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Decentralized Jointly Sparse Optimization by Reweighted 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. A brief introduction to my research interest. - PowerPoint PPT Presentation
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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!