Lazy Queries Can Reduce Variance in Zeroth-order Optimization

A major challenge of applying zeroth-order (ZO) methods is the high query complexity, especially when queries are costly. We propose a novel gradient estimation technique for ZO methods based on adaptive lazy queries that we term as LAZO. Different from the classic one-point or two-point gradient estimation methods, LAZO develops two alternative ways to check the usefulness of old queries from previous iterations, and then adaptively reuses them to construct the low-variance gradient estimates. We rigorously establish that through judiciously reusing the old queries, LAZO can reduce the variance of stochastic gradient estimates so that it not only saves queries per iteration but also achieves the regret bound for the symmetric two-point method. We evaluate the numerical performance of LAZO, and demonstrate the low-variance property and the performance gain of LAZO in both regret and query complexity relative to several existing ZO methods. The idea of LAZO is general, and can be applied to other variants of ZO methods.