  The central objective function of a variational autoencoder (VAE) is its
variational lower bound (the ELBO). Here we show that for standard (i.e.,
Gaussian) VAEs the ELBO converges to a value given by the sum of three
entropies: the (negative) entropy of the prior distribution, the expected
(negative) entropy of the observable distribution, and the average entropy of
the variational distributions (the latter is already part of the ELBO). Our
derived analytical results are exact and apply for small as well as for
intricate deep networks for encoder and decoder. Furthermore, they apply for
finitely and infinitely many data points and at any stationary point (including
local maxima and saddle points). The result implies that the ELBO can for
standard VAEs often be computed in closed-form at stationary points while the
original ELBO requires numerical approximations of integrals. As a main
contribution, we provide the proof that the ELBO for VAEs is at stationary
points equal to entropy sums. Numerical experiments then show that the obtained
analytical results are sufficiently precise also in those vicinities of
stationary points that are reached in practice. Furthermore, we discuss how the
novel entropy form of the ELBO can be used to analyze and understand learning
behavior. More generally, we believe that our contributions can be useful for
future theoretical and practical studies on VAE learning as they provide novel
information on those points in parameters space that optimization of VAEs
converges to.
