Learning and Testing Submodular Functions

Grigory Yaroslavtsev

Columbia UniversityOctober 26, 2012

With Sofya Raskhodnikova (SODA’13)

+ Work in progress with Rocco Servedio

Submodularity• Discrete analog of convexity/concavity, law of

diminishing returns• Applications: optimization, algorithmic game

theory

Let • Discrete derivative:

• Submodular function:

Exact learning• Q: Reconstruct a submodular with poly(|X|) queries (for all

arguments)?• A: Only-approximation (multiplicative) possible [Goemans, Harvey,

Iwata, Mirrokni, SODA’09]

• Q: O-fraction of points (PAC-style learning with membership queries under

uniform distribution)?

• A: Almost as hard [Balcan, Harvey, STOC’11].

Pr𝑟𝑎𝑛𝑑𝑜𝑚𝑛𝑒𝑠𝑠 𝑜𝑓 𝑨 [ Pr

𝑺∼𝑈 (2𝑋)[ 𝑨 (𝑺 )= 𝒇 (𝑺 ) ]≥1−𝜖]≥ 12

Approximate learning• PMAC-learning (Multiplicative), with poly(|X|) queries :

(over arbitrary distributions [BH’11])

• PAAC-learning (Additive)

• Running time: [Gupta, Hardt, Roth, Ullman, STOC’11]

• Running time: poly [Cheraghchi, Klivans, Kothari, Lee, SODA’12]

Pr𝑟𝑎𝑛𝑑𝑜𝑚𝑛𝑒𝑠𝑠 𝑜𝑓 𝑨 [ Pr

𝑺∼𝑈 (2𝑋)[¿ 𝒇 (𝑺 )− 𝑨 (𝑺 )∨≤ 𝜷 ]≥1−𝜖 ]≥ 12

Learning

Goemans,Harvey,Iwata,Mirrokni

Balcan,Harvey

Gupta,Hardt,Roth,Ullman

Cheraghchi,Klivans,Kothari,Lee

Our result with Sofya

Learning -approximationEverywhere

PMAC Multiplicative

PAACAdditive

PAC

(bounded integral range )

Time Poly(|X|) Poly(|X|)

Extrafeatures

Under arbitrary distribution

Tolerant queries

SQ- queries,Agnostic

Agnostic

• For all algorithms

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)

Discrete convexity• Monotone convex

• Convex

1 2 3 … <=R … … … … … … … … n02468

1 2 3 … <=R … … … … >= n-R

… … … n02468

Discrete submodularity

• Monotone submodular

𝑋

∅|𝑺|≤𝑹

• Submodular

𝑋

∅|𝑺|≤𝑹

|𝑺|≥|𝑿|−𝑹

• Case study: = 1 (Boolean submodular functions )Monotone submodular = (monomial)Submodular = (2-term CNF)

Discrete monotone submodularity

≥𝒎𝒂𝒙 ( 𝒇 (𝑺𝟏 ) , 𝒇 (𝑺𝟐))

|𝑺|≤𝑹

• Monotone submodular

Discrete monotone submodularity• Theorem: for monotone submodular f• (by monotonicity)

|𝑺|≤𝑹

𝑇

Discrete monotone submodularity

• S smallest subset of such that • we have => Restriction of on is monotone increasing =>

|𝑺|≤𝑹

𝑇

𝑆 ′

𝑆 ′

Representation by a formula

• Theorem: for monotone submodular f

• Notation switch: ,

• (Monotone)

(no negations)

• (Monotone) submodular can be represented as a (monotone) pseudo-Boolean 2R-DNF with constants

Discrete submodularity• Submodular can be represented as a

pseudo-Boolean 2R-DNF with constants • Hint [Lovasz] (Submodular monotonization): Given submodular define

Then is monotone and submodular.

𝑋

∅|𝑺|≤𝑹

|𝑺|≥|𝑿|−𝑹

Learning pB-formulas and k-DNF• = class of pB-DNF of width with • i-slice defined as

• If its i-slices are -DNF and:

• PAC-learning

(

Pr𝑟𝑎𝑛𝑑 (𝑨) [ Pr

𝑺∼𝑈( {0,1 }𝑛)[ 𝑨 (𝑺 )= 𝒇 (𝑺 ) ]≥1−𝜖]≥ 12

iff

Learning pB-formulas and k-DNF• Learn every i-slice on fraction of arguments • Learning -DNF () (let Fourier sparsity = )

– Kushilevitz-Mansour (Goldreich-Levin): queries/time. – ``Attribute efficient learning’’: queries– Lower bound: () queries to learn a random -junta ( -DNF) up to

constant precision.• Optimizations:

– Slightly better than KM/GL by looking at the Fourier spectrum of (see SODA paper: switching lemma => bound)

– Do all R iterations of KM/GL in parallel by reusing queries

Property testing• Let C be the class of submodular • How to (approximately) test, whether a given is in C?• Property tester: (Randomized) algorithm for distinguishing:

1. -far):

• Key idea: -DNFs have small representations:– [Gopalan, Meka,Reingold CCC’12] (using quasi-sunflowers [Rossman’10]), -DNF formula F there exists:-DNF formula F’ of size such that

Testing by implicit learning• Good approximation by juntas => efficient property

testing [surveys: Ron; Servedio]

– -approximation by -junta– Good dependence on :

• [Blais, Onak] sunflowers for submodular functions

– Query complexity: (independent of n)– Running time: exponential in (we think can be reduced it to

O(– We have lower bound for testing -DNF (reduction from Gap

Set Intersection: distinguishing a random -junta vs + O(log )-junta requires queries)

Previous work on testing submodularity

[Parnas, Ron, Rubinfeld ‘03, Seshadhri, Vondrak, ICS’11]:

• U. • Lower bound: Special case: coverage functions [Chakrabarty, Huang, ICALP’12].

Gap in query complexity

Thanks!