  This study investigates imposing hard inequality constraints on the outputs
of convolutional neural networks (CNN) during training. Several recent works
showed that the theoretical and practical advantages of Lagrangian optimization
over simple penalties do not materialize in practice when dealing with modern
CNNs involving millions of parameters. Therefore, constrained CNNs are
typically handled with penalties. We propose *log-barrier extensions*, which
approximate Lagrangian optimization of constrained-CNN problems with a sequence
of unconstrained losses. Unlike standard interior-point and log-barrier
methods, our formulation does not need an initial feasible solution. The
proposed extension yields an upper bound on the duality gap -- generalizing the
result of standard log-barriers -- and yielding sub-optimality certificates for
feasible solutions. While sub-optimality is not guaranteed for non-convex
problems, this result shows that log-barrier extensions are a principled way to
approximate Lagrangian optimization for constrained CNNs via implicit dual
variables. We report weakly supervised image segmentation experiments, with
various constraints, showing that our formulation outperforms substantially the
existing constrained-CNN methods, in terms of accuracy, constraint satisfaction
and training stability, more so when dealing with a large number of
constraints.

Computer Vision and Pattern Recognition