# Federated Learning with Imbalanced and Agglomerated Data Distribution for
Medical Image Classification

Nannan Wu, Li Yu, Xin Yang, Kwang-Ting Cheng, , and Zengqiang Yan N. Wu, L.
Yu, X. Yang, and Z. Yan are with the School of Electronic Information and
Communications, Huazhong University of Science and Technology, Wuhan, China
(e-mail: z.yan@connect.ust.hk).K. -T. Cheng is with the School of Engineering,
Hong Kong University of Science and Technology, Kowloon, Hong
Kong.Corresponding author: Zengqiang Yan

###### Abstract

Federated learning (FL), training deep models from decentralized data without
privacy leakage, has drawn great attention recently. Two common issues in FL,
namely data heterogeneity from the local perspective and class imbalance from
the global perspective have limited FL’s performance. These two coupling
problems are under-explored, and existing few studies may not be sufficiently
realistic to model data distributions in practical sceneries (e.g. medical
sceneries). One common observation is that the overall class distribution
across clients is imbalanced (e.g. common vs. rare diseases) and data tend to
be agglomerated to those more advanced clients (i.e., the data agglomeration
effect), which cannot be modeled by existing settings. Inspired by real
medical imaging datasets, we identify and formulate a new and more realistic
data distribution denoted as L2 distribution where global class distribution
is highly imbalanced and data distributions across clients are imbalanced but
forming a certain degree of data agglomeration. To pursue effective FL under
this distribution, we propose a novel privacy-preserving framework named
FedIIC that calibrates deep models to alleviate bias caused by imbalanced
training. To calibrate the feature extractor part, intra-client contrastive
learning with a modified similarity measure and inter-client contrastive
learning guided by shared global prototypes are introduced to produce a
uniform embedding distribution of all classes across clients. To calibrate the
classification heads, a softmax cross entropy loss with difficulty-aware logit
adjustment is constructed to ensure balanced decision boundaries of all
classes. Experimental results on publicly-available datasets demonstrate the
superior performance of FedIIC in dealing with both the proposed realistic
modeling and the existing modeling of the two coupling problems.

###### Index Terms:

Federated Learning, Class Imbalance, Contrastive Learning, Data Agglomeration

##  I Introduction

Despite the remarkable success of deep learning (DL) in computer vision [1],
natural language processing [2], etc., it has reached the consensus that DL is
a data-driven technology and collecting a large amount of training data is
often the first and the most significant step. However, it’s infeasible in
practice to fuse multiple data sources for large-scale data collection due to
privacy concerns, especially in medical or financial scenarios. Fortunately,
federated learning (FL), allowing multiple clients to jointly train a unified
deep learning model without data sharing [3, 4], is recently proposed to
address this issue. In vanilla FL, each participating client first trains a
model locally based on its data following standard DL and then collaborates
with others through parameter/gradient aggregation for global/federated model
update [5]. In this way, only clients’ model parameters/gradients, instead of
raw training data, are shared, which is privacy-preserving.

Figure 1:  Illustration of different data distributions with data imbalance,
including global data distribution over seven classes and local data
distributions across five clients. Top: A realistic dermatology data
distribution across five medical data sources [6]. Middle: Two typical
modelings of data imbalance in previous studies of federated learning [7, 8],
denoted as U and L distributions respectively. In the U distribution, global
data is assumed to be balanced and each class/client is considered equally in
data partitioning (implemented by following a Dirichlet distribution of each
class and randomly assigning them to clients with equal probabilities). In the
L distribution, global data distribution is long-tailed and each class/client
is of equal importance in data partitioning in the same way as the U
distribution. Bottom: The proposed L2 distribution. The global distribution
can be viewed as a special long-tailed distribution with a noticeable gap
(i.e., the red arrow) between different groups of classes, i.e., the majority
classes (i.e., the first five classes) and the minority classes (i.e., the
last two classes). In terms of local distributions, in addition to non-IID,
there exists a data agglomeration phenomenon (i.e., clients/classes are not
randomly partitioned with equal probabilities ).

Given that data is individually collected by each client in FL, there widely
exists data heterogeneity [9] in both feature space (e.g., varying intensity,
contrast, etc.) [10, 11, 12, 13, 14, 15] and class space (e.g. varying class
distributions) [7, 16, 17, 18] across different clients. In this paper, we
focus on the latter one and analyze different data heterogeneity settings from
the following two perspectives.

Global: Global data, i.e. the aggregation of all local data, is FL usually is
assumed to be balanced like CIFAR-10 [19], CIFAR-100 [19], Tiny-Imagenet [20],
etc. However, in practice, it is infeasible to ensure uniform global
distributions, which is under-explored. Few most-rent works propose to address
data imbalance by formulating global data into a long-tailed class
distribution with varying class probabilities [8].

Local: In addition to global class imbalance, local class and data
distributions across clients can also be imbalanced. To simulate this, each
class follows the same Dirichlet distribution (e.g. $Dir(\alpha=1.0)$, where
$\alpha$ defines the concentration of each sampled distribution) for data
partitioning to clients [7, 16, 8, 21]. By repeating the above data allocation
process for each class independently (i.e., data partitioning of classes can
be different but share similar concentration degrees), local class and data
distributions become non-IID (i.e., non-independent identically distribution).
This data distribution implicitly assumes all classes and clients are equally
important.

Based on the above global and local distributions, we reformulate existing
data heterogeneity settings as follows:

  1. 1.

U distribution. Global: uniform class distribution. Local: non-IID class and
data distributions following a Dirichlet distribution as illustrated in the
second row of Fig. 1.

  2. 2.

L distribution. Global: long-tailed class distribution. Local: non-IID local
class and data distributions following a Dirichlet distribution (i.e., the
same as U distribution) as illustrated in the third row of Fig. 1.

Unfortunately, both U and L distributions may not always be realistic due to
the following observations in clinical or financial scenarios:

  1. 1.

Global: Following a uniform or a standard long-tailed distribution can hardly
model complicated class distributions. In clinical scenarios, diseases can
differ dramatically in morbidity (e.g., common vs. rare diseases), resulting
in highly-imbalanced class distributions as illustrated in the first row
(i.e., the global part) of Fig. 1.

  2. 2.

Local: Classes and clients are not equally important. One common observation
in clinical or financial scenarios is the data agglomeration phenomenon that
the more advanced a medical/financial institution is, the more
patients/customers it owns of a specific class. Furthermore, rare classes can
be more concentrated than common classes, as the identification of rare
classes highly depends on clients’ expertise. Consequently, common classes can
distribute to nearly all clients while rare classes only exist in few top
institutions as illustrated in the first row (i.e., the local part) of Fig. 1.

In this paper, we, for the first time, identify and formally formulate the new
imbalanced distribution in FL as L2 distribution: non-uniformly sampled long-
tailed global class distribution and non-IID local class and data
distributions coupled with data agglomeration as illustrated in the fourth row
of Fig. 1. To pursue effective FL under this distribution, we propose a novel
FL framework FedIIC to construct a balanced and unified objective for all
clients to simultaneously address data and class imbalance. Specifically, we
first introduce supervised contrastive learning (SCL) [22] with an adaptive
temperature re-weighting the cosine similarity according to the data amount of
each two contrast categories for balanced learning of each local client. Then,
a set of class-wise global embeddings are constructed by aggregating and fine-
tuning the knowledge of all clients and sent to each client to assist SCL
between each sample embedding and global embeddings. In this way, each client
calibrates its own embedding space through the knowledge from others for
balanced learning. Furthermore, we propose a difficulty-aware logit adjustment
method to calibrate the linear classifier of each local model. It should be
noted that the establishment of global embeddings in FedIIC does not require
clients to upload extra information like previous studies [16, 23, 8], which
helps protect data privacy and reduce communication overhead.

The main contributions can be summarized as follows:

  1. 1.

We identify the special data distribution in medical or financial scenarios as
L2 distribution. Compared to both U and L distributions, L2 distribution
discussed in this paper is more realistic.

  2. 2.

We present a novel privacy-preserving framework FedIIC for imbalanced
federated learning which is effective for not only the L2 distribution but
also the L distribution [8].

  3. 3.

Superior performance against the state-of-the-art approaches on federated
learning with imbalanced data for medical and natural image classification.

##  II Related Work

###  II-A Federated Learning

Federated Learning (FL) [5] has drawn great attention due to its privacy-
preserving nature [3, 4]. In FL, one major concern is data heterogeneity
across clients affecting federated convergence [24, 25]. To address this,
FedProx [24] introduced a loss term measuring the distance between the global
model and each local model to avoid the local training process being dominant
by local data. SCAFFOLD [25] introduced a control variate to correct the
update direction of local models. MOON [7] employed contrastive learning (CL)
in the model level to constrain the updated model to be close to the global
model while away from the previous local model. Mu et al. [16] created a set
of prototypes for each class which was used to guide the update for
heterogeneous clients. Zhang et al. [17] re-formulated this problem from the
perspective of class imbalance and established uniform optimization objectives
for all clients via logit calibration. Li et al. [18] focused on an extreme
non-IID case that each client could only own partial categories of data, and
presented a restricted softmax function. Except for the heterogeneity of class
prior probabilities, prior studies also focused on the heterogeneity of cross-
client training images and minimized image heterogeneity through either
feature-level alignment or image-level alignment. Feature-level alignment
among cross-client imaging data can be implemented by generative adversarial
networks (GAN) [26] and local batch normalization [27]. As for image-level
alignment, typical solutions mainly are based on synthesis [28] in the image
space and augmentation in the frequency domain [29, 30].

###  II-B Imbalanced Learning

Given the prevalence of imbalance in real-world datasets, imbalanced learning
has been extensively studied in computer vision. Conventional solutions to
deal with this problem mainly focus on re-balancing the imbalanced data via
re-sampling [31, 32, 33] and re-weighting [34, 35, 36] to increase the
sampling frequency and loss weights of minority classes respectively, which
improves the performance of minority classes at the cost of degrading majority
classes [37]. In addition to re-balancing, some studies suggested setting
different margins for different classes in cross-entropy (CE) losses [38, 39].
Margin is a monotonically decreasing function of class prior probabilities,
enforcing the network to emphasize more on the tail classes. Another type of
solution toward imbalanced learning is decoupling [37, 40]. For instance, Kang
et al. [37] proposed to decouple an integral learning process into two
separate stages: representation learning and classifier learning. BBN [40]
used a bilateral-branch network to focus on the majority classes and minority
classes respectively. Both of these trained relatively balanced classifiers
with clever sampling methods instead of exploring how to obtain better
representations to boost performance even further. Fortunately, thanks to the
superior performance of supervised contrastive learning (SCL) [22] in
representation learning, it has been proven to be effective to handle class-
imbalance [41, 42, 43, 44, 45]. Hybrid-SC [42] introduced a two-branch network
and performed feature learning and classifier learning successively. PaCo [43]
introduced a set of learnable class-wise centers to make SCL more friendly to
imbalanced learning. KCL [41] sampled a uniform number of positives for each
class to avoid SCL being biased to head classes. The two most-recent
approaches TSC [44] and BCL [45] inspire us the most, both of which created a
set of independent features within each mini-batch to help SCL with imbalanced
data.

###  II-C Federated Learning with Imbalanced Data

Class imbalance is common yet under-explored in real-world FL applications.
Wang et al. [46] presented a ratio loss (i.e., a weighted form of CE loss)
depending on a balanced auxiliary dataset for the server to calculate weights,
which was infeasible in most cases. Yang et al. [47] presented an algorithm
based on a multi-arm bandit to select clients for imbalance minimization.
Jiang et al. [48] expanded the problem to the paradigm of semi-supervised
learning and addressed it via a dual bank learning scheme. CReFF [8] and CLIMB
[49] are the most-recent approaches. Inspired by decoupling [37], CReFF
retrained a new classifier with balanced synthetic features in the server to
improve the overall performance of classification. However, feature synthesis
required clients to upload the gradients of the classifiers during training,
which may cause privacy leakage under well-designed attacks. CLIMB assigned
larger weights to clients more likely to own minority classes via a meta-
algorithm. However, it may fail to deal with more general data distributions,
as the basic unit of weighting is a client instead of a class or a sample.

##  III Methodology

Figure 2: Overview of the proposed FedIIC framework.

###  III-A Background and Definitions

Given $K$ participants in FL, we assume each participant owns a private
dataset $D_{k}=\\{(x_{i},y_{i})\\}_{i=1}^{N_{k}},k\in\\{1,2,...,K\\}$, where
$N_{k}$ is the data amount of $D_{k}$, and denote each image-label pair as
$(x_{i}\in X\sim\mathbb{R}^{d},y_{i}\in Y=\\{1,2,...,L\\})$. In addition, we
denote the global dataset as $D_{g}=\\{(x_{i},y_{i})\\}_{i=1}^{\sum N_{k}}$
for distribution analysis which is the fusion of all local datasets though it
is infeasible due to data privacy in FL.

Definition 1 (Global and Local Class Imbalance). Following [46], we define two
types of class imbalance in FL. For both the global dataset and each local
dataset whose class prior probability is denoted as
$p(y)=[p^{1},p^{2},...,p^{L}],\sum_{i}p^{i}=1$, it is regarded as imbalanced
if $p(y)$ is not uniform and the degree of imbalance is expressed as
$\Gamma=\frac{{\rm max}(p(y))}{{\rm min}(p(y))}$.

Definition 2 (Data Concentration) Assume all $K$ clients follow a descending
order in data amount, i.e. $N_{1}\leq N_{2}\leq...\leq N_{K}$. For any
category/class $c\in Y$, assuming samples are distributed to $K$ clients with
probability $p^{c}=[p_{1}^{c},p_{2}^{c},...,p_{K}^{c}]$, data distribution is
concentrated if $p_{1}^{c}\leq p_{2}^{c}\leq...\leq p_{N}^{c}$.

Definition 3 (Degree of Data Concentration) If samples of any class $c$ are
concentrated in just few clients (i.e., $p_{i}^{c}\in p^{c}$ differs
dramatically), the probability distribution
$[p_{1}^{c},p_{2}^{c},...,p_{N}^{c}]$ is of low entropy. Following this, the
degree of data concentration is defined as ${\rm Conc}^{c}=1/{\rm
entropy}([p_{1}^{c},p_{2}^{c},...,p_{N}^{c}])$.

Base on the above definitions, U, L, and L2 distributions are re-defined as
follows:

U distribution: With local imbalance, but without global imbalance and data
concentration [7];

L distribution: With both global and local imbalance, but without data
concentration [8];

L2 distribution: With both global and local imbalance and data concentration.
Degrees of data concentration can vary across classes.

From the perspective of data heterogeneity, L2 distribution can be viewed as a
specialized form of L distribution which is more realistic, especially in
clinical scenarios.

###  III-B Overview

For robust federated learning with L2 distribution, we propose FedIIC as
illustrated in Fig. 2. The key idea is to calibrate both the feature extractor
and the classification head which may be badly affected by data imbalance. To
calibrate the feature extractor, we introduce 1) intra-client contrastive
learning with a modified similarity measure to alleviate local class imbalance
and 2) inter-client contrastive learning with shared global prototypes to
produce a uniform embedding space for all classes across clients. To calibrate
the classification head, we propose a difficulty-aware logit adjustment
algorithm to produce balanced decision boundaries for all classes. Details are
presented in the following.

###  III-C Intra-Client Contrastive Learning

The goal in FL is to obtain a well-performed global classification model
$\mathcal{F}\leftarrow f(g({X}))$, where $g(\cdot)$ and $f(\cdot)$ represent
the feature extractor and the linear classifier respectively. One
straightforward solution is to pursue well-performing models during the local
training phase before aggregation. Unfortunately, as local data is often
limited and imbalanced, simply imposing a cross entropy loss may be sub-
optimal. Therefore, we introduce supervised contrastive learning (SCL) to
assist local training, which is proven to be effective for representation
learning [50]. The basic loss function of SCL can be formulated as

$\mathcal{L}_{SCL}=\sum_{i\in I}\frac{-1}{\left|P(i)\right|}\sum_{j\in
P(i)}\log\frac{\exp(z_{i}\cdot z_{j}/\tau)}{\sum_{a\in A(i)}\exp(z_{i}\cdot
z_{a}/\tau)},$ (1)

where $I$ denotes the index set of the multiviewed batch, $\left|\cdot\right|$
represents the number of elements in a set,
$A(i)=I\backslash\\{i\\},P(i)=\\{s\in A(i)|y_{s}=y_{i}\\}$, $\tau$ represents
the temperature, and $z$ denotes the $l_{2}$-normalized embedding of a sample.
Note that in this paper, we use a 2-layer MLP $h(\cdot)$ to obtain $z$ before
it is normalized as [51], i.e. $z=\frac{h(g(x))}{\lVert h(g(x))\rVert_{2}}$.
The key idea of SCL is to keep the embeddings of the same class closer
(measured by the cosine similarity) while pushing the embeddings of different
classes further away, producing a compact embedding space for the downstream
task (e.g. image classification).

However, SCL can not perfectly address class imbalance from the perspective of
optimization. Assuming there are both majority classes and minority classes in
the dataset, the fastest paths to minimize $\mathcal{L}_{SCL}$ are:

  1. 1.

Increasing the intra-class similarity of a majority class;

  2. 2.

Decreasing the inter-class similarity of any two different majority classes.

Therefore, SCL tends to focus more on the majority classes like traditional
training methods (e.g. the CE loss). To overcome this problem, instead of
either choosing a specific/fixed number of positive samples of each class for
contrastive learning [41] (i.e., balancing the numerator in Eq. 1) or
adjusting the ordinary sum to a class-weighted sum in the denominator [45] in
Eq. 1, we modify the calculation of similarity in SCL with a zoom factor $P$.
Specifically, the similarity between two embeddings $z_{i}$ and $z_{j}$ is
calculated as

$\frac{z_{i}\cdot z_{j}}{(P\tau)}=\frac{z_{i}\cdot
z_{j}}{(p^{i}p^{j})^{t}\tau},$ (2)

where $p^{i}$ is the prior probability of class $i$ in the local dataset and
$t$ is a parameter set as 0.5 by default. Through $P$, sample pairs of the
minority classes are more emphasized than those of the majority classes,
leading to better balance. Denoting a new dynamic temperature factor
$\tau^{\prime}$ as $P\tau$, the loss function of intra-client contrastive
learning is defined as

$\mathcal{L}_{Intra}=\sum_{i\in I}\frac{-1}{\left|P(i)\right|}\sum_{j\in
P(i)}\log\frac{\exp(z_{i}\cdot z_{j}/\tau^{\prime})}{\sum_{a\in
A(i)}\exp(z_{i}\cdot z_{a}/\tau^{\prime})}.$ (3)

###  III-D Inter-Client Contrastive Learning

Figure 3: Illustrations of the embedding space under different data
distributions. In each subfigure, dots denote training samples and shadow
areas represent the decision boundaries of different classes.

The ideal embedding space is balanced where each class occupies a space
uniformly as illustrated in Fig. 3 (a). However, due to class imbalance or
class missing, the occupation of classes becomes non-uniform, where the
majority classes occupy larger space than the minority classes as shown in
Fig. 3 (b) and (c) [52]. In FL, the degrees of local imbalance $\Gamma$ of
different clients can vary dramatically and may even be infinity due to class
missing. As a result, each client may own a non-uniform and different
embedding space occupation across clients as illustrated in Fig. 3, which in
turn produces parameter difference and affects the convergence of FL.

To address this, the key idea is to create a uniform embedding space
occupation for all clients. Assuming a set of class-wise prototypes
$V=\\{v^{1},v^{2},...,v^{L}\\}$, we can adopt inter-client contrastive
learning similar to PCL in FedProc [16] to calibrate the embedding space
trained by

$\mathcal{L}_{Inter}=\sum_{i\in
I}\frac{-1}{\left|P(i)\right|}\log\frac{\exp(z_{i}\cdot
v^{y_{i}}/\tau)}{\sum_{j=1}^{L}\exp(z_{i}\cdot v^{j}/\tau)},$ (4)

where $y_{i}$ is the label of sample $i$. When minimizing
$\mathcal{L}_{Inter}$, the embedding of each sample will get close to the
prototype of the same class while away from the prototypes of different
classes. As prototypes are shared for all clients, clients tend to share a
similar embedding space occupation.

To this end, how to produce high-quality prototypes is the key to inter-client
contrastive learning. In previous studies, one common method to generate
prototypes or global features in FL is uploading and aggregating extra
information. For example, FedProc [16] and FedIRM [23] uploaded features to
the server directly and CReFF [8] uploaded gradients to calculate features
indirectly. However, it may cause privacy leakage under well-designed attacks
and will introduce extra communication costs.

In FedIIC, we propose a new method to generate global prototypes without
uploading extra information. Considering that the essence of linear
classification is to calculate similarity based on vector inner product, the
weights of a well-trained linear classifier are nearly co-linear with the
features vectors of different classes [53, 54, 45]. Therefore, the weights of
a linear classifier denoted as $W=[w^{1},w^{2},...,w^{L}]$, can represent the
class features to some extent and be used as prototypes. Specifically, given a
global model $[f_{g}(\cdot),g_{g}(\cdot),h_{g}(\cdot)]$ (i.e., $f_{g}(\cdot)$,
$g_{g}(\cdot)$, and $h_{g}(\cdot)$ are the feature extractor, the linear
classifier, and the projection head for contrastive learning respectively)
after model aggregation in the server, the weights of $g_{g}(\cdot)$ are fed
to $h_{g}(\cdot)$ to calculate the initial prototypes
$\widetilde{V}=\\{\widetilde{v}^{1},\widetilde{v}^{2},...,\widetilde{v}^{L}\\}$.
Considering that the ideal embeddings of different classes in SCL should be
uniformly distributed on a hypersphere [44], we fine-tune $\widetilde{V}$ via
gradient descent by

$\widetilde{V}\leftarrow\widetilde{V}-\nabla\sum_{i\in Y}\max_{j\in Y,j\neq
i}(\frac{\widetilde{v}^{i}}{\lVert\widetilde{v}^{i}\rVert_{2}}\cdot\frac{\widetilde{v}^{j}}{\lVert\widetilde{v}^{j}\rVert_{2}}).$
(5)

According to Eq. 5, the cosine similarity of any
$({\widetilde{v}^{i}},{\widetilde{v}^{j}})$ pair in $\widetilde{V}$ is
minimized to be equal, resulting in a uniform embedding distribution of all
classes. Finally, the class-wise prototypes $V$ are defined as the element-
wise $l_{2}$-normalization of $\widetilde{V}$.

###  III-E Difficulty-Aware Logit Adjustment

In addition to the calibration of $g(\cdot)$ through inter-client contrastive
learning, we further adopt Logit Adjustment (LA) to calibrate the linear
classifier $f(\cdot)$. Before moving on to LA, we first revisit the multiclass
classification problem with $X\sim\mathbb{R}^{d}$ and $Y\sim\mathbb{R}^{L}$.
Given a sample set $(x,y)$, for unknown probability $p(y|x)$ over $X\times Y$,
a multiclass classification task is to learn a function
$\mathcal{F}:X\to\mathbb{R}^{L}$ by minimizing the misclassification error
$p_{x,y}(y\neq{\arg\max}_{y^{\prime}\in Y}\mathcal{F}_{y^{\prime}}(x))$.
Typically, one minimizes a surrogate loss
$\mathcal{L}:Y\times\mathbb{R}^{L}\rightarrow\mathbb{R}$, such as the softmax
cross-entropy loss:

$\begin{split}\mathcal{L}_{CE}(y,\mathcal{F}(x))&=-\log\frac{e^{\mathcal{F}_{y}(x)}}{\sum_{y^{\prime}\in
Y}e^{\mathcal{F}_{y^{\prime}}(x)}}\\\ &=\log[1+\sum_{{y^{\prime}}\neq
y}e^{\mathcal{F}_{y^{\prime}}(x)-\mathcal{F}_{y}(x)}].\end{split}$ (6)

Under data imbalance where $p(y)$ is highly imbalanced, $\mathcal{L}_{CE}$
tends to focus on the majority classes as a trivial predictor which classifies
all samples to the majority classes will attain a low misclassification error.
Consequently, classes can have imbalanced decision boundaries. To address
this, one solution is to add a per-class margin into the softmax cross
entropy, denoted as Logit Adjustment (LA) defined as

$\mathcal{L}_{LA}(y,\mathcal{F}(x))=\log[1+\sum_{{y^{\prime}}\neq
y}e^{\delta}e^{\mathcal{F}_{y^{\prime}}(x)-\mathcal{F}_{y}(x)}].$ (7)

In quantity-based approaches, $\delta_{y}$ is calculated based on the prior
class probabilities, e.g. $\delta\propto p(y)^{-1/4}$ [38],
$\delta\propto\log(\frac{p(y^{\prime})}{p(y)})$ [39], and
$\delta\propto[\frac{p(y)}{p(y^{\prime})}]^{-1/4}$ [17]. In this way, minority
classes (i.e., with relatively smaller $p(y)$) are upweighted to learn more
balanced decision boundaries.

In FedIIC, the per-class margin is calculated based on not only the prior
class probabilities but also difficulties, denoted as Difficulty-Aware Logit
Adjustment (DALA) inspired by [55]. For instance, in clinical scenarios, some
disease types/classes may have large intra-class variations and are difficult
to diagnose even with a large data amount. Therefore, the above quantity-based
margin-setting approaches may not be appropriate, and the variations in
classification difficulty among classes should be included for margin
adjustment. In FedIIC, we utilize the loss of each class which directly
reflects the difficulty in optimization for margin setting. Specifically, we
set
$\delta\propto\log(\frac{p(y^{\prime})}{p(y)}[\frac{\overline{l}_{ce}(y)}{\overline{l}_{ce}(y^{\prime})}]^{q})$,
where $\overline{l}_{ce}(y)$ is the average CE loss of all samples belonging
to class $y$ in any round and $q$ is a hyper-parameter set as 0.25 by default.
$\overline{l}_{ce}(y)$ is calculated as follows. At any round $r$, the total
sample number of class $y$, denoted as $N^{y}_{r}$, belonging to clients of
communication is calculated. After receiving the global model from the server
and before local training, each client $i$ uploads ${{l}^{i}_{ce}(y)}$, the
total loss of class $y$, to the server. Finally, $\overline{l}_{ce}(y)$ is
calculated as $\frac{1}{N^{y}_{r}}\sum_{i}{l}^{i}_{ce}(y)$. In Section V.D, we
prove that this process has no risk of privacy leakage in data distribution.
Based on this form of $\delta$, it will assign a larger margin due to either
low prior probability or high training loss for a specific class $y$. By
replacing $\delta$ in Eq. 7 with
$\log(\frac{p(y^{\prime})}{p(y)}[\frac{\overline{l}_{ce}(y)}{\overline{l}_{ce}(y^{\prime})}]^{q})=\log(\frac{p(y^{\prime})}{[\overline{l}_{ce}(y^{\prime})]^{q}})-\log(\frac{p(y)}{[\overline{l}_{ce}(y)]^{q}})$,
the softmax cross entropy loss with difficulty-aware logit adjustment
$\mathcal{L}_{DALA}$ is written as

$\begin{split}&\mathcal{L}_{DALA}(y,\mathcal{F}(x))\\\
&=\log[1+\sum_{{y^{\prime}}\neq
y}e^{\log(\frac{p(y^{\prime})}{[\overline{l}_{ce}(y^{\prime})]^{q}})-\log(\frac{p(y)}{[\overline{l}_{ce}(y)]^{q}})}e^{\mathcal{F}_{y^{\prime}}(x)-\mathcal{F}_{y}(x)}]\\\
&=\log[1+\sum_{{y^{\prime}}\neq
y}e^{[\mathcal{F}_{y^{\prime}}(x)+\log(\frac{p(y^{\prime})}{[\overline{l}_{ce}(y^{\prime})]^{q}})]-[\mathcal{F}_{y}(x)+\log(\frac{p(y)}{[\overline{l}_{ce}(y)]^{q}})]}],\end{split}$
(8)

and DALA is written as

$\mathcal{F}_{y}(x)\leftarrow\mathcal{F}_{y}(x)+\log(\frac{p(y)}{[\overline{l}_{ce}(y)]^{q}}).$
(9)

Note that the calculation of $\mathcal{L}_{DALA}$ does not rely on the multi-
viewed batch like $\mathcal{L}_{Intra}$ and $\mathcal{L}_{Inter}$. For a fair
comparison with other methods based on CE loss, only the first view of the
multi-viewed batch is used to calculate $\mathcal{L}_{DALA}$.

Finally, the overall loss function is written as

$\mathcal{L}=\mathcal{L}_{DALA}+k_{1}\mathcal{L}_{Intra}+k_{2}\mathcal{L}_{Inter},$
(10)

where $k_{1}$ and $k_{2}$ are trade-off hyper-parameters. After minimizing
$\mathcal{L}$ during the local training phase of each client, the global model
is updated by FedAvg [5].

##  IV Evaluation

###  IV-A Dataset

####  IV-A1 Real Multi-Source Dermoscopic Image Datasets (denoted as Real)

TABLE I: Data distributions of five data sources from three datasets (i.e.,
the imbalanced distribution in the first row of Fig. 1).

Source | Category
---|---
MEL | NV | BCC | AK | BKL | DF | VASC | Total
PH2 | 40 | 160 | 0 | 0 | 0 | 0 | 0 | 200
ViDIR | 67 | 350 | 5 | 0 | 10 | 4 | 3 | 439
(Legacy)
Atlas | 268 | 575 | 42 | 0 | 69 | 20 | 29 | 1003
Rosendahl | 342 | 803 | 296 | 295 | 490 | 30 | 3 | 2259
ViDIR | 680 | 1832 | 211 | 32 | 475 | 51 | 82 | 3363
(Current)
test | 500 | 500 | 500 | 500 | 500 | 154 | 165 | 2819

Five data sources from three datasets, including PH2 [56], Atlas [57], and
HAM10000 [6], are adopted for realistic evaluation where each source is
considered as an individual client in FL. Details of different sources are
summarized in Table I. For evaluation, we construct a separate test set by
selecting data from the training set of ISIC 2019 and ensure that the test set
has no overlap with the above five data sources. Note that the extra class
named SCC in ISIC 2019 is considered a sub-class of AK when constructing the
test set. Among the seven classes, DF and VASC are defined as minor and others
are major.

####  IV-A2 Intracranial Hemorrhage Classification (denoted as ICH)

The RNSA ICH dataset [58], containing five ICH subtypes, is adopted for
realistic evaluation. Data is pre-processed in the same way as [48, 23], and
in total 67969 images with only one single hemorrhage type are selected. Among
the five classes, Epidural, with the least data amount (i.e., 2.20%) is
defined as minor and others are major. Following [48, 23], data is split
according to 7:1:2 for training, validation, and testing respectively.

####  IV-A3 Skin Lesion Classification (denoted as ISIC)

The training data of ISIC 2019 [6, 59], containing eight classes of 25331
images, for skin lesion classification is used for realistic evaluation. Among
the eight classes, Dermatofibroma (DF) and Vascular Lesion (VASC), accounting
for about 0.94% and 0.99% of the dataset, are defined as the minority classes
and others are the majority classes. Following [48, 23], we split the dataset
by 7:1:2 for training, validation, and testing respectively.

###  IV-B Evaluation Metrics

As the test sets are imbalanced, balanced accuracy (BACC), i.e. the official
metric of the ISIC 2019 Challenge, is adopted for evaluation, defined as

$BACC=\frac{1}{C}\sum_{c=1}^{C}\frac{TP_{c}}{TP_{c}+FN_{c}},$ (11)

where $TP_{c}$ and $FN_{c}$ represent true positives and false negatives of
class $c$ respectively. BACC treats all classes equally instead of being
biased toward the majority classes. For the evaluation of Real, the average
and the standard deviation of BACC of the last five rounds are reported for
comparison.

###  IV-C Data Partition

Figure 4: Data distribution of all clients under the ICH and ISIC settings. In
each subfigure, the horizontal axis represents class IDs while the vertical
axis represents client IDs. Each column represents the number of samples of a
specific class in a specific client.

Following [7, 16, 8, 21], we use a Dirichlet distribution $Dir(\alpha)$ to
divide the training data for each client, where the only hyper-parameter is
$\alpha$. With a larger $\alpha$, the sampled probability distribution $p\sim
Dir(\alpha)$ is more likely to be even. As analyzed in Section III, in L2
distribution, the degree of data concentration of the minority classes is
greater than that of the majority classes, i.e. ${\rm Conc}^{minor}\gg{\rm
Conc}^{major}$. To simulate this, we use two Dirichlet distributions with a
large $\alpha_{1}$ and a small $\alpha_{2}$ for data partitioning of the
minority and the majority classes respectively. To simulate data agglomeration
in L2 distribution, for each class, the sampled probabilities produced by
$Dir(\alpha)$ are sorted first in descending order before assigning samples to
the corresponding clients. In this way, samples of classes are relatively more
concentrated to some clients as illustrated in Fig. 1. Specifically, we set
$\alpha_{1}=50$, $\alpha_{2}=0.1$, and $K=20$ for ICH and $\alpha_{1}=50$,
$\alpha_{2}=0.5$, and $K=10$ for ISIC, resulting in the generated data
distributions as shown in Fig. 4.

TABLE II:  Quantitative results of different methods conducted under the Real, ICH, and ISIC settings. For Real, following [18], the average and the standard deviation of BACC (%) of the last 5 rounds are reported. For ICH and ISIC, BACC (%) of the test set is reported. In addition, BACC (%) of the minority classes, the majority classes, and all classes on the test sets are reported. Bold and underlined numbers represent the best and the second-best results respectively.  Method | Real | ICH | ISIC
---|---|---|---
Minority | Majority | Avg. | Minority | Majority | Avg. | Minority | Majority | Avg.
FedAvg [5] | 47.99$\pm$2.84 | 44.10$\pm$0.72 | 45.21$\pm$0.99 | 15.01 | 86.53 | 72.22 | 76.13 | 76.77 | 76.61
FedProx [24] | 48.24$\pm$2.86 | 44.15$\pm$0.79 | 45.32$\pm$1.08 | 35.78 | 87.66 | 77.29 | 78.63 | 79.25 | 79.10
MOON [7] | 44.07$\pm$2.16 | 42.90$\pm$1.37 | 43.23$\pm$1.43 | 6.07 | 86.47 | 70.39 | 73.18 | 74.58 | 74.23
FedProc [16] | 25.33$\pm$1.78 | 39.26$\pm$0.89 | 35.28$\pm$0.86 | 4.47 | 84.31 | 68.34 | 72.27 | 75.94 | 75.02
FedFocal [60] | 48.44$\pm$1.23 | 42.23$\pm$1.08 | 44.00$\pm$1.00 | 5.43 | 85.53 | 69.51 | 80.22 | 64.06 | 68.10
FedRS [18] | 48.57$\pm$0.69 | 43.90$\pm$0.75 | 45.23$\pm$0.93 | 11.18 | 86.65 | 71.56 | 75.00 | 76.91 | 76.43
FedLC [17] | 51.51$\pm$2.68 | 44.81$\pm$0.60 | 46.73$\pm$0.83 | 37.69 | 86.37 | 76.63 | 78.86 | 77.05 | 77.50
CLIMB [49] | 47.09$\pm$2.04 | 45.66$\pm$0.87 | 46.07$\pm$0.94 | 0.00 | 83.21 | 66.56 | 77.04 | 73.63 | 74.48
CReFF [8] | 68.53$\pm$0.50 | 46.97$\pm$0.26 | 53.13$\pm$0.18 | 70.92 | 85.60 | 82.66 | 90.90 | 76.61 | 80.19
FedIIC (Ours) | 71.41$\pm$0.11 | 48.71$\pm$0.60 | 55.20$\pm$0.24 | 71.56 | 87.97 | 84.69 | 88.63 | 81.56 | 83.33

###  IV-D Experimental Details

EfficientNet-B0 [61], pre-trained by ImageNet [20], is adopted as the backbone
for classification trained by an Adam optimizer with betas as 0.9 and 0.999,
weight decay as 5e-4, a constant learning rate as 1e-4 for Real and 3e-4 for
both ICH and ISIC, and a batch size of 32. For ICH, the multiviewed batch for
contrastive learning is generated by following [48, 23]. For both Real and
ISIC, the multiviewed batch is generated by 1) RandAug [62] and 2) SimAugment
[51]. The hyper-parameters $k_{1}$ and $k_{2}$ in Eq. 10 are set as 2.0 for
ICH and ISIC and are set as 1.2 and 1.0 respectively for Real. For federated
training, the local training epoch is set as 1 and the complete framework is
trained for 200 rounds for ICH and ISIC and 30 rounds for Real. At each
training round, we evaluate the global model on the validation set and save
the best model for testing for ICH and ISIC. At each round, all clients (i.e.,
100%) are included for model aggregation.

###  IV-E Quantitative Comparison

Nine approaches are included for comprehensive comparison and evaluation,
including FedAvg [5], FedProx [24] addressing data heterogeneity, MOON [7] and
FedProc [16] utilizing contrastive learning in FL, [16], FedFocal [60]
utilizing focal loss for balancing, FedRS [18] addressing the class-missing
problem, FedLC [17] addressing the non-IID issue by logits calibration, and
CLIMB [49] and CReFF [8] addressing the global imbalance in FL.

Quantitative results under the Real, ICH, and ISIC settings are summarized in
Table II. Among comparison approaches, FedIIC achieves the best overall
performance. Furthermore, separate evaluation on the classification of the
minority and the majority classes is conducted to analyze how approaches
perform with varying data amounts of different classes. Under both the Reals
and ICH settings, FedIIC performs the best on both the minority and the
majority classes, while achieving the best and the second-best results on the
majority and the minority of ISIC respectively. Though FedProx achieves the
second-best results on the majority classes of both ICH and ISIC, it is at the
cost of the training process being dominated by the majority classes,
resulting in poor performance for the minority classes. Comparatively, CReFF
achieves the best overall performance among comparison approaches with more
balanced results on both the minority and the majority classes.

###  IV-F Ablation Study

TABLE III: Ablation study of components in FedIIC. Bold numbers indicate the best results. FedAvg | DALA | Intra | Inter | Real | ICH | ISIC
---|---|---|---|---|---|---
✓ |  |  |  | 45.21$\pm$0.99 | 72.22 | 76.61
✓ | ✓ |  |  | 54.66$\pm$0.30 | 81.85 | 79.70
✓ | ✓ | ✓ |  | 54.57$\pm$0.28 | 84.49 | 82.40
✓ | ✓ |  | ✓ | 54.62$\pm$0.19 | 84.10 | 82.10
✓ | ✓ | ✓ | ✓ | 55.20$\pm$0.24 | 84.69 | 83.33

To validate the effectiveness of each component in FedIIC, a series of
ablation studies are conducted on Real, ICH, and ISIC following the same
experimental details described in Section IV. Quantitative results are
summarized in Table. III. Under severe global imbalance, FedAvg is struggling.
With the introduction of DALA, the performance of FedAvg is effectively
improved (i.e., an average increase of 9.45%, 9.63%, and 3.09% for Real, ICH,
and ISIC respectively). It is consistent with the analysis in [37, 8] that the
bias caused by class imbalance mainly exists in classifiers, and calibrating
the classifiers is the most efficient way for balanced learning. Based on
this, introducing either intra- or inter-client contrastive learning is
helpful, indicating their benefits for better representation learning under
data imbalance. It should be noted that separately introducing intra- and
inter-client contrastive learning to Real is less effective compared to both
ICH and ISIC, mainly due to highly limited training data in total.
Comparatively, jointly adopting both intra- and inter-client contrastive
learning is beneficial across all settings. By combining all the components,
FedIIC achieves the best overall performance, outperforming FedAvg with large
margins.

##  V Discussion

TABLE IV: Quantitative results of different methods conducted on CIFAR-10 and CIFAR-100. Following [18], the average and the standard deviation of Top-1 accuracy (%) of the last 50 rounds are reported for comparison. Bold and underlined numbers represent the best and the second-best results respectively. Method | CIFAR-10 | CIFAR-100
---|---|---
$\Gamma$=10 |  $\Gamma$=50 |  $\Gamma$=100 |  $\Gamma$=10 |  $\Gamma$=50 |  $\Gamma$=100
FedAvg [5] | 79.92$\pm$1.80 | 68.34$\pm$1.54 | 62.94$\pm$1.26 | 52.82$\pm$0.69 | 47.34$\pm$0.47 | 45.71$\pm$0.85
FedProx [24] | 80.23$\pm$1.69 | 68.42$\pm$1.45 | 63.27$\pm$1.40 | 52.99$\pm$0.93 | 47.43$\pm$0.63 | 45.79$\pm$0.79
MOON [7] | 80.00$\pm$1.61 | 69.41$\pm$2.33 | 64.59$\pm$1.98 | 51.26$\pm$1.05 | 45.94$\pm$0.63 | 44.37$\pm$1.14
FedProc [16] | 71.39$\pm$3.13 | 57.99$\pm$1.19 | 57.21$\pm$1.27 | 45.50$\pm$0.90 | 41.23$\pm$1.99 | 41.42$\pm$1.05
FedFocal [60] | 75.72$\pm$1.74 | 63.92$\pm$1.36 | 59.70$\pm$1.20 | 51.47$\pm$0.84 | 46.02$\pm$0.49 | 44.33$\pm$0.73
FedRS [18] | 81.07$\pm$1.44 | 70.35$\pm$1.38 | 65.32$\pm$1.46 | 53.41$\pm$0.83 | 48.07$\pm$0.65 | 47.17$\pm$0.67
FedLC [17] | 81.67$\pm$1.48 | 71.75$\pm$1.40 | 67.01$\pm$1.71 | 54.57$\pm$0.67 | 49.09$\pm$0.78 | 47.08$\pm$0.65
CLIMB [49] | 79.11$\pm$1.82 | 68.57$\pm$1.65 | 63.42$\pm$1.48 | 52.46$\pm$1.18 | 46.32$\pm$0.68 | 45.17$\pm$0.66
CReFF [8] | 82.59$\pm$1.01 | 75.94$\pm$1.58 | 73.01$\pm$1.49 | 47.94$\pm$1.40 | 46.14$\pm$0.73 | 38.71$\pm$0.85
FedIIC (Ours) | 85.69$\pm$1.03 | 81.90$\pm$1.57 | 80.12$\pm$1.57 | 59.81$\pm$0.74 | 53.67$\pm$0.77 | 51.30$\pm$0.73

###  V-A Extension to Natural Image Classification

####  V-A1 Datasets and Evaluation Metric

Following previous studies, two widely-used datasets CIFAR-10 and CIFAR-100
for image recognition [19] are adopted for evaluation. The former contains 10
classes while the latter contains 100 classes, and both of them contain 50000
images for training and 10000 images for testing. Considering these two
datasets are balanced, two new imbalanced datasets are sub-sampled from them
to simulate global imbalance. Specifically, inspired by [38], we use an
exponential decay function to determine the data amounts of different classes.
To simulate the L2 distribution as illustrated in Fig. 1, especially the gap
between the majority classes (i.e., the first seven classes in CIFAR-10, and
the first 70 classes in CIFAR-100) and the minority classes (i.e., the last
three classes in CIFAR-10, and the last 30 classes in CIFAR-100) in the global
distribution, data of classes is not evenly sampled in the decay function like
[38]. The exact data amount of each class is determined by

${\rm num}(c)=\left\\{\begin{aligned} &{\rm
num_{max}}\times\Gamma^{\frac{c-1}{10L-1}}&c\mbox{ is major}\\\ &{\rm
num_{max}}\times\Gamma^{\frac{c-1+9L}{10L-1}}&c\mbox{ is minor}\\\
\end{aligned}\right.$ (12)

where $L$ is the total number of classes, ${\rm num_{max}}$ is the max size of
all classes (i.e., 5000 in CIFAR-10 and 500 in CIFAR-100) and $\Gamma$ is the
imbalance factor defined in Section III. For both datasets, Top-1 accuracy is
used for evaluation due to the available balanced test set.

Figure 5: Data distribution of all clients across different datasets under
various settings. In each subfigure, the horizontal axis represents class IDs
while the vertical axis represents client IDs. Each column indicates the
number of samples of a specific class in a specific client.

####  V-A2 Experimental Details

Following [8], $\Gamma$ is set as 10, 50, and 100 separately to create
imbalanced datasets. As for data partitioning, we set $\alpha_{1}=10$,
$\alpha_{2}=0.2$, and $K=100$ for CIFAR-10 and $\alpha_{1}=10$,
$\alpha_{2}=0.5$, and $K=50$ for CIFAR-100. The generated data distributions
of CIFAR-10 and CIFAR-100 are illustrated in Fig. 5 (a)-(f). For both
datasets, ResNet-8 [63] is adopted as the backbone for classification and
trained by the SGD optimizer with momentum as 0.9, weight decay as 5e-4, a
constant learning rate as 0.03, and a batch size of 64. In terms of both
intra- and inter-client contrastive learning, the multiviewed batch is
generated by 1) random crop and flip and 2) SimAugment [51]. The hyper-
parameters $k_{1}$ and $k_{2}$ in Eq. 10 are set as 1.0 and 0.6 respectively.
For federated training, the local training epoch is set as 1 and the complete
framework is trained for 1500 rounds. At each round, 20% of clients are
randomly selected for model aggregation and then performance evaluation.

####  V-A3 Quantitative Comparison

Quantitative comparison results are summarized in TABLE IV. Across all
settings, FedIIC achieves the best performance with a large margin compared to
the second-best results. FedProx and MOON, though designed for addressing the
non-IID problem, can hardly outperform FedAvg under global imbalance. The
failure of FedProc demonstrates the necessity of designing appropriate
prototypes in addition to contrastive learning. Similarly, FedFocal performing
worse than FedAvg shows the limitation of focal loss on imbalanced learning.
Given the class-missing problem in L2 distribution, FedRS achieves noticeable
performance improvement compared to FedAvg by limiting the update of missing
classes’ weights.

One interesting observation is that CLIMB fails to outperform FedAvg on both
datasets. In L2 distribution, data imbalance among clients becomes more severe
than both U and L distributions, making CLIMB struggle to find optimal
clients’ weights for balanced learning. Among all comparison approaches, CReFF
achieves the best results on CIFAR-10 but encounters significant performance
degradation on CIFAR-100. It is because that samples belonging to the minority
classes can be extremely limited in L2 distribution, generating poor federated
features and further misleading the retraining of the classifier in CReFF.

###  V-B Evaluation on L Distributions

TABLE V: Quantitative results of different approaches on dealing with data imbalance under L distributions. For CIFAR-10-LT, the average and the standard deviation of Top-1 accuracy (%) of the last 10 rounds are reported. For ICH, BACC (%) of the test set is reported. Bold numbers indicate the best results. Method | CIFAR-10-LT (IF=10) | ICH
---|---|---
Rounds | 40 | 200 | 40 | 200
FedAvg [5] | 56.53$\pm$6.03 | 60.15$\pm$6.61 | 72.71 | 73.74
CLIMB [49] | 59.01$\pm$4.33 | 64.86$\pm$5.96 | 71.73 | 71.73
CReFF [8] | 71.24$\pm$3.41 | 76.89$\pm$4.55 | 75.00 | 82.21
FedIIC (Ours) | 77.26$\pm$1.35 | 79.44$\pm$1.61 | 78.02 | 84.23

In addition to the proposed L2 distribution, we further extend FedIIC to a
more widely-studied L distribution and conduct comparison experiments by
following the non-IID setting in [8] on both the CIFAR-10-LT (IF=10) [38] and
ICH. Specifically, for CIFAR-10-LT, we set $\alpha=0.5$, $K=20$,
$local\\_epoch=10$, and $online\\_clients=8$ by following [8] and follow other
settings described in Section V.A. For ICH, we set $\alpha=1.0$ and follow
other settings described in Section IV. CLIMB [49] and CReFF [8] designed for
FL with data imbalance are included for comparison.

Quantitative results are summarized in Table. V. Among the three baseline
approaches, CReFF achieves the best overall performance, which is consistent
with the results in Tables II and IV. One interesting observation is that
CLIMB outperforms FedAvg on CIFAR-10-LT under the L distribution. However, it
still fails to surpass FedAvg under the ICH setting, indicating the limitation
of client-wise re-weighting in CLIMB. Compared to the baseline approaches,
FedIIC achieves the best overall performance across the two settings with a
large margin, validating the extend-ability of FedIIC on handling other
imbalanced data distributions in addition to L2 distributions.

###  V-C Hyper-Parameter Discussion in DALA

TABLE VI: Ablation study of the hyper-parameter $q$ in DALA on ISIC. BACC (%) of the test set is reported. Bold numbers indicate the best results. Rounds | $q$
---|---
0 | 0.25 | 0.5 | 1 | 2
100 | 79.41 | 81.33 | 80.71 | 81.22 | 80.08
200 | 82.09 | 83.33 | 82.52 | 81.72 | 81.69

In FedIIC, one hyper-parameter is $q$ in difficulty-aware logit adjustment
(DALA), which determines the importance of class difficulties in logit
adjustment. If $q$ is large, logit adjustment is mainly dominated by class
difficulties. Otherwise, DALA is dominated by the prior class probability when
$q$ approaches zero. To quantify the value of $q$ in FedIIC, we conduct a
series of ablation studies on the ISIC dataset with varying $q$ as summarized
in Table. VI. Compared to the baseline logit adjustment ignoring class
difficulties, increasing $q$ is beneficial. But it does not mean that setting
a larger $q$ would necessarily lead to better results as stated in Table VI.
With the increase of $q$, there exists noticeable performance degradation. It
is because emphasizing too much on difficult classes may result in
instabilities in training. According to the quantitative results in Table VI,
setting $q$ as 0.25 achieves the best performance on the ISIC dataset. For
different tasks/datasets, $q$ should be tuned accordingly based on data
distributions and class difficulties.

###  V-D Communication Efficiency

TABLE VII: Communication Efficiency. / means the method fails to achieve the same performance as FedAvg. Bold numbers indicate the best results. Method | ICH | ISIC
---|---|---
Rounds | Speedup | Rounds | Speedup
FedAvg [5] | 175 | 1.00 | 195 | 1.00
FedProx [24] | 42 | 4.17 | 110 | 1.77
MOON [7] | / | / | / | /
FedProc [16] | / | / | / | /
FedFocal [60] | / | / | / | /
FedRS [18] | 196 | 0.89 | 194 | 1.01
FedLC [17] | / | / | / | /
CReFF [8] | 33 | 5.30 | 107 | 1.82
FedIIC (Ours) | 20 | 8.75 | 78 | 2.50

Communication efficiency is one of the most significant metrics to evaluate an
FL algorithm as it measures how fast an algorithm converges. Following [7],
the communication efficiency of each algorithm is defined as the number of
training rounds when it achieves the same performance as FedAvg. Specifically,
for ICH and ISIC, as the best model of FedAvg is selected based on the
validation sets, the performance of each algorithm at round $r$ is evaluated
by the BACC score on the corresponding validation sets.

Quantitative results of communication efficiency are summarized in Table VII.
Across all settings, FedIIC achieves the same performance as FedAvg with the
fewest training rounds, proving that FedIIC is more communication-efficient
than other FL frameworks. One inspiring and promising observation is that
FedIIC reaches the performance of FedAvg with only 20 and 78 training rounds
on the two real medical datasets, accelerating the convergence of FedAvg by
8.75x and 2.50x respectively. The above results further demonstrate the value
of FedIIC in real FL applications.

###  V-E Privacy

Compared to FedAvg, FedIIC requires each client to upload its data size
$N^{y}_{r}$ and the CE loss $\overline{l}_{CE}(y)$ at each round $r$. In this
subsection, we will prove that FedIIC is privacy-preserving under the existing
secure multi-party computation framework. This problem equals to “How can we
compute the sum of a set of numbers without revealing any of them?” which can
be solved by homomorphic encryption (HE) [49]. HE can be simplified as

$D(E(a)\cdot E(b))=a+b,$ (13)

where $a$ and $b$ are two numbers and $D$ and $E$ represent the decryption and
encryption operations respectively. In FedIIC, each client transmits its
encrypted number (e.g. $E(a)$) to the server and the server multiplies
clients’ ciphertexts without decryption due to the lack of keys and sends the
results (e.g. $E(a)\cdot E(b)$) to clients. Then, each client decrypts the
downloaded results (e.g. $D(E(a)\cdot E(b))=a+b$). In this way, either the
server or the clients in FedIIC would have no access to clients’ privacy.

##  VI Conclusion

In this paper, we, for the first time, identify and formulate a special and
realistic imbalanced class distribution in federated learning, denoted as L2
distribution, and propose a novel framework named FedIIC accordingly. The key
idea behind FedIIC is to calibrate both the feature extractor and the
classification head against divergence caused by learning from imbalanced
data. Specifically, we calibrate the feature extractor by both intra- and
inter-client contrastive learning and thus produce a uniform embedding space
for all classes across clients. To calibrate the classification head, a
difficulty-aware logit adjustment algorithm is deployed to produce balanced
decision boundaries for all classes. Extensive experimental results
demonstrate the effectiveness of FedIIC against the state-of-the-art federated
imbalanced learning approaches. Furthermore, FedIIC is proven to be effective
in dealing with data imbalance under other widely-studied distributions. We
believe the newly-identified imbalanced distribution, together with FedIIC,
would inspire future federated imbalanced learning work in realistic
scenarios.

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CHAPTER: DECLARATION OF AUTHORSHIP

I, Dominic Hosler, declare that the work presented in this thesis, except where otherwise stated, is based on my own research and has not been submitted previously for a degree in this or any other university.
Parts of the work reported in this thesis have been published as follows:

§ PUBLICATIONS

* Dominic Hosler, Carsten van de Bruck and Pieter Kok. “Information gap for classical and quantum communication in a Schwarzschild spacetime.” In: Physical Review A 85.4 (2012-04-24), page 042312.

* Eduardo Martín-Martínez, Dominic Hosler and Miguel Montero. “Fundamental limitations to information transfer in accelerated frames.” In: Physical Review A 86.6 (2012-12-07), page 062307.

* Dominic Hosler and Pieter Kok. “Parameter estimation over a relativistic quantum channel.” (2013-06-13) arXiv: 1306.3144. Submitted to: Physical Review A.

CHAPTER: ABSTRACT

In this Ph.D. thesis, I investigate the communication abilities of non-inertial observers and the precision to which they can measure parametrized states.
I introduce relativistic quantum field theory with field quantisation, and the definition and transformations of mode functions in Minkowski, Schwarzschild and Rindler spaces.
I introduce information theory by discussing the nature of information, defining the entropic information measures, and highlighting the differences between classical and quantum information.
I review the field of relativistic quantum information.
We investigate the communication abilities of an inertial observer to a relativistic observer hovering above a Schwarzschild black hole, using the Rindler approximation.
We compare both classical communication and quantum entanglement generation of the state merging protocol, for both the single and dual rail encodings.
We find that while classical communication remains finite right up to the horizon, the quantum entanglement generation tends to zero.
We investigate the observers' abilities to precisely measure the parameter of a state that is communicated between Alice and Rob.
This parameter was encoded to either the amplitudes of a single excitation state or the phase of a NOON state.
With NOON states the dual rail encoding provided greater precision, which is different to the results for the other situations.
The precision was maximum for a particular number of excitations in the NOON state.
We calculated the bipartite communication for Alice-Rob and Alice-AntiRob beyond the single mode approximation.
Rob and AntiRob are causally disconnected counter-accelerating observers.
We found that Alice must choose in advance with whom, Rob or AntiRob she wants to create entanglement using a particular setup.
She could communicate classically to both.

CHAPTER: COPYRIGHT NOTICE

Dominic Hosler (2013)

This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.

The author, Dominic Hosler, reserves his moral right to be identified as the author of this work.
This is reflected in the attribution clause of the applied license.

CHAPTER: ACKNOWLEDGEMENTS

Firstly I would like to thank my supervisor, Pieter Kok for all the support and direction he has given me throughout my studies.
I have really enjoyed working with someone who has such an enthusiasm for physics.
We have shared many interesting discussions.
It is safe to say I would not have achieved all that I have without him.

I have been fortunate enough during my studies towards my Ph.D. to work with some really talented physicists.
Carsten van de Bruck, with whom I collaborated when working on communication near a black hole.
During my trip to Madrid, I was hosted by Juan León with whom I worked on interesting photon localization problems.
The same trip started a collaboration with Eduardo Martín-Martínez and Miguel Montero, I am grateful to be able to work with both of them.

My thanks also goes to my friends and co-workers in the office, who have all supported me.
These were, Carlos Pérez-Delgado, Marcin Zwierz, Samantha Walker, Mark Pearce, Sam Coveney, Michael Woodhouse and Matt Hewitt.
We have shared many in-depth discussions about related and non-related subjects.
They also provided plenty of welcome relief from study.

I would like to thank my family.
My wife Sascha for her continual support while I was studying, her understanding for the nature of my chosen career path and her encouragement when I hit stumbling blocks.
She is always there for me, to listen and help with anything she can.
My daughter Caoimhe for being a source of inspiration and motivation.

I would like to thank my uncle, Akin Kumoluyi for making the possibility of getting a Ph.D. seem more achievable.
I remember his Ph.D. graduation ceremony when I was about 7, it was then I decided that I too could get a Ph.D. if I chose.
I would like to extend gratitude to my grandparents, Mary and Charles Hosler for loaning me their spare room to use as an office whilst writing up.

Finally, I wish to express sincere gratitude to my parents Adam and Sandra Hosler, who dedicated a lot of their time to home educating me and making me believe I could achieve anything.
Without them I would not be where I am today.

For the publication “Information gap for classical and quantum communication in a Schwarzschild spacetime”, Pieter Kok, Carsten van de Bruck and I would like to thank M. Wilde for valuable comments on the manuscript, and J. Dunningham and V. Palge for stimulating discussions.

For the publication “Parameter estimation using NOON states over a relativistic quantum channel”, Pieter Kok, and I would like to thank Michael Skotiniotis for helpful discussions.

For the publication “Fundamental limitations to information transfer in accelerated frames”, Eduardo Martín-Martínez, Miguel Montero and I would like to thank Pieter Kok for useful discussions and Mark Wilde for helpful suggestions.
During the research and preparation of this manuscript, I visited Madrid and I would like to thank, IFF, CSIC, Madrid for their hospitality.
I also acknowledge funding support for the trip from a Santander Research Mobility Award.

The stick people used in a number of my figures are inspired by characters from the web-comic xkcd[http://xkcd.com].

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