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Closed-Form MSE Performance of the Distributed LMS Algorithm

Closed-Form MSE Performance of the Distributed LMS Algorithm. Gonzalo Mateos, Ioannis Schizas and Georgios B. Giannakis ECE Department, University of Minnesota Acknowledgment: ARL/CTA grant no. DAAD19-01-2-0011 USDoD ARO grant no. W911NF-05-1-0283. Motivation.

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Closed-Form MSE Performance of the Distributed LMS Algorithm

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  1. Closed-Form MSE Performance of the Distributed LMS Algorithm Gonzalo Mateos, Ioannis Schizas and Georgios B. Giannakis ECE Department, University of Minnesota Acknowledgment: ARL/CTA grant no. DAAD19-01-2-0011 USDoD ARO grant no. W911NF-05-1-0283

  2. Motivation • Estimation using ad hoc WSNs raises exciting challenges • Communication constraints • Limited power budget • Lack of hierarchy / decentralized processing Consensus • Unique features • Environment is constantly changing (e.g., WSN topology) • Lack of statistical information at sensor-level • Bottom line: algorithms are required to be • Resource efficient • Simple and flexible • Adaptive and robust to changes Single-hop communications

  3. Prior Works • Single-shot distributed estimation algorithms • Consensus averaging [Xiao-Boyd ’05, Tsitsiklis-Bertsekas ’86, ’97] • Incremental strategies [Rabbat-Nowak etal ’05] • Deterministic and random parameter estimation [Schizas etal ’06] • Consensus-based Kalman tracking using ad hoc WSNs • MSE optimal filtering and smoothing [Schizas etal ’07] • Suboptimal approaches [Olfati-Saber ’05],[Spanos etal ’05] • Distributed adaptive estimation and filtering • LMS and RLS learning rules [Lopes-Sayed ’06 ’07]

  4. Problem Statement • Ad hoc WSN with sensors • Single-hop communications only. Sensor ‘s neighborhood • Connectivity information captured in • Zero-mean additive (e.g., Rx) noise • Goal: estimate a signal vector • Each sensor , at time instant • Acquires a regressor and scalar observation • Both zero-mean and spatially uncorrelated • Least-mean squares (LMS) estimation problem of interest

  5. Power Spectrum Estimation • Find spectral peaks of a narrowband (e.g., seismic) source • AR model: • Source-sensor multi-path channels modeled as FIR filters • Unknown orders and tap coefficients • Observation at sensor is • Define: • Challenges • Data model not completely known • Channel fades at the frequencies occupied by

  6. Proposition [Schizas etal’06]: For satisfying 1)-2) and the WSN is connected, then A Useful Reformulation • Introduce the bridge sensor subset • For all sensors , such that • For , a path connecting them devoid of edges linking two sensors • Consider the convex, constrained optimization

  7. Algorithm Construction • Associated augmented Lagrangian • Two key steps in deriving D-LMS • Resort to the alternating-direction method of multipliers Gain desired degree of parallelization • Apply stochastic approximation ideas Cope with unavailability of statistical information

  8. Steps 1,2: Step 3: Tx Rx Tx to from to Bridge sensor Sensor Rx from D-LMS Recursions and Operation • In the presence of communication noise, for and • Simple, distributed, only single-hop exchanges needed Step 1: Step 2: Step 3:

  9. Error-form D-LMS • Study the dynamics of • Local estimation errors: • Local sum of multipliers: (a1) Sensor observations obey where the zero-mean white noise has variance • Introduce and Lemma: Under (a1), for then where and consists of the blocks and with

  10. MSD EMSE Local Global Performance Metrics • Local (per-sensor) and global (network-wide) metrics of interest (a2) is white Gaussian with covariance matrix (a3) and are independent • Define • Customary figures of merit

  11. Proposition:Under (a2)-(a4), the covariance matrix of obeys with . Equivalently, after vectorization where Tracking Performance (a4) Random-walk model: where is zero-mean white with covariance ; independent of and • Let where • Convenient c.v.:

  12. Proposition: Under (a1)-(a4), the D-LMS algorithm achieves consensus in the mean, i.e., provided the step-size is chosen such that with Proposition: Under (a1)-(a4), the D-LMS algorithm is MSE stable for sufficiently small Stability and S.S. Performance • MSE stability follows • Intractable to obtain explicit bounds on • From stability, has bounded entries • The fixed point of is • Enables evaluation of all figures of merit in s.s.

  13. Step-size Optimization • If optimum minimizing EMSE • Not surprising • Excessive adaptation MSE inflation • Vanishing tracking ability lost • Recall • Hard to obtain closed-form , but easy numerically (1-D).

  14. Regressors: w/ ; i.i.d.; w/ Observations: linear data model, WGN w/ Time-invariant parameter: Random-walk model: Simulated Tests node WSN, Rx AWGN w/ , , D-LMS:

  15. Concluding Summary • Developed a distributed LMS algorithm for general ad hoc WSNs • Detailed MSE performance analysis for D-LMS • Stationary setup, time-invariant parameter • Tracking a random-walk • Analysis under the simplifying white Gaussian setting • Closed-form, exact recursion for the global error covariance matrix • Local and network-wide figures of merit for and in s.s. • Tracking analysis revealed minimizing the s.s. EMSE • Simulations validate the theoretical findings • Results extend to temporally-correlated (non-) Gaussian sensor data

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