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Decentralized Jointly Sparse Optimization byReweighted Lq Minimization

Qing Ling Department of Automation

University of Science and Technology of China

Joint work with Zaiwen Wen (SJTU) and Wotao Yin (RICE)

2012/09/05

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A brief introduction to my research interest

optimization and control in networked multi-agent systems

autonomous agents- collect data- process data- communicate

problem: how to efficiently accomplish in-network optimization and control tasks through collaboration of agents?

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Large-scale wireless sensor networks: decentralized signal processing, node localization, sensor selection …

how to fuse big sensory data?e.g. structural health monitoring

how to localize blinds with anchors?

blind anchor

how to assign sensors to targets?

difficulty in data transmission→ decentralized optimization without any fusion center

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Computer/server networks with big data: collaborative data mining

new challenges in the big data era- big data is stored in distributed computers/servers- data transmission is prohibited due to bandwidth/privacy/…- computers/servers collaborate to do data mining

distributed/decentralized optimization

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Wireless sensor and actuator networks: with application in large-scale greenhouse control

decentralized control system design

wireless sensing- temperature- humidity- …

wireless actuating- circulating fan- wet curtain- …

disadvantages of traditional centralized control- communication burden in collecting distributed sensory data- lack of robustness due to packet-loss, time-delay, …

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Recent works

wireless sensor networks- decentralized signal processing with application in SHM- decentralized node localization using SDP and SOCP- decentralized sensor node selection for target tracking

collaborative data mining- decentralized approaches to jointly sparse signal recovery- decentralized approaches to matrix completion

wireless sensor and actuator networks- modeling, hardware design, controller design, prototype

theoretical issues- convergence and convergence rate analysis

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Decentralized Jointly Sparse Optimization byReweighted Lq Minimization

Qing Ling Department of Automation

University of Science and Technology of China

Joint work with Zaiwen Wen (SJTU) and Wotao Yin (RICE)

2012/09/05

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Outline

Background decentralized jointly sparse optimization with applications

Roadmap nonconvex versus convex, difficulty in decentralized computing

Algorithm development successive linearization, inexact average consensus

Simulation and conclusion

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Background (I): jointly sparse optimization

Structured signals A sparse signal: only few elements are nonzero Jointly sparse signals: sparse, with the same nonzero supports

Jointly sparse optimization: to recover X from linear measurements

nonzeros

zeros

measurement matrix measurement noise

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Background (II): decentralized jointly sparse optimization

Decentralized computing in a network Distributed data in distributed agents & no fusion center Consideration of privacy, difficulty in data collection, etc

Goal: agent i has y(i) and A(i), to recover x(i) through collaboration Decentralized jointly sparse optimization

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Background (III): applications

Cooperative spectrum sensing [1][2] Cognitive radios sense jointly sparse spectra {x(i)} Measure from time domain [1] or frequency selective filter [2] Decentralized recovery from {y(i)=A(i)x(i)}

[1] F. Zeng, C. Li, and Z. Tian, “Distributed compressive spectrum sensing in cooperative multi-hop wideband cognitive networks,” IEEE Journal of Selected Topics in Signal Processing, vol. 5, pp. 37–48, 2011[2] J. Meng, W. Yin, H. Li, E. Houssain, and Z. Han, “Collaborative spectrum sensing from sparse observations for cognitive radio networks,” IEEE Journal on Selected Areas on Communications, vol. 29, pp. 327–337, 2011[3] N. Nguyen, N. Nasrabadi, and T. Tran, “Robust multi-sensor classification via joint sparse representation,” submitted to Journal of Advance in Information Fusion

Decentralized event detection [3] Sensors {i} sense few targets represented by jointly sparse {x(i)} Decentralized recovery from {y(i)=A(i)x(i)}

Collaborative data mining, distributed human action recognition, etc

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Roadmap (I): nonconvex versus convex

Convex model: group lasso or L21 norm minimization

Nonconvex versus convex Convex: with global convergence guarantee Nonconvex: often with better recovery performance

Look back on nonconvex models to recover a single sparse signal

Reweighted L1/L2 norm minimization [4][5] Reweighted algorithms for jointly sparse optimization?

[4] E. Candes, M. Wakin, and S. Boyd, “Enhancing sparsity by reweighted L1 minimization,” Journal of Fourier Analysis and Applications, vol. 14, pp. 877–905, 2008[5] R. Chartrand and W. Yin, “Iteratively reweighted algorithms for compressive sensing,” In: Proceedings of ICASSP, 2008

regularization parameter

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Roadmap (II): difficulty in decentralized computing

A popular decentralized computing technique: consensus

common optimization variableobjective function in agent i

local copy in agent i neighboring copies are equal Obviously, two problems are equivalent for a connected network

Efficient algorithms (ADM, SGD, etc) for if it is convex [6] Nothing for consensus in jointly sparse optimization! Signals are different; common supports bring nonconvexity

[6] D. Bertsekas and J. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Second Edition, Athena Scientific, 1997

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Roadmap (III): solution overview

Nonconvex model + convex decentralized computing subproblem Nonconvex model -> successive linearization -> reweighted Lq Natural decentralized computing, one nontrivial subproblem Inexactly solving the subproblem still leads to good recovery

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Algorithm (I): successive linearization

Nonconvex model (q=1 or 2)

regularization parameter

smoothing parameter

“Successive linearization” to the joint sparsity term at t

Actually a majorization minimization approach

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Algorithm (II): reweighted algorithm

Centralized reweighted Lq minimization algorithm Updating weight vector

weight vector u=[u1; u2; uN] Updating signals

From a decentralized implementation perspective … Natural decentralized computing in x-update One subproblem needs decentralized solution in u-update

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Algorithm (III): average consensus

Check u-update: average consensus problem

Rewrite to more familiar forms

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Algorithm (IV): inexact average consensus

Solve the average consensus problem with ADM (time t, slot s/S)

Updating weight vectors (local copies)

Updating Lagrange multipliers (c is a positive constant)

Exact average consensus versus inexact average consensus Exact average consensus: exact implementation of reweighted Lq Introducing inner loops: cost of coordination & communication Inexact average consensus: one iteration in the inner loop

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Algorithm (V): decentralized reweighted Lq

Algorithm outline Updating weight vectors (local copies)

Updating Lagrange multipliers (c is a positive constant)

Updating signals

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Simulation (I): simulation settings

Network settings L=50 agents, randomly deployed in 100×100 area Communication range=30, bidirectionally connected

Measurement settings Signal dimension N=20, signal sparsity K=2 Measurement dimension M=10 Random measurement matrices and random measurement

noise Parameter settings

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Simulation (II): recovery performance

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Simulation (III): convergence rate

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Conclusion

Decentralized jointly sparse optimization problem Jointly sparse signal recovery in a distributed network Reweighted Lq minimization algorithms Feature #1: nonconvex model <- successive linearization Feature #2: decentralized computing <- inexact average consensus

Outlook: many open questions in decentralized optimization

Good news and bad news Local convergence of the centralized algorithms Excellent performance of the decentralized algorithms No theoretical performance guarantee (recovery and

convergence)

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Thanks for your attention!