We consider the problem of function estimation by a multi-agent system consisting of two agents and a fusion center. Each agent receives data comprising of samples of an independent variable (input) and the corresponding values of the dependent variable (output). The data remains local and is not shared with other members in the system. The objective of the system is to collaboratively estimate the function from the input to the output. To this end, we present an iterative distributed algorithm for this function estimation problem. Each agent solves a local estimation problem in a Reproducing Kernel Hilbert Space (RKHS) and uploads the function to the fusion center. At the fusion center, the functions are fused by first estimating the data points that would have generated the uploaded functions and then subsequently solving a least squares estimation problem using the estimated data from both functions. The fused function is downloaded by the agents and is subsequently used for estimation at the next iteration along with incoming data. This procedure is executed sequentially and stopped when the difference between consecutively estimated functions becomes small enough. With respect to the algorithm, we prove existence of basis functions for suitable representation of estimated functions and present closed form solutions to the estimation problems at the agents and the fusion center.

We consider the problem of estimation of a function by a system consisting of two agents and a fusion center. The two agents collect data comprising of samples of an independent variable and the corresponding value of a dependent variable. The objective of the system is to collaboratively estimate the function without any exchange of data among the members of the system. To this end, we propose the following framework. The agents are given a set of features using which they construct suitable function spaces to formulate and solve the estimation problems locally. The estimated functions are uploaded to a fusion space where an optimization problem is solved to fuse the estimates (also known as meta-learning) to obtain the system estimate of the mapping. The fused function is then downloaded by the agents to gather knowledge about the other agents estimate of the function. With respect to the framework, we present the following: a systematic construction of fusion space given the features of the agents; the derivation of an uploading operator for the agents to upload their estimated functions to a fusion space; the derivation of a downloading operator for the fused function to be downloaded. Through an example on least squares regression, we illustrate the distributed estimation architecture that has been developed.

A given region in 2-D Euclidean space is divided by a unknown linear classifier in to two sets each carrying a label. An agent with known dynamics traversing the given region is able to measure the true label perfectly at its position. By following a trajectory, the agent collects data points comprising of its true position and the label at that position. The objective of the agent is to plan a trajectory across the given region to identify the true classifier with high accuracy while minimizing the control cost across the trajectory. We present the following: (i) the classifier identification problem formulated as a control problem; (ii) geometric interpretation of the control problem resulting in one step modified control problems; (iii) control algorithm that results in a data set which is used to identify the true classifier with high accuracy; (iv) convergence of estimated classifier to the true classifier when observed label is not corrupted by noise; (iv) numerical example demonstrating the utility of the control algorithm.

We consider the problem of learning functions by two agents and a fusion center from noisy data. True data comprises of samples of an independent variable (input) and the corresponding value of a dependent variable (output) collectively labeled as (input, output) data. The data received by the agents, both the input and output data, are corrupted by noise. The objective of the agents is to learn a mapping from the true input to the true output. We formulate a general regression problem for the agents followed by the least squares regression (LS) problem. We prove a stochastic representer theorem for the general regression problem and subsequently solve the LS problem. The functions learned by the agents are transmitted to the fusion center where an optimization problem is formulated to fuse the functions together, which is then declared as the mapping. As an example, the methodology developed has been applied to the data generated from a transcendental function.

We consider the problem of estimation of an unknown real valued function with real valued input by an agent. The agent exists in 3D Euclidean space. It is able to traverse in a 2D plane while the function is depicted in a 2D plane perpendicular to the plane of traversal. By viewing the function from a given position, the agent is able to collect a data point lying on the function. By traversing through the plane while paying a control cost, the agent collects a finite set of data points. The set of data points are used by the agent to estimate the function. The objective of the agent is to find a control law which minimizes the control cost while estimating the function optimally. We formulate a control problem for the agent incorporating an inference cost and the control cost. The control problem is relaxed by finding a lower bound for the cost function. We present a kernel based linear regression model to approximate the cost-to-go and use the same in a control algorithm to solve the relaxed optimization problem. We present simulation results comparing the proposed approach with greedy algorithm based exploration.