CreINNs: Credal-Set Interval Neural Networks for Uncertainty Estimation in Classification Tasks

TL;DR

CreINNs address the challenge of uncertainty estimation in classification by representing model uncertainty as intervals over weights and deriving probability-interval predictions that form a credal set. The method introduces Interval SoftMax to produce valid class-probability intervals, a backward-compatible training objective using intersection probabilities, Interval Batch Normalization for deep networks, and an ensemble strategy to bolster calibration. Empirical results on standard multiclass and binary tasks, as well as interval-input cases, show CreINNs provide competitive or superior uncertainty quantification compared to variational BNNs and deep ensembles, while reducing inference cost relative to sampling-based Bayesian methods. The work extends uncertainty quantification to interval data and demonstrates practical impact for safer, more reliable classification in real-world settings.

Abstract

Effective uncertainty estimation is becoming increasingly attractive for enhancing the reliability of neural networks. This work presents a novel approach, termed Credal-Set Interval Neural Networks (CreINNs), for classification. CreINNs retain the fundamental structure of traditional Interval Neural Networks, capturing weight uncertainty through deterministic intervals. CreINNs are designed to predict an upper and a lower probability bound for each class, rather than a single probability value. The probability intervals can define a credal set, facilitating estimating different types of uncertainties associated with predictions. Experiments on standard multiclass and binary classification tasks demonstrate that the proposed CreINNs can achieve superior or comparable quality of uncertainty estimation compared to variational Bayesian Neural Networks (BNNs) and Deep Ensembles. Furthermore, CreINNs significantly reduce the computational complexity of variational BNNs during inference. Moreover, the effective uncertainty quantification of CreINNs is also verified when the input data are intervals.

Figures (12)

• Figure 1: Illustration of the proposed CreINN model for a three-class classification task. CreINN follows the conventional INN architecture, representing inputs $[\underline{\boldsymbol{x}}, \overline{\boldsymbol{x}}]$, node outputs, weights, and biases (i.e., $[\underline{a}_i^l, \overline{a}_i^l]$ and $[\underline{w}_{ji}^l, \overline{w}_{ji}^l]$ for the $i^{th}$ node of $l^{th}$ layer and ${[\underline{\boldsymbol{b}}, \overline{\boldsymbol{b}}]}^l$ for the $l^{th}$ layer, respectively) as deterministic intervals. Using the proposed Interval SoftMax activation, a set of probability intervals $[\underline{\boldsymbol{q}}, \overline{\boldsymbol{q}}]\!:=\!{\{[\underline{q}_k, \overline{q}_k]\}}_{k=1}^{k=3}$ can derived from the outputted deterministic output interval vector. Through redundancy reduction, the resulting reachable probability interval $[\underline{\boldsymbol{q}}^*, \overline{\boldsymbol{q}}^*]$ (shown as parallel dashed lines) can define a credal set $\mathbb{Q}$ for uncertainty estimation, depicted as the light orange convex hull within the probability simplex (a triangle representing all probability distributions over the target space). In addition, an intersection probability $\boldsymbol{q}_{\text{int}}$ can be computed from these probability intervals for class classification purposes. Model training involves minimizing the cross-entropy (CE) loss with constraints that guarantee valid weight and bias intervals. Moreover, the proposed CreINN can handle both interval and standard format data.
• Figure 2: Redundant probability intervals define a credal set $\mathbb{Q}$ (the convex hull in light orange) in a 2D probability simplex by incorporating interval constraints while some probability bounds (e.g., the upper probability $\overline{q}_B$) may not be reachable.
• Figure 3: Determining an intersection probability from the probability interval systems on target space of $\mathbb{Y} \!=\!$ {A, B, D}.
• Figure 4: Monitoring the standard training processes of CreINN, SNN, BNN-F, and BNN-R (a) and AR curves using AU (b), EU (c), and TU (d) estimates, averaged over 15 experimental runs.
• Figure 5: Heat map of a slice of $\boldsymbol{W}_r$, which has a shape of (7, 7, 3, 64) and is derived from the first convolution layer of a trained CreINN.