Conformal Prediction via Regression-as-Classification: A Flexible Approach for Handling Complex Output Distributions

Core Concepts

A novel approach to conformal prediction for regression that converts the regression problem into a classification problem, allowing the use of flexible classification-based conformal prediction techniques to handle complex output distributions such as heteroscedasticity and bimodality.

Abstract

The paper presents a new method called Regression-to-Classification Conformal Prediction (R2CCP) that addresses the challenges of conformal prediction for regression, especially when the output distribution is heteroscedastic, multimodal, or skewed.
Key highlights:

Regression problems are converted into classification problems by discretizing the output space into bins, treating each bin as a distinct class.
A new loss function is designed to preserve the ordering of the continuous output space, penalizing the density on bins far from the true output value while using entropy regularization to facilitate variability.
The resulting method can adapt to heteroscedasticity, bimodality, or both in the label distribution, as demonstrated on synthetic and real datasets.
Empirical results show that R2CCP achieves the shortest prediction intervals compared to other conformal prediction baselines.
The paper first provides background on conformal prediction and its challenges for regression. It then introduces the R2CCP approach, detailing the classification-based framework and the custom loss function. Extensive experiments are conducted to showcase the method's ability to handle complex output distributions and its superior performance over existing conformal prediction techniques.

Stats

The label distribution can exhibit heteroscedasticity, where the variance changes with the input. (Figure 1a)
The label distribution can be bimodal, with two distinct peaks. (Figure 1b)

Quotes

"Conformal Prediction (CP) (Vovk et al., 2005) has recently gained popularity and has been used successfully in applications such as breast cancer detection (Lambrou et al., 2009), stroke risk prediction (Lambrou et al., 2010), and drug discovery (Cortés-Ciriano & Bender, 2020)."
"Despite its popularity, CP for regression can be challenging, especially when the output distribution is heteroscedastic, multimodal, or skewed (Lei & Wasserman, 2014)."

Key Insights Distilled From

Conformal Prediction via Regression-as-Classification

by Etas... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08168.pdf

Deeper Inquiries

How can the R2CCP method be extended to handle even more complex output distributions, such as those with multiple modes or heavy tails

The R2CCP method can be extended to handle more complex output distributions by incorporating additional techniques to capture the nuances of these distributions. One approach could involve using more sophisticated neural network architectures, such as mixture density networks, which can model multimodal distributions effectively. By allowing the neural network to output multiple modes or heavy tails, the R2CCP method can better capture the complexity of the output distribution. Additionally, incorporating techniques like Bayesian neural networks or ensemble methods can provide a more robust estimation of the distribution, especially in cases of heavy tails or multiple modes. These methods can help improve the flexibility and adaptability of the R2CCP method to handle a wider range of output distributions.

What are the potential drawbacks or limitations of the regression-as-classification approach, and how can they be addressed

One potential drawback of the regression-as-classification approach is the discretization of the output space, which may lead to information loss and reduced accuracy, especially in cases where the output distribution is continuous and complex. This discretization can limit the model's ability to capture the nuances of the output distribution, particularly in scenarios with multiple modes or heavy tails. To address this limitation, techniques like adaptive binning or dynamic bin creation based on the data distribution can be implemented. This would allow for a more fine-grained representation of the output space, enabling the model to better capture the complexity of the distribution.
Another limitation is the reliance on a single loss function, which may not always capture the full complexity of the relationship between inputs and outputs. To address this, a combination of loss functions or a more sophisticated loss function that considers the ordinal structure of the output space could be explored. Additionally, incorporating regularization techniques to prevent overfitting and improve generalization can help mitigate the limitations of the regression-as-classification approach.

How can the insights from this work on conformal prediction for regression be applied to other areas of machine learning, such as uncertainty quantification or out-of-distribution detection

The insights from this work on conformal prediction for regression can be applied to other areas of machine learning, such as uncertainty quantification and out-of-distribution detection. In uncertainty quantification, the techniques developed for estimating the distribution over the output space can be leveraged to provide more accurate uncertainty estimates for machine learning models. By incorporating conformal prediction methods, models can provide prediction intervals that reflect the uncertainty in the predictions, leading to more reliable and trustworthy results.
For out-of-distribution detection, the principles of conformal prediction can be used to determine whether a new data point belongs to the same distribution as the training data. By constructing prediction sets based on the conformity scores, models can identify instances that deviate significantly from the training data distribution, indicating potential out-of-distribution samples. This can help improve the robustness and reliability of machine learning models in detecting anomalies and outliers.