optimal budget allocation: theoretical guarantee and efficient algorithm

Optimal Budget Allocation:Theoretical Guarantee and Efficient Algorithm

Tasuku Soma Naonori Kakimura Kazuhiro Inaba Ken-ichi Kawarabayashi

Univ. of Tokyo Univ. of Tokyo Google NII, JST, ERATO

Kawarabayashi LargeGraphProject

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Budget Allocation Problem (Alon et al. ’11)

A mathematical model for company’s ads.

source1

source2

source3

Total: 6

alloc:2→ cap:3

alloc:3→ cap:5

alloc:1→ cap:2

1st trial: Pr 0.32nd trial: Pr 0.73rd trial: Pr 0.2

Influence Probs

success

successsuccess

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source1

source2

source3

Total: 6

alloc:2→ cap:3

alloc:3→ cap:5

alloc:1→ cap:2

Influence Probs

success

successsuccess

2 / 16

source1

source2

source3

Total: 6

alloc:2→ cap:3

alloc:3→ cap:5

alloc:1→ cap:2

Influence Probs

success

successsuccess

2 / 16

source1

source2

source3

Total: 6

alloc:2→ cap:3

alloc:3→ cap:5

alloc:1→ cap:2

Influence Probs

success

successsuccess

2 / 16

source1

source2

source3

Total: 6

alloc:2→ cap:3

alloc:3→ cap:5

alloc:1→ cap:2

Influence Probs

success

successsuccess

2 / 16

source1

source2

source3

Total: 6

alloc:2→ cap:3

alloc:3→ cap:5

alloc:1→ cap:2

Influence Probs

success

successsuccess

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Budget Allocation Problem (formal)G = (S,T ;E): Bipartite graphFor each source s ∈ S:

• capacity c(s)

• Pr of success of the ith trial p(i)s (i = 1, . . . , c(s))

B ∈ Z+: Total budgetf(b) := the expected # of influenced nodes by budget allocation b ∈ ZS ,which satisfies:

• x ≤ y =⇒ f(x) ≤ f(y) (monotonicity)

• f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y) (∀x, y ∈ ZS) (submodularity)

Maximize f(b)

subject to 0 ≤ b(s) ≤ c(s) (s ∈ S)∑s∈S

b(s) ≤ B

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Budget Allocation Problem (formal)G = (S,T ;E): Bipartite graphFor each source s ∈ S:

• capacity c(s)

• Pr of success of the ith trial p(i)s (i = 1, . . . , c(s))

B ∈ Z+: Total budgetf(b) := the expected # of influenced nodes by budget allocation b ∈ ZS ,which satisfies:

• x ≤ y =⇒ f(x) ≤ f(y) (monotonicity)

• f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y) (∀x, y ∈ ZS) (submodularity)

Maximize f(b)

subject to 0 ≤ b(s) ≤ c(s) (s ∈ S)∑s∈S

b(s) ≤ B

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Previous Work

• NP-hard

• If P , NP, (1 − 1/e)-approximation is best possible

• Previous algorithm (Alon et al. ’11) is polytime, but impracticaldue to the heavy time complexity

• What about more complicated real scenarios?

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Our Results

1 Submodular Function Maximization over Integer Lattice:• A gerenal framework including more complicated scenarios

• (1 − 1/e)-approximation algorithm

2 Faster Algorithm for Nonincreasing Influence Probabilities:• Speeding up Alon et al.’s algorithm under natural assumption

• Almost linear time for graph size

• Numerical experiments for real & big data

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1 Submodular Function Maximization over Integer Lattice

2 Faster Algorithm for Nonincreasing Influence Probabilities

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Submodular Func. Maximization over ZS+

f : ZS+ → R ... monotone submodular function over integer lattice

• x ≤ y =⇒ f(x) ≤ f(y)

• f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y)(∀x, y ∈ ZS)

(x ∨ y : elem-wise max, x ∧ y : elem-wise min)

cf. submodularity forset functions:f(X) + f(Y) ≥

f(X ∪ Y) + f(X ∩ Y)(∀X ,Y ⊆ S)

c ∈ ZS+, w ∈ ZS

+, B ∈ Z≥0

Maximize f(b)

subject to 0 ≤ b ≤ c∑s∈S

w(s)b(s) ≤ B (knapsack constraint)

b ∈ ZS

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• x ≤ y =⇒ f(x) ≤ f(y)

• f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y)(∀x, y ∈ ZS)

f(X ∪ Y) + f(X ∩ Y)(∀X ,Y ⊆ S)

c ∈ ZS+, w ∈ ZS

+, B ∈ Z≥0

Maximize f(b)

b ∈ ZS

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• x ≤ y =⇒ f(x) ≤ f(y)

• f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y)(∀x, y ∈ ZS)

f(X ∪ Y) + f(X ∩ Y)(∀X ,Y ⊆ S)

c ∈ ZS+, w ∈ ZS

+, B ∈ Z≥0

Maximize f(b)

b ∈ ZS

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General Framework Including:

• Optimal budget allocation with various unit costs• Optimal budget allocation with competitor• Maximum coverage, Sensor placement, Text summarization, ...

Pseudo Polytime (1 − 1/e)-Approximation Algorithm:

Theorem

A (1 − 1/e)-approximate solution can be found in O(B5|S |4θ) time(θ: the running time of oracle for f ) for a monotone submodular functionmaximization over the integer lattice subject to knapsack constraint.

• Extends the algorithm for optimal budget allocation (Alon et al. ’11)

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General Framework Including:

• Optimal budget allocation with various unit costs• Optimal budget allocation with competitor• Maximum coverage, Sensor placement, Text summarization, ...

Pseudo Polytime (1 − 1/e)-Approximation Algorithm:

Theorem

A (1 − 1/e)-approximate solution can be found in O(B5|S |4θ) time(θ: the running time of oracle for f ) for a monotone submodular functionmaximization over the integer lattice subject to knapsack constraint.

• Extends the algorithm for optimal budget allocation (Alon et al. ’11)

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1 Submodular Function Maximization over Integer Lattice

2 Faster Algorithm for Nonincreasing Influence Probabilities

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Nonincreasing Influence Probabilities

Under a natural assumption, Alon et al.’s algorithm can be accelerated.

Assumption:Influence probabilities of each ad source are nonincreasing:

p(1)s ≥ p(2)

s ≥ · · · ≥ p(c(s))s (∀s ∈ S)

(i.e., Effectiveness of each ad is nonincreasing with time)

Alon et al.’s algorithm: O(B6|S |5|T |) time

Our algorithm: O(B(|S |+ |T |+ |E |)) time (almost linear for graph size)

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Nonincreasing Influence Probabilities

Under a natural assumption, Alon et al.’s algorithm can be accelerated.

Assumption:Influence probabilities of each ad source are nonincreasing:

p(1)s ≥ p(2)

s ≥ · · · ≥ p(c(s))s (∀s ∈ S)

(i.e., Effectiveness of each ad is nonincreasing with time)

Alon et al.’s algorithm: O(B6|S |5|T |) time

Our algorithm: O(B(|S |+ |T |+ |E |)) time (almost linear for graph size)

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Algorithm

Under the assumption, f satisfies the following diminishing marginalreturn property:

f(b + 2es) − f(b + es) ≤ f(b + es) − f(b) (b ∈ ZS , s ∈ S)

Note: This property is NOT implied by submodularity!

Algorithm1: b := 02: while

∑s∈S b(s) < B do

3: Among s with b(s) < c(s), choose one of f(b +es)− f(b) maximum.4: Set b := b + es .5: end while6: return b

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Algorithm

Under the assumption, f satisfies the following diminishing marginalreturn property:

f(b + 2es) − f(b + es) ≤ f(b + es) − f(b) (b ∈ ZS , s ∈ S)

Note: This property is NOT implied by submodularity!

Algorithm1: b := 02: while

∑s∈S b(s) < B do

3: Among s with b(s) < c(s), choose one of f(b +es)− f(b) maximum.4: Set b := b + es .5: end while6: return b

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Algorithm

One more trick:For optimal budget allocation, f(b + es) − f(b) can be computed inO(1) time.

TheoremA (1 − 1/e)-approximate solution can be found in O(B(|V |+ |E |)) time ifthe influence probabilities of each ad source are nonincreasing.

• If B = O(1), runs in linear time for graph size.

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ExperimentsOur algorithm for nonincreasing probabilities vs Heuristics.

Graphs:

• Real Data (Yahoo! Webscope Dataset)≈ 10,000 nodes & 50,000 edges

• Random Graph ≈ 2M nodes & 8M edges

Heuristics:

• Degree-prob ... allocate to nodes with higher degree and prob.

• Degree ... allocate to nodes with higher degree

• Random ... allocate at random (baseline)

Machine: Xeon E5-2690 2.9GHz CPU, 64GB RAM

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Real Data (10k nodes & 50k edges)

0 100 200 300 400 500 600 700 800 900 1000

Influenced n

odes: f (b

Budget: B

Greedy (Ours)Degree-prob

DegreeRandom

• Maximum 15% outperforming14 / 16

Random Graph (2M nodes & 8M edges)

200000

400000

600000

800000

1000000

1200000

1400000

0 100 200 300 400 500 600 700 800 900 1000

Influenced n

odes: f (b

Budget: B

Greedy (Ours)Degree-prob

DegreeRandom

• Our algorithm finds an approx solution in a few seconds15 / 16

Our Results

1 Submodular Function Maximization over Integer Lattice:• A gerenal framework including more complicated scenarios

• (1 − 1/e)-approximation algorithm

2 Faster Algorithm for Nonincreasing Influence Probabilities:• Speeding up Alon et al.’s algorithm under natural assumption

• Almost linear time for graph size

• Numerical experiments for real & big data

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optimal budget allocation: theoretical guarantee and efficient algorithm

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