CoCoNet: Convex Conformal Network for Multi-Dimensional Uncertainty Regions

Alireza Moradi; Thomas Bruys; Pascal Van Hentenryck; Reza Zandehshahvar

stat.MLOct 16, 2026

CoCoNet: Convex Conformal Network for Multi-Dimensional Uncertainty Regions

Authors: Alireza Moradi, Thomas Bruys, Pascal Van Hentenryck, Reza Zandehshahvar

Organizations: H. Milton Stewart School of Industrial and Systems Engineering (ISyE), Georgia Institute of Technology · NSF AI Institute for Advances in Optimization (AI4OPT), Georgia Institute of Technology · School of Electrical and Computer Engineering (ECE), Georgia Institute of Technology

Abstract

Conformal prediction (CP) provides distribution-free coverage guarantees under exchangeability, but the efficiency of its prediction regions depends critically on the choice of the conformity score. This is particularly important in multivariate regression, where dependencies across output dimensions induce complex joint uncertainty structures. This paper introduces CoCoNet (Convex Conformal Network) for learning convex conformity scores that induce flexible, data-adaptive prediction regions. CoCoNet learns the score from held-out data by optimizing prediction-region efficiency while enforcing convexity with respect to the residual. CoCoNet then applies split-conformal calibration to retain finite-sample coverage guarantees. The paper presents two variants of the method: CoCoNet $_{\mathrm{Spline}}$ , based on piecewise-linear scores, and CoCoNet $_{\mathrm{ICNN}}$ , based on Input Convex Neural Networks. Experiments compare CoCoNet with other CP baselines over synthetic and real-world datasets with output dimensions ranging from 2 to 16. The results demonstrate the flexibility of CoCoNet in learning different response types. Across 8 models and 30 multi-dimensional regression tasks, CoCoNet $_\text{ICNN}$ and CoCoNet $_{\mathrm{Spline}\text{-}\ell_1}$ achieve the best efficiencies with average ranks of 1.45 and 2.75, respectively. In particular, CoCoNet $_\text{ICNN}$ ranks first on 22 tasks out of 30 in terms of efficiency.

Figures & tables

Figure 1: Overview of CoCoNet . Given covariates $\mathbf{x}$ , point predictions $\widehat{\mathbf{y}}$ , and actual responses $\mathbf{y}$ , residuals $\mathbf{r}=\mathbf{y}-\widehat{\mathbf{y}}$ are constructed while $h(\cdot)$ encodes $\mathbf{x}$ and $\widehat{\mathbf{y}}$ into an input-dependent representation. The residuals and encoded representation are then passed through the scoring network, where they determine a conformity score $S_{\theta}$ that is convex in the residual space. A held-out calibration set determines the conformal threshold $\widehat{\tau}_{\alpha}$ . The corresponding $\widehat{\tau}_{\alpha}$ -sublevel set defines a convex region in residual space, which is translated by the point prediction $\widehat{\mathbf{y}}$ to obtain the final prediction region in the response space with finite-sample marginal coverage under exchangeability.

Figure 2: Illustration of the learned rotation, coordinate-wise scores, and induced geometry. The input-dependent transformation $\mathbf{z}=\mathbf{Q}^{\top}\mathbf{r}$ rotates the residual coordinates to align them with learned directions of variation (left). Each rotated coordinate is scored by convex piecewise-linear branches $G_{+}$ and $G_{-}$ for positive and negative deviations, with knots $\kappa_{\pm,k}$ (center). The coordinate-wise scores are then aggregated with the $\ell_{1}$ or $\ell_{\infty}$ norm: the $\ell_{\infty}$ aggregation induces a rotated hyperrectangle and the $\ell_{1}$ aggregation an asymmetric polytope in the original residual space (right).

Figure 3: Synthetic uncertainty geometries. Calibrated

90\%

prediction regions are shown for four two-dimensional residual distributions with known population-optimal regions: an ellipsoid, rectangle, triangle, and trapezoid. The comparison illustrates the progression from geometries well matched by structured families to more challenging asymmetric polygonal shapes, where the greater geometric flexibility of

\textit{CoCoNet}{}_{\mathrm{ICNN}}

becomes increasingly important.

		Baselines					CoCoNet
Shape	Metric	Abs. Res.	Emp. Cop.	G-Ellip.	L-Ellip.	MVCS	Spline- $\ell_{\infty}$	Spline- $\ell_{1}$	ICNN
Ellipsoid	IoU $\uparrow$	$0.35_{\pm 0.00}$	$0.79_{\pm 0.01}$	$\color[rgb]{0.0781,0.3125,0.707}\mathbf{0.97_{\pm 0.01}}$	$0.88_{\pm 0.02}$	$\color[rgb]{0.0781,0.3125,0.707}\mathbf{0.97_{\pm 0.02}}$	$0.79_{\pm 0.01}$	$0.83_{\pm 0.03}$	$0.94_{\pm 0.01}$
	$V/V^{\star}$ $\downarrow$	$2.67_{\pm 0.03}$	$1.23_{\pm 0.01}$	$\color[rgb]{0.0781,0.3125,0.707}\mathbf{1.03_{\pm 0.01}}$	$1.12_{\pm 0.02}$	$\color[rgb]{0.0781,0.3125,0.707}\mathbf{1.03_{\pm 0.02}}$	$1.22_{\pm 0.01}$	$1.18_{\pm 0.04}$	$1.06_{\pm 0.02}$
	Cov. %	$89.5_{\pm 0.9}$	$90.6_{\pm 1.1}$	$90.7_{\pm 1.6}$	$90.6_{\pm 1.5}$	$90.6_{\pm 1.7}$	$90.6_{\pm 1.4}$	$90.7_{\pm 1.7}$	$90.5_{\pm 1.5}$
Rectangle	IoU $\uparrow$	$0.53_{\pm 0.01}$	$0.57_{\pm 0.01}$	$0.75_{\pm 0.01}$	$0.73_{\pm 0.01}$	$0.76_{\pm 0.01}$	$\color[rgb]{0.0781,0.3125,0.707}\mathbf{0.90_{\pm 0.02}}$	$0.70_{\pm 0.01}$	$\underline{0.86_{\pm 0.01}}$
	$V/V^{\star}$ $\downarrow$	$1.80_{\pm 0.02}$	$1.68_{\pm 0.03}$	$1.29_{\pm 0.02}$	$1.33_{\pm 0.02}$	$1.29_{\pm 0.02}$	$\color[rgb]{0.0781,0.3125,0.707}\mathbf{1.10_{\pm 0.02}}$	$1.37_{\pm 0.02}$	$\underline{1.15_{\pm 0.01}}$

Table 1: Recovery of population-optimal

90\%

convex regions. IoU and

V/V^{\star}

compare the learned region with the population minimum-area region. Subscripts are standard deviations over five seeds; best in blue , second-best underlined . Gray rows report the empirical test coverage in percent.

Figure 4: Input-adaptive prediction-region geometry on CASP. Four test instances compare

\textit{CoCoNet}{}_{\mathrm{ICNN}}

with Local Ellipsoid, MVCS, and

\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{\infty}}

and

\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{1}}

, using the same fixed point predictor. The cross denotes the point prediction and the star denotes the observed response. Local Ellipsoid adapts orientation and scale while retaining an elliptical boundary; MVCS,

\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{\infty}}

, and

\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{1}}

adapt within their respective norm-based, rotated-hyperrectangular, and polytope families. In contrast,

\textit{CoCoNet}{}_{\mathrm{ICNN}}

can also adapt the boundary geometry itself, producing qualitatively different convex regions across inputs.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Domain	$\bm{n}$	$\bm{d_{x}}$	$\bm{p}$
Univariate benchmarks
Beijing PM2.5 ( Liang et al., 2015 )	Air quality	41,757	14	1
Bike Sharing ( Fanaee-T, 2013 )	Transportation	17,379	60	1
CCPP ( Tüfekci, 2014 )	Energy	9,568	4	1
Concrete ( Yeh, 1998 )	Civil engineering	1,030	8	1
Kin8nm ( Rasmussen et al., 1996 )	Robotics	8,192	8	1

Appendix

Table 3: Datasets used in the reported experiments.

n

denotes the number of observations in the source dataset,

d_{x}

the number of input features, and

p

the response dimension. Multiple values of

p

denote tasks constructed at different response dimensions from the same source dataset.

Component	Configuration
Random seeds	{0,1,2,3,4}
Data partition	25/30/25/20%
$\mathcal{D}_{\mathrm{pred}}$ internal split	85/15% train/validation
$\mathcal{D}_{\mathrm{score}}$ internal split	80/20% train/validation
Point predictor	MLP
Hidden dimensions	[64,32]

Appendix

Table 4: Shared data-partitioning, preprocessing, and point-predictor configuration.

Component	Hyperparameter	Value / Search Space
Shared	Coverage temperature $T$	$0.1$
	Optimizer	Adam
	Gradient clipping	5.0
	$\lambda_{\mathrm{cov}}$	{0, 0.1, 1}
	$\lambda_{\mathrm{miss}}$	{0.01, 0.03, 1, 10}
	Training threshold $\tau_{\mathrm{tr}}$	$1$

Appendix

Table 5: Training configuration and hyperparameter search spaces for the proposed CoCoNet score families. The context encoder

\mathbf{h(x)}

and loss-weight search spaces are shared across both families. Hyperparameter search is performed using only the validation portion of

\mathcal{D}_{\mathrm{score}}

; values in braces are searched, while single values are fixed.

Method	Hyperparameter	Value / Search Space
Global Ellipsoid	Ridge	1e-6
Local Ellipsoid	Number of neighbors	$\max(2p+5,\lceil 0.05n\rceil)$
	Local / global covariance weight	0.95 / 0.05
	Ridge	1e-6
Empirical Copula	Normalizer hidden dimensions	[64,32]
	Scale offset	0.1

Appendix

Table 6: Configurations and hyperparameter search spaces for the baseline methods. Search spaces are evaluated using only the validation portion of

\mathcal{D}_{\mathrm{score}}

Figure 5: Univariate conformity scores and prediction intervals on Beijing PM2.5 (one test observation). Left: conformity scores as functions of the residual

\mathbf{r}=\mathbf{y}-F(\mathbf{x})

, normalized by the calibrated threshold; gray ticks denote calibration residuals. Right: the corresponding

90\%

prediction intervals, with the forecast (cross) and actual observation (star). The CoCoNet variants learn asymmetric intervals, while MVCS and Absolute Residual remain symmetric; LW-SC gives the narrowest interval but misses the observation.

	Baselines						CoCoNet (this work)
Dataset	Abs. Res.		LW-SC		MVCS		CoCoNet Spline		CoCoNet ICNN
	Cov. %	Size	Cov. %	Size	Cov. %	Size	Cov. %	Size	Cov. %	Size
Beijing PM2.5 ( $\times 10^{2}$ )	$91.1_{\pm 0.8}$	$2.10_{\pm 0.05}$	$90.6_{\pm 1.1}$	$2.05_{\pm 0.12}$	$90.5_{\pm 1.1}$	$1.89_{\pm 0.07}$	$90.3_{\pm 0.6}$	$\color[rgb]{0.0781,0.3125,0.707}\mathbf{1.64_{\pm 0.02}}$	$90.7_{\pm 0.9}$	$\underline{1.68_{\pm 0.02}}$
Bike Sharing ( $\times 10^{2}$ )	$90.6_{\pm 0.6}$	$1.49_{\pm 0.05}$	$90.2_{\pm 1.2}$	$1.75_{\pm 0.23}$	$90.0_{\pm 1.4}$	$1.29_{\pm 0.02}$	$90.0_{\pm 0.9}$	$\color[rgb]{0.0781,0.3125,0.707}\mathbf{1.22_{\pm 0.03}}$	$90.5_{\pm 0.8}$	$\underline{1.28_{\pm 0.03}}$
CCPP ( $\times 10^{1}$ )	$90.1_{\pm 0.9}$	$1.28_{\pm 0.02}$	$90.1_{\pm 0.9}$	$1.28_{\pm 0.02}$	$90.4_{\pm 1.0}$	$1.25_{\pm 0.02}$	$90.2_{\pm 0.7}$	$\underline{1.24_{\pm 0.03}}$	$90.3_{\pm 0.7}$	$\color[rgb]{0.0781,0.3125,0.707}\mathbf{1.22_{\pm 0.01}}$
Concrete ( $\times 10^{1}$ )	$90.2_{\pm 1.5}$	$2.52_{\pm 0.21}$	$91.4_{\pm 2.8}$	$2.60_{\pm 0.46}$	$91.7_{\pm 1.9}$	$2.52_{\pm 0.06}$	$91.0_{\pm 1.7}$	$\color[rgb]{0.0781,0.3125,0.707}\mathbf{2.34_{\pm 0.15}}$	$92.4_{\pm 2.4}$	$\underline{2.49_{\pm 0.13}}$

Appendix

Table 7: Univariate regression results at

90\%

target coverage. Each method reports, in gray, the empirical test coverage in percent, followed by the prediction-region size (lower is better); subscripts are standard deviations over five seeds. Where a factor (

\times 10^{k}

) follows the dataset name, the sizes in that row are in units of

10^{k}

. Best and second-best sizes are shown in blue and underline .

Figure 6: Univariate conformity scores and prediction intervals on Protein (one test observation). Left: conformity scores as functions of the residual

r=\mathbf{y}-F(\mathbf{x})

, normalized by the calibrated threshold; gray ticks denote calibration residuals. Right: the corresponding

90\%

prediction intervals, with the forecast (cross) and actual observation (star).

\textit{CoCoNet}{}_{\mathrm{ICNN}}

and

\textit{CoCoNet}{}_{\mathrm{Spline}}

produce the smallest intervals in this example.

Figure 7: Two-dimensional projections of three-dimensional prediction regions on Household Power (one test observation). Each panel shows the exact projection of the $90\%$ prediction region onto one pair of response coordinates, together with the point forecast (cross) and the observed response (star). The $\textit{CoCoNet}{}_{\mathrm{ICNN}}$ region adapts its shape to each pair of coordinates and is asymmetric around the forecast, whereas the MVCS region is centrally symmetric about the forecast and the $\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{\infty}}$ region is a rotated, asymmetric hyperrectangle whose projections are parallelograms; the $\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{1}}$ region is an asymmetric polytope.

Figure 8: Two-dimensional projections of four-dimensional prediction regions on QM8 (one test observation). Each panel shows the exact projection of the $90\%$ prediction region onto one pair of response coordinates, together with the point forecast (cross), the observed response (star), and the calibration residuals centered at the point forecast (gray points). The $\textit{CoCoNet}{}_{\mathrm{ICNN}}$ region adapts its shape to each pair of coordinates and is asymmetric around the forecast, whereas the MVCS region is centrally symmetric about the forecast and the $\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{\infty}}$ region is a rotated, asymmetric hyperrectangle whose projections are polygons with parallel opposite edges; the $\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{1}}$ region is an asymmetric polytope.

Figure 9: Distribution of prediction-region efficiency ranks across 30 multivariate regression tasks. Lower rank indicates greater efficiency. Boxplots show the distribution of per-dataset ranks, diamonds denote mean rank, and the dashed curve reports the number of tasks on which each method ranks first.

Figure 10: Learned conformity scores and prediction regions on Online Retail II. Each row shows one method at the same test input. Left: calibration residuals $\mathbf{r}=\mathbf{y}-F(\mathbf{x})$ with the point forecast (cross, at the origin) and the actual (star). Middle: the learned conformity score over the residual space, normalized by its calibrated threshold, so that the $90\%$ prediction region is the sublevel set at height one; the region is traced on the surface and projected, with the score level curves, onto the floor. Right: level curves of the score together with the boundary of the prediction region. The residuals are strongly skewed; the $\textit{CoCoNet}{}_{\mathrm{ICNN}}$ score bends its level sets toward the mass of the residuals and yields an asymmetric region, whereas the $\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{\infty}}$ region is a rotated rectangle, the $\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{1}}$ region an asymmetric polygon, and the MVCS and Local Ellipsoid regions remains centrally symmetric about the forecast, so both must cover low-density areas to reach the target coverage.

Figure 11: Learned conformity scores and prediction regions on Argoverse2 Motion. Each row shows one method at the same test input. Left: calibration residuals $\mathbf{r}=\mathbf{y}-F(\mathbf{x})$ with the point forecast (cross, at the origin) and the realized residual (star). Middle: the learned conformity score over the residual space, normalized by its calibrated threshold, so that the $90\%$ prediction region is the sublevel set at height one; the region is traced on the surface and projected, with the score level curves, onto the floor. Right: level curves of the score together with the boundary of the prediction region. Here the residuals are close to elliptical and the three regions are similar in extent; the $\textit{CoCoNet}{}_{\mathrm{ICNN}}$ region departs from an ellipse only where the residual density does, while the $\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{\infty}}$ and Local Ellipsoid regions keep the rectangular and elliptical shapes imposed by their score families, and the $\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{1}}$ region stays polygonal.

Method	Input-dependent	Asymmetric	Convex guarantee
MVCS q≥1	Yes	No	Yes
MVCS ${}_{\mathrm{free}\ q}$	Yes	No	No
MVCS ${}_{\mathrm{multi},\,q\geq 1}$	No	Yes	Yes
MVCS ${}_{\mathrm{multi},\,\mathrm{free}\ q}$	No	Yes	No
CoCoNet ICNN	Yes	Yes	Yes

Appendix

Table 9: Structural properties of the MVCS variants and

\textit{CoCoNet}{}_{\mathrm{ICNN}}

. Single-norm MVCS remains centrally symmetric for both constrained and free

q

, whereas the multi-norm formulation can represent asymmetric regions. The evaluated multi-norm variants use the non-input-dependent formulation of MVCS Braun et al. (2026) . The

q\geq 1

variants preserve convexity, while

\textit{CoCoNet}{}_{\mathrm{ICNN}}

is input-dependent, asymmetric, and convex by construction.

Figure 12: MVCS variants versus CoCoNet . Left: mean rank of

\textit{CoCoNet}{}_{\mathrm{ICNN}}

\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{\infty}}

\textit{CoCoNet}{}_{\mathrm{Spline}\text{-}\ell_{1}}

, and the corresponding MVCS variant across the 30 multivariate regression tasks, with one MVCS variant included at a time in the primary eight-method comparison (lower is better). Right: dimension-normalized prediction-region size of each MVCS variant relative to

\textit{CoCoNet}{}_{\mathrm{ICNN}}

on each task (log scale); the horizontal line marks equal size, boxes show quartiles, whiskers extend to

1.5

times the interquartile range, and points denote individual tasks. The label

q\geq 1

denotes variants whose norm exponents are constrained to preserve convexity.

Figure 13: Efficiency-rank distribution for the

\textit{CoCoNet}{}_{\mathrm{ICNN}}

contextual-ablation variants. Lower rank indicates smaller dimension-normalized prediction-region size. The comparison isolates the effects of the contextual encoder, the information provided to it, and input-dependent versus global scoring.

\mathbf{h}[\cdot]

denotes the contextual encoder applied to the listed input; the other variants pass that input to the score model directly, and Glob. ICNN uses a single input-independent score. Diamonds denote mean rank, and the dashed curve reports the number of datasets on which each configuration ranks first.

Dataset

p

\mathbf{h}[x,F(x)]

\mathbf{h}[F(x)]

\mathbf{h}[x]

[x,F(x)]

F(x)

x

Global

Cov. %

Size

Cov. %

Size

Cov. %

Size

Cov. %

Size

Cov. %

Size

Cov. %

Size

Cov. %

Size

ENB

89.4_{\pm 3.5}

\underline{7.55_{\pm 0.13}}

90.0_{\pm 3.2}

9.10_{\pm 1.57}

89.6_{\pm 4.4}

7.64_{\pm 0.29}

92.5_{\pm 2.5}

\color[rgb]{0.0781,0.3125,0.707}\mathbf{7.40_{\pm 0.59}}

90.5_{\pm 5.4}

9.53_{\pm 1.02}

92.6_{\pm 3.9}

8.14_{\pm 0.81}

87.5_{\pm 4.1}

11.05_{\pm 1.86}

Bias-Corr. Temp.

90.4_{\pm 0.5}

\color[rgb]{0.0781,0.3125,0.707}\mathbf{3.88_{\pm 0.13}}

89.9_{\pm 1.4}

4.03_{\pm 0.10}

90.5_{\pm 1.0}

\underline{3.89_{\pm 0.10}}

90.5_{\pm 0.9}

\underline{3.89_{\pm 0.09}}

90.2_{\pm 1.0}

4.05_{\pm 0.05}

90.3_{\pm 0.8}

\underline{3.89_{\pm 0.12}}

90.1_{\pm 1.3}

4.15_{\pm 0.07}

Online Retail II (

\times 10^{1}

)

91.2_{\pm 1.1}

\underline{2.02_{\pm 0.29}}

90.9_{\pm 0.8}

\underline{2.02_{\pm 0.18}}

90.7_{\pm 0.8}

\color[rgb]{0.0781,0.3125,0.707}\mathbf{1.93_{\pm 0.19}}

90.9_{\pm 0.8}

2.33_{\pm 0.16}

90.9_{\pm 0.6}

2.18_{\pm 0.14}

91.0_{\pm 0.6}

2.36_{\pm 0.20}

90.6_{\pm 1.0}

2.81_{\pm 0.96}

SCPF

89.0_{\pm 3.2}

\color[rgb]{0.0781,0.3125,0.707}\mathbf{5.16_{\pm 0.68}}

91.3_{\pm 2.2}

\color[rgb]{0.0781,0.3125,0.707}\mathbf{5.16_{\pm 0.40}}

90.4_{\pm 2.4}

\underline{5.84_{\pm 0.87}}

90.8_{\pm 1.9}

7.12_{\pm 1.26}

91.1_{\pm 1.6}

5.93_{\pm 0.61}

90.9_{\pm 2.2}

7.64_{\pm 1.37}

89.5_{\pm 3.2}

7.21_{\pm 0.64}

QM8 (

\times 10^{-1}

)

90.6_{\pm 1.1}

\underline{1.10_{\pm 0.03}}

90.2_{\pm 0.7}

\color[rgb]{0.0781,0.3125,0.707}\mathbf{1.08_{\pm 0.03}}

90.3_{\pm 0.6}

1.19_{\pm 0.04}

90.3_{\pm 0.7}

1.21_{\pm 0.02}

89.9_{\pm 0.9}

1.17_{\pm 0.09}

90.2_{\pm 0.5}

1.26_{\pm 0.02}

90.0_{\pm 1.1}

1.44_{\pm 0.04}

Appendix

Table 10: Contextual-encoder ablation for

\textit{CoCoNet}{}_{\mathrm{ICNN}}

at 90% target coverage. The comparison isolates the effects of input dependence, the contextual encoder, and the information supplied to the score model.

\mathbf{h}[\cdot]

denotes the contextual encoder applied to the listed input; the columns without

\mathbf{h}

pass that input to the score model directly, and Global uses a single input-independent

\textit{CoCoNet}{}_{\mathrm{ICNN}}

score. Each configuration reports empirical test coverage in gray followed by dimension-normalized prediction-region size (lower is better); subscripts denote standard deviations over five seeds. Where a factor (

\times 10^{k}

) follows the dataset name, reported sizes are scaled by

10^{k}

. Best and second-best sizes are highlighted in blue and by underlining, respectively.

Conformal Prediction

Conformal prediction wraps predictive models with calibrated sets or intervals that provide statistical coverage guarantees.

See all 342 papers

Mar 2, 2026

Co-optimization for Adaptive Conformal Prediction

Conformal prediction (CP) provides finite-sample, distribution-free marginal coverage, but standard conformal regression intervals can be inefficient under heteroscedasticity and skewness. In particular, popular constructions such as conformalized quantile regression (CQR) often inherit a fixed notion of center and enforce equal-tailed errors, which can displace the interval away from high-density regions and produce unnecessarily wide sets. We propose Co-optimization for Adaptive Conformal Prediction (CoCP), a framework that learns prediction intervals by jointly optimizing a center

m(x)

and a radius

h(x)

.CoCP alternates between (i) learning

h(x)

via quantile regression on the folded absolute residual around the current center, and (ii) refining

m(x)

with a differentiable soft-coverage objective whose gradients concentrate near the current boundaries, effectively correcting mis-centering without estimating the full conditional density. Finite-sample marginal validity is guaranteed by split-conformal calibration with a normalized nonconformity score. Theory characterizes the population fixed point of the soft objective and shows that, under standard regularity conditions, CoCP asymptotically approaches the length-minimizing conditional interval at the target coverage level as the estimation error and smoothing vanish. Experiments on synthetic and real benchmarks demonstrate that CoCP yields consistently shorter intervals and achieves state-of-the-art conditional-coverage diagnostics.

Xiaoyi Su, Zhixin Zhou, Rui Luo

Department of Systems Engineering, City University of Hong Kong, Hong Kong SAR, China · Alpha Benito Research, Los Angeles, USA

Conformal Prediction Regression Prediction Intervals

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Sep 15, 2026

Scaling-Score Conformal Prediction for Multi-Target Regression

Multi-target regression requires a model to simultaneously predict several related outputs. Conformal prediction provides distribution-free, finite-sample marginal coverage guarantees, but extending these to joint multi-dimensional regions in a model-agnostic, sample-efficient manner remains challenging: max-aggregation ignores scale differences, copula-based methods are only asymptotically valid, rectangular methods typically split the calibration set, and quantile or density-based methods require training a specialised model beyond a plain point predictor. We propose the scaling-score conformal method, which is model-agnostic (requires only component-wise absolute residuals), uses a single calibration set, and yields four nested output types: an outer rectangle (SCO) with valid joint coverage, the exact set R

α

, a staircase (SC 2 ) over approximation of R

α

, and an inner rectangle (SCI). A single hyperparameter

γ

\in

(0, 1) controls the base-rectangle quantile level independently of

α

. We prove downward-closedness and a rectangular sandwich bound and derive a closed-form outer rectangle. Experiments on 29 realworld datasets confirm valid joint coverage; SC 2 with

γ

= 1-

α

consistently achieves competitive volume relative to baselines, with the advantage growing with output dimension d.

Sylvain Rousseau, Soundouss Messoudi

Heudiasyc · Université de Technologie de Compiègne, CNRS, Heudiasyc, Compiègne, France

Conformal Prediction Regression Prediction Intervals

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Nov 19, 2025

Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings

Conformal prediction (CP) constructs uncertainty sets for model outputs with finite-sample coverage guarantees. Yet ranking scores is straightforward only when they are scalar-valued, limiting CP to real-valued scores or ad-hoc one-dimensional reductions. Vector-valued scores arise naturally in multi-output regression and model aggregation, where each predictor in an ensemble provides its own score. Optimal transport (OT) defines vector ranks and center-outward multivariate quantile regions, though generally with asymptotic coverage guarantees. Applying a fixed transport map learned from calibration data to a new point introduces an uncontrolled approximation error. We restore finite-sample, distribution-free coverage by conformalizing vector-valued OT quantile regions. Each candidate's rank is defined by transporting the calibration scores augmented with that candidate's score, preserving the symmetry needed for validity. This appears to require a continuum of OT problems. However, we prove that the optimal assignment is piecewise constant across a fixed polyhedral partition of score space. This lets us characterize the entire prediction set in

O(n^3)

time, matching the cost of a single assignment solve. It also addresses a limitation of prediction sets: they indicate which outcomes are plausible, but not their relative likelihood. In one dimension, conformal predictive distributions (CPDs) fill this gap by producing a predictive distribution with finite-sample calibration. Extending CPDs beyond one dimension remained an open problem. We construct, to our knowledge, the first multivariate CPDs with finite-sample calibration: a center-outward predictive distribution whose derived uncertainty regions have conformal coverage. We present both conservative and exact randomized versions; the latter generalizes the classical Dempster-Hill procedure.

Eugene Ndiaye

Apple

Uncertainty Quantification Conformal Prediction Optimal Transport Set-Valued Prediction

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Abstract

Conformal Prediction

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