Prize-Collecting Data Fusion for Cost-Performance Tradeoff in - - PowerPoint PPT Presentation

▶
prize collecting data fusion for cost performance
SMART_READER_LITE
LIVE PREVIEW

Prize-Collecting Data Fusion for Cost-Performance Tradeoff in - - PowerPoint PPT Presentation

Jun 10, 2023 151 likes •677 views

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

slide-1
SLIDE 1

Prize-Collecting Data Fusion for Cost-Performance Tradeoff in Distributed Inference

Anima Anandkumar1

Meng Wang1 Lang Tong1 Ananthram Swami2

1School of ECE, Cornell University, Ithaca, NY. 2Army 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

slide-2
SLIDE 2

Distributed Statistical Inference

Sensor Network Applications: Statistical Inference

Sensors: take measurements, e.g., Target, Temperature Fusion center: make a final decision

Fusion center

Wireless sensor networks for inference

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

slide-3
SLIDE 3

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

slide-4
SLIDE 4

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

slide-5
SLIDE 5

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

slide-6
SLIDE 6

Outline

1

Introduction

2

Problem Formulation and Main Results Network and Inference Model Cost-Performance Tradeoff Main Results Simplification of the Problem

3

Special Case: IID Measurements

4

General Correlation Cases: Two Selection Heuristics

5

Simulation

6

Conclusion

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 4 / 22

slide-7
SLIDE 7

Network and Inference Model

Network Model

Fixed location V∆ =(1, · · · , n). Feasible links with cost C(i, j) for link (i, j).

  • Network graph

i j

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 5 / 22

slide-8
SLIDE 8

Network and Inference Model

Network Model

Fixed location V∆ =(1, · · · , n). Feasible links with cost C(i, j) for link (i, j).

  • Network graph

i j

Inference model

Sensor measurements YV. Binary hypothesis: H0 vs. H1: Hk : YV ∼ f(yv|Hk)

  • Dependency graph

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 5 / 22

slide-9
SLIDE 9

Optimal Cost-Performance Tradeoff

Problem Statement

Select Vs ⊂ V and design a fusion scheme Γ(Vs). Minimize the total routing costs C(Γ(Vs)) plus a penalty π based on the error prob. PM(Vs). π(V \Vs)∆ = log PM(Vs) PM(V ) > 0

  • Fusion policy graph

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 6 / 22

slide-10
SLIDE 10

Optimal Cost-Performance Tradeoff

Problem Statement

Select Vs ⊂ V and design a fusion scheme Γ(Vs). Minimize the total routing costs C(Γ(Vs)) plus a penalty π based on the error prob. PM(Vs). π(V \Vs)∆ = log PM(Vs) PM(V ) > 0

  • Fusion policy graph

min

Vs⊂V,Γ(Vs)

  • C(Γ(Vs)) + µπ(V \Vs)
  • , µ > 0

Prize-Collecting Data Fusion

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 6 / 22

slide-11
SLIDE 11

Main Results

min

Vs⊂V,Γ(Vs)

  • C(Γ(Vs))+µ log PM(Vs)

PM(V ) )

  • , µ > 0

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

slide-12
SLIDE 12

Simplification of the Problem

Simplification of the fusion scheme

Minimal sufficient statistic for Vs ⊂ V

Log-Likelihood Ratio: LLR(YVs) = log f(YVs; H0) f(YVs; H1)

Limit to schemes delivering LLR(YVs)

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 8 / 22

slide-13
SLIDE 13

Simplification of the Problem

Simplification of the fusion scheme

Minimal sufficient statistic for Vs ⊂ V

Log-Likelihood Ratio: LLR(YVs) = log f(YVs; H0) f(YVs; H1)

Limit to schemes delivering LLR(YVs)

LLR(YVs) =

c∈C

Ψc(Yc) C : the set of maximal cliques

dependency graph

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 8 / 22

slide-14
SLIDE 14

Simplification of the Problem

Simplification of the fusion scheme

Minimal sufficient statistic for Vs ⊂ V

Log-Likelihood Ratio: LLR(YVs) = log f(YVs; H0) f(YVs; H1)

Limit to schemes delivering LLR(YVs)

LLR(YVs) =

c∈C

Ψc(Yc) C : the set of maximal cliques

dependency graph

Simplification of the penalty function

π(V \Vs)∆ = log PM(Vs) PM(V ) Error exponent approx. in a large network D∆ = − lim

|V |→∞

1 |V | log PM(V )

Error prob. Number of samples

exp(−|V |D)

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 8 / 22

slide-15
SLIDE 15

Outline

1

Introduction

2

Problem Formulation and Main Results Network and Inference Model Cost-Performance Tradeoff Main Results Simplification of the Problem

3

Special Case: IID Measurements

4

General Correlation Cases: Two Selection Heuristics

5

Simulation

6

Conclusion

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 9 / 22

slide-16
SLIDE 16

PCDF: IID case

min

Vs⊂V,Γ(Vs)

  • C(Γ(Vs)) + µ log PM(Vs)

PM(V )

  • , µ > 0

Simplifications of IID measurements

Hk : YV ∼

i∈V

fk(Yi) LLR(YVs) =

i∈Vs

log f(Yi;H0)

f(Yi;H1) = i∈Vs

LLR(Yi) Error exponent D = D(f(Y ; H0)||f(Y ; H1))

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 10 / 22

slide-17
SLIDE 17

PCDF: IID case

min

Vs⊂V,Γ(Vs)

  • C(Γ(Vs)) + µ log PM(Vs)

PM(V )

  • , µ > 0

Simplifications of IID measurements

Hk : YV ∼

i∈V

fk(Yi) LLR(YVs) =

i∈Vs

log f(Yi;H0)

f(Yi;H1) = i∈Vs

LLR(Yi) Error exponent D = D(f(Y ; H0)||f(Y ; H1))

Modified cost-performance tradeoff for IID

min

Vs⊂V,Γ(Vs)

  • C(Γ(Vs)) + µ[|V | − |Vs|]D
  • 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

slide-18
SLIDE 18

Prize Collecting Steiner Tree (PCST)

Definition

Tree with minimum sum edge costs plus node penalties not spanned T∗ = arg min

T=(V ′,E′)

  • e∈E′

ce+

  • i/

∈V ′

πi

  • .

NP-hard, Goemans-Williamson algorithm has approx. ratio of 2 −

1 |V |−1

  • Approx. PCST

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 11 / 22

slide-19
SLIDE 19

Prize Collecting Steiner Tree (PCST)

Definition

Tree with minimum sum edge costs plus node penalties not spanned T∗ = arg min

T=(V ′,E′)

  • e∈E′

ce+

  • i/

∈V ′

πi

  • .

NP-hard, Goemans-Williamson algorithm has approx. ratio of 2 −

1 |V |−1

  • Fusion of IID measurements

q1 = LLR(Y1) q2 = LLR(Y2) LLR(YVs) =

  • i∈Vs

LLR(Yi)

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 11 / 22

slide-20
SLIDE 20

Prize Collecting Steiner Tree (PCST)

Definition

Tree with minimum sum edge costs plus node penalties not spanned T∗ = arg min

T=(V ′,E′)

  • e∈E′

ce+

  • i/

∈V ′

πi

  • .

NP-hard, Goemans-Williamson algorithm has approx. ratio of 2 −

1 |V |−1

  • Fusion of IID measurements

q3 = LLR(Y3) +

2

  • i=1

qi LLR(YVs) =

  • i∈Vs

LLR(Yi)

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 11 / 22

slide-21
SLIDE 21

Prize Collecting Steiner Tree (PCST)

Definition

Tree with minimum sum edge costs plus node penalties not spanned T∗ = arg min

T=(V ′,E′)

  • e∈E′

ce+

  • i/

∈V ′

πi

  • .

NP-hard, Goemans-Williamson algorithm has approx. ratio of 2 −

1 |V |−1

  • Fusion of IID measurements

q4 = LLR(Y4) LLR(YVs) =

  • i∈Vs

LLR(Yi)

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 11 / 22

slide-22
SLIDE 22

Outline

1

Introduction

2

Problem Formulation and Main Results Network and Inference Model Cost-Performance Tradeoff Main Results Simplification of the Problem

3

Special Case: IID Measurements

4

General Correlation Cases: Two Selection Heuristics

5

Simulation

6

Conclusion

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 12 / 22

slide-23
SLIDE 23

Structure of Fusion Schemes for fixed selection

LLR(YVs) =

  • c∈C

Ψc(Yc)

c3 c2 c1

Forwarding graph Dependency graph Aggregation graph Processor Fusion center

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 13 / 22

slide-24
SLIDE 24

Structure of Fusion Schemes for fixed selection

LLR(YVs) =

  • c∈C

Ψc(Yc)

  • Forwarding graph

Dependency graph Aggregation graph Processor Fusion center

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 13 / 22

slide-25
SLIDE 25

Structure of Fusion Schemes for fixed selection

LLR(YVs) =

  • c∈C

Ψc(Yc)

  • Forwarding graph

Dependency graph Aggregation graph Processor Fusion center

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 13 / 22

slide-26
SLIDE 26

Structure of Fusion Schemes for fixed selection

LLR(YVs) =

  • c∈C

Ψc(Yc)

  • Forwarding graph

Dependency graph Aggregation graph Processor Fusion center

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 13 / 22

slide-27
SLIDE 27

Structure of Fusion Schemes for fixed selection

LLR(YVs) =

  • c∈C

Ψc(Yc)

  • LLR(Yc1)

Forwarding graph Dependency graph Aggregation graph Processor Fusion center

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 13 / 22

slide-28
SLIDE 28

Structure of Fusion Schemes for fixed selection

LLR(YVs) =

  • c∈C

Ψc(Yc)

  • LLR(Yc1)

Forwarding graph Dependency graph Aggregation graph Processor Fusion center

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 13 / 22

slide-29
SLIDE 29

Structure of Fusion Schemes for fixed selection

LLR(YVs) =

  • c∈C

Ψc(Yc)

  • LLR(YV ) =

2

  • i=1

LLR(Yci) Forwarding graph Dependency graph Aggregation graph Processor Fusion center

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 13 / 22

slide-30
SLIDE 30

Structure of Fusion Schemes for fixed selection

LLR(YVs) =

  • c∈C

Ψc(Yc)

  • LLR(YV ) =

2

  • i=1

LLR(Yci) Forwarding graph Dependency graph Aggregation graph Processor Fusion center

How to select useful groups? Dependency graph of Vs may be different!

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 13 / 22

slide-31
SLIDE 31

Prize-Collecting Steiner tree (PCST) Reduction

1 2 3 4 5 6 Fusion Center Node selection and data fusion via PCST reduction

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 14 / 22

slide-32
SLIDE 32

Prize-Collecting Steiner tree (PCST) Reduction

1 2 3 4 5 6 Fusion Center v12 v34 v56 D1,2 D3,4 D5,6 Node selection and data fusion via PCST reduction

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 14 / 22

slide-33
SLIDE 33

Prize-Collecting Steiner tree (PCST) Reduction

1 2 3 4 5 6 Fusion Center v12 v34 v56 D1,2 D3,4 D5,6 Groups Node selection and data fusion via PCST reduction

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 14 / 22

slide-34
SLIDE 34

Prize-Collecting Steiner tree (PCST) Reduction

1 2 3 4 5 6 Fusion Center v12 v34 v56 Node selection and data fusion via PCST reduction

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 14 / 22

slide-35
SLIDE 35

Prize-Collecting Steiner tree (PCST) Reduction

1 2 3 4 5 6 Fusion Center v12 v34 v56 Node selection and data fusion via PCST reduction

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 14 / 22

slide-36
SLIDE 36

Prize-Collecting Steiner tree (PCST) Reduction

Proc(1, 2) 1 2 3 4 5 6 Fusion CenterProc(5,6) v12 v34 v56 Node selection and data fusion via PCST reduction

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 14 / 22

slide-37
SLIDE 37

Prize-Collecting Steiner tree (PCST) Reduction

FG FG 1 2 3 4 5 6 Fusion Center Node selection and data fusion via PCST reduction

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 14 / 22

slide-38
SLIDE 38

Prize-Collecting Steiner tree (PCST) Reduction

FG FG AG AG AG AG 1 2 3 4 5 6 Fusion Center Node selection and data fusion via PCST reduction

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 14 / 22

slide-39
SLIDE 39

Prize-Collecting Steiner tree (PCST) Reduction

1 2 3 4 5 6

LLR(Y1, Y2, Y5, Y6)

Node selection and data fusion via PCST reduction

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 14 / 22

slide-40
SLIDE 40

Two Selection Heuristics

Component Selection Heuristic

Groups = components. No new cliques. Approximation guarantee.

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 15 / 22

slide-41
SLIDE 41

Two Selection Heuristics

Component Selection Heuristic

Groups = components. No new cliques. Approximation guarantee.

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 15 / 22

slide-42
SLIDE 42

Two Selection Heuristics

Component Selection Heuristic

Groups = components. No new cliques. Approximation guarantee.

Clique Selection Heuristic

Groups = cliques. New produced cliques. No approximation guarantee.

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 15 / 22

slide-43
SLIDE 43

Two Selection Heuristics

Component Selection Heuristic

Groups = components. No new cliques. Approximation guarantee.

Clique Selection Heuristic

Groups = cliques. New produced cliques. No approximation guarantee.

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 15 / 22

slide-44
SLIDE 44

Two Selection Heuristics

Component Selection Heuristic

Groups = components. No new cliques. Approximation guarantee.

Clique Selection Heuristic

Groups = cliques. New produced cliques. No approximation guarantee. Component selection = clique selection for disjoint cliques.

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 15 / 22

slide-45
SLIDE 45

Outline

1

Introduction

2

Problem Formulation and Main Results Network and Inference Model Cost-Performance Tradeoff Main Results Simplification of the Problem

3

Special Case: IID Measurements

4

General Correlation Cases: Two Selection Heuristics

5

Simulation

6

Conclusion

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 16 / 22

slide-46
SLIDE 46

Simulation: IID Measurements

Objective Value obj vs. Tradeoff Factor

50 100 150 200 250 300 0.5 1 1.5

Tradeoff factor µ

  • Avg. Cost-Perf. Tradeoff:

1 nobj

Approx PCDF No Fusion

Significant saving compared with no data fusion

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 17 / 22

slide-47
SLIDE 47

Simulation: Correlated Measurements

Component/Clique Selection n = 50

80 100 120 140 160 180 200 220 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

  • Avg. Cost-Perf. Tradeoff:

1 nobj

Tradeoff factor µ

  • Comp. selection heuristic

Clique selection heuristic

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 18 / 22

slide-48
SLIDE 48

Simulation: Correlated Measurements (cont.)

5 10 15 5 10 15 5 10 15 5 10 15

Uniform Placement Cluster Process

Component Selection, µ = 140, n = 200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

Dependency Disk Radius δ

  • Avg. Cost-Perf. Tradeoff:

1 nobj

Uniform Cluster

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 19 / 22

slide-49
SLIDE 49

Outline

1

Introduction

2

Problem Formulation and Main Results Network and Inference Model Cost-Performance Tradeoff Main Results Simplification of the Problem

3

Special Case: IID Measurements

4

General Correlation Cases: Two Selection Heuristics

5

Simulation

6

Conclusion

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 20 / 22

slide-50
SLIDE 50

Conclusion

Summary of Results

Prize-Collecting Data Fusion for cost-performance tradeoff PCST for IID data Component and clique selection heuristics for correlated data

Future directions

Local selection and coordination Realtime measures and delay

http://acsp.ece.cornell.edu/members/anima/pubs/Anandkumar09TR.pdf

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 21 / 22

slide-51
SLIDE 51

Dependency Graph and Markov Random Field

Consider an undirected graph G(V), each vertex Vi ∈ V is associated with a random variable Yi For any disjoint sets A, B, C such that C separates A and B, A B C

Yi Yk Yj

A B C A B C YA ⊥ ⊥ YB|YC

Anandkumar, Wang, Tong, Swami PCDF for Cost-Performance Tradeoff INFOCOM 09 22 / 22