A Unified Theory of Conditional Coverage in Conformal Prediction with Applications

Changliang Zou; Liuhua Peng; Yinjie Min

Conditional miscoverage in conformal prediction decomposes non-asymptotically into score-estimation error, finite-sample calibration error, and intrinsic conditional-mismatch error.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.3

2026-06-30 22:31 UTC pith:QD7A2C6B

load-bearing objection The decomposition of conditional miscoverage into three components is the main new piece and could organize the literature, but the conditions for it to hold non-asymptotically are not clear from the abstract. the 1 major comments →

arxiv 2605.11602 v3 pith:QD7A2C6B submitted 2026-05-12 stat.ME

A Unified Theory of Conditional Coverage in Conformal Prediction with Applications

Yinjie Min , Liuhua Peng , Changliang Zou This is my paper

classification stat.ME

keywords conformal predictionconditional coveragemiscoverage decompositioncovariate shiftgraph-structured datahierarchical datamodel selection

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a single framework that explains why conformal prediction methods succeed or fail at giving coverage that adapts to individual points or groups. It breaks down the total conditional miscoverage into three separate parts whose sizes can be tracked separately. This breakdown shows how asymptotic conditional validity arises and lets different procedures be compared on the same terms. The same lens yields rules for choosing models that target conditional coverage and new localized procedures that work under covariate shift. The framework also covers structured data such as graphs and hierarchies.

Core claim

Conditional miscoverage can be decomposed non-asymptotically into three interpretable components: score-estimation error, finite-sample calibration error, and intrinsic conditional-mismatch error. This decomposition clarifies the mechanisms behind asymptotic conditional validity and places existing methods within a common analytical lens. Building on the decomposition, principled guidance follows for conditional-coverage-oriented model selection, localized methods with asymptotic conditional guarantees under covariate shift are developed, and the framework extends to structured data with applications to graph-structured and hierarchical settings.

What carries the argument

The non-asymptotic decomposition of conditional miscoverage into score-estimation error, finite-sample calibration error, and intrinsic conditional-mismatch error.

Load-bearing premise

A standard conformal prediction setup exists with a score function and calibration set, and the three error components can be isolated without further regularity conditions on the data or estimator.

What would settle it

A numerical check in which the observed conditional miscoverage rate does not equal the sum of the three estimated error components when each component is computed separately from the same data and score function.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Existing conditional-coverage methods can be compared and analyzed through the shared decomposition.
Model selection can be guided by balancing the three error components rather than by marginal coverage alone.
Localized procedures can be constructed to achieve asymptotic conditional coverage under covariate shift.
The decomposition extends directly to graph-structured and hierarchical data settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

New score functions could be designed to shrink one specific component while leaving the others unchanged.
The decomposition might be applied to sequential or dependent data by redefining the calibration error term.
Practitioners could monitor the three components separately during deployment to diagnose coverage failures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The decomposition of conditional miscoverage into three components is the main new piece and could organize the literature, but the conditions for it to hold non-asymptotically are not clear from the abstract.

read the letter

The one thing to know is that this paper puts forward a non-asymptotic decomposition of conditional miscoverage into three parts: score-estimation error, finite-sample calibration error, and intrinsic conditional-mismatch error. They argue this decomposition explains asymptotic conditional validity and organizes the existing case-by-case methods.

What the paper does well is develop this framework and then use it for model selection guidance and for new localized methods that achieve asymptotic conditional guarantees under covariate shift. The extensions to graph-structured and hierarchical data show they have thought about how the ideas apply outside standard settings. This kind of organizing work can help the field move past ad-hoc fixes.

The soft spots are around the conditions for the decomposition. The abstract does not list explicit regularity conditions on the data or the score function, so it is not clear how general the clean separation of the three errors really is. If the proof relies on assumptions that are stronger than the standard conformal setup, that would narrow the scope. Also, since the experiments are only said to corroborate without any description of the setups or metrics, it is hard to assess how well the theory holds in practice.

This paper is for researchers in statistical machine learning who focus on conformal prediction and conditional coverage guarantees. A reader looking for a way to compare different conditional conformal methods or to extend them to shifted or structured data would get value from the unified lens.

It deserves serious peer review because the central claim addresses a genuine gap in the literature with a specific analytical tool. Referees should check the proofs for the decomposition and the validity of the extensions.

Referee Report

1 major / 0 minor

Summary. The paper develops a unified framework for conformal prediction methods that target conditional coverage guarantees. Its central contribution is a claimed non-asymptotic decomposition of conditional miscoverage into three components—score-estimation error, finite-sample calibration error, and intrinsic conditional-mismatch error—which is used to analyze existing methods, derive guidance for model selection, construct localized procedures with asymptotic conditional guarantees under covariate shift, and extend the theory to graph-structured and hierarchical data settings. Numerical experiments are referenced to support the claims.

Significance. If the decomposition is rigorously established with explicit conditions and the components provably isolate as described, the work would supply a useful general lens for diagnosing sources of conditional miscoverage and comparing procedures, moving beyond case-by-case analyses. The extensions to covariate shift and structured data, together with model-selection guidance, could have practical value for applications requiring adaptive coverage.

major comments (1)

[Abstract / central contribution] The central claim (abstract) asserts a non-asymptotic decomposition into three cleanly separable components without additional regularity conditions on the data distribution or score estimator. The provided abstract does not supply the explicit definitions of the three error terms, the precise statement of the decomposition, or the conditions under which separation holds; without these, it is impossible to verify that the math supports the claim that the components can be isolated in the standard conformal setup.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for identifying the need for greater precision in how the central contribution is presented. We address the major comment below and outline planned revisions.

read point-by-point responses

Referee: [Abstract / central contribution] The central claim (abstract) asserts a non-asymptotic decomposition into three cleanly separable components without additional regularity conditions on the data distribution or score estimator. The provided abstract does not supply the explicit definitions of the three error terms, the precise statement of the decomposition, or the conditions under which separation holds; without these, it is impossible to verify that the math supports the claim that the components can be isolated in the standard conformal setup.

Authors: We agree that the abstract, as currently written, is too high-level to allow immediate verification of the decomposition. The explicit definitions appear in Definitions 2.1–2.3, the precise non-asymptotic decomposition is stated in Theorem 3.1, and the proof shows that the three terms isolate under the standard conformal setup (exchangeable calibration and test points, no further regularity on the distribution or score estimator). The abstract therefore omits the necessary pointers and conditions. We will revise the abstract to include a concise statement of the decomposition, reference to Theorem 3.1, and an explicit note that no additional regularity conditions are imposed beyond the usual conformal assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract presents the central contribution as a non-asymptotic decomposition of conditional miscoverage into three components derived from the standard conformal prediction setup. No equations, self-citations, fitted parameters renamed as predictions, or ansatzes are supplied in the text that would allow identification of a reduction by construction. The derivation is therefore treated as self-contained against external benchmarks of conformal theory, consistent with the default expectation that most papers exhibit no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the decomposition is stated to clarify asymptotic conditional validity, implying reliance on standard conformal prediction assumptions such as exchangeability or i.i.d. data (with extensions noted for covariate shift). No free parameters or invented entities are mentioned.

axioms (1)

domain assumption Standard conformal prediction setup (score function and calibration set) allows isolation of the three miscoverage components
Invoked by the central contribution statement in the abstract; required for the non-asymptotic decomposition to hold.

pith-pipeline@v0.9.1-grok · 5719 in / 1389 out tokens · 31505 ms · 2026-06-30T22:31:35.054271+00:00 · methodology

0 comments

read the original abstract

Conformal prediction provides prediction sets with finite-sample marginal coverage, but many applications require coverage guarantees that adapt to individual test points, a subpopulation, or a structural component of the data. Existing methods targeting conditional coverage are largely analyzed case by case, leaving limited general theory for understanding where conditional miscoverage comes from, how different procedures should be compared, and how such guarantees can be extended beyond i.i.d.~data. We address these gaps through a unified framework and theory for conformal methods targeting conditional coverage. Our central contribution is a non-asymptotic decomposition of conditional miscoverage into three interpretable components: score-estimation error, finite-sample calibration error, and intrinsic conditional-mismatch error. This decomposition clarifies the mechanisms behind asymptotic conditional validity and places existing methods within a common analytical lens. Building on this framework, we derive principled guidance for conditional-coverage-oriented model selection, and develop localized methods with asymptotic conditional guarantees under covariate shift. Finally, we extend the framework to structured data, with concrete applications to graph-structured and hierarchical settings. Numerical experiments corroborate the theory and demonstrate the effectiveness of the proposed procedures.

Figures

Figures reproduced from arXiv: 2605.11602 by Changliang Zou, Liuhua Peng, Yinjie Min.

**Figure 1.** Figure 1: Comparison of selection strategies under DGP1–DGP3. Violin plots show the [PITH_FULL_IMAGE:figures/full_fig_p033_1.png] view at source ↗

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 2 canonical work pages · 2 internal anchors

[1]

Theoretical Foundations of Conformal Prediction

Angelopoulos, A. N., Barber, R. F. & Bates, S. (2024), ‘Theoretical foundations of confor- mal prediction’,arXiv preprint arXiv:2411.11824. Barber, R. F. & Tibshirani, R. J. (2026), ‘Unifying different theories of conformal predic- tion’,Electronic Journal of Statistics20(1), 1428–1474. 35 Bates, S., Cand` es, E., Lei, L., Romano, Y. & Sesia, M. (2023), ‘...

work page internal anchor Pith review Pith/arXiv arXiv 2024
[2]

& van de Geer, S

Lederer, J. & van de Geer, S. (2014), ‘New concentration inequalities for suprema of em- pirical processes’,Bernoulli20(4), 2020 –

2014
[3]

Lei, J., G’Sell, M., Rinaldo, A., Tibshirani, R. J. & Wasserman, L. (2018), ‘Distribution- free predictive inference for regression’,Journal of the American Statistical Association 113(523), 1094–1111. Lei, J., Robins, J. & Wasserman, L. (2013), ‘Distribution-free prediction sets’,Journal of the American Statistical Association108(501), 278–287. Li, Y., L...

2018
[4]

Conformal Prediction Assessment: A Framework for Conditional Coverage Evaluation and Selection

Sen, P., Namata, G., Bilgic, M., Getoor, L., Galligher, B. & Eliassi-Rad, T. (2008), ‘Col- lective classification in network data’,AI magazine29(3), 93–93. Shafer, G. & Vovk, V. (2008), ‘A tutorial on conformal prediction.’,Journal of Machine Learning Research9(3). Shen, X. & Meinshausen, N. (2025), ‘Engression: extrapolation through the lens of distri- b...

work page internal anchor Pith review Pith/arXiv arXiv 2008
[5]

Then there exists a constantC >0such that for everyt∈ X, Pr Yn+1 ∈ bCLCP(Xn+1)|X n+1 =t −(1−α) ≤C n (nhd)−1/2 log1/2(n) +hlog(1/h) o

holds for the base scorev(·,·)and the kernelK(·,·;h), withβ=din that assumption. Then there exists a constantC >0such that for everyt∈ X, Pr Yn+1 ∈ bCLCP(Xn+1)|X n+1 =t −(1−α) ≤C n (nhd)−1/2 log1/2(n) +hlog(1/h) o . 45 Proof.Sinces ⋆(X, Y) is uniformly distributed on [0,1] conditional onX=tfor anyt∈ X, Assumption 4 holds directly with constants independen...

2023
[6]

is designed to achieve group-conditional coverage over a finite collection of possibly intersecting groups. It can be viewed as a direct extension of quantile regression, which takes as input an arbitrary non-conformity scorev(x, y) and a finite collection of subgroup indicatorsH, solves a single convex minimization problem, and returns a score adjustment...

2023
[7]

Hence the finite-sample calibration error satisfies Γn(w) =O(n −1/2 log1/2 n)

63 For the weightw(x) =r X(x), condition (v) impliesM w ≤ M,B w =E{r X(X1)}= 1, andσ 2 w =E{r 2 X(X1)} ≤ M 2 , whereX 1 ∼P X,1. Hence the finite-sample calibration error satisfies Γn(w) =O(n −1/2 log1/2 n). Conditions (ii)–(iv) ensure Assumption 4 for the oracle score,F s⋆|X=t, andF rX ◦s⋆. Un- like methods whose oracle scores remove conditional heterogen...

2025
[8]

Here we view them as another consequence of the same weighted conformal framework used for prediction sets

and discussed in (Shafer 74 & Vovk 2008), are based on the relative rank of conformity scores. Here we view them as another consequence of the same weighted conformal framework used for prediction sets. This perspective is useful in many localized settings such as outlier detection (Bates et al. 2023), two-sample testing (Hu & Lei 2024), and conditional t...

2008
[9]

Our goal is to show that the weighted calibration used for prediction sets gives an analogous construction forp-values

are one representative example. Our goal is to show that the weighted calibration used for prediction sets gives an analogous construction forp-values. To cover both vector data and structured data, we formulate thep-value directly under the weighted SymmPI framework. Following the notation in Section S3, for an observed samplez obs and any completion zsa...

2025
[10]

Under stability and convergence conditions adapted to the hierarchical setting, we obtain the following specialization

Once the hierarchical construction is expressed in the SymmPI notation, Theorem 6 can be applied. Under stability and convergence conditions adapted to the hierarchical setting, we obtain the following specialization. Theorem S5.Suppose Assumption 13 holds. Assume there exists a functionδ n(ε)such that, Pr |s(x, y;Z (k), Z)−s ⋆ k(x, y)|> ε|P 1, . . . , PK...

2005
[11]

The next lemma shows that this assumption follows from pointwise convergence of the learned score under mild conditions

S5.2 Pointwise Convergence Implies Averaged Convergence Theorem 3 is stated under an averaged quantile-approximation assumption. The next lemma shows that this assumption follows from pointwise convergence of the learned score under mild conditions. Lemma S1.SupposeP X,1 =P X,2 =P X andP T,1 =P T,2 =P T . LetZ n+1 =Zexplicitly specify the dependence ofZon...

2018
[12]

By the definition ofQ ◦ α = arg minf∈F R(f), assume thatf κ◦ ∈ F satisfiesQ ◦ α =f κ◦

that there existsc >0 such that [E{h(X, v(X, Y);f κ)}]1/2 ≥cE|f κ(X)−Q ◦ α(X)| holds for allf κ ∈ F. By the definition ofQ ◦ α = arg minf∈F R(f), assume thatf κ◦ ∈ F satisfiesQ ◦ α =f κ◦. DefineJ 0 =∥κ ◦∥2 2 ∨1≤B 2 ∨1. 124 Based on Theorem 2 of (Li et al. 2007), together with the properties of finite-dimensional linear reproducing kernel Hilbert spaces (R...

2007
[13]

The proof therefore reduces to combining a standard VC/Rademacher bound forF (0) a with the empirical-process inequality in (Lederer & van de Geer 2014)

Then we cannot findusuch thatv 1 > a(x 1)1(ς 1 ≤u) andv 2 ≤a(x 2)1(ς 2 ≤ u) hold simultaneously, which indicates thatF (0) a cannot scatter these two points. The proof therefore reduces to combining a standard VC/Rademacher bound forF (0) a with the empirical-process inequality in (Lederer & van de Geer 2014). Define the empirical Rademacher complexity of...

2014
[14]

It remains to boundR n(F (0) a )

yields Pr sup fa∈F(0) a n−1 nX i=1 fa(X(0) i )−E{f a(X(0))} ≤2R n(F (0) a ) +ε ! ≥1−exp − nε2 2M 2 a . It remains to boundR n(F (0) a ). First we notice that E ( sup fa∈F(0) a n−1 nX i=1 τifa(X(0) i ) |X (0) 1 , . . . , X(0) n ) =E ( sup f∈F (0) n−1 nX i=1 τia(X(0) i )f(X (0) i ) |X (0) 1 , . . . , X(0) n ) where for any givenX (0) i and anyf∈ F (0), the ...

2014
[15]

Raising both sides to the power 2/pyields ∥Qα − bQ∥2 PX ,p ≤ ∥1/L∥PX ,p/(2−p)E n L(X)(Q α(X)− bQ(X))2 o

Applying H¨ older’s inequality to |Qα(X)− bQ(X)| p = n L(X)(Q α(X)− bQ(X))2 op/2 {1/L(X)} p/2 gives E Qα(X)− bQ(X) p ≤ h E n L(X)(Q α(X)− bQ(X))2 oi1/rh E n L(X) −pr′/2 oi1/r′ = h E n L(X)(Q α(X)− bQ(X))2 oip/2 E L(X) −p/(2−p) (2−p)/2 . Raising both sides to the power 2/pyields ∥Qα − bQ∥2 PX ,p ≤ ∥1/L∥PX ,p/(2−p)E n L(X)(Q α(X)− bQ(X))2 o . Combining this...

2025
[16]

The target quantile 150 level is 0.9, and the regularization coefficient is 0.01

•CQR-LR:Linear quantile regression is fitted with anL 2 penalty. The target quantile 150 level is 0.9, and the regularization coefficient is 0.01. •CQR-RF (Meinshausen & Ridgeway 2006):Quantile random forest is fitted with minimum split size 2 and maximum tree depth

2006
[17]

•EffSize baseline:selects the candidate with the smallest average prediction-set size on the calibration set

•AvgRankLoss:ranks the candidates separately for the three losses, averages the ranks, and selects the candidate with the smallest average rank. •EffSize baseline:selects the candidate with the smallest average prediction-set size on the calibration set. •Rand baseline:randomly selects a candidate from the pool. The candidate conformal sets differ in the ...

2025

[1] [1]

Theoretical Foundations of Conformal Prediction

Angelopoulos, A. N., Barber, R. F. & Bates, S. (2024), ‘Theoretical foundations of confor- mal prediction’,arXiv preprint arXiv:2411.11824. Barber, R. F. & Tibshirani, R. J. (2026), ‘Unifying different theories of conformal predic- tion’,Electronic Journal of Statistics20(1), 1428–1474. 35 Bates, S., Cand` es, E., Lei, L., Romano, Y. & Sesia, M. (2023), ‘...

work page internal anchor Pith review Pith/arXiv arXiv 2024

[2] [2]

& van de Geer, S

Lederer, J. & van de Geer, S. (2014), ‘New concentration inequalities for suprema of em- pirical processes’,Bernoulli20(4), 2020 –

2014

[3] [3]

Lei, J., G’Sell, M., Rinaldo, A., Tibshirani, R. J. & Wasserman, L. (2018), ‘Distribution- free predictive inference for regression’,Journal of the American Statistical Association 113(523), 1094–1111. Lei, J., Robins, J. & Wasserman, L. (2013), ‘Distribution-free prediction sets’,Journal of the American Statistical Association108(501), 278–287. Li, Y., L...

2018

[4] [4]

Conformal Prediction Assessment: A Framework for Conditional Coverage Evaluation and Selection

Sen, P., Namata, G., Bilgic, M., Getoor, L., Galligher, B. & Eliassi-Rad, T. (2008), ‘Col- lective classification in network data’,AI magazine29(3), 93–93. Shafer, G. & Vovk, V. (2008), ‘A tutorial on conformal prediction.’,Journal of Machine Learning Research9(3). Shen, X. & Meinshausen, N. (2025), ‘Engression: extrapolation through the lens of distri- b...

work page internal anchor Pith review Pith/arXiv arXiv 2008

[5] [5]

Then there exists a constantC >0such that for everyt∈ X, Pr Yn+1 ∈ bCLCP(Xn+1)|X n+1 =t −(1−α) ≤C n (nhd)−1/2 log1/2(n) +hlog(1/h) o

holds for the base scorev(·,·)and the kernelK(·,·;h), withβ=din that assumption. Then there exists a constantC >0such that for everyt∈ X, Pr Yn+1 ∈ bCLCP(Xn+1)|X n+1 =t −(1−α) ≤C n (nhd)−1/2 log1/2(n) +hlog(1/h) o . 45 Proof.Sinces ⋆(X, Y) is uniformly distributed on [0,1] conditional onX=tfor anyt∈ X, Assumption 4 holds directly with constants independen...

2023

[6] [6]

is designed to achieve group-conditional coverage over a finite collection of possibly intersecting groups. It can be viewed as a direct extension of quantile regression, which takes as input an arbitrary non-conformity scorev(x, y) and a finite collection of subgroup indicatorsH, solves a single convex minimization problem, and returns a score adjustment...

2023

[7] [7]

Hence the finite-sample calibration error satisfies Γn(w) =O(n −1/2 log1/2 n)

63 For the weightw(x) =r X(x), condition (v) impliesM w ≤ M,B w =E{r X(X1)}= 1, andσ 2 w =E{r 2 X(X1)} ≤ M 2 , whereX 1 ∼P X,1. Hence the finite-sample calibration error satisfies Γn(w) =O(n −1/2 log1/2 n). Conditions (ii)–(iv) ensure Assumption 4 for the oracle score,F s⋆|X=t, andF rX ◦s⋆. Un- like methods whose oracle scores remove conditional heterogen...

2025

[8] [8]

Here we view them as another consequence of the same weighted conformal framework used for prediction sets

and discussed in (Shafer 74 & Vovk 2008), are based on the relative rank of conformity scores. Here we view them as another consequence of the same weighted conformal framework used for prediction sets. This perspective is useful in many localized settings such as outlier detection (Bates et al. 2023), two-sample testing (Hu & Lei 2024), and conditional t...

2008

[9] [9]

Our goal is to show that the weighted calibration used for prediction sets gives an analogous construction forp-values

are one representative example. Our goal is to show that the weighted calibration used for prediction sets gives an analogous construction forp-values. To cover both vector data and structured data, we formulate thep-value directly under the weighted SymmPI framework. Following the notation in Section S3, for an observed samplez obs and any completion zsa...

2025

[10] [10]

Under stability and convergence conditions adapted to the hierarchical setting, we obtain the following specialization

Once the hierarchical construction is expressed in the SymmPI notation, Theorem 6 can be applied. Under stability and convergence conditions adapted to the hierarchical setting, we obtain the following specialization. Theorem S5.Suppose Assumption 13 holds. Assume there exists a functionδ n(ε)such that, Pr |s(x, y;Z (k), Z)−s ⋆ k(x, y)|> ε|P 1, . . . , PK...

2005

[11] [11]

The next lemma shows that this assumption follows from pointwise convergence of the learned score under mild conditions

S5.2 Pointwise Convergence Implies Averaged Convergence Theorem 3 is stated under an averaged quantile-approximation assumption. The next lemma shows that this assumption follows from pointwise convergence of the learned score under mild conditions. Lemma S1.SupposeP X,1 =P X,2 =P X andP T,1 =P T,2 =P T . LetZ n+1 =Zexplicitly specify the dependence ofZon...

2018

[12] [12]

By the definition ofQ ◦ α = arg minf∈F R(f), assume thatf κ◦ ∈ F satisfiesQ ◦ α =f κ◦

that there existsc >0 such that [E{h(X, v(X, Y);f κ)}]1/2 ≥cE|f κ(X)−Q ◦ α(X)| holds for allf κ ∈ F. By the definition ofQ ◦ α = arg minf∈F R(f), assume thatf κ◦ ∈ F satisfiesQ ◦ α =f κ◦. DefineJ 0 =∥κ ◦∥2 2 ∨1≤B 2 ∨1. 124 Based on Theorem 2 of (Li et al. 2007), together with the properties of finite-dimensional linear reproducing kernel Hilbert spaces (R...

2007

[13] [13]

The proof therefore reduces to combining a standard VC/Rademacher bound forF (0) a with the empirical-process inequality in (Lederer & van de Geer 2014)

Then we cannot findusuch thatv 1 > a(x 1)1(ς 1 ≤u) andv 2 ≤a(x 2)1(ς 2 ≤ u) hold simultaneously, which indicates thatF (0) a cannot scatter these two points. The proof therefore reduces to combining a standard VC/Rademacher bound forF (0) a with the empirical-process inequality in (Lederer & van de Geer 2014). Define the empirical Rademacher complexity of...

2014

[14] [14]

It remains to boundR n(F (0) a )

yields Pr sup fa∈F(0) a n−1 nX i=1 fa(X(0) i )−E{f a(X(0))} ≤2R n(F (0) a ) +ε ! ≥1−exp − nε2 2M 2 a . It remains to boundR n(F (0) a ). First we notice that E ( sup fa∈F(0) a n−1 nX i=1 τifa(X(0) i ) |X (0) 1 , . . . , X(0) n ) =E ( sup f∈F (0) n−1 nX i=1 τia(X(0) i )f(X (0) i ) |X (0) 1 , . . . , X(0) n ) where for any givenX (0) i and anyf∈ F (0), the ...

2014

[15] [15]

Raising both sides to the power 2/pyields ∥Qα − bQ∥2 PX ,p ≤ ∥1/L∥PX ,p/(2−p)E n L(X)(Q α(X)− bQ(X))2 o

Applying H¨ older’s inequality to |Qα(X)− bQ(X)| p = n L(X)(Q α(X)− bQ(X))2 op/2 {1/L(X)} p/2 gives E Qα(X)− bQ(X) p ≤ h E n L(X)(Q α(X)− bQ(X))2 oi1/rh E n L(X) −pr′/2 oi1/r′ = h E n L(X)(Q α(X)− bQ(X))2 oip/2 E L(X) −p/(2−p) (2−p)/2 . Raising both sides to the power 2/pyields ∥Qα − bQ∥2 PX ,p ≤ ∥1/L∥PX ,p/(2−p)E n L(X)(Q α(X)− bQ(X))2 o . Combining this...

2025

[16] [16]

The target quantile 150 level is 0.9, and the regularization coefficient is 0.01

•CQR-LR:Linear quantile regression is fitted with anL 2 penalty. The target quantile 150 level is 0.9, and the regularization coefficient is 0.01. •CQR-RF (Meinshausen & Ridgeway 2006):Quantile random forest is fitted with minimum split size 2 and maximum tree depth

2006

[17] [17]

•EffSize baseline:selects the candidate with the smallest average prediction-set size on the calibration set

•AvgRankLoss:ranks the candidates separately for the three losses, averages the ranks, and selects the candidate with the smallest average rank. •EffSize baseline:selects the candidate with the smallest average prediction-set size on the calibration set. •Rand baseline:randomly selects a candidate from the pool. The candidate conformal sets differ in the ...

2025