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Blackwell's Approachability for Sequential Conformal Inference

Guillaume Principato, Gilles Stoltz

TL;DR

This work reframes sequential conformal inference in non-exchangeable environments as a repeated two-player vector game and uses Blackwell's approachability to simultaneously address validity (coverage) and efficiency (predictive set length). It introduces an opportunistic, calibration-based strategy (BOACI) that leverages potential structure in the adversary's play, such as drift or regime switching, to achieve improved efficiency while preserving asymptotic validity. The analysis unifies exchangeable, adversarial, and regime-switching settings through a Q-restricted framework and a general target set S_Q, providing pathwise guarantees under mild assumptions. Practically, this yields principled guidance for designing conformal prediction procedures in sequential and non-stationary contexts, with theoretical guarantees and a clear tradeoff between robustness and sharpness. The calibration-based approach offers a scalable route to near-exchangeable performance without requiring full knowledge of the environment.

Abstract

We study conformal inference in non-exchangeable environments through the lens of Blackwell's theory of approachability. We first recast adaptive conformal inference (ACI, Gibbs and Candès, 2021) as a repeated two-player vector-valued finite game and characterize attainable coverage--efficiency tradeoffs. We then construct coverage and efficiency objectives under potential restrictions on the adversary's play, and design a calibration-based approachability strategy to achieve these goals. The resulting algorithm enjoys strong theoretical guarantees and provides practical insights, though its computational burden may limit deployment in practice.

Blackwell's Approachability for Sequential Conformal Inference

TL;DR

This work reframes sequential conformal inference in non-exchangeable environments as a repeated two-player vector game and uses Blackwell's approachability to simultaneously address validity (coverage) and efficiency (predictive set length). It introduces an opportunistic, calibration-based strategy (BOACI) that leverages potential structure in the adversary's play, such as drift or regime switching, to achieve improved efficiency while preserving asymptotic validity. The analysis unifies exchangeable, adversarial, and regime-switching settings through a Q-restricted framework and a general target set S_Q, providing pathwise guarantees under mild assumptions. Practically, this yields principled guidance for designing conformal prediction procedures in sequential and non-stationary contexts, with theoretical guarantees and a clear tradeoff between robustness and sharpness. The calibration-based approach offers a scalable route to near-exchangeable performance without requiring full knowledge of the environment.

Abstract

We study conformal inference in non-exchangeable environments through the lens of Blackwell's theory of approachability. We first recast adaptive conformal inference (ACI, Gibbs and Candès, 2021) as a repeated two-player vector-valued finite game and characterize attainable coverage--efficiency tradeoffs. We then construct coverage and efficiency objectives under potential restrictions on the adversary's play, and design a calibration-based approachability strategy to achieve these goals. The resulting algorithm enjoys strong theoretical guarantees and provides practical insights, though its computational burden may limit deployment in practice.
Paper Structure (35 sections, 24 theorems, 84 equations, 1 algorithm)

This paper contains 35 sections, 24 theorems, 84 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related work
  4. Contributions
  5. Settings and notation
  6. Notation.
  7. Reminder on Conformal Inference
  8. Adaptive conformal inference as a repeated two--player finite game
  9. Direct formulation.
  10. Reformulation as a finite game.
  11. Blackwell's approachability theory
  12. Definition of the objectives and preliminary results
  13. Objectives in the exchangeable setting
  14. Objectives in the adversarial setting
  15. $Q$--restricted opponent’s play and typical examples
  16. ...and 20 more sections

Key Result

Lemma 1

The following equalities define a scalar $b_t \in [0,1]$, and we actually have $b_t \in \mathcal{B}$. As a consequence, $\mathbf{1}_{y_t \notin C_{a_t}(x_t)} = \mathbf{1}_{b_t \leqslant a_t}$.

Theorems & Definitions (54)

  • Lemma 1
  • Example 1
  • Lemma 2
  • proof
  • Theorem 1
  • Remark 1
  • Theorem 2: zaffran2022adaptive
  • Corollary 1
  • Remark 2
  • Lemma 3
  • ...and 44 more