We study the problem of approximating and learning coverage functions. A function c: 2[n] → R+ is a coverage function, if there exists a universe U with non-negative weights w(u) for each u ∈ U and subsets A1, A2,..., An of U such that c(S) = u∈∪i∈SAi w(u). Alternatively, coverage functions can be described as non-negative linear combinations of monotone disjunctions. They are a natural subclass of submodular functions and arise in a number of applications. We give an algorithm that for any γ, δ> 0, given random and uniform examples of an unknown coverage function c, finds a function h that approximates c within factor 1 + γ on all but δ-fraction of the points in time poly(n, 1/γ, 1/δ). This is the first fully-polynomial algorithm for learning an interesting class of functions in the demanding PMAC model of Balcan and Harvey [2012]. Our algorithms are based on several new structural properties of coverage functions. Using the results in [Feldman and Kothari, 2014], we also show that coverage functions are learnable agnostically with excess `1-error over all product and symmetric distributions in time nlog(1/). In contrast, we show that, without assumptions on the distribution, learning coverage functions is at least as hard as learning polynomial-size disjoint DNF formulas, a class of functions for which the best known algorithm runs in time 2Õ(n 1/3) [Klivans and Servedio, 2004]. As an application of our learning results, we give simple differentially-private algorithms for releasing monotone conjunction counting queries with low average error. In particular, for any k ≤ n, we obtain private release of k-way marginals with average error α ̄ in time nO(log(1/ᾱ)). 1

Abstract In the past few years, differential privacy has become a standard concept in the area of privacy. One of the most important problems in this field is to answer queries while preserving differential privacy. In spite of extensive studies, most existing work on differentially private query answering assumes the data are discrete (i.e., in {0, 1} d ) and focuses on queries induced by Boolean functions. In real applications however, continuous data are at least as common as binary data. Thus, in this work we explore a less studied topic, namely, differential privately query answering for continuous data with continuous function. As a first step 1. Part of this work has been appeared in Wang et al. towards the continuous case, we study a natural class of linear queries on continuous data which we refer to as smooth queries. A linear query is said to be K-smooth if it is specified by a function defined on [−1, 1] d whose partial derivatives up to order K are all bounded. We develop two -differentially private mechanisms which are able to answer all smooth queries. The first mechanism outputs a summary of the database and can then give answers to the queries. The second mechanism is an improvement of the first one and it outputs a synthetic database. The two mechanisms both achieve an accuracy of O(n − K 2d+K / ). Here we assume that the dimension d is a constant. It turns out that even in this parameter setting (which is almost trivial in the discrete case), using existing discrete mechanisms to answer the smooth queries is difficult and requires more noise. Our mechanisms are based on L ∞ -approximation of (transformed) smooth functions by low-degree even trigonometric polynomials with uniformly bounded coefficients. We also develop practically efficient variants of the mechanisms with promising experimental results.