Unlocking the Potential of Model Calibration in Federated Learning

## Abstract

Abstract Over the past several years, various federated learning (FL) methodologies have been developed to improve model accuracy, a primary performance metric in machine learning. However, to utilize FL in practical decision-making scenarios, beyond considering accuracy, the trained model must also have a reliable confidence in each of its predictions, an aspect that has been largely overlooked in existing FL research. Motivated by this gap, we propose Non-Uniform Calibration for Federated Learning ( NUCFL ), a generic framework that integrates FL with the concept of model calibration . The inherent data heterogeneity in FL environments makes model calibration particularly difficult, as it must ensure reliability across diverse data distributions and client conditions. Our NUCFL addresses this challenge by dynamically adjusting the model calibration objectives based on statistical relationships between each client’s local model and the global model in FL. In particular, NUCFL assesses the similarity between local and global model relationships, and controls the penalty term for the calibration loss during client-side local training. By doing so, NUCFL effectively aligns calibration needs for the global model in heterogeneous FL settings while not sacrificing accuracy. Extensive experiments show that NUCFL offers flexibility and effectiveness across various FL algorithms, enhancing accuracy as well as model calibration.

## 1 Introduction

Federated learning (FL) has rapidly gained traction as a prominent distributed machine learning approach, enabling collaborative model training across a network of clients by periodically aggregating their local models at a central server [Konecný et al., ; McMahan et al., ].
Its significance extends to various critical applications, including medical diagnostics, self-driving cars, and multilingual systems, where data privacy and decentralized training are paramount.
FL has been widely studied with a focus on improving local model updates Reddi et al. []; Sahu et al. [], refining global aggregation methods Ji et al. []; Wang et al. [], and enhancing communication efficiency Sattler et al. []; Diao et al. [].
Most of these works in FL consider accuracy as the main performance metric.

Motivation. However, beyond accuracy, in various decision-making scenarios where an incorrect prediction may result in high risk (e.g., medical applications or autonomous driving), it is also crucial for the users to determine whether to rely on the FL model’s prediction or not for each decision. In particular, users should rely on the neural network’s decision only when the prediction is likely to be correct. Otherwise, they may need to consider alternatives such as human decision. To achieve this, the trained FL model should have a reliable confidence in each of its predictions, meaning that the confidence of the neural network matches well with its actual accuracy. In centralized settings, recent studies including Guo et al. [] have uncovered that neural networks are often miscalibrated, indicating that the prediction confidence of the model does not accurately reflect the probability of correctness. Handling this miscalibration issue is even more important in many FL use-cases, where an overconfident global model could lead to misinformed decisions with potentially severe consequences for each client.

In centralized training settings, research generally follows two paths to address this miscalibration issue.
First, train-time calibration methods Hebbalaguppe et al. []; Liang et al. []; Liu et al. []; Mukhoti et al. []; Kumar et al. []; Müller et al. []; Lin et al. [] incorporate explicit regularizers during the training process to adjust neural networks, effectively scaling back over/under-confident predictions.
On the other hand, post-hoc calibration Guo et al. []; Hendrycks & Gimpel []; Hinton et al. []; Kull et al. [] transforms the network’s output vector to align the confidence of the predicted label with the actual likelihood of that label for the sample.
Post-hoc calibration methods are applied to the already trained model to improve calibration using an additional holdout dataset.
However, an appropriate holdout dataset may not be available in many privacy-sensitive applications, making post-hoc calibration strategies often impractical for FL.

Figure: Figure 1: Reliability diagrams and calibration errors for centralized training and non-IID FL (using FedAvg) trained with various calibration methods. Our method ensures well-calibrated FL, evidenced by a notably smaller calibration error and a smaller gap (red region) between confidence and accuracy.
Refer to caption: x1.png

Goals and observations. In this paper, we aim to incorporate model calibration into the FL training process, a problem that has been largely underexplored in existing research. This setting introduces new challenges to improving model calibration, as the heterogeneous data distributions across different clients must be carefully considered. We first observe whether existing train-time calibration methods applied during client-side local training can effectively address calibration needs in FL. To gain insights, in Fig. 1, we report accuracy, confidence, and the expected calibration error (ECE) of different methods in different settings. Figs. 1(a) to  1(b) illustrate the capability of train-time calibration methods to narrow the gap between model confidence and accuracy in centralized learning.
A comparison between Figs. 1(a) and  1(c) reveals that FL experiences more significant model miscalibration than centralized learning. This disparity is likely due to data heterogeneity across distributed FL clients, causing each client’s local data to have a different impact on calibration performance. When applying auxiliary-based calibration methods, such as DCA Liang et al. [] and MDCA Hebbalaguppe et al. [] – which add a penalty term to the original classification loss – to FL’s local training (Fig. 1(d)), a better ECE is achieved compared to FL without calibration (Fig. 1(c)). However, we observe that the model in Fig. 1(d) is still not well-calibrated by neglecting global calibration needs in heterogeneous FL settings, resulting in overconfident predictions. We thus pose the following question:

How can we unlock the potential of model calibration in FL while not sacrificing accuracy?

Contributions.
In this paper, we propose Non-Uniform Calibration for Federated Learning (NUCFL), a versatile and generic model calibration framework for FL that integrates seamlessly with any existing FL algorithms, such as FedProx Sahu et al. [], Scaffold Karimireddy et al. [], FedDyn Acar et al. [], and FedNova Wang et al. [].
Designed to work with any train-time calibration auxiliary loss, NUCFL is a straightforward yet effective solution that dynamically adjusts calibration penalties based on the relationship between global and local distribution. Specifically, our intuition is that if each client’s local model closely resembles the global model, it is likely that the local model and data represent the global calibration needs well. Therefore, our idea is to choose a large penalty term to improve model calibration for the clients that have a high similarity, while imposing a small penalty for other clients to prioritize accuracy. This non-uniform calibration approach
considering data heterogeneity across FL clients significantly minimizes calibration errors without accuracy degradation, as seen in Figure 1(e). Importantly, the flexibility of NUCFL allows for continuous improvement as new FL algorithms or calibration losses are developed. Our key contributions are summarized as follows:

- •
Our work is among the early efforts to conduct a systematic study on model calibration in FL. We study the question: How does model calibration impact the behavior of federated optimization methods?
- •
We propose NUCFL, a non-uniform model calibration approach tailored to FL. Our NUCFL dynamically adjusts calibration penalties based on the nuanced relationship between each local model and the global model, effectively capturing data heterogeneity.
- •
We conduct extensive experiments demonstrating that NUCFL is highly flexible, seamlessly integrating with 5 different FL algorithms (e.g., FedProx, Scaffold, FedDyn) and various auxiliary-based train-time calibration methods to improve both calibration performance and accuracy.

## 2 Related Work

Federated learning. FedAvg McMahan et al. [] is the pioneering FL algorithm, with numerous adaptations proposed to mitigate the accuracy drop experienced with non-IID data.
Much of the research has focused on designing better global aggregation methods to enhance model convergence and performance.
Some researchers He et al. []; Lin et al. [] replaced weight averaging with model ensemble and distillation, others Yurochkin et al. []; Wang et al. [] matched local model weights before averaging, and another group Ji et al. [] introduced an attention mechanism to optimize global aggregation.
Another significant area has concentrated on optimizing local training protocols to optimize computation and data usage on clients’ devices.
Some researchers Sahu et al. []; Acar et al. [] employed regularization toward the global model, while [Karimireddy et al., ] utilized server statistics to correct local gradients.
Additionally, various studies have addressed challenges specific to FL, such as ensuring fairness across all participants Li et al. [], enhancing communication efficiency to minimize bandwidth usage Ko et al. [], and addressing privacy concerns inherent in distributed computations Truong et al. []. However, the main focus of existing FL research is on improving accuracy. The mismatch between confidence and actual accuracy in these works leads to overconfident FL models, making FL inapplicable in high-risk decision-making scenarios. Our work bridges this gap by investigating an overlooked aspect of federated learning: model calibration.

Model calibration. Model calibration techniques can be categorized into post-hoc calibration and train-time calibration methods.
Post-hoc methods adjust the model using a hold-out dataset after training.
Common examples include Temperature Scaling Guo et al. [], which adjusts logits by a scalar learned from a hold-out set before softmax, and Dirichlet Calibration Kull et al. [], which modifies log-transformed class probabilities through an additional network layer. However, in FL applications, an additional validation set is often not available due to privacy issues, making post-hoc calibration methods inapplicable.
On the other hand, train-time calibration integrates calibration directly within the training process.
Muller et al. Müller et al. [] enhance calibration using label smoothing on soft targets, while Lin et al. Lin et al. [] use focal loss to minimize the KL divergence between predicted and target distributions.
Recently, augmenting standard cross-entropy loss with additional penalty loss terms has become a popular train-time calibration method Liang et al. []; Hebbalaguppe et al. []; Kumar et al. [].
This auxiliary loss penalizes the model when there is a reduction in cross-entropy loss without a corresponding increase in accuracy, often indicative of overfitting. However, as seen in Fig. 1, a direct application of these methods still faces limitations, as they are not able to meet the calibration needs considering data heterogeneity across FL clients. Our work fills this gap by taking a non-uniform calibration approach considering the characteristics of FL.

#### Calibration and Federated Learning

Peng et al. [] proposed FedCal, a calibration scaler aggregated solely from local clients, and introduced model calibration for FL.
However, FedCal does not consider the interaction between global and local calibration needs, which can lead to a calibration bias towards local heterogeneity.
Our model calibration framework addresses this issue by considering the interaction between client and server, thereby better accommodating broader global calibration requirements.
Furthermore, while FedCal has shown effectiveness only with FedAvg, our framework aims to provide robust calibration across various FL configurations, enhancing flexibility beyond just FedAvg.
It is also worth mentioning that Zhang et al. []; Luo et al. [] also use the term “calibration” in the FL setup. However, we stress that the context of calibration in their work is different from the one in the model calibration literature that considers confidence of the prediction. Zhang et al. []; Luo et al. [] focuses on calibrating the classifier to improve the accuracy itself, which can be categorized into conventional FL works that do not consider the error between model confidence and accuracy.

## 3 Preliminaries

### 3.1 Federated Learning

In Federated Learning (FL), the training data is gathered from a set of clients $M$ and each client $m$ possesses a training set $\mathcal{D}_{m}=\{x_{i},y_{i}\}^{|\mathcal{D}_{m}|}_{i=1}$, where $x$ represents the input data and $y$ denotes the corresponding true label.
We consider the following standard optimization formulation of federated training and seek to find model parameters $w$ that solve the problem:

$$ $\min_{w}\mathcal{L}(w)=\sum_{m\in M}\frac{|\mathcal{D}_{m}|}{|\mathcal{D}|} \mathcal{L}_{m}(w),$ (1) $$

where $w$ represents the model parameter, $\mathcal{D}=\cup_{m}\mathcal{D}_{m}$ denotes the aggregated training set from all clients, and $\mathcal{L}_{m}$ measures the average loss of a model on the training data of the $m$-th client.
The objective is to find a model that fits all clients’ data well on (weighted) average.

FL methods involve parallel local training at clients and global aggregation at a server over multiple communication rounds to address the aforementioned problem.
Most federated optimization methods fit within a general framework outlined by [Reddi et al., ], as shown in Algorithm 1.
At round $t$, the server sends its previous global model ${w}^{(t-1)}$ to all clients as initialization.
Each client $m$ then performs $E$ epochs of local training using ClientOPT depending on the FL algorithm, and subsequently produces a local model ${w}_{m}^{(t,E)}$.
Then each client communicates the difference $\delta^{(t)}_{m}$ between their learned local model and the server model, denoted as $\delta^{(t)}_{m}={w}^{(t-1)}-{w}^{(t,E)}_{m}$.
The server computes a weighted average of the client updates, denoted as $\delta^{(t)}$, and then updates the global model ${w}^{(t)}$ using ServerOPT, which also depends on FL designs.
For example, FedAvg employs standard stochastic gradient descent updates for ClientOPT, and its ServerOPT is formulated as $w^{(t)}=w^{(t-1)}-\delta^{(t)}$.

Figure: Algorithm 1 General FL Framework

### 3.2 Model Calibration

In supervised multi-class classification, the input $x\in\mathcal{X}$ and label $y\in\mathcal{Y}=\{1,...,K\}$ are random variables following a ground truth joint distribution $\pi(x,y)=\pi(y|x)\pi(x)$.
Let $s$ be the confidence score vector in $\mathbb{R}^{K}$, with $s[y]=f(x)$ representing the confidence that the neural network $f(\cdot)$ predicts for a class $y$ given input $x$.
The predicted class $\hat{y}$ is computed as: $\hat{y}=\arg\max_{y}s[y]$, with the corresponding confidence $\hat{s}=\max_{y}s[y]$.
A model is considered perfectly calibrated Guo et al. [] if:

$$ $\mathbb{P}(\hat{y}=y|\hat{s}=p)=p,\forall p\in[0,1].$ (2) $$

That is to say, we expect the confidence estimate of the prediction to reflect the true probability of the prediction being accurate.
For example, given 100 predictions, each with a confidence of $0.8$, we expect that 80 should be correctly classified.
One concept of miscalibration involves the difference between confidence and accuracy, which can be described as:

$$ $\mathbb{E}_{\hat{s}}[|(\hat{y}=y|\hat{s}=p)-p|].$ (3) $$

The primary goal of model calibration, whether via train-time or post-hoc methods, is to minimize discrepancy (3) to ensure the model’s confidence accurately reflects its prediction accuracy.

## 4 Proposed Model Calibration for Federated Learning

### 4.1 Bridging Federated Learning and Model Calibration

Our approach aims to implement calibration on each FL client’s local training.
Note that in conventional FL algorithms, the loss $\mathcal{L}_{m}$ of client $m$ in (1) can be expressed as follows:

$$ $\mathcal{L}_{m}({w}_{m}^{(t,e)})=\frac{1}{|\mathcal{D}_{m}|}\sum^{|\mathcal{D} _{m}|}_{i=1}\ell({w}_{m}^{(t,e)};x_{i},y_{i}),$ (4) $$

where $w_{m}$ is the local model, $t$ the current global round, $e$ the local iteration index, and $\ell(\cdot)$ the loss function (e.g., cross-entropy) applied to a data instance.
One can also modify the loss in (4), i.e., by adding a proximal term as in FedProx Sahu et al. [].
Recent studies on calibration Liang et al. []; Hebbalaguppe et al. [] suggest that incorporating an auxiliary loss, which captures the disparity between accuracy and confidence (as in equation (3)), and penalizing the model accordingly, leads to superior performance compared to direct loss type adjustments Müller et al. []; Brier []; Lin et al. []. These auxiliary loss methods maintain the original classification loss and seamlessly add a penalty, minimizing interference with FL performance, especially since some FL algorithms are specifically designed around their classification loss strategies. This makes the auxiliary loss methods easily applicable and effective across various FL settings.

Proposed framework. Therefore, for client $m$, we propose to integrate the calibration loss with the FL loss as follows:

$$ $\mathcal{L}_{m}^{cal}({w}_{m}^{(t,e)})=\frac{1}{|\mathcal{D}_{m}|}\sum^{| \mathcal{D}_{m}|}_{i=1}\left[\ell({w}_{m}^{(t,e)};x_{i},y_{i})+\beta_{m}\ell_{ cal}({w}_{m}^{(t,e)};x_{i},y_{i})\right],$ (5) $$

where $\ell_{cal}(\cdot)$ is the auxiliary calibration loss function, and $\beta_{m}$ is a scalar weight for each client $m$. In the centralized calibration case, this scalar weight is manually selected to refine the auxiliary calibration loss. However, in our design, we automatically determine $\beta_{m}$ based on the relationship between the server and clients in a FL setup, taking into account the heterogeneity as detailed in Section 4.2. The auxiliary calibration loss $\ell_{cal}$ can be derived from various calibration designs. For instance, [Liang et al., ] directly adds the difference between confidence and accuracy (DCA) as an auxiliary loss, which can be represented as:

$$ $\ell_{cal}\Big{(}\{c_{i}\}_{1\leq i\leq N},\{\hat{s}_{i}\}_{1\leq i\leq N}\Big {)}=\ell_{dca}=\left|\frac{1}{N}\sum_{i=1}^{N}c_{i}-\frac{1}{N}\sum_{i=1}^{N} \hat{s}_{i}\right|,$ (6) $$

where $N$ denotes the total number of training samples, $\hat{s}_{i}$ represents the confidence score of the model’s predicted label $\hat{y}_{i}$ for sample $i$, and $c_{i}=1$ if $\hat{s}_{i}$ equals the true label $y_{i}$; otherwise, $c_{i}=0$.
On the other hand, Hebbalaguppe et al. [] provide a more detailed analysis of (6) and propose a Multi-class DCA (MDCA), which offers a more granular examination by considering all classes instead of focusing solely on the predicted label. The proposed auxiliary loss can be written as:

$$ $\ell_{cal}\Big{(}\{c_{i}\}_{1\leq i\leq N},\{{s}_{i}\}_{1\leq i\leq N}\Big{)}= \ell_{mdca}=\frac{1}{K}\sum_{j=1}^{K}\left|\frac{1}{N}\sum_{i=1}^{N}c_{i}[j]- \frac{1}{N}\sum_{i=1}^{N}s_{i}[j]\right|,$ (7) $$

where $K$ is the number of classes and $N$ the number of samples. $c_{i}[j]=1$ shows that label $j$ is the ground truth for sample $i$, and $c_{i}[j]$ gives the confidence score for the $j$-th class of sample $i$.

Advantages and challenges. The differentiability of these auxiliary loss terms ((6) and (7)) facilitates their integration with other application-specific loss functions, allowing them to enhance calibration without significantly impacting the primary classification loss, $\ell$.
This minimizes any adverse effects on performance while correcting calibration errors.
For instance, FL algorithms like FedProx Sahu et al. [] and FedNova Wang et al. [] incorporate custom loss functions within their local modeling function, ClientOPT (outlined in Algorithm 1), to address deviations from the global model due to non-IID data.
Despite these enhancements, it is critical to note that these advanced FL algorithms still fundamentally require a standard classification loss, which they fine-tune for specific objectives.
When integrating calibration into the ClientOPT of FL algorithms, we will exclusively calibrate the standard classification loss, maintaining the integrity of other design elements, such as global aggregation, to ensure their original objectives are not compromised. Choosing an appropriate $\beta_{m}$ tailored to each client $m$ is the remaining challenge.

### 4.2 Non-Uniform Calibration for Federated Learning ( NUCFL )

Motivation: Limitation of uniform penalty. In Equation (5), we promote calibration during federated local training by targeting the confidence-accuracy gap in individual local models via auxiliary calibration loss.
The main aim of FL is to train a global model capable of effectively handling the global data distribution.
Therefore, we aim for calibration to account for the heterogeneity inherent in FL setups, marking a departure from calibration approaches used in centralized learning.
Direct application of auxiliary loss calibration methods without adapting to FL characteristics would result in uniform calibration weights, such that $\beta_{1}=\beta_{2}=...=\beta_{m}$. This leads to local models being calibrated solely to their respective datasets $D_{m}$, with uniformly small weights biasing calibration toward local heterogeneity, and uniformly large weights potentially neglecting accuracy improvements from classification. Such discrepancies can result in a global model that, while comprising well-calibrated local models, does not achieve optimal performance across the broader distribution. This fixed calibration penalty approach, therefore, risks neglecting the broader global calibration needs necessary for optimal performance across the entire global distribution.

Figure: Figure 2: Idea of proposed NUCFL.
Refer to caption: x2.png

Non-uniform penalty design (NUCFL). Given that each client receives the global model from the previous round as their starting point for each new federated round, this offers an opportunity to indirectly adjust local calibration to mirror global model characteristics.
To leverage this, we propose non-uniform calibration for federated learning (NUCFL), which provides a dynamic calibration penalty to local models to better reflect global calibration needs.
Drawing inspiration from the concept of similarity characterization in FL techniques Sahu et al. []; Tan et al. [], we hypothesize that a local model closely resembling the global model is likely to represent global characteristics well, suggesting that the penalty appropriately reflects the calibration needs of the global model.
Conversely, a dissimilar/heterogeneous local model suggests a focus on local objectives (e.g., to improve accuracy) at the expense of global alignment.
By adjusting penalties at each local epoch based on the similarity between local and global models, NUCFL ensures that local training enhances both the calibration and accuracy of the global model. As illustrated in Fig. 2, we specifically set

$$ $\beta_{m}=\texttt{sim}(\delta^{(t-1)},\delta_{m}^{(t,e)}),$ (8) $$

where $\delta^{(t-1)}$ is the accumulated gradient from the previous round $t-1$, representing the received global model, while $\delta_{m}^{(t,e)}={w}^{(t-1)}-{w}^{(t,e)}_{m}$ calculates the gradient for client $m$ at local epoch $e$ during round $t$.
The function sim($\cdot$) measures similarity between local and global models, generating values between 0 and 1.
Specifically, it can use general cosine similarity or more advanced measures like Centered Kernel Alignment (CKA) Kornblith et al. [].

For instance, if the learned local model is similar to the received global model, indicated by a higher similarity score, it suggests that the local model’s characteristics align well with those of the global model. In such cases, the calibration penalty applied to the local model is increased to reflect the global calibration needs. Otherwise, i.e., if the similarity score is low, we choose a smaller penalty and instead focus on improving the accuracy.
Based on this non-uniform penalty method tailored to FL, our scheme strategically improves model calibration as well as the accuracy.

Remark 1. NUCFL can integrate any auxiliary-based calibration loss and adapt to any FL algorithm regardless of aggregation method or local loss function.
To the best of our knowledge, this is the first work to address model calibration specifically for FL, introducing a tailored calibration method.
Similar to most calibration research Lin et al. []; Müller et al. []; Brier []; Kumar et al. []; Mukhoti et al. []; Liu et al. []; Liang et al. []; Hebbalaguppe et al. [; ], our work is an empirical study that explores the practical impacts of our methods.
This area of research remains challenging, particularly in providing theoretical guarantees for calibration methods, even in centralized settings; we instead substantiate the effectiveness of our method through extensive experiments.
We will demonstrate how our strategic approach effectively calibrates FL in Section 5.2.

Remark 2 (Compatibility with post-hoc calibration). Although our NUCFL is a train-time calibration approach that is applied during FL training, the proposed technique complements post-hoc methods whenever a holdout dataset is available. For example, temperature scaling can be applied to the trained global FL model for further refinement, as empirically demonstrated in Section 5.2.

## 5 Experiments

### 5.1 Experimental Setup

Dataset and model.
We conduct experiments using four image classification datasets commonly utilized in FL research Caldas et al. []; McMahan et al. []; Mohri et al. []: MNIST LeCun et al. [], FEMNIST Cohen et al. [], CIFAR-10 Krizhevsky [], and CIFAR-100 Krizhevsky [].
For the MNIST and FEMNIST datasets, we use a CNN LeCun & Bengio [] with two 5x5 convolution layers, each followed by 2x2 max pooling, and fully connected layers with ReLU activation. We use AlexNet Krizhevsky et al. [] with five convolution layers for the CIFAR-10/100 datasets.

FL data distribution.
In the IID setup, data samples from each class are distributed equally to $M=50$ clients.
To simulate non-IID conditions across clients, we follow Hsu et al. []; Nguyen et al. []; Chen et al. [] to partition the training set into $M=50$ clients using a Dirichlet distribution with $\alpha\in[0.5,0.1,0.05]$.

Calibration baselines. We compare our method against models trained using Cross-Entropy (Uncal.), representing training without calibration, and others such as Focal Loss (FL) Lin et al. [], Label Smoothing (LS) Müller et al. [], Brier Score (BS) Brier [], MMCE Kumar et al. [], FLSD Mukhoti et al. [], MbLS Liu et al. [], DCA Liang et al. [], and MDCA Hebbalaguppe et al. [].
Detailed parameters for each calibration baseline are provided in the Appendix A.1.

FL algorithms and setups.
We conduct model calibration on five well-known FL algorithms, including FedAvg McMahan et al. [], FedProx Sahu et al. [], Scaffold Karimireddy et al. [], FedDyn Acar et al. [], and FedNova Wang et al. [].
We run each FL algorithm for 100 rounds, evaluating the final global model, with 5 epochs for each local training. We use the SGD optimizer with a learning rate of $10^{-3}$, weight decay of $10^{-4}$, and momentum of $0.9$.
For additional details on the training specifics of each algorithm, please see Appendix A.2.

Auxiliary loss and similarity measures for NUCFL. We implement our framework using two auxiliary-based calibration losses, $\ell_{dca}$ and $\ell_{mdca}$. We utilize three similarity measurements–cosine similarity (COS), linear centered kernel alignment (L-CKA) Kornblith et al. [], and RBF-CKA Kornblith et al. []–which output a similarity score ranging from 0 (not similar at all) to 1 (identical) for evaluating the similarity between the learned local model and the global model.
We refer to configurations using our technique as “NUCFL($\ast$ + $\circledcirc$),” where $\ast$ denotes auxiliary calibration losses $\ell_{dca}$/$\ell_{mdca}$ and $\circledcirc$ denotes similarity measurements COS/L-CKA/RBF-CKA, resulting in six configurations for our methods.

Evaluation metrics. To evaluate performance, we measure the accuracy (%) of the final global model on the test set and assess calibration using standard metrics such as Expected Calibration Error (ECE%) and Static Calibration Error (SCE%).
ECE Naeini et al. [], a commonly used metric, approximates calibration error (3) by dividing predictions into $I$ bins and calculating a weighted average of the discrepancies between accuracy and confidence across these bins.
Predictions are categorized into $I$ interval bins based on their probabilities: $\text{ECE}=\sum_{i=1}^{I}\frac{B_{i}}{N}\left|A_{i}-C_{i}\right|$,
where $N$ represents the total number of samples, with weighting based on the proportion of samples within each confidence bin/interval.
Each $i$-th bin covers the interval $(\frac{i-1}{I},\frac{i}{I}]$ within the confidence range, with $B_{i}$ indicating the number of samples in the $i$-th bin.
$A_{i}=\frac{1}{|B_{i}|}\sum_{j\in B_{i}}\mathbb{I}(\hat{y}_{j}=y_{j})$ calculates the accuracy within bin $B_{i}$, and $C_{i}=\frac{1}{|B_{i}|}\sum_{j:\hat{s}_{j}\in B_{i}}\hat{s}_{j}$ gives the average predicted confidence for samples where $\hat{s}_{j}\in B_{i}$.
The recently proposed SCE Nixon et al. []extending ECE for multi-class settings, groups predictions into bins by class probability, computes calibration errors per bin, and averages these across all bins:
$\text{SCE}=\frac{1}{K}\sum_{i=1}^{I}\sum_{j=1}^{K}\frac{B_{i,j}}{N}\left|A_{i,
j}-C_{i,j}\right|$,
where $K$ represents the number of classes, and $B_{i,j}$ denotes the count of $j$-th class samples in the $i$-th bin. $A_{i,j}=\frac{1}{|B_{i,j}|}\sum_{k\in B_{i,j}}\mathbb{I}(j=y_{k})$ measures accuracy, while $C_{i,j}=\frac{1}{|B_{i,j}|}\sum_{k\in B_{i,j}}s_{k}[j]$ gives the average confidence for the $j$-th class in the $i$-th bin.
We set the number of bins, $I=20$, for ECE and SCE, and evaluate the final global model.

### 5.2 Experimental Results

Can train-time calibration apply to federated learning?
We first compare train-time calibration in traditional centralized training versus non-IID federated learning. We train a model on the entire CIFAR-100 training dataset for 100 epochs with various train-time calibration methods, and present the results in Table 6 of Appendix B.1.
Results show that most baselines effectively calibrate in centralized settings, maintaining accuracy and exhibiting lower calibration errors. Yet, when these train-time calibration techniques are applied to FL, as depicted in Table 1, these methods often underperform in calibration and accuracy.
Notably, methods like DCA and MDCA that includes an auxiliary loss for calibration show robustness in federated settings.
They maintain the primary classification objective while penalizing miscalibration, preserving confidence without affecting classification, distinguishing them in FL environments.

The effects of confidence calibration on FL.
Tables 1 and 2 show performance and calibration metrics for various calibration applied to FL algorithms under non-IID scenarios using CIFAR-100 and FEMNIST datasets.
Among these baselines, auxiliary-based methods like DCA and MDCA consistently show superior performance, with comparable accuracy and lower calibration error.
However, our method outperforms them, as evidenced by the superior performance of NUCFL (DCA/MDCA + $\circledcirc$) over DCA/MDCA, highlighting the importance of our non-uniform penalty, which is specifically tailored to address global calibration needs during local training.
The superior performance of NUCFL, incorporating different similarity measurements and various types of auxiliary loss punishment, demonstrates its flexibility and effectiveness.
Furthermore, the consistent improvement of our framework across various FL algorithms illustrates its adaptability and robustness in diverse settings.
See Appendix B.2 for more results including other datasets and data distributions.

**Table 1: Accuracy (%), calibration measures ECE (%), and SCE (%) of various federated optimization methods with different calibration methods under non-IID ($\alpha=0.5$) scenario on the CIFAR-100 dataset. Underlined values indicate the best calibration across all methods.**
| Calibration<br>Method | FedAvg McMahan et al. [] | FedProx Sahu et al. [] | Scaffold Karimireddy et al. [] | FedDyn Acar et al. [] | FedNova Wang et al. [] |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 72.77 | 9.80 | 3.21 | 72.95 | 4.29 | 2.20 | 72.09 | 5.38 | 3.81 | 73.83 | 11.76 | 7.29 | 73.23 | 6.99 | 3.63 |
| Focal Lin et al. [] | 72.61 | 10.96 | 3.95 | 73.62 | 5.40 | 3.17 | 71.63 | 5.82 | 4.37 | 72.64 | 8.70 | 5.13 | 72.48 | 13.81 | 7.11 |
| LS Müller et al. [] | 71.26 | 13.59 | 3.62 | 73.09 | 6.08 | 3.49 | 72.90 | 4.13 | 2.67 | 73.65 | 7.57 | 4.94 | 72.26 | 5.83 | 3.53 |
| BS Brier [] | 71.26 | 9.98 | 3.42 | 72.72 | 5.81 | 2.76 | 71.00 | 4.59 | 3.04 | 72.83 | 15.06 | 8.43 | 71.26 | 5.77 | 3.49 |
| MMCE Kumar et al. [] | 72.64 | 11.66 | 3.67 | 72.64 | 5.26 | 2.94 | 72.46 | 6.77 | 4.95 | 71.95 | 9.27 | 6.24 | 73.99 | 6.32 | 5.68 |
| FLSD Mukhoti et al. [] | 71.10 | 10.29 | 4.06 | 74.23 | 8.45 | 4.29 | 70.39 | 4.84 | 3.35 | 71.95 | 7.19 | 4.67 | 72.35 | 6.83 | 6.07 |
| MbLS Liu et al. [] | 70.16 | 7.99 | 3.98 | 73.42 | 4.44 | 2.73 | 73.11 | 5.97 | 4.75 | 72.58 | 9.69 | 5.69 | 72.38 | 5.16 | 3.44 |
| DCA Liang et al. [] | 72.39 | 6.95 | 3.07 | 73.05 | 4.07 | 2.07 | 72.10 | 4.59 | 3.08 | 73.77 | 8.63 | 4.96 | 73.88 | 5.73 | 3.45 |
| NUCFL (DCA+COS) | 72.97 | 5.85 | 2.99 | 72.84 | 4.02 | 1.89 | 72.53 | 3.87 | 2.58 | 73.48 | 7.03 | 4.35 | 73.84 | 4.62 | 3.04 |
| NUCFL (DCA+L-CKA) | 73.73 | 5.14 | 2.87 | 73.16 | 3.99 | 1.85 | 71.95 | 3.95 | 2.93 | 73.49 | 6.79 | 4.05 | 73.87 | 4.11 | 2.98 |
| NUCFL (DCA+RBF-CKA) | 72.20 | 6.66 | 3.10 | 72.95 | 4.09 | 1.95 | 73.05 | 3.84 | 2.41 | 73.23 | 6.80 | 4.03 | 73.69 | 4.15 | 2.94 |
| MDCA Hebbalaguppe et al. [] | 71.47 | 7.69 | 3.19 | 72.97 | 4.11 | 2.13 | 72.15 | 4.99 | 3.37 | 73.61 | 7.51 | 4.93 | 72.95 | 5.08 | 3.15 |
| NUCFL (MDCA+COS) | 72.24 | 5.24 | 3.03 | 72.78 | 3.82 | 2.04 | 72.07 | 3.96 | 2.89 | 74.06 | 6.99 | 4.33 | 73.93 | 4.28 | 2.91 |
| NUCFL (MDCA+L-CKA) | 73.64 | 5.20 | 2.96 | 73.69 | 3.79 | 1.78 | 72.64 | 4.01 | 3.01 | 73.28 | 7.10 | 4.81 | 72.62 | 5.15 | 3.37 |
| NUCFL (MDCA+RBF-CKA) | 72.71 | 6.59 | 2.98 | 73.31 | 3.97 | 1.90 | 72.10 | 4.14 | 3.14 | 73.27 | 6.95 | 4.15 | 73.04 | 4.03 | 2.79 |

**Table 2: Accuracy (%), calibration measures ECE (%), and SCE (%) of various FL algorithms with different calibration methods under non-IID ($\alpha=0.5$) scenario on the FEMNIST dataset.**
| Calibration<br>Method | FedAvg McMahan et al. [] | FedProx Sahu et al. [] | Scaffold Karimireddy et al. [] | FedDyn Acar et al. [] | FedNova Wang et al. [] |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 91.98 | 4.38 | 1.83 | 92.57 | 4.53 | 1.74 | 92.17 | 5.33 | 2.44 | 93.03 | 5.11 | 2.39 | 92.56 | 4.68 | 1.76 |
| Focal Lin et al. [] | 91.71 | 4.69 | 1.91 | 92.06 | 4.93 | 2.02 | 92.25 | 5.59 | 2.70 | 91.13 | 5.28 | 2.55 | 91.42 | 4.92 | 2.27 |
| LS Müller et al. [] | 91.89 | 3.94 | 1.55 | 92.51 | 4.29 | 1.63 | 92.11 | 5.17 | 2.31 | 93.34 | 4.83 | 2.26 | 92.63 | 4.37 | 1.70 |
| BS Brier [] | 91.99 | 5.06 | 2.03 | 92.51 | 5.53 | 2.58 | 89.53 | 4.03 | 1.49 | 90.90 | 5.18 | 2.39 | 90.94 | 4.23 | 1.47 |
| MMCE Kumar et al. [] | 91.65 | 4.93 | 2.17 | 90.66 | 3.86 | 1.32 | 92.25 | 4.95 | 1.99 | 92.80 | 4.70 | 1.93 | 92.94 | 4.91 | 2.30 |
| FLSD Mukhoti et al. [] | 90.97 | 5.21 | 2.89 | 91.73 | 4.89 | 1.99 | 92.91 | 5.26 | 2.27 | 91.90 | 4.98 | 2.30 | 91.45 | 4.89 | 2.25 |
| MbLS Liu et al. [] | 91.93 | 4.19 | 1.75 | 92.43 | 4.42 | 1.65 | 93.05 | 5.04 | 2.25 | 93.10 | 5.13 | 2.39 | 91.74 | 4.44 | 1.69 |
| DCA Liang et al. [] | 92.04 | 3.99 | 1.59 | 92.90 | 4.23 | 1.61 | 92.95 | 4.38 | 1.68 | 93.20 | 4.52 | 1.77 | 92.56 | 4.20 | 1.39 |
| NUCFL (DCA+COS) | 92.37 | 3.30 | 1.35 | 92.86 | 3.76 | 1.37 | 92.80 | 4.16 | 1.49 | 93.10 | 4.13 | 1.55 | 92.96 | 4.09 | 1.17 |
| NUCFL (DCA+L-CKA) | 91.89 | 3.44 | 1.39 | 92.56 | 3.66 | 1.29 | 92.23 | 4.11 | 1.45 | 93.37 | 4.10 | 1.51 | 92.56 | 3.98 | 1.13 |
| NUCFL (DCA+RBF-CKA) | 91.93 | 3.61 | 1.43 | 92.61 | 3.69 | 1.33 | 92.31 | 4.07 | 1.44 | 93.59 | 4.20 | 1.59 | 92.34 | 3.90 | 1.13 |
| MDCA Hebbalaguppe et al. [] | 91.78 | 3.66 | 1.44 | 93.06 | 3.98 | 1.45 | 93.23 | 4.45 | 1.70 | 93.10 | 4.58 | 1.77 | 92.73 | 4.17 | 1.24 |
| NUCFL (MDCA+COS) | 92.03 | 3.75 | 1.46 | 92.74 | 3.71 | 1.37 | 92.52 | 4.09 | 1.44 | 92.95 | 4.25 | 1.59 | 92.36 | 4.05 | 1.17 |
| NUCFL (MDCA+L-CKA) | 92.11 | 3.19 | 1.20 | 91.96 | 3.84 | 1.40 | 92.47 | 4.17 | 1.50 | 92.57 | 4.16 | 1.55 | 91.59 | 4.01 | 1.15 |
| NUCFL (MDCA+RBF-CKA) | 91.99 | 3.37 | 1.35 | 92.91 | 3.79 | 1.47 | 92.84 | 4.15 | 1.50 | 93.30 | 4.33 | 1.62 | 92.66 | 3.98 | 1.14 |

Mitigating over/under-confidence.
The results in Tables 1 and 2 show that our method improves SCE and ECE over the baselines, but they highlight whether they correct for over-confidence or under-confidence.
To clarify this, we present reliability diagrams Degroot & Fienberg []; Niculescu-Mizil & Caruana [], which visually compare predicted probabilities to actual outcomes, plotting expected accuracy against confidence.
If the model is perfectly calibrated (i.e. if $\mathbb{P}(\hat{y}=y^{*}|s[\hat{y}])=s[\hat{y}]$), then the diagram should plot the identity function.
Any deviation from a perfect diagonal represents miscalibration.
In Fig. 3, we present reliability histograms for non-IID FL using various calibration methods, where each FL method yields over-confident predictions.
This means that, for example, if a model predicts the probability of a class $k$ from $\{1,...,K\}$ as 0.7, we would expect it to be correct in 70 out of 100 cases.
However, if the actual number of correct predictions are less than 70, the model is considered over-confident.
Such over-confidence can be problematic, especially in critical applications like medical diagnostics, where it could lead to under-treatment and serious health risks.
Our methods show a smaller gap (red region) and lower ECE, effectively mitigating over-confidence.
For details on how our method also tackles under-confidence, please refer to Appendix B.3.

Effectiveness of our design.
We explore model calibration on various FL settings, analyzing the effects of different non-IID degrees $\alpha$, numbers of clients $M$, and percentages of participating clients per round.
Fig. 4 shows the accuracy and calibration results of FedAvg across various configurations on the MNIST dataset, where we adjust one variable at a time from the default setting: $M=50$, $\alpha=0.5$, 100% participation.
Across all settings, our method consistently demonstrates robust calibration.
Importantly, our NUCFL maintains effectiveness in more challenging conditions, such as with lower $\alpha$ values indicating higher non-IID disparity or reduced participant numbers per round.
This demonstrates the adaptability of our approach in varied and complex FL environments.

**Table 3: Destructive experiment for our method.**
| Calibration<br>Method | Non-IID FedAvg ($\alpha=0.5$) |  |  |
| --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| NUCFL (DCA+L-CKA) | 73.73 | 5.14 | 2.87 |
| NUCFL (MDCA+L-CKA) | 73.64 | 5.20 | 2.96 |
| Reversed NUCFL (DCA+L-CKA) | 72.19 | 10.17 | 3.88 |
| Reversed NUCFL (MDCA+L-CKA) | 71.86 | 9.64 | 3.21 |

Calibration strategy validation.
Our calibration strategy applies a higher auxiliary penalty loss to local models that significantly deviate from the received global model, addressing global calibration needs during local training.
To validate the soundness of our design, we experiment with an opposite logic, where a higher penalty is applied to local clients more similar to the received global model.
We specifically adjust the weight $\beta_{m}$ for each client $m$ to $\texttt{sim}(\delta^{(t-1)},\delta_{m}^{(t,e)})^{-1}$.
We conducted this destructive experiment to assess whether this approach affects FL and to validate our original calibration logic.
In Table 3, we applied a reverse penalty strategy to optimal configurations of our method (i.e., NUCFL (DCA+L-CKA) and (NDCA+L-CKA)) on the CIFAR-100 dataset.
The results show that using reversed logic negatively impacts the system, leading to a calibration penalty biased towards local heterogeneity and ignoring global calibration needs, thus degrading the FL system’s performance.
This even resulted in poorer performance compared to uncalibrated FL as shown in Table 1.
This experiment confirms the soundness of our design logic, which strategically prioritizes global calibration needs during local training and suggests the benefits of adhering to our approach to avoid disadvantageous effects.

Figure: Figure 3: Reliability diagrams for non-IID FedAvg ($\alpha=0.5$) with different calibration methods using the CIFAR-10 dataset. The lower ECE and smaller gap show the effectiveness of our method.
Refer to caption: x3.png

Figure: Figure 4: Comparison of confidence calibration across different FL settings using the MNIST dataset. The default setting is 50 clients, $\alpha=0.5$, 100% participation, with one variable changed at a time.
Refer to caption: x4.png

**Table 4: Comparison with SOTA baseline.**
| Calibration Method | Non-IID FedAvg ($\alpha=0.5$) |  |  |
| --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 72.77 | 9.80 | 3.21 |
| FedCal Peng et al. [] | 72.77 | 5.97 | 3.03 |
| NUCFL (DCA+L-CKA) | 73.73 | 5.14 | 2.87 |
| NUCFL (MDCA+L-CKA) | 73.64 | 5.20 | 2.96 |

Comparison with the existing model calibration baseline for FL.
We now conduct comparisons with the only existing model calibration framework for FL, FedCal Peng et al. []. Table 4 displays the calibration results of our method alongside the baseline using FedAvg under non-IID conditions ($\alpha=0.5$) on CIFAR-100 datatset. For the scaler used in their work, we adhered to their design, employing an MLP with layer sizes of 100-64-64-100. We see that our proposed NUCFL has shown superior performance compared to FedCal. While the use of a scaler aggregated solely from local clients can help reduce global ECE, it often overlooks the essential interaction between global and local calibration needs. Our method addresses this by considering these interactions, thus avoiding biases toward local heterogeneity and better accommodating the broader global calibration requirements, a challenge that the aggregation methods might not fully resolve.
More comparisons using other datasets are available in Appendix B.4.

**Table 5: Comparison with post-hoc calibration.**
| Calibration<br>Method | Non-IID FedAvg ($\alpha=0.5$) |  |  |
| --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 72.77 | 9.80 | 3.21 |
| DCA Liang et al. [] | 72.39 | 6.95 | 3.07 |
| MDCA Hebbalaguppe et al. [] | 71.47 | 7.69 | 3.19 |
| NUCFL (DCA+L-CKA) | 73.73 | 5.14 | 2.87 |
| TS Guo et al. [] | 72.77 | 7.28 | 3.26 |
| DCA+TS | 72.39 | 6.10 | 2.83 |
| MDCA+TS | 71.47 | 7.33 | 3.21 |
| NUCFL (DCA+L-CKA)+TS | 73.73 | 4.98 | 2.65 |

Compatibility with post-hoc calibration.
Temperature scaling (TS) Guo et al. [] is one of the most effective post-hoc calibration methods, which adjusts a model’s logits by a scaling factor to calibrate confidence scores without affecting accuracy.
Like all post-hoc methods, TS requires a hold-off dataset to tune the model and learn the calibration parameter (i.e., temperature).
However, this need conflicts with the data privacy policies of FL, making TS impractical in such settings.
For this analysis, we set aside these privacy concerns to compare with TS, assuming that data from each client is available at the server.
Under these assumptions, we conduct non-IID ($\alpha=0.5$) FedAvg and sample five images from each client to create a server dataset for post-hoc calibration.
Table 5 compares train-time and post-hoc calibration methods using the CIFAR-100 dataset.
Results show TS outperforming some train-time methods, like MDCA, while our method still provides superiority in both accuracy and calibration error.
Recall from Remark 2 that our NUCFL can also be complemented by post-hoc methods when a holdout dataset is available for refining the global model.
We apply post-hoc calibration on the global model using TS after train-time calibration during local training.
This combined approach shows improved performance over train-time calibration alone.
Notably, integrating TS with NUCFL yields the best results, significantly reducing error.

Further experiments. Additional results on varying degrees of non-IIDness, evaluations across datasets, and discussions on how our method tackles the over/under-confidence problem are available in Appendix B.

## 6 Conclusion

We conducted one of the earliest systematic study on model calibration for FL and introduced a novel framework, NUCFL, designed to ensure a well-calibrated global model.
NUCFL measures the similarity between client and server models and dynamically adjusts calibration penalties for each client, addressing both heterogeneity and global calibration needs effectively.
Through extensive experiments, NUCFL demonstrated significant improvements over existing calibration methods, proving its adaptability with various types of auxiliary calibration penalties and compatibility across different FL algorithms. Our research seeks to deepen understanding of the underlying effects of model calibration on FL, enhancing the trustworthiness of FL systems.

## Appendix A Detailed Settings for Calibration Baselines and Federated Algorithms

### A.1 Calibration Baselines

We offer an overview of each compared calibration method, including the hyperparameter settings utilized during training.
Focal loss Lin et al. [] is defined as $\text{FL}=-\sum^{N}_{i=1}(1-p_{i,y_{i}})^{\gamma}\cdot\log(p_{i,y_{i}})$, where $p_{i,y_{i}}$ is the predicted confidence score for sample $i$ and $\gamma$ is a hyper-parameter. We trained models with $\gamma\in\{1,2\}$. Label Smoothing (LS) Müller et al. [] is defined as $\text{LS}=-\sum^{N}_{i=1}\sum^{K}_{j=1}q_{i,j}\cdot\log(p_{i,j})$, where $p_{i,j}$ is the predicted confidence score for class $j$ of sample $i$, and $q_{i}$ is the soft target vector for sample $i$, such that $q_{i,j}=\frac{\alpha}{K-1}$ if $j\neq y_{i}$, else $q_{i,j}=(1-\alpha)$. $\alpha$ is a hyperparameter set at 0.1, following established literature settings Hebbalaguppe et al. [].
For the Brier Score (BS) Brier [], we train using a loss defined as the squared difference between the predicted probability vector and the one-hot target vector.
For Maximum Mean Calibration Error (MMCE) Kumar et al. [], we use the weighted MMCE, with $\lambda$ selected from $\{2,4\}$, as a regularization term along with cross-entropy. For FLSD Mukhoti et al. [], we use $\gamma=3$, reported as the best configuration in their paper.
For Margin-based Label Smoothing (MbLS) Liu et al. [], we set the margin at 10 and $\mu=0.1$, configurations reported as optimal in their paper.
For DCA Liang et al. [], we train using a loss composed of cross-entropy plus $\beta\cdot\ell_{dca}$, where $\beta$ is a hyperparameter selected from $\{1,5,10\}$ as demonstrated in their paper.
Similarly, for MDCA Hebbalaguppe et al. [], we apply the cross-entropy loss enhanced by $\beta\cdot\ell_{mdca}$, with $\beta$ also chosen from $\{1,5,10\}$ as well.
For all baselines, we report results that show the lowest ECE among their hyperparameter settings in our paper.

### A.2 Federated Algorithms

For all FL algorithms, we distributed the training data across $M=50$ clients and performed 100 rounds of training. Each client conducted 5 local training epochs per round. The batch size was set at 64, and we explored learning rates within the range [1e-2, 5e-3, 1e-3, 5e-4].
For FedAvg McMahan et al. [], clients perform standard stochastic gradient descent (SGD) updates as ClientOPT and upload their gradients to the central server. On the server side, the global model is updated using SGD as ServerOPT with the received gradients from the clients.
FedProx Sahu et al. [] introduces a proximal term during local client training as their ClientOPT, to address data heterogeneity issues by preventing significant drifts in client model updates. It employs a weighted average for aggregation and updates the global model using SGD, similar to ServerOPT in FedAvg. We set the proximal term coefficient parameter $\mu=1$ for the proximal term coefficient in FedProx.
Scaffold Karimireddy et al. [] employs client-variance reduction as its ClientOPT and corrects model drifts in local updates by introducing control variates. For ServerOPT, Scaffold incorporates the clients’ gradients and control variates for aggregation, and updates the global model using SGD.
FedDyn Acar et al. [] dynamically adjusts the local objectives by incorporating two regularizers as ClientOPT to ensure that the local optimum aligns asymptotically with the stationary points of the global objective. We set the parameter $\alpha=0.1$ as commonly used in the literature, and the local model is updated using SGD. Additionally, they introduce their server update as ServerOPT through the use of server state vectors. FedNova Wang et al. [] uses normalized stochastic gradients as ClientOPT to compensate for data imbalances across clients, and then introduces a weighted rescaling process as their ServerOPT. We run all experiments on a 3-GPU cluster of Tesla V100 GPUs, with each GPU having 32GB of memory.

## Appendix B Additional Experiments and Analyses

### B.1 Calibration on Centralized Learning

We present train-time calibration methods for traditional centralized training in Table 6. We train AlexNet using the SGD optimizer on the entire CIFAR-100 training dataset for 100 epochs with various train-time calibration methods. We found that most of these methods effectively calibrate in centralized settings, maintaining accuracy and exhibiting lower calibration errors.

**Table 6: Performance of centeralized training with various calibration methods.**
| Calibration<br>Method | Centralized Training |  |  |
| --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 76.53 | 4.338 | 2.597 |
| Focal Lin et al. [] | 75.70 | 4.079 | 2.240 |
| LS Müller et al. [] | 76.63 | 4.145 | 2.466 |
| BS Brier [] | 75.52 | 5.464 | 3.190 |
| MMCE Kumar et al. [] | 75.31 | 4.329 | 2.690 |
| FLSD Mukhoti et al. [] | 75.49 | 5.162 | 3.145 |
| MbLS Liu et al. [] | 76.57 | 4.011 | 2.428 |
| DCA Liang et al. [] | 77.24 | 3.950 | 2.167 |
| MDCA Hebbalaguppe et al. [] | 76.69 | 3.856 | 2.040 |

### B.2 Additional Results of Calibration on FL

This section explores the performance of each calibration method on different datasets and data distributions for various FL algorithms. To ensure consistency in our experiments, most of the configuration remains unchanged. Specifically, we set the number of clients ($M$) to 50 and the total number of federated rounds ($T$) to 100, with each client performing $E=5$ epochs of local training. Additionally, the hyperparameter search space and the details for each FL algorithm are consistent, as detailed in Appendix A.
Tables 7,  8,  9, and 10 display the performance and calibration metrics of various FL algorithms on the MNIST dataset across IID and non-IID conditions, with $\alpha$ values of 0.5, 0.1, and 0.05, respectively. Similarly, Tables 11,  12,  13, and  14 present the performance and calibration metrics of different FL algorithms on the CIFAR-10 dataset under IID and non-IID conditions, with corresponding $\alpha$ values of 0.5, 0.1, and 0.05, respectively.
Across these experimental results, which consider various data distribution setups and different datasets, our generic framework NUCFL consistently demonstrates superiority over the baseline by achieving lower calibration error. The performance improvements of NUCFL with different configurations, such as various similarity measurements and different auxiliary calibration losses, highlight the effectiveness and flexibility of our method.

**Table 7: Accuracy (%), calibration measures ECE (%), and SCE (%) of various federated optimization methods with different calibration methods under IID scenario on the MNIST dataset. Values in boldface indicate the best calibration across all methods, achieved without compromising accuracy compared to the uncalibrated model.**
| Calibration<br>Method | FedAvg McMahan et al. [] | FedProx Sahu et al. [] | Scaffold Karimireddy et al. [] | FedDyn Acar et al. [] | FedNova Wang et al. [] |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 97.19 | 0.64 | 0.26 | 97.50 | 0.92 | 0.30 | 97.59 | 0.69 | 0.23 | 99.04 | 0.93 | 0.39 | 97.92 | 0.74 | 0.27 |
| Focal Lin et al. [] | 97.70 | 3.84 | 0.83 | 97.35 | 4.70 | 0.99 | 97.45 | 2.76 | 0.62 | 99.01 | 3.48 | 0.58 | 97.76 | 3.92 | 0.82 |
| LS Müller et al. [] | 97.97 | 13.42 | 2.71 | 97.72 | 13.79 | 2.77 | 97.87 | 13.22 | 2.67 | 94.38 | 4.38 | 0.52 | 97.98 | 13.39 | 2.70 |
| BS Brier [] | 95.93 | 2.72 | 0.64 | 95.24 | 3.03 | 0.71 | 96.29 | 2.09 | 0.51 | 91.72 | 8.28 | 0.83 | 96.39 | 2.92 | 0.63 |
| MMCE Kumar et al. [] | 98.48 | 5.40 | 1.12 | 98.19 | 5.51 | 1.14 | 95.10 | 15.22 | 0.68 | 98.08 | 1.51 | 0.29 | 98.49 | 5.41 | 1.12 |
| FLSD Mukhoti et al. [] | 97.23 | 11.74 | 2.33 | 96.68 | 13.21 | 2.62 | 97.12 | 9.66 | 1.91 | 97.28 | 1.48 | 0.39 | 97.20 | 11.95 | 2.37 |
| MbLS Liu et al. [] | 98.01 | 2.93 | 0.66 | 97.75 | 3.39 | 0.74 | 97.91 | 2.65 | 0.61 | 98.32 | 2.39 | 0.49 | 98.07 | 2.97 | 0.67 |
| DCA Liang et al. [] | 98.13 | 0.31 | 0.19 | 97.83 | 0.36 | 0.22 | 97.71 | 0.51 | 0.21 | 98.93 | 0.57 | 0.23 | 98.11 | 0.26 | 0.19 |
| MDCA Hebbalaguppe et al. [] | 97.87 | 0.60 | 0.25 | 97.58 | 0.91 | 0.30 | 97.68 | 0.48 | 0.23 | 99.01 | 0.61 | 0.24 | 97.81 | 0.63 | 0.24 |
| NUCFL (DCA+COS) | 98.97 | 0.56 | 0.15 | 98.50 | 0.42 | 0.18 | 97.40 | 0.33 | 0.20 | 98.88 | 0.54 | 0.27 | 99.02 | 0.24 | 0.16 |
| NUCFL (MDCA+COS) | 98.94 | 0.54 | 0.15 | 98.55 | 0.27 | 0.15 | 96.97 | 0.35 | 0.21 | 98.83 | 0.53 | 0.21 | 98.99 | 0.49 | 0.15 |
| NUCFL (DCA+L-CKA) | 98.99 | 0.59 | 0.16 | 98.77 | 0.41 | 0.14 | 96.84 | 0.32 | 0.28 | 98.63 | 0.33 | 0.16 | 98.08 | 0.25 | 0.15 |
| NUCFL (MDCA+L-CKA) | 98.99 | 0.57 | 0.15 | 98.89 | 0.31 | 0.16 | 97.11 | 0.32 | 0.28 | 98.69 | 0.31 | 0.15 | 98.99 | 0.23 | 0.15 |
| NUCFL (DCA+RBF-CKA) | 99.01 | 0.60 | 0.15 | 98.91 | 0.42 | 0.16 | 96.93 | 0.33 | 0.27 | 98.75 | 0.38 | 0.19 | 98.96 | 0.29 | 0.16 |
| NUCFL (MDCA+RBF-CKA) | 99.09 | 0.60 | 0.16 | 98.77 | 0.37 | 0.15 | 97.54 | 0.31 | 0.23 | 98.86 | 0.32 | 0.15 | 98.28 | 0.26 | 0.16 |

**Table 8: Accuracy (%), calibration measures ECE (%), and SCE (%) of various federated optimization methods with different calibration methods under non-IID ($\alpha=0.5$) scenario on the MNIST dataset. Values in boldface indicate the best calibration across all methods, achieved without compromising accuracy compared to the uncalibrated model.**
| Calibration<br>Method | FedAvg McMahan et al. [] | FedProx Sahu et al. [] | Scaffold Karimireddy et al. [] | FedDyn Acar et al. [] | FedNova Wang et al. [] |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 96.52 | 1.59 | 0.60 | 96.05 | 1.74 | 0.60 | 95.65 | 6.24 | 0.96 | 98.37 | 1.48 | 0.38 | 96.55 | 1.24 | 0.49 |
| Focal Lin et al. [] | 95.88 | 5.96 | 1.44 | 94.94 | 6.67 | 1.58 | 95.01 | 5.15 | 1.36 | 98.28 | 1.71 | 0.26 | 96.06 | 5.88 | 1.33 |
| LS Müller et al. [] | 97.53 | 20.50 | 4.13 | 96.65 | 2.41 | 4.14 | 96.75 | 2.16 | 4.40 | 98.80 | 5.15 | 3.06 | 97.48 | 1.91 | 3.84 |
| BS Brier [] | 94.26 | 3.52 | 1.07 | 93.58 | 4.28 | 1.06 | 94.27 | 3.94 | 1.06 | 94.68 | 5.31 | 0.53 | 93.96 | 3.15 | 0.88 |
| MMCE Kumar et al. [] | 98.43 | 7.32 | 1.51 | 94.74 | 9.06 | 1.85 | 86.48 | 15.38 | 4.52 | 97.48 | 1.58 | 0.49 | 98.30 | 7.25 | 1.47 |
| FLSD Mukhoti et al. [] | 93.95 | 14.96 | 3.38 | 93.29 | 16.31 | 3.52 | 93.41 | 13.88 | 3.13 | 95.84 | 1.99 | 0.39 | 94.93 | 15.07 | 3.15 |
| MbLS Liu et al. [] | 97.22 | 5.49 | 1.23 | 96.46 | 6.34 | 1.40 | 96.55 | 5.70 | 1.31 | 98.71 | 2.45 | 0.55 | 97.24 | 5.17 | 1.12 |
| DCA Liang et al. [] | 96.94 | 0.64 | 0.44 | 96.04 | 0.77 | 0.51 | 96.42 | 0.45 | 0.60 | 98.47 | 1.14 | 0.22 | 97.13 | 0.64 | 0.38 |
| MDCA Hebbalaguppe et al. [] | 96.92 | 1.48 | 0.56 | 96.02 | 1.63 | 0.63 | 95.82 | 0.96 | 0.61 | 98.93 | 1.13 | 0.21 | 96.98 | 1.09 | 0.43 |
| NUCFL (DCA+COS) | 98.86 | 0.51 | 0.19 | 98.16 | 0.46 | 0.32 | 96.70 | 0.44 | 0.58 | 98.43 | 0.94 | 0.21 | 97.92 | 0.48 | 0.17 |
| NUCFL (MDCA+COS) | 98.80 | 0.52 | 0.17 | 98.18 | 0.35 | 0.25 | 96.17 | 0.41 | 0.42 | 98.14 | 1.03 | 0.21 | 97.83 | 0.52 | 0.18 |
| NUCFL (DCA+L-CKA) | 98.97 | 0.48 | 0.16 | 98.58 | 0.39 | 0.19 | 96.43 | 0.48 | 0.63 | 98.57 | 0.53 | 0.13 | 97.79 | 0.48 | 0.16 |
| NUCFL (MDCA+L-CKA) | 98.95 | 0.52 | 0.16 | 98.61 | 0.41 | 0.18 | 95.99 | 0.44 | 0.61 | 98.60 | 0.46 | 0.11 | 97.50 | 0.55 | 0.16 |
| NUCFL (DCA+RBF-CKA) | 99.07 | 0.44 | 0.16 | 98.67 | 0.32 | 0.18 | 96.85 | 0.45 | 0.59 | 98.53 | 0.51 | 0.12 | 97.94 | 0.57 | 0.17 |
| NUCFL (MDCA+RBF-CKA) | 98.94 | 0.55 | 0.17 | 98.58 | 0.37 | 0.20 | 96.78 | 0.43 | 0.63 | 98.16 | 0.48 | 0.12 | 97.89 | 0.53 | 0.16 |

**Table 9: Accuracy (%), calibration measures ECE (%), and SCE (%) of various federated optimization methods with different calibration methods under non-IID ($\alpha=0.1$) scenario on the MNIST dataset. Values in boldface indicate the best calibration across all methods, achieved without compromising accuracy compared to the uncalibrated model.**
| Calibration<br>Method | FedAvg McMahan et al. [] | FedProx Sahu et al. [] | Scaffold Karimireddy et al. [] | FedDyn Acar et al. [] | FedNova Wang et al. [] |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 92.18 | 1.96 | 1.04 | 91.76 | 3.21 | 0.96 | 91.50 | 5.03 | 1.32 | 97.74 | 3.28 | 0.50 | 93.80 | 1.95 | 0.69 |
| Focal Lin et al. [] | 90.75 | 9.02 | 2.28 | 90.76 | 10.93 | 2.35 | 88.74 | 16.49 | 5.38 | 97.53 | 2.38 | 0.47 | 93.50 | 8.32 | 1.82 |
| LS Müller et al. [] | 95.01 | 42.28 | 8.53 | 92.76 | 34.48 | 6.92 | 92.34 | 40.26 | 8.14 | 98.46 | 25.86 | 5.20 | 94.55 | 28.00 | 5.76 |
| BS Brier [] | 87.59 | 5.06 | 1.89 | 89.00 | 8.58 | 1.88 | 89.56 | 14.45 | 3.03 | 96.76 | 3.20 | 0.53 | 89.09 | 3.23 | 1.69 |
| MMCE Kumar et al. [] | 95.72 | 9.20 | 1.86 | 91.48 | 13.28 | 2.67 | 89.37 | 5.49 | 3.49 | 96.86 | 4.95 | 0.54 | 97.38 | 7.89 | 1.63 |
| FLSD Mukhoti et al. [] | 87.07 | 19.26 | 4.59 | 89.15 | 24.00 | 4.90 | 89.87 | 26.66 | 5.45 | 97.87 | 1.67 | 0.37 | 91.63 | 19.97 | 4.01 |
| MbLS Liu et al. [] | 93.48 | 14.22 | 3.08 | 91.97 | 12.60 | 2.76 | 89.32 | 8.29 | 4.58 | 98.02 | 6.16 | 1.32 | 94.34 | 9.21 | 2.12 |
| DCA Liang et al. [] | 92.89 | 1.06 | 0.81 | 92.45 | 1.62 | 0.70 | 92.34 | 3.49 | 1.12 | 97.75 | 2.18 | 0.41 | 94.34 | 0.87 | 0.49 |
| MDCA Hebbalaguppe et al. [] | 92.44 | 1.93 | 0.98 | 91.81 | 2.78 | 0.91 | 92.38 | 3.34 | 1.13 | 97.58 | 2.03 | 0.39 | 94.24 | 1.81 | 0.73 |
| NUCFL (DCA+COS) | 94.50 | 0.45 | 0.20 | 93.47 | 0.71 | 0.35 | 93.35 | 3.43 | 1.25 | 97.68 | 1.17 | 0.37 | 95.35 | 0.61 | 0.22 |
| NUCFL (MDCA+COS) | 95.39 | 0.42 | 0.21 | 93.78 | 1.45 | 0.48 | 93.44 | 3.29 | 1.03 | 97.87 | 1.25 | 0.33 | 95.83 | 0.45 | 0.20 |
| NUCFL (DCA+L-CKA) | 94.91 | 0.32 | 0.18 | 93.32 | 0.53 | 0.31 | 92.98 | 2.84 | 0.98 | 97.80 | 1.11 | 0.36 | 95.80 | 0.40 | 0.14 |
| NUCFL (MDCA+L-CKA) | 95.36 | 0.39 | 0.21 | 93.09 | 0.41 | 0.27 | 92.67 | 3.09 | 1.04 | 97.83 | 0.97 | 0.27 | 94.86 | 0.59 | 0.21 |
| NUCFL (DCA+RBF-CKA) | 95.61 | 0.50 | 0.19 | 93.46 | 0.31 | 0.25 | 92.68 | 3.26 | 1.16 | 97.19 | 1.22 | 0.32 | 94.87 | 0.52 | 0.19 |
| NUCFL (MDCA+RBF-CKA) | 95.85 | 0.48 | 0.19 | 93.69 | 0.57 | 0.23 | 92.18 | 3.04 | 0.91 | 97.32 | 1.31 | 0.43 | 94.18 | 0.53 | 0.18 |

**Table 10: Accuracy (%), calibration measures ECE (%), and SCE (%) of various federated optimization methods with different calibration methods under non-IID ($\alpha=0.05$) scenario on the MNIST dataset. Values in boldface indicate the best calibration across all methods, achieved without compromising accuracy compared to the uncalibrated model.**
| Calibration<br>Method | FedAvg McMahan et al. [] | FedProx Sahu et al. [] | Scaffold Karimireddy et al. [] | FedDyn Acar et al. [] | FedNova Wang et al. [] |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 90.51 | 5.22 | 1.33 | 85.02 | 5.60 | 2.26 | 84.20 | 13.96 | 3.85 | 89.51 | 4.52 | 1.57 | 88.11 | 1.68 | 1.49 |
| Focal Lin et al. [] | 88.95 | 15.43 | 3.23 | 83.22 | 14.26 | 3.83 | 81.50 | 24.37 | 5.74 | 88.49 | 3.24 | 1.46 | 88.23 | 9.74 | 2.65 |
| LS Müller et al. [] | 88.48 | 50.62 | 10.57 | 77.52 | 34.49 | 9.32 | 75.89 | 41.50 | 10.87 | 84.95 | 6.20 | 4.66 | 80.84 | 26.31 | 7.87 |
| BS Brier [] | 84.36 | 12.65 | 2.90 | 74.96 | 9.44 | 4.14 | 69.21 | 18.55 | 7.03 | 83.05 | 5.85 | 3.87 | 78.29 | 2.79 | 3.34 |
| MMCE Kumar et al. [] | 93.99 | 13.18 | 2.74 | 83.32 | 16.74 | 3.58 | 77.48 | 29.33 | 6.48 | 88.49 | 3.48 | 1.16 | 89.48 | 9.90 | 2.34 |
| FLSD Mukhoti et al. [] | 87.42 | 29.42 | 5.78 | 81.52 | 28.11 | 6.46 | 77.73 | 37.03 | 8.76 | 86.86 | 5.87 | 1.81 | 87.00 | 21.39 | 4.76 |
| MbLS Liu et al. [] | 89.96 | 21.51 | 4.71 | 79.66 | 13.59 | 4.80 | 80.73 | 22.71 | 6.25 | 89.40 | 5.05 | 2.03 | 82.47 | 6.62 | 4.27 |
| DCA Liang et al. [] | 91.48 | 3.83 | 1.01 | 85.30 | 2.68 | 1.91 | 84.39 | 9.75 | 3.18 | 89.52 | 3.95 | 1.07 | 89.57 | 0.99 | 1.35 |
| MDCA Hebbalaguppe et al. [] | 90.36 | 5.41 | 1.38 | 84.13 | 4.55 | 2.24 | 83.52 | 12.51 | 3.65 | 90.30 | 4.06 | 1.21 | 89.60 | 1.92 | 1.34 |
| NUCFL (DCA+COS) | 92.16 | 1.38 | 0.83 | 89.07 | 1.03 | 0.91 | 87.40 | 2.28 | 1.99 | 89.62 | 2.48 | 1.04 | 90.73 | 0.66 | 0.27 |
| NUCFL (MDCA+COS) | 92.20 | 1.30 | 0.74 | 89.29 | 1.89 | 1.10 | 86.94 | 3.37 | 1.80 | 90.90 | 2.72 | 0.79 | 90.76 | 1.25 | 0.30 |
| NUCFL (DCA+L-CKA) | 91.40 | 1.20 | 0.87 | 90.86 | 0.90 | 0.76 | 84.93 | 3.65 | 1.33 | 90.61 | 2.39 | 0.90 | 90.80 | 0.89 | 0.26 |
| NUCFL (MDCA+L-CKA) | 92.33 | 1.63 | 0.80 | 90.51 | 0.86 | 0.67 | 85.03 | 3.19 | 1.67 | 90.89 | 2.15 | 0.70 | 90.47 | 0.60 | 0.19 |
| NUCFL (DCA+RBF-CKA) | 91.06 | 1.38 | 0.78 | 90.31 | 0.59 | 0.58 | 84.93 | 3.14 | 1.76 | 90.13 | 2.49 | 0.84 | 89.35 | 0.71 | 0.23 |
| NUCFL (MDCA+RBF-CKA) | 92.15 | 1.77 | 0.81 | 90.00 | 0.71 | 0.65 | 85.71 | 3.79 | 2.11 | 90.76 | 2.18 | 0.73 | 90.73 | 0.55 | 0.19 |

**Table 11: Accuracy (%), calibration measures ECE (%), and SCE (%) of various federated optimization methods with different calibration methods under IID scenario on the CIFAR-10 dataset. Values in boldface indicate the best calibration across all methods, achieved without compromising accuracy compared to the uncalibrated model.**
| Calibration<br>Method | FedAvg McMahan et al. [] | FedProx Sahu et al. [] | Scaffold Karimireddy et al. [] | FedDyn Acar et al. [] | FedNova Wang et al. [] |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 79.23 | 9.60 | 2.13 | 78.52 | 7.41 | 2.52 | 78.27 | 15.91 | 5.45 | 77.65 | 10.11 | 8.17 | 79.06 | 9.88 | 2.82 |
| Focal Lin et al. [] | 78.46 | 4.79 | 1.51 | 78.37 | 11.74 | 2.96 | 78.78 | 12.36 | 5.69 | 73.91 | 6.12 | 5.49 | 78.37 | 11.61 | 4.40 |
| LS Müller et al. [] | 78.78 | 4.30 | 1.73 | 78.08 | 13.38 | 3.83 | 78.25 | 14.49 | 5.19 | 76.49 | 8.20 | 10.07 | 78.63 | 10.37 | 4.00 |
| BS Brier [] | 74.90 | 5.27 | 1.50 | 72.97 | 8.82 | 3.03 | 76.54 | 9.32 | 6.06 | 77.41 | 15.87 | 9.85 | 76.06 | 5.04 | 2.01 |
| MMCE Kumar et al. [] | 78.49 | 4.82 | 1.23 | 78.54 | 7.69 | 2.33 | 78.26 | 14.61 | 5.21 | 75.03 | 8.60 | 6.30 | 78.22 | 8.02 | 3.01 |
| FLSD Mukhoti et al. [] | 76.76 | 8.52 | 2.04 | 78.01 | 15.46 | 3.73 | 77.26 | 14.43 | 5.36 | 76.09 | 8.60 | 6.30 | 76.50 | 8.54 | 2.03 |
| MbLS Liu et al. [] | 78.73 | 9.17 | 2.00 | 78.30 | 9.01 | 2.75 | 78.57 | 16.01 | 5.06 | 76.65 | 8.09 | 8.28 | 79.47 | 8.63 | 1.94 |
| DCA Liang et al. [] | 79.33 | 5.37 | 1.71 | 78.51 | 6.35 | 2.16 | 78.23 | 12.02 | 5.09 | 77.40 | 7.81 | 5.46 | 79.39 | 8.56 | 2.19 |
| MDCA Hebbalaguppe et al. [] | 79.40 | 7.04 | 2.03 | 78.52 | 8.84 | 3.67 | 77.41 | 12.33 | 6.04 | 77.12 | 8.16 | 7.18 | 79.42 | 7.73 | 2.00 |
| NUCFL (DCA+COS) | 79.29 | 5.09 | 1.35 | 78.51 | 5.26 | 2.09 | 78.22 | 10.19 | 4.95 | 77.54 | 6.00 | 4.60 | 79.84 | 6.76 | 1.90 |
| NUCFL (MDCA+COS) | 79.60 | 6.79 | 1.99 | 78.51 | 5.67 | 2.15 | 78.82 | 10.97 | 5.02 | 77.69 | 6.10 | 4.74 | 79.15 | 6.91 | 2.02 |
| NUCFL (DCA+L-CKA) | 79.50 | 5.85 | 2.08 | 79.61 | 5.34 | 1.93 | 78.35 | 11.35 | 5.08 | 77.18 | 6.03 | 4.07 | 79.57 | 6.69 | 1.99 |
| NUCFL (MDCA+L-CKA) | 79.27 | 5.48 | 2.08 | 79.61 | 5.69 | 2.06 | 78.32 | 11.25 | 5.08 | 76.81 | 5.76 | 4.29 | 79.26 | 7.03 | 2.03 |
| NUCFL (DCA+RBF-CKA) | 79.42 | 5.59 | 2.01 | 79.61 | 5.44 | 1.83 | 78.59 | 9.05 | 4.73 | 76.66 | 5.91 | 4.19 | 79.49 | 6.70 | 2.03 |
| NUCFL (MDCA+RBF-CKA) | 79.87 | 5.68 | 2.04 | 79.60 | 5.57 | 2.03 | 78.59 | 10.06 | 4.97 | 76.88 | 5.44 | 4.69 | 79.46 | 7.15 | 2.02 |

**Table 12: Accuracy (%), calibration measures ECE (%), and SCE (%) of various federated optimization methods with different calibration methods under non-IID ($\alpha=0.5$) scenario on the CIFAR-10 dataset. Values in boldface indicate the best calibration across all methods, achieved without compromising accuracy compared to the uncalibrated model.**
| Calibration<br>Method | FedAvg McMahan et al. [] | FedProx Sahu et al. [] | Scaffold Karimireddy et al. [] | FedDyn Acar et al. [] | FedNova Wang et al. [] |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 75.33 | 9.69 | 2.31 | 74.87 | 5.36 | 2.84 | 75.19 | 7.99 | 2.98 | 74.00 | 18.84 | 5.88 | 75.37 | 10.67 | 2.34 |
| Focal Lin et al. [] | 73.60 | 5.04 | 1.63 | 74.92 | 9.71 | 3.16 | 74.04 | 9.44 | 3.04 | 71.26 | 16.35 | 5.27 | 74.07 | 5.40 | 3.70 |
| LS Müller et al. [] | 75.00 | 3.87 | 2.04 | 74.84 | 11.26 | 4.21 | 75.90 | 8.38 | 2.47 | 74.63 | 15.35 | 4.37 | 75.50 | 5.80 | 2.68 |
| BS Brier [] | 71.22 | 5.40 | 1.76 | 73.89 | 9.27 | 2.71 | 73.03 | 6.03 | 2.20 | 73.22 | 16.23 | 5.33 | 71.70 | 6.34 | 2.41 |
| MMCE Kumar et al. [] | 74.65 | 9.08 | 3.72 | 75.07 | 9.29 | 2.81 | 75.16 | 6.37 | 2.95 | 71.34 | 14.53 | 5.17 | 74.16 | 6.82 | 3.65 |
| FLSD Mukhoti et al. [] | 70.74 | 8.81 | 2.62 | 74.52 | 13.24 | 4.24 | 73.50 | 6.37 | 2.63 | 73.39 | 18.96 | 6.78 | 70.75 | 8.21 | 2.37 |
| MbLS Liu et al. [] | 75.57 | 8.70 | 2.10 | 74.96 | 6.31 | 2.63 | 75.68 | 8.67 | 3.96 | 74.16 | 16.62 | 5.90 | 75.59 | 9.03 | 2.63 |
| DCA Liang et al. [] | 75.34 | 5.45 | 1.67 | 74.99 | 4.14 | 2.42 | 75.12 | 6.23 | 2.55 | 73.88 | 10.00 | 4.86 | 74.43 | 6.22 | 2.32 |
| MDCA Hebbalaguppe et al. [] | 75.42 | 6.44 | 2.25 | 74.92 | 6.86 | 2.68 | 74.16 | 6.20 | 3.07 | 75.70 | 9.27 | 4.57 | 75.05 | 6.49 | 2.24 |
| NUCFL (DCA+COS) | 75.39 | 5.10 | 1.42 | 74.93 | 3.51 | 2.13 | 75.13 | 5.54 | 1.76 | 74.98 | 6.17 | 4.14 | 75.91 | 5.74 | 2.22 |
| NUCFL (MDCA+COS) | 75.47 | 6.03 | 2.09 | 74.97 | 3.47 | 2.08 | 75.36 | 5.60 | 1.71 | 73.10 | 5.03 | 4.06 | 75.33 | 5.54 | 2.03 |
| NUCFL (DCA+L-CKA) | 76.96 | 4.76 | 1.22 | 75.76 | 3.66 | 2.15 | 75.34 | 5.23 | 1.41 | 74.68 | 5.42 | 4.24 | 75.29 | 4.99 | 1.70 |
| NUCFL (MDCA+L-CKA) | 76.34 | 4.76 | 1.22 | 75.74 | 3.27 | 1.91 | 75.57 | 4.17 | 1.30 | 74.67 | 5.27 | 4.13 | 76.85 | 6.01 | 2.24 |
| NUCFL (DCA+RBF-CKA) | 76.74 | 4.43 | 1.13 | 75.70 | 3.70 | 2.00 | 76.11 | 4.91 | 1.76 | 74.75 | 5.38 | 4.06 | 76.74 | 5.60 | 1.91 |
| NUCFL (MDCA+RBF-CKA) | 76.88 | 4.95 | 1.25 | 75.73 | 3.81 | 2.38 | 76.82 | 4.94 | 1.57 | 74.41 | 5.28 | 4.14 | 76.27 | 6.29 | 2.26 |

**Table 13: Accuracy (%), calibration measures ECE (%), and SCE (%) of various federated optimization methods with different calibration methods under non-IID ($\alpha=0.1$) scenario on the CIFAR-10 dataset. Values in boldface indicate the best calibration across all methods, achieved without compromising accuracy compared to the uncalibrated model.**
| Calibration<br>Method | FedAvg McMahan et al. [] | FedProx Sahu et al. [] | Scaffold Karimireddy et al. [] | FedDyn Acar et al. [] | FedNova Wang et al. [] |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 54.46 | 8.62 | 4.42 | 48.77 | 10.99 | 4.34 | 51.34 | 7.95 | 6.04 | 54.00 | 8.35 | 5.88 | 52.57 | 13.46 | 6.22 |
| Focal Lin et al. [] | 50.18 | 3.21 | 4.91 | 48.25 | 11.18 | 4.28 | 49.48 | 6.42 | 5.90 | 51.26 | 5.03 | 5.27 | 52.71 | 16.29 | 11.48 |
| LS Müller et al. [] | 52.13 | 2.92 | 4.75 | 47.23 | 6.99 | 5.28 | 51.94 | 7.27 | 6.79 | 54.12 | 4.09 | 3.68 | 52.22 | 12.66 | 6.02 |
| BS Brier [] | 47.18 | 5.21 | 5.02 | 42.16 | 5.21 | 3.36 | 48.09 | 7.42 | 5.69 | 53.62 | 9.23 | 6.33 | 50.25 | 16.54 | 6.40 |
| MMCE Kumar et al. [] | 52.62 | 8.25 | 3.95 | 48.76 | 6.91 | 4.93 | 51.90 | 8.02 | 6.02 | 53.99 | 8.53 | 6.17 | 51.50 | 14.95 | 9.13 |
| FLSD Mukhoti et al. [] | 48.31 | 5.21 | 5.65 | 48.21 | 9.25 | 4.45 | 49.03 | 8.53 | 7.43 | 54.69 | 10.53 | 6.70 | 50.58 | 14.67 | 9.42 |
| MbLS Liu et al. [] | 53.06 | 12.77 | 6.98 | 46.92 | 5.03 | 4.30 | 49.95 | 8.91 | 7.41 | 54.16 | 7.37 | 5.90 | 52.64 | 11.69 | 5.88 |
| DCA Liang et al. [] | 54.10 | 4.32 | 3.01 | 48.23 | 5.47 | 4.43 | 51.99 | 6.90 | 5.76 | 53.88 | 7.00 | 4.14 | 52.09 | 11.11 | 5.54 |
| MDCA Hebbalaguppe et al. [] | 55.02 | 3.93 | 3.19 | 47.48 | 6.20 | 4.29 | 49.55 | 8.77 | 7.12 | 55.70 | 7.56 | 5.37 | 52.88 | 11.51 | 5.83 |
| NUCFL (DCA+COS) | 54.60 | 3.94 | 3.08 | 48.24 | 4.74 | 3.15 | 51.47 | 5.05 | 5.05 | 56.98 | 5.17 | 3.74 | 52.54 | 8.13 | 5.50 |
| NUCFL (MDCA+COS) | 55.25 | 3.74 | 2.89 | 48.54 | 4.37 | 3.13 | 51.62 | 5.38 | 5.16 | 56.03 | 5.03 | 4.06 | 52.68 | 9.54 | 5.11 |
| NUCFL (DCA+L-CKA) | 55.77 | 3.43 | 2.42 | 48.03 | 4.42 | 3.35 | 51.99 | 5.27 | 4.93 | 56.88 | 5.14 | 4.04 | 52.41 | 8.15 | 4.97 |
| NUCFL (MDCA+L-CKA) | 54.97 | 3.27 | 2.43 | 48.31 | 5.01 | 3.71 | 51.94 | 4.63 | 4.20 | 56.75 | 5.38 | 3.81 | 52.50 | 7.12 | 4.84 |
| NUCFL (DCA+RBF-CKA) | 55.60 | 3.39 | 2.25 | 48.88 | 4.70 | 3.71 | 52.17 | 5.14 | 5.09 | 56.77 | 5.27 | 3.90 | 52.20 | 7.19 | 4.18 |
| NUCFL (MDCA+RBF-CKA) | 55.97 | 3.46 | 2.21 | 48.32 | 4.84 | 3.74 | 52.04 | 5.06 | 4.66 | 56.88 | 5.44 | 3.99 | 52.45 | 7.36 | 5.02 |

**Table 14: Accuracy (%), calibration measures ECE (%), and SCE (%) of various federated optimization methods with different calibration methods under non-IID ($\alpha=0.05$) scenario on the CIFAR-10 dataset. Values in boldface indicate the best calibration across all methods, achieved without compromising accuracy compared to the uncalibrated model.**
| Calibration<br>Method | FedAvg McMahan et al. [] | FedProx Sahu et al. [] | Scaffold Karimireddy et al. [] | FedDyn Acar et al. [] | FedNova Wang et al. [] |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ | Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |
| Uncal. | 48.84 | 11.30 | 4.86 | 42.50 | 13.03 | 4.91 | 44.99 | 9.61 | 11.80 | 45.00 | 8.35 | 5.88 | 52.57 | 13.45 | 6.22 |
| Focal Lin et al. [] | 39.86 | 5.10 | 4.63 | 41.91 | 9.68 | 4.84 | 42.23 | 7.79 | 9.34 | 41.26 | 16.35 | 13.66 | 52.71 | 16.29 | 11.48 |
| LS Müller et al. [] | 43.13 | 7.08 | 5.82 | 41.91 | 6.72 | 5.38 | 43.30 | 9.03 | 10.88 | 45.12 | 7.35 | 4.37 | 52.22 | 12.68 | 6.02 |
| BS Brier [] | 31.80 | 14.48 | 6.67 | 41.75 | 6.98 | 6.96 | 41.20 | 7.42 | 9.48 | 44.62 | 9.23 | 6.33 | 50.25 | 16.54 | 6.40 |
| MMCE Kumar et al. [] | 43.72 | 7.59 | 3.91 | 42.96 | 8.52 | 3.93 | 42.50 | 7.69 | 9.72 | 43.39 | 9.78 | 8.69 | 51.50 | 14.95 | 9.13 |
| FLSD Mukhoti et al. [] | 34.83 | 5.22 | 6.42 | 42.22 | 6.01 | 5.39 | 42.65 | 9.15 | 10.69 | 44.63 | 9.60 | 6.80 | 50.58 | 14.67 | 9.42 |
| MbLS Liu et al. [] | 44.73 | 15.81 | 6.76 | 42.09 | 8.28 | 4.63 | 44.27 | 9.51 | 11.48 | 46.13 | 7.37 | 5.90 | 52.64 | 11.69 | 5.88 |
| DCA Liang et al. [] | 48.94 | 10.07 | 3.91 | 42.28 | 6.57 | 3.40 | 44.78 | 7.07 | 7.82 | 46.95 | 7.60 | 5.56 | 52.07 | 11.11 | 5.54 |
| MDCA Hebbalaguppe et al. [] | 48.95 | 9.30 | 4.80 | 42.33 | 8.18 | 4.25 | 45.20 | 15.26 | 14.90 | 46.99 | 7.92 | 5.39 | 52.80 | 11.51 | 5.83 |
| NUCFL (DCA+COS) | 48.46 | 8.07 | 3.87 | 42.32 | 5.84 | 3.14 | 44.76 | 6.53 | 7.03 | 49.86 | 6.17 | 4.01 | 52.54 | 8.13 | 5.50 |
| NUCFL (MDCA+COS) | 48.67 | 8.53 | 4.17 | 43.20 | 5.01 | 2.75 | 44.37 | 6.52 | 7.14 | 50.31 | 6.03 | 4.06 | 52.68 | 9.54 | 5.11 |
| NUCFL (DCA+L-CKA) | 49.00 | 7.62 | 3.16 | 42.87 | 5.03 | 2.77 | 45.29 | 6.06 | 6.94 | 50.68 | 6.15 | 4.18 | 52.41 | 8.15 | 4.97 |
| NUCFL (MDCA+L-CKA) | 50.49 | 7.50 | 2.95 | 43.01 | 3.24 | 2.14 | 45.38 | 5.15 | 5.17 | 50.75 | 5.81 | 3.98 | 52.50 | 7.12 | 4.84 |
| NUCFL (DCA+RBF-CKA) | 49.50 | 7.76 | 3.09 | 42.90 | 4.90 | 2.61 | 45.99 | 6.14 | 6.48 | 49.67 | 6.27 | 4.31 | 52.19 | 7.19 | 4.18 |
| NUCFL (MDCA+RBF-CKA) | 49.32 | 7.72 | 3.07 | 43.08 | 3.44 | 2.13 | 45.69 | 5.39 | 5.59 | 49.89 | 6.18 | 4.28 | 52.45 | 7.36 | 5.02 |

### B.3 Mitigating Over/Under-Confidence

Figure: Figure 5: Reliability diagrams for non-IID FedAvg ($\alpha=0.5$) with different calibration methods using the MNIST dataset. The lower ECE and smaller gap show the effectiveness of our method.
Refer to caption: x5.png

In addition to the over-confidence examples discussed in Section 5.2, we also observe cases of under-confidence in FL algorithms. Figure 5 presents the reliability histograms for non-IID ($\alpha=0.5$) FedAvg using the MNIST dataset, where the results show a gap that exceeds the diagonal line, indicating under-confidence issues. While over-confidence is often viewed as more problematic in deep neural networks for real-world applications, under-confidence can also pose challenges in specific scenarios. For example, in a medical diagnosis scenario, a threshold of 0.95 for negative predictions means that any instance classified as negative with a probability $\leq$ 0.95 requires manual review. An underconfident model may classify an instance as negative with a probability of 0.51, suggesting a true probability of correctness that is at least 0.51, yet this does not clarify whether it surpasses the threshold. This leads to compulsory manual inspection regardless of the actual correctness, thus increasing the workload. Among all the figures, we see that our proposed method effectively calibrates FedAvg and exhibits a smaller gap (red region), demonstrating improved performance over other approaches.

### B.4 Comparison with FedCal

This section presents additional experimental results and comparisons with FedCal Peng et al. [] from Section 5.2.
Table 15 provides complete comparisons across different datasets.
The results in Table 15 indicate that our proposed NUCFL outperforms FedCal.
FedCal improves local client calibration using a local scaler design and aggregates these scalers to create a global scaler for the global model.
However, relying solely on an aggregated scaler from local clients can lower global ECE but may neglect crucial interactions between global and local calibration needs.
Our method accounts for these interactions, mitigating biases toward local heterogeneity and more effectively meeting global calibration requirements.

**Table 15: Comparison with SOTA baseline on different datasets.**
| Dataset | Calibration Method | Non-IID FedAvg ($\alpha=0.5$) |  |  |
| --- | --- | --- | --- | --- |
| Acc $\uparrow$ | ECE $\downarrow$ | SCE $\downarrow$ |  |  |
| MNIST | UnCal. | 96.52 | 1.59 | 0.60 |
| FedCal | 96.52 | 0.94 | 0.41 |  |
| NUCFL(DCA+L-CKA) | 98.97 | 0.48 | 0.16 |  |
| FEMNIST | UnCal. | 91.98 | 4.38 | 1.83 |
| FedCal | 91.98 | 3.92 | 1.56 |  |
| NUCFL(DCA+L-CKA) | 91.89 | 3.44 | 1.39 |  |
| CIFAR-10 | UnCal. | 75.33 | 9.69 | 2.31 |
| FedCal | 75.33 | 7.11 | 1.90 |  |
| NUCFL(DCA+L-CKA) | 76.96 | 4.76 | 1.22 |  |
| CIFAT-100 | UnCal. | 72.77 | 9.80 | 3.21 |
| FedCal | 72.77 | 5.97 | 3.03 |  |
| NUCFL(DCA+L-CKA) | 73.73 | 5.14 | 2.87 |  |

## Peer Reviews

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: - Could the authors provide a comparison between NUCFL and Zhang et al. [2022] and Luo et al. [2021], according to the introduced metrics?
- Does NUCFL increase computational costs on the client-side?
- How could NUCFL be adapted to unsupervised/semi-supervised settings?
- How sensible if NUCFL to the hyperparam $\beta$?
soundness: 3
strengths: - The paper underlines a gap in the current FL research and proposes an effective method to address it
-  NUCFL can be easily added on top of existing methods
- The paper extensively shows the efficacy of NUCFL across various settings, model architectures and methods
- NUCFL is extensively studied through in-depth and well-carried analyses
- NUCFL does not increase communication costs
summary: The paper addresses the issue of model calibration in federated learning (FL), with a specific focus on heterogeneous scenarios, characterized by distribution shifts across clients. The paper argues that most works have focused on improving the accuracy of the proposed methods so far, but almost no attention has been given to the model's confidence in its predictions. This highlights a gap in the current literature: FL models should hava reliable confidence in their predictions to be deployed in real-world use cases. To fill this gap, this work introduces Non-Uniform Calibration for Federated Learning (NUCFL), a novel method to integrate FL with model calibration, that can be easily combined with existing approaches.

The work discusses the various benefits of NUCFL, studying its application with several calibration approaches and FL methods, showing its effectiveness in improving the model's confidence in its predictions, according to multiple metrics.
weaknesses: - Difficulties in proving NUCFL's convergence and theoretical properties, as also pointed out by the authors
- The paper argues that works like Zhang et al. [2022] and Luo et al. [2021] misuse the term "calibration" and do not further analyze the comparison with those methods. However, I believe it would still be relevant to the FL community to understand how NUCFL compares to them. 
- The application of NUCFL seems limited to supervised scenarios

Comments (did not affect my rating):
- I believe the paper would benefit from the outlined Algorithm of NUCFL for better understanding
- Please modify the citation style to improve the paper's readibility (the citation are not enclosed in brackets)
- Line 136: typo ".."

---

confidence: 5
contribution: 2
initial_rating: 3
presentation: 2
questions: Please see the weakness.
soundness: 2
strengths: The motivation makes sense.

The paper is well-written and easy to follow.
summary: This paper presents a model calibration method for federated learning, which adaptively adjusts penalties based on the relationship between local and global models.
weaknesses: The contribution of this paper is limited. The proposed method seems a simple incremental work of the existing FL model calibration method.

Although I understand the author's motivation, I need the author to explain why the proposed metric is more important than accuracy. The proposed metrics are not intuitive enough. Typically, accuracy and the proposed metric are positively correlated. The author should add some detailed experiments to validate its motivation.

The experimental results show that The proposed method is only slightly optimized for federated learning. In addition, the proposed method may cause a decrease in accuracy in some scenarios.

Lake of comparison with SOTA FL optimization methods [1-5]. These methods achieve better model performance by guiding the model to optimize towards the flat region. I think these methods can still reduce the measures proposed by the author. I wonder if the author can compare these methods or integrate the proposed method into them.

[1] FedSpeed: Larger local interval, less communication round, and higher generalization accuracy ICLR 2023

[2] Is Aggregation the Only Choice? Federated Learning via Layer-wise Model Recombination, KDD 2024

[3] Generalized Federated Learning via Sharpness Aware Minimization, ICML 2022

[4] Dynamic regularized sharpness aware minimization in federated learning: Approaching global consistency and smooth landscape. ICML 2023

[5] FedCross: Towards Accurate Federated Learning via Multi-Model Cross-Aggregation, ICDE 2024

---

confidence: 2
contribution: 2
initial_rating: 5
presentation: 2
questions: N/A
soundness: 3
strengths: [1] The paper takes an important step to explore FL from the perspective of model calibration, which is important to various decision-making scenarios. 

[2] Extensive experiments are carried out to show that NUCFL can seamlessly integrate with existing FL methods and improve the original performance of those methods.
summary: This paper aims to ensure the reliability of FL in real-world applications. It proposes Non-Uniform Calibration for Federated Learning (NUCFL) that integrates FL with model calibration to achieve the goal. NUCFL dynamically adjusts the model calibration objectives by measuring the relationships between the local and the global model. Experiments show the superiority of NUCFL compared with existing FL methods.
weaknesses: [1] It is unclear whether there is a trade-off between accuracy and reliability. If so, please give a detailed illustration of this aspect. If not, the advantage of not sacrificing accuracy seems an irrelevant contribution. 

[2] The relation between model calibration and enhancing the confidence of FL outputs is weak. It requires further demonstration and analysis of why model calibration can realize that. 

[3] It is better to provide a further investigation into how existing centralized calibration methods can adapt to the FL setting, for example, simply combining them with FedAvg framework. Empirical studies can also involve this part.

---

confidence: 4
contribution: 2
initial_rating: 8
presentation: 3
questions: 1. Can the authors provide any insight on how would the proposed approach perform under different local training conditions? For example with minimal local training (≤ 1 epoch or with very low learning rates) or extensive local training (e.g., training the local model to convergence or near convergence every round).
2. Have the authors considered the use of auxiliary local models or personalized layers, to compute similarity? 
3. Why did the authors decide to use raw similarity/relevance values rather than a softmaxed version with a normalization factor constructed based on the entire distribution of client similarities?
soundness: 3
strengths: - Comprehensive Evaluation: Experiments are extensive and demonstrate consistent improvements across datasets, models, and non-IID settings.
- Writing and Presentation: The paper is well-written, with clear diagrams and formatting that aid readability.
- Practical Relevance: The method holds strong practical implications, bringing effective calibration methods to FL.
summary: This paper contributes to a well-established line of research that adapts standard machine learning (ML) techniques for federated settings by leveraging the similarity between global and local models to weight client contributions. Here, the ML technique under adaptation is model calibration, achieved through an auxiliary loss function. The paper’s primary contribution is a proposal to weight this auxiliary calibration loss based on a customizable similarity metric between global and local models, such as cosine similarity or other advanced measures. In this context, the “local” model refers to the locally trained replica of the global model. The authors provide an extensive empirical evaluation on standard federated learning datasets for computer vision, using LDA to generate non-IID partitions for IID datasets and some naturally heterogeneous datasets as well (FEMNIST). Their results indicate improved performance over baseline methods across all datasets, models, and levels of data heterogeneity.
weaknesses: I must mention that I am much more familiar with FL than model calibration and thus focus on this side of the work. 

- Limited Novelty in Core Insight: Although the similarity-based weighting approach is practical, it is not particularly novel, e.g.,[1,2,3]. The authors do acknowledge the influence of prior FL works that leverage model similarity metrics. 
- Potential Sensitivity to Hyperparameters: The method might be sensitive to local training hyperparameters like learning rate and epoch count, which could impact the similarity measure’s effectiveness. In cases closer to FedSGD with only one or a few steps performed, data heterogeneity may minimally impact the model replica, potentially diminishing the relevance of the similarity function. Similarly, scenarios involving extensive local training or very high learning rates may affect the similarity measure’s utility as all models may diverge very far from the global model regardless of the degree of local data heterogeneity. 
- No Exploration of Alternative Similarity References: Using only the client’s local model replica as the similarity reference could limit the applicability; other approaches (e.g., auxiliary persistent local models or specific personalized layers) might better capture client-specific characteristics.




[1] Tian Li, Shengyuan Hu, Ahmad Beirami, Virginia Smith:
Ditto: Fair and Robust Federated Learning Through Personalization. ICML 2021: 6357-6368

[2] Tao Yu, Eugene Bagdasaryan, Vitaly Shmatikov:
Salvaging Federated Learning by Local Adaptation. CoRR abs/2002.04758 (2020)

[3] Yutao Huang, Lingyang Chu, Zirui Zhou, Lanjun Wang, Jiangchuan Liu, Jian Pei, Yong Zhang:
Personalized Cross-Silo Federated Learning on Non-IID Data. AAAI 2021: 7865-7873

Meta-review: The paper introduces Non-Uniform Calibration for Federated Learning (NUCFL), a novel framework addressing model calibration in FL environments. It highlights a critical gap in FL research—ensuring reliable model predictions (calibration) across heterogeneous data distributions. The key innovation lies in dynamically adjusting calibration penalties based on the similarity between local and global models, thus aligning local training with global calibration needs. NUCFL is versatile and compatible with various FL algorithms. The authors provided extensive emprical validation in the manuscripts and during rebuttal, demonstrating the effectiveness of NUCFL on wide range of FL algorithms and datasets. Despite some theoretical and exploratory gaps, I lean to recommend acceptance.

Decision: Accept (Poster)