Improve Single-Point Zeroth-Order Optimization Using High-Pass and Low-Pass Filters

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.