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).
@cite_25: Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications that have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we use a learning theoretic angle for studying submodular functions. We provide algorithms for learning submodular functions, as well as lower bounds on their learnability. In doing so, we uncover several novel structural results revealing both extremal properties as well as regularities of submodular functions, of interest to many areas.
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 .