Nonconvex Zeroth-Order Stochastic ADMM Methods with Lower Function Query Complexity

Zeroth-order (gradient-free) method is a class of powerful optimization tool for many machine learning problems because it only needs function values (not gradient) in the optimization. In particular, zeroth-order method is very suitable for many complex problems such as black-box attacks and bandit feedback, whose explicit gradients are difficult or infeasible to obtain. Recently, although many zeroth-order methods have been developed, these approaches still exist two main drawbacks: 1) high function query complexity; 2) not being well suitable for solving the problems with complex penalties and constraints. To address these challenging drawbacks, in this paper, we propose a novel fast zeroth-order stochastic alternating direction method of multipliers (ADMM) method (\emph{i.e.}, ZO-SPIDER-ADMM) with lower function query complexity for solving nonconvex problems with multiple nonsmooth penalties. Moreover, we prove that our ZO-SPIDER-ADMM has the optimal function query complexity of $O(dn + dn^{\frac{1}{2}}\epsilon^{-1})$ for finding an $\epsilon$-approximate local solution, where $n$ and $d$ denote the sample size and dimension of data, respectively. In particular, the ZO-SPIDER-ADMM improves the existing best nonconvex zeroth-order ADMM methods by a factor of $O(d^{\frac{1}{3}}n^{\frac{1}{6}})$. Moreover, we propose a fast online ZO-SPIDER-ADMM (\emph{i.e.,} ZOO-SPIDER-ADMM). Our theoretical analysis shows that the ZOO-SPIDER-ADMM has the function query complexity of $O(d\epsilon^{-\frac{3}{2}})$, which improves the existing best result by a factor of $O(\epsilon^{-\frac{1}{2}})$. Finally, we utilize a task of structured adversarial attack on black-box deep neural networks to demonstrate the efficiency of our algorithms.