  We present a variational method for online state estimation and parameter
learning in state-space models (SSMs), a ubiquitous class of latent variable
models for sequential data. As per standard batch variational techniques, we
use stochastic gradients to simultaneously optimize a lower bound on the log
evidence with respect to both model parameters and a variational approximation
of the states' posterior distribution. However, unlike existing approaches, our
method is able to operate in an entirely online manner, such that historic
observations do not require revisitation after being incorporated and the cost
of updates at each time step remains constant, despite the growing
dimensionality of the joint posterior distribution of the states. This is
achieved by utilizing backward decompositions of this joint posterior
distribution and of its variational approximation, combined with Bellman-type
recursions for the evidence lower bound and its gradients. We demonstrate the
performance of this methodology across several examples, including
high-dimensional SSMs and sequential Variational Auto-Encoders.
