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# A multi-modal distributed learning algorithm in reproducing kernel Hilbert spaces

*Proceedings of the 6th Annual Learning for Dynamics & Control Conference*, PMLR 242:1241-1252, 2024.

#### Abstract

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.