We prove that any submodular function f : {0, 1} n → {0, 1, ..., k} can be represented as a pseudoBoolean 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 kDNFs, 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/ǫ) , 1/ǫ 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 ǫ 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 ǫ.
Raskhodnikova and Yaroslavtsev consider learning and testing of submodular functions taking values in the range {0, 1, . . . , k} (referred to as pseudo-Boolean) REF .
Learning pseudo-Boolean k-DNF and Submodular Functions
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