用于模型校准的邻域平滑

# Neighborhood Smoothing for Calibration
Source: [https://arxiv.org/html/2610.09020](https://arxiv.org/html/2610.09020)
Idan Horowitz & Avigdor GalAffiliation:Faculty of Data and Decision SciencesAffiliation:TechnionAffiliation:Haifa, 3200003, ILAffiliation:idanhorowitz@campus\.technion\.ac\.il, avigal@technion\.ac\.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 NeSCal, 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, NeSCal improves predictive quality, and the improvement is complementary to post\-hoc calibration: after temperature scaling, NeSCal 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\.

## 1Introduction

Confidence calibration in classification refers to the alignment between the empirical correctness of a model’s predictions and its confidence in these predictions\. A well\-calibrated confidence serves as a reliable indication of prediction correctness and an essential component of downstream decisions\. Modern deep neural networks tend to be over\-confident\([Guo et al\., 2017](https://arxiv.org/html/2610.09020#bib.bib2)\)\. Approaches to addressing miscalibration are typically either post\-hoc, learning a function on a withheld data set to reshape confidence values\([Guo et al\., 2017](https://arxiv.org/html/2610.09020#bib.bib2);[Kull et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib13);[Platt, 2000](https://arxiv.org/html/2610.09020#bib.bib16);[Zadrozny and Elkan, 2002](https://arxiv.org/html/2610.09020#bib.bib15)\), or train\-time, modifying the learning objective to encourage the model to be better calibrated, either by some explicit, calibration\-aware penalty term\([Hebbalaguppe et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib19);[Karandikar et al\., 2021](https://arxiv.org/html/2610.09020#bib.bib17);[Krishnan and Tickoo, 2020](https://arxiv.org/html/2610.09020#bib.bib18);[Kumar et al\., 2018](https://arxiv.org/html/2610.09020#bib.bib8)\), or implicitly, through a loss shown to improve calibration\([Mukhoti et al\., 2020](https://arxiv.org/html/2610.09020#bib.bib20);[Müller et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib21)\)\.

Existing train\-time calibration methods largely modify the learning objective based primarily on model outputs and labels, without explicitly exploiting the geometry of the representations learned by the model\. Yet these representations encode information about which samples the model regards as similar\. Nearby samples in representation space should, when they correspond to similar underlying inputs, receive similar predictions, providing a natural structure over which predictive behavior can be regularized\.

In this work, we investigate model regularization to encourage adjacent samples in a neighborhood graph over its representation space to receive similar predictive distributions, a principle we term*graph smoothing*\. Penalizing the variation of the learned function in such a graph is the basis of manifold regularization\([Belkin et al\., 2006](https://arxiv.org/html/2610.09020#bib.bib12)\), where it has been used to improve classification accuracy under limited supervision\. Applied to calibration, suppressing predictive disagreement constrains local confidence variation: a well\-calibrated prediction can constrain neighboring calibration errors when the connected samples have similar true probabilities of correctness\. By exploiting this additional structure during training, graph smoothing can also improve the learned predictor in ways that remain beneficial when combined with subsequent post\-hoc recalibration\.

![Refer to caption](https://arxiv.org/html/2610.09020v1/figs/nescal-overview.png)Figure 1:Overview of NeSCal training\.We instantiate graph smoothing as NeSCal \(NeighborhoodSmoothing forCalibration\), a train\-time regularizer that penalizes Jensen–Shannon \(JS\) divergence between neighboring predictive distributions, added to a task loss through a single weighting term at no inference\-time cost\.

The paper’s contributions are quintuple\. \(i\) We introduce*graph smoothing*as a general principle for train\-time calibration, in which a model is regularized to encourage similar predictive distributions for samples adjacent in a neighborhood graph over representation space \(Section[3](https://arxiv.org/html/2610.09020#S3)\)\. \(ii\) We demonstrate the relationship between graph\-based predictive smoothness and calibration, showing that the variation of the predictive distribution translates into a bound on the propagation of pointwise calibration error along paths in the graph \(Proposition[1](https://arxiv.org/html/2610.09020#Thmproposition1); Theorem[1](https://arxiv.org/html/2610.09020#Thmtheorem1)\)\. \(iii\) We characterize when graph smoothing improves calibration and when it cannot, identifying the variation of true correctness probability across an edge as the quantity governing both \(Propositions[2](https://arxiv.org/html/2610.09020#Thmproposition2)and[3](https://arxiv.org/html/2610.09020#Thmproposition3)\)\. \(iv\) We instantiate the principle as a train\-time regularizer NeSCal, penalizing the Jensen–Shannon divergence between the predictive distributions of neighbors in akk\-nearest\-neighbor graph[4](https://arxiv.org/html/2610.09020#S4)\. \(v\) We show empirically that NeSCal improves negative log\-likelihood \(NLL\) in all eleven main benchmarks and ranks first or second among the evaluated methods in seven of eight image and tabular settings\. After temperature scaling, it attains the lowest NLL in seven of those eight settings \(Section[6](https://arxiv.org/html/2610.09020#S6)\)\.

## 2Background

Consider a genericKK\-class classification problem, with a modelffthat receives inputsx∈Xx\\in Xand produces softmax probability vectorsp⁡\(x\):=f⁡\(x\)∈ΔK−1p\(x\):=f\(x\)\\in\\Delta^\{K\-1\}, the\(K−1\)\(K\{\-\}1\)\-dimensional probability simplex\. Writepi:=p⁡\(xi\)p\_\{i\}:=p\(x\_\{i\}\)and letpi,kp\_\{i,k\}denote theiith sample’s class\-kkprobability\. Each inputxxhas an associated true labely∈\{1,…,K\}y\\in\\\{1,\\ldots,K\\\}, withYYthe corresponding label random variable\. The model’s predicted label and its associated confidence forxxarey^​\(x\)=arg⁡maxk​pk​\(x\)\\hat\{y\}\(x\)=\\arg\\max\_\{k\}p\_\{k\}\(x\)andc⁡\(x\)=maxk⁡pk​\(x\)c\(x\)=\\max\_\{k\}p\_\{k\}\(x\)\.

We write the true conditional class\-probability vector \(the Bayes posterior\)η⁡\(x\):=\(P⁡\(Y=1∣X=x\),…,P⁡\(Y=K∣X=x\)\)∈ΔK−1\\eta\(x\):=\\big\(P\(Y=1\\mid X=x\),\\ldots,P\(Y=K\\mid X=x\)\\big\)\\in\\Delta^\{K\-1\}\. For the predicted class, the conditional probability of correctness isa⁡\(x\):=P⁡\(Y=y^​\(x\)∣X=x\)=ηy^​\(x\)​\(x\)a\(x\):=P\(Y=\\hat\{y\}\(x\)\\mid X=x\)=\\eta\_\{\\hat\{y\}\(x\)\}\(x\), a single coordinate ofη⁡\(x\)\\eta\(x\)\. Thus,c⁡\(x\)c\(x\)is the model’s predicted probability of being correct, whilea⁡\(x\)a\(x\)is the corresponding true conditional probability of correctness\. We defineℰcal​\(x\):=\|a⁡\(x\)−c⁡\(x\)\|\\mathcal\{E\}\_\{\\mathrm\{cal\}\}\(x\):=\|a\(x\)\-c\(x\)\|as the pointwise calibration error atxx, writingℰcali:=ℰcal​\(xi\)\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{i\}\}:=\\mathcal\{E\}\_\{\\mathrm\{cal\}\}\(x\_\{i\}\)for samples\. Perfect pointwise calibration is then achieved ifc⁡\(x\)=a⁡\(x\)c\(x\)=a\(x\)for allx∈𝒳x\\in\\mathcal\{X\}\. The posteriorη⁡\(x\)\\eta\(x\)is a property of the data distribution anda⁡\(x\)a\(x\)also depends on the model’s predicted label; neither is directly observed from a single labeled example, so empirical calibration metrics assess aggregated notions of calibration\. The most widely used metric is Expected Calibration Error \(ECE\)[Naeini et al\. \(2015\)](https://arxiv.org/html/2610.09020#bib.bib1), which measures*confidence calibration*\([Guo et al\., 2017](https://arxiv.org/html/2610.09020#bib.bib2)\)by binning predictions according to confidence and comparing confidence means with empirical accuracy within each bin\. ECE is a binning\-based estimator, it is thus sensitive to the binning scheme and to finite\-sample bias\([Nixon et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib6);[Roelofs et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib22);[Zhang et al\., 2020](https://arxiv.org/html/2610.09020#bib.bib9)\)\. Moreover, since ECE aggregates across predictions, pointwise over\- and under\-confidence can cancel\. A low ECE therefore does not imply that predictions are calibrated at a specific point\. A discussion of the alternative metrics can be found in Appendix[D\.6](https://arxiv.org/html/2610.09020#A4.SS6)\.

## 3Graph Smoothing for Confidence Calibration

### 3\.1Problem Setup

We adopt the setting of Section[2](https://arxiv.org/html/2610.09020#S2), with a dataset𝒟=\{\(xi,yi\)\}i=1n\\mathcal\{D\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}of i\.i\.d\. draws fromP⁡\(X,Y\)P\(X,Y\)withX⊆ℝdX\\subseteq\\mathbb\{R\}^\{d\}, and a probabilistic classifierfθ:X→ΔK−1f\_\{\\theta\}:X\\rightarrow\\Delta^\{K\-1\}with softmax output layer, trained to minimize𝔼\(X,Y\)∼P​\[ℒ⁡\(fθ​\(X\),Y\)\]\\mathbb\{E\}\_\{\(X,Y\)\\sim P\}\[\\mathcal\{L\}\(f\_\{\\theta\}\(X\),Y\)\]for a task lossℒ\\mathcal\{L\}\. Writepi:=fθ​\(xi\)p\_\{i\}:=f\_\{\\theta\}\(x\_\{i\}\),ai:=a⁡\(xi\)a\_\{i\}:=a\(x\_\{i\}\)andci:=c⁡\(xi\)c\_\{i\}:=c\(x\_\{i\}\), so thatℰcali=\|ai−ci\|\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{i\}\}=\|a\_\{i\}\-c\_\{i\}\|\. Our analysis targets pointwise calibration: the top\-label guarantee is required at each point rather than on average, without necessitating the full predictive vector to match the posterior\.

Leth:𝒳→𝒵⊆ℝlh:\\mathcal\{X\}\\to\\mathcal\{Z\}\\subseteq\\mathbb\{R\}^\{l\}produce representationszi=h⁡\(xi\)z\_\{i\}=h\(x\_\{i\}\)\. A neighborhood graph𝒢=\(V,E\)\\mathcal\{G\}=\(V,E\)has nodesV=\{zi\}i=1n⊂𝒵V=\\\{z\_\{i\}\\\}\_\{i=1\}^\{n\}\\subset\\mathcal\{Z\}and connects nearby representations under a chosen distance\. Its adjacency matrixWWassigns nonnegative weightsWi​jW\_\{ij\}to edges\(i,j\)∈E\(i,j\)\\in E\. LetD\(pi∥pj\)\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)be a nonnegative divergence between the predictive probability distributions at observationsiiandjj\.

We defineRn\(p\):=∑i,jWi​jD\(pi∥pj\)R\_\{n\}\(p\):=\\sum\_\{i,j\}W\_\{ij\}\\,\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)as the*neighborhood consistency regularizer*and consider the penalized formulation

minfθ⁡ℒ⁡\(θ\)=ℒtask​\(θ\)\+λ​Rn​\(p\),\\min\_\{f\_\{\\theta\}\}\\;\\mathcal\{L\}\(\\theta\)=\\mathcal\{L\}\_\{\\mathrm\{task\}\}\(\\theta\)\+\\lambda\\,R\_\{n\}\(p\),\(1\)whereℒtask\\mathcal\{L\}\_\{\\mathrm\{task\}\}is the task loss, andλ\>0\\lambda\>0controls the regularization strength\.

### 3\.2Formal Guarantees

We relate predictive disagreement to confidence variation and pointwise calibration\-error propagation on the training graph, at a fixed predictor\. Held\-out calibration is assessed empirically in Section[6](https://arxiv.org/html/2610.09020#S6)\. Proofs and the local quadratic interpretation of the regularizer appear in Appendix[A](https://arxiv.org/html/2610.09020#A1)\. We make the standing requirement thatD\(p∥q\)≥CD∥p−q∥12\\mathrm\{D\}\(p\\\|q\)\\geq C\_\{\\mathrm\{D\}\}\\\|p\-q\\\|\_\{1\}^\{2\}for allp,q∈ΔK−1p,q\\in\\Delta^\{K\-1\}, withCD\>0C\_\{\\mathrm\{D\}\}\>0\. This is a Pinsker\-type condition that implies\|ci−cj\|≤12​‖pi−pj‖1≤12​CD​D\(pi∥pj\)\|c\_\{i\}\-c\_\{j\}\|\\leq\\tfrac\{1\}\{2\}\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{1\}\\leq\\frac\{1\}\{2\\sqrt\{C\_\{\\mathrm\{D\}\}\}\}\\sqrt\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\}, which bounds the propagation of pointwise calibration error:

###### Proposition 1\(Calibration Propagation\)

Under the standing domination condition, for any edge\(i,j\)∈E\(i,j\)\\in E,

ℰcalj≤ℰcali\+12​CD​D\(pi∥pj\)\+\|ai−aj\|\\mathcal\{E\}\_\{\{\\mathrm\{cal\}\}\_\{j\}\}\\;\\leq\\;\\ \\mathcal\{E\}\_\{\{\\mathrm\{cal\}\}\_\{i\}\}\+\\frac\{1\}\{2\\sqrt\{C\_\{\\mathrm\{D\}\}\}\}\\sqrt\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\}\+\|a\_\{i\}\-a\_\{j\}\|\(2\)

The bound decomposes propagation into three terms: the calibration error already present atii, a smoothness term controlled directly by the regularizer throughD\(pi∥pj\)\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\), and the local variation\|ai−aj\|\|a\_\{i\}\-a\_\{j\}\|of the true correctness probability\. Only the second is directly penalized\. With a fixed graph, the correctness gap depends onθ\\thetaonly through the predicted labels\. A calibrated prediction therefore constrains neighboring errors when connected points have similar correctness probabilities\. The bound, however, does not detail how a calibrated prediction arises in the first place\. This is left to the task loss\.

Theorem[1](https://arxiv.org/html/2610.09020#Thmtheorem1)extends Proposition[1](https://arxiv.org/html/2610.09020#Thmproposition1)from a single edge to paths through the graph:

###### Theorem 1\(Pathwise Calibration Propagation\)

Under the standing domination condition, letP=\(v0,v1,…,vr\)P=\(v\_\{0\},v\_\{1\},\\ldots,v\_\{r\}\)be a simple directed path in𝒢\\mathcal\{G\}fromv0=sv\_\{0\}=stovr=vv\_\{r\}=v\. WriteDℓ:=D\(pvℓ−1∥pvℓ\)D\_\{\\ell\}:=\\mathrm\{D\}\(p\_\{v\_\{\\ell\-1\}\}\\\|p\_\{v\_\{\\ell\}\}\)\. Then

ℰcalv≤ℰcals\+r4​CD​∑ℓ=1rDℓ\+\|av−as\|\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{v\}\}\\leq\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{s\}\}\+\\sqrt\{\\frac\{r\}\{4C\_\{\\mathrm\{D\}\}\}\\sum\_\{\\ell=1\}^\{r\}D\_\{\\ell\}\}\+\|a\_\{v\}\-a\_\{s\}\|\(3\)

A well\-calibrated prediction transfers to points connected by short paths along which both the predictive divergence and the variation of the true correctness probability remain small\. The bound grows with path length and thus the guarantee is most informative locally\. Additional disagreement bounds appear in Appendix[A](https://arxiv.org/html/2610.09020#A1)\(Propositions[5](https://arxiv.org/html/2610.09020#Thmproposition5)and[6](https://arxiv.org/html/2610.09020#Thmproposition6), Corollary[1](https://arxiv.org/html/2610.09020#Thmcorollary1)\)\.

### 3\.3When Smoothing Improves Calibration

Section[3\.2](https://arxiv.org/html/2610.09020#S3.SS2)bounds the propagation of calibration error, given that it is small somewhere\. We next identify an obstruction to calibrating both endpoints of an edge and a condition under which idealized pairwise averaging reduces their summed expected squared calibration error\.

###### Proposition 2\(Smoothing Obstruction\)

Under the standing domination condition, for every edge\(i,j\)∈E\(i,j\)\\in E,

ℰcali\+ℰcalj≥\|ai−aj\|−12​CD​D\(pi∥pj\)\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{i\}\}\+\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{j\}\}\\geq\|a\_\{i\}\-a\_\{j\}\|\-\\frac\{1\}\{2\\sqrt\{C\_\{\\mathrm\{D\}\}\}\}\\sqrt\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\}\(4\)

For fixed predicted labels, reducing predictive disagreement raises the lower bound toward the correctness gap\|ai−aj\|\|a\_\{i\}\-a\_\{j\}\|\. A rising floor, however, does not establish that the realized error rose, as the error may already have exceeded it\. Proposition[2](https://arxiv.org/html/2610.09020#Thmproposition2)therefore identifies the regime in which smoothing is provably obstructed from calibrating both endpoints\. The next result characterizes when averaging reduces expected squared error\.

To analyze the error removed by averaging, we fix an edge and randomize over training\. Let𝒯\\mathcal\{T\}denote randomness from initialization, data order, and stochastic optimization\. We analyze idealized pairwise averaging on a single edge, in which both endpoints adopt the mixturem=12​\(pi\+pj\)m=\\tfrac\{1\}\{2\}\(p\_\{i\}\+p\_\{j\}\)\. Both samples must predict the same fixed label almost surely under𝒯\\mathcal\{T\}, so thataia\_\{i\}andaja\_\{j\}are fixed targets, and then averaging the predictive distributions is equivalent to averaging the confidences,maxk⁡mk=12​\(ci\+cj\)\\max\_\{k\}m\_\{k\}=\\tfrac\{1\}\{2\}\(c\_\{i\}\+c\_\{j\}\)\. The regularizer has access to neitheraanor the labels; it only draws predictions toward one another\. The question is when doing so reduces the summed expected squared calibration error\.

###### Proposition 3\(Smoothing threshold\)

Let\(i,j\)\(i,j\)be an edge withy^i=y^j=k∗\\hat\{y\}\_\{i\}=\\hat\{y\}\_\{j\}=k^\{\\ast\}almost surely under𝒯\\mathcal\{T\}, and writeei:=ci−aie\_\{i\}:=c\_\{i\}\-a\_\{i\},ej:=cj−aje\_\{j\}:=c\_\{j\}\-a\_\{j\}for the pointwise errors\. Replacing both confidences by their average strictly reduces the summed expected squared calibration error of the edge iff

\(ai−aj\)2<𝔼𝒯​\[\(ei−ej\)2\]\.\(a\_\{i\}\-a\_\{j\}\)^\{2\}\\;<\\;\\mathbb\{E\}\_\{\\mathcal\{T\}\}\\bigl\[\(e\_\{i\}\-e\_\{j\}\)^\{2\}\\bigr\]\.\(5\)

A bias–variance decomposition and correlation\-dependent specialization appear in Appendix[A\.9](https://arxiv.org/html/2610.09020#A1.SS9)\. Only the differenceei−eje\_\{i\}\-e\_\{j\}enters equation[5](https://arxiv.org/html/2610.09020#S3.E5)\. Averaging can reduce error that differs between neighbors, whether random or systematic, but cannot touch the component they share, and systematic overconfidence is precisely the error deep classifiers are known to exhibit\([Guo et al\., 2017](https://arxiv.org/html/2610.09020#bib.bib2)\)\. This motivates testing whether gains from neighborhood smoothing remain useful after a global confidence adjustment\. Section[6\.2](https://arxiv.org/html/2610.09020#S6.SS2)evaluates this empirically by applying the same post\-hoc calibrators to every train\-time method\.

## 4NeSCal

### 4\.1The Regularizer

NeSCal instantiates the graph\-smoothing objective of Equation[1](https://arxiv.org/html/2610.09020#S3.E1)with the Jensen–Shannon divergence and akk\-nearest\-neighbor graph over learned representations:

minfθℒ\(θ\)=ℒCE\(θ\)\+λ∑i,jWi​jJS\(pi∥pj\)\\min\_\{f\_\{\\theta\}\}\\;\\mathcal\{L\}\(\\theta\)=\\mathcal\{L\}\_\{\\mathrm\{CE\}\}\(\\theta\)\+\\lambda\\,\\sum\_\{i,j\}W\_\{ij\}\\,\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)\(6\)
whereℒCE\\mathcal\{L\}\_\{\\mathrm\{CE\}\}is the cross\-entropy \(CE\) loss andJS\(pi∥pj\)\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)is the Jensen–Shannon divergence\([Endres and Schindelin, 2003](https://arxiv.org/html/2610.09020#bib.bib10);[Fuglede and Topsoe, 2004](https://arxiv.org/html/2610.09020#bib.bib11)\),JS\(pi∥pj\)=12DKL\(pi∥m\)\+12DKL\(pj∥m\)\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)=\\tfrac\{1\}\{2\}D\_\{\\mathrm\{KL\}\}\(p\_\{i\}\\\|m\)\+\\tfrac\{1\}\{2\}D\_\{\\mathrm\{KL\}\}\(p\_\{j\}\\\|m\), withm=12​\(pi\+pj\)m=\\tfrac\{1\}\{2\}\(p\_\{i\}\+p\_\{j\}\)andDKL\(u∥v\)=∑kuklog\(uk/vk\)D\_\{\\mathrm\{KL\}\}\(u\\\|v\)=\\sum\_\{k\}u\_\{k\}\\log\(u\_\{k\}/v\_\{k\}\)\. We use base\-22logarithms in the JS divergence, so thatJS\(pi∥pj\)∈\[0,1\]\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)\\in\[0,1\]\. Cross\-entropy and reported NLL use natural logarithms\.

The adjacency matrixW∈\{0,1\}n×nW\\in\\\{0,1\\\}^\{n\\times n\}encodes akk\-nearest\-neighbor graph \(kNN\) in the representation space𝒵\\mathcal\{Z\}, withWi​j=1⇔j∈kNN⁡\(i\)W\_\{ij\}=1\\iff j\\in\\mathrm\{kNN\}\(i\)\. For optimization, we average the penalty over the directed edge setE=\{\(i,j\):Wi​j=1\}E=\\\{\(i,j\):W\_\{ij\}=1\\\}, usingR¯n​\(p\)=Rn​\(p\)/\|E\|\\bar\{R\}\_\{n\}\(p\)=R\_\{n\}\(p\)/\|E\|\.

JS divergence satisfies the total\-variation domination condition withCJS=1/\(8​ln⁡2\)C\_\{\\mathrm\{JS\}\}=1/\(8\\ln 2\), allowing the general confidence and propagation bounds to apply\. Its symmetry and bounded range make it a sufficient and practical \(rather than uniquely optimal\) choice for comparing neighboring predictive distributions, while comparison of the full vectors captures disagreement across all classes\. The induced local quadratic geometry is described in Appendix[B](https://arxiv.org/html/2610.09020#A2)\.

### 4\.2Design Choices

#### Neighborhood Graph\.

Neighbors are identified by cosine distance in𝒵\\mathcal\{Z\}\. The graph is constructed once at activation and held fixed, limiting the construction cost and keeping the neighborhoods unchanged during regularized training\. We usek=10k=10by default\. Section[6\.3](https://arxiv.org/html/2610.09020#S6.SS3)examines sensitivity to this choice\.

#### Activation and Schedule\.

The activation epoch is a hyperparameter, swept jointly withλ\\lambdaon validation NLL\. Proposition[2](https://arxiv.org/html/2610.09020#Thmproposition2)motivates delaying graph construction until the representation matures and neighborhoods become semantically meaningful\. If the across\-run dispersion of neighboring errors shrinks as training converges, the tolerable correctness gap narrows \(Proposition[3](https://arxiv.org/html/2610.09020#Thmproposition3)\), motivating a reduction in regularization strength\. NeSCal accordingly activates the regularizer after an initial warm\-up and decaysλ\\lambdaover the remainder of training on a cosine schedule\. Further discussion appears in Appendix[C\.4](https://arxiv.org/html/2610.09020#A3.SS4)\.

#### Task Loss\.

NeSCal uses cross\-entropy as its task loss\. Cross\-entropy is a strictly proper scoring rule: its expected loss is uniquely minimized at the true conditional distribution\. In contrast, the regularizerRnR\_\{n\}is minimized whenever neighboring predictions agree, regardless of their correctness\. Train\-time calibration methods are frequently combined with label smoothing[Szegedy et al\. \(2016\)](https://arxiv.org/html/2610.09020#bib.bib23)or focal loss[Lin et al\. \(2017\)](https://arxiv.org/html/2610.09020#bib.bib24), neither of which is a proper scoring rule since population minimizers generally differ from the true conditional distribution\. Replacing cross\-entropy with either would therefore remove the strictly proper scoring rule that anchors the supervised term to that distribution\. The smoothing penalty, which does not itself identify the correct posterior, would then be susceptible to propagating a shared bias\. We therefore retain a strictly proper task loss\.

### 4\.3Implementation

Figure[1](https://arxiv.org/html/2610.09020#S1.F1)presents an overview of the NeSCal training process\. Training uses minibatch updates with detached neighbor targets: current predictions for neighbors in the minibatch and cached predictions otherwise\. Inference is unchanged\. Algorithm[1](https://arxiv.org/html/2610.09020#alg1)and the full implementation and complexity analysis appear in Appendix[C](https://arxiv.org/html/2610.09020#A3), while measured costs are reported in Appendix[E\.10](https://arxiv.org/html/2610.09020#A5.SS10)\.

## 5Experimental Setup

We compare NeSCal with train\-time calibration methods across image, text, and tabular classification, evaluating each trained model both directly and after post\-hoc recalibration\. This section states the protocol; full implementation details are deferred to Appendix[D](https://arxiv.org/html/2610.09020#A4)\.

### 5\.1Datasets and Architectures

Image\.CIFAR\-10 and CIFAR\-100\([Krizhevsky and Hinton, 2009](https://arxiv.org/html/2610.09020#bib.bib35)\)with ResNet\-50\([He et al\., 2016](https://arxiv.org/html/2610.09020#bib.bib36)\)and WideResNet\-28\-10\([Zagoruyko and Komodakis, 2016](https://arxiv.org/html/2610.09020#bib.bib37)\), and Tiny\-ImageNet\([Le and Yang, 2015](https://arxiv.org/html/2610.09020#bib.bib38)\)with ResNet\-50\.

Text\.20 Newsgroups\([Lang, 1995](https://arxiv.org/html/2610.09020#bib.bib39)\)\(20NG\) with the global\-pooling CNN \(GPCNN\)\([Lin et al\., 2013](https://arxiv.org/html/2610.09020#bib.bib55)\), and Amazon Reviews\([Hou et al\., 2024](https://arxiv.org/html/2610.09020#bib.bib42)\)\(binary and five\-class\) with a multilayer perceptron \(MLP\) head on fine\-tuned and then frozen RoBERTa\-base\([Liu et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib41)\)representations\. In the frozen\-encoder setting, NeSCal regularizes only the head with the encoder representation remaining fixed during training\.

Tabular\.Helena, Jannis\([Guyon et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib45)\)and a subsample of Covertype\([Blackard, 1998](https://arxiv.org/html/2610.09020#bib.bib44)\), with an MLP classifier\.

### 5\.2Compared Methods

We compare with CE and six train\-time calibration methods: focal loss \(FL\)\([Mukhoti et al\., 2020](https://arxiv.org/html/2610.09020#bib.bib20);[Lin et al\., 2017](https://arxiv.org/html/2610.09020#bib.bib24)\), label smoothing \(LS\)\([Müller et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib21);[Szegedy et al\., 2016](https://arxiv.org/html/2610.09020#bib.bib23)\), and margin\-based label smoothing \(MbLS\)\([Liu et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib29)\), which modify the training objective; and MMCE\([Kumar et al\., 2018](https://arxiv.org/html/2610.09020#bib.bib8)\), MDCA\([Hebbalaguppe et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib19)\), and s\-AvUC\([Karandikar et al\., 2021](https://arxiv.org/html/2610.09020#bib.bib17)\), which introduce explicit calibration\-related penalties\. Every method is tuned under the protocol below\. The main comparison uses NeSCal with cosine decay; the constant schedule is examined separately\.

To assess composition, we apply five post\-hoc calibrators to every train\-time method\. Temperature scaling \(TS\)\([Guo et al\., 2017](https://arxiv.org/html/2610.09020#bib.bib2)\)and Dirichlet calibration\([Kull et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib13)\)provide established output\-based adjustments, ranging from a single temperature to a multiclass calibration map\. LoCal Nets \(LoCal\)\([Barbera et al\., 2026](https://arxiv.org/html/2610.09020#bib.bib33)\), LoRe\([Luo et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib14)\), and DAC\([Tomani et al\., 2023](https://arxiv.org/html/2610.09020#bib.bib31)\)use local or density information in representation space\. Together, they test whether the gains from training persist under both global output recalibration and recalibration informed by representations\.

### 5\.3Model Selection and Evaluation

All methods are trained for a fixed epoch budget, except the frozen\-encoder heads, which use early stopping\. Checkpoints and hyperparameters are selected by validation NLL, a strictly proper scoring rule for the full predictive distribution\([Gneiting and Raftery, 2007](https://arxiv.org/html/2610.09020#bib.bib47)\)\. ECE measures agreement between confidence and empirical correctness after aggregation and can therefore be low when predictions are uninformative\. This distinction matters when selecting regularization strength: strong smoothing can reduce ECE while degrading accuracy and NLL\. Selection by a proper score remains sensitive to this loss of predictive information, although it can select a configuration with higher ECE\. The shared criterion gives every method a common selection target, although this may differ from its original protocol\. Appendix[E\.9](https://arxiv.org/html/2610.09020#A5.SS9)examines the resulting trade\-offs\.

The primary evaluation metric is also NLL, for the reasons stated above\. We additionally report the multiclass Brier score\([Brier, 1950](https://arxiv.org/html/2610.09020#bib.bib48)\), accuracy and ECE, alongside secondary calibration metrics and the calibration–refinement decomposition\([Kull and Flach, 2015](https://arxiv.org/html/2610.09020#bib.bib46)\)in Appendix[D\.6](https://arxiv.org/html/2610.09020#A4.SS6)\. We assess consistency across settings rather than significance within them \(Appendix[E\.2](https://arxiv.org/html/2610.09020#A5.SS2)\)\.

## 6Experimental Results

We address three questions: whether NeSCal improves calibration and predictive quality relative to CE and the competing train\-time methods \(Section[6\.1](https://arxiv.org/html/2610.09020#S6.SS1)\); whether these gains persist after post\-hoc recalibration \(Section[6\.2](https://arxiv.org/html/2610.09020#S6.SS2)\); and whether they depend on the neighborhood structure the theory relies on, and in which settings they arise \(Sections[6\.3](https://arxiv.org/html/2610.09020#S6.SS3)and[6\.4](https://arxiv.org/html/2610.09020#S6.SS4)\)\. Tables report test\-set means, with standard deviations where shown\. Unless a caption states otherwise, bold and italic identify the best and second\-best distinct means, with ties at displayed precision sharing emphasis\. NeSCal denotes the cosine schedule on real\-data benchmarks; synthetic results use the constant schedule\. R50 and WRN denote ResNet\-50 and WideResNet\-28\-10, respectively\.

### 6\.1Predictive quality and calibration

NeSCal improves NLL over CE in all eleven main benchmark settings and Brier score in ten \(Table[1](https://arxiv.org/html/2610.09020#S6.T1); Appendix Table[6](https://arxiv.org/html/2610.09020#A5.T6)\)\. These gains generally accompany improved accuracy, which increases in nine settings, including seven of the eight image and tabular settings \.

Against the other train\-time methods, NeSCal ranks first or second \(including ties\) on NLL in seven of the eight image and tabular settings\. Because NLL rewards both calibration and accuracy, we assess calibration improvement by asking whether any method is better on both at once\. A method is Pareto\-optimal if no other method is at least as accurate and has no higher ECE, with strict improvement in at least one\. NeSCal is Pareto\-optimal in five of the eight image and tabular settings both before and after TS\. Base\-model ECE is where NeSCal is least competitive; Section[6\.2](https://arxiv.org/html/2610.09020#S6.SS2)shows that this ranking changes once confidence scale is corrected\. On the text benchmarks, NeSCal’s improvements over CE are smaller and do not yield the lowest NLL against other methods\. Section[6\.4](https://arxiv.org/html/2610.09020#S6.SS4)characterizes this variation across settings\.

Table 1:Test NLL before and after validation\-fitted temperature scaling; lower is better\. Values are means±\\pmSD over seeds, with seed counts specified in Appendix[D](https://arxiv.org/html/2610.09020#A4)\.
### 6\.2Composition Under Post\-hoc Recalibration

An important assessment of a train\-time calibration method is whether its gains survive the cheapest post\-hoc correction\. After the same validation\-fitted temperature scaling is applied to every trained model, NeSCal retains lower NLL than CE in all eleven settings and has the lowest NLL among the evaluated methods in seven of the eight image and tabular settings \(Table[1](https://arxiv.org/html/2610.09020#S6.T1)\)\. On CIFAR\-10 and CIFAR\-100 with ResNet\-50, its NLL is respectively6\.3%6\.3\\%and6\.0%6\.0\\%below the strongest evaluated baseline after TS\. TS compresses the spread of base\-model ECE across methods on CIFAR from2\.22\.2–6\.36\.3to0\.50\.5–1\.11\.1percentage points, after which NeSCal ranks first or second in five of the eight image and tabular settings \(Table[7](https://arxiv.org/html/2610.09020#A5.T7)\)\. Much of the base\-model ECE advantage of methods that lower confidence directly is therefore recoverable by a single temperature, whereas NeSCal retains its NLL advantage after recalibration\.

This pattern is consistent with the distinction motivated by Proposition[3](https://arxiv.org/html/2610.09020#Thmproposition3): neighborhood smoothing can reduce variation in neighboring prediction errors, while temperature scaling adjusts confidence globally\. On the two CIFAR settings with ResNet\-50, NeSCal reduces the reported proxy for neighboring\-error dispersion and the across\-seed spread of individual confidences \(Appendix[E\.11](https://arxiv.org/html/2610.09020#A5.SS11)\)\. Failure\-prediction AUROC and AURC also improve over cross\-entropy in all five image settings \(Appendix[E\.5](https://arxiv.org/html/2610.09020#A5.SS5)\)\. These observations support a change in predictive quality beyond confidence scale, without requiring the two interventions to act on strictly independent error components\.

The broader composition analysis supports the same conclusion: NeSCal attains the lowest best\-observed post\-hoc NLL in six of eight tabular and image settings when each training method is evaluated with all five post\-hoc calibrators \(Appendix[E\.4](https://arxiv.org/html/2610.09020#A5.SS4)\)\.

### 6\.3Design choices, cost and robustness

If NeSCal’s gains came from indiscriminate smoothing, a random graph of the same degree would reproduce them\. It does not: kNN graphs yield lower NLL than random graphs in all three evaluated controls \(Appendix[E\.7](https://arxiv.org/html/2610.09020#A5.SS7), Table[14](https://arxiv.org/html/2610.09020#A5.T14)\)\. This is consistent with the role of the correctness gap\|ai−aj\|\|a\_\{i\}\-a\_\{j\}\|in Section[3](https://arxiv.org/html/2610.09020#S3): random edges connect points whose correctness probabilities differ, where enforcing agreement conflicts with calibration\.

The validation\-selected cosine configurations improve NLL, Brier score, and accuracy over constant regularization in all four CIFAR settings \(Table[15](https://arxiv.org/html/2610.09020#A5.T15)\)\. Yet constant regularization yields lower final edge JS on both evaluated ResNet\-50 graphs \(Table[19](https://arxiv.org/html/2610.09020#A5.T19)\): stronger final agreement does not necessarily imply better predictive quality\. Sensitivity to neighborhood size is smaller than the NLL gain over CE in six of eight ablated settings\.

No divergence uniformly dominates in the stable, scale\-matched comparisons and activation rules do not transfer consistently across settings, so activation epoch and regularization strength remain validation\-selected hyperparameters \(Appendices[E\.7](https://arxiv.org/html/2610.09020#A5.SS7)and[E\.8](https://arxiv.org/html/2610.09020#A5.SS8)\)\.

The regularizer is inexpensive per run: on CIFAR, the reported training\-time increase is below1%1\\%, with one\-time graph construction taking1717–2727seconds per run\. Inference time is unchanged \(Appendix[E\.10](https://arxiv.org/html/2610.09020#A5.SS10)\)\.

Finally, we test whether the gains survive shift and perturbation\. Under input corruption, NeSCal improves NLL and ECE over CE in four of five image settings, with higher accuracy in all five\. On those same 4, it leads all other methods in accuracy\. Under label noise, ECE improves in all four evaluated conditions, while NLL improves in only one\. The gains therefore depend on the evaluation conditions and metric \(Appendices[E\.6](https://arxiv.org/html/2610.09020#A5.SS6)and[E\.12](https://arxiv.org/html/2610.09020#A5.SS12)\)\.

### 6\.4When does NeSCal help?

Proposition[3](https://arxiv.org/html/2610.09020#Thmproposition3)motivates two requirements for useful smoothing: neighboring errors must differ enough for averaging to help, and the connected points must have sufficiently similar correctness probabilities\. Table[2](https://arxiv.org/html/2610.09020#S6.T2)describes these conditions through mean edge JS and edge agreement \(\|E\|−1∑\(i,j\)∈E𝟏\{y^i=y^j\}\|E\|^\{\-1\}\\sum\_\{\(i,j\)\\in E\}\\mathbf\{1\}\\\{\\hat\{y\}\_\{i\}=\\hat\{y\}\_\{j\}\\\}\), measured at activation before regularized training\. These diagnostics characterize the available disagreement and neighborhood coherence, although neither directly measures the true correctness gap\. The small gains on Covertype, Jannis, and the two Amazon settings are consistent with limited disagreement to remove: their mean edge JS is only0\.0030\.003–0\.0130\.013, and their relative NLL improvements are0\.40\.4–0\.8%0\.8\\%\. Tiny\-ImageNet presents a different limitation\. Its high disagreement occurs on the least coherent graph, increasing the risk that smoothing suppresses genuine differences between neighbors as well as estimation error\. Its relative NLL gain is3\.9%3\.9\\%, compared with8\.68\.6–11\.7%11\.7\\%across the four CIFAR settings\.

Helena and 20NG show why graph diagnostics alone are insufficient\. Although their numbers overlap those of the CIFAR settings, their gains are much smaller\. A plausible additional factor is how the model responds to the consistency constraint\. The tabular MLPs learn representations, but through much shallower networks than the image ResNets\. The 20NG model learns convolutional features over frozen word embeddings, while the Amazon models retain a frozen sentence encoder\. These architectures offer different scopes for adapting features under the regularizer, including learning rate schedules\. If neighboring predictions share errors rooted in the available features, making them agree will not correct those errors; the constraint would be satisfied by changing output probabilities\. The observed gains are consistent with this limitation: they are3\.93\.9–11\.7%11\.7\\%in the image settings and0\.40\.4–1\.3%1\.3\\%in the tabular and text settings\. Local consistency is therefore most useful when disagreement is both removable and supported by an informative, learnable representation\.

Table 2:Mean edge JS and Edge Agreement at activation and improvement in test NLL over CE, sorted by edge JS\. PositiveΔ\\DeltaNLL indicates improvement; the relative column is as a fraction of CE’s NLL\.

## 7Related Work

#### Train\-time calibration\.

Train\-time approaches encourage calibration either explicitly or implicitly, through some modification to training\. Explicit objectives include accuracy\-uncertainty alignment\([Krishnan and Tickoo, 2020](https://arxiv.org/html/2610.09020#bib.bib18)\), matching average class probabilities to empirical frequencies\([Hebbalaguppe et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib19)\), and penalizing confidence\-accuracy discrepancies through soft binning\([Karandikar et al\., 2021](https://arxiv.org/html/2610.09020#bib.bib17)\)or a Huber loss\([Patra et al\., 2023](https://arxiv.org/html/2610.09020#bib.bib27)\)\. Other approaches construct differentiable calibration objectives using kernel methods\([Kumar et al\., 2018](https://arxiv.org/html/2610.09020#bib.bib8);[Popordanoska et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib26)\)\. Distribution matching\([Marx et al\., 2023](https://arxiv.org/html/2610.09020#bib.bib28)\)provides a flexible calibration objective whose conditioning kernels can explicitly target local calibration\. Implicit approaches improve calibration without directly optimizing a calibration discrepancy, including FL\([Lin et al\., 2017](https://arxiv.org/html/2610.09020#bib.bib24);[Mukhoti et al\., 2020](https://arxiv.org/html/2610.09020#bib.bib20)\), LS\([Szegedy et al\., 2016](https://arxiv.org/html/2610.09020#bib.bib23);[Müller et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib21)\), and MbLS\([Liu et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib29)\)\. SVLS\([Islam and Glocker, 2021](https://arxiv.org/html/2610.09020#bib.bib34)\)constructs soft targets from neighboring ground\-truth pixel labels and NACL\([Murugesan et al\., 2025](https://arxiv.org/html/2610.09020#bib.bib30)\)penalizes logit deviations from a neighborhood\-derived prior; both utilize local pixel structure specifically in image segmentation tasks\. Mixup\([Zhang et al\., 2018](https://arxiv.org/html/2610.09020#bib.bib53)\)is also known to improve calibration\([Thulasidasan et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib54)\), but it is an augmentation protocol rather than an objective: it synthesizes inputs and interpolates labels between randomly paired samples\. In comparison, NeSCal uses representation\-space neighborhoods to regularize agreement between full predictive distributions, with no label access required by the smoothing penalty\.

#### Local post\-hoc calibration\.

Several post\-hoc methods exploit locality to recalibrate pretrained models\. LoRe\([Luo et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib14)\)estimates correctness through kernel\-weighted local accuracy, while KCal\([Lin et al\., 2023](https://arxiv.org/html/2610.09020#bib.bib32)\)constructs class probabilities using kernel density estimates in a learned metric space and DAC\([Tomani et al\., 2023](https://arxiv.org/html/2610.09020#bib.bib31)\)uses density proxies from hidden representations to determine sample\-dependent temperatures\. LoCal\([Barbera et al\., 2026](https://arxiv.org/html/2610.09020#bib.bib33)\)learns a representation projection and new logits, aligning predictions with kernel\-estimated local class frequencies using Jensen–Shannon distance\. LoCal is the post\-hoc method most closely related to NeSCal , but the objectives differ: LoCal matches predictions to local label\-frequency estimates in a separate calibration stage, directly targeting multiclass local calibration, whereas NeSCal encourages agreement between neighboring predictive distributions during classifier training, connecting predictive smoothness to confidence variation and the propagation of pointwise confidence calibration\.

#### Graph regularization\.

Our approach builds on graph\-based manifold regularization, which encourages functions to vary smoothly over the data geometry\([Belkin et al\., 2006](https://arxiv.org/html/2610.09020#bib.bib12)\)\. Similarly, NCR\([Iscen et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib56)\)penalizes the KL divergence between each example’s prediction and the similarity\-weighted average of its neighbors’ predictions, so that neighbors can override noisy labels\. A mixture target can be met while individual neighbors disagree sharply; NeSCal instead penalizes every edge, as its bounds require\. NCR also draws neighbors from the current minibatch, whereas NeSCal fixes a kNN graph over the full training set, and NCR targets accuracy under label noise rather than calibration\.

## 8Conclusion

We introduced graph smoothing for train\-time calibration and instantiated it as NeSCal, a JS neighborhood regularizer with no inference\-time cost\. Our analysis bounds pointwise calibration\-error propagation and identifies both an obstruction and a condition for improvement under idealized averaging\. Across eleven benchmarks, NeSCal improves NLL over CE; after TS, it attains the lowest NLL among the evaluated train\-time methods in seven of eight image and tabular settings\. Three findings characterize the method’s behavior: representation\-based neighborhoods yield lower NLL than random neighborhoods; stronger final neighborhood agreement does not necessarily imply better predictive quality; and gains persist under a common temperature\-scaling procedure\. Together, these findings support neighborhood smoothing as a framework for train\-time calibration and predictive improvement\.

Future work should consider further development of the graph smoothing framework, and may focus on encouraging smoothing in the regimes where it helps, either through the penalty or through hyperparameters, and suppressing it on harmful edges\. It would also be worthwhile to investigate alternative graph\-based penalties or underlying graph structures, and explore graph smoothing on benchmarks where the graph is inherent rather than induced\.

## Reproducibility Statement

We provide all materials necessary to reproduce and verify our results throughout the paper and appendices\. Appendix[D](https://arxiv.org/html/2610.09020#A4)specifies data processing, splits, architectures, optimization, search spaces, selection, metrics, and hardware\. Appendix[C](https://arxiv.org/html/2610.09020#A3)and Algorithm[1](https://arxiv.org/html/2610.09020#alg1)specify NeSCal; Appendices[A](https://arxiv.org/html/2610.09020#A1)and[B](https://arxiv.org/html/2610.09020#A2)provide assumptions and proofs\. Seed counts and split conventions, including exceptions, are documented in Appendix[D](https://arxiv.org/html/2610.09020#A4)\. Run manifests identify selected configurations and reported aggregates\. Reproduction code is available at[https://github\.com/idanh8/nescal\_paper\_code](https://github.com/idanh8/nescal_paper_code)\.

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## Appendix AFurther Theoretical Results and Proofs

### A\.1Graph Laplacian Approximation

###### Definition 1

Given an unweighted, symmetric adjacency matrixW∈\{0,1\}n×nW\\in\\\{0,1\\\}^\{n\\times n\}, the graph Laplacian isL:=diag⁡\(d1,…,dn\)−WL:=\\operatorname\{diag\}\(d\_\{1\},\\ldots,d\_\{n\}\)\-W, wheredi:=∑jWi​jd\_\{i\}:=\\sum\_\{j\}W\_\{ij\}\. For any functionu:V→ℝu:V\\to\\mathbb\{R\}on the nodes, the Laplacian quadratic form

u⊤​L​u=12​∑i,jWi​j​\(ui−uj\)2u^\{\\top\}Lu=\\frac\{1\}\{2\}\\sum\_\{i,j\}W\_\{ij\}\\left\(u\_\{i\}\-u\_\{j\}\\right\)^\{2\}\(7\)measures the smoothness ofuuover the graph, growing asuuvaries more sharply between neighboring nodes\.

###### Proposition 4\(Graph Laplacian Approximation\)

LetD:ΔK−1×ΔK−1→ℝ≥0\\mathrm\{D\}:\\Delta^\{K\-1\}\\times\\Delta^\{K\-1\}\\to\\mathbb\{R\}\_\{\\geq 0\}satisfyD\(p∥p\)=0\\mathrm\{D\}\(p\\\|p\)=0and, for someε\>0\\varepsilon\>0, letD\(p∥⋅\)\\mathrm\{D\}\(p\\\|\\cdot\)be three times continuously differentiable on a neighborhood ofΔε:=\{u∈ΔK−1:mink⁡uk≥ε\}\\Delta\_\{\\varepsilon\}:=\\\{u\\in\\Delta^\{K\-1\}:\\min\_\{k\}u\_\{k\}\\geq\\varepsilon\\\}inℝK\\mathbb\{R\}^\{K\}, with third derivatives jointly continuous in\(p,q\)\(p,q\)onΔε×Δε\\Delta\_\{\\varepsilon\}\\times\\Delta\_\{\\varepsilon\}; writeM3\(ε\):=supp,q∈Δεsup‖u‖2=1\|∇q3D\(p∥q\)\[u,u,u\]\|M\_\{3\}\(\\varepsilon\):=\\sup\_\{p,q\\in\\Delta\_\{\\varepsilon\}\}\\sup\_\{\\\|u\\\|\_\{2\}=1\}\\bigl\|\\nabla\_\{q\}^\{3\}\\mathrm\{D\}\(p\\\|q\)\[u,u,u\]\\bigr\|, which is finite by compactness\. Suppose the predictive distributions obey the probability floormini,k⁡pi,k≥ε\\min\_\{i,k\}p\_\{i,k\}\\geq\\varepsilon\. Define the induced local form:GD\(p\):=12∇q2D\(p∥q\)\|q=pG\_\{\\mathrm\{D\}\}\(p\):=\\tfrac\{1\}\{2\}\\nabla\_\{q\}^\{2\}\\mathrm\{D\}\(p\\\|q\)\\big\|\_\{q=p\}\. Then for any edge\(i,j\)∈E\(i,j\)\\in E,

D\(pi∥pj\)=\(pi−pj\)𝖳GD\(pi\)\(pi−pj\)\+ρi​j\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)=\(p\_\{i\}\-p\_\{j\}\)^\{\\mathsf\{T\}\}G\_\{\\mathrm\{D\}\}\(p\_\{i\}\)\(p\_\{i\}\-p\_\{j\}\)\+\\rho\_\{ij\}\(8\)where\|ρi​j\|≤16​M3​\(ε\)​‖pi−pj‖23\|\\rho\_\{ij\}\|\\leq\\tfrac\{1\}\{6\}M\_\{3\}\(\\varepsilon\)\\,\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{2\}^\{3\}\. Consequently,

Rn​\(p\)=∑i,jWi​j​\(pi−pj\)𝖳​GD​\(pi\)​\(pi−pj\)\+ρ,\|ρ\|≤16​M3​\(ε\)​∑i,jWi​j​‖pi−pj‖23\.R\_\{n\}\(p\)=\\sum\_\{i,j\}W\_\{ij\}\\,\(p\_\{i\}\-p\_\{j\}\)^\{\\mathsf\{T\}\}G\_\{\\mathrm\{D\}\}\(p\_\{i\}\)\\,\(p\_\{i\}\-p\_\{j\}\)\+\\rho,\\qquad\|\\rho\|\\leq\\tfrac\{1\}\{6\}M\_\{3\}\(\\varepsilon\)\\sum\_\{i,j\}W\_\{ij\}\\,\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{2\}^\{3\}\.\(9\)

ExpandingD\\mathrm\{D\}on each edge about the source prediction shows thatRn​\(p\)R\_\{n\}\(p\)is, up to an explicitly bounded remainder, a weighted graph\-Laplacian quadratic form, giving it a geometric interpretation as a smoothness penalty over some representation manifoldℳ⊆𝒵\\mathcal\{M\}\\subseteq\\mathcal\{Z\}\. The result holds for any sufficiently regular divergence; the specific weighting is determined by the divergence’s local curvature\. The expansion is anchored at the source predictionpip\_\{i\}, weighting edge\(i,j\)\(i,j\)byGD​\(pi\)G\_\{\\mathrm\{D\}\}\(p\_\{i\}\)\. Sincepi=fθ​\(xi\)p\_\{i\}=f\_\{\\theta\}\(x\_\{i\}\), minimizingRnR\_\{n\}acts as a smoothness penalty onfθf\_\{\\theta\}\. Its magnitude depends on both the predictive distributions and the edge weights\.

Proposition[4](https://arxiv.org/html/2610.09020#Thmproposition4)expressesRnR\_\{n\}as a weighted graph\-Laplacian\-like smoothness penalty and determines the weightingGD​\(pi\)G\_\{\\mathrm\{D\}\}\(p\_\{i\}\), but no result below depends on it; the guarantees follow instead from Proposition[5](https://arxiv.org/html/2610.09020#Thmproposition5)\. The quadratic form is an accurate approximation only where\|pi,k−pj,k\|≪pi,k\|p\_\{i,k\}\-p\_\{j,k\}\|\\ll p\_\{i,k\}; it is an interpretive device, not a bound we optimize\.

#### Proof\.

Fix an edge\(i,j\)\(i,j\)and writep:=pip:=p\_\{i\},q:=pjq:=p\_\{j\}, andδ:=q−p\\delta:=q\-p\. Sincep,q∈ΔK−1p,q\\in\\Delta^\{K\-1\}are probability vectors, they satisfy∑k=1Kpk=∑k=1Kqk=1\\sum\_\{k=1\}^\{K\}p\_\{k\}=\\sum\_\{k=1\}^\{K\}q\_\{k\}=1, so𝟏⊤​δ=∑kδk=0\\mathbf\{1\}^\{\\top\}\\delta=\\sum\_\{k\}\\delta\_\{k\}=0\. The expansion below is Taylor’s theorem with Lagrange remainder applied tot↦D\(p∥p\+tδ\)t\\mapsto\\mathrm\{D\}\(p\\\|p\+t\\delta\); it is exact for anyδ\\delta\.

We take the multivariate Taylor expansion ofD\(p∥q\)D\(p\\\|q\)aboutq=pq=pin its second argument:

D\(p∥q\)=D\(p∥p\)\+∇qD\(p∥q\)\|q=p⊤δ\+12δ⊤∇q2D\(p∥q\)\|q=pδ\+ρi​j\\mathrm\{D\}\(p\\\|q\)=\\mathrm\{D\}\(p\\\|p\)\+\\nabla\_\{q\}\\mathrm\{D\}\(p\\\|q\)\\big\|\_\{q=p\}^\{\\top\}\\delta\+\\tfrac\{1\}\{2\}\\delta^\{\\top\}\\nabla\_\{q\}^\{2\}\\mathrm\{D\}\(p\\\|q\)\\big\|\_\{q=p\}\\,\\delta\+\\rho\_\{ij\}\(10\)
*Zeroth order\.*By hypothesisD\(p∥p\)=0\\mathrm\{D\}\(p\\\|p\)=0\.

*First order\.*For fixedpp,D\(p∥q\)\\mathrm\{D\}\(p\\\|q\)is nonnegative in its second argument and equals00atq=pq=p, soq=pq=pminimizesD\(p∥⋅\)\\mathrm\{D\}\(p\\\|\\cdot\)over the simplex\. Sincepplies in the interior, the only binding constraint at this minimum is that probabilities sum to one,𝟏⊤​q=1\\mathbf\{1\}^\{\\top\}q=1\. Stationarity then forces the gradient to be constant across coordinates,∇qD\(p∥q\)\|q=p=μ𝟏\\nabla\_\{q\}\\mathrm\{D\}\(p\\\|q\)\\big\|\_\{q=p\}=\\mu\\mathbf\{1\}for someμ∈ℝ\\mu\\in\\mathbb\{R\}\. Becausepip\_\{i\}andpjp\_\{j\}are both probability vectors, their difference satisfies𝟏⊤​δ=∑kδk=0\\mathbf\{1\}^\{\\top\}\\delta=\\sum\_\{k\}\\delta\_\{k\}=0, and hence the first\-order term vanishes:

∇qD\(p∥q\)\|q=p⊤δ=μ𝟏⊤δ=0\.\\nabla\_\{q\}\\mathrm\{D\}\(p\\\|q\)\\big\|\_\{q=p\}^\{\\top\}\\delta=\\mu\\mathbf\{1\}^\{\\top\}\\delta=0\.
*Second order\.*By the definition ofGDG\_\{\\mathrm\{D\}\},

12δ⊤∇q2D\(p∥q\)\|q=pδ=δ⊤GD\(p\)δ=\(pi−pj\)⊤GD\(pi\)\(pi−pj\),\\frac\{1\}\{2\}\\delta^\{\\top\}\\nabla\_\{q\}^\{2\}\\mathrm\{D\}\(p\\\|q\)\\big\|\_\{q=p\}\\delta=\\delta^\{\\top\}G\_\{\\mathrm\{D\}\}\(p\)\\delta=\(p\_\{i\}\-p\_\{j\}\)^\{\\top\}G\_\{\\mathrm\{D\}\}\(p\_\{i\}\)\(p\_\{i\}\-p\_\{j\}\),\(11\)with the last equality holding because the quadratic form is even inδ\\delta\. Sinceq=pq=pminimizesD\(p∥⋅\)\\mathrm\{D\}\(p\\\|\\cdot\)over the simplex, the second\-order necessary condition givesδ⊤​GD​\(p\)​δ≥0\\delta^\{\\top\}G\_\{\\mathrm\{D\}\}\(p\)\\delta\\geq 0for every admissible differenceδ\\delta\(those with∑kδk=0\\sum\_\{k\}\\delta\_\{k\}=0\); that is,GD​\(p\)G\_\{\\mathrm\{D\}\}\(p\)is positive semidefinite on the tangent space of the simplex, though not necessarily positive definite as a matrix\.

*Remainder\.*By Taylor’s theorem with Lagrange remainder,ρi​j=16∇q3D\(p∥q\)\|q=ξ\[δ,δ,δ\]\\rho\_\{ij\}=\\tfrac\{1\}\{6\}\\nabla\_\{q\}^\{3\}\\mathrm\{D\}\(p\\\|q\)\\big\|\_\{q=\\xi\}\[\\delta,\\delta,\\delta\]for someξ\\xion the segment betweenpip\_\{i\}andpjp\_\{j\}\. Both endpoints lie inΔε\\Delta\_\{\\varepsilon\}, which is convex, soξ∈Δε\\xi\\in\\Delta\_\{\\varepsilon\}and\|ρi​j\|≤16​M3​\(ε\)​‖δ‖23\|\\rho\_\{ij\}\|\\leq\\tfrac\{1\}\{6\}M\_\{3\}\(\\varepsilon\)\\\|\\delta\\\|\_\{2\}^\{3\}\. For log\-based divergences the third derivatives grow as a coordinate approaches zero, which is why the floor is required\. Summing over edges withWi​j≥0W\_\{ij\}\\geq 0gives the bound onρ\\rho\.□\\square

### A\.2Predictive Smoothness

###### Definition 2\(Quadratic total\-variation domination\)

Forp,q∈ΔK−1p,q\\in\\Delta^\{K\-1\}, letTV⁡\(p,q\):=12​‖p−q‖1\\operatorname\{TV\}\(p,q\):=\\tfrac\{1\}\{2\}\\\|p\-q\\\|\_\{1\}denote the total variation distance\. A divergenceD\\mathrm\{D\}onΔK−1\\Delta^\{K\-1\}*quadratically dominates total variation*with constantCD\>0C\_\{\\mathrm\{D\}\}\>0if

D\(p∥q\)≥4CDTV\(p,q\)2∀p,q∈ΔK−1,\\mathrm\{D\}\(p\\\|q\)\\;\\geq\\;4\\,C\_\{\\mathrm\{D\}\}\\operatorname\{TV\}\(p,q\)^\{2\}\\qquad\\forall\\,p,q\\in\\Delta^\{K\-1\},\(12\)equivalentlyD\(p∥q\)≥CD∥p−q∥12\\mathrm\{D\}\(p\\\|q\)\\geq C\_\{\\mathrm\{D\}\}\\\|p\-q\\\|\_\{1\}^\{2\}\.

This is a Pinsker\-type condition requiringD\\mathrm\{D\}to grow at least quadratically in total variation, which rules out divergences that vanish faster than‖p−q‖12\\\|p\-q\\\|\_\{1\}^\{2\}near the diagonal\.

###### Lemma 1\(Predictive\-distribution smoothness\)

LetD\\mathrm\{D\}dominate squared total variation with constantCDC\_\{\\mathrm\{D\}\}according to Definition[2](https://arxiv.org/html/2610.09020#Thmdefinition2)\. Then, for every edge\(i,j\)∈E\(i,j\)\\in E,

‖pi−pj‖1≤1CD​D\(pi∥pj\)\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{1\}\\leq\\frac\{1\}\{\\sqrt\{C\_\{\\mathrm\{D\}\}\}\}\\sqrt\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\}\(13\)

#### Proof\.

By Definition[2](https://arxiv.org/html/2610.09020#Thmdefinition2),D\(pi∥pj\)≥CD∥pi−pj∥12\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\\geq C\_\{\\mathrm\{D\}\}\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{1\}^\{2\}\. Since both sides are non\-negative, taking square roots and rearranging gives

‖pi−pj‖1≤1CD​D\(pi∥pj\)\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{1\}\\leq\\frac\{1\}\{\\sqrt\{C\_\{\\mathrm\{D\}\}\}\}\\sqrt\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\}\(14\)□\\square

### A\.3Confidence Smoothness

###### Proposition 5\(Confidence Smoothness\)

Under the conditions of Lemma[1](https://arxiv.org/html/2610.09020#Thmlemma1), for every edge\(i,j\)∈E\(i,j\)\\in E,

\|ci−cj\|≤12​‖pi−pj‖1≤12​CD​D\(pi∥pj\)\.\|c\_\{i\}\-c\_\{j\}\|\\leq\\tfrac\{1\}\{2\}\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{1\}\\leq\\frac\{1\}\{2\\sqrt\{C\_\{\\mathrm\{D\}\}\}\}\\sqrt\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\}\.\(15\)

#### Proof\.

The maximum operator is11\-Lipschitz with respect to the supremum norm\. Letki∈arg⁡maxk⁡pi,kk\_\{i\}\\in\\arg\\max\_\{k\}p\_\{i,k\}\. Then

ci−cj=pi,ki−maxk⁡pj,k≤pi,ki−pj,ki≤‖pi−pj‖∞\.c\_\{i\}\-c\_\{j\}=p\_\{i,k\_\{i\}\}\-\\max\_\{k\}p\_\{j,k\}\\leq p\_\{i,k\_\{i\}\}\-p\_\{j,k\_\{i\}\}\\leq\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{\\infty\}\.\(16\)Exchanging the roles ofiiandjjyieldscj−ci≤‖pj−pi‖∞c\_\{j\}\-c\_\{i\}\\leq\\\|p\_\{j\}\-p\_\{i\}\\\|\_\{\\infty\}, and hence

\|ci−cj\|≤‖pi−pj‖∞\|c\_\{i\}\-c\_\{j\}\|\\leq\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{\\infty\}\(17\)Sincepi,pj∈ΔK−1p\_\{i\},p\_\{j\}\\in\\Delta^\{K\-1\}, we have

∑k=1K\(pi,k−pj,k\)=∑k=1Kpi,k−∑k=1Kpj,k=0\.\\sum\_\{k=1\}^\{K\}\(p\_\{i,k\}\-p\_\{j,k\}\)=\\sum\_\{k=1\}^\{K\}p\_\{i,k\}\-\\sum\_\{k=1\}^\{K\}p\_\{j,k\}=0\.\(18\)
Letk⋆∈arg⁡maxk​\|pi,k−pj,k\|k^\{\\star\}\\in\\arg\\max\_\{k\}\|p\_\{i,k\}\-p\_\{j,k\}\|\. Then∑k≠k⋆\(pi,k−pj,k\)=−\(pi,k⋆−pj,k⋆\)\\sum\_\{k\\neq k^\{\\star\}\}\(p\_\{i,k\}\-p\_\{j,k\}\)=\-\(p\_\{i,k^\{\\star\}\}\-p\_\{j,k^\{\\star\}\}\)and we have:

∑k≠k⋆\|pi,k−pj,k\|\\displaystyle\\sum\_\{k\\neq k^\{\\star\}\}\|p\_\{i,k\}\-p\_\{j,k\}\|≥\|∑k≠k⋆\(pi,k−pj,k\)\|=\|pi,k⋆−pj,k⋆\|\\displaystyle\\geq\\left\|\\sum\_\{k\\neq k^\{\\star\}\}\(p\_\{i,k\}\-p\_\{j,k\}\)\\right\|=\|p\_\{i,k^\{\\star\}\}\-p\_\{j,k^\{\\star\}\}\|⟹‖pi−pj‖1\\displaystyle\\Longrightarrow\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{1\}=\|pi,k⋆−pj,k⋆\|\+∑k≠k⋆\|pi,k−pj,k\|\\displaystyle=\|p\_\{i,k^\{\\star\}\}\-p\_\{j,k^\{\\star\}\}\|\+\\sum\_\{k\\neq k^\{\\star\}\}\|p\_\{i,k\}\-p\_\{j,k\}\|≥2​\|pi,k⋆−pj,k⋆\|=2​‖pi−pj‖∞\.\\displaystyle\\geq 2\|p\_\{i,k^\{\\star\}\}\-p\_\{j,k^\{\\star\}\}\|=2\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{\\infty\}\.and thus

‖pi−pj‖∞≤12​‖pi−pj‖1\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{\\infty\}\\leq\\frac\{1\}\{2\}\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{1\}\(19\)
Combining these inequalities with Lemma[1](https://arxiv.org/html/2610.09020#Thmlemma1)gives

\|ci−cj\|≤‖pi−pj‖∞≤12​‖pi−pj‖1≤12​CD​D\(pi∥pj\),\|c\_\{i\}\-c\_\{j\}\|\\leq\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{\\infty\}\\leq\\frac\{1\}\{2\}\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{1\}\\leq\\frac\{1\}\{2\\sqrt\{C\_\{\\mathrm\{D\}\}\}\}\\sqrt\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\},\(20\)□\\square

### A\.4Local Disagreement Bound

###### Proposition 6\(Local Disagreement Bound\)

Letℒ⁡\(θ\)=ℒtask​\(θ\)\+λ​Rn​\(p\)\\mathcal\{L\}\(\\theta\)=\\mathcal\{L\}\_\{\\mathrm\{task\}\}\(\\theta\)\+\\lambda R\_\{n\}\(p\)withℒtask≥0\\mathcal\{L\}\_\{\\mathrm\{task\}\}\\geq 0, and sample an edge\(i,j\)\(i,j\)with probabilityWi​j/\|E\|W\_\{ij\}/\|E\|, where\|E\|=∑i,jWi​j\|E\|=\\sum\_\{i,j\}W\_\{ij\}\. Then for anyt\>0t\>0,

Pr\(D\(pi∥pj\)≥t\)≤Rn​\(p\)\|E\|​t≤ℒ⁡\(θ\)λ​\|E\|​t\\Pr\\left\(\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\\geq t\\right\)\\leq\\frac\{R\_\{n\}\(p\)\}\{\|E\|\\,t\}\\leq\\frac\{\\mathcal\{L\}\(\\theta\)\}\{\\lambda\\,\|E\|\\,t\}\(21\)

#### Proof\.

Sinceℒtask​\(θ\)≥0\\mathcal\{L\}\_\{\\mathrm\{task\}\}\(\\theta\)\\geq 0, rearrangingℒ⁡\(θ\)=ℒtask​\(θ\)\+λ​Rn​\(p\)\\mathcal\{L\}\(\\theta\)=\\mathcal\{L\}\_\{\\mathrm\{task\}\}\(\\theta\)\+\\lambda R\_\{n\}\(p\)gives

Rn​\(p\)=ℒ​\(θ\)−ℒtask​\(θ\)λ≤ℒ⁡\(θ\)λ\.R\_\{n\}\(p\)=\\frac\{\\mathcal\{L\}\(\\theta\)\-\\mathcal\{L\}\_\{\\mathrm\{task\}\}\(\\theta\)\}\{\\lambda\}\\;\\leq\\;\\frac\{\\mathcal\{L\}\(\\theta\)\}\{\\lambda\}\.\(22\)LetX:=D\(pi∥pj\)X:=\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)for an edge sampled with probabilityWi​j/\|E\|W\_\{ij\}/\|E\|\. SinceD≥0\\mathrm\{D\}\\geq 0,XXis a nonnegative random variable with

𝔼\[X\]=1\|E\|∑i,jWi​jD\(pi∥pj\)=Rn​\(p\)\|E\|≤ℒ⁡\(θ\)λ​\|E\|\.\\mathbb\{E\}\[X\]=\\frac\{1\}\{\|E\|\}\\sum\_\{i,j\}W\_\{ij\}\\,\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)=\\frac\{R\_\{n\}\(p\)\}\{\|E\|\}\\;\\leq\\;\\frac\{\\mathcal\{L\}\(\\theta\)\}\{\\lambda\\,\|E\|\}\.\(23\)Markov’s inequality then gives, for anyt\>0t\>0,

Pr⁡\(X≥t\)≤𝔼⁡\[X\]t=Rn​\(p\)\|E\|​t≤ℒ⁡\(θ\)λ​\|E\|​t\\Pr\(X\\geq t\)\\leq\\frac\{\\mathbb\{E\}\[X\]\}\{t\}=\\frac\{R\_\{n\}\(p\)\}\{\|E\|t\}\\leq\\frac\{\\mathcal\{L\}\(\\theta\)\}\{\\lambda\|E\|t\}\(24\)
□\\square

### A\.5Probabilistic Confidence Smoothness Bound

###### Corollary 1\(Probabilistic Confidence Smoothness\)

Under the conditions of Proposition[5](https://arxiv.org/html/2610.09020#Thmproposition5)and Proposition[6](https://arxiv.org/html/2610.09020#Thmproposition6), for anyt\>0t\>0, with probability at least1−ℒ⁡\(θ\)λ​\|E\|​t1\-\\frac\{\\mathcal\{L\}\(\\theta\)\}\{\\lambda\|E\|t\}over a sampled edge\(i,j\)\(i,j\),

\|ci−cj\|≤12​tCD\|c\_\{i\}\-c\_\{j\}\|\\;\\leq\\;\\frac\{1\}\{2\}\\sqrt\{\\frac\{t\}\{C\_\{\\mathrm\{D\}\}\}\}\(25\)

#### Proof\.

By Proposition[6](https://arxiv.org/html/2610.09020#Thmproposition6), the event\{D\(pi∥pj\)<t\}\\\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)<t\\\}holds with probability at least1−ℒ⁡\(θ\)/\(λ​\|E\|​t\)1\-\\mathcal\{L\}\(\\theta\)/\(\\lambda\|E\|t\)\. On this event, Proposition[5](https://arxiv.org/html/2610.09020#Thmproposition5)gives

\|ci−cj\|≤12​CD​D\(pi∥pj\)<12​t/CD\|c\_\{i\}\-c\_\{j\}\|\\leq\\frac\{1\}\{2\\sqrt\{C\_\{\\mathrm\{D\}\}\}\}\\sqrt\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\}<\\frac\{1\}\{2\}\\sqrt\{t/C\_\{\\mathrm\{D\}\}\}\(26\)□\\square

### A\.6Proof of Proposition[1](https://arxiv.org/html/2610.09020#Thmproposition1)

By the triangle inequality,

\|aj−cj\|≤\|aj−ai\|\+\|ai−ci\|\+\|ci−cj\|\.\|a\_\{j\}\-c\_\{j\}\|\\;\\leq\\;\|a\_\{j\}\-a\_\{i\}\|\+\|a\_\{i\}\-c\_\{i\}\|\+\|c\_\{i\}\-c\_\{j\}\|\.\(27\)The second term isℰcali\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{i\}\}and the third is bounded by Proposition[5](https://arxiv.org/html/2610.09020#Thmproposition5)\.□\\square

### A\.7Proof of Theorem[1](https://arxiv.org/html/2610.09020#Thmtheorem1)

By the triangle inequality applied at the endpoints ofPP,

ℰcalv=\|av−cv\|≤\|av−as\|\+\|as−cs\|\+\|cs−cv\|=ℰcals\+\|cs−cv\|\+\|av−as\|\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{v\}\}=\|a\_\{v\}\-c\_\{v\}\|\\leq\|a\_\{v\}\-a\_\{s\}\|\+\|a\_\{s\}\-c\_\{s\}\|\+\|c\_\{s\}\-c\_\{v\}\|=\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{s\}\}\+\|c\_\{s\}\-c\_\{v\}\|\+\|a\_\{v\}\-a\_\{s\}\|\(28\)Telescoping the confidence term alongPPand applying Proposition[5](https://arxiv.org/html/2610.09020#Thmproposition5)to each edge,

\|cs−cv\|≤∑ℓ=1r\|cvℓ−1−cvℓ\|≤12​CD​∑ℓ=1rDℓ\.\|c\_\{s\}\-c\_\{v\}\|\\;\\leq\\;\\sum\_\{\\ell=1\}^\{r\}\\bigl\|c\_\{v\_\{\\ell\-1\}\}\-c\_\{v\_\{\\ell\}\}\\bigr\|\\;\\leq\\;\\frac\{1\}\{2\\sqrt\{C\_\{\\mathrm\{D\}\}\}\}\\sum\_\{\\ell=1\}^\{r\}\\sqrt\{D\_\{\\ell\}\}\.\(29\)By Cauchy–Schwarz,∑ℓ=1rDℓ≤r​∑ℓ=1rDℓ\\sum\_\{\\ell=1\}^\{r\}\\sqrt\{D\_\{\\ell\}\}\\leq\\sqrt\{r\\sum\_\{\\ell=1\}^\{r\}D\_\{\\ell\}\}, so\|cs−cv\|≤r4​CD​∑ℓDℓ\|c\_\{s\}\-c\_\{v\}\|\\leq\\sqrt\{\\tfrac\{r\}\{4C\_\{\\mathrm\{D\}\}\}\\sum\_\{\\ell\}D\_\{\\ell\}\}\. Substituting yields the stated bound\.□\\square

### A\.8Proof of Proposition[2](https://arxiv.org/html/2610.09020#Thmproposition2)

By the triangle inequality,

\|ai−aj\|≤\|ai−ci\|\+\|ci−cj\|\+\|cj−aj\|=ℰcali\+\|ci−cj\|\+ℰcalj\|a\_\{i\}\-a\_\{j\}\|\\;\\leq\\;\|a\_\{i\}\-c\_\{i\}\|\+\|c\_\{i\}\-c\_\{j\}\|\+\|c\_\{j\}\-a\_\{j\}\|=\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{i\}\}\+\|c\_\{i\}\-c\_\{j\}\|\+\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{j\}\}\(30\)Rearranging gives

ℰcali\+ℰcalj≥\|ai−aj\|−\|ci−cj\|\.\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{i\}\}\+\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{j\}\}\\geq\|a\_\{i\}\-a\_\{j\}\|\-\|c\_\{i\}\-c\_\{j\}\|\.\(31\)Substituting\|ci−cj\|≤12​CD​D\(pi∥pj\)\|c\_\{i\}\-c\_\{j\}\|\\leq\\frac\{1\}\{2\\sqrt\{C\_\{\\mathrm\{D\}\}\}\}\\sqrt\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\}from Proposition[5](https://arxiv.org/html/2610.09020#Thmproposition5)yields equation[4](https://arxiv.org/html/2610.09020#S3.E4)\.□\\square

Perfect calibration at both endpoints forces\|ci−cj\|=\|ai−aj\|\|c\_\{i\}\-c\_\{j\}\|=\|a\_\{i\}\-a\_\{j\}\|and hence, by Proposition[5](https://arxiv.org/html/2610.09020#Thmproposition5),D\(pi∥pj\)≥4CD\(ai−aj\)2\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\\geq 4C\_\{\\mathrm\{D\}\}\(a\_\{i\}\-a\_\{j\}\)^\{2\}: no predictor with divergence below this level can be perfectly calibrated at both endpoints unlessai=aja\_\{i\}=a\_\{j\}\.

### A\.9Proposition[3](https://arxiv.org/html/2610.09020#Thmproposition3): Bias–Variance Decomposition and Proof

Under the fixed\-class assumption of Proposition[3](https://arxiv.org/html/2610.09020#Thmproposition3),aia\_\{i\}andaja\_\{j\}are fixed under𝒯\\mathcal\{T\}\. Decompose

ci=ai\+bi\+εi,bi:=𝔼𝒯​\[ci\]−ai,𝔼𝒯​\[εi\]=0,c\_\{i\}=a\_\{i\}\+b\_\{i\}\+\\varepsilon\_\{i\},\\qquad b\_\{i\}:=\\mathbb\{E\}\_\{\\mathcal\{T\}\}\[c\_\{i\}\]\-a\_\{i\},\\qquad\\mathbb\{E\}\_\{\\mathcal\{T\}\}\[\\varepsilon\_\{i\}\]=0,\(32\)and analogously atjj, so that the expected squared pointwise calibration error decomposes as𝔼𝒯​\[\(ci−ai\)2\]=bi2\+Var⁡\(εi\)\\mathbb\{E\}\_\{\\mathcal\{T\}\}\[\(c\_\{i\}\-a\_\{i\}\)^\{2\}\]=b\_\{i\}^\{2\}\+\\mathrm\{Var\}\(\\varepsilon\_\{i\}\)into a systematic and a random part\. Hence the right\-hand side of equation[5](https://arxiv.org/html/2610.09020#S3.E5)is

𝔼𝒯​\[\(ei−ej\)2\]=\(bi−bj\)2\+Var⁡\(εi−εj\)\.\\mathbb\{E\}\_\{\\mathcal\{T\}\}\\bigl\[\(e\_\{i\}\-e\_\{j\}\)^\{2\}\\bigr\]=\(b\_\{i\}\-b\_\{j\}\)^\{2\}\+\\mathrm\{Var\}\(\\varepsilon\_\{i\}\-\\varepsilon\_\{j\}\)\.\(33\)In particular, ifbi=bjb\_\{i\}=b\_\{j\},Var⁡\(εi\)=Var⁡\(εj\)=σ2\\mathrm\{Var\}\(\\varepsilon\_\{i\}\)=\\mathrm\{Var\}\(\\varepsilon\_\{j\}\)=\\sigma^\{2\}andCorr⁡\(εi,εj\)=ρ\\mathrm\{Corr\}\(\\varepsilon\_\{i\},\\varepsilon\_\{j\}\)=\\rho, the condition is\|ai−aj\|<σ​2​\(1−ρ\)\|a\_\{i\}\-a\_\{j\}\|<\\sigma\\sqrt\{2\(1\-\\rho\)\}\.

#### Proof\.

WriteΔ:=ai−aj\\Delta:=a\_\{i\}\-a\_\{j\}andc¯:=12​\(ci\+cj\)\\bar\{c\}:=\\tfrac\{1\}\{2\}\(c\_\{i\}\+c\_\{j\}\), withci=pi,k∗c\_\{i\}=p\_\{i,k^\{\\ast\}\}andcj=pj,k∗c\_\{j\}=p\_\{j,k^\{\\ast\}\}\. For the mixturem=12​\(pi\+pj\)m=\\tfrac\{1\}\{2\}\(p\_\{i\}\+p\_\{j\}\)we havemk∗=c¯m\_\{k^\{\\ast\}\}=\\bar\{c\}, while for every coordinatekk,mk=12​\(pi,k\+pj,k\)≤c¯m\_\{k\}=\\tfrac\{1\}\{2\}\(p\_\{i,k\}\+p\_\{j,k\}\)\\leq\\bar\{c\}\. Hencemaxk⁡mk=c¯\\max\_\{k\}m\_\{k\}=\\bar\{c\}, the predicted class is preserved, andai,aja\_\{i\},a\_\{j\}are unchanged\. Before averaging, the summed expected squared calibration error is𝔼⁡\[ei2\]\+𝔼⁡\[ej2\]\\mathbb\{E\}\[e\_\{i\}^\{2\}\]\+\\mathbb\{E\}\[e\_\{j\}^\{2\}\]\. After, both endpoints receivec¯\\bar\{c\}, with errorsc¯−ai=12​\(ei\+ej\)−12​Δ\\bar\{c\}\-a\_\{i\}=\\tfrac\{1\}\{2\}\(e\_\{i\}\+e\_\{j\}\)\-\\tfrac\{1\}\{2\}\\Deltaandc¯−aj=12​\(ei\+ej\)\+12​Δ\\bar\{c\}\-a\_\{j\}=\\tfrac\{1\}\{2\}\(e\_\{i\}\+e\_\{j\}\)\+\\tfrac\{1\}\{2\}\\Delta, whose squares sum to12​\(ei\+ej\)2\+12​Δ2\\tfrac\{1\}\{2\}\(e\_\{i\}\+e\_\{j\}\)^\{2\}\+\\tfrac\{1\}\{2\}\\Delta^\{2\}\. Taking expectations and subtracting, the change is

12​Δ2\+12​𝔼​\[\(ei\+ej\)2\]−𝔼⁡\[ei2\]−𝔼⁡\[ej2\]=12​Δ2−12​𝔼​\[\(ei−ej\)2\],\\tfrac\{1\}\{2\}\\Delta^\{2\}\+\\tfrac\{1\}\{2\}\\mathbb\{E\}\[\(e\_\{i\}\+e\_\{j\}\)^\{2\}\]\-\\mathbb\{E\}\[e\_\{i\}^\{2\}\]\-\\mathbb\{E\}\[e\_\{j\}^\{2\}\]=\\tfrac\{1\}\{2\}\\Delta^\{2\}\-\\tfrac\{1\}\{2\}\\mathbb\{E\}\[\(e\_\{i\}\-e\_\{j\}\)^\{2\}\],\(34\)strictly negative if and only ifΔ2<𝔼⁡\[\(ei−ej\)2\]\\Delta^\{2\}<\\mathbb\{E\}\[\(e\_\{i\}\-e\_\{j\}\)^\{2\}\]\. Since𝔼⁡\[εi\]=𝔼⁡\[εj\]=0\\mathbb\{E\}\[\\varepsilon\_\{i\}\]=\\mathbb\{E\}\[\\varepsilon\_\{j\}\]=0,𝔼⁡\[\(ei−ej\)2\]=\(bi−bj\)2\+Var⁡\(εi−εj\)\\mathbb\{E\}\[\(e\_\{i\}\-e\_\{j\}\)^\{2\}\]=\(b\_\{i\}\-b\_\{j\}\)^\{2\}\+\\mathrm\{Var\}\(\\varepsilon\_\{i\}\-\\varepsilon\_\{j\}\), which equals2​σ2​\(1−ρ\)2\\sigma^\{2\}\(1\-\\rho\)under the symmetric assumptions\.□\\square

#### Interpretation\.

For fixed predicted labels, Proposition[2](https://arxiv.org/html/2610.09020#Thmproposition2)limits how small the summed edge calibration error can be at a given predictive divergence\. Under Proposition[3](https://arxiv.org/html/2610.09020#Thmproposition3)’s assumptions, pairwise averaging removes12​𝔼​\[\(ei−ej\)2\]\\tfrac\{1\}\{2\}\\mathbb\{E\}\[\(e\_\{i\}\-e\_\{j\}\)^\{2\}\]from summed expected squared error while adding\(ai−aj\)2/2\(a\_\{i\}\-a\_\{j\}\)^\{2\}/2due to the correctness gap\. Improvement occurs exactly when the former exceeds the latter\.

Under the equal\-bias, equal\-variance specialization, the tolerable correctness gap scales with1−ρ\\sqrt\{1\-\\rho\}and vanishes atρ=1\\rho=1, so predictions that err together cannot correct one another\. Since neighboring predictions from a single network are plausibly positively correlated,2​σ\\sqrt\{2\}\\sigmais a ceiling rather than an estimate of the operative threshold, and whatever benefit smoothing confers is a reduction of within\-neighborhood error dispersion, not a correction of the neighborhood’s mean error\.

The threshold depends on the training stage through𝔼𝒯​\[\(ei−ej\)2\]\\mathbb\{E\}\_\{\\mathcal\{T\}\}\[\(e\_\{i\}\-e\_\{j\}\)^\{2\}\], which includes systematic differences between neighboring errors as well as random variation\. If this quantity decreases, the tolerable correctness gap narrows, motivating weaker smoothing later in training\.

## Appendix BNeSCal\-Specific Guarantees

Here we instantiate the results of Section[3\.2](https://arxiv.org/html/2610.09020#S3.SS2)\. For base\-22JS divergence, the domination constant isCJS=1/\(8​ln⁡2\)C\_\{\\mathrm\{JS\}\}=1/\(8\\ln 2\)\. Substituting this constant into the general bounds yields the following guarantees\.

###### Corollary 2\(Confidence smoothness and propagation under NeSCal \)

For every edge\(i,j\)∈E\(i,j\)\\in E,

\|ci−cj\|≤2ln\(2\)JS\(pi∥pj\)\|c\_\{i\}\-c\_\{j\}\|\\leq\\sqrt\{2\\ln\(2\)\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)\}\(35\)and consequently

ℰcalj≤ℰcali\+2ln\(2\)JS\(pi∥pj\)\+\|ai−aj\|\.\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{j\}\}\\leq\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{i\}\}\+\\sqrt\{2\\ln\(2\)\\,\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)\}\+\|a\_\{i\}\-a\_\{j\}\|\.\(36\)Along any simple directed pathP=\(v0,…,vr\)P=\(v\_\{0\},\\ldots,v\_\{r\}\)in𝒢\\mathcal\{G\},

ℰcalvr≤ℰcalv0\+2ln\(2\)r∑ℓ=1rJS\(pvℓ−1∥pvℓ\)\+\|avr−av0\|\.\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{v\_\{r\}\}\}\\leq\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{v\_\{0\}\}\}\+\\sqrt\{2\\ln\(2\)\\,r\\sum\_\{\\ell=1\}^\{r\}\\mathrm\{JS\}\(p\_\{v\_\{\\ell\-1\}\}\\\|p\_\{v\_\{\\ell\}\}\)\}\+\\bigl\|a\_\{v\_\{r\}\}\-a\_\{v\_\{0\}\}\\bigr\|\.\(37\)

#### Proof\.

Writem=\(p\+q\)/2m=\(p\+q\)/2, so thatJSln\(p∥q\)=12DKL\(p∥m\)\+12DKL\(q∥m\)\\mathrm\{JS\}\_\{\\ln\}\(p\\\|q\)=\\tfrac\{1\}\{2\}D\_\{\\mathrm\{KL\}\}\(p\\\|m\)\+\\tfrac\{1\}\{2\}D\_\{\\mathrm\{KL\}\}\(q\\\|m\)\. Sincep−m=12​\(p−q\)p\-m=\\tfrac\{1\}\{2\}\(p\-q\)andq−m=−12​\(p−q\)q\-m=\-\\tfrac\{1\}\{2\}\(p\-q\), we have‖p−m‖1=‖q−m‖1=12​‖p−q‖1\\\|p\-m\\\|\_\{1\}=\\\|q\-m\\\|\_\{1\}=\\tfrac\{1\}\{2\}\\\|p\-q\\\|\_\{1\}\. Pinsker’s inequalityDKL\(u∥v\)≥12∥u−v∥12D\_\{\\mathrm\{KL\}\}\(u\\\|v\)\\geq\\tfrac\{1\}\{2\}\\\|u\-v\\\|\_\{1\}^\{2\}\(in nats\) then gives

DKL\(p∥m\),DKL\(q∥m\)≥12⋅14∥p−q∥12=18∥p−q∥12D\_\{\\mathrm\{KL\}\}\(p\\\|m\),\\;D\_\{\\mathrm\{KL\}\}\(q\\\|m\)\\geq\\tfrac\{1\}\{2\}\\cdot\\tfrac\{1\}\{4\}\\\|p\-q\\\|\_\{1\}^\{2\}=\\tfrac\{1\}\{8\}\\\|p\-q\\\|\_\{1\}^\{2\}\(38\)thusJSln\(p∥q\)≥18∥p−q∥12\\mathrm\{JS\}\_\{\\ln\}\(p\\\|q\)\\geq\\tfrac\{1\}\{8\}\\\|p\-q\\\|\_\{1\}^\{2\}\. Converting to base22viaJS=JSln/ln⁡2\\mathrm\{JS\}=\\mathrm\{JS\}\_\{\\ln\}/\\ln 2yieldsJS\(p∥q\)≥18​ln⁡2∥p−q∥12\\mathrm\{JS\}\(p\\\|q\)\\geq\\frac\{1\}\{8\\ln 2\}\\\|p\-q\\\|\_\{1\}^\{2\}, i\.e\.CJS=1/\(8​ln⁡2\)C\_\{\\mathrm\{JS\}\}=1/\(8\\ln 2\)\. Substituting into Proposition[5](https://arxiv.org/html/2610.09020#Thmproposition5)gives12​CJS=12​8​ln⁡2=2​ln⁡2\\frac\{1\}\{2\\sqrt\{C\_\{\\mathrm\{JS\}\}\}\}=\\tfrac\{1\}\{2\}\\sqrt\{8\\ln 2\}=\\sqrt\{2\\ln 2\}, which is the stated confidence bound\.

The remaining claims follow by the same substitution\. In Proposition[1](https://arxiv.org/html/2610.09020#Thmproposition1),12CD−1/2D\(pi∥pj\)\\tfrac\{1\}\{2\}C\_\{\\mathrm\{D\}\}^\{\-1/2\}\\sqrt\{\\mathrm\{D\}\(p\_\{i\}\\\|p\_\{j\}\)\}becomes2ln\(2\)JS\(pi∥pj\)\\sqrt\{2\\ln\(2\)\\,\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)\}; in Theorem[1](https://arxiv.org/html/2610.09020#Thmtheorem1),r4​CD\\tfrac\{r\}\{4C\_\{\\mathrm\{D\}\}\}becomes2​ln⁡\(2\)​r2\\ln\(2\)rinside the first radical; and in Proposition[2](https://arxiv.org/html/2610.09020#Thmproposition2)the same replacement gives the NeSCal\-specific obstruction\.□\\square

Proposition[6](https://arxiv.org/html/2610.09020#Thmproposition6)bounds large disagreement at a fixed predictor: for an edge sampled with probabilityWi​j/\|E\|W\_\{ij\}/\|E\|andt\>0t\>0,Pr\(JS\(pi∥pj\)≥t\)≤ℒ\(θ\)/\(λ\|E\|t\)\\Pr\(\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)\\geq t\)\\leq\\mathcal\{L\}\(\\theta\)/\(\\lambda\|E\|t\)\. Instantiating Proposition[2](https://arxiv.org/html/2610.09020#Thmproposition2)gives

ℰcali\+ℰcalj≥\|ai−aj\|−2ln\(2\)JS\(pi∥pj\)\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{i\}\}\+\\mathcal\{E\}\_\{\\mathrm\{cal\}\_\{j\}\}\\geq\|a\_\{i\}\-a\_\{j\}\|\-\\sqrt\{2\\ln\(2\)\\,\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)\}\(39\)
Taken together, these delimit what NeSCal can and cannot achieve\. The penaltyJS\(pi∥pj\)\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)appears in every bound as the quantity the regularizer directly controls, andλ\\lambdagoverns how strongly it is suppressed\. The residual\|ai−aj\|\|a\_\{i\}\-a\_\{j\}\|appears in each with the opposite sign; it is not penalized, and is determined by which points the frozen graph connects and by the predicted labels\. NeSCal is therefore expected to improve calibration to the extent that its graph connects points which genuinely share a true probability of correctness \- a condition the design choices of Section[4\.2](https://arxiv.org/html/2610.09020#S4.SS2)are intended to promote and which Section[6](https://arxiv.org/html/2610.09020#S6)examines empirically\.

### B\.1JS\-Specific Laplacian Approximation Lemma

###### Corollary 3\(JS\-induced weighting\)

ForD=JS\\mathrm\{D\}=\\mathrm\{JS\}\(base22\), under the probability floormini,k⁡pi,k≥ε\\min\_\{i,k\}p\_\{i,k\}\\geq\\varepsilonof Proposition[4](https://arxiv.org/html/2610.09020#Thmproposition4), the induced local form is diagonal,

GJS​\(pi\)=H⁡\(pi\)=diag⁡\(18​ln⁡\(2\)​pi,1,…,18​ln⁡\(2\)​pi,K\)G\_\{\\mathrm\{JS\}\}\(p\_\{i\}\)=H\(p\_\{i\}\)=\\mathrm\{diag\}\\left\(\\frac\{1\}\{8\\ln\(2\)\\,p\_\{i,1\}\},\\ldots,\\frac\{1\}\{8\\ln\(2\)\\,p\_\{i,K\}\}\\right\)\(40\)which is positive definite sincepi,k\>0p\_\{i,k\}\>0for softmax outputs\. Consequently,

Rn​\(p\)=∑i,jWi​j​\(pi−pj\)𝖳​H​\(pi\)​\(pi−pj\)\+ρ,\|ρ\|≤112​ln⁡2​ε2​∑i,jWi​j​‖pi−pj‖23\.R\_\{n\}\(p\)=\\sum\_\{i,j\}W\_\{ij\}\\,\(p\_\{i\}\-p\_\{j\}\)^\{\\mathsf\{T\}\}H\(p\_\{i\}\)\(p\_\{i\}\-p\_\{j\}\)\+\\rho,\\qquad\|\\rho\|\\leq\\frac\{1\}\{12\\ln 2\\,\\varepsilon^\{2\}\}\\sum\_\{i,j\}W\_\{ij\}\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{2\}^\{3\}\.\(41\)

#### Proof\.

TheJS\\mathrm\{JS\}divergence satisfiesJS\(p∥p\)=0\\mathrm\{JS\}\(p\\\|p\)=0, and its formula is smooth on the positive orthant with third derivatives jointly continuous in\(p,q\)\(p,q\), so under the probability floor of Proposition[4](https://arxiv.org/html/2610.09020#Thmproposition4)its conditions hold\. By Proposition[4](https://arxiv.org/html/2610.09020#Thmproposition4), the zeroth\- and first\-order terms of the expansion ofJS\(pi∥pj\)\\mathrm\{JS\}\(p\_\{i\}\\\|p\_\{j\}\)aboutpi=pjp\_\{i\}=p\_\{j\}vanish, and the weighting is determined byGJS\(p\)=12∇q2JS\(p∥q\)\|q=pG\_\{\\mathrm\{JS\}\}\(p\)=\\tfrac\{1\}\{2\}\\nabla\_\{q\}^\{2\}\\mathrm\{JS\}\(p\\\|q\)\|\_\{q=p\}\. It therefore suffices to extract the second\-order coefficient\. Writeδk=qk−pk\\delta\_\{k\}=q\_\{k\}\-p\_\{k\}andmk=pk\+δk2m\_\{k\}=p\_\{k\}\+\\tfrac\{\\delta\_\{k\}\}\{2\}, and expandJS\\mathrm\{JS\}in natural\-log units,

JSln\(p∥q\)\\displaystyle\\mathrm\{JS\_\{\\ln\}\}\(p\\\|q\)=∑k12​\[pk​ln⁡pk\+qk​ln⁡qk−\(pk\+qk\)​ln⁡mk\]\\displaystyle=\\sum\_\{k\}\\frac\{1\}\{2\}\\Bigl\[p\_\{k\}\\ln p\_\{k\}\+q\_\{k\}\\ln q\_\{k\}\-\(p\_\{k\}\+q\_\{k\}\)\\ln m\_\{k\}\\Bigr\]=∑k12​\[pk​ln⁡pk\+\(pk\+δk\)​ln⁡\(pk\+δk\)−\(2​pk\+δk\)​ln⁡\(pk\+δk2\)\]\\displaystyle=\\sum\_\{k\}\\frac\{1\}\{2\}\\Bigl\[p\_\{k\}\\ln p\_\{k\}\+\(p\_\{k\}\+\\delta\_\{k\}\)\\ln\(p\_\{k\}\+\\delta\_\{k\}\)\-\(2p\_\{k\}\+\\delta\_\{k\}\)\\ln\(p\_\{k\}\+\\frac\{\\delta\_\{k\}\}\{2\}\)\\Bigr\]Using the second\-order Taylor expansions

ln⁡\(pk\+δk\)\\displaystyle\\ln\(p\_\{k\}\+\\delta\_\{k\}\)=ln⁡pk\+δkpk−δk22​pk2\+O⁡\(δ3\),\\displaystyle=\\ln p\_\{k\}\+\\tfrac\{\\delta\_\{k\}\}\{p\_\{k\}\}\-\\tfrac\{\\delta\_\{k\}^\{2\}\}\{2p\_\{k\}^\{2\}\}\+O\(\\delta^\{3\}\),ln⁡\(pk\+δk2\)\\displaystyle\\ln\\left\(p\_\{k\}\+\\tfrac\{\\delta\_\{k\}\}\{2\}\\right\)=ln⁡pk\+δk2​pk−δk28​pk2\+O⁡\(δ3\),\\displaystyle=\\ln p\_\{k\}\+\\tfrac\{\\delta\_\{k\}\}\{2p\_\{k\}\}\-\\tfrac\{\\delta\_\{k\}^\{2\}\}\{8p\_\{k\}^\{2\}\}\+O\(\\delta^\{3\}\),and retaining only the second\-order terms \(the lower orders cancel by Proposition[4](https://arxiv.org/html/2610.09020#Thmproposition4)\) gives

∑k12​\[δk2pk−δk22​pk\+δk24​pk−δk22​pk\]=∑kδk28​pk\.\\sum\_\{k\}\\tfrac\{1\}\{2\}\\left\[\\tfrac\{\\delta\_\{k\}^\{2\}\}\{p\_\{k\}\}\-\\tfrac\{\\delta\_\{k\}^\{2\}\}\{2p\_\{k\}\}\+\\tfrac\{\\delta\_\{k\}^\{2\}\}\{4p\_\{k\}\}\-\\tfrac\{\\delta\_\{k\}^\{2\}\}\{2p\_\{k\}\}\\right\]=\\sum\_\{k\}\\frac\{\\delta\_\{k\}^\{2\}\}\{8p\_\{k\}\}\.\(42\)
HenceGJS,ln​\(p\)=diag⁡\(1/\(8​pk\)\)G\_\{\\mathrm\{JS,\\ln\}\}\(p\)=\\mathrm\{diag\}\\left\(1/\(8p\_\{k\}\)\\right\)\. For the remainder,JSln\(p∥q\)=∑kφk\(qk\)\\mathrm\{JS\}\_\{\\ln\}\(p\\\|q\)=\\sum\_\{k\}\\varphi\_\{k\}\(q\_\{k\}\)is separable, withφk′′′​\(q\)=12​\(−q−2\+14​m−2\)\\varphi\_\{k\}^\{\\prime\\prime\\prime\}\(q\)=\\tfrac\{1\}\{2\}\\bigl\(\-q^\{\-2\}\+\\tfrac\{1\}\{4\}m^\{\-2\}\\bigr\),m=\(pk\+q\)/2m=\(p\_\{k\}\+q\)/2, so\|φk′′′\|≤1/\(2​ε2\)\|\\varphi\_\{k\}^\{\\prime\\prime\\prime\}\|\\leq 1/\(2\\varepsilon^\{2\}\)on the floor; since∑k\|uk\|3≤‖u‖23\\sum\_\{k\}\|u\_\{k\}\|^\{3\}\\leq\\\|u\\\|\_\{2\}^\{3\},M3​\(ε\)≤1/\(2​ε2\)M\_\{3\}\(\\varepsilon\)\\leq 1/\(2\\varepsilon^\{2\}\)in natural\-log units, and Proposition[4](https://arxiv.org/html/2610.09020#Thmproposition4)gives\|ρi​j\|≤‖pi−pj‖23/\(12​ε2\)\|\\rho\_\{ij\}\|\\leq\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{2\}^\{3\}/\(12\\varepsilon^\{2\}\), which becomes‖pi−pj‖23/\(12​ln⁡2​ε2\)\\\|p\_\{i\}\-p\_\{j\}\\\|\_\{2\}^\{3\}/\(12\\ln 2\\,\\varepsilon^\{2\}\)in base22\. Converting to base22via the exact identityJS=JSln/ln⁡2\\mathrm\{JS\}=\\mathrm\{JS\}\_\{\\ln\}/\\ln 2\(eachDKLD\_\{\\mathrm\{KL\}\}term rescales identically under change of base\) givesH⁡\(p\)=diag⁡\(1/\(8​ln⁡\(2\)​pk\)\)H\(p\)=\\mathrm\{diag\}\\left\(1/\(8\\ln\(2\)\\,p\_\{k\}\)\\right\), and substitutingδk=pi,k−pj,k\\delta\_\{k\}=p\_\{i,k\}\-p\_\{j,k\}yields the matrix form\.□\\square

Up to the constant1/\(8​ln⁡2\)1/\(8\\ln 2\),H⁡\(pi\)H\(p\_\{i\}\)is the Fisher information metric of the categorical distribution\([Amari and Nagaoka, 2000](https://arxiv.org/html/2610.09020#bib.bib25)\), which everyff\-divergence induces up to12​f′′​\(1\)\\tfrac\{1\}\{2\}f^\{\\prime\\prime\}\(1\)\. PenalizingJS\\mathrm\{JS\}therefore measures disagreement in the intrinsic geometry of the simplex rather than the Euclidean geometry of the probability vector\. The weighting is a property of the geometry rather than a performance guarantee, and it holds only where the expansion is accurate,\|pi,k−pj,k\|≪pi,k\|p\_\{i,k\}\-p\_\{j,k\}\|\\ll p\_\{i,k\}\.

## Appendix CNeSCal Implementation and Computational Cost

### C\.1Objective Normalization

The objective in Section[4\.1](https://arxiv.org/html/2610.09020#S4.SS1)uses the summed penaltyRn​\(p\)R\_\{n\}\(p\); optimization uses the directed\-edge meanR¯n​\(p\)=Rn​\(p\)/\|E\|\\bar\{R\}\_\{n\}\(p\)=R\_\{n\}\(p\)/\|E\|\. This rescales the coefficient:λmean=\|E\|​λsum\\lambda\_\{\\mathrm\{mean\}\}=\|E\|\\lambda\_\{\\mathrm\{sum\}\}\. Algorithm[1](https://arxiv.org/html/2610.09020#alg1)and all reported regularization strengths use the edge\-mean convention\.

### C\.2Training Procedure and Cost

Algorithm 1NeSCal1:Training set

𝒟=\{\(xi,yi\)\}i=1n\\mathcal\{D\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}; model

fθf\_\{\\theta\}with representation map

hh; number of neighbors

kk; regularization strength

λ0\\lambda\_\{0\}; activation epoch

TaT\_\{a\}; total epochs

TT
2:Initialize model parameters

θ\\theta
3:for

t=1t=1to

TTdo

4:if

t=Tat=T\_\{a\}then

5:Compute

zi←h⁡\(xi\)z\_\{i\}\\leftarrow h\(x\_\{i\}\)for all

ii⊳\\trianglerightno gradient

6:

W←kNNGraph​\(\{zi\}i=1n,k,cosine\)W\\leftarrow\\textsc\{kNNGraph\}\(\\\{z\_\{i\}\\\}\_\{i=1\}^\{n\},k,\\mathrm\{cosine\}\)⊳\\trianglerightfixed thereafter

7:

p^i←stopgrad⁡\(fθ​\(xi\)\)\\widehat\{p\}\_\{i\}\\leftarrow\\operatorname\{stopgrad\}\(f\_\{\\theta\}\(x\_\{i\}\)\)for all

ii⊳\\trianglerightinitialize cache

8:endif

9:if

t≥Tat\\geq T\_\{a\}then

10:

λt←λ02​\[1\+cos⁡\(π​t−TaT−Ta\)\]\\displaystyle\\lambda\_\{t\}\\leftarrow\\frac\{\\lambda\_\{0\}\}\{2\}\\left\[1\+\\cos\\left\(\\pi\\frac\{t\-T\_\{a\}\}\{T\-T\_\{a\}\}\\right\)\\right\]
11:else

12:

λt←0\\lambda\_\{t\}\\leftarrow 0
13:endif

14:for allminibatches

B⊆𝒟B\\subseteq\\mathcal\{D\}do

15:

pi←fθ​\(xi\)p\_\{i\}\\leftarrow f\_\{\\theta\}\(x\_\{i\}\)for all

i∈Bi\\in B
16:

ℒCE←1\|B\|​∑i∈BCE⁡\(pi,yi\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{CE\}\}\\leftarrow\\frac\{1\}\{\|B\|\}\\sum\_\{i\\in B\}\\mathrm\{CE\}\(p\_\{i\},y\_\{i\}\)
17:if

t≥Tat\\geq T\_\{a\}then

18:

p^i←stopgrad⁡\(pi\)\\widehat\{p\}\_\{i\}\\leftarrow\\operatorname\{stopgrad\}\(p\_\{i\}\)for all

i∈Bi\\in B⊳\\trianglerightcurrent targets for neighbors in this batch

19:

EB←\{\(i,j\):i∈B,Wi​j=1\}\\displaystyle E\_\{B\}\\leftarrow\\\{\(i,j\):i\\in B,\\;W\_\{ij\}=1\\\}
20:

R←1\|EB\|∑\(i,j\)∈EBJS\(pi∥p^j\)\\displaystyle R\\leftarrow\\frac\{1\}\{\|E\_\{B\}\|\}\\sum\_\{\(i,j\)\\in E\_\{B\}\}\\mathrm\{JS\}\\\!\\left\(p\_\{i\}\\,\\\|\\,\\widehat\{p\}\_\{j\}\\right\)⊳\\trianglerightp^j\\widehat\{p\}\_\{j\}detached

21:else

22:

R←0R\\leftarrow 0
23:endif

24:

θ←θ−α​∇θ\(ℒCE\+λt​R\)\\displaystyle\\theta\\leftarrow\\theta\-\\alpha\\nabla\_\{\\theta\}\\left\(\\mathcal\{L\}\_\{\\mathrm\{CE\}\}\+\\lambda\_\{t\}R\\right\)
25:endfor

26:endfor

Algorithm[1](https://arxiv.org/html/2610.09020#alg1)summarizes the training procedure\. Training proceeds by standard cross\-entropy minimization until the activation epochTaT\_\{a\}, at which point a single inference pass supplies the embeddings from which thekk\-nearest\-neighbor graph is constructed, and a second seeds a cache of predictive distributions for every training sample\. FromTaT\_\{a\}onward each minibatch incurs the neighborhood\-consistency penalty in addition to the task loss, withλ\\lambdaannealed over the remaining epochs and the cache refreshed with the predictions computed in that step\. The graph itself is never rebuilt\.

Computing live predictions for every neighbor would require additional network evaluations\. Neighbor targets are therefore detached current\-step predictions when the neighbor belongs to the minibatch, and cached predictions otherwise\. Each cached prediction is refreshed when its sample is visited, once per epoch\. The predictive bounds are evaluated using current predictions, as described in Appendix[E\.11](https://arxiv.org/html/2610.09020#A5.SS11)\.

The graph construction is a one\-off cost\. Thekk\-nearest\-neighbor search is exact, computed as a chunked cosine similarity, and is thereforeO⁡\(n2​l\)O\(n^\{2\}l\)in the number of samplesnnand the embedding dimensionll\. Because the graph is built once and held fixed \(Section[4\.2](https://arxiv.org/html/2610.09020#S4.SS2)\), this cost is incurred a single time and amortized over the remainingT−TaT\-T\_\{a\}epochs\.

The recurring cost is the penalty itself\. It requires one divergence evaluation per edge, eachO⁡\(K\)O\(K\)in the number of classes, givingO⁡\(n​k​K\)O\(nkK\)per epoch against theO⁡\(n​C\)O\(nC\)cost of the forward and backward passes, whereCCis the per\-sample cost of the network\. SinceC≫k​KC\\gg kKfor most of the architectures considered in Section[5](https://arxiv.org/html/2610.09020#S5), the relative overhead is small\. Memory overhead isO⁡\(n​k\)O\(nk\)for the edge list, held on host memory, andO⁡\(n​K\)O\(nK\)for the cached predictions\. Measured graph\-construction times, peak memory, and training wall\-clock relative to the baselines are reported in Appendix[E\.10](https://arxiv.org/html/2610.09020#A5.SS10)\.

### C\.3Directed Neighborhoods

Because thekk\-nearest\-neighbor relation is not symmetric \(jjmay be among thekknearest neighbors ofiiwithout the opposite being true\),𝒢\\mathcal\{G\}is a directed graph, andWWis not in general symmetric\. Nothing in Section[3](https://arxiv.org/html/2610.09020#S3)requires symmetry\. Larger neighborhoods can reduce averaging variance while admitting more heterogeneous points\.

### C\.4Activation and Schedule Rationale

Activation epoch andλ\\lambdaare selected jointly by validation NLL; the theory motivates the schedule without determining it\. Under Proposition[3](https://arxiv.org/html/2610.09020#Thmproposition3)’s assumptions, a decrease in𝔼⁡\[\(ei−ej\)2\]\\mathbb\{E\}\[\(e\_\{i\}\-e\_\{j\}\)^\{2\}\]narrows the correctness gap that averaging can tolerate, motivating decay\. Proposition[2](https://arxiv.org/html/2610.09020#Thmproposition2)also motivates delaying graph construction until neighborhoods are informative: smoothing across a large correctness gap can preclude calibration at both endpoints\. Delaying may stabilize predicted labels, which fixes the corresponding correctness probabilities\. Activation balances these considerations\.

NeSCal activates after a warm\-up and then decaysλ\\lambdaon a cosine schedule\. Appendix[E\.7](https://arxiv.org/html/2610.09020#A5.SS7)compares constant and cosine schedules; Appendix[E\.8](https://arxiv.org/html/2610.09020#A5.SS8)evaluates diagnostic activation rules and their transfer across settings\.

## Appendix DTraining and Implementation Details

Unless stated otherwise, every method in a given setting shares the architecture, optimizer, schedule, epoch budget, batch size, data splits and seeds; only the training objective differs\.

### D\.1Datasets, Splits and Preprocessing

Table 3:Datasets\. Splits are train/val/testImage\.Standard augmentation \(random crop with 4\-pixel padding, random horizontal flip\) and per\-channel normalization; evaluation uses normalization only\. The validation split is a seeded10%10\\%partition of the training set, fixed across seeds\.

Tabular\.Following[Gorishniy et al\. \(2021\)](https://arxiv.org/html/2610.09020#bib.bib43), features are quantile\-transformed to a normal output distribution, except for Helena, which is standardized\. The transform is fitted on the training partition only\. The test and validation splits are fixed across seeds\.

Text\.The 20 Newsgroups pipeline follows the MMCE reference implementation: vocabulary of20,00020\{,\}000, sequence length1,0001\{,\}000, GloVe\-100d initialization\([Pennington et al\., 2014](https://arxiv.org/html/2610.09020#bib.bib40)\)\. Frozen\-encoder embeddings are mean\-pooled over tokens with the attention mask and extracted once, so the head sees a fixed vector dataset\. The test split is fixed across seeds in every setting; the validation split is redrawn per seed on the 20NG setting, and for the frozen\-encoder Amazon settings it is held fixed so that the same encoder embeddings are reused\.

### D\.2Architectures and Optimization

Table 4:Training configuration per setting\.Image runs use mixed\-precision training; all others use full precision\. Tabular and text settings use a constant learning rate by design, which removes any interaction between theλ\\lambdaschedule and LR decay \- a useful control for the schedule comparison\.

### D\.3NeSCal Configuration and Selection

Unless ablated,k=10k=10, cosine distance, JS divergence, a directed unweighted kNN graph over penultimate features, frozen at the activation epoch\. The penalty uses the directed\-edge meanR¯n=Rn/\|E\|\\bar\{R\}\_\{n\}=R\_\{n\}/\|E\|, removing its direct scaling with the number of edges\. Cached neighbor predictions are refreshed when their samples are visited \(Appendix[C](https://arxiv.org/html/2610.09020#A3)\)\. All reported regularization strengths use the edge\-mean convention \(Appendix[C\.1](https://arxiv.org/html/2610.09020#A3.SS1)\)\.

Activation epoch andλ\\lambdaare selected jointly\. A single cross\-entropy warm\-up run per seed is checkpointed at every candidate onset; each grid point resumes from the corresponding checkpoint, so all branches share a bit\-identical prefix and differ only in the intervention under study\. Resumption restores optimizer state and the scheduler’s absolute epoch\. This checkpoint\-resumption setup enables a wider grid search space at lower cost\. Within each schedule, activation epoch andλ\\lambdaare selected by mean validation NLL over seeds1818and4242\. Evaluation of the selected configuration is on seeds18,42,61318,42,613and20262026\. The same selection\-evaluation protocol applies to all methods, with the following exceptions: for 20NG, CE and NeSCal additionally include evaluation on seeds77and12341234; s\-AvUC for Tiny\-ImageNet uses three evaluation seeds, excluding20262026\. Table[5](https://arxiv.org/html/2610.09020#A4.T5)gives the search grids\. Divergent configurations are excluded from selection; Helena omitsλ=0\.5\\lambda=0\.5\.

Table 5:NeSCal search grids\. Onsets are epochs of the total budget\.
### D\.4Competitor Search Spaces

Each competitor is tuned over setting\-dependent candidate ranges informed by the search ranges or parameter\-sensitivity analyses reported in its original paper\. Configurations are selected by seed\-averaged validation NLL, using the same selection criterion as NeSCal\. For compactness, the values below report the widest range of candidate values considered for each method across settings; individual settings may use subsets of these values, dependent on author recommendations under parallel settings\. We use each method’s own notation\. FL is searched over fixedγ∈\{2,3\}\\gamma\\in\\\{2,3\\\}and the adaptive, sample\-dependent variant withγhigh∈\{2,3\}\\gamma\_\{\\mathrm\{high\}\}\\in\\\{2,3\\\}, selecting once across both families\. LS usesα∈\{0\.05,0\.1\}\\alpha\\in\\\{0\.05,0\.1\\\}; MMCEλ∈\{1,3,7,10,15,20\}\\lambda\\in\\\{1,3,7,10,15,20\\\}at the Laplacian kernel width of[Kumar et al\. \(2018\)](https://arxiv.org/html/2610.09020#bib.bib8), owing to their reported insensitivity toλ\\lambdain that range; MDCAβ∈\{1,5,10,15,20,25\}\\beta\\in\\\{1,5,10,15,20,25\\\}; MbLSλ∈\{0,0\.05,0\.1,0\.2,0\.3\}\\lambda\\in\\\{0,0\.05,0\.1,0\.2,0\.3\\\}with marginm∈\{4,6,8,10,12\}m\\in\\\{4,6,8,10,12\\\}\. s\-AvUC is searched sequentially in the orderκ∈\{0\.1,0\.5,0\.9\}\\kappa\\in\\\{0\.1,0\.5,0\.9\\\}, temperatureT∈\{0\.5,1,2\}T\\in\\\{0\.5,1,2\\\}, penalty weightβ∈\{0\.1,1,3\}\\beta\\in\\\{0\.1,1,3\\\}, matching the authors’ own recommendation, as a full grid is prohibitive\. Note thatλ\\lambdadenotes the penalty weight for MMCE, MbLS and NeSCal, but theℓ2\\ell\_\{2\}coefficient in the notation of[Karandikar et al\. \(2021\)](https://arxiv.org/html/2610.09020#bib.bib17)\.

### D\.5Post\-hoc Calibrators

Post\-hoc calibrators are fitted on the validation split used for base\-model selection and evaluated on the test split, following the protocol of[Guo et al\. \(2017\)](https://arxiv.org/html/2610.09020#bib.bib2)\. TS uses one\-dimensional numerical minimization of validation NLL\. Dirichlet calibration uses the full matrix parameterization with off\-diagonal and intercept regularization \(ODIR\), with regularization strength selected on validation\. LoCal uses its local calibration objective and early stopping\. Feature preprocessing and other departures from the reference methods are described below\.

### D\.6Metrics

The limitations of ECE have motivated debiased and binning\-free alternatives\([Arrieta\-Ibarra et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib4);[Błasiok et al\., 2023](https://arxiv.org/html/2610.09020#bib.bib3)\)\. Other notions retain different information: classwise calibration requiresℙ⁡\(Y=k∣pk​\(X\)\)=pk​\(X\)\\mathbb\{P\}\(Y=k\\mid p\_\{k\}\(X\)\)=p\_\{k\}\(X\)for each class, while canonical calibration requiresℙ⁡\(Y=k∣p⁡\(X\)\)=pk​\(X\)\\mathbb\{P\}\(Y=k\\mid p\(X\)\)=p\_\{k\}\(X\)\([Kull et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib13);[Vaicenavicius et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib7);[Widmann et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib5)\)\. Local calibration assesses reliability within neighborhoods in representation space\([Luo et al\., 2022](https://arxiv.org/html/2610.09020#bib.bib14)\)\.

We report accuracy, ECE\([Naeini et al\., 2015](https://arxiv.org/html/2610.09020#bib.bib1)\), an equal\-mass\-binned ECE variant\([Nixon et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib6)\), classwise ECE\([Kull et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib13)\), the local calibration error \(LCE\), using the classwise extension of[Barbera et al\. \(2026\)](https://arxiv.org/html/2610.09020#bib.bib33)to the confidence\-based measure of[Luo et al\. \(2022\)](https://arxiv.org/html/2610.09020#bib.bib14), and the binned calibration–refinement decomposition of the top\-label Brier score\([Brier, 1950](https://arxiv.org/html/2610.09020#bib.bib48);[Kull and Flach, 2015](https://arxiv.org/html/2610.09020#bib.bib46)\)\.

ECE and MCE use1515equal\-width bins of top\-label confidence, the last bin closed; the equal\-mass variant uses1515equal\-count bins\. Classwise ECE applies the same binning to each class probability and averages over classes\. LCE uses a Gaussian kernel \(γ=10\\gamma=10\) in a5050\-dimensional PCA projection of the penultimate features, fitted on the validation split, excluding bins with at most2020samples\. The calibration–refinement decomposition bins top\-label confidence into the same1515bins; because it ignores within\-bin variation, its two terms need not sum exactly to the top\-label Brier score\.

### D\.7Additional Experiments and Ablations

We ablate the components of Section[4\.2](https://arxiv.org/html/2610.09020#S4.SS2): neighborhood size, random versus nearest\-neighbor graphs, graphs built on PCA\- and UMAP\-reduced representations, and alternative divergences at matched initial penalty magnitude\. We also compare regularization schedules and diagnostic activation rules\. Further studies assess input corruption, out\-of\-distribution detection, class imbalance, and label noise\. Results appear in Appendix[E](https://arxiv.org/html/2610.09020#A5)\.

### D\.8Theoretical Diagnostics

The quantities of Sections[3\.2](https://arxiv.org/html/2610.09020#S3.SS2)and Appendix[B](https://arxiv.org/html/2610.09020#A2)are computed on the training nodes of the frozen graph, at the selected weights, from a full inference pass with current predictions rather than from the cached distributions used during training\. We report whether bounds hold on every edge, the median per\-edge tightness of Corollary[2](https://arxiv.org/html/2610.09020#Thmcorollary2)and the fraction of edges satisfying the small\-disagreement condition under which the Laplacian expansion of Proposition[4](https://arxiv.org/html/2610.09020#Thmproposition4)is valid\. We do not report an aggregate ratio of the Laplacian quadratic form toRnR\_\{n\}: it is dominated by a small number of edges carrying near\-zero class probabilities, where the third\-derivative remainder diverges, and is therefore uninformative about the graph as a whole\. This is consistent with the lemma being an interpretive device rather than a bound we optimize\.

The regime diagnostics of Table[2](https://arxiv.org/html/2610.09020#S6.T2)are logged by the training runs at the activation epoch and averaged over seeds, on the same frozen graph\.

### D\.9Robustness and Perturbation Studies

Corruption robustness is evaluated on CIFAR\-10\-C, CIFAR\-100\-C, and Tiny\-ImageNet\-C using the 15 standard corruption types \(14 on Tiny\-ImageNet\-C\) at all five severity levels\([Hendrycks and Dietterich, 2019](https://arxiv.org/html/2610.09020#bib.bib49)\); out\-of\-distribution \(OOD\) separation uses SVHN\([Netzer et al\., 2011](https://arxiv.org/html/2610.09020#bib.bib50)\)as the OOD dataset for models trained on CIFAR\-10, CIFAR\-100, and Tiny\-ImageNet\. SVHN images are resized where necessary to match the model input resolution and normalized using the corresponding in\-distribution statistics\. Detection is based on maximum softmax probability\([Hendrycks and Gimpel, 2017](https://arxiv.org/html/2610.09020#bib.bib51)\)and evaluated using AUROC\. We additionally train under long\-tailed class imbalance \(imbalance factors1010,5050,100100\) and symmetric label noise \(20%20\\%,40%40\\%\), reusing the clean\-selected configuration\.

### D\.10Cost Measurement

We report wall\-clock training time excluding evaluation, one\-off graph construction time, and peak memory, measured on the same hardware for all methods within a setting\. The reported NeSCal wall\-clock includes the warm\-up prefix, so the comparison against baselines is like\-for\-like\. All settings ran on either a single NVIDIA RTX 5000 Ada Gen\. or NVIDIA RTX A6000 GPU, with the GPU model kept consistent for every run in that setting\.

### D\.11Deviations from Reference Implementations

We note the following deviations from the original implementations and protocols\.

*Protocol\.*All methods share a validation split carved from the training set \(4545k/55k on CIFAR\), are evaluated at the epoch of best validation NLL rather than at the final epoch, and are tuned on validation NLL; several of the compared methods tune on a calibration metric in their own papers\. CIFAR budgets \(Table[4](https://arxiv.org/html/2610.09020#A4.T4)\) are shorter than the350350\-epoch schedule of[Mukhoti et al\. \(2020\)](https://arxiv.org/html/2610.09020#bib.bib20)\.

*Architectures\.*ResNet\-50 uses the standard small\-input stem \(3×33\{\\times\}3convolution, no max\-pooling\) on both CIFAR and Tiny\-ImageNet\. The 20 Newsgroups model follows the canonical global\-pooling CNN of[Kumar et al\. \(2018\)](https://arxiv.org/html/2610.09020#bib.bib8), with the GloVe embedding matrix held frozen rather than fine\-tuned, and a5050\-epoch budget\. Tabular models use a fixed3×2563\{\\times\}256MLP with a fixed epoch budget, rather than the per\-dataset tuned architectures and early stopping of[Gorishniy et al\. \(2021\)](https://arxiv.org/html/2610.09020#bib.bib43); accuracies are correspondingly0\.50\.5–1\.51\.5points below their tuned reference values\. Covertype is subsampled to100100k rows\.

*Baselines\.*MDCA is evaluated as cross\-entropy plus MDCA, without the focal or label\-smoothing compositions reported by its authors\. Two further departures are shared across methods rather than specific to one\. All CIFAR runs use batch size128128, following[Mukhoti et al\. \(2020\)](https://arxiv.org/html/2610.09020#bib.bib20)and[Kumar et al\. \(2018\)](https://arxiv.org/html/2610.09020#bib.bib8), rather than the512512of[Karandikar et al\. \(2021\)](https://arxiv.org/html/2610.09020#bib.bib17)\. MDCA is trained end\-to\-end on the standard architecture rather than on the frozen\-backbone head with additional convolutional and fully connected layers used by[Hebbalaguppe et al\. \(2022\)](https://arxiv.org/html/2610.09020#bib.bib19)\. MbLS on 20 Newsgroups uses the5050\-epoch constant\-rate schedule shared by all methods in that setting rather than the100100\-epoch decayed schedule of[Liu et al\. \(2022\)](https://arxiv.org/html/2610.09020#bib.bib29)\. Calibration metrics use1515bins throughout, following three of the four reference papers;[Kumar et al\. \(2018\)](https://arxiv.org/html/2610.09020#bib.bib8)use2020\.

*Post\-hoc\.*LoRe is applied in the same standardized PCA feature space and Gaussian kernel as our locality metric rather than at its published Laplacian bandwidth, which was tuned on different features\. LoCal is run at its authors’ published configuration in every setting, without per\-setting tuning\.

## Appendix EAdditional Results

### E\.1Full per\-setting metrics

Tables[6](https://arxiv.org/html/2610.09020#A5.T6)–[8](https://arxiv.org/html/2610.09020#A5.T8)report the metrics accompanying the NLL comparison in the main text\. NeSCal improves Brier score over CE in ten of eleven settings and accuracy in nine\.

NeSCal attains the lowest or second\-lowest LCE in five of eight image and tabular settings, counting ties at displayed precision\. Improvements are not uniform across settings or calibration measures\. Classwise ECE values are small in absolute terms on the many\-class datasets and should be interpreted alongside the other metrics\.

Table 6:Brier score \(top, lower is better\) and accuracy in percent \(bottom, higher is better\)\. Conventions as in Table[1](https://arxiv.org/html/2610.09020#S6.T1)\.Table 7:ECE in percent for the base model \(top\) and after temperature scaling \(bottom\); lower is better\.Table 8:Classwise ECE, LCE, and equal\-mass ECE, all in percent; lower is better\.
### E\.2Across\-setting comparisons

We summarize pairwise wins across setting means and report exploratory Wilcoxon signed\-rank comparisons in Table[9](https://arxiv.org/html/2610.09020#A5.T9), which counts, for each baseline, the settings in which NeSCal attains the lower mean, and applies an exact two\-sided Wilcoxon signed\-rank test to the per\-setting mean differences\.pp\-values are uncorrected; with eight settings the smallest attainable is0\.0080\.008\. Several settings share datasets, thus the uncorrectedpp\-values are interpreted as exploratory summaries\. For NLL after temperature scaling NeSCal wins in all eight image and tabular settings against FL, LS, MMCE and MbLS, and in seven of eight against MDCA and s\-AvUC\.

Table 9:NeSCal against each baseline across settings\. “All” covers the eleven completed settings, “I\+T” the eight image and tabular settings\.
### E\.3Calibration–refinement decomposition

Table[10](https://arxiv.org/html/2610.09020#A5.T10)reports the binned refinement component of the base\-model top\-label Brier decomposition\. This component measures within\-bin variation in correctness and is lower for NeSCal than for CE in ten of eleven real\-data settings\. It provides a complementary diagnostic under the specified binning scheme; recalibration can change bin membership and therefore the estimate\.

Table 10:Reported binned refinement component of the base\-model top\-label Brier decomposition; lower is better\.
### E\.4Post\-hoc composition

Table[11](https://arxiv.org/html/2610.09020#A5.T11)reports each train\-time method’s best observed NLL and ECE across the five calibrators, selected separately and retrospectively on test means\. NeSCal attains the lowest NLL in six of eleven real\-data settings, all of them image or tabular\.

Composition does not improve every metric: on Helena, TS slightly lowers NeSCal’s NLL while increasing ECE from1\.25%1\.25\\%to2\.35%2\.35\\%\.

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\.
### E\.5Failure prediction and selective classification

We assess how well confidence separates correct from incorrect predictions using AUROC\([Hendrycks and Gimpel, 2017](https://arxiv.org/html/2610.09020#bib.bib51)\)and area under the risk–coverage curve \(AURC\)\([Geifman et al\., 2019](https://arxiv.org/html/2610.09020#bib.bib52)\)\. Temperature scaling preserves predicted labels but can change confidence rankings across multiclass examples, so we report both base\-model and recalibrated results\.

NeSCal attains the best or second\-best AUROC \(higher is better\) and AURC \(lower is better\) on all CIFAR settings, winning outright in three, both on the base model and after TS\. Relative to CE, base\-model AURC decreases by approximately15\.5%15\.5\\%on CIFAR\-10/ResNet\-50 and13\.5%13\.5\\%on CIFAR\-100/ResNet\-50\. \(Table[12](https://arxiv.org/html/2610.09020#A5.T12)\)\.

Table 12:Failure\-prediction AUROC \(higher is better\) and AURC \(×103\\times 10^\{3\}, lower is better\), before and after temperature scaling\. Means over seeds; SDs omitted\.
### E\.6Robustness

Table[13](https://arxiv.org/html/2610.09020#A5.T13)reports corruption robustness and SVHN detection\. NeSCal improves corrupted NLL and ECE over CE in four of five image settings and accuracy in all five, attaining the highest corrupted accuracy among the methods in Table[13](https://arxiv.org/html/2610.09020#A5.T13)in four settings\. OOD detection results are mixed: AUROC improves over CE on both CIFAR\-100 settings and Tiny\-ImageNet, but decreases on both CIFAR\-10 settings\. On Tiny\-ImageNet, NeSCal attains the highest reported SVHN AUROC,0\.92210\.9221\.

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\.
### E\.7Ablations

Component ablations use seeds1818and4242, the cosine vs constant schedule comparison matches the headline in seed numbers, with activation epoch and regularization strength selected separately for each schedule\. The Fisher\-weighted arm usesDF​\(pi,pj\)=18​ln⁡2​∑k\(pi,k−pj,k\)2max⁡\(pi,k,10−2\)D\_\{\\mathrm\{F\}\}\(p\_\{i\},p\_\{j\}\)=\\frac\{1\}\{8\\ln 2\}\\sum\_\{k\}\\frac\{\(p\_\{i,k\}\-p\_\{j,k\}\)^\{2\}\}\{\\max\(p\_\{i,k\},10^\{\-2\}\)\}\. The floor applies only to the source probabilities\. This is a stabilized version of the local quadratic weighting in Proposition[4](https://arxiv.org/html/2610.09020#Thmproposition4)\.

#### Graph construction\.

Projection\-based graphs retain gains over CE in some of the evaluated settings, without a clear, consistent winner emerging\. The choice of feature transformation is therefore not uniformly neutral\. Table[14](https://arxiv.org/html/2610.09020#A5.T14)displays the random\-graph results discussed in Section[6\.3](https://arxiv.org/html/2610.09020#S6.SS3)\.

Table 14:Test NLL for neighborhood controls; lower is better\. Random uses headlineλ\\lambda; Random \(scaled\) matches penalty magnitude\.
#### Constant versus cosine regularization\.

With each schedule selected by validation NLL, cosine decay improves NLL, Brier score, and accuracy in all four CIFAR settings; ECE improves in three \(Table[15](https://arxiv.org/html/2610.09020#A5.T15)\)\. On the two ResNet\-50 settings with graph diagnostics, constant regularization produces lower final edge JS, whereas cosine decay produces lower across\-seed error dispersion \(Table[19](https://arxiv.org/html/2610.09020#A5.T19)\)\. These observations support relaxing the consistency constraint during training: stronger agreement at the selected weights does not necessarily imply better predictive quality\. They motivate cosine decay as the default without implying uniform improvement across calibration metrics\.

Table 15:Constant versus cosine regularization on CIFAR\.
#### Neighborhood structure\.

At matched initial penalty magnitude, random graphs provide little improvement over CE in the three evaluated settings\. Using the headlineλ\\lambdaproduces worse NLL than CE in all three \(Table[16](https://arxiv.org/html/2610.09020#A5.T16)\), supporting the importance of which samples the graph connects\.

#### Divergence sensitivity\.

Divergence arms are at matched initial penalty magnitude, with no divergence uniformly dominating among the stable, scale\-matched runs\. The common penalty target over\-regularizes some settings, so cells marked†\\daggerand divergent runs are excluded from the comparison of divergence quality\.

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

### E\.8Activation rules

If the activation epoch could be placed from a quantity observable during the warm\-up, NeSCal would lose one swept hyperparameter\. We tested three such rules on the cross\-entropy trajectory:

- •*entropy drop*: mean predictive entropy falls below a fraction of its initial value;
- •*CWCC*: class\-weighted cross\-class connectivity falls below a fraction of its initial value;
- •*probe*: kNN\-probe accuracy reaches a fraction of its maximum over the warm\-up\.

For these rule experiments, diagnostics are logged every1010epochs and activation gates are rounded down to a separate2020\-epoch checkpoint grid\. Table[17](https://arxiv.org/html/2610.09020#A5.T17)reports each rule’s regret, the test\-NLL difference against the grid\-selected onset at the sameλ\\lambda\.

No rule transfers consistently across settings\. Probe\-rule regret ranges from−0\.039\-0\.039to\+0\.072\+0\.072nats\. Because this rule uses the maximum probe accuracy over the warm\-up trajectory, it is a retrospective diagnostic rather than an online activation rule\.

Table 17:Regret of activation rules: test NLL relative to the grid\-selected onset at the sameλ\\lambda; positive is worse\. –: rule not evaluated\.
### E\.9Selection criterion

Every train\-time method is selected by validation NLL across checkpoints and hyperparameters\. Selecting by ECE can favor less informative predictions\. Across the eleven settings, ECE\-based selection improves NeSCal’s test ECE by0\.850\.85percentage points on average but worsens NLL by0\.0170\.017nats\. On CIFAR\-100/R50, for example, ECE improves from3\.95%3\.95\\%to2\.47%2\.47\\%while NLL worsens from0\.80260\.8026to0\.8710\.871\. Similar trade\-offs occur among the baselines\. These results support the shared proper\-score criterion while confirming that it does not optimize each method’s ECE\.

### E\.10Training cost

Table[18](https://arxiv.org/html/2610.09020#A5.T18)reports training\-loop time, one\-off graph\-construction time, and peak GPU memory\. Peak GPU memory is the maximum allocated at any point in the run, including the one\-off graph construction\. Where NeSCal ’s peak clearly exceeds CE’s \(Tiny\-ImageNet, CIFAR\-100/WRN and the tabular settings\), the maximum occurs during the chunked similarity computation of graph construction; differences of a few hundredths of a GB are within run\-to\-run allocator variation\. Relative overhead is small on the image models and larger on the smaller tabular and text models, where the absolute increase in the reported training time is at most27\.427\.4seconds\. Peak memory increases vary across settings\. Inference is unchanged\.

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 \(one seed\)\.
### E\.11Theory diagnostics

We evaluate bound tightness and related diagnostics on the frozen training graphs for CIFAR\-10 and CIFAR\-100 with ResNet\-50, using CE and both NeSCal schedules over four seeds \(Table[19](https://arxiv.org/html/2610.09020#A5.T19)\)\.

All evaluated bounds hold on every edge in all evaluated configurations\. As expected of worst\-case inequalities, they are conservative: median confidence\-bound tightness is0\.020\.02–0\.040\.04on CIFAR\-10 and0\.100\.10–0\.130\.13on CIFAR\-100, where neighboring disagreement is larger\. We therefore use them to identify which quantities govern calibration propagation rather than to predict its magnitude\. The Markov bound of Proposition[6](https://arxiv.org/html/2610.09020#Thmproposition6)is informative when the penalty is small relative to the threshold: on CIFAR\-10 it bounds the fraction of edges withJS≥0\.25\\mathrm\{JS\}\\geq 0\.25by0\.520\.52, against an observed0\.150\.15; on CIFAR\-100, where mean disagreement is higher, it is not informative at this threshold\.

The diagnostics bearing on Proposition[3](https://arxiv.org/html/2610.09020#Thmproposition3)are consistent with its mechanism\. Cosine\-scheduled NeSCal reduces the across\-seed dispersion of neighboring errors and of individual confidences on both settings, and both schedules reduce the NLL gap to a test\-fitted temperature\. Final edge agreement behaves differently across schedules: the constant schedule lowers mean edge JS relative to CE on both settings, whereas under cosine decayλt\\lambda\_\{t\}approaches zero by the end of training, and edge JS at the selected checkpoint is close to CE’s\. Better test NLL therefore need not coincide with the smallest final penalty, consistent with the regularizer acting on the dispersion of neighboring errors rather than on end\-state agreement alone\. Finally, the quadratic expansion of Corollary[3](https://arxiv.org/html/2610.09020#Thmcorollary3)describes the geometry near agreement; at the trained weights, neighboring predictions lie outside this regime on nearly all edges, so we treat the Fisher\-type weighting as an interpretation of the penalty’s local geometry rather than a description of training, in line with the ablations, where Fisher weighting shows no consistent advantage\.

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](https://arxiv.org/html/2610.09020#Thmproposition3), not the full expected squared error difference\.CIFAR\-10 / R50CIFAR\-100 / R50QuantityCENeSCal \(const\)NeSCal \(cos\)CENeSCal \(const\)NeSCal \(cos\)Bound violations \(L1, confidence\)000000Confidence\-bound tightness, median0\.0190\.0380\.0190\.1310\.1170\.101Edges within the expansion’s regime0%0\\%0%0\\%0%0\\%0\.005%0\.005\\%0\.005%0\.005\\%0\.005%0\.005\\%Across\-seed dispersion proxyσ​2​\(1−ρ\)\\sigma\\sqrt\{2\(1\-\\rho\)\}0\.0550\.0610\.0380\.1020\.1020\.088Across\-seed confidence spreadσc\\sigma\_\{c\}0\.0590\.0590\.0350\.1030\.1120\.076NLL gap to the test\-fitted oracle temperature0\.0200\.0050\.0150\.0300\.0250\.008Mean edge JS at the selected weights0\.1340\.1020\.1290\.3170\.3070\.323Reciprocal edges–0\.4350\.435–0\.4390\.439#### Synthetic setup\.

Each scenario draws equiprobable Gaussian classes inℝ20\\mathbb\{R\}^\{20\}with identity covariance, whose centers differ only in the first two coordinates\. In the label\-noise scenario, the centers lie on a circle of radius88\. In the geometric scenario, half lie on that circle and half on a circle of radius2\.82\.8centered at\(12\.8,0\)\(12\.8,0\), so the second group’s classes are more closely spaced\. In both scenarios, the labels of half the classes are redrawn uniformly at random with probabilityH∈\{0,0\.2,0\.4,0\.6\}H\\in\\\{0,0\.2,0\.4,0\.6\\\}\. Data are split60/20/2060/20/20into training, validation and test sets, and the model is an MLP with two hidden layers of width6464and a1616\-dimensional representation\. NeSCal usesk=10k=10and constant regularization, with activation epoch and strength selected by validation NLL and held fixed across the heterogeneity sweep\.

Evaluation compares predictions with pointwise reference distributionsqiq\_\{i\}and reports the top\-label reference error1n​∑i\(ci−qi,y^i\)2\\frac\{1\}\{n\}\\sum\_\{i\}\\bigl\(c\_\{i\}\-q\_\{i,\\hat\{y\}\_\{i\}\}\\bigr\)^\{2\}, the mean squared pointwise calibration error of Section[2](https://arxiv.org/html/2610.09020#S2)withai=qi,y^ia\_\{i\}=q\_\{i,\\hat\{y\}\_\{i\}\}\. In the label\-noise scenario,qiq\_\{i\}is the analytical noisy\-label posterior\. In the geometric scenario,qiq\_\{i\}mixes the clean posterior with the uniform distribution at weightHHwhen sampleii’s generating class is noisy, and equals the clean posterior otherwise\. This coincides with the noisy\-label posterior wherever the class posterior is concentrated within either the noisy or the clean group, and differs from it only where classes from the two groups overlap\. The comparator labeled “oracle” chooses, on the test set, the temperature minimizing this error among4747candidates \(six equally spaced in\[0\.5,1\)\[0\.5,1\)and4141in\[1,5\]\[1,5\]\); it is a test\-informed diagnostic rather than a deployable calibrator\.

#### The synthetic regimes\.

Figure[2](https://arxiv.org/html/2610.09020#A5.F2)compares NeSCal with CE and the test\-informed temperature search\. In the label\-noise scenario, the regional confidence biases have opposite signs, which no single temperature can correct together, and NeSCal attains lower top\-label reference error than the searched temperatures at every evaluatedH\>0H\>0\. In the geometric scenario, the biases share a sign: the temperature search is best, and NeSCal has higher reference error than CE\. Both outcomes are consistent with the idealized analysis of Proposition[3](https://arxiv.org/html/2610.09020#Thmproposition3)\. Averaging an edge changes its summed expected squared error by12​\(ai−aj\)2−12​𝔼​\[\(ei−ej\)2\]\\tfrac\{1\}\{2\}\(a\_\{i\}\-a\_\{j\}\)^\{2\}\-\\tfrac\{1\}\{2\}\\mathbb\{E\}\[\(e\_\{i\}\-e\_\{j\}\)^\{2\}\]\(Appendix[A\.9](https://arxiv.org/html/2610.09020#A1.SS9)\), so smoothing helps only when neighboring errors differ more than neighboring correctness probabilities do\. When neighbors share their error, as under a common same\-sign bias, the second term is small, and any correctness gap, which the more closely spaced classes of the geometric scenario plausibly create, makes smoothing increase error\. A global temperature acts on the shared component and is therefore the appropriate correction\. The geometric scenario therefore exhibits the failure mode the analysis predicts, and the label\-noise scenario the regime in which smoothing and rescaling are not interchangeable\.

![Refer to caption](https://arxiv.org/html/2610.09020v1/figs/synthetic_oracle.png)Figure 2:Mean squared top\-label reference error across heterogeneity strengthsHH, per scenario, on CE, CE with the test\-informed temperature \(*oracle*\), and NeSCal\.

### E\.12Perturbation studies

These studies test neighborhood smoothing under class imbalance, which can make neighborhoods less representative for rare classes and label noise, which weakens the supervised anchor\. Both studies use CIFAR\-100 with the clean\-selected configurations of R50 and WRN \(Appendix[D\.9](https://arxiv.org/html/2610.09020#A4.SS9)\)\.

#### Label noise\.

NeSCal lowers ECE relative to CE in all four architecture–noise conditions, but improves NLL and accuracy in only one \(Table[20](https://arxiv.org/html/2610.09020#A5.T20)\)\. These results show that improved aggregate confidence calibration under label noise need not accompany improved predictive quality\.

Table 20:Symmetric label noise on CIFAR\-100, without re\-tuningλ\\lambda\.
#### Class imbalance\.

Under long\-tailed training, the JS penalty improves NLL over CE in three of six settings\. SquaredL2L\_\{2\}attains the lowest NLL in all six\. Because these arms use equalλ\\lambdarather than matched penalty magnitude, the effect of divergence weighting \(Table[21](https://arxiv.org/html/2610.09020#A5.T21)\) is not isolated\.

Table 21:Long\-tailed CIFAR\-100: test NLL for CE and three divergences at equalλ\\lambda\.