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
Jan 05, 2016
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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
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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
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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]
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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
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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
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A Useful Reformulation
Introduce the bridge sensor subset1) For all sensors , such that2) For , a path connecting them devoid of edges linking
two sensors
Consider the convex, constrained optimization
Proposition [Schizas etal’06]: For satisfying 1)-2) and the WSN is connected, then
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Algorithm Construction Associated augmented Lagrangian
Two key steps in deriving D-LMS1) Resort to the alternating-direction method of multipliers
Gain desired degree of parallelization
2) Apply stochastic approximation ideasCope with unavailability of statistical
information
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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:
Sensor
Rxfrom
Tx
toBridge sensor
Txto
Rx
from
Steps 1,2:
Step 3:
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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
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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
EMSEMSD
Local
Global
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Tracking Performance(a4) Random-walk model: where is zero-mean
white with covariance ; independent of and
Let where Convenient c.v.:
Proposition: Under (a2)-(a4), the covariance matrix of obeys
with . Equivalently, after vectorization
where
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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.
Proposition: Under (a1)-(a4), the D-LMS algorithm is MSE stable for sufficiently small
Proposition: Under (a1)-(a4), the D-LMS algorithm achieves consensus in the mean, i.e., provided the step-size is chosen such that with
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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).
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, D-LMS:
Simulated Tests node WSN, Rx AWGN w/ ,
Random-walk model:Time-invariant parameter:
Regressors: w/
; i.i.d.; w/
Observations: linear data model, WGN w/
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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