator that can be computed from a small number of function evaluations. This estimator is constructed randomly, and its expectation matches the gradient of a smooth approximation of the objective function whose quality improves as the underlying smoothing parameter [math] is reduced. Gradient estimators requiring a smaller number of function evaluations are preferable from a computational point of view. While estimators based on a single function evaluation can be obtained by use of the divergence theorem from vector calculus, their variance explodes as [math] tends to 0. Estimators based on multiple function evaluations, on the other hand, suffer from numerical cancellation when [math] tends to 0. To combat both effects simultaneously, we extend the objective function to the complex domain and construct a gradient estimator that evaluates the objective at a complex point whose coordinates have small imaginary parts of the order [math]. As this estimator requires only one function evaluation, it is immune to cancellation. In addition, its variance remains bounded as [math] tends to 0. We prove that zeroth-order algorithms that use our estimator offer the same theoretical convergence guarantees as the state-of-the-art methods. Numerical experiments suggest, however, that they often converge faster in practice.