  Integrating logical reasoning within deep learning architectures has been a
major goal of modern AI systems. In this paper, we propose a new direction
toward this goal by introducing a differentiable (smoothed) maximum
satisfiability (MAXSAT) solver that can be integrated into the loop of larger
deep learning systems. Our (approximate) solver is based upon a fast coordinate
descent approach to solving the semidefinite program (SDP) associated with the
MAXSAT problem. We show how to analytically differentiate through the solution
to this SDP and efficiently solve the associated backward pass. We demonstrate
that by integrating this solver into end-to-end learning systems, we can learn
the logical structure of challenging problems in a minimally supervised
fashion. In particular, we show that we can learn the parity function using
single-bit supervision (a traditionally hard task for deep networks) and learn
how to play 9x9 Sudoku solely from examples. We also solve a "visual Sudok"
problem that maps images of Sudoku puzzles to their associated logical
solutions by combining our MAXSAT solver with a traditional convolutional
architecture. Our approach thus shows promise in integrating logical structures
within deep learning.
