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

Learning Robust Object Recognition Using Composed Scenes from Generative Models

Recurrent feedback connections in the mammalian visual system have been hypothesized to play a role in synthesizing input in the theoretical framework of analysis by synthesis. The comparison of internally synthesized representation with that of the input provides a validation mechanism during perceptual inference and learning. Inspired by these ideas, we proposed that the synthesis machinery can compose new, unobserved images by imagination to train the network itself so as to increase the robustness of the system in novel scenarios. As a proof of concept, we investigated whether images composed by imagination could help an object recognition system to deal with occlusion, which is challenging for the current state-of-the-art deep convolutional neural networks. We fine-tuned a network on images containing objects in various occlusion scenarios, that are imagined or self-generated through a deep generator network. Trained on imagined occluded scenarios under the object persistence constraint, our network discovered more subtle and localized image features that were neglected by the original network for object classification, obtaining better separability of different object classes in the feature space. This leads to significant improvement of object recognition under occlusion for our network relative to the original network trained only on un-occluded images. In addition to providing practical benefits in object recognition under occlusion, this work demonstrates the use of self-generated composition of visual scenes through the synthesis loop, combined with the object persistence constraint, can provide opportunities for neural networks to discover new relevant patterns in the data, and become more flexible in dealing with novel situations.Comment: Accepted by 14th Conference on Computer and Robot Visio

Data-Driven Approach to Simulating Realistic Human Joint Constraints

Modeling realistic human joint limits is important for applications involving physical human-robot interaction. However, setting appropriate human joint limits is challenging because it is pose-dependent: the range of joint motion varies depending on the positions of other bones. The paper introduces a new technique to accurately simulate human joint limits in physics simulation. We propose to learn an implicit equation to represent the boundary of valid human joint configurations from real human data. The function in the implicit equation is represented by a fully connected neural network whose gradients can be efficiently computed via back-propagation. Using gradients, we can efficiently enforce realistic human joint limits through constraint forces in a physics engine or as constraints in an optimization problem.Comment: To appear at ICRA 2018; 6 pages, 9 figures; for associated video, see https://youtu.be/wzkoE7wCbu

Formal Verification of Input-Output Mappings of Tree Ensembles

Recent advances in machine learning and artificial intelligence are now being considered in safety-critical autonomous systems where software defects may cause severe harm to humans and the environment. Design organizations in these domains are currently unable to provide convincing arguments that their systems are safe to operate when machine learning algorithms are used to implement their software. In this paper, we present an efficient method to extract equivalence classes from decision trees and tree ensembles, and to formally verify that their input-output mappings comply with requirements. The idea is that, given that safety requirements can be traced to desirable properties on system input-output patterns, we can use positive verification outcomes in safety arguments. This paper presents the implementation of the method in the tool VoTE (Verifier of Tree Ensembles), and evaluates its scalability on two case studies presented in current literature. We demonstrate that our method is practical for tree ensembles trained on low-dimensional data with up to 25 decision trees and tree depths of up to 20. Our work also studies the limitations of the method with high-dimensional data and preliminarily investigates the trade-off between large number of trees and time taken for verification