Conditional Conformal Prediction Guarantees

• The paper advances conformal prediction by introducing a spectrum of conditional guarantees that balance valid coverage with computational efficiency.
• It leverages group-conditional and covariate-shift approaches via augmented quantile regression, dynamic programming, and RKHS regularization.
• Empirical evidence from both synthetic and real-world scenarios validates these methods under data heterogeneity and varying subpopulation shifts.

Conformal prediction with conditional guarantees refers to a collection of distribution-free methodologies aiming to endow predictive sets with statistical validity not just in the marginal sense (i.e., averaged over all test points), but uniformly or approximately across subpopulations, regions of covariate space, or even at the level of each individual test point. This research addresses the impossibility of exact, universally-valid, finite-sample conditional coverage and has resulted in an extensive taxonomy of relaxations and algorithmic frameworks characterized by key trade-offs in statistical validity, computational complexity, and prediction set efficiency. Conditional guarantees are now understood as a spectrum—ranging from exact group-conditional coverage in finite VC-dimension settings, to approximate guarantees controlled by smoothness, regularization, or reweighting schemes over user-specified shift classes.

1. Formal Definitions and Fundamental Limits

Given i.i.d. observations $(X_i, Y_i)$ from an unknown joint $P$, conformal prediction constructs set-valued predictors $\widehat C(x)$ such that the marginal coverage property

$\Pr(Y_{n+1} \in \widehat C(X_{n+1})) \geq 1-\alpha$

is universally met. Conditional coverage, defined as

$$\Pr(Y_{n+1} \in \widehat C(X_{n+1}) \mid X_{n+1}=x) \geq 1-\alpha~\text{for almost all }x,$$

is generally impossible in a distribution-free, finite-sample regime due to the non-atomicity and high dimensionality of $\mathcal X$ (Gao et al., 23 Feb 2025, Gibbs et al., 2023, Bellotti, 2021). Consequently, research pivots to meaningful relaxations (group, local, weighted, or approximate conditional coverage) and to constructing sets within restricted classes or averages over covariate shifts.

Marginal and conditional coverage are related in that exact marginal coverage is an average over the conditional coverage probabilities. The impossibility of finite-sample conditional guarantees at the pointwise level leads to alternative objectives: (i) group-conditional (e.g., coverage within subpopulations), (ii) weighted or covariate-shifted coverage, and (iii) approximate or local conditional coverage quantified by deviation bounds.

2. Methodological Spectrum

2.1 Finite-Dimensional Group and Covariate-Conditional Guarantees

Modern frameworks reformulate conditional coverage as uniform validity under a class $\mathcal F$ of covariate reweightings or indicator functions for subgroups: $$\mathbb E[f(X)(1\{Y \in \widehat C(X)\} - (1-\alpha))] \approx 0$$ for all $f\in\mathcal F$, where finite-dimensional $\mathcal F$ (such as group indicators, low-dimensional projections, or RKHS balls) admit exact finite-sample coverage via augmented quantile regression (Gibbs et al., 2023, Bairaktari et al., 24 Feb 2025, Jung et al., 28 Sep 2025).

Kandinsky Conformal Prediction constructs randomization-augmented quantile regressions over a user-chosen linear class $\mathcal W=\{w_\beta(x,y)\}$, producing prediction sets that possess, with high probability, weighted coverage error $O(\sqrt{d/n})$ uniformly over $\mathcal W$ (Bairaktari et al., 24 Feb 2025). This framework generalizes Mondrian (partition-based) conformal, class-conditional, and fully overlapping or fractional group-weighted coverage, enabling minimax-optimal high-probability conditional validity.

2.2 Structured Prediction Set Classes and Volume Optimality

Addressing the size (volume) of prediction sets, recent work defines conditional restricted volume-optimality on a structured class—e.g., $\mathcal C_k$ of unions of $k$ intervals—in each outcome space. By designing a split-conformal score through dynamic programming over a nest of unions-of-intervals that capture level sets of a fitted conditional CDF, one obtains (Gao et al., 23 Feb 2025):

• exact marginal coverage,
• approximate conditional coverage accurate up to a slack $O(\delta)$ controlled by the CDF estimate,
• near-optimal volume among all $\mathcal C_k$ sets satisfying the desired (approximate) coverage.

This volume-optimality holds distribution-free and adapts to the multimodality of $Y \mid X$, outperforming classical interval-based and DCP approaches under substantial model mismatch and data heterogeneity.

3. Algorithmic Frameworks and Practical Implementation

Framework	Target Guarantee	Key Ingredient
Augmented Quantile Regression (Gibbs et al., 2023, Bairaktari et al., 24 Feb 2025)	Group/shift-conditional, finite-sample	Convex quantile regression over $\mathcal W$, randomization
Dynamic Programming (DP) (Gao et al., 23 Feb 2025)	Conditional volume-optimality	Nested unions-of-intervals, DP for structure
RKHS Regularization (Jung et al., 28 Sep 2025)	Kernel/local shift-conditional	RKHS quantile regression, regularization path-tracing
Partition Learning (Kiyani et al., 26 Apr 2024)	Data-adaptive group conditional	Gradient-descent for learned partition and multi-threshold
Iterative Feedback (Bellotti, 2021)	Approx. object-conditional	Feedback min-L2 conditional coverage bias

Efficient algorithms emerge for each guarantee type, often with solution path or alternating minimization routines for hyperparameter selection and sensitivity adjustment (Jung et al., 28 Sep 2025, Bairaktari et al., 24 Feb 2025, Kiyani et al., 26 Apr 2024). Computational implementations include convex optimization for pinball loss, dynamic programming for set construction, and bootstrap cross-validation for regularization tuning.

4. Theoretical Guarantees and Quantitative Bounds

For finite-dimensional weighted/covariate shift classes, exact finite-sample guarantees hold (Gibbs et al., 2023, Bairaktari et al., 24 Feb 2025), with error at most $O(d/(n+1))$ (where $d = \dim(\Phi)$), and worst-case deviation controlled by the chosen regularization in infinite-dimensional settings (Jung et al., 28 Sep 2025). In practice, with RKHS or Lipschitz constraints, one obtains

$$| \mathbb E[f(X)(1\{Y \in \widehat{C}(X)\} - (1-\alpha))] | \leq O(\lambda) + O(1/\sqrt{n})$$

where $\lambda$ is the regularization parameter. SpeedCP accelerates the computation of these kernel-based conformal sets to match a single quantile fit while providing explicit finite-sample error control (Jung et al., 28 Sep 2025).

For conditional volume-optimality, if the estimator $\widehat F(y\,|\,x)$ is uniformly accurate on unions-of-$k$-intervals up to error $\delta$, then the constructed set

$$\lambda(\widehat{C}(X_{2n+1})) \leq OPT_k(F(\cdot|X_{2n+1}),\,1-\alpha+\tfrac{1}{n}+4\delta+\gamma)$$

with coverage

$$\Pr[Y_{2n+1}\in \widehat{C}(X_{2n+1})\mid X_{2n+1}=x] \geq 1-\alpha-3\delta$$

for most $x$, with high probability over the calibration data (Gao et al., 23 Feb 2025).

Empirical process and Rademacher complexity tools yield minimax-optimal rates for split-conformal and weighted-coverage approaches, matching lower bounds up to logarithmic factors (Duchi, 28 Feb 2025). Uniformity across subpopulations, groups, and "cells" of the covariate space is achieved with error controlled by the calibration sample size and model class complexity.

5. Applications and Empirical Evidence

These conditional-coverage methodologies have been validated across diverse settings:

• Synthetic multivariate mixtures: DP-based conformal sets recover the true multimodal structure, yielding prediction set volumes near the theoretical optimum (Gao et al., 23 Feb 2025).
• Group-fair prediction: Kandinsky and RKHS-based conformal methods deliver uniform coverage across overlapping demographic or discovery-driven subgroups (Bairaktari et al., 24 Feb 2025, Jung et al., 28 Sep 2025).
• Large-class classification: Clustered conformal prediction outperforms standard and class-wise conformal for class-conditional coverage, especially in many-class, limited-sample regimes (Ding et al., 2023).
• High-dimensional embeddings: SpeedCP with low-rank latent embeddings achieves near-nominal bin-wise or admixture-conditional coverage on molecular, text, and medical imaging datasets (Jung et al., 28 Sep 2025).
• Practical high-dimensional regression: Partition learning via PLCP enables data-adaptive grouping and robust coverage, matching or outperforming baseline methods in both in-domain and out-of-domain (OOD) scenarios (Kiyani et al., 26 Apr 2024).

Empirical analyses demonstrate that approximate conditional validity can be tightly controlled with sufficient calibration data and accurate conditional models or embeddings. The DP method is notably robust to tuning parameters such as $k$ and grid granularity.

6. Limitations and Ongoing Developments

Exact finite-sample, pointwise conditional coverage remains provably unattainable without triviality. All current relaxations trade the strength of the guarantee for practicality: (i) coverage for finite-dimensional classes of shifts or groupings, (ii) "near-conditional" coverage up to estimation or regularization slack, or (iii) smooth error bounds decaying with calibration size (Gao et al., 23 Feb 2025, Gibbs et al., 2023, Duchi, 28 Feb 2025). For infinite-dimensional classes, one must carefully regularize or accept explicit error bounds.

Computationally, kernel and optimization-based approaches can be expensive in high dimension; recent algorithmic innovations (e.g., SpeedCP, path-tracing) alleviate but do not remove these issues (Jung et al., 28 Sep 2025). For generic multimodal or highly-heteroskedastic regimes, performance is sensitive to the quality of conditional CDF/quantile estimators and tuning of structure parameters ($k$, kernel bandwidth, group size).

7. Connections and Broader Impact

Conditional conformal prediction unifies and extends classical marginal, group/mondrian, and local conformal paradigms under a single distribution-free statistical umbrella, linking to RKHS and quantile regression, dynamic programming, and partition learning—a convergence that is driving both methodological understanding and computational tractability.

Collectively, these advances enable practitioners to design uncertainty sets that are adaptive to heterogeneity and subpopulation structure, carrying explicit statistical guarantees that are interpretable, tunable, and empirically validated. As the field progresses, the focus will be on extending computational efficiency, understanding the fundamental gap to full conditional validity, and constructing diagnostic tools for practitioners to quantify and communicate the scope of their coverage guarantees (Gao et al., 23 Feb 2025, Bairaktari et al., 24 Feb 2025, Jung et al., 28 Sep 2025, Gibbs et al., 2023).