Exact and Approximate Conformal Inference for Multi-Output Regression

TL;DR

This paper develops exact and approximate conformal inference methods for multi-output regression, enabling uncertainty quantification with improved computational efficiency for linear and nonlinear models.

Contribution

It provides exact conformal p-value derivations for linear multi-output models and introduces exttt{unionCP} and multivariate exttt{rootCP} for efficient approximation of prediction regions.

Findings

Exact conformal p-values derived for linear multi-output models

Proposed methods are computationally efficient for diverse predictors

Empirical results demonstrate effectiveness on real and simulated data

Abstract

It is common in machine learning to estimate a response $y$ given covariate information $x$. However, these predictions alone do not quantify any uncertainty associated with said predictions. One way to overcome this deficiency is with conformal inference methods, which construct a set containing the unobserved response $y$ with a prescribed probability. Unfortunately, even with a one-dimensional response, conformal inference is computationally expensive despite recent encouraging advances. In this paper, we explore multi-output regression, delivering exact derivations of conformal inference $p$-values when the predictive model can be described as a linear function of $y$. Additionally, we propose \texttt{unionCP} and a multivariate extension of \texttt{rootCP} as efficient ways of approximating the conformal prediction region for a wide array of multi-output predictors, both linear and nonlinear, while preserving computational advantages. We also provide both theoretical and empirical evidence of the effectiveness of these methods using both real-world and simulated data.