The document proposes an algorithm for distributed optimization with mobile nodes that do not know the cost function beforehand. Each node estimates the gradient vector to update its location. The proposed algorithm improves upon an existing algorithm by relying on information-rich nodes in the neighborhood instead of a linear combination of neighbors' estimates. It also uses a variable step size to increase the probability of finding information-rich nodes early in the iterations. Simulation results show the proposed algorithm achieves better performance than the existing algorithm and a non-cooperative scheme. The algorithm has applications in sensor networks, environmental monitoring, and other domains.

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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