  In this paper, we consider variational autoencoders (VAE) for general state
space models. We consider a backward factorization of the variational
distributions to analyze the excess risk associated with VAE. Such backward
factorizations were recently proposed to perform online variational learning
and to obtain upper bounds on the variational estimation error. When
independent trajectories of sequences are observed and under strong mixing
assumptions on the state space model and on the variational distribution, we
provide an oracle inequality explicit in the number of samples and in the
length of the observation sequences. We then derive consequences of this
theoretical result. In particular, when the data distribution is given by a
state space model, we provide an upper bound for the Kullback-Leibler
divergence between the data distribution and its estimator and between the
variational posterior and the estimated state space posterior
distributions.Under classical assumptions, we prove that our results can be
applied to Gaussian backward kernels built with dense and recurrent neural
networks.
