Distributed Learning and Function Fusion in Reproducing Kernel Hilbert Space

We consider the problem of function estimation by a multi-agent system comprising 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 develop 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. To analyze the algorithm, we define learning operators for the agents, fusion center and the system. We study the asymptotic properties of the norm of the learning operators and find sufficient conditions under which they converge to $1$. Given a sequence of data points, we define and prove the existence of the learning operator for the system. We prove that the porposed learning algorithm is consistent and demonstrate the same using an example. The paper has been submitted to L4DC 2024

The Role of Information in Multi-Agent Decision Making

Networked multi-agent systems have become an integral part of many engineering systems. Collaborative decision making in multi-agent systems poses many challenges. In this thesis, we study the impact of information and its availability to agents on collaborative decision making in multi-agent systems. We consider the problem of detecting Markov and Gaussian models from observed data using two observers. We consider two Markov chains and two observers. Each observer observes a different function of the state of the true unknown Markov chain. Given the observations, the aim is to find which of the two Markov chains has generated the observations. We formulate block binary hypothesis testing problem for each observer and show that the decision for each observer is a function of the local likelihood ratio. We present a consensus scheme for the observers to agree on their beliefs and the asymptotic convergence of the consensus decision to the true hypothesis is proven. A similar problem framework is considered for the detection of Gaussian models using two observers. Sequential hypothesis testing problem is formulated for each observer and solved using local likelihood ratio. We present a consensus scheme taking into account the random and asymmetric stopping time of the observers. The notion of ``value of information" is introduced to understand the ``usefulness" of the information exchanged to achieve consensus. Next, we consider the binary hypothesis testing problem with two observers. There are two possible states of nature. There are two observers which collect observations that are statistically related to the true state of nature. The two observers are assumed to be synchronous. Given the observations, the objective of the observers is to collaboratively find the true state of nature. We consider centralized and decentralized approaches to solve the problem. In each approach there are two phases: (1) probability space construction: the true hypothesis is known, observations are collected to build empirical joint distributions between hypothesis and the observations; (2) given a new set of observations, hypothesis testing problems are formulated for the observers to find their individual beliefs about the true hypothesis. Consensus schemes for the observers to agree on their beliefs about the true hypothesis are presented. The rate of decay of the probability of error in the centralized approach and rate of decay of the probability of agreement on the wrong belief in the decentralized approach are compared. Numerical results comparing the centralized and decentralized approaches are presented. All propositions from the set of events for an agent in a multi-agent system might not be simultaneously verifiable. We study the concepts of \textit{event-state-operation structure} and \textit{relationship of incompatibility} from literature and use them as a tool to study the structure of the set of events. We present an example from multi-agent hypothesis testing where the set of events do not form a boolean algebra, but form an ortholattice. A possible construction of a 'noncommutative probability space', accounting for \textit{incompatible events} (events which cannot be simultaneously verified) is discussed. As a possible decision-making problem in such a probability space, we consider the binary hypothesis testing problem. We present two approaches to this decision-making problem. In the first approach, we represent the available data as coming from measurements modeled via projection valued measures (PVM) and retrieve the results of the underlying detection problem solved using classical probability models. In the second approach, we represent the measurements using positive operator valued measures (POVM). We prove that the minimum probability of error achieved in the second approach is the same as in the first approach. Finally, we consider the binary hypothesis testing problem with learning of empirical distributions. The true distributions of the observations under either hypothesis are unknown. Empirical distributions are estimated from observations. A sequence of detection problems is solved using the sequence of empirical distributions. The convergence of the information state and optimal detection cost under empirical distributions to the information state and optimal detection cost under the true distribution are shown. Numerical results on the convergence of optimal detection cost are presented

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Not AvailableThe use of ocean colour remote sensing to facilitate the monitoring of phytoplankton biomass in coastal waters is hampered by the high variability in absorption and scattering from substances other than phytoplankton. The eastern Arabian Sea coastal shelf is influenced by river run-off, winter convection and monsoon upwelling. Bio-optical parameters were measured along this coast from March 2009 to June 2011, to characterise the optical water type and validate three Chlorophyll-a (Chla) algorithms applied to Moderate Resolution Imaging Spectroradiometer on Aqua (MODIS-Aqua) data against in situ measurements. Ocean Colour 3 band ratio (OC3M), Garver–Siegel–Maritorena Model (GSM) and Generalized Inherent Optical Property (GIOP) Chla algorithms were evaluated. OC3M performed better than GSM and GIOP in all regions and overall, was within 11% of in situ Chla. GSM was within 24% of in situ Chla and GIOP on average was 55% lower. OC3M was less affected by errors in remote sensing reflectance Rrs(λ) and by spectral variations in absorption coefficient (aCDOM(λ)) of coloured dissolved organic material (CDOM) and total suspended matter (TSM) compared to the other algorithms. A nine year Chla time series from 2002 to 2011 was generated to assess regional differences between OC3M and GSM. This showed that in the north eastern shelf, maximum Chla occurred during the winter monsoon from December to February, where GSM consistently gave higher Chla compared to OC3M. In the south eastern shelf, maximum Chla occurred in June to July during the summer monsoon upwelling, and OC3M yielded higher Chla compared to GSM. OC3M currently provides the most accurate Chla estimates for the eastern Arabian Sea coastal waters.Not Availabl