Compressed Decentralized Learning of Conditional Mean Embedding Operators in Reproducing Kernel Hilbert Spaces

Conditional mean embedding (CME) operators encode conditional probability distributions within reproducing kernel Hilbert spaces (RKHS). In this paper, we present a decentralized algorithm for a collection of agents to cooperatively approximate CME over a network. Communication constraints limit the agents from sending all data to their neighbors; we only allow sparse representations of covariance operators to be exchanged among agents, compositions of which defines CME. Using a coherence-based compression scheme, we present a consensus-type algorithm that preserves the average of the approximations of the covariance operators across the network. We theoretically prove that the iterative dynamics in RKHS is stable. We then empirically study our algorithm to estimate CMEs to learn spectra of Koopman operators for Markovian dynamical systems and to execute approximate value iteration for Markov decision processes (MDPs).