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

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