Distributed Inference: High Dimensional Consensus

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Carnegie Mellon Distributed Inference: High Dimensional Consensus José M. F. Moura Work with: Usman A. Khan (Upenn), Soummya Kar (Princeton) The Australian National University RSISE Systems and Control Series Seminar Canberra, Australia, July 30, 2010 Acknowledgements: AFOSR grant # FA95501010291, NSF grant # CCF1011903, ONR MURI N000140710747

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Distributed Inference: High Dimensional Consensus. Jos é M. F. Moura Work with: Usman A. Khan ( Upenn ), Soummya Kar (Princeton) The Australian National University RSISE Systems and Control Series Seminar Canberra, Australia, July 30, 2010. - PowerPoint PPT Presentation

Transcript of Distributed Inference: High Dimensional Consensus

Page 1: Distributed Inference: High Dimensional Consensus

Carnegie Mellon

Distributed Inference: High Dimensional Consensus

José M. F. MouraWork with: Usman A. Khan (Upenn), Soummya Kar

(Princeton)

The Australian National UniversityRSISE Systems and Control Series Seminar

Canberra, Australia, July 30, 2010

Acknowledgements: AFOSR grant # FA95501010291, NSF grant # CCF1011903, ONR MURI N000140710747

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Outline

Motivation for networked systems and distributed algorithms

Identify main characteristics of networked systems and distributed algorithms

Consensus algorithms and emerging behavior Example: Localization Conclusions

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Motivation Networked systems: agents, sensors

Applications: inference (detection, estimation, filtering, …) Distributed algorithms:

Consensus:

More general algorithms – High dimensional consensus

Realistic conditions: Randomness: infrastructure (link failures), random protocols (gossip),

communication noise Quantization effects Measurement updates

Issues: convergence – design topology to speed convergence; prove Applications

Localization

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Example

1 2

3

W12 =W21 =0

W =

2

4W11 0 W13

0 W22 W23

W31 W32 W33

3

5

• If link not available,• W is symmetric, sparse• W reflects the topology of network• Neighborhoods:

Wi j =Wj i =0

­ 3 = f1;2g;­ 1 = f3g;­ 2 = f3g

xn(i +1) =Wnnxn(i) +P

l2 ­ nWnlxl(i)

x(i) = [x1(i) ¢¢¢xN (i)]T

x(i +1) =Wx(i)

In matrix form, consensus is:

Consensus is linear & iterative – issues: convergence and rate of convergence

Networked Systems: Consensus

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Consensus: Optimization

Consensus

Convergence

Limit

Spectral condition

1 2

3

Kar & Moura, Transactions Signal Processing, vol. 56, no. 6, June 2008

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Topology Design

Speed convergence by making small

Choose where nonzero entries of W are and the actual values of the nonzero entries of W

W =

2

4W11 0 W13

0 W22 W23

W31 W32 W33

3

51 2

3

1 2

3

Kar & Moura, Transactions Signal Processing, vol. 56, no. 6, June 2008

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Topology Design

Design Laplacean to minimize

Equal weights : weight α (Xiao and Boyd, CDC, Dec 2003)

Graph design:

subject to constraints, e.g., number of edges M, structure of graph

Kar & Moura, Transactions Signal Processing, vol. 56, no. 6, June 2008

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Topology Design

Nonrandom topology: (topology static or fixed) Class 1: Noiseless communication

Class 2: Noisy communication Random topology: Class 3 links may fail intermittently

Random topology with communication costs and budget constraint: Class 4 Communication in link (i,j) has cost Link (i,j) fails with probability Average comm. network budget constraint per iteration

x(i +1) =Wx(i)

x(i +1) =Wx(i) +n(i)

random

Kar & Moura, Transactions Signal Processing, vol. 56, no. 6, June 2008

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Topology Design – Class 1: Ramanujan (LPS)

Fig.1. A non-bipartite LPS Ramanujan graph with degree k = 6, and number of vertices N = 62 (Figure constructed using software Pajek)

Lubotzky, Phillips, Sarnak (LPS) (1988) and Margulis (1988)

(22/12/1887 – 26/4/1920)

Kar & Moura, Transactions Signal Processing, vol. 56, no. 6, June 2008

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Comparison Studies

Convergencespeed, Sc, of P ne (i) ! Pe

Performance Metric

LPS Ramanujan(We use a non-bipartite Ramanujan graph construction from LPS and call it LPS-II.)

Regular Ring Lattice (RRL)Highly-structured regular graphswith nearest-neighbor connectivity

Erdos-Renyi (ER)Random networks

Watts-Strogatz (WS-I)Small-world networks usingWatts-Strogatz construction

vs vs vs

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LPS Ramanujan vs Regular Ring Lattice (RRL)

m1 = ¡ m0 = ¹ 1; K =¾2I ; SNR = 10log³¹ 2

¾2

´= ¡ 25 db

1000 1200 1400 1600 1800 2000 2200200

300

400

500

600

700

800

900

N

Sc(L

PS¡II)

Sc(R

ingLatt

ice)

Fig.2. Ratio of convergence speed of LPS-II graphs to RRL graphs for different values of N and k = 18.

Regular graph

Ratio

spe

ed c

onve

rgen

ce

Ramanujan

Kar & Moura, Transactions Signal Processing, vol. 56, no. 6, June 2008

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LPS-II vs Erdös-Renýi (ER)

The top blue line corresponds to the LPS-II graphs. The LPS-II graphs perform much better than the best ER graphs.

The relative performance of the LPS-II graphs over the ER graphs increases steadily with increasing N.

m1 = ¡ m0 = ¹ 1; K =¾2I ; SNR = 10log³¹ 2

¾2

´= ¡ 25 db

1000 1500 2000 2500 3000 35000.1

0.12

0.14

0.16

0.18

0.2

0.22

N

Sc

LPS-IIER(max)ER(avg)ER(min)

Fig.3. Comparison of convergence speed between LPS-II graphs and ER graphs for varying N and kavg = 18.

Kar & Moura, Transactions Signal Processing, vol. 56, no. 6, June 2008

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LPS-II vs Watts-Strogatz (WS-I)

0.2 0.4 0.6 0.8 10.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

pw

Sc

LPS-IIWS-I(max)WS-I(avg)WS-I(min)

SmallWorld

Random

Fig. 4. Study of convergence speed Sc of WS-I graphs with N = 6038 and kavg = 18. The top blue line denotes the LPS-II graph with N = 6038 and k = 18.

Kar & Moura, Transactions Signal Processing, vol. 56, no. 6, June 2008

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Topology Design–Class 4: Communication Costs Communication in link (i,j) has cost Link (i,j) fails with probability Average comm. network budget constraint per iteration

Convex optimization (SDP):

Kar & Moura, Transactions Signal Processing, vol. 56:7, July 2008

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Random Topology w/ Comm. Cost

Fig. 4. Per step convergence gain Sg: N = 80 and |E| = 9N=720

Kar & Moura, Transactions Signal Processing, vol. 56:7, July 2008

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High Dimensional Consensus

LOCAL INTERACTIONS: The local updates are given as

GLOBAL BEHAVIOR: Under whatconditions does HDC converge:

for some appropriate function, wl

limt! 1

xl(t+1) =wl (x(0);u(0)) ; (1)

16Khan, Kar, Moura, ICASSP ‘09, ‘10, ASILOMAR ‘09, IEEE TSP ‘10.

uk(t +1) = uk(t); k 2 · ;xl ;j (t +1) = f l

¡xK ­ (l);j (t)

¢+gl

¡uK · (l);j (t)

¢; l 2 ­ ;

(1)xl(t +1) =X

j 2K ­ (l)[ f lg

pl j xj (t) +X

k2K · (l)

blkuk(t): (1)

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Distributed Localization

m = 2-D plane

Localize M sensors with unknown locations in m-dimensional Euclidean space [1]

Minimal number, n=m+1 , of anchors with known locations

Sensors only communicate in a neighborhood

Only local distances in the neighborhood are known to the sensor

There is no central fusion center[1] Khan, Kar, Moura, “Distributed Sensor Localization in Random Environments using Minimal Number of Anchor Nodes,” IEEE Tr. on Sign. Pr., 57(5), pp. 2000-2016, May 2009.

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Distributed Sensor Localization

Assumptions Sensors lie in convex hull of anchors Anchors not on a hyper-plane Sensors find m+1 neighbors so they lie in their

convex hull Only local distances available

Distributed localization (DILOC) algorithm Sensor updates position estimate as convex l.c. of n=m+1 neighbors Weights of l.c. are barycentric coordinates Barycentric coordinates: ratio of generalized volumes Barycentric coordinates: Cayley-Menger determinants (local distances)

xl(t +1) = pl lxl(t) +X

j 2K ­ (l)

pl j xj (t) +X

k2K · (l)

blkuk(0):

(TRIANGULATION)

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Barycentric Coord. & Cayley-Menger Det.

pl3 =A1l2

A123

1

32

l

Barycentric coordinates: Example 2D:

Cayley-Menger determinants:

plk =Af lg[ £ l nf kg

A£ l

A2· =

1sm+1

¯¯¯¯

0 1Tm+11m+1 ¡

¯¯¯¯;

where ¡ = fd2lj g; l; j 2 · ; is the matrix of squared distances, dl j , among them+1 points in · and

sm =2m(m!)2

(¡ 1)m+1 ; m= f0;1;2;: : :g:

Its ¯rst few terms are ¡ 1;2;¡ 16;288;¡ 9216;460800;: : : :

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Set-up phase: Triangulation

Test to find a triangulation set, Convex hull inclusion test: based on the following observation.

The test becomes

A l12+A l13+A l23 =A123 A l12+A l13+A l23 >A123

32 32

1

l

1

l

l 2 C(£ l ); ifX

k2£ l

A£ l [ f lgnf kg = A£ l ;

l =2 C(£ l); ifX

k2£ l

A · [ f lgnf kg >A£ l

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Distributed Localization Distributed localization algorithm (DILOC)

K anchors and M sensors (K+M=N) in m dimensions:

Matrix form:

xl(t +1) = pl lxl(t) +X

j 2K ­ (l)

pl j xj (t) +X

k2K · (l)

blkuk(0):

U =£u1(t) ¢¢¢um(t)

¤; U : K £ m

X (t) =£x1(t) ¢¢¢xm(t)

¤; X (t) : M £ m

·U

X (t+1)

¸=

·I 0B P

¸ ·U

X (t)

¸

limt! 1

X (t) = [I ¡ P ]¡ 1BU

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Distributed Localization: DILOC

DILOC:

Assume:

·U

X (t+1)

¸=

·I 0B P

¸ ·U

X (t)

¸

{𝚼§ k2­ l plk +§ k2· l blk = 1

(Triangulation)

(Barycentric Coordinates)

limt! 1

X (t) = [I ¡ P ]¡ 1BU

Theorem [Convergence]: Under above assumptions:1. The underlying Markov chain with transition probability is

absorbing.2. DILOC converges to the exact sensor coordinates:

𝚼

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Distributed Localization: Simulations N=7 node network in 2-d plane

M= 4 sensors, K = m+1 = 3 anchors

M = 497 sensors

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Convergence of DILOC

Theorem [Convergence]:

Connected on average

Random network

Persistence cond.

Noisy communication vj (t)

Distributed distance localization algorithm converges

Errors in intersensor distances P¡¹dt

¢and B

¡¹dt¢

1t t t d d d,

Khan, Kar, and Moura, “DILAND: An Algorithm for Distributed Sensor Localiz. with Noisy Distance Meas.,” IEEE Tr. Signal Pr., 58:3, 1940-1947, March 2010

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Proof of Theorem Proof: Cannot use standard stochastic approx. techniques because

function of past measurements, non Markovian

Study path behavior of error process wrt idealized update

Define error process wrt idealized update:

Dynamics of error process:

Error goes to zero:

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Conclusion

High Dimensional Consensus Optimization: Topology design Distributed localization (DILOC):

Linear iterative Local communications Barycentric coordinates Cayley-Menger determinants

Convergence: Deterministic networks (protocols): standard Markov chain

arguments Random networks: structural (link) failures, noisy comm, quantized

data − standard stochastic approximation algorithms not sufficient to prove convergence

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Bibliography Soummya Kar, Saeed Aldosari, and José M. F. Moura, “

Topology for Distributed Inference on Graphs,” IEEE Transactions on Signal Processing, volume 56 number 6, pp. 2609-2613, June 2008.

Soummya Kar and José M. F. Moura, “Sensor Networks with Random Links: Topology Design for Distributed Consensus,” IEEE Transactions on Signal Processing, 56:7, pp. 3315-3326, July 2008.

U. A. Khan, S. Kar, and J. M. F. Moura, “Distributed Sensor Localization in Random Environments using Minimal Number of Anchor Nodes,” IEEE Transactions on Signal Processing, 57: 5, pp. 2000-2016, May 2009; DOI:10.1109/TSP.2009.2014812.

U. A. Khan, S. Kar, and J. M. F. Moura, “DILAND: An Algorithm for Distributed Sensor Localization with Noisy Distance Measurements,” IEEE Transactions on Signal Processing, Vol. 58:3, pp.:1940-1947, March 2010.

U. A. Khan, S. Kar, and J. M. F. Moura, “Higher dimensional consensus: Learning in large-scale Networks,” IEEE Transactions on Signal Processing, Vol. 58:5, pp. 2836 -2849, May 2010.

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