An Algorithm with Optimal Dimension-Dependence for Zero-Order Nonsmooth Nonconvex Stochastic Optimization
This solves an open question in stochastic optimization for researchers, offering optimal dimension-dependence and convergence rates, though it is incremental as it builds on prior first-order advancements.
The paper tackles the problem of optimizing nonsmooth nonconvex stochastic functions using only noisy zero-order evaluations, refuting a prior conjecture by providing an algorithm with complexity O(dδ^{-1}ε^{-3}), which is optimal in dimension and accuracy parameters, and shows nonsmooth optimization is as easy as smooth in this setting.
We study the complexity of producing $(δ,ε)$-stationary points of Lipschitz objectives which are possibly neither smooth nor convex, using only noisy function evaluations. Recent works proposed several stochastic zero-order algorithms that solve this task, all of which suffer from a dimension-dependence of $Ω(d^{3/2})$ where $d$ is the dimension of the problem, which was conjectured to be optimal. We refute this conjecture by providing a faster algorithm that has complexity $O(dδ^{-1}ε^{-3})$, which is optimal (up to numerical constants) with respect to $d$ and also optimal with respect to the accuracy parameters $δ,ε$, thus solving an open question due to Lin et al. (NeurIPS'22). Moreover, the convergence rate achieved by our algorithm is also optimal for smooth objectives, proving that in the nonconvex stochastic zero-order setting, nonsmooth optimization is as easy as smooth optimization. We provide algorithms that achieve the aforementioned convergence rate in expectation as well as with high probability. Our analysis is based on a simple yet powerful lemma regarding the Goldstein-subdifferential set, which allows utilizing recent advancements in first-order nonsmooth nonconvex optimization.