Distributed Regression by Two Agents from Noisy Data

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.