  We propose a self-supervised learning approach for solving the following
constrained optimization task in log-linear models or Markov networks. Let $f$
and $g$ be two log-linear models defined over the sets $\mathbf{X}$ and
$\mathbf{Y}$ of random variables respectively. Given an assignment $\mathbf{x}$
to all variables in $\mathbf{X}$ (evidence) and a real number $q$, the
constrained most-probable explanation (CMPE) task seeks to find an assignment
$\mathbf{y}$ to all variables in $\mathbf{Y}$ such that $f(\mathbf{x},
\mathbf{y})$ is maximized and $g(\mathbf{x}, \mathbf{y})\leq q$. In our
proposed self-supervised approach, given assignments $\mathbf{x}$ to
$\mathbf{X}$ (data), we train a deep neural network that learns to output
near-optimal solutions to the CMPE problem without requiring access to any
pre-computed solutions. The key idea in our approach is to use first principles
and approximate inference methods for CMPE to derive novel loss functions that
seek to push infeasible solutions towards feasible ones and feasible solutions
towards optimal ones. We analyze the properties of our proposed method and
experimentally demonstrate its efficacy on several benchmark problems.
