Neighborhood Smoothing for Calibration

Idan Horowitz; Avigdor Gal

cs.LGOct 6, 2026

Neighborhood Smoothing for Calibration

Authors: Idan Horowitz, Avigdor Gal

Organizations: Faculty of Data and Decision Sciences Technion Haifa, 3200003, IL

Abstract

Modern neural networks are often miscalibrated, with a tendency to overconfidence. Existing train-time calibration methods largely modify task losses or calibration penalties, leaving neighborhood structure in learned representations underexploited. We introduce graph smoothing as a general principle for train-time calibration, which encourages similar predictive distributions across neighboring samples in representation space. We analyze the effects of graph smoothing, deriving bounds that connect predictive divergence between neighboring samples to local confidence variation and to the propagation of pointwise calibration error, and characterize the conditions under which smoothing can or cannot improve calibration. In light of this analysis, we propose \modelNoSpace, a graph-based train-time regularizer that penalizes the Jensen--Shannon divergence between predictive distributions of neighboring samples. We present a thorough empirical analysis, showing that across standard calibration benchmarks, \model improves predictive quality, and the improvement is complementary to post-hoc calibration: after temperature scaling, \model attains the lowest NLL of all evaluated train-time methods in seven of the eight image and tabular settings. These findings demonstrate the value of graph smoothing over learned representations for neural network calibration.

Figures & tables

Base model
Setting	CE	FL	LS	MMCE	MDCA	s-AvUC	MbLS	NeSCal
CIFAR-10 / R50	0.2065 $\pm$ 0.0091	0.1821 $\pm$ 0.0045	0.2285 $\pm$ 0.0064	0.2031 $\pm$ 0.0043	0.2089 $\pm$ 0.0074	0.2065 $\pm$ 0.0052	0.2227 $\pm$ 0.0052	0.1843 $\pm$ 0.0058
CIFAR-10 / WRN	0.1972 $\pm$ 0.0077	0.1572 $\pm$ 0.0033	0.1838 $\pm$ 0.0034	0.1791 $\pm$ 0.0083	0.1951 $\pm$ 0.0071	0.2090 $\pm$ 0.0102	0.1791 $\pm$ 0.0025	0.1752 $\pm$ 0.0028
CIFAR-100 / R50	0.8782 $\pm$ 0.0249	0.8463 $\pm$ 0.0044	0.8966 $\pm$ 0.0186	0.8935 $\pm$ 0.0393	0.8889 $\pm$ 0.0139	0.8763 $\pm$ 0.0170	0.8614 $\pm$ 0.0174	0.8026 $\pm$ 0.0155
CIFAR-100 / WRN	0.8785 $\pm$ 0.0049	0.7681 $\pm$ 0.0058	0.8388 $\pm$ 0.0086	0.8838 $\pm$ 0.0219	0.8786 $\pm$ 0.0139	0.8691 $\pm$ 0.0043	0.8206 $\pm$ 0.0117	0.7760 $\pm$ 0.0173
Tiny-ImageNet	1.4704 $\pm$ 0.0254	1.4960 $\pm$ 0.0101	1.4816 $\pm$ 0.0254	1.4797 $\pm$ 0.0344	1.4406 $\pm$ 0.0225	1.4403 $\pm$ 0.0317	1.4287 $\pm$ 0.0251	1.4126 $\pm$ 0.0254

Table 1: Test NLL before and after validation-fitted temperature scaling; lower is better. Values are means

\pm

SD over seeds, with seed counts specified in Appendix D .

Setting	Edge JS	Edge agreement	$\Delta$ NLL	Relative
Covertype	0.003	0.958	0.0017	$0.7\%$
Amazon (binary)	0.004	0.987	0.0014	$0.6\%$
Jannis	0.009	0.935	0.0059	$0.8\%$
Amazon (5-class)	0.013	0.909	0.0028	$0.4\%$
CIFAR-10 / WRN	0.026	0.901	0.0220	$11.2\%$
20NG	0.027	0.748	0.0104	$1.3\%$

Table 2: Mean edge JS and Edge Agreement at activation and improvement in test NLL over CE, sorted by edge JS. Positive

\Delta

NLL indicates improvement; the relative column is as a fraction of CE’s NLL.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Modality	Classes	Size	Split
CIFAR-10 / CIFAR-100	image	10 / 100	60,000	45k / 5k / 10k
Tiny-ImageNet	image	200	110,000	90k / 10k / 10k
20 Newsgroups	text	20	18,846	13,570 / 1,507 / 3,769
Amazon (binary) (frozen RoBERTa)	text	2	11,349	7,264 / 1,816 / 2,269
Amazon (5-class) (frozen RoBERTa)	text	5	22,815	14,602 / 3,650 / 4,563
Helena	tabular	100	65,196	41,724 / 10,432 / 13,040

Appendix

Table 3: Datasets. Splits are train/val/test

Setting	Optimizer	LR	LR schedule	Epochs	Batch
CIFAR-10/100, ResNet-50 & WRN-28-10	SGD ( $0.9$ , wd $5\!\times\!10^{-4}$ )	$0.1$	$\times 0.1$ at $50\%,75\%$	200	128
Tiny-ImageNet, ResNet-50	SGD ( $0.9$ , wd $5\!\times\!10^{-4}$ )	$0.1$	$\times 0.1$ at $40\%,60\%$	100	64
20 Newsgroups, GPCNN	Adam	$10^{-3}$	constant	50	128
Frozen encoder + MLP head	SGD ( $0.9$ , wd $10^{-4}$ )	$10^{-3}$	constant	50 (early stopping)	256
Tabular, MLP $3\times 256$ (dropout $0.1$ )	AdamW (wd $10^{-5}$ )	$10^{-3}$	constant	100	512

Appendix

Table 4: Training configuration per setting.

Setting	Candidate onsets	$\lambda$ grid (cosine / constant)
CIFAR-10, ResNet-50	$\{50,65,80,95,110,125\}$	$\{1,3,5,10,20\}$ / $\{1,3,5\}$
CIFAR-10, WRN-28-10	$\{50,65,80,95,110,125\}$	$\{5,10,20,30,50\}$ / $\{3,5,10\}$
CIFAR-100, ResNet-50	$\{50,65,80,95,100,110,125\}$	$\{5,10,20,30,50\}$ / $\{3,5,8,10\}$
CIFAR-100, WRN-28-10	$\{50,65,80,95,100,110,125\}$	$\{5,10,20,30,40\}$ / $\{3,5,8,10\}$
Tiny-ImageNet	$\{20,30,38,40,50,58\}$	$\{5,10,20,30,40\}$ / $\{5,10,20\}$
20NG	$\{3,5,8,12,20\}$	$\{0.5,1,2,5\}$ / $\{0.5,1,2,5\}$

Appendix

Table 5: NeSCal search grids. Onsets are epochs of the total budget.

Brier score
Setting	CE	FL	LS	MMCE	MDCA	s-AvUC	MbLS	NeSCal
CIFAR-10 / R50	0.0953 $\pm$ 0.0045	0.0866 $\pm$ 0.0029	0.0872 $\pm$ 0.0035	0.0988 $\pm$ 0.0021	0.0943 $\pm$ 0.0087	0.0949 $\pm$ 0.0054	0.0948 $\pm$ 0.0057	0.0872 $\pm$ 0.0015
CIFAR-10 / WRN	0.0782 $\pm$ 0.0051	0.0733 $\pm$ 0.0016	0.0750 $\pm$ 0.0019	0.0854 $\pm$ 0.0030	0.0852 $\pm$ 0.0066	0.0831 $\pm$ 0.0052	0.0807 $\pm$ 0.0052	0.0785 $\pm$ 0.0018
CIFAR-100 / R50	0.3447 $\pm$ 0.0081	0.3296 $\pm$ 0.0019	0.3430 $\pm$ 0.0083	0.3481 $\pm$ 0.0128	0.3482 $\pm$ 0.0041	0.3443 $\pm$ 0.0057	0.3372 $\pm$ 0.0045	0.3178 $\pm$ 0.0050
CIFAR-100 / WRN	0.3321 $\pm$ 0.0033	0.3024 $\pm$ 0.0022	0.3162 $\pm$ 0.0015	0.3396 $\pm$ 0.0066	0.3353 $\pm$ 0.0058	0.3337 $\pm$ 0.0025	0.3161 $\pm$ 0.0060	0.3057 $\pm$ 0.0061
Tiny-ImageNet	0.4778 $\pm$ 0.0057	0.4991 $\pm$ 0.0030	0.4757 $\pm$ 0.0063	0.4819 $\pm$ 0.0079	0.4696 $\pm$ 0.0071	0.4691 $\pm$ 0.0096	0.4651 $\pm$ 0.0073	0.4693 $\pm$ 0.0064

Appendix

Table 6: Brier score (top, lower is better) and accuracy in percent (bottom, higher is better). Conventions as in Table 1 .

Base model
Setting	CE	FL	LS	MMCE	MDCA	s-AvUC	MbLS	NeSCal
CIFAR-10 / R50	2.46 $\pm$ 0.49	1.14 $\pm$ 0.04	3.98 $\pm$ 0.21	0.83 $\pm$ 0.10	2.56 $\pm$ 0.40	2.55 $\pm$ 0.53	2.08 $\pm$ 0.49	2.29 $\pm$ 0.11
CIFAR-10 / WRN	2.75 $\pm$ 0.30	0.78 $\pm$ 0.07	1.64 $\pm$ 0.31	1.13 $\pm$ 0.25	2.66 $\pm$ 0.14	3.00 $\pm$ 0.20	2.10 $\pm$ 0.51	2.39 $\pm$ 0.18
CIFAR-100 / R50	5.90 $\pm$ 0.58	2.73 $\pm$ 0.50	1.39 $\pm$ 0.18	6.02 $\pm$ 0.49	6.08 $\pm$ 0.24	5.83 $\pm$ 0.94	3.84 $\pm$ 0.51	3.95 $\pm$ 0.67
CIFAR-100 / WRN	8.19 $\pm$ 0.25	1.87 $\pm$ 0.12	3.41 $\pm$ 0.57	7.45 $\pm$ 0.38	7.91 $\pm$ 0.18	7.03 $\pm$ 0.61	3.62 $\pm$ 0.29	4.60 $\pm$ 0.62
Tiny-ImageNet	5.75 $\pm$ 0.29	3.12 $\pm$ 0.23	4.63 $\pm$ 0.29	4.56 $\pm$ 0.35	6.00 $\pm$ 0.25	6.27 $\pm$ 0.47	2.69 $\pm$ 0.63	2.02 $\pm$ 0.36

Appendix

Table 7: ECE in percent for the base model (top) and after temperature scaling (bottom); lower is better.

Classwise ECE
Setting	CE	FL	LS	MMCE	MDCA	s-AvUC	MbLS	NeSCal
CIFAR-10 / R50	0.58 $\pm$ 0.08	0.43 $\pm$ 0.02	0.83 $\pm$ 0.02	0.40 $\pm$ 0.05	0.58 $\pm$ 0.08	0.58 $\pm$ 0.07	0.57 $\pm$ 0.04	0.54 $\pm$ 0.02
CIFAR-10 / WRN	0.61 $\pm$ 0.06	0.45 $\pm$ 0.04	0.53 $\pm$ 0.04	0.42 $\pm$ 0.02	0.61 $\pm$ 0.03	0.68 $\pm$ 0.04	0.50 $\pm$ 0.07	0.59 $\pm$ 0.03
CIFAR-100 / R50	0.21 $\pm$ 0.00	0.18 $\pm$ 0.00	0.19 $\pm$ 0.00	0.21 $\pm$ 0.01	0.21 $\pm$ 0.01	0.21 $\pm$ 0.01	0.19 $\pm$ 0.01	0.18 $\pm$ 0.01
CIFAR-100 / WRN	0.24 $\pm$ 0.00	0.18 $\pm$ 0.00	0.20 $\pm$ 0.01	0.23 $\pm$ 0.01	0.24 $\pm$ 0.01	0.22 $\pm$ 0.01	0.20 $\pm$ 0.00	0.19 $\pm$ 0.00
Tiny-ImageNet	0.14 $\pm$ 0.00	0.14 $\pm$ 0.00	0.14 $\pm$ 0.00	0.14 $\pm$ 0.00	0.14 $\pm$ 0.00	0.15 $\pm$ 0.00	0.14 $\pm$ 0.00	0.13 $\pm$ 0.00

Appendix

Table 8: Classwise ECE, LCE, and equal-mass ECE, all in percent; lower is better.

	NLL (all)		NLL (I+T)		NLL+TS (all)		NLL+TS (I+T)		Brier (all)
vs.	wins	$p$	wins	$p$	wins	$p$	wins	$p$	wins	$p$
CE	11/11	0.001	8/8	0.008	11/11	0.001	8/8	0.008	10/11	0.002
FL	7/11	0.067	5/8	0.109	10/11	0.014	8/8	0.008	7/11	0.083
LS	7/11	0.175	7/8	0.016	10/11	0.042	8/8	0.008	5/11	0.878
MMCE	8/11	0.206	8/8	0.008	8/11	0.083	8/8	0.008	8/11	0.240
MDCA	10/11	0.019	8/8	0.008	9/11	0.005	7/8	0.016	11/11	0.001

Appendix

Table 9: NeSCal against each baseline across settings. “All” covers the eleven completed settings, “I+T” the eight image and tabular settings.

Setting	CE	FL	LS	MMCE	MDCA	s-AvUC	MbLS	NeSCal
CIFAR-10 / R50	0.0424 $\pm$ 0.0020	0.0470 $\pm$ 0.0046	0.0381 $\pm$ 0.0012	0.0439 $\pm$ 0.0012	0.0420 $\pm$ 0.0036	0.0424 $\pm$ 0.0026	0.0424 $\pm$ 0.0027	0.0388 $\pm$ 0.0006
CIFAR-10 / WRN	0.0346 $\pm$ 0.0024	0.0377 $\pm$ 0.0038	0.0326 $\pm$ 0.0008	0.0382 $\pm$ 0.0011	0.0377 $\pm$ 0.0030	0.0365 $\pm$ 0.0023	0.0359 $\pm$ 0.0023	0.0352 $\pm$ 0.0010
CIFAR-100 / R50	0.1242 $\pm$ 0.0014	0.1229 $\pm$ 0.0015	0.1249 $\pm$ 0.0007	0.1239 $\pm$ 0.0032	0.1248 $\pm$ 0.0013	0.1237 $\pm$ 0.0019	0.1231 $\pm$ 0.0016	0.1160 $\pm$ 0.0019
CIFAR-100 / WRN	0.1185 $\pm$ 0.0018	0.1151 $\pm$ 0.0010	0.1069 $\pm$ 0.0005	0.1203 $\pm$ 0.0016	0.1197 $\pm$ 0.0018	0.1189 $\pm$ 0.0009	0.1161 $\pm$ 0.0026	0.1118 $\pm$ 0.0014
Tiny-ImageNet	0.1463 $\pm$ 0.0014	0.1558 $\pm$ 0.0011	0.1497 $\pm$ 0.0016	0.1484 $\pm$ 0.0022	0.1448 $\pm$ 0.0016	0.1434 $\pm$ 0.0029	0.1468 $\pm$ 0.0019	0.1453 $\pm$ 0.0030
Helena	0.1777 $\pm$ 0.0008	0.1789 $\pm$ 0.0013	0.1772 $\pm$ 0.0006	0.1770 $\pm$ 0.0010	0.1767 $\pm$ 0.0009	0.1762 $\pm$ 0.0013	0.1772 $\pm$ 0.0006	0.1773 $\pm$ 0.0009

Appendix

Table 10: Reported binned refinement component of the base-model top-label Brier decomposition; lower is better.

NLL $\downarrow$
Setting	CE	FL	LS	MMCE	MDCA	s-AvUC	MbLS	NeSCal
CIFAR-10 / R50	0.1863 $\pm$ 0.0084 T	0.1790 $\pm$ 0.0052 L	0.2108 $\pm$ 0.0046 D	0.1993 $\pm$ 0.0061 L	0.1858 $\pm$ 0.0144 T	0.1855 $\pm$ 0.0121 L	0.2156 $\pm$ 0.0064 A	0.1698 $\pm$ 0.0046 T
CIFAR-10 / WRN	0.1495 $\pm$ 0.0129 L	0.1508 $\pm$ 0.0043 L	0.1637 $\pm$ 0.0028 L	0.1726 $\pm$ 0.0063 L	0.1618 $\pm$ 0.0115 L	0.1540 $\pm$ 0.0117 L	0.1599 $\pm$ 0.0095 L	0.1444 $\pm$ 0.0041 L
CIFAR-100 / R50	0.8489 $\pm$ 0.0267 T	0.8452 $\pm$ 0.0038 T	0.8944 $\pm$ 0.0190 A	0.8631 $\pm$ 0.0402 T	0.8577 $\pm$ 0.0137 T	0.8478 $\pm$ 0.0137 T	0.8558 $\pm$ 0.0168 T	0.7944 $\pm$ 0.0191 T
CIFAR-100 / WRN	0.8070 $\pm$ 0.0061 T	0.7681 $\pm$ 0.0057 T	0.8334 $\pm$ 0.0049 D	0.8309 $\pm$ 0.0151 T	0.8141 $\pm$ 0.0155 T	0.8212 $\pm$ 0.0099 T	0.8173 $\pm$ 0.0103 T	0.7613 $\pm$ 0.0178 T
Tiny-ImageNet	1.4415 $\pm$ 0.0246 T	1.4950 $\pm$ 0.0099 T	1.4563 $\pm$ 0.0255 T	1.4614 $\pm$ 0.0319 T	1.4096 $\pm$ 0.0228 T	1.4066 $\pm$ 0.0307 T	1.4208 $\pm$ 0.0246 T	1.4098 $\pm$ 0.0255 T

Appendix

Table 11: Best observed post-hoc results for each train-time method. NLL and ECE are minimized separately over TS (T), LoCal (L), Dirichlet (D), DAC (A), and LoRe (R); ECE is in percent. Best method selection is retrospective on test means. Calibrator ties at reported precision use the order T, L, D, A, R.

Setting	Metric	CE	FL	LS	MMCE	MDCA	s-AvUC	MbLS	NeSCal
CIFAR-10 / R50	AUROC	0.9357	0.9278	0.8537	0.9359	0.9351	0.9352	0.8995	0.9392
	AUROC+TS	0.9352	0.9270	0.8563	0.9361	0.9349	0.9349	0.9001	0.9395
	AURC	6.60	8.57	19.08	7.25	6.53	6.53	11.21	5.58
	AURC+TS	6.64	8.66	18.69	7.24	6.55	6.55	11.11	5.57
CIFAR-10 / WRN	AUROC	0.9343	0.9290	0.8951	0.9393	0.9379	0.9350	0.9340	0.9384
	AUROC+TS	0.9316	0.9295	0.8957	0.9391	0.9362	0.9322	0.9331	0.9370

Appendix

Table 12: Failure-prediction AUROC (higher is better) and AURC (

\times 10^{3}

, lower is better), before and after temperature scaling. Means over seeds; SDs omitted.

Corrupted NLL
Setting	CE	FL	LS	MMCE	MDCA	s-AvUC	MbLS	NeSCal
CIFAR-10 / R50	1.2363	0.9956	1.0565	0.9828	1.2799	1.2882	1.0554	1.2483
CIFAR-10 / WRN	1.3416	1.0069	1.0651	0.9919	1.3808	1.4140	1.1209	1.2342
CIFAR-100 / R50	2.6843	2.4157	2.4532	2.6950	2.7057	2.6799	2.5766	2.4443
CIFAR-100 / WRN	3.1114	2.5448	2.6913	2.9448	3.1199	2.9370	2.6783	2.5604
Tiny-ImageNet	4.1629	4.0258	3.8927	4.1066	4.1212	4.1864	3.8153	3.9413

Appendix

Table 13: Robustness: corruption metrics averaged over five severity levels and 15 corruption types for CIFAR-10/100-C or 14 for Tiny-ImageNet-C; SVHN detection AUROC uses maximum softmax probability.

Setting	CE	kNN	Random	Random (scaled)
CIFAR-10 / R50	0.2065	0.1843	0.2156	0.2058
CIFAR-100 / R50	0.8782	0.8026	1.0386	0.8785
Helena	2.5651	2.5534	2.5890	2.5639

Appendix

Table 14: Test NLL for neighborhood controls; lower is better. Random uses headline

\lambda

; Random (scaled) matches penalty magnitude.

Setting	Schedule	NLL $\downarrow$	Brier $\downarrow$	Accuracy $\uparrow$	ECE $\downarrow$
CIFAR-10 / R50	Constant	0.1917 $\pm$ 0.0068	0.0941 $\pm$ 0.0033	0.9358 $\pm$ 0.0026	0.0155 $\pm$ 0.0012
	Cosine	0.1843 $\pm$ 0.0058	0.0872 $\pm$ 0.0015	0.9428 $\pm$ 0.0015	0.0229 $\pm$ 0.0011
CIFAR-10 / WRN	Constant	0.1966 $\pm$ 0.0092	0.0897 $\pm$ 0.0037	0.9413 $\pm$ 0.0028	0.0252 $\pm$ 0.0021
	Cosine	0.1752 $\pm$ 0.0028	0.0785 $\pm$ 0.0018	0.9494 $\pm$ 0.0011	0.0239 $\pm$ 0.0018
CIFAR-100 / R50	Constant	0.8728 $\pm$ 0.0238	0.3423 $\pm$ 0.0062	0.7541 $\pm$ 0.0052	0.0539 $\pm$ 0.0067
	Cosine	0.8026 $\pm$ 0.0155	0.3178 $\pm$ 0.0050	0.7719 $\pm$ 0.0052	0.0395 $\pm$ 0.0067

Appendix

Table 15: Constant versus cosine regularization on CIFAR.

Arm	C10/R50	C10/WRN	C100/R50	C100/WRN	Helena	Jannis	Covertype	20NG
Headline ( $k{=}10$ , JS, raw features)	0.1843	0.1752	0.8026	0.7760	2.5534	0.7066	0.2499	0.8023
Cross-entropy (no regularizer)	0.2065	0.1972	0.8782	0.8785	2.5651	0.7125	0.2516	0.8127
Random graph, same $\lambda$	0.2156	–	1.0386	–	2.5890	–	–	–
Random graph, matched penalty	0.2058	–	0.8785	–	2.5639	–	–	–
$k=2$	0.1830	0.1736	0.8368	0.7936	2.5554	0.7069	0.2487	0.7466
$k=5$	0.1789	0.1702	0.8148	0.7985	2.5553	0.7072	0.2531	0.7650

Appendix

Table 16: Ablations: test NLL, lower is better. – : not run. † : the common penalty target over-regularizes this setting; div.: diverged.

Setting	Entropy drop	CWCC	Probe (hindsight)
CIFAR-10 / R50	$+0.0222$	$-0.0018$	$+0.0178$
CIFAR-10 / WRN	$-0.0012$	$+0.0298$	$+0.0170$
CIFAR-100 / R50	$+0.0940$	$+0.2987$	$+0.0123$
CIFAR-100 / WRN	$+0.0439$	$-0.0427$	$-0.0388$
Tiny-ImageNet	$+0.0686$	–	$+0.0718$
Helena	–	–	$+0.0075$

Appendix

Table 17: Regret of activation rules: test NLL relative to the grid-selected onset at the same

\lambda

; positive is worse. –: rule not evaluated.

Setting	CE time	NeSCal time	Overhead	Graph build	CE memory	NeSCal memory
CIFAR-10 / R50	3430	3456	$+0.7\%$	20.4	2.05	2.05
CIFAR-10 / WRN	5028	5053	$+0.5\%$	26.9	2.29	2.26
CIFAR-100 / R50	2702	2725	$+0.9\%$	17.3	2.11	2.15
CIFAR-100 / WRN	5068	5094	$+0.5\%$	27.1	2.59	3.05
Tiny-ImageNet	11036	11085	$+0.4\%$	146.2	3.89	6.89
Helena	36.5	50.1	$+37\%$	1.49	0.04	1.44

Appendix

Table 18: Training cost: wall-clock training time (s), relative overhead, one-off graph construction (s), and peak GPU memory (GB). Means follow headline seeds except CIFAR-10/ResNet-50 NeSCal (one seed).

	CIFAR-10 / R50			CIFAR-100 / R50
Quantity	CE	NeSCal (const)	NeSCal (cos)	CE	NeSCal (const)	NeSCal (cos)
Bound violations (L1, confidence)	0	0	0	0	0	0
Confidence-bound tightness, median	0.019	0.038	0.019	0.131	0.117	0.101
Edges within the expansion’s regime	$0\%$	$0\%$	$0\%$	$0.005\%$	$0.005\%$	$0.005\%$
Across-seed dispersion proxy $\sigma\sqrt{2(1-\rho)}$	0.055	0.061	0.038	0.102	0.102	0.088
Across-seed confidence spread $\sigma_{c}$	0.059	0.059	0.035	0.103	0.112	0.076

Appendix

Table 19: Theory diagnostics at the selected weights on the frozen training graph. Tightness concerns the observable predictive and confidence bounds. The dispersion proxy corresponds to the equal-bias, equal-variance specialization of Proposition 3 , not the full expected squared error difference.

Figure 2: Mean squared top-label reference error across heterogeneity strengths $H$ , per scenario, on CE, CE with the test-informed temperature ( oracle ), and NeSCal.

		NLL $\downarrow$		ECE (%) $\downarrow$		Accuracy (%) $\uparrow$
Architecture	Noise	CE	NeSCal	CE	NeSCal	CE	NeSCal
ResNet-50	$20\%$	1.2521	1.3123	6.42	4.32	68.20	67.64
	$40\%$	1.7125	1.9218	15.96	9.66	61.01	56.95
WRN-28-10	$20\%$	1.1743	1.1528	2.92	1.70	70.19	70.44
	$40\%$	1.5154	1.6212	9.58	4.85	64.35	61.74

Appendix

Table 20: Symmetric label noise on CIFAR-100, without re-tuning

\lambda

Architecture	Imbalance	Cross-entropy	JS	Fisher-weighted	Uniform ( $L_{2}$ )
ResNet-50	$10$	1.8104	1.8018	1.8403	1.7629
	$50$	2.9898	3.0154	3.0717	2.8144
	$100$	3.4013	3.4364	3.5764	3.2381
WRN-28-10	$10$	1.7965	1.6838	1.6325	1.5680
	$50$	2.5238	2.4804	2.6973	2.3800
	$100$	2.9397	2.9967	3.2001	2.8844

Appendix

Table 21: Long-tailed CIFAR-100: test NLL for CE and three divergences at equal

\lambda

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Computer Science and Applied Mathematics Laboratory (LIMA), Faculty of Science and Technology, Chadli Bendjedid University, El Tarf 36000, Algeria
FedLAS: Feature-Modulated Bidirectional Label Smoothing for Neural Network Calibration
Jun 26, 2026Thiru Thillai Nadarasar Bahavan, Sachith Seneviratne, Saman HalgamugeModel Calibration Image Classification
The University of Melbourne, Parkville, Australia

Apr 22, 2026, cs.LGJ/K move · Enter open · S save

Too Sharp, Too Sure: When Calibration Follows Curvature

Alessandro Morosini, Matea Gjika, Tomaso Poggio, Pierfrancesco Beneventano

Massachusetts Institute of Technology, Cambridge, MA, USA.

Model Calibration Neural Network Training Dynamics Statistical Learning Theory

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Modern neural networks can achieve high accuracy while remaining poorly calibrated, producing confidence estimates that do not match empirical correctness. Yet calibration is often treated as a post-hoc attribute. We take a different perspective: we study calibration as a training-time phenomenon on small vision tasks, and ask whether calibrated solutions can be obtained reliably by intervening on the training procedure. We identify a tight coupling between calibration, curvature, and margins during training of deep networks under multiple gradient-based methods. Empirically, Expected Calibration Error (ECE) closely tracks curvature-based sharpness throughout optimization. Mathematically, we show that both ECE and Gauss--Newton curvature are controlled, up to problem-specific constants, by the same margin-dependent exponential tail functional along the trajectory. Guided by this mechanism, we introduce a margin-aware training objective that explicitly targets robust-margin tails and local smoothness, yielding improved out-of-sample calibration across optimizers without sacrificing accuracy.

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Too Sharp, Too Sure: When Calibration Follows Curvature

When does distribution shift break graph neural networks calibration?

FedLAS: Feature-Modulated Bidirectional Label Smoothing for Neural Network Calibration