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We suggest a general approach to quantification of different forms ofaleatoric uncertainty in regression tasks performed by artificial neuralnetworks. It is based on the simultaneous training of two neural networks witha joint loss function and a specific hyperparameter $\lambda>0$ that allows forautomatically detecting noisy and clean regions in the input space andcontrolling their {\em relative contribution} to the loss and its gradients.After the model has been trained, one of the networks performs predictions andthe other quantifies the uncertainty of these predictions by estimating thelocally averaged loss of the first one. Unlike in many classical uncertaintyquantification methods, we do not assume any a priori knowledge of the groundtruth probability distribution, neither do we, in general, maximize thelikelihood of a chosen parametric family of distributions. We analyze thelearning process and the influence of clean and noisy regions of the inputspace on the loss surface, depending on $\lambda$. In particular, we show thatsmall values of $\lambda$ increase the relative contribution of clean regionsto the loss and its gradients. This explains why choosing small $\lambda$allows for better predictions compared with neural networks without uncertaintycounterparts and those based on classical likelihood maximization. Finally, wedemonstrate that one can naturally form ensembles of pairs of our networks andthus capture both aleatoric and epistemic uncertainty and avoid overfitting.

Pairing an arbitrary regressor with an artificial neural network estimating aleatoric uncertainty