TY - GEN

T1 - Learning pseudo-Boolean k-DNF and submodular functions

AU - Raskhodnikova, Sofya

AU - Yaroslavtsev, Grigory

PY - 2013

Y1 - 2013

N2 - 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, kO(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 nO(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 ∈.

AB - 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, kO(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 nO(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 ∈.

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U2 - 10.1137/1.9781611973105.98

DO - 10.1137/1.9781611973105.98

M3 - Conference contribution

AN - SCOPUS:84876038221

SN - 9781611972511

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1356

EP - 1368

BT - Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013

PB - Association for Computing Machinery

T2 - 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013

Y2 - 6 January 2013 through 8 January 2013

ER -