A SYMMETRIC ADAPTIVE ALGORITHM FOR SPEEDING-UP CONSENSUS Daniel Thai, Elizabeth Bodine-Baron, and Babak Hassibi Department of Electrical Engineering California Institute of Technology thai,eabodine,hassibi@caltech.edu ABSTRACT Performing distributed consensus in a network has been an important research problem for several years, and is directly applicable to sensor networks, autonomous vehicle formation, etc. While there exists a wide variety of algorithms that can be proven to asymptotically reach consensus, in applications in- volving time-varying parameters and tracking, it is often cru- cial to reach consensus “as quickly as possible”. In [1] it has been shown that, with global knowledge of the network topol- ogy, it is possible to optimize the convergence time in dis- tributed averaging algorithms via solving a semi-definite pro- gram (SDP) to obtain the optimal averaging weights. Unfor- tunately, in most applications, nodes do not have knowledge of the full network topology and cannot implement the re- quired SDP in a distributed fashion. In this paper, we present a symmetric adaptive weight algorithm for distributed consen- sus averaging on bi-directional noiseless networks. The algo- rithm uses an LMS (Least Mean Squares) approach to adap- tively update the edge weights used to calculate each node’s values. The derivation shows that global error can be mini- mized in a distributed fashion and that the resulting adaptive weights are symmetric—symmetry being critical for conver- gence to the true average. Simulations show that convergence time is nearly equal to that of a non-symmetric adaptive al- gorithm developed in [2], and significantly better than that of the non-adaptive Metropolis-Hastings algorithm. Most im- portantly, our symmetric adaptive algorithm converges to the sample mean, whereas the method of [2] converges to an ar- bitrary value and results in significant error. Index Terms— Adaptive Consensus, LMS algorithm, Sensor Network 1. INTRODUCTION Consensus is an important problem and has received much attention in the literature on sensor networks and distributed algorithms (see e.g., [2, 1, 3, 4, 5, 6, 7] and the references This work was supported in parts by the National Science Foundation under grant CCF 0729203, by the Office of Naval Research under grant N00014-08-1-0747, by the David and Lucille Packard Foundation, and by Caltech’s Lee Center for Advanced Networking. therein). A basic sensor network is made up of n nodes, each one of which is separated by a certain distance, and takes a measurement of some value. We assume that each sensor’s reading is independently corrupted by noise. It is often costly to transmit all n readings of the sensor network to the user, and in many cases, the readings must be aggregated into a sin- gle value. Therefore, we wish to compute the average value of all of the sensors’ readings for transmission to a base station. Several methods exist to perform this task. One of the simplest is to designate a ‘super node.’ In this method, all other nodes transmit all of the values they have recorded to the ‘super node.’ [1]. The ‘super node’ receives all of the information, performs the averaging, and transmits to the user this single value. (This assumes that the transmitter is based at the ’super node’.) However, there are several problems with this method. The first is that the system can be easily rendered inoperative by destroying the ‘super node’. A second is that the system cannot easily react to additions and deletions of nodes; the ‘super node’ needs to know how many nodes there are. To solve the problems associated with a central controller, sev- eral papers have proposed various methods for performing distributed averaging [2, 1, 3, 4, 5, 6, 7]. While there exists a wide variety of algorithms that can be proven to asymptotically reach consensus, in many appli- cations (especially those involving time-varying parameters) time-to-convergence is also important. For the distributed av- eraging problem, minimizing the convergence time reduces to minimizing the second largest eigenvalue of the weight ma- trix used by the network to perform consensus. When the network topology is known to all nodes in the network, this minimization can be done via solving a semi-definite program (SDP) [1]. Unfortunately, in most applications, this is an un- realistic assumption and distributed methods, which rely only on a node’s local knowledge of the network, need to be de- veloped. In [2] an adaptive weight update method was pro- posed and shown to have faster convergence time than fixed- weight methods. Unfortunately, since the method of [2] re- sults in non-symmetric weights it generally reaches consen- sus to a value unequal to the desired sample mean. In this paper, we propose a symmetric adaptive weight algorithm for distributed consensus averaging. Simulations show that con-