Subdifferentially polynomially bounded functions and Gaussian smoothing-based zeroth-order optimization

We study the class of subdifferentially polynomially bounded (SPB) functions, which is a rich class of locally Lipschitz functions that encompasses all Lipschitz functions, all gradient- or Hessian-Lipschitz functions, and even some non-smooth locally Lipschitz functions. We show that SPB functions are compatible with Gaussian smoothing (GS), in the sense that the GS of any SPB function is well-defined and satisfies a descent lemma akin to gradient-Lipschitz functions, with the Lipschitz constant replaced by a polynomial function. Leveraging this descent lemma, we propose GS-based zeroth-order optimization algorithms with an adaptive stepsize strategy for minimizing SPB functions, and analyze their convergence rates with respect to both relative and absolute stationarity measures. Finally, we also establish the iteration complexity for achieving a $(\delta, \epsilon)$-approximate stationary point, based on a novel quantification of Goldstein stationarity via the GS gradient that could be of independent interest.