2019/12/02 by Nilsen, Geir K., Munthe-Kaas, Antonella Z., Skaug, Hans J. +1
#FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML)
paper · doi:10.48550/arxiv.1912.00832
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 L2-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 L2-regularization rate, the approximation error is close to zero even when K≪ 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.