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eCP: Informative uncertainty quantification via Equivariantized Conformal Prediction with pre-trained models

Nikolaos Bousias, Lars Lindemann, George Pappas

TL;DR

This work addresses the inefficiency of conformal prediction in long-horizon uncertainty by introducing Equivariantized Conformal Prediction (eCP), which leverages known geometric symmetries via a group-averaging operator applied to a pretrained predictor. By contracting the nonconformity score distribution in increasing convex order, eCP yields sharper, planner-friendly conformal sets while preserving finite-sample coverage. The authors prove a suite of results—risk improvements under increasing convex order, quantile contraction, Chernoff bound tightening, rate-function dominance, and Hoeffding bound improvements—and show asymptotic and extreme-quantile consequences. Empirical evaluations on pedestrian trajectory benchmarks demonstrate substantial reductions in 95% conformal radii without compromising validity, confirming practical gains in symmetry-rich, long-horizon forecasting tasks.

Abstract

We study the effect of group symmetrization of pre-trained models on conformal prediction (CP), a post-hoc, distribution-free, finite-sample method of uncertainty quantification that offers formal coverage guarantees under the assumption of data exchangeability. Unfortunately, CP uncertainty regions can grow significantly in long horizon missions, rendering the statistical guarantees uninformative. To that end, we propose infusing CP with geometric information via group-averaging of the pretrained predictor to distribute the non-conformity mass across the orbits. Each sample now is treated as a representative of an orbit, thus uncertainty can be mitigated by other samples entangled to it via the orbit inducing elements of the symmetry group. Our approach provably yields contracted non-conformity scores in increasing convex order, implying improved exponential-tail bounds and sharper conformal prediction sets in expectation, especially at high confidence levels. We then propose an experimental design to test these theoretical claims in pedestrian trajectory prediction.

eCP: Informative uncertainty quantification via Equivariantized Conformal Prediction with pre-trained models

TL;DR

This work addresses the inefficiency of conformal prediction in long-horizon uncertainty by introducing Equivariantized Conformal Prediction (eCP), which leverages known geometric symmetries via a group-averaging operator applied to a pretrained predictor. By contracting the nonconformity score distribution in increasing convex order, eCP yields sharper, planner-friendly conformal sets while preserving finite-sample coverage. The authors prove a suite of results—risk improvements under increasing convex order, quantile contraction, Chernoff bound tightening, rate-function dominance, and Hoeffding bound improvements—and show asymptotic and extreme-quantile consequences. Empirical evaluations on pedestrian trajectory benchmarks demonstrate substantial reductions in 95% conformal radii without compromising validity, confirming practical gains in symmetry-rich, long-horizon forecasting tasks.

Abstract

We study the effect of group symmetrization of pre-trained models on conformal prediction (CP), a post-hoc, distribution-free, finite-sample method of uncertainty quantification that offers formal coverage guarantees under the assumption of data exchangeability. Unfortunately, CP uncertainty regions can grow significantly in long horizon missions, rendering the statistical guarantees uninformative. To that end, we propose infusing CP with geometric information via group-averaging of the pretrained predictor to distribute the non-conformity mass across the orbits. Each sample now is treated as a representative of an orbit, thus uncertainty can be mitigated by other samples entangled to it via the orbit inducing elements of the symmetry group. Our approach provably yields contracted non-conformity scores in increasing convex order, implying improved exponential-tail bounds and sharper conformal prediction sets in expectation, especially at high confidence levels. We then propose an experimental design to test these theoretical claims in pedestrian trajectory prediction.
Paper Structure (23 sections, 21 theorems, 104 equations, 4 figures, 3 tables)

This paper contains 23 sections, 21 theorems, 104 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Safe Planning with Conformal Prediction
  4. Symmetries in Machine Learning
  5. Why symmetrizing predictors enhances uncertainty quantification
  6. Preliminaries
  7. Group Theory & Equivariant Functions
  8. Conformal Prediction
  9. Symmetries-infused Conformal Prediction
  10. Symmetry-infused CP with equivariantized pre-trained model
  11. Risk improvement via increasing convex ordering
  12. Quantile contraction under equivariantization of pre-trained models
  13. Chernoff bound improvement via MGF ordering
  14. Rate-function dominance and asymptotic consequences
  15. Asymptotic consequence (under standard tightness).
  16. ...and 8 more sections

Key Result

Lemma 1

Under Assumption as:G-invariant_score, the symmetrization operator eq:symmetrization_operator_invariance is a $G$-invariant projection in the sense that Moreover, if $f\in F_G$ then $\Pi_G[s;f](x,y)=s\!(f(x),y)$ for all $(x,y)$ (idempotence on $G$-equivariant models).

Figures (4)

  • Figure 1: Caption
  • Figure 2: Non-conformity score distributions for SocialVAE and Eq${_{\text{SO2}}}$SocialVAE with $95^{\text{th}}$-quantile in dashed lines.
  • Figure 3: Set size reduction via symmetrization of Conformal prediction in multi-step prediction in ETH-UCY and NBA Rebound/Score datasets.
  • Figure 4: Ablation studies on the group size and approximate equivariance.

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: Group-invariant distribution
  • Definition 5: $G^{n+1}$-Exchangeability
  • Definition 6: Symmetrization Operator
  • Lemma 1: Invariance and projection of $\Pi_G$
  • proof
  • Lemma 2
  • proof
  • ...and 36 more