Building Conformal Prediction Intervals with Approximate Message Passing (2410.16493v1)

Abstract: Conformal prediction has emerged as a powerful tool for building prediction intervals that are valid in a distribution-free way. However, its evaluation may be computationally costly, especially in the high-dimensional setting where the dimensionality and sample sizes are both large and of comparable magnitudes. To address this challenge in the context of generalized linear regression, we propose a novel algorithm based on Approximate Message Passing (AMP) to accelerate the computation of prediction intervals using full conformal prediction, by approximating the computation of conformity scores. Our work bridges a gap between modern uncertainty quantification techniques and tools for high-dimensional problems involving the AMP algorithm. We evaluate our method on both synthetic and real data, and show that it produces prediction intervals that are close to the baseline methods, while being orders of magnitude faster. Additionally, in the high-dimensional limit and under assumptions on the data distribution, the conformity scores computed by AMP converge to the one computed exactly, which allows theoretical study and benchmarking of conformal methods in high dimensions.

• The paper employs an AMP algorithm to approximate leave-one-out residuals in generalized linear models, accelerating the computation of conformal prediction intervals.
• It introduces Taylor-AMP, a method that approximates residuals through derivative estimations, significantly reducing computational demands.
• Experimental results demonstrate that AMP-based methods offer prediction intervals with comparable accuracy and improved efficiency in high-dimensional settings.

Building Conformal Prediction Intervals with Approximate Message Passing

The paper presents an innovative approach to constructing conformal prediction intervals utilizing Approximate Message Passing (AMP) within the context of generalized linear regression. Conformal prediction serves as a reliable method for forming prediction intervals without reliance on distribution assumptions. However, the computational demands of conformal prediction escalate in high-dimensional settings where both the dimensionality and sample sizes are substantial. This paper introduces an AMP-based algorithm to enhance the computational efficiency of full conformal prediction by estimating conformity scores more swiftly.

Key Contributions

1. Applications of AMP in Generalized Linear Regression: The authors employ the AMP algorithm to approximate the conformity scores required for full conformal prediction within generalized linear models. By simultaneously estimating the leave-one-out (LOO) residuals for all samples, AMP significantly accelerates the process, addressing the costly nature of traditional methods.
2. Introduction of Taylor-AMP: Building on AMP, the paper proposes Taylor-AMP, a novel method that further expedites the calculation of prediction intervals. Taylor-AMP does this by approximating the residuals through derivative estimations around a reference value, eliminating the need to compute estimators individually for each possible label.
3. High-Dimensional Study with Gaussian Data: The research demonstrates that AMP effectively approximates the LOO residuals in high-dimensional spaces, converging to exact scores under certain distributional assumptions. This allows for theoretical exploration and benchmarking of conformal methods within high-dimensional limits.

Numerical Results

The effectiveness of the proposed methods is corroborated through experiments on synthetic and real datasets. The results demonstrate that Taylor-AMP yields prediction intervals comparable to traditional methods while achieving substantial computational gains. Particularly noteworthy are the intervals' sizes and coverage on Gaussian synthetic data and various real datasets, revealing the AMP-based methods' practical viability.

Implications and Future Directions

The implications of this research are twofold: it promotes the adoption of AMP in high-dimensional statistical challenges and provides a framework for swiftly assessing conformal prediction in applied settings. Moreover, the discussion hints at extensions of AMP methods toward Bayesian frameworks and applications beyond generalized linear models, suggesting potential avenues for further exploration.

In summary, the work establishes a vital link between conformal prediction and AMP, yielding substantial improvements in computation without compromising prediction accuracy. This advancement opens new possibilities for robust and efficient uncertainty quantification in high-dimensional data analysis.