1. Learning and Testing Submodular Functions Grigory Yaroslavtsev http://grigory.us Slides at http://grigory.us/cis625/lecture3.pdf http://grigory.us/cis625/lecture3.pdf CIS 625: Computational Learning Theory

2. Submodularity

3. Approximating everywhere

4. Approximate learning

5. Goemans, Harvey, Iwata, Mirrokni Balcan, Harvey Gupta, Hardt, Roth, Ullman Cheraghchi, Klivans, Kothari, Lee Raskhodnikova, Y. Learning TimePoly(|X|) Extra features Under arbitrary distribution Tolerant queries SQ- queries, Agnostic

6. Learning: Bigger picture XOS = Fractionally subadditive Subadditive Submodular Gross substitutes OXS [Badanidiyuru, Dobzinski, Fu, Kleinberg, Nisan, Roughgarden,SODA’12] Additive (linear) Coverage (valuations)

7. Discrete convexity

8. Monotone submodular Submodular

9. Discrete monotone submodularity

12. Representation by a formula

13. Discrete submodularity

14. Proof

16. Coverage by monotone lower bounds

17. Learning pB-formulas and k-DNF

18. Learning Fourier coefficients

19. Property testing

20. Testing by implicit learning

21. Previous work on testing submodularity

22. Directions