Submodular Functions Are Noise Stable. Cheraghchi, M., Klivans, A., Kothari, P., & Lee, H. K. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1586–1592, 2012.
Submodular Functions Are Noise Stable [link]Link  Submodular Functions Are Noise Stable [link]Paper  abstract   bibtex   
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 $\{-1,1\}^n$ (for any constant accuracy parameter $ε$). 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). Additionally we give simple algorithms that efficiently release differentially private answers to all Boolean conjunctions and to all halfspaces with constant average error, subsuming and improving recent work due to Gupta, Hardt, Roth and Ullman (STOC 2011).
@INPROCEEDINGS{ref:CKKL12,
  author =	 {Cheraghchi, Mahdi and Klivans, Adam and Kothari,
                  Pravesh and Lee, Homin K.},
  title =	 {Submodular Functions Are Noise Stable},
  year =	 2012,
  booktitle =	 {Proceedings of the 23rd {Annual ACM-SIAM Symposium
                  on Discrete Algorithms (SODA)}},
  pages =	 {1586--1592},
  numpages =	 7,
  url_Link =	 {https://dl.acm.org/doi/10.5555/2095116.2095242},
  url_Paper =	 {https://eccc.weizmann.ac.il//report/2011/090/},
  abstract =	 {We show that all non-negative submodular functions
                  have high \textit{noise-stability}.  As a
                  consequence, we obtain a polynomial-time learning
                  algorithm for this class with respect to any product
                  distribution on $\{-1,1\}^n$ (for any constant
                  accuracy parameter $\epsilon$).  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).  Additionally we give
                  simple algorithms that efficiently release
                  differentially private answers to all Boolean
                  conjunctions and to all halfspaces with constant
                  average error, subsuming and improving recent work
                  due to Gupta, Hardt, Roth and Ullman (STOC 2011).}
}

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