Epistemic Uncertainty Quantification in Deep Learning Classification by the Delta Method

TL;DR

This paper introduces a computationally efficient method for quantifying epistemic uncertainty in deep neural networks using a modified Delta method based on the Fisher information matrix, demonstrated on image classification tasks.

Contribution

It proposes a low-cost Delta method variant for deep networks leveraging eigenpairs of the Fisher information matrix, with error bounds and practical implementation details.

Findings

Meaningful uncertainty rankings for images were obtained.

False positives exhibit higher epistemic uncertainty than true positives.

The method is effective on MNIST and CIFAR-10 datasets.

Abstract

The Delta method is a classical procedure for quantifying epistemic uncertainty in statistical models, but its direct application to deep neural networks is prevented by the large number of parameters $P$. We propose a low cost variant of the Delta method applicable to $L_2$-regularized deep neural networks based on the top $K$ eigenpairs of the Fisher information matrix. We address efficient computation of full-rank approximate eigendecompositions in terms of either the exact inverse Hessian, the inverse outer-products of gradients approximation or the so-called Sandwich estimator. Moreover, we provide a bound on the approximation error for the uncertainty of the predictive class probabilities. We observe that when the smallest eigenvalue of the Fisher information matrix is near the $L_2$-regularization rate, the approximation error is close to zero even when $K\ll P$. A demonstration of the methodology is presented using a TensorFlow implementation, and we show that meaningful rankings of images based on predictive uncertainty can be obtained for two LeNet-based neural networks using the MNIST and CIFAR-10 datasets. Further, we observe that false positives have on average a higher predictive epistemic uncertainty than true positives. This suggests that there is supplementing information in the uncertainty measure not captured by the classification alone.