Adaptive Confidence Sets for Nonparametric Density Estimation under Local Differential Privacy

Abstract

In nonparametric density estimation without privacy constraints, it is known that a confidence set whose coverage holds uniformly over a `rough'class of regularity $r\geq0$ can simultaneously attain, in $L^2$-diameter, the minimax rate over a smaller, smoother class of regularity $s>r$ if and only if $s \le 2r$. We study such adaptive honest confidence sets under non-interactive $\alpha$-local differential privacy (LDP) constraints. Our main result is that, for bounded $\alpha$, such `smooth-rate adaptation'is now possible if and only if $s \le 4r/3$. In this smaller range we construct an honest private confidence set whose diameter is of the private minimax order on each of the two classes. As in the non-private case, the threshold arises because an honest set cannot, at a smooth density, be narrower than the separation rate arising in a related testing problem. Under LDP we prove that this private testing rate is of order $(n\alpha^2)^{-2r/(4r+3)}$, which is at most the private estimation rate $(n\alpha^2)^{-s/(2s+2)}$ over the smooth class exactly when $s \le 4r/3$. For $s>4r/3$, however, we show that smooth-rate adaptation becomes possible once one discards from the rough class the densities that lie outside the smooth class but within an $L^2$-radius of the order of the private testing rate.