Learnable Conformal Prediction

• Learnable conformal prediction is a framework where components of the prediction pipeline are trained on data to target efficiency, set geometry, or local validity.
• Different approaches, from constrained ERM to adaptive interval geometry and context-aware nonconformity, allow dynamic calibration of prediction sets in diverse applications.
• LCP techniques are applied in classification, regression, time series, and robotics, offering smoother optimization objectives while preserving distribution-free marginal coverage.

Learnable conformal prediction (LCP) denotes conformal prediction procedures in which some component of the conformal pipeline is learned from data rather than fixed a priori. Depending on the formulation, the learned object may be the prediction-set parameterization, the nonconformity function, a local partition of the calibration space, an object-dependent monotone score transform, a center-radius interval geometry, or a horizon-level aggregation of multi-step errors. Across these variants, the central division of labor is stable: learning targets efficiency, set geometry, or local validity, while conformal calibration on a held-out calibration set preserves marginal coverage under exchangeability (Bai et al., 2022, Stutz et al., 2021, Kiyani et al., 2024, Su et al., 2 Mar 2026).

1. Foundational formulation

Conformal prediction (CP) constructs prediction sets by thresholding a nonconformity score with a calibrated empirical quantile. In multiclass classification, with calibration set $\mathcal{D}_{\mathrm{cal}}=\{(X_i,Y_i)\}_{i=1}^m$, calibration scores $S_{f,i}=S_f(X_i,Y_i)$, and empirical quantile $\widehat Q$ given by the $\lceil(1-\alpha)(m+1)\rceil$-th smallest score, the conformal set is

$$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$

and under exchangeability,

$$\mathbb{P}\{\,Y\in \widehat C_f(X)\,\}\ge 1-\alpha.$$

This validity statement holds irrespective of how $f$ is trained; in LCP, training is redirected toward predictive efficiency rather than validity itself (Jia et al., 22 Nov 2025).

The same structural principle appears in split conformal regression and more general set prediction. The population objective considered in general-function-class formulations is to minimize expected efficiency subject to coverage,

$$\min_\theta \mathbb{E}[\mathrm{cost}(S_\theta(X))]\quad \text{subject to}\quad \mathbb{P}(Y\in S_\theta(X))\ge 1-\alpha,$$

with empirical constrained ERM and subsequent reconformalization used to recover exact finite-sample coverage (Bai et al., 2022).

A persistent distinction in the LCP literature is between marginal and conditional validity. Split conformal guarantees finite-sample marginal coverage, but full conditional coverage is typically unattainable in a distribution-free manner with finite calibration data. This impossibility motivates learned relaxations such as partitions, local thresholds, or context-aware scores that target “local” or approximate conditional behavior without abandoning the exchangeability-based conformal mechanism (Kiyani et al., 2024).

2. Main design patterns

The term LCP covers several technically distinct constructions. What they share is that the conformal predictor is no longer a fixed post-hoc wrapper around a frozen score.

Approach	Learnable component	Representative paper
Constrained ERM	Prediction-set parameters $\theta$ and thresholds $t$	(Bai et al., 2022)
Conformal training	Model parameters through smooth calibration and smooth set prediction	(Stutz et al., 2021)
Cost-sensitive conformal training	Rank-weighted cross-entropy based on true-label rank	(Jia et al., 22 Nov 2025)
Learned calibration structure	Soft partition $S_{f,i}=S_f(X_i,Y_i)$0 and region thresholds $S_{f,i}=S_f(X_i,Y_i)$1	(Kiyani et al., 2024)
Locally adaptive score transform	Object-dependent monotone transform $S_{f,i}=S_f(X_i,Y_i)$2	(Colombo, 2023)
Adaptive interval geometry	Center $S_{f,i}=S_f(X_i,Y_i)$3 and radius $S_{f,i}=S_f(X_i,Y_i)$4	(Su et al., 2 Mar 2026)
Time-series aggregation	Horizon weights $S_{f,i}=S_f(X_i,Y_i)$5 in a multi-step score	(Cleaveland et al., 2023)
Context-aware nonconformity	Lightweight neural score $S_{f,i}=S_f(X_i,Y_i)$6	(Kumar et al., 26 Sep 2025)

These formulations differ primarily in where learning intervenes. Some methods learn the prediction set directly, some learn the score used for calibration, and some learn calibration-side structure from the calibration data itself. A plausible implication is that “learnability” in CP is best understood as a family resemblance rather than a single algorithmic template.

3. Classification-oriented LCP

A major strand of LCP in classification seeks to reduce expected prediction set size while preserving post-hoc conformal coverage. In conformal training (ConfTr), conformalization is simulated inside each mini-batch during SGD: the batch is split into calibration and prediction halves, hard thresholding is replaced by a temperature-controlled sigmoid, and calibration uses differentiable sorting or quantile approximations. This enables end-to-end optimization of set-level objectives such as inefficiency, class-conditional inefficiency, coverage confusion, and group-specific mis-coverage; at test time, the smooth surrogate is discarded and exact CP is re-applied on a held-out calibration set (Stutz et al., 2021).

A broader formulation treats LCP as constrained ERM over a function class of prediction sets. In this view, regression intervals, multi-output level sets, and classification label sets are all instances of a common optimization problem with efficiency cost and empirical coverage constraint. Because the empirical coverage term contains the hard indicator $S_{f,i}=S_f(X_i,Y_i)$7, the training problem is relaxed with smooth surrogates and a Lagrangian, followed by reconformalization on a second split. The resulting guarantees are approximate valid population coverage and near-optimal efficiency within class under uniform concentration, and exact finite-sample coverage after reconformalization (Bai et al., 2022).

Recent classification-specific work has focused on the mismatch between surrogate set-size objectives and the true conformal objective. In “Cost-Sensitive Conformal Training with Provably Controllable Learning Bounds” (Jia et al., 22 Nov 2025), prior LCP methods based on surrogate indicators such as Sigmoid or Gaussian error function are criticized because they “lack uniform approximation error bounds to the discontinuous indicator” and can produce a persistent large gap between soft set size and actual set size. The proposed alternative is rank-weighted cross-entropy (RWCE), built on the true-label rank

$S_{f,i}=S_f(X_i,Y_i)$8

The paper proves that expected conformal set size is upper bounded by expected true-label rank up to a model-independent slack,

$S_{f,i}=S_f(X_i,Y_i)$9

and then upper-bounds $\widehat Q$0 by the RWCE objective

$\widehat Q$1

This yields a controllable differentiable bound on expected set size and, empirically, an average $\widehat Q$2 reduction in APSS, including a Caltech-101, ResNet, APS example in which APSS is reduced from $\widehat Q$3 to $\widehat Q$4 while retaining coverage $\widehat Q$5 (Jia et al., 22 Nov 2025).

4. Learned locality and conditional structure

One route to LCP targets conditional validity more directly by learning uncertainty-aware structure from the calibration data. Partition Learning Conformal Prediction (PLCP) learns a soft partition $\widehat Q$6 and region-specific thresholds $\widehat Q$7 by minimizing a pinball-loss objective,

$\widehat Q$8

The learned $\widehat Q$9 approximate local conditional quantiles, while $\lceil(1-\alpha)(m+1)\rceil$0 groups points into uncertainty-homogeneous regions. PLCP is implemented with alternating gradient descent, provides both population and finite-sample MSCE bounds, and was reported to improve coverage and length in both classification and regression, including ID and OOD settings (Kiyani et al., 2024).

A conceptually different route is to redefine the conformity measure itself through a trainable, object-dependent change of variables,

$\lceil(1-\alpha)(m+1)\rceil$1

where $\lceil(1-\alpha)(m+1)\rceil$2 is strictly increasing in $\lceil(1-\alpha)(m+1)\rceil$3 and has an $\lceil(1-\alpha)(m+1)\rceil$4-independent codomain. Under these conditions, split CP with transformed scores remains marginally valid because exchangeability applies to the transformed ranks. In regression with $\lceil(1-\alpha)(m+1)\rceil$5, the resulting interval is

$\lceil(1-\alpha)(m+1)\rceil$6

so interval width becomes explicitly $\lceil(1-\alpha)(m+1)\rceil$7-dependent while validity is preserved (Colombo, 2023).

The locally adaptive transform framework is notable because its training objective avoids the usual non-differentiable quantile bottleneck. Instead of optimizing a single confidence level, it averages interval size over all $\lceil(1-\alpha)(m+1)\rceil$8, yielding a smooth surrogate involving sums over transformed calibration scores and implicit differentiation through $\lceil(1-\alpha)(m+1)\rceil$9. This differentiability claim is explicit: “Contrary to other CP-aware training methods, the objective function is smooth and can be minimized through standard gradient methods without approximations” (Colombo, 2023).

5. Regression, time series, and robotics

In regression, LCP has been extended from learned widths to learned interval geometry. Co-optimization for Adaptive Conformal Prediction (CoCP) parameterizes intervals as

$$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$0

and alternates between learning $$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$1 as the $$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$2-quantile of folded residuals $$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$3 and refining $$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$4 with a differentiable soft-coverage objective. Split conformal calibration uses the normalized score

$$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$5

leading to the calibrated interval

$$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$6

The theory characterizes a $$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$7-soft oracle and shows asymptotic convergence to the oracle highest-density interval under standard regularity conditions; empirically, CoCP yields shorter intervals and improved conditional-coverage diagnostics on synthetic and real benchmarks (Su et al., 2 Mar 2026).

In long-horizon time series, LCP appears as learning a parameterized multi-step nonconformity function rather than calibrating each horizon step separately. The learned aggregator is

$$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$8

with $$\widehat C_f(X)=\{\,y\in\mathcal{Y}:S_f(X,y)\le \widehat Q\,\},$$9 and $$\mathbb{P}\{\,Y\in \widehat C_f(X)\,\}\ge 1-\alpha.$$0. Calibrating a single threshold on this aggregated score yields

$$\mathbb{P}\{\,Y\in \widehat C_f(X)\,\}\ge 1-\alpha.$$1

thereby avoiding the conservatism of union-bound constructions that require confidence $$\mathbb{P}\{\,Y\in \widehat C_f(X)\,\}\ge 1-\alpha.$$2 at each step. The learning problem is encoded as a MILCP and relaxed to an LCP, and the paper proves that the relaxed LCP has the same optimal cost as the original MILCP (Cleaveland et al., 2023).

In robotics, LCP has been instantiated as a learned context-aware nonconformity function $$\mathbb{P}\{\,Y\in \widehat C_f(X)\,\}\ge 1-\alpha.$$3 built from geometric, semantic, and task-specific features. The same template is used across classification, object detection, and path planning: train $$\mathbb{P}\{\,Y\in \widehat C_f(X)\,\}\ge 1-\alpha.$$4 on a proper training split, freeze $$\mathbb{P}\{\,Y\in \widehat C_f(X)\,\}\ge 1-\alpha.$$5, calibrate a scalar threshold on a disjoint calibration set, and then form CP sets with the learned score. Reported results include set-size reduction in classification, tighter detection intervals, and improved planning safety, while maintaining the standard split-CP marginal coverage theorem because validity applies to any fixed measurable score (Kumar et al., 26 Sep 2025).

6. Guarantees, limitations, and nomenclature

A recurrent feature of LCP is that learning usually does not alter the fundamental validity theorem. When a score or set family is learned on one split and calibrated on a disjoint split, exchangeability of calibration and test points still yields marginal coverage. The learned component affects efficiency, local adaptivity, or interval geometry, not the basic split-conformal rank argument. This is explicit in classification RWCE, where “the learned $$\mathbb{P}\{\,Y\in \widehat C_f(X)\,\}\ge 1-\alpha.$$6 affects efficiency (set size), not validity,” and in context-aware robotics LCP, where validity holds for any measurable nonconformity function fixed before calibration (Jia et al., 22 Nov 2025, Kumar et al., 26 Sep 2025).

The main limitation is that conditional coverage remains elusive. PLCP states that “it is impossible to construct nontrivial prediction sets with full conditional coverage guarantees” in the fully distribution-free finite-sample setting, so learned partitions or local scores only approximate local validity through relaxations such as MSCE or related diagnostics (Kiyani et al., 2024). A plausible implication is that many empirical gains attributed to LCP should be interpreted as improved allocation of uncertainty across the covariate space rather than resolution of the conditional-coverage impossibility.

Another limitation concerns objective alignment. Surrogate-indicator methods in conformal training are differentiable, but the 2025 cost-sensitive analysis argues that Sigmoid and Gaussian error function surrogates “do not have a uniform error bound to the indicator function,” leading to “uncontrollable learning bounds” and persistent train-test objective mismatch (Jia et al., 22 Nov 2025). By contrast, monotone-transform LCP emphasizes smooth objectives obtained without indicator approximation, and CoCP uses a soft-coverage objective whose gradients concentrate near interval boundaries rather than a direct surrogate for set size (Colombo, 2023, Su et al., 2 Mar 2026).

Exchangeability is also a strict boundary of the standard theory. Several formulations note that CP validity can fail under distribution shift unless specialized procedures are introduced. This is why online threshold adaptation in robotics is described as heuristic, and why local or auxiliary-data methods frame their stronger results as asymptotic or approximate conditional guarantees rather than exact finite-sample conditional validity (Kumar et al., 26 Sep 2025, Min et al., 7 Jun 2026).

Finally, the acronym “LCP” is overloaded. In the time-series work, LCP appears inside “MILCP” and “LCP” as “linear complementarity program,” explicitly “not to be confused with Learnable Conformal Prediction” (Cleaveland et al., 2023). In another nearby literature, “LCP” denotes “localized conformal prediction,” and “ELCP” denotes “Enhanced Localized Conformal Prediction,” a method that incorporates auxiliary data to refine localized prediction sets while preserving finite-sample marginal coverage (Min et al., 7 Jun 2026). Within the learnable-CP literature proper, however, the dominant meaning is the one shared by constrained ERM, conformal training, learned local structure, and learned score design: conformal prediction with learnable components calibrated post hoc to retain distribution-free marginal validity.