Distributed parametric and nonparametric regression with on-line performance bounds computation

In this paper we focus on collaborative multi-agent systems, where agents are distributed over a region of interest and collaborate to achieve a common estimation goal. In particular, we introduce two consensus-based distributed linear estimators. The first one is designed for a Bayesian scenario, where an unknown common finite-dimensional parameter vector has to be reconstructed, while the second one regards the nonparametric reconstruction of an unknown function sampled at different locations by the sensors. Both of the algorithms are characterized in terms of the trade-off between estimation performance, communication, computation and memory complexity. In the finite-dimensional setting, we derive mild sufficient conditions which ensure that a distributed estimator performs better than the local optimal ones in terms of estimation error variance. In the nonparametric setting, we introduce an on-line algorithm that allows the agents to simultaneously compute the function estimate with small computational, communication and data storage efforts, as well as to quantify its distance from the centralized estimate given by a Regularization Network, one of the most powerful regularized kernel methods. These results are obtained by deriving bounds on the estimation error that provide insights on how the uncertainty inherent in a sensor network, such as imperfect knowledge on the number of agents and the measurement models used by the sensors, can degrade the performance of the estimation process. Numerical experiments are included to support the theoretical findings.