Conditional Validity of Conformal Regression Predictors for Heteroskedastic Data

Core Concepts

Conformal prediction offers a distribution-free approach to estimating prediction intervals with statistical guarantees. This paper investigates how conformal predictors can be constructed to adapt to heteroskedastic noise in the data, while maintaining conditional validity with respect to the level of heteroskedasticity.

Abstract

The paper discusses the problem of conditional uncertainty quantification in the regression setting, where the goal is to construct prediction intervals that adapt to the heteroskedasticity of the underlying process. It introduces the concept of conditional validity and explores different approaches to achieve this, including normalized conformal prediction and Mondrian conformal prediction.
The key insights are:

Theoretical conditions are derived for attaining conditional validity of non-Mondrian conformal predictors. This is related to the notion of pivotal quantities from classical statistics.

It is shown that a large class of common distributions, including normal, Laplace, and uniform distributions, give rise to conditionally valid normalized conformal predictors in a natural way.

Experiments on synthetic data are used to analyze the impact of misspecification and contamination on the conditional validity of different conformal predictors. This provides practical diagnostic tools to assess when a non-Mondrian conformal predictor can be expected to be conditionally valid.

Overall, the paper provides a theoretical and empirical investigation of how conformal prediction can be adapted to handle heteroskedastic regression problems, while maintaining rigorous statistical guarantees.

Stats

The data-generating process for the synthetic experiments follows the form y(x,s) ~ N(μ(x,s), σ(x,s)^2), where s is a dummy variable labeling subgroups with different noise levels.
The misspecification of the mean and variance estimates is simulated by adding Gaussian noise to the true values.

Quotes

"Conformal prediction has a probabilistic validity guarantee, but this only holds w.r.t. the full data distribution, i.e. on average over the whole instance space. Consequently, the algorithm is allowed to attain the claimed validity by solely focusing on the 'easy' parts of the data, which are often more abundant, while ignoring the more difficult parts."
"A key concept in statistics is heteroskedasticity, where the conditional distributions for different values of the conditioning variable have a different variance. In this paper, the focus lies on modelling heteroskedastic noise with guarantees conditional on the level of heteroskedasticity, i.e. where the data set is divided based on an estimate of the residual variance."

Key Insights Distilled From

Conditional validity of heteroskedastic conformal regression

by Nicolas Dewo... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2309.08313.pdf

Deeper Inquiries

How can the theoretical results on conditional validity be extended to other types of conformal predictors beyond the normalized and Mondrian approaches

The theoretical results on conditional validity presented in the paper can be extended to other types of conformal predictors by considering the underlying assumptions and conditions that lead to conditional validity. The key aspect to consider is the relationship between the nonconformity measure, the taxonomy function, and the data-generating distribution. By analyzing how these components interact and ensuring that the nonconformity measure is pivotal for the conditional distributions, it is possible to generalize the concept of conditional validity to different types of conformal predictors. This extension would involve examining the specific characteristics and requirements of each conformal prediction method to determine their conditional validity in various scenarios.

What are the implications of the findings in this paper for the practical application of conformal prediction in real-world regression problems with heterogeneous noise

The findings in this paper have significant implications for the practical application of conformal prediction in real-world regression problems with heterogeneous noise. Understanding the conditions under which conformal predictors can maintain conditional validity in the presence of heteroskedastic noise is crucial for ensuring the reliability and accuracy of prediction intervals. By considering the impact of misspecification and contamination on the performance of conformal predictors, practitioners can make informed decisions about the suitability of different methods for handling heterogeneous noise in regression problems. This knowledge can guide the selection of appropriate nonconformity measures and taxonomy functions to improve the robustness and effectiveness of conformal prediction in practical applications.

Can the insights from this work be applied to develop new conformal prediction methods that are specifically tailored for heteroskedastic data, going beyond the adaptations considered here

The insights from this work can be leveraged to develop new conformal prediction methods that are specifically tailored for heteroskedastic data. By building on the theoretical framework established in the paper, researchers can explore novel approaches to address the challenges posed by heteroskedastic noise in regression problems. This could involve designing nonconformity measures and taxonomy functions that are optimized for capturing the variability in the data and adapting to the specific characteristics of heteroskedastic processes. Additionally, incorporating the concept of pivotal quantities and standardized measures can provide a foundation for developing innovative conformal prediction techniques that offer improved performance and reliability in the presence of heterogeneous noise.