Epistemic Uncertainty Quantification in Deep Learning Classification by the Delta Method

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 neu...

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Bibliographic Details
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
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Summary: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.
ISSN:2331-8422