Epistemic Uncertainty Quantification in Deep Learning Classification by the Delta Method

• Published in: arXiv.org 2021-02
• Main Authors: Nilsen, Geir K, Munthe-Kaas, Antonella Z, Skaug, Hans J, Brun, Morten
• Format: Article
• Language: English
• Subjects: Approximation; Artificial neural networks; Deep learning; Eigenvalues; Hessian matrices; Machine learning; Neural networks; Statistical analysis; Statistical models; Uncertainty
• ISSN: 2331-8422

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.