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. DAAD USDoD ARO grant no. W911NF

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 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 Proposition [Schizas etal ’ 06]: For satisfying 1)-2) and the WSN is connected, then

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 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 Rx from Tx to Bridge sensor Tx to Rx from Steps 1,2: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 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

11 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

12 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

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, 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/

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