Learning Coverage Functions

We study the problem of approximating and learning coverage functions. A function $c: 2^{[n]} \rightarrow \R^{+}$ is a coverage function, if there exists a universe $U$ with non-negative weights $w(u)$ for each $u \in U$ and subsets $A_1, A_2,..., A_n$ of $U$ such that $c(S) = \sum_{u \in \cup_{i \in S} A_i} w(u)$. 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 show that over the uniform distribution coverage functions with range [0,1] are PAC learnable to $\ell_1$-error of $\eps$ in $poly(n,1/\eps)$ time and using $poly(\log n,1/\eps)$ random examples. We also show a proper learning algorithm for coverage functions whose running time is polynomial in the size of the universe over which the coverage function is defined. Our algorithm is based on several new structural properties of the Fourier spectrum of coverage functions and, in particular, we prove that any coverage function can be $\eps$-approximated in $\ell_1$ by a coverage function that depends only on $1/\eps^2$ variables. In contrast, we show that, without assumptions on the distribution, learning coverage is at least as hard as learning polynomial-size disjoint DNF formulas. Our PAC learning algorithm on the uniform distribution implies the first polynomial-time differentially private algorithm for releasing monotone disjunction queries with low average error over the uniform distribution on disjunctions. This problem was first considered by Gupta et al. (2011) and the best previous algorithm runs in time $n^{O(\log(1/\alpha))}$, where $\alpha$ is the accuracy of release (Cheraghchi et al., 2012). Further, our proper learning algorithm implies that the queries can be released using a synthetic database in time $poly(n) \log(1/\alpha)^{O(\log(1/\alpha))}$.