Title: Tube Loss: A Novel Approach for Prediction Interval Estimation and Probabilistic Forecasting
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: Uncertainty Quantification, Prediction Interval Estimation, Loss function, Regression

Abstract: This paper introduces a new loss function termed Tube Loss, designed for simultaneously estimating the lower and upper bounds of a Prediction Interval (PI) in regression tasks, while also facilitating probabilistic forecasting in auto-regressive settings.  The PIs derived by minimizing the empirical risk with Tube Loss demonstrate superior performance compared to those from existing approaches. Notably, the method ensures that the generated PI asymptotically achieve a user-defined target confidence level $t \in (0,1)$, a result supported by theoretical proof.  Additionally, Tube Loss offers flexibility by allowing users to shift the PI vertically using a tunable parameter $r$, which enables the intervals to better capture high-density regions of the response variable's distribution, particularly useful for skewed distributions, resulting in narrower and more informative intervals. The approach also facilitates a balance between interval coverage and average width within the same optimization process through parameter $\delta$ that allows for further width reduction via recalibration.  Unlike some existing techniques, the empirical risk under Tube Loss can be efficiently minimized using gradient descent. Comprehensive experiments validate the effectiveness of the proposed method across various models, including kernel machines, neural networks, deep learning architectures, and in probabilistic forecasting applications.