The Evidence Lower Bound of Variational Autoencoders Converges to a Sum of Three Entropies

Abstract

The central objective function of a variational autoencoder (VAE) is its variational lower bound. Here we show that for standard VAEs the variational bound is at convergence equal to the sum of three entropies: the (negative) entropy of the latent distribution, the expected (negative) entropy of the observable distribution, and the average entropy of the variational distributions. Our derived analytical results are exact and apply for small as well as complex neural networks for decoder and encoder. Furthermore, they apply for finite and infinitely many data points and at any stationary point (including local and global maxima). As a consequence, we show that the variance parameters of encoder and decoder play the key role in determining the values of variational bounds at convergence. Furthermore, the obtained results can allow for closed-form analytical expressions at convergence, which may be unexpected as neither variational bounds of VAEs nor log-likelihoods of VAEs are closed-form during learning. As our main contribution, we provide the proofs for convergence of standard VAEs to sums of entropies. Furthermore, we numerically verify our analytical results and discuss some potential applications. The obtained equality to entropy sums provides novel information on those points in parameter space that variational learning converges to. As such, we believe they can potentially significantly contribute to our understanding of established as well as novel VAE approaches.