maximizing submodular function over the integer lattice

Maximizing Submodular Functionover the Integer Lattice

Tasuku Soma(Univ. Tokyo)

Joint work with:Yuichi Yoshida (NII, Tokyo)

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1 Monotone Submodular Function Maximization on ZS+

2 Algorithms

3 DR-Submodular Cover

4 Summary

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2 Algorithms

4 Summary

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Monotone Submodular Func Maximization

E: finite set, f : 2E → R ... monotone submodular

maximize f (X ) subject to X ∈ F

|X | ≤ k,∑

i∈X w (i) ≤ 1, etc

• NP-hard in general• O(1)-appriximable for various constraints (cardinality,knapsack, matroid, etc) and efficient algorithms

• Powerful model for machine learning

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Monotone Submodular Func Maximization

E: finite set, f : 2E → R ... monotone submodular

maximize f (X ) subject to X ∈ F

|X | ≤ k,∑

i∈X w (i) ≤ 1, etc

• NP-hard in general• O(1)-appriximable for various constraints (cardinality,knapsack, matroid, etc) and efficient algorithms

• Powerful model for machine learning

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Limitation of Set Function

Some real scenarios cannot be captured by a set function.

Budget Allocation [Alon–Gamzu–Tennenholtz ’13,S.–Kakimura–Inaba–Kawarabayashi ’14]:We want to decide how much budget set aside for eachad source.

Generalized Sensor Placement:We can put more than one sensors in each spot.

Can we generalize these set functionmodels?

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Definitions of Submodularity on {0, 1}E

E: finite set

f : {0, 1}E → R is submodularf (X ) + f (Y ) ≥ f (X ∪ Y ) + f (X ∩ Y ) (∀X,Y ⊆ E)

Diminishing Returnf (X ∪ e) − f (X ) ≥ f (Y ∪ e) − f (Y )

(X ⊆ Y ⊆ E, e ∈ E \ Y )

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Definitions of Submodularity on {0, 1}E

E: finite set

f : {0, 1}E → R is submodularf (X ) + f (Y ) ≥ f (X ∪ Y ) + f (X ∩ Y ) (∀X,Y ⊆ E)

Diminishing Returnf (X ∪ e) − f (X ) ≥ f (Y ∪ e) − f (Y )

(X ⊆ Y ⊆ E, e ∈ E \ Y )

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Definitions of Submodularity on ZE

f : ZE → R is lattice submodular:f (x) + f (y) ≥ f (x ∨ y) + f (x ∧ y) (∀x, y ∈ ZE)

coord-wise max coord-wise min

⇑ 6⇓

f is diminishing return submodular (DR-submodular):f (x + ei) − f (x) ≥ f (y + ei) − f (y)

(x ≤ y ∈ ZE, i ∈ E)

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⇑ 6⇓

(x ≤ y ∈ ZE, i ∈ E)

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⇑ 6⇓

(x ≤ y ∈ ZE, i ∈ E)

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Monotone Submod Func Maximization on ZE+

f : ZE+ → R+ ... monotone lattice/DR-submodular func

(with f (0) = 0), r ∈ Z+

Maximize f (x)

subject to 0 ≤ x ≤ r1, x ∈ ZE+ ∩ F

• F = {x : x(E) ≤ k} (cardinality)• F = P(+) (ρ) (polymatroid)• F = {x : w>x ≤ 1} (knapsack)

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Can We Reduce it to Set Function?YES, if f is DR-submodular.

r copies

←→

Drawback:• The new ground set has r |E | size, pseudopoly• Does not work for lattice-submodular function

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r copies

←→

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r copies

←→

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r copies

←→

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Our Results

Theorem (S. and Yoshida ’15)For any ε > 0, (1 − 1/e − ε )-appriximate polytimealgorithms for various constaints.

DR-submodular lattice submodular

cardinality X(deterministic) X(deterministic)

polymatroid X(random) open

knapsack X(random) only pseudopoly

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2 Algorithms

4 Summary

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Algorithms

Naive Approach:Choosing the best coordinate and step size in everyiteration?

k∗, i∗ ∈ argmaxk,i

f (kei | x)k

where f (kei | x) = f (kei + x) − f (x).

• Use “Decreasing Threshold Greedy”[Badanidiyuru–Vondrák ’14] to determine step size

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Algorithms

Naive Approach:Choosing the best coordinate and step size in everyiteration?

k∗, i∗ ∈ argmaxk,i

f (kei | x)k

where f (kei | x) = f (kei + x) − f (x).

• Use “Decreasing Threshold Greedy”[Badanidiyuru–Vondrák ’14] to determine step size

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Cardinality/DR-Submodular

DecresingThresholdGreedy1: x := 0, d := maxi∈E f (ei), θ := d2: while θ ≥ ε d

r :3: for each i ∈ E :4: Find the largest k s.t.

f (kei | x)k

≥ θ and x + kei

feasible.5: x := x + kei6: θ := (1 − ε )θ7: return x

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Cardinality/DR-Submodular

f is concave along each coordinate.

f (· | x)

step size k

slopeθ

Such k can be found in O(log r) time with binary search.

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Cardinality/Lattice-SubmodularIdea: Devide the range into polynomially many regions.

(1 − ε )fmax

(1 − ε )2fmax k

LemmaIf there exists k with f (kei | x) ≥ kθ, we can find k′ withf (k′ei | x) ≥ (1 − ε )k′θ.

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(1 − ε )fmax

(1 − ε )2fmax

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(1 − ε )fmax

(1 − ε )2fmax k

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Polymatroid/DR-Submodular

Idea: Mimic Continuous Greedy Algorithm

Multilinear Extension of f : 2E → R

F (x) =∑X⊆E

f (X )∏i∈X

x(i)∏i<X

(1 − x(i)) (x ∈ [0, 1]E)

Key Facts• F is monotone if f is monotone• F is concave along positive direction if f is submodular

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Idea: Gluing the multilinear extensions on eachhypercube.

fill by the multilinear ext off̃ (X ) = f (1 + 1X )

The resulting extension shares the same property?→ YES, if f is DR-submodular.

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2 Algorithms

4 Summary

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Submodular Cover [Wolsey ’82]

Somewhat “dual” problem of maximization

f, c : 2S → R+ monotone submodular, α > 0

minimize c(X ) subject to f (X ) ≥ α

c : cost, f : quality, α : worst guarantee

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Examples in ML:• Efficient Sensor Placement

[Krause&Guestrin ’05, Krause&Leskovec ’08]

• Text Summarization [Lin & Bilmes ’10]

• Object Finding [Song et al. ’14, Chen et al. ’14]

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Algorithmic Results:• For c(X ) = |X |, O(log d/β)-approx [Wolsey ’82]

• For integral f, c, O(ρ log d)-approx [Wan et al. ’09]

d = maxs f (s), ρ: curvature of c,β := min{f (s | X ) : s ∈ S,X ⊆ S, f (s | X ) > 0}

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DR-Submodular Cover

f, c : ZS → R+ monotone DR-submodular, α > 0, r ∈ Z+

minimize c(x)subject to f (x) ≥ α

0 ≤ x ≤ r1

Theorem (S.–Yoshida, to appear in NIPS ’15)An algorithm for finding a (nearly) feasible solution ofO(ρ log d/β) approx in O (n log nr log r) time.

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DR-Submodular Cover

f, c : ZS → R+ monotone DR-submodular, α > 0, r ∈ Z+

minimize c(x)subject to f (x) ≥ α

0 ≤ x ≤ r1

Theorem (S.–Yoshida, to appear in NIPS ’15)An algorithm for finding a (nearly) feasible solution ofO(ρ log d/β) approx in O (n log nr log r) time.

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2 Algorithms

4 Summary

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Summary

Our Results• Useful genealizations of monotone submodular funcmaximization and submodular cover

• Various polytime approximation algorithms

Recent Work• Online monotone submodular func maximization onZE+ [Avigdor-Elgrabli et al. ’15]

• Nonmonotone submodular func maximization on ZE+

[Gottschalk–Peis ’15]

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maximizing submodular function over the integer lattice

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