  Differentially private (stochastic) gradient descent is the workhorse of DP
private machine learning in both the convex and non-convex settings. Without
privacy constraints, second-order methods, like Newton's method, converge
faster than first-order methods like gradient descent. In this work, we
investigate the prospect of using the second-order information from the loss
function to accelerate DP convex optimization. We first develop a private
variant of the regularized cubic Newton method of Nesterov and Polyak, and show
that for the class of strongly convex loss functions, our algorithm has
quadratic convergence and achieves the optimal excess loss. We then design a
practical second-order DP algorithm for the unconstrained logistic regression
problem. We theoretically and empirically study the performance of our
algorithm. Empirical results show our algorithm consistently achieves the best
excess loss compared to other baselines and is 10-40x faster than DP-GD/DP-SGD.
