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 .
Gupta al @cite_5 design an algorithm that learns a submodular function with the range @math within a given additive error @math on all but @math fraction of the probability mass (according to a specified product distribution on the domain). Their algorithm requires membership queries, but works even when these queries are answered with additive error @math . It takes @math time.
@cite_5
2950933581 1755884204 2951171906 2099119387
We investigate the approximability of several classes of real-valued functions by functions of a small number of variables ( juntas ). Our main results are tight bounds on the number of variables required to approximate a function @math within @math -error @math over the uniform distribution: 1. If @math is submodular, then it is @math -close to a function of @math variables. This is an exponential improvement over previously known results. We note that @math variables are necessary even for linear functions. 2. If @math is fractionally subadditive (XOS) it is @math -close to a function of @math variables. This result holds for all functions with low total @math -influence and is a real-valued analogue of Friedgut's theorem for boolean functions. We show that @math variables are necessary even for XOS functions. As applications of these results, we provide learning algorithms over the uniform distribution. For XOS functions, we give a PAC learning algorithm that runs in time @math . For submodular functions we give an algorithm in the more demanding PMAC learning model (Balcan and Harvey, 2011) which requires a multiplicative @math factor approximation with probability at least @math over the target distribution. Our uniform distribution algorithm runs in time @math . This is the first algorithm in the PMAC model that over the uniform distribution can achieve a constant approximation factor arbitrarily close to 1 for all submodular functions. As follows from the lower bounds in (, 2013) both of these algorithms are close to optimal. We also give applications for proper learning, testing and agnostic learning with value queries of these classes. We show that all non-negative submodular functions have high noise-stability . As a consequence, we obtain a polynomial-time learning algorithm for this class with respect to any product distribution on @math (for any constant accuracy parameter @math ). Our algorithm also succeeds in the agnostic setting. Previous work on learning submodular functions required either query access or strong assumptions about the types of submodular functions to be learned (and did not hold in the agnostic setting). Let @math be a finite lattice and let @math be a positive integer. A function @math is said to be submodular if @math for all @math . In this paper we study submodular functions when @math is a diamond. Given oracle access to @math we are interested in finding @math such that @math as efficiently as possible. We establish a min--max theorem, which states that the minimum of the submodular function is equal to the maximum of a certain function defined over a certain polyhedron; and a good characterisation of the minimisation problem, i.e., we show that given an oracle for computing a submodular @math and an integer @math such that @math , there is a proof of this fact which can be verified in time polynomial in @math and @math ; and a pseudo-polynomial time algorithm for the minimisation problem, i.e., given an oracle for computing a submodular @math one can find @math in time bounded by a polynomial in @math and @math . We study the performance of a family of randomized parallel coordinate descent methods for minimizing the sum of a nonsmooth and separable convex functions. The problem class includes as a special case L1-regularized L1 regression and the minimization of the exponential loss ("AdaBoost problem"). We assume the input data defining the loss function is contained in a sparse @math matrix @math with at most @math nonzeros in each row. Our methods need @math iterations to find an approximate solution with high probability, where @math is the number of processors and @math for the fastest variant. The notation hides dependence on quantities such as the required accuracy and confidence levels and the distance of the starting iterate from an optimal point. Since @math is a decreasing function of @math , the method needs fewer iterations when more processors are used. Certain variants of our algorithms perform on average only @math arithmetic operations during a single iteration per processor and, because @math decreases when @math does, fewer iterations are needed for sparser problems.