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).
The other recent work @cite_25 on learning submodular functions was motivated by bundle pricing and used passive supervised learning as a model for learning consumer valuations of added options. In particular, they have a polynomial-time algorithm that can learn (using random examples only) monotone , non-negative, submodular functions within a multiplicative factor of @math over distributions. As our machinery breaks down over non-product distributions, none of our results hold in this setting. For product distributions, Balcan and Harvey gave the first @math -time algorithm that can learn (using random examples only) monotone , non-negative, Lipschitz submodular functions with minimum value @math within a multiplicative factor of @math .
@cite_25
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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). In this work, we present a new algorithm for maximizing a non-monotone submodular function subject to a general constraint. Our algorithm finds an approximate fractional solution for maximizing the multilinear extension of the function over a down-closed polytope. The approximation guarantee is 0.372 and it is the first improvement over the 1 e approximation achieved by the unified Continuous Greedy algorithm [, FOCS 2011]. 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. This paper presents a polynomial-time @math -approximation algorithm for maximizing nonnegative @math -submodular functions. This improves upon the previous @math -approximation by Ward and Z ivn 'y (SODA'14), where @math . We also show that for monotone @math -submodular functions there is a polynomial-time @math -approximation algorithm while for any @math a @math -approximation algorithm for maximizing monotone @math -submodular functions would require exponentially many queries. In particular, our hardness result implies that our algorithms are asymptotically tight. We also extend the approach to provide constant factor approximation algorithms for maximizing skew-bisubmodular functions, which were recently introduced as generalizations of bisubmodular functions.