Dimension Independent Generalization of DP-SGD for Overparameterized Smooth Convex Optimization

Abstract: 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.