Distributed Optimization in Sensor
Networks
Mike Rabbat & Rob Nowak
Monday, April 26, 2004
IPSN’04, Berkeley, CA
A Motivating Example
“What is µ(x)?”
(E.g.,
“It’s
)
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!”
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Two Extreme Approaches
n sensors distributed uniformly
over a square region
1) Transmit Data
2) Transmit A Result
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compute
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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
1. Sensors transmit data, result computed at destination
2. Sensors process in-network, result transmitted to destination
Distributed Iterative Optimization
n sensors
Minimize w.r.t. µ
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E.g., fi(µ; xi) = (µ – xi)2
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Strategy:
• Cycle over each sensor
• Update using previous
value and local data
Distributed Iterative Optimization
n sensors
Minimize w.r.t. µ
#1(1)
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Energy used:
E.g., fi(µ; xi) = (µ – xi)2
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Strategy:
• Cycle over each sensor
• Update using previous
value and local data
Distributed Iterative Optimization
n sensors
Minimize w.r.t. µ
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Energy used:
E.g., fi(µ; xi) = (µ – xi)2
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#1(1)
#2(1)
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Strategy:
• Cycle over each sensor
• Update using previous
value and local data
Distributed Iterative Optimization
n sensors
Minimize w.r.t. µ
#n(1)
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Energy used:
E.g., fi(µ; xi) = (µ – xi)2
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#1(1)
#2(1)
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Strategy:
• Cycle over each sensor
• Update using previous
value and local data
Distributed Iterative Optimization
n sensors
Minimize w.r.t. µ
#n(K)
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Energy used:
Compared with:
E.g., fi(µ; xi) = (µ – xi)2
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#1(K)
#2(K)
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Strategy:
• Cycle over each sensor
• Update using previous
value and local data
• Repeat K times
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
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
Convergence
Theorem (Nedić & Bertsekas, ’01):
Assume the fi are convex on a convex set ,
µ*2 , and krfi(µ)k<C. Define D = diam().
Then after K cycles, with
we are guaranteed that
Comparing Resource Usage
1) Transmit Data
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2) Transmit A Result
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Energy Savings
• When does distributed processing use less energy?
Energy
vs.
n
Robust Estimation
• Estimate the mean (squared
error loss)
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3.5
3
2.5
(z)
• Robust estimate
(robust loss function)
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1.5
1
0.5
0
-2
-1
0
z
1
2
Robust Estimates and “Bad Sensors”
• E.g., Monitoring ozone levels, µ = average level
• Normal sensor measures µ ± 1
200 sensors
10 measurements each
10% “bad”
• Damaged sensor measures µ ± 10
10
• Energy used
(K = 25 iterations)
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Residual Error
compared to
robust loss
squared error loss
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-10
0
50
100
Iteration (k)
150
200
Source Localization
• Isotropic energy source located at µ
• Sensor i at location ri,
measures received signal strength
• Local cost functions
Source Localization
100 sensors
50 £ 50 square
10 measurements
Avg SNR = 3dB
Converged in 45 cycles
Compare with
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.
Ongoing & Future Work
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[email protected]
www.cae.wisc.edu/~rabbat
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