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Learning and Testing Submodular Functions Grigory Yaroslavtsev Columbia University October 26, 2012 With Sofya Raskhodnikova (SODA’13) + Work in progress with Rocco Servedio

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Submodularity

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Exact learning

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Approximate learning

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Goemans, Harvey, Iwata, Mirrokni Balcan, Harvey Gupta, Hardt, Roth, Ullman Cheraghchi, Klivans, Kothari, Lee Our result with Sofya Learning TimePoly(|X|) Extra features Under arbitrary distribution Tolerant queries SQ- queries, Agnostic

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Learning: Bigger picture XOS = Fractionally subadditive Subadditive Submodular Gross substitutes OXS [Badanidiyuru, Dobzinski, Fu, Kleinberg, Nisan, Roughgarden,SODA’12] Additive (linear) Value demand Other positive results: Learning valuation functions [Balcan, Constantin, Iwata, Wang, COLT’12] PMAC-learning (sketching) valuation functions [BDFKNR’12] PMAC learning Lipschitz submodular functions [BH’10] (concentration around average via Talagrand)

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Discrete convexity

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Monotone submodular Submodular

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Discrete monotone submodularity

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Representation by a formula

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Discrete submodularity

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Learning pB-formulas and k-DNF

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Property testing

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Testing by implicit learning

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Previous work on testing submodularity

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