Learning Coverage Functions

We study the problem of approximating and learning coverage functions. A function c : 2 → 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 [BH12]. Our algorithm relies on first solving a simpler problem of learning coverage functions with low l1-error. Our algorithms are based on several new structural properties of coverage functions and, in particular, we prove that any coverage function can be ǫ-approximated in l1 by a coverage function that depends only on O(1/ǫ) variables. This is tight up to a constant factor. In contrast, we show that, without assumptions on the distribution, learning coverage functions is at least as hard as learning polynomialsize disjoint DNF formulas, a class of functions for which the best known algorithm runs in time 2 ) [KS04]. As an application of our result, we give a simple polynomial-time differentially-private algorithm for releasing monotone disjunction counting queries with low average error over the uniform distribution on disjunctions. Work done while the author was at IBM Research Almaden.