  Single-point zeroth-order optimization (SZO) is useful in solving online
black-box optimization and control problems in time-varying environments, as it
queries the function value only once at each time step. However, the vanilla
SZO method is known to suffer from a large estimation variance and slow
convergence, which seriously limits its practical application. In this work, we
borrow the idea of high-pass and low-pass filters from extremum seeking control
(continuous-time version of SZO) and develop a novel SZO method called HLF-SZO
by integrating these filters. It turns out that the high-pass filter coincides
with the residual feedback method, and the low-pass filter can be interpreted
as the momentum method. As a result, the proposed HLF-SZO achieves a much
smaller variance and much faster convergence than the vanilla SZO method and
empirically outperforms the residual-feedback SZO method, which is verified via
extensive numerical experiments.
