16 research outputs found
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 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
\log k / \epsilon)}, 1/\epsilon and log(1/\delta) 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((\log n/ \loglog n)^{1/2}) and constant proximity parameter \epsilon
A polynomial lower bound for testing monotonicity
We show that every algorithm for testing n-variate Boolean functions for monotonicity has query complexity Ω(n1/4). All previous lower bounds for this problem were designed for nonadaptive algorithms and, as a result, the best previous lower bound for general (possibly adaptive) monotonicity testers was only Ω(logn). Combined with the query complexity of the non-adaptive monotonicity tester of Khot, Minzer, and Safra (FOCS 2015), our lower bound shows that adaptivity can result in at most a quadratic reduction in the query complexity for testing monotonicity. By contrast, we show that there is an exponential gap between the query complexity of adaptive and non-adaptive algorithms for testing regular linear threshold functions (LTFs) for monotonicity. Chen, De, Servedio, and Tan (STOC 2015) recently showed that non-adaptive algorithms require almost Ω(n1/2) queries for this task. We introduce a new adaptive monotonicity testing algorithm which has query complexity O(logn) when the input is a regular LTF