1. Distributed Optimization in Sensor Networks
Mike Rabbat & Rob Nowak
Monday, April 26, 2004
IPSN’04, Berkeley, CA

2. A Motivating Example “What is µ(x)?” “It’s !” (E.g., ) x1 x2 x3 x4 x5xn

3. Two Extreme Approaches
n sensors distributed uniformly over a square region
compute
1) Transmit Data
2) Transmit A Result x1 x2 x3 x4 x5 x6 x7 x… xn x1 x2 x3 x4 x5 x6 x7 x… xn

4. Energy-Accuracy Tradeoffs
Multi-hop communication
b(n) = total number of bits
h(n) = avg number of hops per bit
e(n) = avg energy per hop
Total Energy Consumption
Consider two situations
Sensors transmit data, result computed at destination
Sensors process in-network, result transmitted to destination

5. Distributed Iterative Optimization
n sensors
Minimize w.r.t. µ
E.g., fi(µ; xi) = (µ – xi)2
Strategy:
Cycle over each sensor
Update using previous value and local data x… x7 x6 xn x5 x4 x1 x2 x3

6. Distributed Iterative Optimization
n sensors
Minimize w.r.t. µ
E.g., fi(µ; xi) = (µ – xi)2
Strategy:
Cycle over each sensor
Update using previous value and local data
#1(1) x… x7 x6 xn x5 x4 x1 x2 x3
Energy used:

7. Distributed Iterative Optimization
n sensors
Minimize w.r.t. µ
E.g., fi(µ; xi) = (µ – xi)2
Strategy:
Cycle over each sensor
Update using previous value and local data
#1(1) x… x7 x6 xn x5 x4 x1 x2 x3
#2(1)
Energy used:

8. Distributed Iterative Optimization
n sensors
Minimize w.r.t. µ
E.g., fi(µ; xi) = (µ – xi)2
Strategy:
Cycle over each sensor
Update using previous value and local data
#n(1)
#1(1) x… x7 x6 xn x5 x4 x1 x2 x3
#2(1)
Energy used:

9. Distributed Iterative Optimization
n sensors
Minimize w.r.t. µ
E.g., fi(µ; xi) = (µ – xi)2
Strategy:
Cycle over each sensor
Update using previous value and local data
Repeat K times
#n(K)
#1(K) x… x7 x6 xn x5 x4 x1 x2 x3
#2(K)
Energy used:
Compared with:

10. Incremental Subgradient Methods
Have data xi at sensor i, for i=1,2,…,n
Find µ that minimizes
Distributed iterative procedure
small positive step size

11. Incremental Subgradient Methods
Have data xi at sensor i, for i=1,2,…,n
Find µ that minimizes
Distributed iterative procedure
use subgradient of fi is non-differentiable

12. Convergence Theorem (Nedić & Bertsekas, ’01):
Assume the fi are convex on a convex set , µ*2 , and kr fi(µ)k<C. Define D = diam(). Then after K cycles, with
we are guaranteed that

13. Comparing Resource Usage
1) Transmit Data
2) Transmit A Result x1 x2 x3 x4 x5 x6 x7 x… xn x1 x2 x3 x4 x5 x6 x7 x… xn

14. Energy Savings
When does distributed processing use less energy?
vs.

15. Robust Estimation Estimate the mean (squared error loss)
Robust estimate (robust loss function)

16. Robust Estimates and “Bad Sensors”
E.g., Monitoring ozone levels, µ = average level
Normal sensor measures µ ± 1
Damaged sensor measures µ ± 10
Energy used (K = 25 iterations) compared to
200 sensors
10 measurements each
10% “bad”
robust loss
squared error loss

17. Source Localization Isotropic energy source located at µ
Sensor i at location ri, measures received signal strength
Local cost functions

18. Source Localization 100 sensors 50 £ 50 square 10 measurements
Avg SNR = 3dB
Converged in 45 cycles
Compare with

19. In Conclusion When is in-network processing more energy efficient?
Incremental subgradient optimization
Simple to implement
Applicable to a general class of problems
Analyzable rate of convergence
Distributed in-network processing uses less energy:
vs.

20. Ongoing & Future Work xn x… x7 x1 x6 x5 x2 x3 x4 rabbat@cae.wisc.edu