  Differentially private learning algorithms inject noise into the learning
process. While the most common private learning algorithm, DP-SGD, adds
independent Gaussian noise in each iteration, recent work on matrix
factorization mechanisms has shown empirically that introducing correlations in
the noise can greatly improve their utility. We characterize the asymptotic
learning utility for any choice of the correlation function, giving precise
analytical bounds for linear regression and as the solution to a convex program
for general convex functions. We show, using these bounds, how correlated noise
provably improves upon vanilla DP-SGD as a function of problem parameters such
as the effective dimension and condition number. Moreover, our analytical
expression for the near-optimal correlation function circumvents the cubic
complexity of the semi-definite program used to optimize the noise correlation
matrix in previous work. We validate our theory with experiments on private
deep learning. Our work matches or outperforms prior work while being efficient
both in terms of compute and memory.
