Learning pseudo-Boolean k-DNF and submodular functions

We prove that any submodular function f : {0, 1}ⁿ → {0, 1, ..., k} can be represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show that an analog of Håstad's switching lemma holds for pseudo-Boolean k-DNFs if all constants associated with the terms of the formula are bounded. This allows us to generalize Mansour's PAC-learning algorithm for k-DNFs to pseudo-Boolean k-DNFs, and hence gives a PAC-learning algorithm with membership queries under the uniform distribution for submodular functions of the form f : {0, 1}ⁿ → {0, 1, ..., k}. Our algorithm runs in time polynomial in n, k^{O(k log k/∈)} and log(1/δ) and works even in the agnostic setting. The line of previous work on learning submodular functions [Balcan, Harvey (STOC '11), Gupta, Hardt, Roth, Ullman (STOC '11), Cheraghchi, Klivans, Kothari, Lee (SODA '12)] implies only n^O(k) query complexity for learning submodular functions in this setting, for fixed ∈ and δ. Our learning algorithm implies a property tester for submodularity of functions f : {0, 1}ⁿ → {0, . . . , k} with query complexity polynomial in n for k = O((log n/log log n)^1/2) and constant proximity parameter ∈.