Private Zeroth-Order Nonsmooth Nonconvex Optimization

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 $M=\tilde\Omega\left(\frac{d}{\delta\epsilon^3} + \frac{d^{{3/2}}{\rho\delta\epsilon2}\right)$.} 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 $\rho \ge \sqrt{d}\epsilon$.