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1. Abstract

The paper proposes an algorithm for distributed optimization in

mobile nodes. Compared with other works, an important

consideration here is that the nodes do not know the cost

function beforehand. Instead of decision making based on

linear combination of the neighbor estimates, the proposed

algorithm relies on information rich nodes. To quickly find

these nodes, the algorithm adopts an iterative procedure with

an adaptive step size through iterations. Comparative

simulation results are presented to support the proposed

algorithm.

The algorithm can be used in several applications, such as

distributed odor source localization, cooperative prey herding

modeling and mobile robots.

2. Introduction

A

B

3. Proposed Algorithm

Definitions: We denote as the neighborhood of node at time which is defined as the set of nodes that are connected to node at time including itself. We define as the noisy measurement of the cost function at node and time index which is given by where is measurement noise with variance . Assume that at time , each node has access to noisy measurements of the cost function at times and . Using these measurements, the local error signal is

using a first-order Taylor series expansion, we have

where

The vector denotes the direction of motion from to

. According to (3), (4) and (5), we can relate the gradient

vector to local error signal via

where and

When the gradient vector at is available, we can use it to

update from to as

where μ is the step size parameter. Since the cost function is unknown beforehand we can not use (7) to update the local estimates. Thus, the objective for each node becomes that of determining a good estimate for this gradient vector.

5. Simulation Results

we consider a cost function as shown in Fig. 2 (left), where the

value of cost function is given by:

The shape of cost function may change in time. At time index

, nodes are randomly and uniformly distributed over a

40 × 40 rectangular region centered at (0, 0) as shown in Fig.

2 (right). In Fig. 3 we plot the final locations of different nodes

for non-cooperative scheme [4], algorithm in [6] and proposed

algorithm. Obviously, the non-cooperative scheme does not

work well and only a small fraction of the nodes can find the

optimum location. In the proposed algorithm, the most of

nodes have moved from their initial locations to the locations

of peak values. Fig. 4 shows the average location of nodes

per iteration and proves that the proposed algorithm

provides better performance.

References

Contact Information

[1] D. Estrin, G. Pottie, and M. Srivastava, “Intrumenting the world with wireless sensor setworks,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Salt Lake City, UT, May 2001, pp. 2033-2036.

[2] C. G. Lopes and A. H. Sayed, “Distributed processing over adaptive networks,” in Proc. Adaptive Sensor Array Processing Workshop, MIT Lincoln Lab., Lexington, MA, Jun. 2006.

[3] A. H. Sayed, “Diffusion adaptation over networks,” to appear in Ereference Signal Processing, R. Chellapa and S. Theodoridis, editors, Elsevier, 2013.

[4] F. Cattivelli and A. H. Sayed, “Self-organization in bird flight formations using diffusion adaptation,” in Proc. 3rd International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Aruba, Dutch Antilles, Dec. 2009, pp. 4952.

[5] J. Chen, X. Zhao and A. H. Sayed, “Bacterial motility via diffusion adaptation,” in Proc. 44th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, Nov. 2010.

[6] J. Chen and A. H. Sayed, “Bio-inspired cooperative optimization with application to bacteria motility,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Prague, Czech Republic, pp. 5788-5791, May 2011.

Translational System and Signal Processing

Group (TSS)

National University of Singapore http://www.ece.nus.edu.sg/stfpage/eleyangz/index.html

4 Engineering Drive 3

Dept. of ECE, NUS

Singapore -117576

Tel: +65-6515-2262

Email:

eleyangz@nus.edu.sg

Consider a set of nodes as (Fig. 1)

The objective of each node is to estimate the vector

that maximizes a cost function .

where depends only on data available at node k.

Applications: sensor networks, precision agriculture,

environment monitoring, disaster relief management, smart

spaces, target localization, as well as medical applications [1, 2].

Why Adaptive Network: In many applications the statistical

information for the underlying processes of interest is not

available [3].

Basic Assumption In The Reported Adaptive Networks:

There are applications that the nodes do not know the form of

the cost function beforehand where nodes can only sense

variations in the values of the objective function as they diffuse

through the space.

Example (foraging model for bacteria): The bacterial foraging

for food by means of moving towards the direction of

increasing nutrients in response to chemical signaling [5, 6].

Our Work: We consider the problem of form (1) with an

important assumption that the nodes do not know the form of

the cost function beforehand. We interpret the successive local

estimates as location vectors in a 2-D space and propose an

algorithm for adaptation over networks with mobile nodes.

Our Contribution: We improved the performance of algorithm

in [6] can be improved by employing the following ideas:

Using the data related to information-rich node instead of

linear combination of the neighbors’ estimates.

Using variable step-size in the iterative optimization

algorithm. We choose to use larger step sizes in the initial

iterations to increase the probability of finding the

information-rich nodes.

This scheme not only improves the convergence rate, but also

improves the cost function averaged per node:

Paper ID: FrB10.4

Fig. 3. The final locations of different nodes for non-cooperative scheme in (4) (left), the algorithm given in [6] (middle) and the proposed algorithm (right).

Fig. 2. The distribution of the cost function (left); the initial position of nodes and the location of peak values of the cost function at (x1, y1) = (−15,−12), (x2, y2) = (15, 12)

(right)

Fig. 4. The average location of nodes (ηk(i)) per iteration i.

Algorithm[ 6 ]

The form of the cost function is known beforehand.

Fig. 1. A network with N nodes: the neighborhood of node k are distinguished.

Motivation: In a non-cooperative scheme, each node can

estimate the gradient vector at as

Therefore, for non-cooperative scheme, Eq. (7) changes to

where is the indicator function: it is equal to one when and zero otherwise. It is important to note that we can get better result if we allow cooperation among nodes. The algorithm in [6] is a cooperation based one that has the following update equation:

We propose a new algorithm to enhance the performance of (9) under the conditions of the signals are noisy, thus a linear combination of neighboring nodes can’t provide accurate estimate of gradient vector.

Method: The successive are defined as location vectors and thus the neighborhood of node at time is given by

where is some radius value. Since we need to use the data related to information-rich nodes more than other nodes to improve the estimation. An information-rich node in the neighborhood of node has the following property:

We do not have access to thus the noisy measurements are used to find the information-rich node at node as

We get the modified direction vector,

by replacing into . To improve the performance,

we replace the fixed step-size with a variable step-size as

We can modify the update equation in (8) and (9) as follows:

A Bio-Inspired Cooperative Algorithm for Distributed Source

Localization with Mobile Nodes

Azam Khalili1, Amir Rastegarnia1, Md Kafiul Islam2 and Zhi Yang2

1 Dept. of Electrical Engineering, Malayer University, Iran; 2 Dept. of Electrical & Computer Engineering, National University of Singapore, Singapore

Email: a.khalili@ieee.org; a_rastegar@ieee.org; kafiul_islam@nus.edu.sg; eleyangz@nus.edu.sg