Quantifying Model Uncertainty in Inverse Problems via Bayesian Deep Gradient Descent

Recent advances in reconstruction methods for inverse problems leverage powerful data-driven models, e.g., deep neural networks. These techniques have demonstrated state-of-the-art performances for several imaging tasks, but they often do not provide uncertainty on the obtained reconstructions. In this work, we develop a novel scalable data-driven knowledge-aided computational framework to quantify the model uncertainty via Bayesian neural networks. The approach builds on and extends deep gradient descent, a recently developed greedy iterative training scheme, and recasts it within a probabilistic framework. Scalability is achieved by being hybrid in the architecture: only the last layer of each block is Bayesian, while the others remain deterministic, and by being greedy in training. The framework is showcased on one representative medical imaging modality, viz. computed tomography with either sparse view or limited view data, and exhibits competitive performance with respect to state-of-the-art benchmarks, e.g., total variation, deep gradient descent and learned primal-dual.Comment: 8 pages, 6 figure

Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks

Effective training of deep neural networks suffers from two main issues. The first is that the parameter spaces of these models exhibit pathological curvature. Recent methods address this problem by using adaptive preconditioning for Stochastic Gradient Descent (SGD). These methods improve convergence by adapting to the local geometry of parameter space. A second issue is overfitting, which is typically addressed by early stopping. However, recent work has demonstrated that Bayesian model averaging mitigates this problem. The posterior can be sampled by using Stochastic Gradient Langevin Dynamics (SGLD). However, the rapidly changing curvature renders default SGLD methods inefficient. Here, we propose combining adaptive preconditioners with SGLD. In support of this idea, we give theoretical properties on asymptotic convergence and predictive risk. We also provide empirical results for Logistic Regression, Feedforward Neural Nets, and Convolutional Neural Nets, demonstrating that our preconditioned SGLD method gives state-of-the-art performance on these models.Comment: AAAI 201