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).