Correlated Parameters to Accurately Measure Uncertainty in Deep Neural Networks.

In this article, a novel approach for training deep neural networks using Bayesian techniques is presented. The Bayesian methodology allows for an easy evaluation of model uncertainty and, additionally, is robust to overfitting. These are commonly the two main problems classical, i.e., non-Bayesian architectures have to struggle with. The proposed approach applies variational inference in order to approximate the intractable posterior distribution. In particular, the variational distribution is defined as the product of multiple multivariate normal distributions with tridiagonal covariance matrices. Every single normal distribution belongs either to the weights or to the biases corresponding to one network layer. The layerwise a posteriori variances are defined based on the corresponding expectation values, and furthermore, the correlations are assumed to be identical. Therefore, only a few additional parameters need to be optimized compared with non-Bayesian settings. The performance of the new approach is evaluated and compared with other recently developed Bayesian methods. Basis of the performance evaluations are the popular benchmark data sets MNIST and CIFAR-10. Among the considered approaches, the proposed one shows the best predictive accuracy. Moreover, extensive evaluations of the provided prediction uncertainty information indicate that the new approach often yields more useful uncertainty estimates than the comparison methods.

Measuring the Uncertainty of Predictions in Deep Neural Networks with Variational Inference.

We present a novel approach for training deep neural networks in a Bayesian way. Compared to other Bayesian deep learning formulations, our approach allows for quantifying the uncertainty in model parameters while only adding very few additional parameters to be optimized. The proposed approach uses variational inference to approximate the intractable a posteriori distribution on basis of a normal prior. By representing the a posteriori uncertainty of the network parameters per network layer and depending on the estimated parameter expectation values, only very few additional parameters need to be optimized compared to a non-Bayesian network. We compare our approach to classical deep learning, Bernoulli dropout and Bayes by Backprop using the MNIST dataset. Compared to classical deep learning, the test error is reduced by 15%. We also show that the uncertainty information obtained can be used to calculate credible intervals for the network prediction and to optimize network architecture for the dataset at hand. To illustrate that our approach also scales to large networks and input vector sizes, we apply it to the GoogLeNet architecture on a custom dataset, achieving an average accuracy of 0.92. Using 95% credible intervals, all but one wrong classification result can be detected.