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.

翻译：本文考虑一般状态空间模型的变分自编码器（VAE）。我们采用变分分布的向后分解来分析与VAE相关的过剩风险。这种向后分解方法 recently 被提出用于执行在线变分学习并获取变分估计误差的上界。在观测到独立序列轨迹且状态空间模型及变分分布满足强混合假设的条件下，我们给出了一个关于样本数量与观测序列长度的显式预言不等式。随后推导了该理论结果的相关推论。特别地，当数据分布由状态空间模型给定时，我们得到了数据分布与其估计量之间、以及变分后验与估计状态空间后验分布之间库尔贝克-莱布勒散度的上界。在经典假设下，我们证明该结果可应用于由稠密神经网络与循环神经网络构建的高斯向后核。