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Robust Validation: Confident Predictions Even When Distributions Shift

Maxime Cauchois, Suyash Gupta, Alnur Ali, John C. Duchi

TL;DR

This work tackles predictive validity under distribution shift by proposing conformal-inference-based prediction sets that remain valid when test distributions lie within an $f$-divergence neighborhood of the training distribution. It develops a theory linking worst-case coverage to $f$-divergence via the $g_{f,\rho}$ transform and provides practical plug-in estimators $\widehat{C}_{n,f,\rho}$ with finite-sample guarantees. The authors introduce procedures to estimate plausible shifts—including high-probability coverage over shift classes and worst-direction shift estimation—plus a debiased, cross-fit framework to quantify covariate-shift sensitivity using unlabeled data. Empirically, robust conformalization improves coverage on CIFAR-10, MNIST, ImageNet, and a COVID-19 forecasting task without substantial inflation of interval length, suggesting practical utility for uncertainty quantification under distributional change.

Abstract

While the traditional viewpoint in machine learning and statistics assumes training and testing samples come from the same population, practice belies this fiction. One strategy -- coming from robust statistics and optimization -- is thus to build a model robust to distributional perturbations. In this paper, we take a different approach to describe procedures for robust predictive inference, where a model provides uncertainty estimates on its predictions rather than point predictions. We present a method that produces prediction sets (almost exactly) giving the right coverage level for any test distribution in an $f$-divergence ball around the training population. The method, based on conformal inference, achieves (nearly) valid coverage in finite samples, under only the condition that the training data be exchangeable. An essential component of our methodology is to estimate the amount of expected future data shift and build robustness to it; we develop estimators and prove their consistency for protection and validity of uncertainty estimates under shifts. By experimenting on several large-scale benchmark datasets, including Recht et al.'s CIFAR-v4 and ImageNet-V2 datasets, we provide complementary empirical results that highlight the importance of robust predictive validity.

Robust Validation: Confident Predictions Even When Distributions Shift

TL;DR

This work tackles predictive validity under distribution shift by proposing conformal-inference-based prediction sets that remain valid when test distributions lie within an -divergence neighborhood of the training distribution. It develops a theory linking worst-case coverage to -divergence via the transform and provides practical plug-in estimators with finite-sample guarantees. The authors introduce procedures to estimate plausible shifts—including high-probability coverage over shift classes and worst-direction shift estimation—plus a debiased, cross-fit framework to quantify covariate-shift sensitivity using unlabeled data. Empirically, robust conformalization improves coverage on CIFAR-10, MNIST, ImageNet, and a COVID-19 forecasting task without substantial inflation of interval length, suggesting practical utility for uncertainty quantification under distributional change.

Abstract

While the traditional viewpoint in machine learning and statistics assumes training and testing samples come from the same population, practice belies this fiction. One strategy -- coming from robust statistics and optimization -- is thus to build a model robust to distributional perturbations. In this paper, we take a different approach to describe procedures for robust predictive inference, where a model provides uncertainty estimates on its predictions rather than point predictions. We present a method that produces prediction sets (almost exactly) giving the right coverage level for any test distribution in an -divergence ball around the training population. The method, based on conformal inference, achieves (nearly) valid coverage in finite samples, under only the condition that the training data be exchangeable. An essential component of our methodology is to estimate the amount of expected future data shift and build robustness to it; we develop estimators and prove their consistency for protection and validity of uncertainty estimates under shifts. By experimenting on several large-scale benchmark datasets, including Recht et al.'s CIFAR-v4 and ImageNet-V2 datasets, we provide complementary empirical results that highlight the importance of robust predictive validity.

Paper Structure

This paper contains 54 sections, 34 theorems, 236 equations, 10 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background: split conformal inference under exchangeability
  3. Related work
  4. A few motivating examples
  5. Robust predictive inference
  6. Characterizing and computing quantiles over $f$-divergence balls
  7. Achieving coverage with empirical estimates
  8. Procedures for estimating future distribution shift
  9. High-probability coverage over specific classes of shifts
  10. Finding directions of maximal shift
  11. Coverage sensitivity under covariate shifts
  12. Covariate-specific sensitivity analysis
  13. Cross-fit dual estimation of the sensitivity function
  14. Empirical analysis
  15. CIFAR-10, MNIST, and ImageNet datasets
  16. ...and 39 more sections

Key Result

Proposition 1

Define the function $g_{f, \rho} : [0, 1] \to [0, 1]$ by Then the inverse guarantees that for all distributions $P$ on $\mathbb{R}$ and $\alpha \in (0, 1)$,

Figures (10)

  • Figure 1: Empirical coverage for the prediction sets generated by the standard conformal methodology across nine regression data sets and 50 random splits of each data set, with an exponential tilting in $X$ space along the first principal component of $X$. The horizontal axis gives the value of the tilting parameter $a$; the vertical the coverage level. A green line marks the average coverage, a black line marks the median coverage, and the horizontal red line marks the nominal coverage $.95$. The blue bands show the coverage at deciles over 50 splits.
  • Figure 2: Empirical coverage and average size for the prediction sets generated by the standard conformal methodology ("SC") and the chi-squared divergence, across 20 random splits of the CIFAR-10 data. We set $\rho$ according to the sampling ("$\chi^2$-S"), regression ("$\chi^2$-R"), and classification-based ("$\chi^2$-C") strategies for estimating the amount of shift that we describe in Section in \ref{['sec:futureshiftestimation']}. The horizontal red line marks the marginal coverage $.95$.
  • Figure 3: Empirical coverage and average size for the prediction sets generated by the standard conformal methodology ("SC") and the chi-squared divergence, across 20 random splits of the MNIST data. We set $\rho$ according to the sampling ("$\chi^2$-S"), regression ("$\chi^2$-R"), and classification-based ("$\chi^2$-C") strategies for estimating the amount of shift that we describe in Section in \ref{['sec:futureshiftestimation']}. The horizontal red line marks the marginal coverage $.95$.
  • Figure 4: Empirical coverage and average size for the prediction sets generated by the standard conformal methodology ("SC") and the chi-squared divergence, across 20 random splits of the ImageNet data. We set $\rho$ according to the sampling ("$\chi^2$-S"), regression ("$\chi^2$-R"), and classification-based ("$\chi^2$-C") strategies for estimating the amount of shift that we describe in Section in \ref{['sec:futureshiftestimation']}. The horizontal red line marks the marginal coverage $.9$.
  • Figure 5: Empirical coverage for the prediction sets generated by the chi-squared divergence, following the same experimental setup from Section \ref{['sec:motivation-exp']}. The horizontal axis gives the value of the tilting parameter $a$; the vertical the coverage level. A green line marks the average coverage, a black line marks the median coverage, and the horizontal red line marks the nominal coverage $.95$. The blue bands show the coverage at various deciles.
  • ...and 5 more figures

Theorems & Definitions (39)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Corollary 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Definition 3.1
  • Definition 3.2
  • Lemma 4.1: Example 3, DuchiNa21
  • Theorem 2
  • ...and 29 more