  We propose to learn deep undirected graphical models (i.e., MRFs) with a
non-ELBO objective for which we can calculate exact gradients. In particular,
we optimize a saddle-point objective deriving from the Bethe free energy
approximation to the partition function. Unlike much recent work in approximate
inference, the derived objective requires no sampling, and can be efficiently
computed even for very expressive MRFs. We furthermore amortize this
optimization with trained inference networks. Experimentally, we find that the
proposed approach compares favorably with loopy belief propagation, but is
faster, and it allows for attaining better held out log likelihood than other
recent approximate inference schemes.
