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Beyond Adaptive Submodularity: Approximation Guarantees of Greedy Policy with Adaptive Submodularity Ratio Kaito Fujii (UTokyo) & Shinsaku Sakaue (NTT) The 36th I nternational Conference on Machine Learning Jun. 12, 2019 Application: I n fl

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Beyond Adaptive Submodularity: Approximation Guarantees of Greedy Policy with Adaptive Submodularity Ratio

Kaito Fujii (UTokyo) & Shinsaku Sakaue (NTT)

The 36th International Conference on Machine Learning

Jun. 12, 2019

SLIDE 2

Application: Influence maximization

Select a subset of ads to influence as many people as possible 2/ 8

SLIDE 3

Application: Influence maximization

Select a subset of ads to influence as many people as possible Non-adaptive setting Select a subset in advance 2/ 8

SLIDE 4

Application: Influence maximization

Select a subset of ads to influence as many people as possible Non-adaptive setting Select a subset in advance 2/ 8

SLIDE 5

Application: Influence maximization

Select a subset of ads to influence as many people as possible Non-adaptive setting Select a subset in advance 2/ 8

SLIDE 6

Application: Influence maximization

Select a subset of ads to influence as many people as possible Non-adaptive setting Select a subset in advance 2/ 8

SLIDE 7

Application: Influence maximization

Select a subset of ads to influence as many people as possible Adaptive setting Select ads one by one 2/ 8

SLIDE 8

Application: Influence maximization

Select a subset of ads to influence as many people as possible Adaptive setting Select ads one by one 2/ 8

SLIDE 9

Application: Influence maximization

Select a subset of ads to influence as many people as possible Adaptive setting Select ads one by one 2/ 8

SLIDE 10

Application: Influence maximization

Select a subset of ads to influence as many people as possible Adaptive setting Select ads one by one 2/ 8

SLIDE 11

Application: Influence maximization

Select a subset of ads to influence as many people as possible Adaptive setting Select ads one by one 2/ 8

SLIDE 12

Application: Influence maximization

Select a subset of ads to influence as many people as possible Non-adaptive setting Select a subset in advance Adaptive setting Select ads one by one Q1 When does the greedy policy work well? 2/ 8

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Application: Influence maximization

Select a subset of ads to influence as many people as possible Non-adaptive setting Select a subset in advance Adaptive setting Select ads one by one Q1 When does the greedy policy work well? Q2 How different are the non-adaptive and adaptive policies? 2/ 8

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Adaptive submodularity ratio

We propose a new concept called adaptive submodularity ratio Adaptive submodularity ratio

[this study]

Submodularity ratio

[Das–Kempe’11]

3/ 8

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Adaptive submodularity ratio

We propose a new concept called adaptive submodularity ratio Adaptive submodularity ratio

[this study]

submodular functions

Submodularity ratio

[Das–Kempe’11]

3/ 8

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Adaptive submodularity ratio

We propose a new concept called adaptive submodularity ratio Adaptive submodularity ratio

[this study]

γℓ,k = 1 γℓ,k = 0 submodular functions arbitrary monotone functions

Submodularity ratio

[Das–Kempe’11]

3/ 8

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Adaptive submodularity ratio

We propose a new concept called adaptive submodularity ratio

adaptive submodular functions

[Golovin–Krause’11]

Adaptive submodularity ratio

[this study]

γℓ,k = 1 γℓ,k = 0 submodular functions arbitrary monotone functions

Submodularity ratio

[Das–Kempe’11]

3/ 8

SLIDE 18

Adaptive submodularity ratio

We propose a new concept called adaptive submodularity ratio

adaptive submodular functions

[Golovin–Krause’11]

arbitrary adaptive monotone functions

Adaptive submodularity ratio

[this study]

γℓ,k = 1 γℓ,k = 0 submodular functions arbitrary monotone functions

Submodularity ratio

[Das–Kempe’11]

3/ 8

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Adaptive submodularity ratio

Adaptive submodularity ratio γℓ,k ∈ [0, 1] is a parameter that measures the distance to adaptive submodular functions γℓ,k

△

= min

|ψ|≤ℓ, π∈Πk

∑

v∈V Pr(v ∈ E(π, Φ)|Φ ∼ ψ)∆(v|ψ)

∆(π|ψ) 4/ 8

SLIDE 20

Adaptive submodularity ratio

Adaptive submodularity ratio γℓ,k ∈ [0, 1] is a parameter that measures the distance to adaptive submodular functions γℓ,k

△

= min

|ψ|≤ℓ, π∈Πk

∑

v∈V Pr(v ∈ E(π, Φ)|Φ ∼ ψ)∆(v|ψ)

∆(π|ψ)

the expected marginal gain of policy π

4/ 8

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Adaptive submodularity ratio

Adaptive submodularity ratio γℓ,k ∈ [0, 1] is a parameter that measures the distance to adaptive submodular functions γℓ,k

△

= min

|ψ|≤ℓ, π∈Πk

∑

v∈V Pr(v ∈ E(π, Φ)|Φ ∼ ψ)∆(v|ψ)

∆(π|ψ)

the probability that element v is selected by policy π the expected marginal gain of single element v

4/ 8

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Adaptive submodularity ratio

Adaptive submodularity ratio γℓ,k ∈ [0, 1] is a parameter that measures the distance to adaptive submodular functions γℓ,k

△

= min

|ψ|≤ℓ, π∈Πk

∑

v∈V Pr(v ∈ E(π, Φ)|Φ ∼ ψ)∆(v|ψ)

∆(π|ψ) Q1 When does the greedy policy work well? 4/ 8

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Adaptive submodularity ratio

Adaptive submodularity ratio γℓ,k ∈ [0, 1] is a parameter that measures the distance to adaptive submodular functions γℓ,k

△

= min

|ψ|≤ℓ, π∈Πk

∑

v∈V Pr(v ∈ E(π, Φ)|Φ ∼ ψ)∆(v|ψ)

∆(π|ψ) Q1 When does the greedy policy work well? Theorem Adaptive Greedy is (1 − exp(−γk,k))-approximation 4/ 8

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Bounds on adaptivity gaps

A non-adaptive policy approximates an optimal adaptive policy GAPk(f, p)

△

= An optimal non-adaptive policy An optimal adaptive policy 5/ 8

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Bounds on adaptivity gaps

A non-adaptive policy approximates an optimal adaptive policy GAPk(f, p)

△

= Q2 How different are the non-adaptive and adaptive policies? Theorem GAPk(f, p) ≥ β0,kγ0,k

β0,k

△

= min

S⊆V : |S|≤k

E[f(S, Φ)]

∑

v∈S E[f({v}, Φ)]

5/ 8

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Application: Influence maximization

Non-adaptive setting

Select a subset in advance

Adaptive setting

Select ads one by one 6/ 8

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Application: Influence maximization

Non-adaptive setting

Select a subset in advance

Adaptive setting

Select ads one by one Theorem γℓ,k ≥ k + 1 2k

n bipartite graphs with the triggering model

6/ 8

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Application: Adaptive Feature Selection

Select a subset of features to be observed precisely y ≈ A(Φ) w Non-adaptive setting Adaptive setting Select a subset in advance Observe features one by one 7/ 8

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Application: Adaptive Feature Selection

Select a subset of features to be observed precisely y ≈ A(Φ) w Non-adaptive setting Adaptive setting Select a subset in advance Observe features one by one Theorem γℓ,k ≥ min

φ

min

S⊆V : |S|≤ℓ+k λmin(A(φ)⊤ S A(φ)S)

7/ 8

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Summary

Adaptive Submodularity Ratio is applied to Theorem 1 Theorem 2 Application 1 Application 2 Bounds on approximation ratio of Adaptive Greedy Bounds on adaptivity gaps Influence maximization on bipartite graphs Adaptive feature selection