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: [email protected] 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 (η k (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 Khalili 1 , Amir Rastegarnia 1 , Md Kafiul Islam 2 and Zhi Yang 2 1 Dept. of Electrical Engineering, Malayer University, Iran; 2 Dept. of Electrical & Computer Engineering, National University of Singapore, Singapore Email: [email protected]; [email protected]; [email protected]; [email protected]