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 $(\alpha,\alpha\rho^2/2)$-Renyi differential privacy and finds a $(\delta,\epsilon)$-stationary point so long as $M=\tilde\Omega(\frac{d}{\delta\epsilon^3} + \frac{d^{3/2}}{\rho\delta\epsilon^2})$. This matches the optimal complexity found in its non-private zeroth-order analog. Notably, although the objective is not smooth, we have privacy ``for free'' when $\rho \ge \sqrt{d}\epsilon$.