Private Isotonic Regression

Abstract

In this paper, we consider the problem of differentially private (DP) algorithms for isotonic regression. For the most general problem of isotonic regression over a partially ordered set (poset) X and for any Lipschitz loss function, we obtain a pure-DP algorithm that, given n input points, has an expected excess empirical risk of roughly width(X) |X| / n, where width(X) is the width of the poset. In contrast, we also obtain a near-matching lower bound of roughly (width(X) + |X|) / n, that holds even for approximate-DP algorithms. Moreover, we show that the above bounds are essentially the best that can be obtained without utilizing any further structure of the poset. In the special case of a totally ordered set and for _1 and _2^2 losses, our algorithm can be implemented in near-linear running time; we also provide extensions of this algorithm to the problem of private isotonic regression with additional structural constraints on the output function.

Tasks

Reproductions