Beyond Adaptive Submodularity: Approximation Guarantees of Greedy - PowerPoint PPT Presentation

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 uence maximization 2/ 8 Select a subset of ads to in fl uence as many people as possible

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence as many people as possible Non-adaptive setting Select a subset in advance

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence as many people as possible Non-adaptive setting Select a subset in advance

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence as many people as possible Non-adaptive setting Select a subset in advance

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence as many people as possible Non-adaptive setting Select a subset in advance

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence as many people as possible Adaptive setting Select ads one by one

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence as many people as possible Adaptive setting Select ads one by one

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence as many people as possible Adaptive setting Select ads one by one

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence as many people as possible Adaptive setting Select ads one by one

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence as many people as possible Adaptive setting Select ads one by one

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence 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?

Application: I n fl uence maximization 2/ 8 Select a subset of ads to in fl uence 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 di ff erent are the non-adaptive and adaptive policies?

Adaptive submodularity ratio 3/ 8 We propose a new concept called adaptive submodularity ratio Submodularity ratio [Das – Kempe ’ 11] Adaptive submodularity ratio [this study]

Adaptive submodularity ratio 3/ 8 We propose a new concept called adaptive submodularity ratio submodular Submodularity ratio functions [Das – Kempe ’ 11] Adaptive submodularity ratio [this study]

Adaptive submodularity ratio 3/ 8 We propose a new concept called adaptive submodularity ratio arbitrary submodular Submodularity ratio monotone functions [Das – Kempe ’ 11] functions γ ℓ, k = 1 γ ℓ, k = 0 Adaptive submodularity ratio [this study]

Adaptive submodularity ratio 3/ 8 We propose a new concept called adaptive submodularity ratio arbitrary submodular Submodularity ratio monotone functions [Das – Kempe ’ 11] functions γ ℓ, k = 1 γ ℓ, k = 0 adaptive submodular Adaptive submodularity ratio functions [this study] [Golovin – Krause ’ 11]

Adaptive submodularity ratio 3/ 8 We propose a new concept called adaptive submodularity ratio arbitrary submodular Submodularity ratio monotone functions [Das – Kempe ’ 11] functions γ ℓ, k = 1 γ ℓ, k = 0 adaptive arbitrary submodular adaptive Adaptive submodularity ratio functions monotone [this study] [Golovin – Krause ’ 11] functions

Adaptive submodularity ratio 4/ 8 Adaptive submodularity ratio γ ℓ, k ∈ [ 0 , 1 ] is a parameter that measures the distance to adaptive submodular functions ∑ v ∈ V Pr( v ∈ E ( π, Φ) | Φ ∼ ψ )∆( v | ψ ) △ γ ℓ, k = min ∆( π | ψ ) | ψ |≤ ℓ, π ∈ Π k

Adaptive submodularity ratio 4/ 8 Adaptive submodularity ratio γ ℓ, k ∈ [ 0 , 1 ] is a parameter that measures the distance to adaptive submodular functions ∑ v ∈ V Pr( v ∈ E ( π, Φ) | Φ ∼ ψ )∆( v | ψ ) △ γ ℓ, k = min ∆( π | ψ ) | ψ |≤ ℓ, π ∈ Π k the expected marginal gain of policy π

Adaptive submodularity ratio 4/ 8 Adaptive submodularity ratio γ ℓ, k ∈ [ 0 , 1 ] is a parameter that measures the distance to adaptive submodular functions the expected marginal gain of single element v the probability that element v is selected by policy π ∑ v ∈ V Pr( v ∈ E ( π, Φ) | Φ ∼ ψ )∆( v | ψ ) △ γ ℓ, k = min ∆( π | ψ ) | ψ |≤ ℓ, π ∈ Π k

Adaptive submodularity ratio 4/ 8 Adaptive submodularity ratio γ ℓ, k ∈ [ 0 , 1 ] is a parameter that measures the distance to adaptive submodular functions ∑ v ∈ V Pr( v ∈ E ( π, Φ) | Φ ∼ ψ )∆( v | ψ ) △ γ ℓ, k = min ∆( π | ψ ) | ψ |≤ ℓ, π ∈ Π k Q1 When does the greedy policy work well?

Adaptive submodularity ratio 4/ 8 Adaptive submodularity ratio γ ℓ, k ∈ [ 0 , 1 ] is a parameter that measures the distance to adaptive submodular functions ∑ v ∈ V Pr( v ∈ E ( π, Φ) | Φ ∼ ψ )∆( v | ψ ) △ γ ℓ, k = min ∆( π | ψ ) | ψ |≤ ℓ, π ∈ Π k Q1 When does the greedy policy work well? Theorem Adaptive Greedy is ( 1 − exp ( − γ k , k )) -approximation

Bounds on adaptivity gaps 5/ 8 A non-adaptive policy approximates an optimal adaptive policy △ GAP k ( f , p ) = An optimal An optimal non-adaptive policy adaptive policy

Bounds on adaptivity gaps 5/ 8 A non-adaptive policy approximates an optimal adaptive policy △ GAP k ( f , p ) = Q2 How di ff erent are the non-adaptive and adaptive policies? E [ f ( S , Φ)] △ Theorem GAP k ( f , p ) ≥ β 0 , k γ 0 , k β 0 , k = min ∑ v ∈ S E [ f ( { v } , Φ)] S ⊆ V : | S |≤ k

Application: I n fl uence maximization 6/ 8 Non-adaptive setting Select a subset in advance Adaptive setting Select ads one by one

Application: I n fl uence maximization 6/ 8 Non-adaptive setting Select a subset in advance Adaptive setting Select ads one by one k + 1 Theorem γ ℓ, k ≥ on bipartite graphs with the triggering model 2 k

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

Application: Adaptive Feature Selection 7/ 8 Select a subset of features to be observed precisely Non-adaptive setting Select a subset in advance ≈ A (Φ) y w Adaptive setting Observe features one by one S ⊆ V : | S |≤ ℓ + k λ min ( A ( φ ) ⊤ Theorem γ ℓ, k ≥ min S A ( φ ) S ) min φ

Summary 8/ 8 Adaptive Submodularity Ratio is applied to Bounds on approximation ratio of Adaptive Greedy Theorem 1 Bounds on adaptivity gaps Theorem 2 Application 1 I n fl uence maximization on bipartite graphs Application 2 Adaptive feature selection Poster #163 at Paci fi c Ballroom, Wen 6:30 – 9:00 PM