Lower Bounds for Differentially Private ERM: Unconstrained and Non-Euclidean

We consider the lower bounds of differentially private empirical risk minimization (DP-ERM) for convex functions in constrained/unconstrained cases with respect to the general ℓ p norm beyond the ℓ 2 norm considered by most of the previous works. We provide a simple black-box reduction approach which can generalize lower bounds in constrained case to unconstrained case. For ( ǫ, δ )-DP, we achieve Ω( √ d log(1 /δ ) ǫn ) lower bounds for both constrained and unconstrained cases and any ℓ p geometry where p ≥ 1 by introducing a novel biased mean property for fingerprinting codes, where n is the size of the data-set and d is the dimension.