Learning pseudo-Boolean k-DNF and Submodular Functions

Abstract

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 Hastad'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 k / ), 1/ and log(1/ ) 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 epsilon and delta. 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(( n/ n)^1/2) and constant proximity parameter .

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