Constrained Deep Networks: Lagrangian Optimization via Log-Barrier Extensions

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.