Differentially Private Top-k Selection via Canonical Lipschitz Mechanism

Selecting the top-$k$ highest scoring items under differential privacy (DP) is a fundamental task with many applications. This work presents three new results. First, the exponential mechanism, permute-and-flip and report-noisy-max, as well as their oneshot variants, are unified into the Lipschitz mechanism, an additive noise mechanism with a single DP-proof via a mandated Lipschitz property for the noise distribution. Second, this new generalized mechanism is paired with a canonical loss function to obtain the canonical Lipschitz mechanism, which can directly select k-subsets out of $d$ items in $O(dk+d \log d)$ time. The canonical loss function assesses subsets by how many users must change for the subset to become top-$k$. Third, this composition-free approach to subset selection improves utility guarantees by an $\Omega(\log k)$ factor compared to one-by-one selection via sequential composition, and our experiments on synthetic and real-world data indicate substantial utility improvements.

Maximizing approximately k-submodular functions

We introduce the problem of maximizing approximately $k$-submodular functions subject to size constraints. In this problem, one seeks to select $k$-disjoint subsets of a ground set with bounded total size or individual sizes, and maximum utility, given by a function that is "close" to being $k$-submodular. The problem finds applications in tasks such as sensor placement, where one wishes to install $k$ types of sensors whose measurements are noisy, and influence maximization, where one seeks to advertise $k$ topics to users of a social network whose level of influence is uncertain. To deal with the problem, we first provide two natural definitions for approximately $k$-submodular functions and establish a hierarchical relationship between them. Next, we show that simple greedy algorithms offer approximation guarantees for different types of size constraints. Last, we demonstrate experimentally that the greedy algorithms are effective in sensor placement and influence maximization problems.