Contraction and Statistical Inference under Privacy for Uniformly Bounded Distributions

Abstract

We investigate $c$-interior pointwise maximal leakage (PML) as a tool for contraction analyses and disclosure control. Based on the strong adversarial threat models from maximal leakage, $c$-interior PML generalizes local differential privacy (LDP) to data-generating distributions with densities uniformly bounded away from zero by $c>0$. Viewing $c$-interior PML as an algebraic constraint on a kernel yields more flexible (and often tighter) contraction analyses than standard LDP. We provide tight bounds on the Dobrushin coefficient, and bound the contraction coefficient of the Hockeystick-divergence. We further derive strong data processing inequalities on $f$-divergences under $c$-interior PML constraints when the input distributions to the divergence are restricted to be in the $c$-interior. These results extend beyond the regime of pure LDP to cover a larger class of kernels, including, e.g., arbitrary stochastic matrices. We apply the results to minimax theory and provide asymptotically optimal strategies under $c$-interior PML constraints for binary hypothesis testing and mean estimation. The results show that disclosure control with PML allows analysts to reason about systems in a more differentiated manner: For example, it allows us to quantify the privacy leakage of deterministic systems, and can give precise adversarial guarantees with respect to arbitrary distributional assumptions. Interestingly, a recurring theme in the disclosure analyses is that if the privacy problem is relatively regular (if the density bound $c$ is large), private inference can be possible without incurring any additional cost in terms of sample complexity.