Distributed Optimization in Sensor Networks
Distributed Optimization in Sensor Networks Michael Rabbat, Robert Nowak. Information Processing in Sensor Networks, 2004
October 14, 2011
Meng Guo
Automatic Control Lab, KTH
Introduction
Introduction
Wireless sensor Network (WSN) collects enormous amount of data over space and time. Objective: to derive an optimal estimate of certain parameters from these “raw” data. How? To minimize the energy consumption. Simple example: n sensors uniformly distributed over 1 m2 . Each collects m measurements. Task: what is the average value?
Meng Guo
Automatic Control Lab, KTH
Introduction
Possible Approaches
Possible Approaches
1
Sensors transmit all data to a central processor. O(mn) bits over O(1) meter.
2
Sensors first compute a local average and then transmit to the central unit.O(n) bits over O(1) meter.
3
Construct a path passing through all sensors once. Each node adds its local average to the total along the way. An accumulation process from start node to finish node. O(n) bits over O(n−1/2 ) meter. Far less energy and communication needed for large (m, n).
Meng Guo
Automatic Control Lab, KTH
General Setup
General Setup
So-called Incremental Subgradient Optimization. An estimate of the parameter is passed from node to node. Each node adjusts the estimate based on local data, then passes the update to the next node. Several “cycles” through the network may be needed.
It is distributed, “in-network” optimization!
Meng Guo
Automatic Control Lab, KTH
Formulation
Formulation A network of n sensors in which each sensor collects m measurements. Denote xi,j the jth measurement at the ith sensor. Goal: 1 θˆ = arg min θ∈Θ n θ∈
Rd
n X
fi ({xi,j }m j=1 , θ)
i=1
is the set of parameters to be estimated.
Cost function fi : Rd → R, are convex and Θ is a nonempty, closed, convex subset of Rd . Denote fi (θ) = fi ({xi,j }m j=1 , θ), which depends on sensor i’s measurements.
Meng Guo
Automatic Control Lab, KTH
Formulation
Centralized Method
Centralized Version Centralized subgradient descent approach: θˆ(k+1) = θˆ(k) − α
n X
gi,k ,
i=1
where gi,k ∈ ∂fi (θˆ(k) ), α is a positive step size, and k is the iteration number. Note! each update step in this approach uses data from all of the sensors. ∂f (θ0 ) is the subdifferential of f at θ0 , the set of all subgradients of f at θ0 . The subgradient of f at θ0 is any direction g satisfying f (θ) ≥ f (θ0 ) + (θ − θ0 )T g ,
Meng Guo
∀θ ∈ Θ
Automatic Control Lab, KTH
Formulation
Decenralized Incremental Method
Decenralized Incremental Method
Each update: a cycle of n subiterations (k)
θi is the estimate of θ at sensor i during the kth cycle. At each subiteration sensor i minimize the cost function fi (θ), based on {xi,j }m j=1 : (k)
θi (k)
(k)
where gi,k ∈ ∂fi (θi−1 ) and θ0 Assumptions: 1 2
(k)
= θi−1 − αgi,k (k−1)
= θn
optimal solution θ∗ exists. kgi,k k ≤ ζ for all fi (θ) and θ ∈ Θ
Convergence guaranteed.
Meng Guo
Automatic Control Lab, KTH
Formulation
Convergence Speed
Convergence Speed After K cycles, K =b
kθˆ(0) − θ∗ k , α2 ζ 2
it is guaranteed that min f (θˆ(0) ) ≤ f (θ∗ ) + αζ 2
0≤k≤K
Set = αζ 2 . Convergence to -ball of the optimal value after at most K≤
c0 ζ 2 = O(−2 ) 2
Time varying or fixed step size α.
Meng Guo
Automatic Control Lab, KTH
Formulation
Energy Features
Energy Features
Energy for communication >> local computation. How the amount of communication required scales with the size (m, n)of the network? Total energy used for communication grows linearly with the size of the network: Eincr (n) ≤ c2 n −2 e(n) Energy saving ratio (w.r.t centralized data fusion) R=
Eincr (n) = c3 m n1/d ε2 Ecent (n)
Meng Guo
Automatic Control Lab, KTH
Applications
Example 1
Example 1 Problem: Identify and discard “bad” measurements from the estimation process. Solution: Incremental algorithm with modified local cost function (instead of kx − θk2 ). Less weight to points that deviate greatly from the parameter θ: ( kx − θk2 /2, for kx − θk ≤ r ρh (x, θ) = 2 γkx − θk − γ /2, for kx − θk > r Distributed cost function: m
fi (θ) =
1 X ρ(xi, j , θ) m j=1
Meng Guo
Automatic Control Lab, KTH
Applications
Meng Guo
Example 1
Automatic Control Lab, KTH
Applications
Example 2
Example 2
Problem: Estimate the location of an acoustic source using the signal strength received at each sensor. Based on the isotropic energy propagation model, the local cost function is m 1 X A fi (θ) = (xi,j − ) m kθ − ri kβ j=1
Fits into the general incremental subgradient framework.
Meng Guo
Automatic Control Lab, KTH
Applications
Meng Guo
Example 2
Automatic Control Lab, KTH
Applications
Example 3
Example 3 Problem: Cluster measurements from a mixture of Gaussian distributions with unknown means and covariances (different weights). Distributed Expectation-Maximization (DEM) Algorithms fi (θ) =
n X m X
J X log( αi,k N (yi,j |µk , Σk )))
i=1 j=1
k=1
Generally hard: 1
2
Run DEM to estimate the mean and covariance of each distribution (how many). Determine which class it belongs to.
More assumptions needed to implement the Decenralized Incremental Method
Meng Guo
Automatic Control Lab, KTH
Applications
Meng Guo
Example 3
Automatic Control Lab, KTH
Conclusion
Summary
Incremental subgradient optimization 1 2 3 4
Determine the cost function Circulate a parameter estimate through the network. Small adjustments at each node based on local data. Terminate after enough cycles
Enormous energy saving with large systems. Constraints? 1 2 3
Path routing, centralized planning? Optimal step size α. Robustness to communication failures?
Meng Guo
Automatic Control Lab, KTH
October 14, 2011
Meng Guo
Automatic Control Lab, KTH
Introduction
Introduction
Wireless sensor Network (WSN) collects enormous amount of data over space and time. Objective: to derive an optimal estimate of certain parameters from these “raw” data. How? To minimize the energy consumption. Simple example: n sensors uniformly distributed over 1 m2 . Each collects m measurements. Task: what is the average value?
Meng Guo
Automatic Control Lab, KTH
Introduction
Possible Approaches
Possible Approaches
1
Sensors transmit all data to a central processor. O(mn) bits over O(1) meter.
2
Sensors first compute a local average and then transmit to the central unit.O(n) bits over O(1) meter.
3
Construct a path passing through all sensors once. Each node adds its local average to the total along the way. An accumulation process from start node to finish node. O(n) bits over O(n−1/2 ) meter. Far less energy and communication needed for large (m, n).
Meng Guo
Automatic Control Lab, KTH
General Setup
General Setup
So-called Incremental Subgradient Optimization. An estimate of the parameter is passed from node to node. Each node adjusts the estimate based on local data, then passes the update to the next node. Several “cycles” through the network may be needed.
It is distributed, “in-network” optimization!
Meng Guo
Automatic Control Lab, KTH
Formulation
Formulation A network of n sensors in which each sensor collects m measurements. Denote xi,j the jth measurement at the ith sensor. Goal: 1 θˆ = arg min θ∈Θ n θ∈
Rd
n X
fi ({xi,j }m j=1 , θ)
i=1
is the set of parameters to be estimated.
Cost function fi : Rd → R, are convex and Θ is a nonempty, closed, convex subset of Rd . Denote fi (θ) = fi ({xi,j }m j=1 , θ), which depends on sensor i’s measurements.
Meng Guo
Automatic Control Lab, KTH
Formulation
Centralized Method
Centralized Version Centralized subgradient descent approach: θˆ(k+1) = θˆ(k) − α
n X
gi,k ,
i=1
where gi,k ∈ ∂fi (θˆ(k) ), α is a positive step size, and k is the iteration number. Note! each update step in this approach uses data from all of the sensors. ∂f (θ0 ) is the subdifferential of f at θ0 , the set of all subgradients of f at θ0 . The subgradient of f at θ0 is any direction g satisfying f (θ) ≥ f (θ0 ) + (θ − θ0 )T g ,
Meng Guo
∀θ ∈ Θ
Automatic Control Lab, KTH
Formulation
Decenralized Incremental Method
Decenralized Incremental Method
Each update: a cycle of n subiterations (k)
θi is the estimate of θ at sensor i during the kth cycle. At each subiteration sensor i minimize the cost function fi (θ), based on {xi,j }m j=1 : (k)
θi (k)
(k)
where gi,k ∈ ∂fi (θi−1 ) and θ0 Assumptions: 1 2
(k)
= θi−1 − αgi,k (k−1)
= θn
optimal solution θ∗ exists. kgi,k k ≤ ζ for all fi (θ) and θ ∈ Θ
Convergence guaranteed.
Meng Guo
Automatic Control Lab, KTH
Formulation
Convergence Speed
Convergence Speed After K cycles, K =b
kθˆ(0) − θ∗ k , α2 ζ 2
it is guaranteed that min f (θˆ(0) ) ≤ f (θ∗ ) + αζ 2
0≤k≤K
Set = αζ 2 . Convergence to -ball of the optimal value after at most K≤
c0 ζ 2 = O(−2 ) 2
Time varying or fixed step size α.
Meng Guo
Automatic Control Lab, KTH
Formulation
Energy Features
Energy Features
Energy for communication >> local computation. How the amount of communication required scales with the size (m, n)of the network? Total energy used for communication grows linearly with the size of the network: Eincr (n) ≤ c2 n −2 e(n) Energy saving ratio (w.r.t centralized data fusion) R=
Eincr (n) = c3 m n1/d ε2 Ecent (n)
Meng Guo
Automatic Control Lab, KTH
Applications
Example 1
Example 1 Problem: Identify and discard “bad” measurements from the estimation process. Solution: Incremental algorithm with modified local cost function (instead of kx − θk2 ). Less weight to points that deviate greatly from the parameter θ: ( kx − θk2 /2, for kx − θk ≤ r ρh (x, θ) = 2 γkx − θk − γ /2, for kx − θk > r Distributed cost function: m
fi (θ) =
1 X ρ(xi, j , θ) m j=1
Meng Guo
Automatic Control Lab, KTH
Applications
Meng Guo
Example 1
Automatic Control Lab, KTH
Applications
Example 2
Example 2
Problem: Estimate the location of an acoustic source using the signal strength received at each sensor. Based on the isotropic energy propagation model, the local cost function is m 1 X A fi (θ) = (xi,j − ) m kθ − ri kβ j=1
Fits into the general incremental subgradient framework.
Meng Guo
Automatic Control Lab, KTH
Applications
Meng Guo
Example 2
Automatic Control Lab, KTH
Applications
Example 3
Example 3 Problem: Cluster measurements from a mixture of Gaussian distributions with unknown means and covariances (different weights). Distributed Expectation-Maximization (DEM) Algorithms fi (θ) =
n X m X
J X log( αi,k N (yi,j |µk , Σk )))
i=1 j=1
k=1
Generally hard: 1
2
Run DEM to estimate the mean and covariance of each distribution (how many). Determine which class it belongs to.
More assumptions needed to implement the Decenralized Incremental Method
Meng Guo
Automatic Control Lab, KTH
Applications
Meng Guo
Example 3
Automatic Control Lab, KTH
Conclusion
Summary
Incremental subgradient optimization 1 2 3 4
Determine the cost function Circulate a parameter estimate through the network. Small adjustments at each node based on local data. Terminate after enough cycles
Enormous energy saving with large systems. Constraints? 1 2 3
Path routing, centralized planning? Optimal step size α. Robustness to communication failures?
Meng Guo
Automatic Control Lab, KTH
