  We study differentially private (DP) algorithms for smooth stochastic minimax
optimization, with stochastic minimization as a byproduct. The holy grail of
these settings is to guarantee the optimal trade-off between the privacy and
the excess population loss, using an algorithm with a linear time-complexity in
the number of training samples. We provide a general framework for solving
differentially private stochastic minimax optimization (DP-SMO) problems, which
enables the practitioners to bring their own base optimization algorithm and
use it as a black-box to obtain the near-optimal privacy-loss trade-off. Our
framework is inspired from the recently proposed Phased-ERM method [22] for
nonsmooth differentially private stochastic convex optimization (DP-SCO), which
exploits the stability of the empirical risk minimization (ERM) for the privacy
guarantee. The flexibility of our approach enables us to sidestep the
requirement that the base algorithm needs to have bounded sensitivity, and
allows the use of sophisticated variance-reduced accelerated methods to achieve
near-linear time-complexity. To the best of our knowledge, these are the first
near-linear time algorithms with near-optimal guarantees on the population
duality gap for smooth DP-SMO, when the objective is
(strongly-)convex--(strongly-)concave. Additionally, based on our flexible
framework, we enrich the family of near-linear time algorithms for smooth
DP-SCO with the near-optimal privacy-loss trade-off.
