A gradient sampling method with complexity guarantees for general Lipschitz functions

Zhang et al. [9] proposed a novel modification of Goldstein’s classical subgradient method for minimizing Lipschitz continuous functions, which has finite-time efficiency guarantees. This work, however, makes use of a nonstandard subgradient oracle model and requires the function to be directionally differentiable. In this note, 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 differentiablity.