Prize-Collecting Data Fusion for Cost-Performance Tradeoff in Distributed Inference Anima Anandkumar 1 Meng Wang 1 Lang Tong 1 Ananthram Swami 2 1 School of ECE, Cornell University, Ithaca, NY. 2 Army Research Laboratory, Adelphi MD. INFOCOM 2009 April 23, 2009 . Supported by Army Research Laboratory CTA. Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 1 / 22
Distributed Statistical Inference Sensor Network Applications: Statistical Inference Sensors: take measurements, e.g., Target, Temperature Fusion center: make a final decision Wireless sensor networks for inference Fusion center Energy constraints Measurement selection, inference accuracy In-network data fusion Sensor selection, routing and fusion policies Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 2 / 22
Optimal Node Selection For Tradeoff � � Network graph Dependency graph � � Cost-Performance Tradeoff Cost ≡ Total cost of routing with fusion Performance degradation ≡ Inference error probability Objective ≡ Cost + µ Performance degradation, µ > 0 Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 3 / 22
Optimal Node Selection For Tradeoff � � Network graph Dependency graph Fusion policy graph � � � Cost-Performance Tradeoff Cost ≡ Total cost of routing with fusion Performance degradation ≡ Inference error probability Objective ≡ Cost + µ Performance degradation, µ > 0 Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 3 / 22
Optimal Node Selection For Tradeoff � � Network graph Dependency graph Fusion policy graph � � � Cost-Performance Tradeoff Cost ≡ Total cost of routing with fusion Performance degradation ≡ Inference error probability Objective ≡ Cost + µ Performance degradation, µ > 0 Challenges Presence of Correlation Multi-Hop Routing & Fusion Optimality: NP-hard, Brute Force? Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 3 / 22
Outline Introduction 1 Problem Formulation and Main Results 2 Network and Inference Model Cost-Performance Tradeoff Main Results Simplification of the Problem Special Case: IID Measurements 3 General Correlation Cases: Two Selection Heuristics 4 Simulation 5 Conclusion 6 Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 4 / 22
Network and Inference Model Network Model Fixed location V ∆ � i j � =(1 , · · · , n ) . Feasible links with cost C ( i, j ) for link ( i, j ) . Network graph � Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 5 / 22
Network and Inference Model Network Model Fixed location V ∆ � i j � =(1 , · · · , n ) . Feasible links with cost C ( i, j ) for link ( i, j ) . Network graph � Inference model Sensor measurements Y V . Binary hypothesis: H 0 vs. H 1 : H k : Y V ∼ f ( y v |H k ) Dependency graph � Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 5 / 22
Optimal Cost-Performance Tradeoff Problem Statement Select V s ⊂ V and design a fusion scheme Γ( V s ) . Minimize the total routing costs C (Γ( V s )) plus a penalty π based on the error prob. P M ( V s ) . Fusion policy graph � = log P M ( V s ) π ( V \ V s ) ∆ P M ( V ) > 0 Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 6 / 22
Optimal Cost-Performance Tradeoff Problem Statement Select V s ⊂ V and design a fusion scheme Γ( V s ) . Minimize the total routing costs C (Γ( V s )) plus a penalty π based on the error prob. P M ( V s ) . Fusion policy graph � = log P M ( V s ) π ( V \ V s ) ∆ P M ( V ) > 0 � � min C (Γ( V s )) + µπ ( V \ V s ) , µ > 0 V s ⊂ V, Γ( V s ) Prize-Collecting Data Fusion Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 6 / 22
Main Results C (Γ( V s ))+ µ log P M ( V s ) � � min P M ( V ) ) , µ > 0 V s ⊂ V, Γ( V s ) IID measurements 2 − ( | V | − 1) − 1 approximation via Prize-Collecting Steiner Tree � � � � � � � � � � � � � � � � PCST Correlated data: component and clique selection heuristics Provable approximation guarantee for special dependency graphs. Substantially better than no data fusion. Performance under different node placements. Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 7 / 22
Simplification of the Problem Simplification of the fusion scheme Minimal sufficient statistic for V s ⊂ V Log-Likelihood Ratio: LLR ( Y V s ) = log f ( Y V s ; H 0 ) f ( Y V s ; H 1 ) Limit to schemes delivering LLR ( Y V s ) Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 8 / 22
Simplification of the Problem Simplification of the fusion scheme Minimal sufficient statistic for V s ⊂ V Log-Likelihood Ratio: LLR ( Y V s ) = log f ( Y V s ; H 0 ) f ( Y V s ; H 1 ) Limit to schemes delivering LLR ( Y V s ) LLR ( Y V s ) = � Ψ c ( Y c ) c ∈ C dependency graph C : the set of maximal cliques Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 8 / 22
Simplification of the Problem Simplification of the fusion scheme Minimal sufficient statistic for V s ⊂ V Log-Likelihood Ratio: LLR ( Y V s ) = log f ( Y V s ; H 0 ) f ( Y V s ; H 1 ) Limit to schemes delivering LLR ( Y V s ) LLR ( Y V s ) = � Ψ c ( Y c ) c ∈ C dependency graph C : the set of maximal cliques Simplification of the penalty function Error prob. = log P M ( V s ) exp( −| V |D ) π ( V \ V s ) ∆ P M ( V ) Error exponent approx. in a large network Number of samples 1 D ∆ = − lim | V | log P M ( V ) | V |→∞ Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 8 / 22
Outline Introduction 1 Problem Formulation and Main Results 2 Network and Inference Model Cost-Performance Tradeoff Main Results Simplification of the Problem Special Case: IID Measurements 3 General Correlation Cases: Two Selection Heuristics 4 Simulation 5 Conclusion 6 Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 9 / 22
PCDF: IID case C (Γ( V s )) + µ log P M ( V s ) � � min , µ > 0 P M ( V ) V s ⊂ V, Γ( V s ) Simplifications of IID measurements H k : Y V ∼ � f k ( Y i ) i ∈ V log f ( Y i ; H 0 ) LLR ( Y V s ) = � f ( Y i ; H 1 ) = � LLR ( Y i ) i ∈ V s i ∈ V s Error exponent D = D ( f ( Y ; H 0 ) || f ( Y ; H 1 )) Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 10 / 22
PCDF: IID case C (Γ( V s )) + µ log P M ( V s ) � � min , µ > 0 P M ( V ) V s ⊂ V, Γ( V s ) Simplifications of IID measurements H k : Y V ∼ � f k ( Y i ) i ∈ V log f ( Y i ; H 0 ) LLR ( Y V s ) = � f ( Y i ; H 1 ) = � LLR ( Y i ) i ∈ V s i ∈ V s Error exponent D = D ( f ( Y ; H 0 ) || f ( Y ; H 1 )) Modified cost-performance tradeoff for IID � � min C (Γ( V s )) + µ [ | V | − | V s | ] D V s ⊂ V, Γ( V s ) Asymptotic convergence to the original problem. The optimal solution is the Prize Collecting Steiner Tree. Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 10 / 22
Prize Collecting Steiner Tree (PCST) Definition Tree with minimum sum edge costs plus node penalties not spanned �� � � T ∗ = arg min c e + π i . T =( V ′ ,E ′ ) e ∈ E ′ i/ ∈ V ′ NP-hard, Goemans-Williamson algorithm has approx. ratio of 1 2 − | V |− 1 � � � � � � � � � � � � � � � � Approx. PCST Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 11 / 22
Prize Collecting Steiner Tree (PCST) Definition Tree with minimum sum edge costs plus node penalties not spanned � q 2 = LLR ( Y 2 ) q 1 = LLR ( Y 1 ) � �� � � T ∗ = arg min c e + π i . T =( V ′ ,E ′ ) e ∈ E ′ i/ ∈ V ′ NP-hard, Goemans-Williamson algorithm has approx. ratio of 1 2 − | V |− 1 � � � � � � � � � � � � � � � � Fusion of IID measurements � LLR ( Y V s ) = LLR ( Y i ) i ∈ V s Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 11 / 22
Prize Collecting Steiner Tree (PCST) Definition Tree with minimum sum edge costs plus node penalties not spanned �� � 2 � T ∗ = arg min c e + π i . � q 3 = LLR ( Y 3 ) + � q i T =( V ′ ,E ′ ) e ∈ E ′ i =1 i/ ∈ V ′ NP-hard, Goemans-Williamson algorithm has approx. ratio of 1 2 − | V |− 1 � � � � � � � � � � � � � � � � Fusion of IID measurements � LLR ( Y V s ) = LLR ( Y i ) i ∈ V s Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 11 / 22
Prize Collecting Steiner Tree (PCST) Definition Tree with minimum sum edge costs plus node penalties not spanned �� � � T ∗ = arg min c e + π i . T =( V ′ ,E ′ ) e ∈ E ′ i/ ∈ V ′ � q 4 = LLR ( Y 4 ) NP-hard, Goemans-Williamson algorithm has approx. ratio of 1 2 − | V |− 1 � � � � � � � � � � � � � � � � Fusion of IID measurements � LLR ( Y V s ) = LLR ( Y i ) i ∈ V s Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 11 / 22
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