  Energy-based models (EBMs) provide an elegant framework for density
estimation, but they are notoriously difficult to train. Recent work has
established links to generative adversarial networks, where the EBM is trained
through a minimax game with a variational value function. We propose a
bidirectional bound on the EBM log-likelihood, such that we maximize a lower
bound and minimize an upper bound when solving the minimax game. We link one
bound to a gradient penalty that stabilizes training, thereby providing
grounding for best engineering practice. To evaluate the bounds we develop a
new and efficient estimator of the Jacobi-determinant of the EBM generator. We
demonstrate that these developments significantly stabilize training and yield
high-quality density estimation and sample generation.
