Co-optimization for Adaptive Conformal Prediction

Xiaoyi Su; Zhixin Zhou; Rui Luo

stat.MLMar 2, 2026

Co-optimization for Adaptive Conformal Prediction

Authors: Xiaoyi Su, Zhixin Zhou, Rui Luo

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

Abstract

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.

Figures & tables

Figure 1: From any interval to HDI via folding and boundary balancing. (a) Under skewness, equal-tailed intervals can be displaced relative to the

(1-\alpha)

-HDI and are typically longer. (b) Folding around a candidate center

m

maps

y

\lvert y-m\rvert

, producing a two-layer view. (c) The folded

(1-\alpha)

-boundary corresponds to two endpoints; translating

m

shortens the interval when the endpoint densities are imbalanced (push–pull). (d) At equilibrium the endpoint densities match, recovering the HDI.

Figure 2: Gradient dynamics of the soft-coverage objective. (a) The soft window serves as a differentiable surrogate for the interval indicator. The underlying data density (dashed line) is imbalanced at the two boundaries. (b) The gradient of the soft window with respect to the center

m

acts as a signed sampling kernel, being negative at the left boundary and positive at the right. The resulting local gradient (the product of the conditional density and the derivative kernel) is asymmetric under skewness; the higher-density boundary generates a dominant directional signal that shifts the center

m

toward the concentration of probability mass.

Figure 3: Overview of the CoCP framework. The pipeline begins by parameterizing the prediction interval into a center

m(x)

and a radius

h(x)

, which are then iteratively refined through an alternating co-optimization process during K-fold cross-fitting. The final adaptive interval is constructed by aggregating the learned components and applying a split-conformal calibration to ensure marginal validity.

Dataset	Method	Coverage	Length ( $\downarrow$ )	ConMAE ( $\downarrow$ )
Normal	Oracle	0.9000 (0.0000)	1.5922 (0.0101)	0.0000 (0.0000)
	Split	0.8995 (0.0050)	1.8735 (0.0404)	0.1119 (0.0030)
	CPL	0.9004 (0.0103)	1.5055 (0.0456)	0.0768 (0.0071)
	Naive CC	0.9001 (0.0033)	1.5902 (0.0159)	0.0093 (0.0021)
	CQR	0.9016 (0.0041)	1.5989 (0.0165)	0.0095 (0.0026)
	CPCP	0.9026 (0.0120)	1.6050 (0.0335)	0.0180 (0.0051)

Table 1: Experimental results on synthetic datasets (mean

\pm

std over 10 splits,

\alpha=0.1

). Lower Length and ConMAE are better. The Oracle row reports the true

(1-\alpha)

-HDI for reference.

Figure 4: Prediction intervals on the synthetic Normal dataset. This visualization compares nine conformal methods under symmetric, heteroscedastic Gaussian noise. The gray area denotes the Oracle $(1-\alpha)$ -HDI. Most adaptive methods, including CoCP, successfully capture the symmetric uncertainty, while the Split baseline (top-left) produces overly conservative, non-adaptive intervals.

Figure 5: Prediction intervals on the synthetic LogNormal dataset. Under skewed conditional noise, the discrepancy between center-invariant methods (e.g., CQR) and the Oracle HDI (gray) becomes prominent. CoCP (bottom-right) demonstrates its ability to shift the interval center and scale the radius simultaneously, resulting in a geometry that closely aligns with the Oracle’s bounds.

Figure 6: Prediction intervals on the synthetic Exponential dataset. This case represents extreme skewness. While methods like RCP remain centered around the conditional mean, CoCP effectively utilizes the boundary-balancing mechanism to push the interval toward the high-density region near the lower bound, achieving significantly shorter lengths while maintaining valid coverage.

Dataset	Description	Samples ( $n$ )	Dim ( $d$ )	Ref.
bike	Hourly bike sharing demand	10,886	18	[ 13 ]
bio	Protein tertiary structure properties	45,730	9	[ 34 ]
blog	Blog feedback volume prediction	52,397	280	[ 5 ]
facebook-1	Facebook comment volume (Variant 1)	40,948	53	[ 40 ]
facebook-2	Facebook comment volume (Variant 2)	81,311	53	[ 40 ]
homes	King County house sales prices	21,613	19	[ 21 ]

Table 2: Overview of the real-world datasets used in our study.

n

and

d

represent the number of samples and features, respectively.

Dataset	Method	Coverage	Length ( $\downarrow$ )	MSCE ( $\downarrow$ )	WSC ( $\uparrow$ )	$\ell_{1}$ -ERT ( $\downarrow$ )	$\ell_{2}$ -ERT ( $\downarrow$ )
bike	Split	0.8980 (0.0111)	0.7700 (0.0360)	0.0041 (0.0016)	0.6974 (0.0259)	0.0663 (0.0063)	0.0080 (0.0022)
	CPL	0.9146 (0.0100)	0.7938 (0.0373)	0.0021 (0.0015)	0.7800 (0.0515)	0.0369 (0.0087)	0.0022 (0.0023)
	CQR	0.8993 (0.0040)	0.7856 (0.1255)	0.0018 (0.0006)	0.7573 (0.0180)	0.0350 (0.0127)	0.0020 (0.0015)
	CPCP	0.9030 (0.0162)	0.7475 (0.0515)	0.0010 (0.0003)	0.7933 (0.0251)	0.0206 (0.0090)	0.0003 (0.0007)
	RCP	0.9014 (0.0096)	0.7497 (0.0474)	0.0013 (0.0006)	0.7787 (0.0158)	0.0250 (0.0125)	0.0008 (0.0009)
	C-HDR	0.8994 (0.0091)	0.7610 (0.0325)	0.0015 (0.0005)	0.7752 (0.0114)	0.0258 (0.0136)	0.0013 (0.0014)

Table 3: Experimental results on real-world datasets (mean

\pm

std over 10 splits,

\alpha=0.1

). We report coverage and average length, plus conditional-coverage diagnostics (MSCE, WSC at

\delta=0.1

, and

\ell_{1}/\ell_{2}

-ERT). Best values per dataset/metric (excluding Coverage) are highlighted in bold.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Convergence dynamics of CoCP over alternating iterations (

T

). The plots visualize the trade-off between interval length (blue, left axis) and conditional coverage error (ConMAE; red, right axis). While symmetric distributions (Normal) are near-optimal at

T=0

, skewed distributions require alternating updates to optimize this trade-off, achieving either significantly lower ConMAE (LogNormal) or drastically reduced interval length (Exponential).

Dataset	$T$	Coverage	Length ( $\downarrow$ )	ConMAE ( $\downarrow$ )	Train Time (s)
Normal	0	0.9013 (0.0039)	1.5966 (0.0187)	0.0077 (0.0021)	14.71 (2.57)
	1	0.9014 (0.0043)	1.5982 (0.0206)	0.0077 (0.0022)	29.37 (3.52)
	2	0.9005 (0.0044)	1.5973 (0.0203)	0.0075 (0.0021)	42.64 (4.18)
	3	0.9007 (0.0041)	1.5982 (0.0170)	0.0072 (0.0015)	56.02 (5.34)
	4	0.9008 (0.0037)	1.5994 (0.0180)	0.0067 (0.0017)	69.49 (6.33)
	5	0.9003 (0.0041)	1.5971 (0.0187)	0.0070 (0.0014)	82.47 (7.57)

Appendix

Table 4: Ablation study on the number of alternating iterations (

T

) on synthetic datasets, including

T=0

. Results are averaged over 10 splits (mean

\pm

std). Training time is reported in seconds.

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Department of Information Management, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Da’an Dist., Taipei, 106216, Taiwan.

Conformal Prediction Temporal Distribution Shift Regression Prediction Intervals

Open paper AskPDF arXiv

Online conformal prediction must balance fast adaptation to distribution shift against stable coverage: feedback-driven methods react quickly but become volatile, while strongly discounted Bayesian methods lag and inflate intervals at tight coverage. We introduce State-Adaptive Bayesian Conformal Prediction (SA-BCP), which forms the predictive quantile as a gated convex combination of long-term temporal inertia and local spatial evidence from a kernel density estimate, controlled by a single interpretable evidence threshold

K

. We establish three results: (i) asymptotic marginal validity of the resulting intervals up to a gate-controlled bias that vanishes as spatial evidence accumulates (exact under recurrent states); (ii) a closed-form expression for the MSE-optimal threshold,

K^*_{\mathrm{MSE}}=α(1-α)/M^{\mathcal{T}}

, trading the coverage-indicator (Bernoulli) variance against the temporal structural bias

M^{\mathcal{T}}

; and (iii) a rolling-origin procedure for selecting

K

online -- consistent under stationarity, with

O(\sqrt{T\log N})

regret against the best fixed

K

and, for a segmented variant, a sublinear dynamic-regret bound under sublinearly many (

B_T=o(T)

) threshold shifts. Across four financial-volatility and weather datasets, three target coverage levels, and eight baselines, SA-BCP attains at-or-above-nominal coverage in most settings while producing substantially sharper intervals -- up to roughly

3\times

lower Winkler score than discounted Bayesian CP at the tightest coverage -- and a coverage-matched audit confirms these efficiency gains are not an artifact of under-coverage. We disclose our principal limitation: a volatility-specialized CF-GARCH competitor remains more efficient on its home volatility-base series, though it does not transfer across domains.

Abstract

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