SEF: A Method for Computing Prediction Intervals by Shifting the Error Function in Neural Networks

TL;DR

SEF introduces a simple, universal approach to neural network uncertainty quantification by constructing prediction intervals via shifting the error function with a displacement constant $\mu$ derived from residuals. The method trains three networks concurrently: a primary regressor ${\text{NN}}_{\text{approx}}$ and two bound-estimators ${\text{NN}}_{\text{lower}}$, ${\text{NN}}_{\text{upper}}$ that target $l_i=y_i-\mu$ and $u_i=y_i+\mu$, respectively. Across synthetic datasets with both homoscedastic and heteroscedastic noise, SEF achieves high coverage $\text{PICP}$ at the specified level $\gamma$ while maintaining competitive or superior interval widths (MPIW, NMPIW) relative to PI3NN and PIVEN. The results indicate that SEF provides robust, computationally light uncertainty quantification without extensive hyperparameter tuning, making it appealing for regression tasks requiring reliable prediction intervals in practice.

Abstract

In today's era, Neural Networks (NN) are applied in various scientific fields such as robotics, medicine, engineering, etc. However, the predictions of neural networks themselves contain a degree of uncertainty that must always be taken into account before any decision is made. This is why many researchers have focused on developing different ways to quantify the uncertainty of neural network predictions. Some of these methods are based on generating prediction intervals (PI) via neural networks for the requested target values. The SEF (Shifting the Error Function) method presented in this paper is a new method that belongs to this category of methods. The proposed approach involves training a single neural network three times, thus generating an estimate along with the corresponding upper and lower bounds for a given problem. A pivotal aspect of the method is the calculation of a parameter from the initial network's estimates, which is then integrated into the loss functions of the other two networks. This innovative process effectively produces PIs, resulting in a robust and efficient technique for uncertainty quantification. To evaluate the effectiveness of our method, a comparison in terms of successful PI generation between the SEF, PI3NN and PIVEN methods was made using two synthetic datasets.

Figures (3)

• Figure 1: Geometrical representation of the lines of shifting for the upper and lower bounds.
• Figure 2: Graphical representations of the alignment of points at the beginning, during, and at the end of the training process.
• Figure 3: Scatter plot illustrating the highest PICP case from the fourth random split.