A gradient sampling method with complexity guarantees for Lipschitz functions in high and low dimensions

Zhang et al. introduced a novel modification of Goldstein's classical subgradient method, with an efficiency guarantee of $O(\varepsilon^{-4})$ for minimizing Lipschitz functions. Their work, however, makes use of a nonstandard subgradient oracle model and requires the function to be directionally differentiable. In this paper, we show that both of these assumptions can be dropped by simply adding a small random perturbation in each step of their algorithm. The resulting method works on any Lipschitz function whose value and gradient can be evaluated at points of differentiability. We additionally present a new cutting plane algorithm that achieves better efficiency in low dimensions: $O(d\varepsilon^{-3})$ for Lipschitz functions and $O(d\varepsilon^{-2})$ for those that are weakly convex.