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Localized Conformal Multi-Quantile Regression

Yuan Lu

TL;DR

<3-5 sentence high-level summary> Localized Conformal Multi-Quantile Regression (LCMQR) addresses the dual challenges of inefficiency and lack of local adaptivity in uncertainty quantification by marrying multi-quantile regression with kernel-based localization within a split-conformal framework. The authors resolve a theoretical inconsistency in CCQR with a consistent Average-then-Max scoring, proving tighter, valid prediction intervals and establishing marginal and asymptotic conditional coverage. They further extend the approach to Group-Calibrated LCMQR (GC-LCMQR), adding stratified calibration to guarantee finite-sample validity within subgroups in heterogeneous populations, demonstrated on both benchmark regression tasks and an ITE setting. Empirical results show superior interval efficiency for LCMQR and robust group-wise validity for GC-LCMQR, including safe uncertainty quantification for high-variance focal groups in treatment-effect scenarios.

Abstract

Standard conformal prediction methods guarantee marginal coverage but often produce inefficient intervals that fail to adapt to local heteroscedasticity, while recent localized approaches often struggle to maintain validity across distinct subpopulations with varying noise profiles. To address these challenges, we introduce Localized Conformal Multi-Quantile Regression (LCMQR), a novel framework that synergizes multi-quantile information with kernel-based localization to construct efficient and adaptive prediction intervals. Theoretically, we resolve an inconsistency in Conformalized Composite Quantile Regression (CCQR) by proving that our consistent Average-then-Max scoring mechanism systematically yields tighter intervals than the Max-then-Average approach used in prior work. For heterogeneous environments, we extend this framework to Group-Calibrated LCMQR (GC-LCMQR) via a stratified calibration step that guarantees finite-sample validity within distinct subgroups. Experiments on benchmark datasets and an Individual Treatment Effect (ITE) task demonstrate that LCMQR achieves superior efficiency on standard benchmarks, while GC-LCMQR uniquely achieves group-level coverage for target subgroups in mixture populations where baselines fail.

Localized Conformal Multi-Quantile Regression

TL;DR

<3-5 sentence high-level summary> Localized Conformal Multi-Quantile Regression (LCMQR) addresses the dual challenges of inefficiency and lack of local adaptivity in uncertainty quantification by marrying multi-quantile regression with kernel-based localization within a split-conformal framework. The authors resolve a theoretical inconsistency in CCQR with a consistent Average-then-Max scoring, proving tighter, valid prediction intervals and establishing marginal and asymptotic conditional coverage. They further extend the approach to Group-Calibrated LCMQR (GC-LCMQR), adding stratified calibration to guarantee finite-sample validity within subgroups in heterogeneous populations, demonstrated on both benchmark regression tasks and an ITE setting. Empirical results show superior interval efficiency for LCMQR and robust group-wise validity for GC-LCMQR, including safe uncertainty quantification for high-variance focal groups in treatment-effect scenarios.

Abstract

Standard conformal prediction methods guarantee marginal coverage but often produce inefficient intervals that fail to adapt to local heteroscedasticity, while recent localized approaches often struggle to maintain validity across distinct subpopulations with varying noise profiles. To address these challenges, we introduce Localized Conformal Multi-Quantile Regression (LCMQR), a novel framework that synergizes multi-quantile information with kernel-based localization to construct efficient and adaptive prediction intervals. Theoretically, we resolve an inconsistency in Conformalized Composite Quantile Regression (CCQR) by proving that our consistent Average-then-Max scoring mechanism systematically yields tighter intervals than the Max-then-Average approach used in prior work. For heterogeneous environments, we extend this framework to Group-Calibrated LCMQR (GC-LCMQR) via a stratified calibration step that guarantees finite-sample validity within distinct subgroups. Experiments on benchmark datasets and an Individual Treatment Effect (ITE) task demonstrate that LCMQR achieves superior efficiency on standard benchmarks, while GC-LCMQR uniquely achieves group-level coverage for target subgroups in mixture populations where baselines fail.

Paper Structure

This paper contains 24 sections, 4 theorems, 17 equations, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Problem Setup
  4. Related Work
  5. Group-Conditional Validity and Application to ITE
  6. Proposed Method
  7. Base Framework: Localized Conformal Multi-Quantile Regression
  8. Theoretical Justification
  9. Extension: Group-Calibrated LCMQR (GC-LCMQR)
  10. Theoretical Analysis
  11. Marginal Coverage Guarantee
  12. Asymptotic Conditional Coverage
  13. Finite-Sample Group Validity (GC-LCMQR)
  14. Experiments
  15. Efficiency on Real-World Benchmarks
  16. ...and 9 more sections

Key Result

Theorem 1

For any data point $(x, y)$ and any set of quantile estimators, the nonconformity score of LCMQR is always less than or equal to that of CCQR. Consequently, for a fixed miscoverage rate $\alpha$, the calibration threshold satisfies $\hat{Q}_{LCMQR} \le \hat{Q}_{CCQR}$, resulting in tighter predictio

Theorems & Definitions (8)

  • Theorem 1: Efficiency via Jensen's Inequality
  • proof
  • Theorem 2: Marginal Coverage
  • proof
  • Theorem 3: Asymptotic Conditional Coverage
  • proof
  • Theorem 4: Finite-Sample Group Coverage
  • proof