  This paper considers the generalization performance of differentially private
convex learning. We demonstrate that the convergence analysis of Langevin
algorithms can be used to obtain new generalization bounds with differential
privacy guarantees for DP-SGD. More specifically, by using some recently
obtained dimension-independent convergence results for stochastic Langevin
algorithms with convex objective functions, we obtain $O(n^{-1/4})$ privacy
guarantees for DP-SGD with the optimal excess generalization error of
$\tilde{O}(n^{-1/2})$ for certain classes of overparameterized smooth convex
optimization problems. This improves previous DP-SGD results for such problems
that contain explicit dimension dependencies, so that the resulting
generalization bounds become unsuitable for overparameterized models used in
practical applications.
