  We propose a general optimization-based framework for computing
differentially private M-estimators and a new method for constructing
differentially private confidence regions. Firstly, we show that robust
statistics can be used in conjunction with noisy gradient descent or noisy
Newton methods in order to obtain optimal private estimators with global linear
or quadratic convergence, respectively. We establish local and global
convergence guarantees, under both local strong convexity and self-concordance,
showing that our private estimators converge with high probability to a nearly
optimal neighborhood of the non-private M-estimators. Secondly, we tackle the
problem of parametric inference by constructing differentially private
estimators of the asymptotic variance of our private M-estimators. This
naturally leads to approximate pivotal statistics for constructing confidence
regions and conducting hypothesis testing. We demonstrate the effectiveness of
a bias correction that leads to enhanced small-sample empirical performance in
simulations. We illustrate the benefits of our methods in several numerical
examples.
