Multivariate Conformal Prediction using Optimal Transport

TL;DR

This paper introduces OTCP, a novel multivariate conformal prediction method using optimal transport to create distribution-free uncertainty sets in high-dimensional spaces, with demonstrated improvements on regression benchmarks.

Contribution

The paper proposes a new multivariate conformal prediction framework based on optimal transport, addressing the challenge of vector ranking without a canonical order.

Findings

Achieves distribution-free coverage guarantees in multivariate settings.

Demonstrates improved predictive sets on multivariate regression benchmarks.

Analyzes computational and statistical trade-offs in OT-based conformity scores.

Abstract

Conformal prediction (CP) quantifies the uncertainty of machine learning models by constructing sets of plausible outputs. These sets are constructed by leveraging a so-called conformity score, a quantity computed using the input point of interest, a prediction model, and past observations. CP sets are then obtained by evaluating the conformity score of all possible outputs, and selecting them according to the rank of their scores. Due to this ranking step, most CP approaches rely on a score functions that are univariate. The challenge in extending these scores to multivariate spaces lies in the fact that no canonical order for vectors exists. To address this, we leverage a natural extension of multivariate score ranking based on optimal transport (OT). Our method, OTCP, offers a principled framework for constructing conformal prediction sets in multidimensional settings, preserving distribution-free coverage guarantees with finite data samples. We demonstrate tangible gains in a benchmark dataset of multivariate regression problems and address computational \& statistical trade-offs that arise when estimating conformity scores through OT maps.