Constrained Deep Networks: Lagrangian Optimization via Log-Barrier Extensions

This study investigates the optimization aspects of imposing hard inequality constraints on the outputs of CNNs. In the context of deep networks, constraints are commonly handled with penalties for their simplicity, and despite their well-known limitations. Lagrangian-dual optimization has been largely avoided, except for a few recent works, mainly due to the computational complexity and stability/convergence issues caused by alternating explicit dual updates/projections and stochastic optimization. Several studies showed that, surprisingly for deep CNNs, the theoretical and practical advantages of Lagrangian optimization over penalties do not materialize in practice. 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. Furthermore, we provide a new technical result, which shows that the proposed extensions yield an upper bound on the duality gap. This generalizes the duality-gap result of standard log-barriers, yielding sub-optimality certificates for feasible solutions. While sub-optimality is not guaranteed for non-convex problems, our result shows that log-barrier extensions are a principled way to approximate Lagrangian optimization for constrained CNNs via implicit dual variables. We report comprehensive weakly supervised segmentation experiments, with various constraints, showing that our formulation outperforms substantially the existing constrained-CNN methods, both in terms of accuracy, constraint satisfaction and training stability, more so when dealing with a large number of constraints

Curriculum semi-supervised segmentation

This study investigates a curriculum-style strategy for semi-supervised CNN segmentation, which devises a regression network to learn image-level information such as the size of a target region. These regressions are used to effectively regularize the segmentation network, constraining softmax predictions of the unlabeled images to match the inferred label distributions. Our framework is based on inequality constraints that tolerate uncertainties with inferred knowledge, e.g., regressed region size, and can be employed for a large variety of region attributes. We evaluated our proposed strategy for left ventricle segmentation in magnetic resonance images (MRI), and compared it to standard proposal-based semi-supervision strategies. Our strategy leverages unlabeled data in more efficiently, and achieves very competitive results, approaching the performance of full-supervision.Comment: Accepted as paper as MICCAI 2O1

Seed, Expand and Constrain: Three Principles for Weakly-Supervised Image Segmentation

We introduce a new loss function for the weakly-supervised training of semantic image segmentation models based on three guiding principles: to seed with weak localization cues, to expand objects based on the information about which classes can occur in an image, and to constrain the segmentations to coincide with object boundaries. We show experimentally that training a deep convolutional neural network using the proposed loss function leads to substantially better segmentations than previous state-of-the-art methods on the challenging PASCAL VOC 2012 dataset. We furthermore give insight into the working mechanism of our method by a detailed experimental study that illustrates how the segmentation quality is affected by each term of the proposed loss function as well as their combinations.Comment: ECCV 201