Tube Loss: A Novel Loss Function for Prediction Interval Estimation

Abstract

This paper proposes a novel loss function called Tube Loss, developed for the simultaneous estimation of the lower and upper bounds of a Prediction Interval (PI) in regression problems, including probabilistic forecasting in autoregressive frameworks. The PIs obtained through empirical risk minimization using Tube Loss exhibit superior performance compared to those derived from existing approaches. A theoretical analysis confirms that the estimated PIs asymptotically attain a user-specified confidence level $1-\alpha$. A distinctive feature of Tube Loss is its ability to shift the PI along the support of the response distribution through a tunable parameter, allowing the intervals to better align with high-density regions of the distribution. This is especially valuable for generating tighter intervals when the response distribution is skewed. Moreover, the method allows further narrowing of PIs through recalibration. Unlike several prior techniques, the empirical risk associated with Tube Loss can be efficiently optimized via gradient descent. Extensive experiments demonstrate the robustness and accuracy of the proposed method in delivering high-quality PIs across a range of models, including kernel machines, neural networks, and probabilistic forecasting frameworks.