Zono-Conformal Prediction: Zonotope-Based Uncertainty Quantification for Regression and Classification Tasks

TL;DR

Zono-conformal prediction is introduced, a novel approach inspired by interval predictor models and reachset-conformant identification that constructs prediction zonotopes with assured coverage and provides probabilistic coverage guarantees and methods for detecting outliers in the identification data.

Abstract

Conformal prediction is a popular uncertainty quantification method that augments a base predictor to return sets of predictions with statistically valid coverage guarantees. However, current methods are often computationally expensive and data-intensive, as they require constructing an uncertainty model before calibration. Moreover, existing approaches typically represent the prediction sets with intervals, which limits their ability to capture dependencies in multi-dimensional outputs. We address these limitations by introducing zono-conformal prediction, a novel approach inspired by interval predictor models and reachset-conformant identification that constructs prediction zonotopes with assured coverage. By placing zonotopic uncertainty sets directly into the model of the base predictor, zono-conformal predictors can be identified via a single, data-efficient linear program. While we can apply zono-conformal prediction to arbitrary nonlinear base predictors, we focus on feed-forward neural networks in this work. Aside from regression tasks, we also construct optimal zono-conformal predictors in classification settings where the output of an uncertain predictor is a set of possible classes. We provide probabilistic coverage guarantees and present methods for detecting outliers in the identification data. In extensive numerical experiments, we show that zono-conformal predictors are less conservative than interval predictor models and standard conformal prediction methods, while achieving a similar coverage over the test data.

Figures (10)

• Figure 1: Boundary and non-boundary points.
• Figure 2: Trade-off between coverage and conservatism for different regression tasks.
• Figure 3: Prediction sets for different regression tasks projected onto the $y_1-y_2$ plane for example data points from the test set. The true outputs are denoted by gray crosses. For the synthetic data sets SD-R1 and SD-R2, we included 100 outputs, which would have been possible for the given input using different uncertainty realizations.
• Figure 4: Trade-off between coverage and conservatism for different classification tasks.
• Figure 5: Prediction sets for classification tasks for example data points from the test set. The $x$-axis represents the output score for the true class, while the $y$-axis corresponds to the output for an arbitrarily chosen incorrect class. The gray shaded region marks the domain where the true class has a higher score than the incorrect class.

Theorems & Definitions (11)

• Definition 1: Zonotopes kuehn1998wrapping
• Definition 2: Interval Norm althoff2023conf
• Example 1
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 6: Zono-Conformal Regression
• Theorem 7: Zono-Conformal Classification
• Proposition 8: Detection of Boundary Points
• Proposition 9: Zono-Conformal Regression with Outlier Removal