We introduce a new zeroth-order algorithm for private stochastic optimization on nonconvex and nonsmooth objectives. Given a dataset of size $M$, our algorithm ensures $(\alpha,\alpha\rho^2/2)$-R\'enyi differential privacy and finds a $(\delta,\epsilon)$-stationary point so long as $M=\tilde\Omega\left(\frac{d}{\delta\epsilon^3} + \frac{d^{3/2}}{\rho\delta\epsilon^2}\right)$. This matches the optimal complexity of its non-private zeroth-order analog. Notably, although the objective is not smooth, we have privacy ``for free'' whenever $\rho \ge \sqrt{d}\epsilon$.

翻译：本文提出了一种针对非凸非光滑目标函数的私有随机优化新零阶算法。给定规模为$M$的数据集，该算法确保$(\alpha,\alpha\rho^2/2)$-R\'enyi差分隐私，并在$M=\tilde\Omega\left(\frac{d}{\delta\epsilon^3} + \frac{d^{3/2}}{\rho\delta\epsilon^2}\right)$条件下找到$(\delta,\epsilon)$-稳定点。这与非私有零阶算法的最优复杂度相匹配。值得注意的是，尽管目标函数非光滑，但当$\rho \ge \sqrt{d}\epsilon$时，我们实现了"免费"的隐私保护。