  We introduce general tools for designing efficient private estimation
algorithms, in the high-dimensional settings, whose statistical guarantees
almost match those of the best known non-private algorithms. To illustrate our
techniques, we consider two problems: recovery of stochastic block models and
learning mixtures of spherical Gaussians. For the former, we present the first
efficient $(\epsilon, \delta)$-differentially private algorithm for both weak
recovery and exact recovery. Previously known algorithms achieving comparable
guarantees required quasi-polynomial time. For the latter, we design an
$(\epsilon, \delta)$-differentially private algorithm that recovers the centers
of the $k$-mixture when the minimum separation is at least $
O(k^{1/t}\sqrt{t})$. For all choices of $t$, this algorithm requires sample
complexity $n\geq k^{O(1)}d^{O(t)}$ and time complexity $(nd)^{O(t)}$. Prior
work required minimum separation at least $O(\sqrt{k})$ as well as an explicit
upper bound on the Euclidean norm of the centers.
