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Closed-Form MSE Performance of the Distributed LMS Algorithm

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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

- 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

- 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]

- 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

- 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

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

- 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

- 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

- Resort to the alternating-direction method of multipliers

Steps 1,2:

Step 3:

Tx

Rx

Tx

to

from

to

Bridge sensor

Sensor

Rx

from

- In the presence of communication noise, for and
- Simple, distributed, only single-hop exchanges needed

Step 1:

Step 2:

Step 3:

- 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

MSD

EMSE

Local

Global

- 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

Proposition:Under (a2)-(a4), the covariance matrix of obeys

with . Equivalently, after vectorization

where

(a4) Random-walk model: where is zero-mean white with covariance ; independent of and

- Let where
- Convenient c.v.:

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

- 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.

- 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).

Regressors: w/

; i.i.d.; w/

Observations: linear data model, WGN w/

Time-invariant parameter:

Random-walk model:

node WSN, Rx AWGN w/ ,

, D-LMS:

- 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