  In recent years, privacy-preserving machine learning algorithms have
attracted increasing attention because of their important applications in many
scientific fields. However, in the literature, most privacy-preserving
algorithms demand learning objectives to be strongly convex and Lipschitz
smooth, which thus cannot cover a wide class of robust loss functions (e.g.,
quantile/least absolute loss). In this work, we aim to develop a fast
privacy-preserving learning solution for a sparse robust regression problem.
Our learning loss consists of a robust least absolute loss and an $\ell_1$
sparse penalty term. To fast solve the non-smooth loss under a given privacy
budget, we develop a Fast Robust And Privacy-Preserving Estimation (FRAPPE)
algorithm for least absolute deviation regression. Our algorithm achieves a
fast estimation by reformulating the sparse LAD problem as a penalized least
square estimation problem and adopts a three-stage noise injection to guarantee
the $(\epsilon,\delta)$-differential privacy. We show that our algorithm can
achieve better privacy and statistical accuracy trade-off compared with the
state-of-the-art privacy-preserving regression algorithms. In the end, we
conduct experiments to verify the efficiency of our proposed FRAPPE algorithm.
