A Comparison of the Delta Method and the Bootstrap in Deep Learning Classification

We validate the recently introduced deep learning classification adapted Delta method by a comparison with the classical Bootstrap. We show that there is a strong linear relationship between the quantified predictive epistemic uncertainty levels obtained from the two methods when applied on two LeNet-based neural network classifiers using the MNIST and CIFAR-10 datasets. Furthermore, we demonstrate that the Delta method offers a five times computation time reduction compared to the Bootstrap.

On the Delta Method for Uncertainty Approximation in Deep Learning

The Delta method is a well known procedure used to quantify uncertainty in statistical models. The method has previously been applied in the context of neural networks, but has not reached much popularity in deep learning because of the sheer size of the Hessian matrix. In this paper, we propose a low cost variant of the method based on an approximate eigendecomposition of the positive curvature subspace of the Hessian matrix. The method has a computational complexity of $O(KPN)$ time and $O(KP)$ space, where $K$ is the number of utilized Hessian eigenpairs, $P$ is the number of model parameters and $N$ is the number of training examples. Given that the model is $L_2$-regularized with rate $\lambda/2$, we provide a bound on the uncertainty approximation error given $K$. We show that when the smallest Hessian eigenvalue in the positive $K/2$-tail of the full spectrum, and the largest Hessian eigenvalue in the negative $K/2$-tail of the full spectrum are both approximately equal to $\lambda$, the error will be close to zero even when $K\ll P$ . We demonstrate the method by a TensorFlow implementation, and show that meaningful rankings of images based on prediction uncertainty can be obtained for a convolutional neural network based MNIST classifier. We also observe that false positives have higher prediction uncertainty than true positives. This suggests that there is supplementing information in the uncertainty measure not captured by the probability alone.

Efficient Computation of Hessian Matrices in TensorFlow

The Hessian matrix has a number of important applications in a variety of different fields, such as optimzation, image processing and statistics. In this paper we focus on the practical aspects of efficiently computing Hessian matrices in the context of deep learning using the Python scripting language and the TensorFlow library.