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Non-Asymptotic Analysis of Efficiency in Conformalized Regression

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
conformal predictionefficiencyconformalized regressionquantile regressionuncertainty quantification
Abstract:

Conformal prediction provides prediction sets with coverage guarantees. The informativeness of conformal prediction depends on its efficiency, typically quantified by the expected size of the prediction set. Prior work on the efficiency of conformalized regression commonly treats the miscoverage level α\alpha as a fixed constant. In this work, we establish non-asymptotic bounds on the deviation of the prediction set length from the oracle interval length for conformalized quantile and median regression trained via SGD, under mild assumptions on the data distribution. Our bounds of order O(1/n+1/(α2n)+1/m+exp⁡(−α2m))\mathcal{O}(1/\sqrt{n} + 1/(\alpha^2 n) + 1/\sqrt{m} + \exp(-\alpha^2 m)) capture the joint dependence of efficiency on the proper training set size nn, the calibration set size mm, and the miscoverage level α\alpha. The results identify phase transitions in convergence rates across different regimes of α\alpha, offering guidance for allocating data to control excess prediction set length. Empirical results are consistent with our theoretical findings.

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Overview

Overall Novelty Assessment

The paper establishes finite-sample bounds on prediction set length for conformalized quantile and median regression, capturing joint dependence on training size n, calibration size m, and miscoverage level α. It resides in the Conformalized Quantile Regression leaf, which contains four papers within the Quantile-Based Efficiency Methods branch. This is a moderately populated research direction within the broader Efficiency Optimization category, indicating active but not overcrowded investigation of quantile-based approaches to tighten prediction intervals while maintaining coverage guarantees.

The taxonomy reveals neighboring work in Unconditional and Localized Quantile Approaches (two papers) and Comparative Analysis of Quantile Methods (one paper), alongside sibling branches pursuing Volume and Length Minimization through direct optimization or adaptive scoring. The paper's focus on finite-sample efficiency bounds connects it to theoretical coverage analysis in the Marginal Coverage Theory leaf, while its SGD training assumptions relate to broader Distribution-Free Frameworks. The scope_note for its leaf emphasizes adaptive interval construction, distinguishing it from volume-based methods that optimize geometric properties rather than quantile-derived intervals.

Among thirteen candidates examined, no contribution was clearly refuted by prior work. The first contribution (finite-sample bounds with joint n-m-α dependence) examined ten candidates with zero refutations, suggesting this specific theoretical characterization may be novel within the limited search scope. The second contribution (bounds for conformalized median regression under homoscedasticity) examined one candidate, and the third (phase transition guidance for data allocation) examined two candidates, both without refutation. These statistics indicate that among the top-K semantic matches retrieved, none provide overlapping finite-sample efficiency analysis with explicit α-dependence.

Based on the limited literature search of thirteen candidates, the work appears to occupy a distinct position within conformalized quantile regression by providing non-asymptotic efficiency bounds that explicitly model miscoverage level α. The analysis does not claim exhaustive coverage of all related theoretical work, and the moderate size of the Conformalized Quantile Regression leaf suggests room for specialized contributions like finite-sample efficiency characterization.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
13
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: efficiency analysis of conformalized regression with finite sample bounds. The field of conformal prediction for regression has matured into a rich taxonomy spanning theoretical guarantees, efficiency optimization, multivariate extensions, robustness considerations, specialized settings, model integration strategies, and diverse applications. Theoretical Foundations and Coverage Guarantees establish the distribution-free validity that underpins the entire framework, ensuring finite-sample coverage without parametric assumptions. Efficiency Optimization and Prediction Set Construction focuses on reducing prediction set size while maintaining coverage, employing quantile-based methods such as Conformalized Quantile[3] and volume-minimization techniques like Volume Sorted Sets[4]. Multivariate and Structured Output Prediction extends these ideas to high-dimensional or geometrically constrained outputs, while Robustness and Adversarial Settings address distributional shifts and worst-case scenarios through works like One Sample Robust[5] and Robust Efficient Sets[6]. Specialized Regression Settings tackle domain-specific challenges, Model Integration combines conformal methods with Bayesian or neural approaches, and Applications demonstrate practical impact across fields from drug discovery to spatial forecasting. Within Efficiency Optimization, a particularly active line of research centers on quantile-based methods that leverage conditional quantile estimators to construct tighter prediction intervals. Efficiency Analysis Conformalized[0] contributes to this cluster by providing finite-sample efficiency bounds for conformalized quantile regression, directly addressing the trade-off between coverage guarantees and interval width. This work sits naturally alongside Conformalized Quantile[3], which introduced the foundational quantile regression framework, and Improved Quantile[14], which refines quantile estimation for better efficiency. Nearby efforts like Thresholded Intervals[18] explore alternative interval construction strategies, while Length Optimization[7] and Efficiency Oriented Selection[22] pursue complementary approaches to minimizing prediction set size. The central tension across these branches involves balancing unconditional coverage guarantees with conditional efficiency, a challenge that motivates ongoing work on adaptive scoring functions, conditional validity, and optimal transport-based methods.

Claimed Contributions

Finite-sample bounds for CQR with joint dependence on n, m, and α

The authors establish non-asymptotic upper bounds on the expected length deviation of conformalized quantile regression prediction sets from oracle intervals. Unlike prior work, their bounds explicitly capture the joint dependence on training set size n, calibration set size m, and miscoverage level α, placing assumptions directly on the data distribution rather than on intermediate quantities.

10 retrieved papers
Finite-sample bounds for CMR under homoscedastic conditions

The authors derive non-asymptotic efficiency bounds for conformalized median regression that parallel those for CQR. Under homoscedasticity assumptions, they show that CMR produces symmetric prediction intervals with length deviation bounds of the same order as CQR.

1 retrieved paper
Theoretical guidance on phase transitions and data allocation

The authors identify phase transitions in convergence rates across different regimes of the miscoverage level α, providing the first analysis that jointly considers training size, calibration size, and miscoverage level. This analysis offers practical guidance for data allocation strategies to control prediction set length deviation.

2 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Finite-sample bounds for CQR with joint dependence on n, m, and α

The authors establish non-asymptotic upper bounds on the expected length deviation of conformalized quantile regression prediction sets from oracle intervals. Unlike prior work, their bounds explicitly capture the joint dependence on training set size n, calibration set size m, and miscoverage level α, placing assumptions directly on the data distribution rather than on intermediate quantities.

Contribution

Finite-sample bounds for CMR under homoscedastic conditions

The authors derive non-asymptotic efficiency bounds for conformalized median regression that parallel those for CQR. Under homoscedasticity assumptions, they show that CMR produces symmetric prediction intervals with length deviation bounds of the same order as CQR.

Contribution

Theoretical guidance on phase transitions and data allocation

The authors identify phase transitions in convergence rates across different regimes of the miscoverage level α, providing the first analysis that jointly considers training size, calibration size, and miscoverage level. This analysis offers practical guidance for data allocation strategies to control prediction set length deviation.

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