CLAPS: Aleatoric-Epistemic Scaling via Last-Layer Laplace for Conformal Regression

Dongseok Kim; Hyoungsun Choi; Mohamed Jismy Aashik Rasool; Gisung Oh

cs.LGDec 1, 2025

CLAPS: Aleatoric-Epistemic Scaling via Last-Layer Laplace for Conformal Regression

Authors: Dongseok Kim, Hyoungsun Choi, Mohamed Jismy Aashik Rasool, Gisung Oh

Organizations: Department of Computer Engineering, Gachon University

Abstract

Conformal regression provides finite-sample marginal coverage, but interval efficiency depends on how uncertainty is allocated across inputs. We propose Conformal Laplace-Aware Predictive Scaling (CLAPS), which combines a learned heteroscedastic scale with last-layer Laplace uncertainty. The learned variance also weights the training features used to construct the posterior, yielding a noise-weighted support correction in the learned representation. The correction contracts as weighted feature information increases and becomes most influential under localized support imbalance. Experiments show near-nominal marginal coverage and modest efficiency improvements over the matched aleatoric baseline on real regression benchmarks.

Figures & tables

W=\mathrm{diag}\left(\frac{1}{h^{2}(x_{1})},\ldots,\frac{1}{h^{2}(x_{n_{\mathrm{tr}}})}\right).

Algorithm 1 Conformal Laplace-Aware Predictive Scaling (CLAPS)

Method	Marginal Cov.	Sparse Cov.	Sparse/Dense Width	Interval Score
Split CP	$0.8986\pm 0.0012$	$0.8402\pm 0.0053$	$1.0000\pm 0.0000$	$2.5451\pm 0.0415$
CV+	$0.9074\pm 0.0012$	$0.9701\pm 0.0014$	$1.0160\pm 0.0010$	$1.9008\pm 0.0068$
LACP	$0.8998\pm 0.0012$	$0.9016\pm 0.0101$	$4.4414\pm 0.1890$	$2.6777\pm 0.0536$
CQR	$0.8984\pm 0.0014$	$0.8162\pm 0.0053$	$1.8488\pm 0.0520$	$2.8165\pm 0.0348$
UACQR-S	$0.8981\pm 0.0013$	$0.8246\pm 0.0048$	$1.8924\pm 0.0532$	$2.7876\pm 0.0313$
EPICSCORE-MDN	$0.8995\pm 0.0016$	$0.9057\pm 0.0032$	$3.9642\pm 0.1020$	$2.5340\pm 0.0236$

Table 1: High-dimensional weak-support mechanism study. Sparse/Dense Width is the ratio of average interval width in the sparse low-noise region to that in the dense low-noise region.

Metric	CLAPS $-$ LACP	95% CI
Marginal Coverage	$-0.0004$	$[-0.0012,\phantom{-}0.0004]$
Sparse Coverage	$-0.0026$	$[-0.0040,-0.0012]$
Sparse/Dense Width Ratio	$-0.0451$	$[-0.0561,-0.0345]$
Interval Score	$-0.0357$	$[-0.0424,-0.0292]$

Table 2: Paired differences between CLAPS and LACP for the primary metrics.

Method	Marginal Cov.	Sparse/Dense Width		Interval Score
Method	Marginal Cov.	Low-noise	High-noise	Interval Score
LACP	$0.9012\pm 0.0011$	$6.0382\pm 0.2147$	$1.9036\pm 0.0559$	$3.6657\pm 0.1137$
Unweighted Laplace	$0.9014\pm 0.0012$	$4.9728\pm 0.1622$	$1.8959\pm 0.0547$	$3.5967\pm 0.1142$
LWCP	$0.9012\pm 0.0013$	$1.0373\pm 0.0017$	$1.0352\pm 0.0016$	$3.9800\pm 0.0846$
CLAPS	$0.9010\pm 0.0011$	$5.9649\pm 0.2094$	$1.9090\pm 0.0559$	$3.6398\pm 0.1121$

Table 3: Isolation of noise-weighted feature geometry. The low- and high-noise columns report the ratio of average interval width in the sparse region to that in the dense region at the same observation-noise level.

Figure 1: Regime study for the last-layer epistemic correction. Panels (a) and (c) vary the total training-set size at fixed sparse-region retention. Panels (b) and (d) vary sparse-region retention at fixed total training size. The top row reports the sparse-to-dense interval-width ratio and CLAPS epistemic fraction; the bottom row reports sparse-region interval score. Error bars show

95\%

bootstrap confidence intervals.

Method	Coverage	Comparison with LACP		Weak-Q5 Score
Method	Coverage	$\Delta$ Width	$\Delta$ Score	Weak-Q5 Score
Split CP	$0.9017\pm 0.0021$	$16.80\pm 2.91\%$	$25.53\pm 3.04\%$	$18.34\pm 3.04\%$
CV+	$0.9164\pm 0.0021$	$18.02\pm 2.76\%$	$18.21\pm 2.29\%$	$\phantom{-}8.79\pm 2.47\%$
LACP	$0.8998\pm 0.0021$	$\phantom{-}0.00\pm 0.00\%$	$\phantom{-}0.00\pm 0.00\%$	$\phantom{-}0.00\pm 0.00\%$
CQR	$0.9000\pm 0.0022$	$40.24\pm 5.84\%$	$31.94\pm 4.79\%$	$30.43\pm 4.26\%$
UACQR-S	$0.8989\pm 0.0023$	$35.54\pm 4.97\%$	$37.27\pm 5.92\%$	$39.65\pm 6.31\%$

Table 4: Real-data benchmark over eight datasets. Width and interval-score changes are relative to LACP. Weak-Q5 Score is the relative interval-score change in the weakest learned-feature support quintile.

Table 5: Paired comparisons between CLAPS and LACP on the real-data benchmarks. All differences are computed as CLAPS minus LACP.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Comparison method	Metric	Mean difference	95% CI
LACP	Marginal Coverage	$-0.0002$	$[-0.0007,\phantom{-}0.0003]$
	Low-noise Sparse/Dense Width	$-0.0733$	$[-0.0907,-0.0571]$
	High-noise Sparse/Dense Width	$\phantom{-}0.0054$	$[\phantom{-}0.0037,\phantom{-}0.0074]$
	Interval Score	$-0.0260$	$[-0.0317,-0.0203]$
Unweighted Laplace	Marginal Coverage	$-0.0004$	$[-0.0019,\phantom{-}0.0010]$
	Low-noise Sparse/Dense Width	$\phantom{-}0.9921$	$[\phantom{-}0.8763,\phantom{-}1.1107]$

Appendix

Table 6: Paired comparisons for Experiment 2. All differences are computed as CLAPS minus the comparison method.

Training size	Method	Marginal Cov.	Sparse/Dense Width	Sparse Interval Score	CLAPS Epi. Frac.
$100$	LACP	$0.9021\pm 0.0014$	$1.0725\pm 0.0451$	$4.4936\pm 0.2258$	—
	Unweighted Laplace	$0.9013\pm 0.0014$	$1.0629\pm 0.0383$	$4.3667\pm 0.1988$	—
	LWCP	$0.9009\pm 0.0011$	$1.2229\pm 0.0156$	$3.9478\pm 0.1293$	—
	CLAPS	$0.9021\pm 0.0014$	$1.0752\pm 0.0440$	$4.4629\pm 0.2194$	$0.0301\pm 0.0090$
$250$	LACP	$0.8984\pm 0.0014$	$1.3236\pm 0.0914$	$5.3503\pm 0.2354$	—
	Unweighted Laplace	$0.8984\pm 0.0015$	$1.2862\pm 0.0778$	$5.2994\pm 0.2291$	—

Appendix

Table 7: Full results for the training-size sweep in Experiment 3. Values are means and standard errors over random seeds.

Sparse retention	Method	Marginal Cov.	Sparse/Dense Width	Sparse Interval Score	CLAPS Epi. Frac.
$0.01$	LACP	$0.8979\pm 0.0014$	$3.4513\pm 0.1974$	$6.1947\pm 0.2428$	—
	Unweighted Laplace	$0.8980\pm 0.0013$	$3.2555\pm 0.1799$	$6.1989\pm 0.2401$	—
	LWCP	$0.8986\pm 0.0012$	$1.1228\pm 0.0074$	$4.8564\pm 0.1338$	—
	CLAPS	$0.8976\pm 0.0013$	$3.4752\pm 0.1998$	$6.1272\pm 0.2418$	$0.0463\pm 0.0058$
$0.025$	LACP	$0.8991\pm 0.0015$	$3.6762\pm 0.2666$	$5.4421\pm 0.2664$	—
	Unweighted Laplace	$0.8988\pm 0.0015$	$3.4596\pm 0.2435$	$5.4045\pm 0.2603$	—

Appendix

Table 8: Full results for the local-support sweep in Experiment 3. Values are means and standard errors over random seeds.

Quintile	$\Delta$ Coverage	$\Delta$ Width	$\Delta$ Score
Q1	$-0.0040\pm 0.0011$	$-2.21\pm 0.30\%$	$-1.02\pm 0.26\%$
Q2	$-0.0041\pm 0.0017$	$-1.89\pm 0.27\%$	$-0.95\pm 0.21\%$
Q3	$-0.0024\pm 0.0012$	$-1.63\pm 0.26\%$	$-1.05\pm 0.24\%$
Q4	$-0.0034\pm 0.0015$	$-0.99\pm 0.25\%$	$-0.57\pm 0.23\%$
Q5	$\phantom{-}0.0065\pm 0.0020$	$\phantom{-}0.73\pm 0.34\%$	$-1.23\pm 0.47\%$

Appendix

Table 9: Raw-input support-stratified comparison between CLAPS and LACP in Experiment 4. All differences are computed as CLAPS minus LACP and reported as means and standard errors. Q1 and Q5 denote the strongest and weakest support, respectively.

$\lambda$	Coverage	Overall		Weak-Q5		Epistemic fraction
$\lambda$	Coverage	$\Delta$ Width	$\Delta$ Score	$\Delta$ Width	$\Delta$ Score	Overall	Weak-Q5
$10^{-5}$	$0.9014\pm 0.0023$	$-2.10\pm 0.41\%$	$-2.20\pm 0.32\%$	$1.45\pm 0.66\%$	$-3.68\pm 0.56\%$	$9.93\pm 0.83\%$	$16.07\pm 0.87\%$
$10^{-4}$	$0.9015\pm 0.0023$	$-2.11\pm 0.41\%$	$-2.19\pm 0.32\%$	$1.41\pm 0.66\%$	$-3.67\pm 0.56\%$	$9.87\pm 0.83\%$	$15.96\pm 0.88\%$
$10^{-3}$	$0.9017\pm 0.0023$	$-2.09\pm 0.41\%$	$-2.15\pm 0.32\%$	$1.29\pm 0.65\%$	$-3.64\pm 0.56\%$	$9.67\pm 0.82\%$	$15.60\pm 0.87\%$
$10^{-2}$	$0.9016\pm 0.0023$	$-2.03\pm 0.41\%$	$-2.01\pm 0.31\%$	$0.89\pm 0.62\%$	$-3.46\pm 0.52\%$	$8.95\pm 0.78\%$	$14.34\pm 0.82\%$
$10^{-1}$	$0.9021\pm 0.0023$	$-1.85\pm 0.37\%$	$-1.70\pm 0.27\%$	$0.20\pm 0.51\%$	$-2.92\pm 0.43\%$	$7.47\pm 0.69\%$	$11.78\pm 0.71\%$

Appendix

Table 10: Prior-precision selection and sensitivity on the eight real-data benchmarks. Panel (a) reports means and standard errors over dataset–seed runs. Changes in width and interval score are relative to matched LACP. Panel (b) reports hierarchical paired bootstrap comparisons between training-only selected CLAPS and LACP.

Feature dimension	Coverage	Overall		Weak-Q5			Epistemic fraction
Feature dimension	Coverage	$\Delta$ Width	$\Delta$ Score	$\Delta$ Coverage	$\Delta$ Width	$\Delta$ Score	Overall	Weak-Q5
$16$	$0.8992\pm 0.0029$	$-0.36\pm 0.12\%$	$-0.33\pm 0.10\%$	$0.0019\pm 0.0010$	$0.26\pm 0.15\%$	$-0.90\pm 0.25\%$	$1.64\pm 0.14\%$	$3.33\pm 0.26\%$
$32$	$0.9000\pm 0.0029$	$-1.09\pm 0.22\%$	$-0.60\pm 0.16\%$	$0.0010\pm 0.0011$	$-0.12\pm 0.25\%$	$-1.39\pm 0.34\%$	$2.59\pm 0.22\%$	$5.35\pm 0.42\%$
$64$	$0.8996\pm 0.0025$	$-0.63\pm 0.22\%$	$-0.82\pm 0.23\%$	$0.0091\pm 0.0020$	$1.11\pm 0.44\%$	$-1.92\pm 0.53\%$	$3.96\pm 0.37\%$	$8.09\pm 0.71\%$
$128$	$0.9016\pm 0.0023$	$-1.18\pm 0.26\%$	$-1.15\pm 0.28\%$	$0.0125\pm 0.0029$	$1.51\pm 0.55\%$	$-2.97\pm 0.71\%$	$4.78\pm 0.43\%$	$10.36\pm 0.82\%$

Appendix

Table 11: Sensitivity to the learned representation dimension. Panel (a) reports predictive performance and posterior contribution. Changes in width and interval score are relative to matched LACP. Panel (b) reports model and last-layer posterior costs.

Variance floor	Coverage	Overall		Weak-Q5			Epistemic fraction
Variance floor	Coverage	$\Delta$ Width	$\Delta$ Score	$\Delta$ Coverage	$\Delta$ Width	$\Delta$ Score	Overall	Weak-Q5
$10^{-6}$	$0.9017\pm 0.0025$	$-0.78\pm 0.27\%$	$-0.87\pm 0.17\%$	$0.0117\pm 0.0025$	$1.04\pm 0.41\%$	$-1.93\pm 0.42\%$	$4.03\pm 0.39\%$	$8.17\pm 0.73\%$
$10^{-5}$	$0.9018\pm 0.0025$	$-0.78\pm 0.26\%$	$-0.89\pm 0.17\%$	$0.0130\pm 0.0024$	$1.08\pm 0.42\%$	$-2.04\pm 0.45\%$	$4.08\pm 0.40\%$	$8.27\pm 0.76\%$
$10^{-4}$	$0.9013\pm 0.0023$	$-0.89\pm 0.26\%$	$-0.82\pm 0.17\%$	$0.0104\pm 0.0024$	$0.92\pm 0.41\%$	$-1.98\pm 0.44\%$	$4.03\pm 0.40\%$	$8.17\pm 0.75\%$
$10^{-3}$	$0.9013\pm 0.0024$	$-0.64\pm 0.24\%$	$-0.81\pm 0.16\%$	$0.0090\pm 0.0020$	$1.13\pm 0.39\%$	$-1.96\pm 0.43\%$	$3.94\pm 0.38\%$	$7.95\pm 0.71\%$

Appendix

Table 12: Sensitivity to the variance floor. Panel (a) reports predictive performance and posterior contribution. Changes in width and interval score are relative to matched LACP. Panel (b) reports variance-head and last-layer numerical diagnostics. Panel (c) reports hierarchical paired bootstrap comparisons with the default floor

\epsilon=10^{-4}

Figure 2: Raw-input and learned-feature distances under the ID reference, feature-visible OOD, and feature-collapsed OOD conditions. The dotted and dash-dotted lines denote the

90

th and

99

th percentiles of the corresponding ID distances.

	kNN distance		Target shift		CLAPS response
Condition	Raw input	Learned feature	Mean shift	Mean RMSE	Epi. fraction	Width ratio
ID reference	$2.5013\pm 0.0034$	$3.2301\pm 0.0445$	$0.0000\pm 0.0000$	$0.6170\pm 0.0141$	$4.97\pm 0.34\%$	$0.9849\pm 0.0013$
Feature-visible OOD	$5.6563\pm 0.0098$	$33.2810\pm 0.8751$	$4.4904\pm 0.0413$	$4.9156\pm 0.0602$	$72.20\pm 2.47\%$	$2.5061\pm 0.1551$
Feature-collapsed OOD	$4.7013\pm 0.0423$	$3.9992\pm 0.0617$	$2.2472\pm 0.0614$	$1.5441\pm 0.0896$	$17.70\pm 1.56\%$	$1.0448\pm 0.0069$

Appendix

Table 13: OOD stress test under a fixed representation. Values are means and standard errors over

30

random seeds. The width ratio compares CLAPS with matched LACP.

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SPACR: Single-Pass Adaptive Training of Uncertainty-Aware Conformal Regressors

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Conformal Prediction (CP) provides robust uncertainty guarantees for predictive models, but is typically applied post hoc, which misaligns model training with the conformal goal of producing efficient (i.e., narrow) intervals. We propose SPACR (Single-Pass Adaptive Conformal Regressor), a novel method for directly training uncertainty-aware regressors within a differentiable loss. SPACR jointly optimizes accuracy, efficiency, and validity without batch-splitting or a predefined confidence level during training. As a result, a single SPACR model yields valid prediction intervals at multiple confidence levels during inference, avoiding the costly retraining required by methods like Directly Optimized Inductive Conformal Regression (DOICR). Experiments on diverse tabular and image datasets show that SPACR consistently gives tighter intervals and better coverage-efficiency trade-offs compared to standard CP and DOICR, while significantly reducing computational costs relative to retraining-dependent baselines.

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