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How Synthetic Labels Improve Conformal Prediction: A Perspective on Conditional Coverage

Qianyi Chen, Bo Li

arXiv:2609.33482v1cs.LGstat.ME

TL;DR

When trusted calibration data are scarce, it is unclear whether abundant covariates labeled by auxiliary models can improve conditional coverage. The paper introduces prediction-powered quantile learning, using synthetic labels with paired trusted corrections and an independent trusted calibration split. Across regression benchmarks and a human-rating study, it reports improved conditional coverage while preserving marginal validity and often producing more compact sets.

  • Problem

    Post-hoc conformal calibration data may be too scarce to learn how uncertainty varies across inputs, even though marginal coverage alone can mask conditional miscoverage.

    Source: §1 · p. 1

  • Method

    Prediction-powered quantile learning estimates pinball risk from synthetic-labeled data, corrects bias with paired trusted and synthetic outcomes, and reserves an independent trusted split for final conformalization.

    Source: §1 · p. 2

  • Results

    Across eight regression benchmarks and HelpSteer2, synthetic-powered quantile learning improves downstream conditional-coverage diagnostics and produces more compact prediction sets.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Final conformalization removes global threshold shifts, so useful synthetic information depends on density-weighted shape error and covariance aligned with the learning error visible at deployment.

    Source: §1 · p. 2, §7 · p. 16

    The HelpSteer2 study uses prompt clusters and frozen LLM outputs, so it is a realistic case study rather than a universal guarantee about LLM-generated labels.

    Source: §6 · p. 16

Abstract

from arXiv · show

Conformal prediction provides distribution-free finite-sample marginal coverage, but post-hoc calibration data may be too scarce to learn how uncertainty varies across inputs. Meanwhile, abundant covariates can often be labeled cheaply by domain models or general-purpose language models. We study whether these synthetic labels can improve conditional coverage when only a small trusted sample is available. Building on score-quantile regression, we introduce prediction-powered quantile learning: a synthetic-labeled pool estimates pinball risk, paired trusted and synthetic outcomes correct its bias, and an independent trusted split performs final conformalization. Profiling pinball risk over scalar corrections reveals that population conditional-coverage error is its functional gradient; the corresponding Hessian removes global shifts and weights remaining shape error by boundary density. Composing this geometry with prediction-powered learning yields a three-resource expansion and a benefit--cost rule for synthetic power. Across eight regression benchmarks, synthetic-powered quantile learning substantially improves downstream conditional coverage while preserving marginal validity and producing more compact prediction sets. A human-rating study finds similar gains from external LLM labels and exposes a quality--quantity--cost tradeoff.

1 INTRODUCTION

With few trusted outcomes, synthetic labels may help learn input-dependent score quantiles while an independent trusted split preserves marginal validity. The paper develops prediction-powered quantile learning and a post-conformal analysis, then evaluates the approach across regression benchmarks and a human-rating study.

1 INTRODUCTION

  • With scarce trusted calibration outcomes, learning heterogeneous thresholds and applying the final conformal correction compete for the same small reservoir.Marginal coverage can hide undercoverage on difficult inputs compensated by overly wide sets on easy ones, especially under heteroscedasticity.Source: §1 · p. 1
  • Prediction-powered quantile learning estimates pinball risk from a large synthetic-labeled sample, corrects its bias with paired trusted and synthetic outcomes, and reserves trusted data for final calibration.The method is applied to conditional outcome and nonconformity-score quantiles and compared with supervised, semi-supervised, transport, pretraining, and positive-augmentation baselines.Source: §1 · p. 2
  • Profiling pinball risk over the scalar correction identifies calibrated coverage error as its functional gradient and a density-weighted Hessian that removes global threshold shifts.Composing this geometry with prediction-powered influence yields a three-resource expansion and a power rule balancing covariance benefit against synthetic noise.Source: §1 · p. 2
  • Across eight regression benchmarks and 30 common splits, synthetic-powered quantile learning improves conditional-coverage diagnostics and produces more compact sets; HelpSteer2 shows similar gains with external LLM labels.The experiments also use a matched permutation control to distinguish input-specific information from generic stabilization.Source: §1 · p. 2

1.1 RELATED WORKS

  • Unlike methods using covariates alone or synthetic scores for calibration, this work asks how outcome-like synthetic labels can improve conditional coverage before an independent trusted split performs conformalization.It uses prediction-powered correction to learn conditional quantiles, where only learning error that survives final correction can improve conditional coverage.Source: §1.1 · p. 3

2 PRELIMINARIES AND PROBLEM SETTING

The preliminaries define split conformal prediction and conditional coverage, then describe RCP's local score thresholds and the trusted, synthetic, and calibration samples used for prediction-powered quantile learning.

2 PRELIMINARIES AND PROBLEM SETTING

  • Split conformal prediction uses an independent calibration sample of nonconformity scores to set a threshold; exchangeability supports its marginal coverage guarantee.For continuous scores, the stated coverage equals k/(m + 1), with the threshold set to +∞ when k = m + 1.Source: §2 · p. 3
  • Marginal coverage averages the conditional coverage curve over inputs, while RCP learns q(x) for local adaptation and uses a scalar residual correction to restore marginal validity.The correction is the kth order statistic of residuals from a separate trusted calibration sample.Source: §2 · p. 3
  • The target q*(x) is the conditional τ-quantile of the score, learned with pinball loss; under the stated uniform Lipschitz condition, pinball excess risk controls pre-conformal coverage error.The proposition's guarantee concerns coverage before the final scalar correction.Source: §2 · pp. 3–4
  • The setting combines a small trusted labeled sample, a much larger same-population covariate pool labeled by an auxiliary model, and an independent trusted sample reserved for final scalar correction.The auxiliary labeler is applied at both trusted and pool covariates; trusted learning data target the risk, the synthetic pool estimates an auxiliary risk, and untouched trusted calibration restores finite-sample coverage.Source: §2 · p. 4

3 METHODOLOGY AND THEORY

The methodology uses prediction-powered quantile learning with an untouched trusted split for final calibration, then analyzes which learned threshold errors survive conformalization. This geometry yields a three-resource expansion and a benefit–cost rule for synthetic power.

3.1 PREDICTION-POWERED QUANTILE LEARNING

  • Prediction-powered learning estimates pinball risk from a synthetic-labeled pool and corrects its bias with paired trusted and synthetic outcomes, while an untouched trusted split preserves exact marginal validity.The correction is unbiased in population for every power, including with a biased labeler; positive augmentation omits the paired subtraction and may bias the threshold.Source: §3.1 · p. 4

3.2 WHAT SURVIVES FINAL CONFORMALIZATION?

  • Final conformalization cancels constant threshold shifts, so deployed conditional coverage responds to threshold shape rather than global level.The profiled risk's functional gradient is population-calibrated conditional-coverage error, and its density-weighted Hessian removes the global component.Source: §3.2 · p. 5
  • The post-conformal influence map composes inverse-Hessian threshold changes with boundary-density weighting to give the deployed coverage perturbation.An unpenalized intercept makes the stationary scalar correction zero; the correction is retained for restricted or penalized learners.Source: §3.2 · pp. 5–6

3.3 WHEN DO SYNTHETIC LABELS HELP?

  • The three-resource expansion separates trusted-learning variance, a paired-covariance benefit, synthetic-variance cost, and an order-1/m final-calibration term that synthetic data cannot remove.The expansion holds under the stated fixed-dimensional regularity conditions and is around the calibrated stationary target.Source: §3.3 · p. 6
  • Unit synthetic power improves on supervision if and only if 2b > (1 + κ)c when clipping is inactive; positive projected covariance alone does not justify unit power.Under exact score-quantile specification with conditionally independent synthetic scores, b = 0 and zero power is a first-order optimum; density variation converts probability-scale error into heterogeneous coverage changes.Source: §3.3 · pp. 6–7

4 EXPERIMENTAL STUDY

Across eight regression benchmarks, prediction-powered quantile learning improves conditional-coverage diagnostics and produces more compact sets while preserving marginal coverage. Controlled comparisons and resource experiments examine where gains come from, and theory-aligned diagnostics test the predicted mechanisms.

4.1 BENCHMARK DESIGN

  • The benchmark uses eight regression datasets, a common audited protocol, and fixed trusted-data roles, with an untouched trusted split reserved for final calibration.The main low-label configuration uses a 30% synthetic pool, 2% base training, and a 1% conformal reservoir; RCP divides that reservoir between quantile learning and final calibration.Source: §4 · p. 7, §4.1 · pp. 7–8

4.2 MAIN CONDITIONAL-COVERAGE BENCHMARK

  • RCP-PPI lowers grouped MSCE on seven of eight datasets, improves every WSC point estimate, and lowers L1-ERT on all eight.On Diamonds, MSCE falls from 7.12 to 3.33, WSC rises from .724 to .816, and L1-ERT falls from .090 to .051.Source: §4.2 · p. 9
  • RCP-PPI’s WEC grouped-MSCE exception and other diagnostic differences qualify the overall gains; marginal coverage remains in [0.882, 0.919], and both PPI methods reduce mean log volume on every dataset versus supervised counterparts.On WEC, RCP-PTFT is strongest in grouped MSCE, while RCP-PPI still improves WSC and L1-ERT; no single diagnostic alone defines conditional coverage.Source: §4.2 · pp. 9–10

4.3 WHERE DO THE GAINS COME FROM?

  • Genuine covariate–score pairing improves both grouped-MSCE estimators and WSC on five prespecified datasets relative to a matched permutation placebo.The placebo preserves the synthetic-score marginal distribution and training setup while destroying covariate–score pairing; generic stabilization also contributes to augmentation gains.Source: §4.3 · p. 10, Figure 1 · p. 11
  • Synthetic-label value does not increase monotonically with teacher accuracy, while gains are largest in ultra-low and low trusted-label regimes and usually shrink as trusted data increase.Oracle labels generally define an upper ceiling, but post-conformal value also depends on boundary-aligned information, learner tangent space, and synthetic variance.Source: §4.3 · p. 10, Table 6 · p. 12

4.4 DIRECT CHECKS OF THE POST-CONFORMAL THEORY

  • A small synthetic pool produces most observed gains, while larger pools often plateau or become mildly nonmonotone; theory-aligned diagnostics also test predicted derivative and three-resource scaling behavior.The diagnostics report constant-shift nullity, near-second-order remainder slopes of 1.981 and 2.009, and resource-error slopes consistent with the predicted components.Source: §4.3 · p. 12, §4.4 · p. 12, Figure 2 · p. 13

5 REALISTIC EXTERNAL-LLM SUPERVISION

The HelpSteer2 study evaluates whether external LLM ratings can improve deployed conditional-coverage behavior and prediction-set size. Results show a quality–quantity–cost tradeoff, while boundary-focused diagnostics reveal information not captured by global rating error.

5 REALISTIC EXTERNAL-LLM SUPERVISION

  • The study uses five human ratings per prompt–response pair, frozen 768-dimensional features, and 20 paired seeds with separate development, final-calibration, and test roles.The dataset contains 21,352 rows with zero prompt overlap across roles.Source: §5 · p. 14
  • At nearly equal spend, 400 Flash labels deliver slightly lower average MSCE and smaller sets than 54.8 Pro labels, while cost-matched Pro has slightly higher minimum-group coverage.At equal label count, the Pro route costs 3.20 times as much as Flash.Source: §5 · p. 14, Table 8 · p. 15
  • 24.6%: Flash-PPI lowers grouped MSCE from 0.00665 to 0.00502; Pro-PPI lowers it to 0.00400 (39.9%).Equal-count Pro-PPI also raises minimum-group coverage from 0.738 to 0.773 and lowers mean log volume from −0.426 to −0.481.Source: Figure 3 · p. 14, §5 · p. 14, Table 8 · p. 15
  • Although only 13.6% of trusted pilot scores exceed the target boundary, Flash and Pro reduce disagreement versus shuffled scores by 32.5% and 39.5%, respectively.Their balanced boundary accuracies are 0.637 and 0.701, despite neither beating the constant-median baseline in global five-rating MAE.Source: §5 · p. 14, Table 10 · p. 15
  • The HelpSteer2 results are descriptive rather than a new row-level i.i.d. conformal guarantee because prompt-cluster dependence remains.The case study is compatible with the profiled-risk perspective but is not presented as direct empirical proof of its population identity.Source: §5 · p. 14

6 DISCUSSION

The discussion distinguishes bias-corrected prediction-powered learning from positive augmentation and gives practical guidance for preserving validity and evaluating synthetic supervision. It also bounds the claims to finite-resolution diagnostics and specified study settings.

6 DISCUSSION

  • Bias-corrected PPI uses paired trusted and synthetic scores as a control variate, with first-order value governed by post-conformal covariance rather than generic prediction accuracy.Positive augmentation can stabilize a high-variance threshold learner, but is not bias robust and need not preserve correct covariate–label pairing.Source: §6 · p. 16
  • Preserve an untouched trusted final-calibration split, compare methods within the same conformal family, and report conditional diagnostics alongside marginal coverage and set size.Labeler choice should reflect task and budget; boundary-aligned information per dollar may matter more than global accuracy or nominal model strength.Source: §6 · p. 16
  • The sharp three-resource theorem is local and fixed dimensional, and grouped MSCE, WSC, and L1-ERT are finite-resolution diagnostics rather than pointwise validity guarantees.The tabular study covers eight datasets, one target coverage level, and fixed labelers; HelpSteer2 is a prompt-clustered case study, and synthetic labels do not replace trusted outcomes.Source: §6 · p. 16

7 CONCLUSION

With many covariates but few trusted labels, prediction-powered quantile learning connects synthetic supervision to improved conditional coverage while retaining marginal validity. Across domain-model and LLM labelers, the paper reports better deployed coverage and often smaller sets, with gains explained by post-conformal shape information and a benefit–cost rule.

7 CONCLUSION

  • Across external labelers, bias-corrected quantile learning improves deployed coverage and often yields smaller prediction sets while preserving the usual marginal guarantee.The conclusion covers both domain models and general-purpose LLMs.Source: §7 · p. 16
  • Final conformalization removes global threshold shifts and retains density-weighted shape information aligned with the trusted learning error visible at deployment.This separates usable synthetic information from synthetic noise and finite-calibration error.Source: §7 · p. 16
  • The post-conformal perspective provides a benefit–cost rule for choosing how strongly synthetic labels should influence quantile learning.Source: §7 · p. 16

A.1 DATASETS AND ROLE PROTOCOL

The appendix specifies the datasets and non-overlapping resource roles, defines the compared baseline families and controls, and describes the diagnostics and statistical reporting used for evaluation.

A.1 DATASETS AND ROLE PROTOCOL

  • The eight datasets span scalar and multivariate regression, with response dimension up to 49; the main experiment uses a 30% synthetic pool, 2% base training, and a 1% conformal reservoir.Each seed uses non-overlapping reservoirs, and ordinary synthetic methods do not access true outcomes in the synthetic pool.Source: §A.1 · p. 18

A.2 DETAILED BASELINE DEFINITIONS

  • Baselines include split conformal, score transport, nearest-neighbor matching, CQR, and RCP variants using pretraining, PPI, augmentation, and ablation controls.RCP-PPI-CV selects its mixture weight by five-fold trusted pinball validation; RCP-Aug lacks paired subtraction and is not bias robust.Source: §A.2 · pp. 18–19

A.3 METRICS AND STATISTICAL REPORTING

  • Evaluation reports grouped coverage diagnostics, WSC, L1-ERT, marginal coverage, set volume, and other outcomes; main results use 30 common splits with paired, dataset-specific bootstrap intervals.The intervals are descriptive and have no multiple-comparison adjustment; no cross-dataset effect is formed.Source: §A.3 · p. 19

A.4 PAIRED RESULTS, MARGINAL COVERAGE, AND VOLUME

The paired-results appendix audits conditional error, marginal coverage, and set volume across the benchmark, then adds interpretation controls and label-quality and trusted-budget comparisons. Its controls and sweeps are described, but the supplied passages do not give their numerical outcomes.

A.4 PAIRED RESULTS, MARGINAL COVERAGE, AND VOLUME

  • The K = 10 raw, debiased, and split-half audits agree with the principal K = 30 direction in nearly all dataset–method cells.The appendix also reports paired grouped-MSCE changes, marginal coverage, mean log volume, and an independent split-half audit; no method produced a full set.Source: §A.4 · p. 19, Table 11 · p. 19, Table 12 · p. 20

A.5 RCP INTERPRETATION CONTROLS

  • Direct pooling can yield low grouped error but assumes synthetic labels are trustworthy without bias correction; RCP-Extra controls for the value of additional trusted outcomes.The matched-permutation contrast tests genuine covariate–score pairing against a placebo that preserves the synthetic-score marginal distribution and other training conditions.Source: §A.5 · p. 21, Table 17 · p. 21

A.6 SENSITIVITY TO LABEL QUALITY AND TRUSTED-LABEL BUDGET

  • The weak, moderate, and oracle labeler comparison and trusted-label budget sweep use common splits, varying only nested trusted-data prefixes in the budget experiment.The supplied passage states that the main text reports these comparisons but gives no cell-level outcomes.Source: §A.6 · p. 22

A.7 SUPPLEMENTARY EXPERIMENTS

Supplementary experiments examine quantile risk, synthetic-pool size, realistic LLM supervision, and theory diagnostics. They report improvements in intermediate risk and audited metrics, alongside checks of deployment geometry and three-resource scaling.

A.7 SUPPLEMENTARY EXPERIMENTS

  • PPI-ERM has lower mean pinball risk and lower across-split variability than supervised learning on every dataset.Both learners are evaluated on the same trusted validation folds used for power selection.Source: §A.7 · p. 22, Table 18 · p. 22
  • Even a 5% synthetic pool improves all three metrics over supervised RCP on the four audited datasets.Further pool growth helps until pool-estimation error becomes small; performance may then plateau or become nonmonotone when usable covariance and power are unchanged.Source: §A.7 · p. 22

A.8 REALISTIC LLM-GENERATED SUPERVISION ON HELPSTEER2

  • HelpSteer2 ratings cover five distinct quality dimensions, and the held-out evaluation uses prompt-level separation across data roles.The study has 21,352 rows; external labelers score 500 pilot rows, split into 100 paired rows and 400 auxiliary-pool rows per seed. Only 13.6% of true pilot scores exceed the target boundary, motivating a shuffled-score reference.Source: §A.8 · pp. 22–23

A.9 THEORY DIAGNOSTICS

  • The controlled diagnostics map theory predictions to experiments, while the direct checks leave the eight-dataset and HelpSteer2 results unchanged.The theory-to-experiment map isolates geometry, information mechanisms, and resource scaling; practical evidence remains in the real-data benchmark and permutation experiment.Source: §A.9 · p. 23, §A.10 · p. 24

A.10 DIRECT CHECKS OF THE DEPLOYMENT DERIVATIVE AND RESOURCE SCALING

  • Constant threshold shifts change deployed thresholds by at most 8.88 × 10−16 and coverage curves by at most 2.22 × 10−16; nonconstant-direction remainder slopes are 1.981 and 2.009.In the joint resource scan, deployment-error slopes are −1.159 for oracle power and −1.171 for supervision; varying pool size gives coefficient 0.0240 [0.00643, 0.04234] versus leading coefficient 0.0192, while the pool-free supervised control is flat.Source: §A.10 · p. 24

A.11 PROFILED-RISK AND SELECTION DIAGNOSTICS

The diagnostics find that coverage-based candidate selection does not consistently beat fixed unit PPI, and that finite-resolution estimates can have coupling-dependent variance. The accompanying theory details describe the regularity conditions and information mechanisms behind these results.

A.11 PROFILED-RISK AND SELECTION DIAGNOSTICS

  • Raw pinball selection changes in 205 of 240 comparisons, while cross-calibrated criteria and deployed thresholds remain invariant up to 8.88 × 10−16 under candidate-specific shifts.The authors treat this as an implementation invariant, not evidence of predictive quality.Source: §A.11 · p. 24
  • The synthetic-assisted coverage-moment selector is worse than fixed unit PPI on Bike, Diamonds, and Transcoding, while the SGEMM interval crosses zero.The study compares six candidates using four selection criteria and retains fixed RCP-PPI; the authors report candidate selection as a negative method result.Source: §A.11 · p. 24, Table 20 · p. 24, Figure 4 · p. 25
  • On HelpSteer2, Flash-PPI and Pro-PPI have worse pinball risk than supervised RCP by 0.002264 and 0.003030, yet improve final grouped MSCE by 0.001571 and 0.002011.These are descriptive paired results because the reconstructed experiment has prompt-cluster dependence, not a new i.i.d. conformal guarantee.Source: §A.11 · p. 25, Table 21 · p. 25
  • Across 32 settings, the quadratic-estimator bias and variance-formula discrepancies are at most 2.10 and 2.36 Monte Carlo standard errors; synthetic correction reduces variance only under positive coupling.It increases variance under independence or negative coupling, so correct estimation does not guarantee better selection.Source: §A.11 · p. 25, Table 22 · p. 25, Figure 5 · p. 26

B.1 REGULARITY AND PROFILED-RISK DETAILS; B.2 INFORMATION MECHANISMS AND CONSEQUENCES

  • The asymptotic analysis assumes local learning and deployment regularity, while the information results distinguish trusted–synthetic covariance that survives post-conformal geometry from covariance in removed shift directions.Under the stated local synthetic-law path, pooling improves on a/n precisely when ∆2 < a(1 + κ); matched paired–pool PPI remains centered at the trusted target.Source: §B.1 · p. 26, §B.2 · pp. 26–27

B.3 FINITE-RESOLUTION POST-CONFORMAL EVALUATION

The appendix develops finite-resolution post-conformal evaluation and supporting proofs for the risk, calibration, geometry, and learning expansions. It explains when synthetic-assisted discrepancy estimation is unbiased, why its variance can move either way, and the assumptions behind the asymptotic results.

B.3 FINITE-RESOLUTION POST-CONFORMAL EVALUATION

  • For finite features, the synthetic-assisted quadratic discrepancy estimator is unbiased when paired and pool synthetic laws match and the evaluation samples are independent.When the projected error is zero, evaluation is intrinsically second order, and synthetic correction can reduce or increase selection noise; the training-stage power rule does not transfer unchanged.Source: §B.3 · p. 27

C.1 PINBALL RISK AND PRE-CONFORMAL COVERAGE

  • The pinball-risk proof obtains its coverage bound by integrating the conditional CDF deviation from the target quantile and then averaging over covariates.The argument treats thresholds above and below the target quantile symmetrically under the stated Lipschitz condition.Source: §C.1 · p. 28

C.2 EXACT MSCE IDENTITIES AND PROBABILITY-SCALE CALIBRATION

  • With continuous calibration residuals, the probability-scale calibration variable has a Beta distribution; shifting the learned threshold by a constant leaves the final prediction set unchanged.The proof uses exchangeable calibration and test residuals for rank validity, with conservative treatment of ties.Source: §C.2 · p. 28

C.3 PROFILED RISK AND POST-CONFORMAL GEOMETRY

  • Profiling over the scalar correction gives a gradient equal to calibrated coverage error, while its Hessian removes constant-shift directions and weights remaining shape changes by boundary density.The resulting geometry can be singular because global shifts are removed.Source: §C.3 · p. 29

C.4 STATIONARY-TARGET PPI ASYMPTOTICS

  • The three-resource expansion assumes a locally regular learning and deployment problem, independent splits, continuous calibration residuals, and a uniform learning expansion.The theorem's contribution is downstream post-conformal geometry conditional on that expansion; unrestricted rich-class signed pinball ERM may be ill posed at λ ≥ 1.Source: §C.4 · pp. 29–30

C.5 INFORMATION BOUNDARY, POWER, AND ALLOCATION

The appendix characterizes when synthetic information can reduce post-conformal learning variance, how to estimate the relevant projected quantities, and where the guarantees stop. It also gives supplementary excess-risk and local coverage results under explicit conditions.

C.5 INFORMATION BOUNDARY, POWER, AND ALLOCATION

  • Under certified correctness, pooled estimating scores have covariance Var(ξ)/(n + N); under exact specification and conditional independence, the optimal synthetic power is zero.In the paired-score oracle check, λ⋆ = 1/(1 + κ) and L(λ⋆) = a/(n + N).Source: §C.5 · p. 31

C.6 CROSS-FITTED ESTIMATION OF PROJECTED BOUNDARY INFORMATION

  • Cross-fitting estimates the projected constants a, b, and c consistently when the nuisance estimates converge in the stated norms and fourth moments are uniformly bounded.The procedure uses fold-specific pilots, boundary-density and geometry estimates, and paired and pool scores.Source: §C.6 · p. 31
  • At exact specification, boundary disagreement gives a coarse upper bound but does not identify the sign of b.The bound is established for the boundary-indicator disagreement diagnostic.Source: §C.6 · p. 31
  • A scalar X-only control removes at most the between-X part of the leading learning constant, with a finite pool scaling the attainable reduction by 1/(1 + κ).The ratio is defined as zero when c = 0.Source: §C.6 · p. 32

C.7 SYNTHETIC-LAW MISMATCH

  • With fixed nonzero paired–pool mismatch, the population objective generally targets a shifted parameter, and increasing either sample size does not restore the trusted target.The local remainder formula is not asserted for arbitrarily large mismatch; the passage points to importance weighting or an explicit transport model in that case.Source: §C.7 · p. 32

C.8 QUADRATIC POST-CONFORMAL DISCREPANCY

  • The quadratic discrepancy construction makes the corrected squared mean unbiased for the squared mean and derives an exact variance formula from orthogonal sample components.The variance argument uses Hoeffding decompositions of second-order inner-product U-statistics and independence of the two samples.Source: §C.8 · p. 32

D.1 A DISTRIBUTION-FREE GLOBAL EXCESS-RISK BOUND

  • The distribution-free global excess-risk bound establishes consistency, not dominance over alternative methods.The appendix describes these supplementary results as complementary to the main paper’s local efficiency comparisons.Source: §D.1 · p. 33, D SUPPLEMENTARY LEARNING-THEORETIC… · p. 33

D.2 A RADIUS-DEPENDENT NONPARAMETRIC EXTENSION

  • Under local curvature, smoothness, and sup-norm localization, the nonparametric risk rate yields population-conformalized conditional error OPr(r⋆); independent finite calibration is governed separately.The localization must hold with high probability for the stated conditional-error consequence.Source: §2 · p. 34
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