  Energy-based models (EBMs) have experienced a resurgence within machine
learning in recent years, including as a promising alternative for
probabilistic regression. However, energy-based regression requires a proposal
distribution to be manually designed for training, and an initial estimate has
to be provided at test-time. We address both of these issues by introducing a
conceptually simple method to automatically learn an effective proposal
distribution, which is parameterized by a separate network head. To this end,
we derive a surprising result, leading to a unified training objective that
jointly minimizes the KL divergence from the proposal to the EBM, and the
negative log-likelihood of the EBM. At test-time, we can then employ importance
sampling with the trained proposal to efficiently evaluate the learned EBM and
produce stand-alone predictions. Furthermore, we utilize our derived training
objective to learn mixture density networks (MDNs) with a jointly trained
energy-based teacher, consistently outperforming conventional MDN training on
four real-world regression tasks within computer vision. Code is available at
https://github.com/fregu856/ebms_proposals.
