CoCoNet: Convex Conformal Network for Multi-dimensional Uncertainty Regions

_{The ICNN conformity score (left of each pair) and its 90% prediction region (right) during training on the four synthetic shapes. The dashed outline is the true smallest 90% region; the curves below track IoU and V/V* over the epochs.}

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Overview

This repository includes codes and data for CoCoNet: Convex Conformal Network for Multi-Dimensional Uncertainty Regions. A BibTeX entry for citation is provided at the end of this page.

CoCoNet learns a conformity score that is convex in the residual r = y − F(x). An encoder maps the input x and the point prediction F(x) to a representation that conditions the score, so the shape of the region can change with x. A calibration set gives the conformal threshold, and the sublevel set of the score at that threshold is the prediction region: convex, input-adaptive, and covering y with the target probability under exchangeability.

Two score families are provided:

• CoCoNet_ICNN: an input-convex neural network. It can represent any convex region, including rounded polygons and asymmetric shapes.
• CoCoNet_Spline: convex piecewise-linear scores along learned rotated axes. Taking their maximum (ℓ_∞, spline) gives a rotated box; summing them (ℓ₁, spline_l1) gives an asymmetric polytope. Both have a closed-form volume.

Every method receives the residuals of the same frozen MLP point predictor per (dataset, seed), is calibrated by split conformal prediction on D_cal, and is scored on D_test by joint coverage and size (mean volume)^1/d.

What is included

🔷 Synthetic shapes	ellipse, rectangle, triangle and trapezoid with a known optimal 90% region
📊 Multivariate benchmark	30 datasets, response dimension d = 2 to 16
📈 Univariate benchmark	6 datasets
⚖️ Extended comparison	MVCS with free exponents, multi-norm MVCS and PBS

All at 90% target coverage over five seeds. The repository ships code and raw data only: no results, fitted models or tuned settings are stored; every number comes from running the experiments.

Methods

Method	--method	Prediction region
CoCoNetICNN	icnn	sublevel set of an input-convex neural network score
CoCoNetSpline-ℓ∞	spline	rotated box from convex piecewise-linear scores
CoCoNetSpline-ℓ1	spline_l1	asymmetric polytope: the same scores summed (multivariate only; it reuses the spline's tuning)
Abs. Res.	absolute_residual	axis-aligned box, s = maxj |rj| on standardized residuals
Emp. Cop.	empirical_copula	axis-aligned box from the empirical copula of normalized scores
G-Ellip.	global_ellipsoid	one Mahalanobis ellipsoid
L-Ellip.	local_ellipsoid	k-nearest-neighbour local covariance ellipsoid
MVCS	mvcs	convex (q ≥ 1) minimum-volume covering set
LW-SC	locally_weighted	locally weighted split-conformal interval (univariate only)

Extended comparison (multivariate only)

Method	--method	Prediction region
MVCS, free q	mvcs_free	the same set with a free exponent (q ≥ 0.1, not convex when q < 1)
MVCS multi-norm, q ≥ 1	mvcs_multi	one global set: after a rotation, a scaled qj-norm ball in each of 4 regions (signs of the first two rotated coordinates), every qj ≥ 1
MVCS multi-norm, free q	mvcs_multi_free	the same with free exponents
PBS	pbs	pullback of a Euclidean ball through a conditional neural optimal-transport map

The two multi-norm variants run the MVCS authors' own code. Fetch it once with bash scripts/get_mvcs.sh, which clones their public repository at a fixed commit into third_party/.

Installation

pip install -r requirements.txt
bash scripts/get_mvcs.sh        # only for mvcs_multi / mvcs_multi_free

Python 3.13. The reported numbers were produced with torch 2.7.1+cu126, numpy 2.3.1, pandas 2.3.1, scipy 1.16.0 and scikit-learn 1.7.0. Raw data live in data/raw/; UCI and OpenML files that are missing are downloaded on first use.

Quick start

Each command runs one method on one dataset or shape, for every seed or for the seeds you give, and writes one folder per seed with its row.json.

# real data: results/<dataset>/<method>/seed<S>/row.json
python run.py --dataset enb --method icnn                     # seeds 0-4
python run.py --dataset enb --method spline --seeds 0 1
python run.py --dataset enb --method spline_l1 --seeds 0 1
python run.py --dataset concrete --method locally_weighted    # univariate baseline

# synthetic shapes: results_synthetic/<shape>/<method>/seed<S>/row.json
python synthetic.py --shape triangle --method icnn            # seeds 42-46
python synthetic.py --shape rectangle --method mvcs

Each row.json holds the test coverage, the mean volume and its size (mean V)^1/d, the conformal threshold and the data hashes, plus the selected loss weights and checkpoint for the learned scores. --help on either script lists every flag.

Experiments

🔷 Synthetic shapes

Four 2-D residual laws whose smallest 90% region is known exactly (coconet/data/synthetic.py). The residual is uniform in a convex body B with probability 0.9 and uniform in the shell 1.6 B \ B otherwise, then rotated and scaled. The core is 14 times denser than the shell, so B itself is the unique smallest region with 90% coverage; its area is V*. The input x is uniform on [−1, 1]², and every method is given the exact mean as its point prediction.

Shapes	ellipse, rectangle, triangle, trapezoid
Methods	absolute_residual, empirical_copula, global_ellipsoid, local_ellipsoid, mvcs, spline, spline_l1, icnn
Splits per seed	2,500 train · 2,000 validation · 8,000 calibration · 1,000 test
Seeds	42–46
Metrics	test coverage · V/V* (mean region area over V*) · IoU with the true region

IoU compares the region at an input with the true region on a 500 × 500 grid, averaged over x = (0, 0) and 24 random inputs. The settings that differ from the real-data runs are in the synthetic block of configs/experiment.yaml: γ_cov ∈ {0, 0.1}, and a fixed ICNN (learning rate 0.002, batch size 32, encoder [64, 32]). The spline is tuned as on real data, and spline_l1 reuses its settings.

📓 Notebook

synthetic.ipynb runs the synthetic experiment from start to finish and is shipped with its outputs, so the results and the figure can be read without running anything.

Step	What it does
Settings	shapes, methods and seeds; by default one seed per shape (the figure seeds), or all five with SEEDS = [42, 43, 44, 45, 46]
1. Run	every method on every shape through synthetic.py; finished runs are not repeated and their saved result is shown
2. Results	test coverage, V/V* and IoU of every run
3. Figure	the calibrated 90% regions of CoCoNetICNN, MVCS, CoCoNetSpline-ℓ∞, CoCoNetSpline-ℓ1 and L-Ellip. against the true region, saved as results_synthetic/synthetic_regions.pdf

jupyter notebook synthetic.ipynb

The figure alone is python plot_synthetic.py. It reads the region.npz file that each run saves (the method's score at x = (0, 0) on the IoU grid).

📊 Real data

python run.py --dataset <key> --method <method>

Multivariate datasets take absolute_residual, empirical_copula, global_ellipsoid, local_ellipsoid, mvcs, spline, spline_l1, icnn and the four extended methods. Univariate datasets take absolute_residual, locally_weighted, mvcs, spline and icnn.

End to end. Each run fits the MLP point predictor of the (dataset, seed) and, for icnn and spline, first tunes their settings (spline_l1 reuses the spline's). The tuned choice is saved in <output-dir>/tuning/ with a hash of the data split and residuals, and later runs of the same (dataset, seed) reuse it. Tuning is the expensive step: a few extra trainings per (dataset, seed). To share point predictors across runs, save them once with python fit_point_predictor.py --dataset enb --out my_predictors and pass --point-predictor-dir my_predictors.

Volumes. On real data, every multivariate region is measured with the same Monte Carlo estimator the ICNN uses (per test point, groups of 30 points, 2¹⁵ directions, doubling until the size changes by less than 0.1%), as in the reported results. --volume closed_form reports each baseline's closed-form volume instead. PBS always reports its own change-of-variables volume, and univariate intervals are exact.

Reproducibility

The CoCoNet_ICNN results were produced on a Quadro RTX 6000.

Protocol

All settings live in configs/experiment.yaml; the datasets are defined in configs/datasets.yaml.

Stage	Setting
Data split	Dpred / Dscore / Dcal / Dtest = 25 / 30 / 25 / 20% from one seeded permutation; large datasets are first capped to a seeded random sample
Preprocessing	features standardized and one-hot encoded, targets standardized, all with statistics of Dpred; volumes are reported on the original target scale
Point predictor	MLP [64, 32] fitted on Dpred and frozen; every method uses its residuals
Objective	(mean V)1/d + λmiss · mean[(S − 1)+] + λcov · (mean σ(10 (1 − S)) − 0.9)2, threshold fixed at 1
Optimisation	Adam, gradient clipping at 5, patience 80; at most 400 epochs (ICNN) or 300 (both splines)
Loss weights	λ = γ × reference scale of the initialised model; γmiss ∈ {0.01, 0.03, 1, 10} × γcov ∈ {0, 0.1, 1}; the coverage-admissible candidate with the smallest validation size wins
Checkpoint	every 5 epochs, by calibrated validation coverage and size
Tuning	per (dataset, seed), at the neutral weights: the ICNN encoder width {[64, 32], [256, 128]}; the spline knots {2, 4, 6, 8}, batch size {32, 64, 128} and encoder width
Calibration	split conformal on Dcal, 90% target coverage
ICNN volume	whitened Monte Carlo estimate at M = 215 and 2M = 216 directions in one pass, 216 reported; converged when doubling changes the size by less than 0.1%; at d = 1 the exact interval length

Baseline	Setting
MVCS	the authors' hyperparameter grid (60 candidates, selected by validation objective); q ≥ 1 (mvcs) or q = max(|q|, 0.1) (mvcs_free)
Multi-norm MVCS	the authors' code and defaults with two splits; the convex variant projects every exponent onto q ≥ 1 after each step
PBS	the five network configurations in the config; the one with the smallest validation coverage error of its plug-in sets is kept
Ellipsoids, copula, LW-SC	the settings listed in the config

Datasets

n is the number of rows in the source data after rows with a missing target are dropped, d_x the number of input features after preprocessing (constant columns dropped, categorical columns one-hot encoded; seed 0), d the response dimension, and "rows used" the number of rows the experiments run on: a seeded random subsample when the source has more rows than max_rows in configs/datasets.yaml, otherwise all of them. Missing features are median-imputed from D_pred. The encoder is fitted on D_pred, so the number of one-hot columns can change with the seed: 36 to 42 for Online Retail II and 60 to 61 for Bike Sharing.

Multivariate: 30 datasets, d = 2 to 16

Key	Label	Source	n	d_x	d	Rows used
argoverse2_2d	Argoverse2-2D	Argoverse 2 motion forecasting: displacement of the focal agent 6 s ahead from summaries of its 5 s history	24,988	28	2	20,000
argoverse2_motion	Argoverse2 Motion	same source, 3 s horizon	24,988	28	2	20,000
bias_corr_temp	Bias-Corr. Temp.	UCI 514: next-day (Tmax, Tmin) from present-day readings and forecasts	7,723	46	2	all
enb	ENB	UCI Energy Efficiency (OpenML 41478): heating and cooling load	768	8	2	all
geographical_music	Geographical Music	UCI 315: (latitude, longitude) of a track from 68 audio features	1,059	68	2	all
casp	CASP	processed CASP arrays, two targets	45,730	8	2	20,000
online_retail_ii	Online Retail II	UCI 502: (Quantity, Price) of an invoice line from country and date	1,067,371	42	2	20,000
qm9_2d	QM9-2D	QM9: first 2 of 12 molecular properties (mu, alpha)	130,831	22	2	20,000
household_power	Household Power	UCI 235: three sub-metering readings from four whole-house quantities	2,049,280	4	3	20,000
scpf	SCPF	See Click Predict Fix (OpenML 41487): views, votes, comments of an issue report	1,137	23	3	all
argoverse2_4d	Argoverse2-4D	displacements at 3 s and 6 s	24,988	28	4	20,000
qm8	QM8	QM8 spectra: the four RI-CC2 quantities from composition and Coulomb-matrix eigenvalues	21,786	14	4	20,000
qm9_4d	QM9-4D	QM9: first 4 properties (+ homo, lumo)	130,831	22	4	20,000
sgemm	SGEMM	UCI 440: four repeated run times of a GPU kernel configuration	241,600	18	4	20,000
beijing_aq_6d	Beijing AQ-6D	UCI 501: six pollutants of one station-hour from its meteorology	383,980	37	6	20,000
communities_crime	Communities Crime	UCI 211: six per-capita crime rates of a community	1,990	124	6	all
atp1d	ATP1D	Mulan ATP1D (OpenML 41475): next-day minimum price of six flight/fare combinations from price-quote history	337	370	6	all
qm9_6d	QM9-6D	QM9: first 6 properties (+ r2, zpve)	130,831	22	6	20,000
sarcos	SARCOS	SARCOS inverse dynamics: 7 joint torques from the 21-D arm state	48,933	21	7	20,000
argoverse2_8d	Argoverse2-8D	displacements at 1.5, 3, 4.5 and 6 s	24,988	28	8	20,000
qm9_8d	QM9-8D	QM9: first 8 properties (+ Cv, U0)	130,831	22	8	20,000
beijing_aq_12d	Beijing AQ-12D	UCI 501: PM2.5 of all 12 stations at one hour from every station's meteorology (hours with all 12 readings)	28,663	256	12	20,000
knee_contact_force	Knee Contact Force	Zenodo 7253458: 12 knee contact-force peaks of a walking trial	2,786	12	12	all
wq	WQ	Mulan water quality (OpenML 41491): 14 species and indicator concentrations from 16 physico-chemical measurements	1,060	16	14	all
qm7b	QM7b	QM7b: the 14 molecular properties from Coulomb-matrix eigenvalues and element counts	7,211	29	14	all
oes10	OES10	OES 2010: 16 occupation employment counts per metropolitan area	403	298	16	all
oes97	OES97	OES 1997	334	263	16	all
bdg2_fox	BDG2-Fox	Building Data Genome 2, site Fox: hourly electricity of 16 buildings from site weather and calendar features	17,537	12	16	all
bdg2_eagle	BDG2-Eagle	same, site Eagle	17,105	12	16	all
bdg2_lamb	BDG2-Lamb	same, site Lamb	17,544	10	16	all

The Argoverse 2 and QM9 tasks of different d are cut from one table each; the columns that are targets of another variant are removed from the features. QM9 properties are taken in the order mu, alpha, homo, lumo, r2, zpve, Cv, U0 (then gap, U, H, G, unused).

Univariate: 6 datasets

Key	Label	Source	n	d_x	Rows used
beijing_pm25	Beijing PM2.5	UCI 381: hourly PM2.5 from concurrent meteorology	41,757	14	10,000
bike_sharing	Bike Sharing	UCI 275: hourly rental count	17,379	60	10,000
ccpp	CCPP	UCI 294: net power output of a combined-cycle plant	9,568	4	all
concrete	Concrete	UCI concrete compressive strength	1,030	8	all
kin8nm	Kin8nm	OpenML 189: forward kinematics of an 8-link arm	8,192	8	all
protein	Protein	UCI 265 (CASP): RMSD of a protein structure	45,730	9	10,000

Repository layout

├── run.py                   one method on one real dataset  → results/<dataset>/<method>/seed<S>/row.json
├── synthetic.py             one method on one synthetic shape → results_synthetic/<shape>/<method>/seed<S>/
├── synthetic.ipynb          the synthetic experiment step by step: runs, per-run results, figure
├── plot_synthetic.py        the figure of synthetic regions, from the saved runs
├── fit_point_predictor.py   fit and save the MLP point predictor of a dataset
├── configs/
│   ├── experiment.yaml      every setting of the experiments
│   └── datasets.yaml        the 36 datasets: loader, target columns, row cap
├── coconet/
│   ├── pipeline.py          data preparation, one run_<method> per method, evaluation
│   ├── tuning.py            per-(dataset, seed) tuning of the learned scores
│   ├── models/              icnn.py, spline.py (the scores; spline.py has the ℓ∞ and ℓ1 aggregations), mlp.py, volume_estimator.py
│   ├── training/            loss, loss-weight grid search, training loop, checkpoint selection
│   ├── conformal/           calibration, ellipsoids, empirical copula, MVCS (single and multi-norm), PBS
│   └── data/                loaders, preprocessing, splits, synthetic shapes
├── scripts/get_mvcs.sh      fetches the MVCS authors' code used by the multi-norm baselines
└── data/raw/                raw dataset files

Citation

If you use this code, please cite:

@article{moradi2026coconet,
  title={CoCoNet: Convex Conformal Network for Multi-Dimensional Uncertainty Regions},
  author={Moradi, Alireza and Bruys, Thomas and Van Hentenryck, Pascal and Zandehshahvar, Reza},
  journal={arXiv preprint arXiv:2610.24246},
  year={2026},
  url={https://arxiv.org/abs/2610.24246}
}