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

Zhou Lu, Daogao Liu

Abstract:

We consider the lower bounds of differentially private empirical risk minimization (DP-ERM) for convex functions in both constrained and unconstrained cases concerning the general $\ell_p$ norm beyond the $\ell_2$ norm considered by most of the previous works.

We provide a simple black-box reduction approach that can generalize lower bounds in constrained to unconstrained cases.
Moreover, for $(\epsilon,\delta)$-DP, we achieve the optimal $\Omega(\frac{\sqrt{d \log(1/\delta)}}{\epsilon n})$ lower bounds for both constrained and unconstrained cases and any $\ell_p$ geometry where $p\geq 1$ by considering $\ell_1$ loss over the $\ell_{\infty}$ ball.

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