Revisiting Randomized Smoothing: Nonsmooth Nonconvex Optimization Beyond Global Lipschitz Continuity

Abstract: Randomized smoothing is a widely adopted technique for optimizing nonsmooth objective functions. However, its efficiency analysis typically relies on global Lipschitz continuity, a condition rarely met in practical applications. To address this limitation, we introduce a new subgradient growth condition that naturally encompasses a wide range of locally Lipschitz functions, with the classical global Lipschitz function as a special case. Under this milder condition, we prove that randomized smoothing yields a differentiable function that satisfies certain generalized smoothness properties. To optimize such functions, we propose novel randomized smoothing gradient algorithms that, with high probability, converge to $(\delta, \epsilon)$-Goldstein stationary points and achieve a sample complexity of $\tilde{\mathcal{O}}(d^{5/2}\delta^{-1}\epsilon^{-4})$. By incorporating variance reduction techniques, we further improve the sample complexity to $\tilde{\mathcal{O}}(d^{3/2}\delta^{-1}\epsilon^{-3})$, matching the optimal $\epsilon$-bound under the global Lipschitz assumption, up to a logarithmic factor. Experimental results validate the effectiveness of our proposed algorithms.