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Carlo Fischione Fall 2011 Principles of Wireless Sensor Networks
Principles of Wireless Sensor Networks
http://www.ee.kth.se/~carlofi/teaching/pwsn-2011/wsn_course.shtml
Lecture 5 Stockholm, October 14, 2011
Carlo Fischione
Royal Institute of Technology - KTH
Stockholm, Sweden e-mail: carlofi@kth.se
Fast-Lipschitz Optimization
Carlo Fischione Fall 2011 Principles of Wireless Sensor Networks
Today’s lecture
Previous lecture Non expansive mappings, agreement, consensus
Today F-Lipschitz optimization
Protocol stack
Phy
MAC
Routing
Transport
Session
Application
Presentation
Carlo Fischione
Today’s learning goals
In today’s lecture, you will learn some of the key aspects of Fast
Lipschitz optmization When a problem is F-Lipschitz? Why should I care of F-Lipschitz? The theory is illustrated by distributed estimation applications Why F-Lipschitz is useful for distributed estimation? Other applications will be mentioned as well
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Optimization over networks
Optimization needs to be computed by fast algorithms of low complexity time-varying networks, little time to compute optimal solutions computations often must be distributed E.g., cross layer protocol design, distributed detection, estimation,
content distribution, routing, ....
Parallel and distributed computation Fundamental theory for optimization over networks Drowback over energy-constrained wireless networks: the cost for communication not considered
An alternative theory is needed
In a number of cases, Fast-Lipschitz optimization
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Outline
• Definition of Fast-Lipschitz optimization
• Computation of the optimal solution
• Problems in canonical form
• Examples of application
• Conclusions
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Network optimization
We have seen that a class of distributed estimation problems needs a fast distributed computation of an optimization
Is there a general formulation of optiomization problmes that can
be solved quickly and efficiently in a distributed manner? In many cases, Fast-Lipschitz optimization
C. Fischione, “F-Lipschitz Optimization with Wireless Sensor Networks Applications”, IEEE Transactions on Automatic Control, Accepted for Publication, to appear, 2011.
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
The Fast-Lipschitz optimization
nonempty compact set “containing” other constraints
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Computation of the solution
Centralized optimization Problem solved by a central processor
Distributed optimization
Decision variables and constraints are associated to nodes that cooperate to compute the solution in parallel
Network of n nodes
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
The F-Lipschitz optimization
Convex Optimization
Geometric Optimization
F-Lipschitz Optimization
Interference Function Optimization
Non-Convex Optimization
F-Lipschitz optimization problems can be convex, geometric, quadratic, interference-function,...
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Pareto Optimal Solution
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Notation
Norm infinity: sum along a row
Norm 1: sum along a column
Gradient
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Functions may be non-convex
F-Lipschitz qualifying properties
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Outline
• Definition of Fast-Lipschitz optimization
• Computation of the optimal solution
• Problems in canonical form
• Examples of application
• Conclusions
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Optimal Solution
The Pareto optimal solution is just given by a set of (in general non-
linear) equations.
Solving a set of equations is much easier than solving an optimization problem by traditional Lagrangian methods.
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Lagrangian methods
Theorem: Consider a feasible F-Lipschitz problem. Then, the KKT conditions are necessary and sufficient.
KKT conditions:
Lagrangian
Lagrangian methods to compute the solution
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Centralized optimization
The optimal solution is given by iterative methods to solve systems of non-linear equations (e.g., Newton methods)
is a matrix to ensure and maximize convergence speed Many other methods are available, e.g., second-oder methods. Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Distributed optimization
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
F-Lipschitz optimization
Inequality constraints satisfy the equality at
the optimum?
Compute the solution by F-
Lipschitz methods
Compute the solution by
Lagrangian methods
yes no
F-Lipschitz optimization: a class of problems for which all the
constraints are active at the optimum
Optimum: the solution to the set of equations given by the constraints
No Lagrangian methods, which are computationally expensive, particularly on wireless networks
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Outline
• Definition of Fast-Lipschitz optimization
• Computation of the optimal solution
• Problems in canonical form
• Examples of application
• Conclusions
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Problems in canonical form
F-Lipschitz form Canonical form Bertsekas, Non Linear Programming, 2004
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Problems in canonical form
Carlo Fischione
The problem is convex, but is also F-Lipschitz:
The solution is given by the constraints at the equality, trivially
Example 1: from canonical to F-Lipschitz
Off-diagonal monotonicity
Diagonal dominance
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Example 2: a hidden F-Lipschitz
Non F-Lipschitz
Simple variable transformation, , F-Lipschitz
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
An F-Lipschitz Matlab Toolbox
M. Leithe, Introducing a Matlab Toolbox for F-Lipschitz optimization, Master Thesis KTH, 2011
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Outline
• Definition of Fast-Lipschitz optimization
• Computation of the optimal solution
• Problems in canonical form
• Examples of application
• Conclusions
Fall 2011 Principles of Wireless Sensor Networks
Carlo Fischione
Distributed estimation
We now study F-Lipschitz optimization for distributed estimation distributed detection distributed radio power control
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Estimation
A. Speranzon, C. Fischione, K. H. Johansson, A. Sangiovanni-Vincentelli, “A Distributed Minimum Variance Estimator for Sensor Networks”, IEEE Journal on Selected Areas in Communications, special issue on Control and Communications, Vol. 26, N. 4, pp. 609—621, May 2008.
A. Speranzon, C. Fischione, K. H. Johansson, “Distributed and Collaborative Estimation over Wireless Sensor Networks”, IEEE CDC 2006.
Centralized estimation: No/little intelligence on nodes
phenomena
1 2 N
Fusion Center
sensors
phenomena
1 2 N
Distributed estimation: no central coordination
sensors
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Nodes perform a noisy measurement of a common time-varying signal
Communication subject to space-time varying packet losses
Network and signals
d(t)
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Distributed estimator
Nodes exchange local measurements and estimates
Goal: find locally the coefficients and that minimize the variance of the estimation error
Local estimate
Global vector of the estimates
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Estimation coefficients
• Estimation coefficients minimizing the average estimation error, under stability constraints
Small Bias
Stable estimation error
Estimation error
• A centralized optimization problem • How to distribute the computation of the optimal solution?
1. Cost function and first constraint easy to distribute 2. Second constraint is difficult to distribute
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
How to distribute the second constraint
Node j
Node i
The 1-norm and max-norm are easy to distribute, but give infeasibility
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
How to distribute the second constraint
A global constraint is translated into a local one by using some thresholds
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Distributed estimator
By using the thresholds: from centralized to distributed optimization
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Estimation coefficients
How to compute the thresholds? What is the performance of the estimator?
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
How to compute the thresholds?
The higher the thresholds the lower the estimation error
A Lipschitz optimization problem See second part of the lecture
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Performance
The error variance of the estimator that makes a simple average of the received measurements is an upper bound to the error variance of the proposed distributed estimator.
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Simulation example
Network with 30 nodes randomly deployed.
Signal to track:
Variance of the additive noise:
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Simulation Example (2)
Packet loss probability
“Laplacian” Estimator
Instantaneous Average Estimator Proposed Distributed Estimator
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Remarks
A class of distributed estimators for WSNs Optimal distributed estimator Stability conditions Performance analysis
Open issues Model-based estimator Estimator for signal with spatial and temporal correlation
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Optimal thresholds
Optimal solution: the fixed point of a contraction mapping
Each node updates asynchronously its threshold after receiving those of neighboring nodes.
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Distributed binary detection
measurements at node i
hypothesis thesting with S
measurements and threshold xi
probability of false alarm probability of misdetection A threshold minimizing the prob. of false alarm maximizes the
prob. of misdetection. How to choose optimally the thresholds when nodes exchange
opinions? Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Threshold optimization in distributed detection
How to solve the problem by parallel and distributed operations among the nodes?
The problem is convex Lagrangian methods (interior point methods) could be applied Drowback: too many message passing (Lagrangian multipliers) among nodes to
compute iteratively the optimal solution
An alternative method: F-Lipschitz optimization
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
5
Distributed detection: F-Lipschitz vs Lagrangian methods
31 36
231
Number of iterations Number of function evaluations
F-Lipschitz Lagrangian methods (interior point)
10 nodes network
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Interference function theory
vector of radio powers e interference that the radio power has to overcome
Properties of the (Type-I) interference function
nodes
Tx
Rx
Foschini, Miljanic, “A simple distributed autonomous power control algorithm and its convergence,” IEEE Trans. Veh. Technol., 1993.
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Example: Radio Power Control with unreliable components
Unreliable transceivers introduce intermodulation powers difficult to compensate
A difficult optimization problem How to distribute the computation?
SINR
Outages
Tx
Rx
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
A change of variables and a redefinition
F-Lipschitz qualifying properties are much more general than the interference function ones.
More radio power control problems can be solved than traditional ones based on the Interference Function.
Power control as an F-Lipschitz problem
Principles of Wireless Sensor Networks Fall 2011
Carlo Fischione
Conclusions
Existing methods for optimization over networks are too
expensive
Studied the Fast-Lipschitz optimization Application to distributed estimation, and other cases
F-Lipschitz optimization is a panacea for many cases, but still
there is a lack of a theory for fast parallel and distributed computations
Principles of Wireless Sensor Networks Fall 2011