Federated Partially Supervised Learning for Decentralized Medical Images

Keywords: enhance :::: the :::: data ::::::::::: government ::::::::: situation :: in ::::::::::::: privacy-critical :::::::::::: infrastructure, :::::::::::: decentralized :::::::: learning ::::: has :::::::: played ::: an :::::::: the :::::::: medical :::::::: domain. ::: As :: a ::::::::: trade-off, :::::::::::: FPSL, ::::: namely, ::::::: federated :::::::: partially :::::::::: supervised :::::::: learning ::::::::: multi-label :::::::::::: classification. ::: To :::::: tackle :::: label :::::::: scarcity ::: and ::::: class ::::::::: imbalance, :: two :::::: major ::::::::::: challenges ::: of :::::: FPSL, we propose a simple yet robust FPSL framework, FedPSL and ::: a ::::::::::::::: prototype-based ::::::::::: multi-label ::::::::: classifier. ::: We ::::::: further :::::::: propose :: a :::::: local ::::::::::: consistency ::::::::::::: regularization ::::::: training :::::: module ::: to :::::::: generate :::::::::::::: high-confidence ::::::: pseudo ::::: labels ::: as ::::: label :::::::: scarcity :::: and :::::: class :::::::::: imbalance. The findings of this study :::: also : pose a new research direction towards label-efficient learning on medical images Partially supervised learning, federated learning, multi-label classification

limit the effectiveness of partially supervised learningdue to the lack of access to decentralized data, leading to a bottleneck for label-efficient learning. As a remedy, this work formulates and discusses a new learning problem federated partially supervised learning (FPSL) for decentralized medical images with partial labels. We illustrate the negative impact of decentralized partially labeled data on neural networks and the necessity of studying FPSL via a set of multi-label thoracic disease classification tasks. Motivated by the empirical observations in this paper, :: 

I. INTRODUCTION

Fig. 1: Illustration of FPSL for a multi-label classification task on chest X-ray images. In each client (data node), each image is partially labeled for only one thoracic disease, e.g. we only know whether each image in the first data node has infiltration but have no knowledge on the other three diseases. Due to data regulations, only model weights and the metadata (e.g. statistics) of the local data can be communicated between each data node and the parameter server (see Sec. ?? :: IV : for a formal description). The goal of FPSL is to utilize four partially labeled datasets stored in different data nodes to train the model of interest in the parameter server. scarcity and high annotation cost. The problem of PSL, also known as the missing annotations problem [?] or partial labels problem [?] in the literature, is a family of learning tasks where the training data are partially labeled. Unlike commonly seen labeled data and unlabeled data, the definition of partially labeled data is associated with multi-task learning :::::: 
(MTL) [?]: given a task of interest that can be decomposed into multiple sub-tasks, an instance is only annotated for a subset of sub-tasks. In the medical domain, the problem of PSL commonly arises from the collection of multiple datasets from different sources for the task of interest, where each dataset is annotated for a specific sub-task as the annotation process usually requires relevant expertise. This makes all these datasets partially labeled when the task of interest includes all these sub-tasks. 
As the datasets are acquired from different sources (e.g. different hospitals), the partially labeled datasets might be stored separately in different locations without direct connections. Thus, it is natural to think about the connection between PSL and federated learning (FL) [?]. FL is a learning paradigm that aims to utilize decentralized data stored separately in different places and has become a topic of active research in medical image analysis [?], [?], [?], [?], [?], [?]. In this work, we extend the problem formulation of PSL to a federated setup and formulate federated partially supervised learning (FPSL) for medical images. As one of the core contributions of this work, a formal problem definition of FPSL is provided in Sec. IV. To the best of our knowledge, to :::: take ::::: into ::::::: account ::: is :::: the :::: data ::::::::::: regulations. Here, data regulations refer to situations where the data stored in the client are not allowed to be transferred to the server or other clients. A detailed explanation is given in Sec. ?? :: 
IV. These regulations might be made and supervised by either the data holder or even the government (e.g. EU General Data Protection Regulation [?] and US Health Insurance Portability and Accountability Act [?]). With this new constraint, some of the previous PSL methods will be infeasible to implement. For example, [?] utilizes partial labels to generate vicinal labels for supervised learning (SL), thus can only be implemented as centralized training. 
A key problem that FPSLaims to address is the overfitting caused by :: 
As ::: the ::::: first ::::: study ::: in :::::: FPSL, :: a ::::::: primary ::::: goal :: in ::: 
this ::::: work :: is ::: to ::::::::::: demonstrate ::: the :::::::: negative :::::: impact :::::: caused ::: by ::::::::::: 
decentralized ::::::: partially ::::::: labeled ::::: data. ::::::::::: Specifically, ::: we :::: aim :: to :::::: 
provide ::: an ::::::::: empirical :::::::::::: understanding ::: on ::: the ::::::: effects :: of :: label scarcity and non-IID data, which has not been addressed in previous PSL studies [?], [?], [?], [?]. In particular, the learning process can be easily dominated by a few major classes when class imbalanceexists [?], :::: class :::::::: imbalance ::::: under :: 
the ::::::::: federated ::::: setup, :::::: which :::: both ::: can ::::: lead :: to ::::::::: overfitting. ::::: Here, :: 
the ::::: term ::::: "label ::::::: scarcity" ::::: refers :: to ::: the :::::::: situation ::: that :::: only :::::: limited ::::: partial :::::: labels ::: are ::::::::: available :: in :::: the ::::::: clients. :::: The ::::: term ::::: "class ::::::::: 
imbalance" :::: has ::: two ::::::::: meanings: : the classes with more partial labels will contribute more to the learned representations, which will lead to overfitting. One such :: can ::::::::: dominate ::: the We theoretically and empirically analyze the advantages of this module over FedAVG [?], a seminal robust baseline for non-IID data ::: and :::: thus :::::::: improve ::: the ::: FL ::::::::::: performance. 
As the first study of FPSL on decentralized medical images, we evidence the contributions of this work by assessing FedPSL in a simulated exemplar PSL task, :::: task, :::::::: federated partially supervised multi-label classification. We evaluate FedPSL against strong baselines in terms of both performance and robustness under various data challenges such as class imbalance and data scarcity :::: label ::::::: scarcity :::: and :::: class ::::::::: imbalance. The empirical results show that FedPSL can consistently outperform the baseline methods and can be used as a robust framework for FPSL. 
The contributions can be summarized as follows: 
1) We formulate and discuss for the first time the problem of FPSL for decentralized medical images, and propose FedPSL, a simple and robust FPSL framework for training :::::::: pipeline :::::::: including :: a :::::::::::::: prototype-based : aggregation module, which can compensate for the cross-node class imbalance :::: aims ::: to ::::::: smooth ::: the :::: FL ::::::: process : and improve the overall performance. 4) We show initial evidence that FPSL is an under-explored problem compared with existing learning paradigms and offer the community the first benchmark MLC task under FPSL : of ::::::::: federated ::::::: partially ::::::::: supervised :::::::::: multi-label :::::::::: 
classification, accompanied with a set of performance evaluations and baseline comparisons. 
The rest of this paper is organized as follows. Sec. II reviews the relevant literature for FL and PSL and Sec. IV formally formulates FPSL, the problem of interest. Sec. V describes the proposed solution in detail. Sec. VI describes the proposed benchmark tasks and provides experimental results and analysis. Section VII summarizes this work. 

II. RELATED WORK


A. Federated Learning

As an emerging research area, there are limited FL studies related to this work. Three related areas are federated unsupervised representation learning (FURL) [?], federated positiveunlabeld (PU) learning [?], and federated semi-supervised learning (FSSL) [?]. FedU [?] discuss a divergence-aware update mechanism for FURL. Different from FedPSL, the divergence-aware module of FedU only considers the local updates instead of global aggregation. As a federated extension of PU learning, FedAwS [?] shares a similar problem formulation as FPSL by assuming that each client only has access to labels of one class. However, FedAwS is designed for multi-class classification only. That is to say, each client will have both fully labeled and unlabeled data, and thus differs from the partial labels problem discussed in this work. As semi-SL has been successfully applied to PSL, FSSL is another related domain to FPSL. FSSL method FedMatch [?], for instance, also adopts a CR-based pseudo-labeling training strategy. However, compared to FedPSL, FedMatch has a completely different problem formulation, which enables the usage of fully labeled data, and does not utilize the link between instance discrimination-based self -SL and CR. 

B. Partially Supervised Learning

Recently, there have been efforts made to utilize multiple partially labeled datasets in the medical domain. [?] requires a fully-labeled dataset to learn the prior to perform expectation-maximization iterations. [?], [?] both rely on complex neural networks for centralized training . [?] proposes a marginal loss and an exclusion loss for multi-class segmentation while a fully-labeled dataset is required in the training. These methods do not directly solve the data scarcity problem and still rely on large amounts of data (even the fully labeled ones). As a comparison, VLUU [?] addresses the label scarcity issue by generating vicinal labels based on human structure similarity. However, VLUU can only work in a centralized training environment. As all these ::::::: 
However, sub-dataset. : For a better illustration of PSL, a common task is presented below as a concrete example. 
2) Multi-Label Classification: As a generalization of multiclass classification, a multi-label classification (MLC) task could be interpreted as C binary classification tasks. In contrast to multi-class classification, the classes in MLC are not mutually exclusive, i.e. each image instance could belong to more than one category at the same time. For example, a chest X-ray image could be diagnosed as cardiomegaly and emphysema simultaneously. Mathematically, given the input image space X , F = {f : X → R C } is a family of functions of interest. For partially supervised multi-label classification :::: [?], each sub-dataset is only annotated for a true subset of C. Here, :: we ::: use :: ⊂ ::::: instead :: of :: = ::::: because :: x :::: might ::: not ::::: contain :: all :::::: classes. 

A. Federated Partially Supervised Learning

IV. ::::::::: PROBLEM ::::::: SETUP Now, we formulate the problem of federated partially supervised learning ::::::::: multi-label :::::::::::: classification(FPSL). Without loss of generality, we consider the most representative case where K = C, each data node (client) only has partial labels for a unique class. Let : , :: an ::::::::: exemplar :::: task :: of :::::: FPSL. datasets :::::::: {U k } K k=1::::: might :::: also ::: be ::::::: available :: in ::::: each ::::: client. : Given a model of interest f θ and an independent fully labeled target dataset T = {(x t i , y t i )} nt i=1 that is unseen during the training, the learning outcome is to find the optimal parameter set θ that minimizes the estimated empirical risk: 
Rθ = 1 n t nt i L(f θ (x t i ), y t i ), (1) 
where L(•, •) is the loss function. 
In this work, we consider a seminal FL setup, where each data node (client) is only connected to a master node (server). The master node does not store any clinical data and could be implemented as a parameter server (PS) [?]. In addition to the standard setup of FL, the data regulations in the medical domain pose a new constraint: the transferring of clinical data between the master node and data nodes is prohibited. That is to say, only model weights and metadata : 
(e.g. :::::::: statistics :: of :::: 
data) [?] should be communicated across nodes. In contrast to the common data privacy issues in FL, the data holders are prevented from exchanging user data in any form due to regulations. For example, a hospital might not be allowed to upload the patients' data stored in its server to another institute. With this new constraint, FPSL on medical images is more challenging than a simple integration of FL and PSL. 
It is worth mentioning that the primary goal of this work is to formulate FPSL on medical images. Thus, the discussion on federated communication and privacy-preserving techniques is beyond the scope of this work. 

V. METHOD

In this section, we present the proposed FedPSL framework, which consists of two stages. The first stage is a federated unsupervised pre-training stage (Sec. V-A). ::: We ::: use framework-agnostic : . :: In ::::: other :::::: words, :::: any self -SL framework, as the backbone framework in each data node. We choose SimSiam for the following two advantages: (i) SimSiam does not require a large batch size [?] or a dynamic dictionary [?] for unbiased estimation [?]; and (ii) SimSiam does not require an additional copy of model weights ( [?]). Thus, SimSiam allows for a lightweight implementation. SimSiam formulates the pretext task based on instance discrimination, by maximizing the similarity between two different views of the same instance. Given an input image x and two augmentations τ ∼ χ, τ ∼ χ that are sampled from an image augmentation distribution χ. τ (x) and τ (x) can be generated as two different views of x and used as the inputs for the same network, separately. As the original data augmentation policy in [?] is not suitable for medical images, we follow the data augmentation policy proposed in [?]. 
As the data nodes share the same network architecture, we use θ to denote the parameters θ k in node k. SimSiam has two networks: an encoder f θ and a predictor h θ3 . Two augmented views τ (x) and τ (x) are both processed by f θ . However, the predictor h θ is applied on one view and a stop-gradient operation [?], [?] is applied on the other view. Concretely, let p 1 = h θ • f θ (τ (x)) and z 2 = f θ (τ (x)). The pretext task is to minimize the negative cosine similarity between the embeddings of the two views: 
L sim (τ (x), τ (x)) = - p 1 p 1 z 2 z 2 ,
where • is the l 2 -norm. Following a similar definition, a symmetrized loss [?], [?] can be defined as 
L θ = 1 2 L sim (τ (x), τ (x)) + 1 2 L sim (τ (x), τ (x)).
The overall framework for unsupervised pre-training is illustrated in Fig. ). {θ k } K k=1 are aggregated into θ 0 in the PS and the data nodes are fully synchronized with the PS. Given the metadata of {S k } K k=1 , we simply aggregate {θ k } K k=1 by FedAvg [?], : : : ::: 
θ 0 = k n k n n k k n k :::::: θ k , (2) 
f θ k :: is ::::::::: initialized ::: by : the federated unsupervised pre-training described in Sec. V-A. In addition to ::: Let ::: g φ :: be ::: the ::::::::: multi-label ::::::: 
classifier ::: on ::: top ::: of : the feature extractor , we introduce ::: f θ k . 
:::: 
From :: a :::: MTL :::::::::: perspective, ::: g φ ::: can ::: be :::::::::: decomposed :::: into C binary predictors {g φ k,j } C j=1 for C, :::::::: classifier. ::::::::: Motivated :: by :::: this, ::: we image :: x ::: can ::: be ::::::::: formulated ::: as :::::::::::::::::::::: for ::::: class : c ::: in ::::: client :: k is performed by minimizing the binary cross-entropy loss 
g φ k • f θ k = φ k T • f θ k (x).
L c k = - 1 |S c k | x∈S c k y c p c (x) + (1 -y c ) log(1 -p c (x)),
::::::::::::::::::::::::::::::::::::::::::: 
( 
) 5 
where y k i denotes the label of x i :: y c :::::: 
denotes :::: the :::::: ground :::: truth :: 
or :::::: pseudo ::::::: partial ::::: label ::: of :: x : with respect to class k and 
p k (τ (x)) = g φ k,k • f θ k (τ (x i ))
ŷjc = 1, if p c (τ (x)) ≥ α 0, if p c (τ (x)) ≤ 1 -α , (6) 
where p j (τ (x)) = g φ k,j • f θ k (τ (x i )) and 0 < α < 1 is a threshold value. The images with undefined pseudo labels (i.e. 1 -α < p j (τ (x)) < α ::::::::::::::::::: 1 -α < p c (τ (x)) < α) will not be used in the training. As shown in [?], [?], α should be large :::::: 
enough : to generate a pseudo label with high confidence or low uncertainty (e.g. α = 0.95 in this work following [?]). 
Analogous to Eq. , we minimize 
L pseudo = - 1 n k j i y j i p j (τ (x))+ (1 -y j i ) log(1 -p j (τ (x))).
The final optimization goal is to minimize the sum of Eq. and Eq. 
L φ = L sup + λL pseudo ,
where λ is a hyperparameter which controls the weight of L pseudo (we set λ = 1 following [?]). The overall framework for CR training in each data node is illustrated in Fig. ??. 
We will describe the proposed global aggregation module in Sec. ?? ::::: Then, ::: we :::: aim ::: to ::::::::: maximize ::: the :::::::::: agreement ::::::: between :: 
the ::::::::: prediction :::::: vector ::: on ::::: τ (x) :::: and ::: the ::::::: pseudo :::: label ::: of :::: τ (x) 
:::::: 
∀c / ∈ C k .
2) Necessity of Two-Stage Training ::::: Global ::::::::::: Aggregation: 
Again, assume only class k is annotated for node k. Without the federated unsupervised pre-training stage, the learning process could be easily dominated by the term L sup when jointly optimizing θ and φ via minimizing Eq. , especially in the early stage of the training . With a large value of α, there will be much fewer pseudo labels than ground truth labels. This situation will inevitably lead to f θ k being overfitted to class k and further influence the training for {g φ k,j } j =k in node k. Given the limited partial labels, where the majority of labels will be negative, f θ k will learn less meaningful representations ( f θ is overfitted to the negative cases) for the downstream tasks. Moreover, for non-IID datasets stored in different nodes, there could be domain shifts across data nodes ( f θj is overfitted to node j) :: At ::: the :::: end :: of ::::: each :::: local and :::::::: metadata. Thus, we utilize the first federated unsupervised pre-training stage to mitigate overfitting. One might argue that the federated mechanism (to be introduced ::::::::: exchanging :::::::: {φ k } K k=0:::::::: between :::: the :::: PS :::: and ::: the ::::: 
clients ::::: does ::: not :::::: violate ::: the :::: data :::::::::: regulations in Sec. ??) could alleviate the overfitting by aggregating knowledge from other nodes. Note, however, when the node is trained with local data independently, the overfitting still exists. In addition, relying on federated aggregation alone is inefficient for two reasons. First, the updates from other nodes are delayed updates [?] ( real-time updates). Second, the communication costs in FL are non-trivial. 
3) Global Aggregation: The training in the second stage consists of a warm-up phase (Phase I) and a regular phase (Phase II), which have different global aggregation methods. 
In Phase I, only the partial labels are utilized for supervised training in each data node, only φ k,k is trained for node ::: 
IV. ::: We :::: use ::: two :::::::: different :::::::::: aggregation ::::::::::: mechanisms :: θ :::: and :: φ. 

::

As ::: θ k ::::::::: represents ::: the ::::::: general :::::::::: knowledge :::::: learned ::::: from ::::: client kfollowing : , ::: we :::::: simply :::::::: aggregate :::::::: {θ k } K k=1 :::: with Eq. . This can be understood as warm-up training to learn reasonable priors for the prediction heads. This design is motivated by the fact that FPSL tasks commonly do not have fully labeled images ( semi-SL [?]) and are more prone to overfitting caused by the class imbalance in each datanode : 2 :::::: where : 
n k = c∈C |S c k |.
::::::::::: 
:: 
In ::::::: contrast :: to :::: θ k , ::: φ k :: is :::::: prone :: to ::: be :::::::::: influenced ::: by :::: local ::: 
data. Note, because each node only has partially labeled data, minimizing Eq. at the beginning will not only learn unreliable prediction heads for classes with no labelsbut also introduce noise to f θ . This design distinguishes this work from standard CR. In the warm-up training, the aggregation of the encoder f θ simply follows Eq. . For the prediction heads, we set φ 0,j = φ j,j , for class j ∈ C : φ :::: can :: be ::::::::::: decomposed :::: into :::::::: prototype ::::::: vectors: 
:::::::::::: 
φ k = {µ c k } C c=1
. :::: The :::::: quality ::: of :: φ :: is ::::::::: determined ::: by ::: the :::::: quality :: 
of :::::::: prototype :::::::: vectors, :::::: which :: is ::::::: further ::::::::: influenced ::: by ::::: local We define the divergence-aware weight matrix A as: 
A j,m = nj,m φ t j,m -φ t-1 0,m 2 2 k n k,m φ t k,m -φ t-1 0,m 2 2 
, where n j,m denotes the number of ground truth or pseudo labels of class m in node j, • 2 2 is the Frobenius norm that measures the weight divergence, φ t j,m denotes the weights Algorithm 1 Federated Consistency Regularization Training Stage. We use t to denote the t th round of federated :::: Local training . Each round can consist of a few local epochs. T w is the number of warm-up rounds. T is the total number of rounds. We use φ t k to denote the model weights stored in node k at the t th round. The update and aggregation of θ are detailed in Sec. ??, thus omitted here :::::::: procedure : for simplicity :::: client :: k. Update :: θ k ::: by :::::::::: descending :::::: ∇ θ k L k : 
Input: φ 0 , {S k } K k=1 ,
Eq. 5 return :: θ, :: φ of the prediction head for class m in node j by the end of the t th round. For the prediction head of class m in the PS, we aggregate the weights of prediction heads {φ k,m } k by: φ 0,m = k A k,m φ k,m . In contrast to classification. :::::: Thus, :: it ::: is :::::::: unstable :: to ::::::::: minimize : Eq. , we aggregate the model weights {θ k } k by : 
θ 0 = k m A k,m m,k A k,m θ k .
Eq. reflects the principle of maximum entropy as 
m A k,m m,k A k,m = m A k,m /C k m A k,m /C
, where each class is assigned with equal weights 1 C . Notably, Eq. is described for general cases. In practice, the calculation of Eq. can be significantly simplified if the binary prediction heads share parameters ( if using a shared MLP as the binary prediction heads, in :: an ::::::::: alternative ::::::: fashion, : illustrated in Algorithm 2 : 1. 
φ t k,m -φ t-1 0,m 2 

4) Theoretical Analysis on Divergence-Aware Aggregation

Module: As a comparison, let us take a closer look at FedAVG, which is known for robustness against non-IID data. Let φ 0 be the model weights stored in the PS and {φ k } K k=1 be the model weights stored in each data node to be aggregated. For class j, we have φ 0,j = K k=1 n k n φ k,j . Assume that node m = j does not have any examples for class j ( caused by class imbalance), the generated positive pseudo labels ( "1" in Eq. ) will be noisy labels [?]. Note m = j implies that node m does not have ground truth for class j. In this case, φ m,j will have decreased performance due to noisy information and n k n can not properly reflect the weight of φ m,j (say n m is large) :::::::: However, :: at ::: the :::: is ::::::::: undefined. Thus, FedAvg might not be an optimal solution for FPSL. 
After discussing the limitation of FedAVG, we analyze the training dynamics of the divergence-aware aggregation module (DAAM). First, we assume that the weight divergences for all data nodes are at the same level, there is no outlier (either too large or too small) among the data nodes. At the beginning of training, for class m, node m will have more labels than node j = m which only have limited pseudo labels due to uncertainty.Thus, n j,m should be smaller than n m,m and :::: Meanwhile, ::::::::: {S c k } c∈C k :::: can :::: only ::::::: generate ::::::::: {µ c k } c∈C k ::: by ::: Eq. :: 3, i.e. φ t m,m should be assigned a larger weight than φ t j,m . An extreme case is n j,m = 0, where the weight for φ t j,m is 0, no contribution. When the federated training proceeds, more pseudo labels will be generated in node j = m and a larger weight will be assigned to φ t j,m accordingly. By neglecting the weight divergences, :::: there ::: are ::::::: missing ::::::::: prototypes :: in :::: each ::::: client. 

VI. EXPERIMENTS

The purposes of the conducted experiments are twofold ::::::: threefold. Firstly, we aim to illustrate that FPSL is an under-explored yet challenging problem compared with standard FL and centralized training. Secondly, we want to discuss several initial solutions to FPSL. :::::: Thirdly, :: 
we ::::: want ::: to :::::::::: demonstrate :::: the ::::::::: robustness ::: of :::::::: FedPSL ::::: against :::: for k = 1, 2 Upload {θ k,t } K k=1 to PS to get θ 0,t Aggregate with Eq. 2 and Eq. Upload {φ k,t } K k=1 to PS to get φ 0,t Aggregate with Eq. 2 and Eq. 8 labeled data. For the ::::::: methods :::::::: including :: unsupervised pretrainingstage : , ::::: there ::: are :::: 200 ::::::: epochs :: of ::::::::::: pre-training :::: with ::: the ::::::: federated ::::::: version ::: of :::::::: SimSiam ::: [?] : . ::::::::: Following :::: [?], we use a standard stochastic gradient descent optimizer with momentum of 0.9 . The ::: 
and ::: the : initial learning rate is 0.05 with cosine annealing, following [?]. For CR-based methods, the . :::: The warm-up period is T w = 30 :: 
for :::::::: FedPSL : , :::::: FedCR : , :::::::: FedCR+, ::: 
and :::::::: FixMatch. 

VII. CONCLUSION

In this paper, we formulate and discuss a new problem FPSL :::::::: federated ::::::: partially :::::::::: supervised ::::::: learning ::::::: (FPSL) : for decentralized partially labeled medical images. We also present FedPSL, a simple yet robust solution to FPSL. We are the first to leverage self -SL to mitigate :::: label ::::::: scarcity :::: and : class imbalance in FPSL. We further propose a CR-based local training module to mitigate label scarcity and a divergence-aware ::::::::::::: prototype-based : global aggregation module to make up for the cross-site class imbalance :::: avoid ::::: large :::::: weight :::::::::: divergence. Finally, we provide an empirical understanding of FPSL and our results indicate a new research direction in label-efficient learning with partial supervision. 

F

  UELED by the advances in deep learn- ing research, partially supervised learning (PSL) [?], [?], [?], [?], [?], [?], [?], [?] :::::::::::::::::::::::::::::::: [?], [?], [?], [?], [?], [?], [?], [?], [?] has become an emerging research direction for label-efficient learning on medical images, considering the practical issues such as data 


:

  :::::: learning :::::: process :::: and ::: for ::::: each ::::: class, ::::: there ::: are ::::: more ::::::: negative :::::::: examples :::: than ::::::: positive ::::::::: examples. ::::::: Without :::: loss :: of ::::::::: generality, ::we :::::::: illustrate :::::: FPSL :::: with :::::::::: multi-label ::::::::::: classification :::::::: (MLC), : a representative task prone to overfitting in a federated setupis multi-label classification (MLC), which we use to illustrate the challenge of FPSL with affordable computational resources. MLC is a :::::::::: fundamental ::: yet : challenging task as it does not have mutually exclusive classes( : . :: In ::::::: contrast :: to ::::::::: multi-class ::::::: settings, we can not utilize the constraint of mutually exclusive classes as prior knowledge in either loss formulation [?] ::::::: [?], [?] or data augmentation [?]) and normally suffers from the class imbalance caused by long-tailed distributions.To fill the aforementioned methodological gaps, we present a simple yet robust FPSL framework FedPSL. : A ::::: direct ::::::::::: consequence ::: of ::::: label :::::::: scarcity :::: and ::::: class ::::::::: imbalance : is :::::::::: overfitting. ::::: With ::::::: limited :::::: labels, :: it ::: is ::::::: difficult :: to ::::: learn :: a ::::: robust :::::: feature :::::::: extractor ::: or ::::: learn :: to :::::::: generate ::::::: reliable :::::: pseudo ::::: labels ::::::: directly. ::::: From :::: the :::::::::: perspective ::: of ::::: MTL, :::: the ::::::: learning :::::: process :::: can ::::: also ::: be :::::::::: dominated ::: by :: a ::::: few :::::::: sub-task :::: with ::::::: relatively :::::: more :::::: labels. ::::::::: Moreover, :::: the :::::: design ::: of ::::::::: multi-label ::::::: classifier :::::: should ::::: also :::::: comply ::: to ::: the :::: FL ::::: setup. ::: To :::: this :::: end, FedPSL consists of two training stages, which are designed to mitigate label scarcity efficiently :::::: utilizes :: a ::::::::::::: prototype-based ::::::::: multi-label :::::::: classifier, :::::: where : a :::::::::::: mathematical :::: link ::: can ::: be :::: built ::::::: between ::: the ::::::::: prototypes :::: and ::: the ::::::: weights ::: of : a :::::::: standard ::::: linear ::::::: classifier. ::: In :::::::: contrast :: to :: a :::::::: standard ::::::::::: classification ::::: layer :::: [?] : , : a :::::::::::::: prototype-based :::::::::::: classification ::::: layer ::::: [?] : is ::::: more :::::: robust :::::: against ::::: label ::::::: scarcity :::: and ::::: class :::::::::: imbalance. :::::: There ::: are :::: two :::::: training :::::: stages ::::: when ::::::::: optimizing ::: the ::::::: feature :::::::: extractor ::: and ::: the :::::::: prototypes. The first training stage is a federated unsupervised pre-training stage, where we leverage self-supervised learning (self -SL) to learn robust and transferable class-agnostic representations from unlabeled data (i.e. in the first training stage, all partially labeled data are deemed as unlabeled data). As unsupervised pre-training has been demonstrated to achieve competitive performance compared to supervised pre-training in representation learning on benchmark tasks [?] , we use self-SL to alleviate overfitting caused by class imbalance :::::::::: Specifically, ::: we :::::::: pre-train :::: the ::::::: feature :::::::: extractor :: of :: the :::::::: classifier ::: of :::::: interest. The second training stage is a novel federated consistency regularization (CR) :::::: partially ::::::::: supervised training stage, which consists of two modules to facilitate the federated training with partial labels. The first module is a local training module that generates pseudo labels based on CR. CR [?] ::::::::: consistency ::::::::::::: regularization :::: [?], :::::: which : was originally proposed to address the semi-supervised multi-class classification, which aims to enforce an agreement between the distributions of the pseudo labels generated for two augmented views of the same unlabeled image. In this work, we adapt CR for the partially supervised MLC ::::::::: consistency :::::::::::: regularization :: for :: the :::::::: partially ::::::::: supervised :::::::::: multi-label ::::::::::: classification :::: [?] task for the first time. Without any fully labeled data, the proposed local module learns to produce pseudo partial labels with high confidence based on the knowledge learned from local ground truth partial labels and knowledge transferred from the other data nodes. It is worth mentioning that the CR training here is easy to implement as it shares a similar workflow with the unsupervised pre-training in the first stage, which is illustrated in Fig. ??. :::::: clients. ::: We :::: also :::::::: leverage :: an ::::::::: alternative :::::: training ::::::: strategy :::::::: between ::: the :::::: feature :::::::: extractor ::: and ::::::::: prototypes :: in :::: each ::::: client :: to :::::: ensure :::::: smooth ::::::: updates. : The second module is a divergence-aware aggregation moduledesigned to attenuate cross-site class imbalance in FL. Unlike the standard setup of FL ::::::::::::: prototype-based :::::::::: aggregation ::::::: module, where the weight for each local model is decided by the relative size of data in each client, both the amount of effective partial labels and the :::::::::: aggregation :::::::::: mechanism :: is :::::: based :: on :::: the ::::::: distance :::::: metric ::::::: between ::: the :::: local ::::::::: prototypes ::: and ::: the :::::: global ::::::::: prototypes :: to :::: avoid :::: large weight divergence [?] between the local model and global model in each client are considered in the aggregation step. 


: 2 )

 2 :::::::: framework ::: for :::::::: federated ::::::: partially ::::::::: supervised : multi-label classification. We empirically demonstrate that federated unsupervised pre-training is a promising direction for FPSL.3) We propose a novel federated CR training pipeline for FPSL including a divergence-aware :::::: partially :::::::::: supervised :::::: 


:

  ::: none :: of ::::: these :::::::: methods ::: are :::::::: designed ::: for ::: the ::::::: situation :::: that ::: the ::::::: partially :::::: labeled :::::::: datasets ::: are :::::::::::: decentralized. :::::::: [?], [?] :::::: address ::the :::::: partial :::::: labels ::::: issue :: by :::::::::: generating :::::: vicinal ::::: labels :::::: based :: on ::::: human :::::::: structure ::::::::: similarity, :::::: which :::: can :::: only ::: be ::::::::::: implemented :: in : a :::::::::: centralized ::::::: training ::::::::::: environment. :::::::: [?], [?] ::: both ::::::: require : a :::: fully :::::: labeled :::::: dataset ::: in ::: the :::::: training :::::::: process. : It :: is :::: less ::::::: practical :: to :::::: assume :::: that :::: fully ::::::: labeled :::: data ::: are :::::::: available :: in ::::: each ::::: client, ::: and :::: only :::::: having :::: one ::::: client :: or :: a ::: few :::::: clients ::::: with :::: fully :::::: labeled ::: data :::: will ::::::::: inevitably :::::: impair ::: the ::::::: learning ::::::: process :: in ::::::: contrast :: to ::::::::: centralized ::::::: training. :::::::: Besides, :::: PSL ::::::: methods :::: that ::: are :::::: based :: on :::: label :::::::::: propagation :::: [?] :::: have ::::::: iterated ::::::: training ::::::::: procedure, ::::: which ::: not :::: only ::::::: increase ::: the ::::::::: complexity :: of : a :::::::: federated ::::::::::::: implementation ::: but :::: also :::: lead :: to ::::::::::: sub-optimal ::::::::::: performance. ::: A ::::::: practical ::::: issue ::: that :: is ::::: often ::::::: ignored :: in :::: the ::::::: medical :::::: domain ::: is ::: that ::::: there ::: are :::: only :::::: limited ::::::: labeled :::: data :::::::: available. ::: In :::: this ::::: work, ::: we :::::: denote :: the :::::::: situation :::: that ::::: only :::::: limited :::::: partial :::::: labels ::: are :::::::: available :: in :::: each ::::: client :: as ::::: label ::::::: scarcity : . ::::: With :::::::::: small-scale :::: local ::::::: training ::: data ::: in ::::: each :::::: client, ::::: PSL :::::::: methods ::::: with :::::::: complex ::::::: network :::::::::: architectures :: or ::::::: training ::::::::: procedures :::::::: [?], [?] ::::::: normally ::::::: perform :::: much :::::: worse :: in :: a :::::::: federated ::::::::::: environment :::: than ::: the :::::::::: counterparts ::: that :::: have :::::: access :: to ::: the ::::::::: large-scale ::::::: training :::: data :: in : a ::::::::: centralized :::::::::: environment :::: [?] : . ::: As ::::::: existing : PSL methods struggle in the problem formulation defined in Sec. ?? :: IV), it is meaningful to develop a FPSL framework :::: study ::: the ::::::: problem ::: of ::::: FPSL ::: and :::::: develop :: a :::::: robust ::::::: solution. III. PROBLEM SETUP :::::::::::::: PRELIMINARIES A. Preliminaries 1) Partially Supervised Learning: Before we formulate the problem of federated partially supervised learning, we briefly review partially supervised learning (PSL). Given a task of interest, suppose there are C > 1 classes of interest indexed by the set C. Let x denote an image instance, y full denote the corresponding complete label of x with the label set S(y full ) ⊂ C, where S(•) is a set operation. 2 Analogous to y full , we define the incomplete label or partial label of x as y part with the label set S(y part ). Here, we require |S(y part )| = ∅ and |S(y part )| ⊆ |S(y full )| :::::::::::::::: |S(y part )| ⊂ |S(y full )|, i.e. 0 < |S(y part )| < |S(y full )|, where | • | is the cardinality. For simplicity, we use y to denote the partial label in the remainder of the paper. Without loss of generality, we assume that the partially labeled dataset S = {(x i , y i )} |S| i=1 can be split into C : K sub-datasets where each sub-dataset contains label information of only one class : a ::: few ::::::: classes, i.e. S = C k S k , where :::::::::: S = K k S k . :::::: Here, : S k = {(x k i , y k i )} |S k | i=1 denotes the partially labeled dataset for class k ∈ C. Note, the assumption here is the most representative case as all other cases are trivial extensions [?]. For example , each partially labeled image could have more than one class annotated. :: in ::: the ::: k th ::::::::: sub-dataset :::: and ::: y k i :: is :::: the :::::: partial ::::: label ::: of ::: the :::::::: example ::: x k i :::: with :::::::::::::: S(y k i ) = C k ⊂ C :::::: where ::: C k :: is ::: the ::::: class ::: set :::: for ::: the ::: k th ::::::::: 


:

  :::::::: Analogous :: to :::: Sec. ::::: III-.1, ::: we ::::: have :: K :::::: clients ::::: (data :::::: nodes) :: in : a ::::::: federated ::::::: system ::: and : S k = {(x k i , y k i )} n k i=1 denote the :::::: denotes :: the :::::::: partially ::::::: labeled : dataset stored in node k ≤ K, where n k = |S k | and y k i is the partial label of x k i with respect to class k ∈ C ::: the ::: k th ::::: client. Following standard practice in FL, we assume S k ∩ S l = ∅ for k = l and {S k } K k=1 are all non-IID data. :: In ::::::: addition ::: to ::: the :::::::: partially :::::: labeled ::::: data, :::::::: unlabeled :::::: 




  self :::-SL :: to ::::: learn :::::::::: meaningful :::: and :::::::::: transferable ::::::::::::: representations ::::: across :::::: clients ::::::: without :::: any ::::::: labels, :::::: which ::: can ::: be ::::::::: efficiently :::: used :: to :::::::: mitigate ::: the ::::: label ::::::: scarcity :::: and :::: class :::::::::: imbalance ::: and ::: 


1 C

 1 also ::::::: provide : a :::::: strong ::::::::::: initialization ::: for ::: the :::::: second ::::: stage. ::::: Given ::the :::::::::: pre-trained :::::: feature :::::::: extractor :::: and ::::::::: prototypes ::::::::: extracted :: by ::the ::::::: feature ::::::::: extractor, ::: an :::::: initial :::::::::::::: prototype-based ::::::::: multi-label ::::::: classifier :::: can ::: be :::::: built. : The second stage is a federated consistency regularization ::::::: partially ::::::::: supervised : training stage, which consists of a local consistency regularization training module (Sec. ?? :::: V-B.1) and a divergence-aware ::::: novel ::::: global aggregation module (Sec. ??). Unsupervised Pre-Training. The first stage is based on self-supervised learning via an instance discrimination pretext task. While both the encoder (green) and predictor MLP (light green) are trained in the first stage, only f θ will be leveraged in the next training stage. Consistency Regularization Training. The second stage is based on semi-supervised learning. The figure only illustrates the self-training procedure with pseudo labels. Illustration of local training modules for (a) the federated unsupervised pre-training stage (Sec. V-A) and (b) the federated consistency regularization training stage (Sec. V-B). Both training stages leverage two-view self-training to maximize mutual agreement between two views. :::::: V-B.2). :::: The ::::: local ::::::: training :::::: module :::: can :::::: utilize :::::: ground :::: truth :::::: partial ::::::: labels :::: and ::::::: further :::::::: generate :::::::::::::: high-confidence :::::: pseudo :::::: partial :::::: labels :: to :::::::: fine-tune :::: the :::::: feature :::::::: extractor :::: and ::::::: improve ::: the ::::::: quality ::: of :::::::::: prototypes. :::: The :::::: global :::::::::: aggregation :::::: module :: is :::::::: designed ::: to ::::: avoid ::::: large :::::: weight ::::::::: divergence ::::: [?] : to ::::::: stabilize ::: the ::::::: training ::::::: process. ::: An ::::::::: end-to-end ::::::: training ::::::: pipeline : is ::::::::: presented :: in :::: Sec. :::::: V-B.3. : A. Stage 1: Federated Unsupervised Pre-TrainingIn contrast to standard SL, PSL has sparser annotations given the same amount of training data , for a multi-label classification task formulated under Sec. ??, there are only of annotations compared to the fully labeled counterpart. ::::: Under ::: the :::::::: federated ::::: setup, ::: the :::::::::::::: decentralization ::: will ::::::::: exacerbate :: the ::::: class ::::::::: imbalance :::::: caused ::: by ::: the ::::::: different ::::::: number ::: of ::::: partial ::::: labels :: in ::::: each ::::: client. :::::: Thus, : a :::::::::::::: straight-forward ::::: local ::::::: partially ::::::::: supervised ::::::: training ::::: might ::: fail ::: to ::::: grasp ::: the ::::::: features ::: in ::::: regard :: to :::::::: unlabeled ::::::: classes :: in :::: that :::::: client. ::: As ::: we ::::::: assume :::: that :::: each :::: client ::::: only ::: has :: a :::: few :::::: classes ::::::: labeled, :::: this ::: can :::: lead ::: to ::::: severe :::::::: overfitting ::: for ::::: other ::::::: clients. :: So :::: far, ::::::::::: unsupervised ::::::::::: pre-training :::: has ::::: been ::::::::::: demonstrated :: to :::::: achieve ::::::::::: competitive ::::::::::: performance ::::::::: compared :: to ::::::::: supervised ::::::::: pre-training :::: in :::::::::::::: representation :::::::: learning :::: on ::::::::::: benchmark :::: tasks :::::::: [?], [?]. : To maximally utilize the available training dataand mitigate the overfitting caused by limited partial labels, :::::: either :::::::: partially ::::::: labeled ::: or ::::::::: unlabeled, we leverage self -SL to learn transferable class-agnostic representations across data nodes during the first stage. 1) Local Training: We use SimSiam [?], a state-of-the-art ::In ::: the :::: first :::::: stage, ::: all :::::::: available ::::: data :::: will ::: be :::::::::: considered :: as ::::::::unlabeled :: as ::: no ::::: labels :::: will :: be :::::::: involved :: in ::: the ::::::: training. ::: As ::: the ::::focus ::: of :::: this ::::: work :: is :::::: FPSL, ::: we :::::: simply ::::: make ::: the :::: first ::::: stage :::::::::::::::: 


??. 1 )

 1 Global Aggregation: The model of interest is f θ and its parameter set. At the beginning of the federated training , ::: -SL ::::::::: framework :::::: should ::: be :::::::: feasible. :: In :::: each :::::::: federated ::::::: training ::::: round, ::: let :::::::: {θ k } K k=1::: be :::: the :::::: model ::::::: weights :::::::: uploaded ::: to : the global model weights θ 0 are randomly initialized in the PS and :: PS :::: by : K copies of θ 0 are distributed to each data node as {θ k } K k=1 , the data nodes are fully synchronized with the PS . Then, node k updates ::::: clients, :::::: where : θ k by training on S k independently. After the same number of local epochs, : is ::::::: updated :: by :::::::::: performing :::::: locally ::: self ::: -SL : (e.g. ::::::: instance :::::::::::: discrimination :::::::::::::: [?], [?], [?], [?] : 




  where n = k n k .The federated unsupervised pre-training stage is depicted in Algorithm 1. The process ::::::::::::: n k = |S k ∪ U k | :: is ::: the ::::: count of local training and global aggregation is repeated for a pre-defined number of rounds (T ) ::::::::examples. Note, only model parameters {θ k } K k=0 and metadata are exchanged between the PS and the data nodes ::the :::::::: metadata :::::::: {n k } K k=1:::: will ::: be :::::::: uploaded :: to ::: the ::: PS ::::along ::::: with ::: the :::::: model :::::: weights. Input: θ 0 0 , {S k } K k=1 , T Output: θ T 0 Federated Unsupervised Pre-Training Stage. We use t to denote the t th round of federated training. Each round consists of a few local epochs. T is the total number of rounds. We use θ t k to denote the model weights stored in node k at the t th round. B. Stage 2: Federated Consistency Regularization ::::::: Partially ::::::::::: Supervised Training Similar to Sec. V-A, the federated consistency regularization (CR) training stage also has a local training module and a global aggregation module. We will present the second stage in four ::: The :::::: second :::::: stage :: is :::::::: presented ::: in ::::: three : steps. Firstly, in :: we :::::::: describe : the local training module , we adapt CR training to PSL in Sec. ??. Secondly, we discuss the necessity of two-stage training in Sec. ??, as a complement to Sec. ??. Thirdly, we introduce the :::: from ::: the :::::::::: perspective :: of ::::: MTL. ::::::::: Secondly, ::: we :::::::: propose :: a ::::: novel : global aggregation module for the second stage in Sec. ??, including a novel divergence-aware method. Fourthly, we theoretically analyze the mechanism of the proposed divergence-aware aggregation modulein Sec. ??. ::::: based ::: on ::::::::: prototypes. ::::::: Thirdly, ::: we ::::::: propose :: an ::::::::: alternative ::::::: training ::::::: strategy ::::: which :::: can :::::: further ::::::: improve ::: the :::: local ::::::: training ::::::: module. : 1) Consistency Regularization :::: Local ::::::: Training with Partial Labels: Given node k and S k , only class k (or task k from a multi-task learning perspective [?]) is annotated. Let f θ k be the feature extractor inherited from ::::: stored :: in ::: the :::: k th :::::client. 


:

  :::::: construct :: a ::::::::::::: prototype-based :::::::::: multi-label :::::::: classifier. ::::: Given ::::: image ::x, :::::: f θ k (x) :: is :: the :::::::: extracted :::::: feature :::::: vector. :::::: Then, :: for ::::: class ::::: c ∈ C, :: we :::::: define ::: the :::::::: prototype ::::: [?] : of ::::: class :: c :: in ::::: client :: k :: as : ::::::::::::::::::::::(3)where g φ k,j denotes the binary prediction head for class j ∈ C in node ::::::::::::: norm(v) = v v ::is :: a :::::::::::: normalization :::::::: operator ::: and ::: S c k :::::: denotes ::: the ::: set :: of :::::: images :::: that ::: are :::::: labeled ::: as :::: class :: c. :: If :::::: c ∈ C k , i.e. :::::: client : k . For example, for a multi-label classification problem, g φ k,j • f θ k is a binary classifier for :: has :::::: partial ::::: labels :: for : class j :: c, ::: the ::::: initial :: S c k:::: just ::::::: contains ::: the :::::: images :::: with :::::: ground :::: truth :::::: partial ::::: labels ::::: with :::::: respect :: to :: c. ::: We ::: use :::: the :::: unit ::::: vector :::::::: following ::: the :::::::: principle :: of ::::::::: maximum ::::::: entropy :::: [?] :::::: because ::: we :::: have :: no ::::: prior ::::::::: knowledge :: on ::::: class :: c. :: S c k:: is ::::::: updated :::::::::: dynamically :: as :: it ::: can ::::: also :::::: contain ::::::: images ::::: with :::: high :::::::::: confidence :::::: pseudo ::::: labels :::: with :::::: respect ::: to :: c, ::::: which :::: will ::: be :::::::: discussed :::::: below. : For class k, the supervised training ::: Let ::::::::: φ ∈ R d×C :::::: denote :: the ::::::: weight :::::: matrix :: of :::: the ::::: linear :::::::: classifier ::: g φ , :::::: where :: d :: is ::: the :::::::: dimension ::: of :::: the :::::: feature ::::::: vector. :::: The :::::: model :::::::::: prediction :: of ::::: 


Fig. 2 :

 2 Fig. 2: ::::::::: Consistency ::::::::::: Regularization :::::::: Training. 




  is the predicted probability of τ (x) for : c. : :: As ::::: each ::::: client :::: has ::::: only ::::::: partially ::::::: labeled ::::: data, ::: S c k ::::: could :: be ::::: small :::: for ::::: some ::::::: c ∈ C k ::: or :::::: empty :::::::: ∀c / ∈ C k . ::: In ::::::: addition :: to ::::::: utilizing :::::::::: prototypes :::::::::: downloaded ::::: from :::: the :::: PS, :: it :: is :::: also :::::::: beneficial :: to ::::::: generate ::::::: pseudo ::::: labels :: to ::::::: enlarge ::: S c k :: to ::::::: improve :: the ::::::: quality :: of :::::::::: prototypes. ::::: After :::::::: defining ::: the :::::::::: prototypes ::: and :::::::::: optimization ::::::: object, ::: we ::::: now :::: get ::::: back :: to ::::: how ::: to :::::: update ::: S c k . :::::::::::: Theoretically, :::: any ::::: semi ::: -SL ::::::: methods ::::: that :::: can ::::::: generate :::: high ::::::::: confidence ::::::: pseudo :::::: labels :::::: could :::::: work. ::: In :::: this ::::: work, :: we :::::: adopt :::::::::: consistency :::::::::::: regularization :::::: training :::: [?] : . ::: We ::::: adapt :::::::: FixMatch :::: [?] : , :: a ::::::::::::: state-of-the-art :::::::: method, ::: to ::: the :::::::: partially ::::::::: supervised ::::::::: multi-label ::::::::::: classification ::::::: problem ::: for ::: the :::: first :::: time. ::: For class k. For class j = k, we :::::: c / ∈ C k , ::: the :::::: image : x :: in ::::: client :: k :::: does ::: not :::: have :::::: ground ::::: truth ::::: partial :::::: labels ::: for ::::: class :: c. ::: We : first generate two augmented views τ (x) and τ (x) for each image x. Similar to Sec. ??, we aim to maximize the agreement between the predictions between two views, ::: as :::::: shown :: in :::: Fig. :: 2. The pseudo label for class j : c : is defined as 


:

  ::::: training ::::: round :: t, ::::::: {θ k } K k=1:::: and :::::::: {φ k } K k=1 :: are :::::::: uploaded :: to ::: the ::: PS. : It :: is :::::: worth :::::::::: mentioning ::: that :::::::: {φ k } K k=1::: are ::::: both ::::: model ::::::: weights ::: 


:

  :::: partial :::::: labels. :: It :: is :::::: highly ::::::: possible :::: that ::: the :::::::::: prototypes :: of ::: the :::: same ::::: class :::::: exhibit ::::: large ::::::::: divergence :::::: across ::::::: different :::::: clients. :: It : is ::::::: critical :: to :::::: ensure ::: the ::::::: stability ::::: when ::::::: updating ::: the ::::::::: prototypes in the PS.During Phase II, both ground truth labels and pseudo labels are used in the training following ::::Take :::: class :: c :: as ::: an ::::::: example. ::: Let :::: µ c 0,t ::: be ::: the :::::: global ::::::::: prototype ::: of :::: class :: c ::::: kept ::::: from ::: last :::::: training :::::: round : t :::: and ::::: µ c 0,t+1 ::: be ::: the ::::::: updated :::::::: prototype ::: of :::: class : c :: to ::: be ::::::::::: synchronized :: to ::: K ::::: clients ::: at ::: the :::::::: beginning ::: of :::::: training ::::: round ::::: t + 1. ::: We ::::::::: aggregate :::::::: {µ c k } K k=1 :::: with : Eq. . We propose a divergence-aware aggregation module (DAAM) based on two motivations. First, given a class, the weight assignment should reflect the contributions of the number of effective labels used in the training. Here, the effective label denotes the ground truth or pseudo partial label for a specific class used in :: 2: : n k = exp sim(µ c 0,t ,µ c k ) j exp sim(µ c 0,t ,µ c j ) , :::::::::::::::::::: (8) ::::: where ::::::: sim(•, •) ::: is :::::::: similarity :::::::: measure : (e.g. :::::: cosine ::::::::: similarity) ::::::: between ::: two ::::::: vectors. ::::::: sim(•, •) ::: can ::: be ::::::::: interpreted :: as ::::::::: measuring :: the ::::::: weight :::::::::: divergence :::::::: [?], [?]. : Eq. . Second, the weight assignment should also take the weight divergence caused by cross-site class imbalance into consideration. Intuitively, a class (or node) with a larger weight divergence implies the model learns more knowledge for this class (or node). Thus, we want to assign a small weight to the model with large weight divergence to balance the learning process. : 8 :::::: implies ::: that ::: the :::: local ::::::::: prototype :::: with :::::: smaller :::::: weight ::::::::: divergence :::::: against :: the :::::: global ::::::::: prototype :::::: should :: be :::::::: assigned :::::: larger :::::: weight :: in ::: the :::::global ::::::::::: aggregation, ::::: which ::::::: ensures : a ::::::: smooth :::::: update ::::::: process. : 


2 :

 2 T w , T :: S c k : ::::::: Labeled :::::: image :: set ::: of :::: class : c : Output: φ T 0 : C: :::: Set :: of :::::: classes : :: E: ::::::: Number ::: of :::::: epochs : 1: function CLIENT UPDATE(θ k , φ k , C) for t = 1, 2, • • • , E do 3: : // :::: With ::: φ k ::::: fixed 4::::::: 


3 )

 3 :::::: Training ::::::: Strategy: :::: Note, ::: the :::::::::: prototypes :: φ :: is ::::::::: dependent ::on :: θ ::: and :: φ :::::: should ::: be :::: fixed ::::: when :::::::::: performing ::::::::::::: prototype-based ::::::::::: 


2

  shares the same value ∀m. An overview of the second stage is : 5 :: by :::::::: updating : θ ::: and :: φ ::::::::::::: simultaneously. :: In :::: each :::::: client, ::: the ::::::: training :: is :::::::: conducted :: 




  early ::::: phase ::: of :::::::: federated ::::::: partially ::::::::: supervised :::::::: training, :: as :::: each :::: client :::: only :::: has ::::::: partially :::::: labeled :::: data, ::: S c k:::: will :: be ::::: empty ::::::: ∀c / ∈ C k ::: and ::: µ c k :: 




  there :: is :: a ::::::::: warm-up ::::::: training :::::: phase :::::: where :::: each :::: client :::: first :::::: trains :::: with ::::: local :::::: ground ::::: truth ::::: partial :::::: labels :: in ::: the ::::::::: supervised ::::::: fashion. ::: By :::::::: replacing ::::::: {S c k } c∈C :::: with ::::::::: {S c k } c∈C k , ::: we ::: can :::::: acquire ::: the ::::: local :::::: training ::::::::: procedure :: in ::: the :::::::: warm-up ::::: phase.::::::::: 


:

  scheme ::: for :::::::: federated ::::::: partially ::::::::: supervised ::::::: training :: is :::::::: illustrated :: in ::::::::: Algorithm :: 2. 




  label ::::::: scarcity :::: and :::::: class :::::::::: imbalance. : We use a multi-label classification (MLC) task on chest X-ray images (CXRs) to evaluate the proposed framework. The labels for a MLC task are usually sparse (e.g. 60% of CXRs in ChestX-ray14 [?] have no findings of thoracic diseases), which makes ::::::: federated partially supervised MLC even more difficult. 


Algorithm 2 :

 2 :::::: Training ::::::: scheme ::: for :::::::: FedPSL. 


S

  k : ::::::: Partially ::::::: labeled :::: data :: in ::::: client :: k : ::: U k : ::::::::: Unlabeled :::: data :: in ::::: client :: k : :::: θ 0,0 : ::::::::: Pre-trained :::::: model ::::::: weights : :::: θ k,t : ::::: Model ::::::: weights ::: of ::::: client :: k :: at ::: the :::: end :: of ::::: round :: t :::::::::::::: φ k,0 = {µ c k } c∈C k : :::::: Initial ::::::::: prototypes ::: of ::::: client :: k :::::::: extracted :: by :::: θ 0,0 : ::::::::::::: φ k,t = {µ c k } c∈C : :::::::::: Prototypes :: of ::::: client : k :: at ::: the :::: end :: of ::::: round : t ::: T w : ::::::: Number ::: of :::::::: warm-up :::::: training ::::::: rounds :: T : ::::::: Number ::: of ::::::: training :::::: rounds 1: for t = 1, 2, • • • , T w do Warm-up 2: 


7

 7 


Fig. 3 :

 3 Fig. 3: Label statistics of four partially labeled training sets :: in ::: four ::::: clients (i.e. : S 1 -S 4 )and : . ::: "No ::::::: Finding" :::::: means ::: that : a fully labeled test set (T ) in two experiments :::: CXR :::::: contains ::::: none :: of :::: eight :::::: diseases : of :::::: interest. 




  , • • • , K do k,t , φ k,t ← Client Update(θ k,t , φ k,t , C k ) 5:Upload {θ k,t } K k=1 and {φ k,Tw } K k=1 to PS to get θ 0,t and φ 0,Tw Aggregate with Eq. 2 and Eq. 7 


	3:	θ k,t ← θ 0,t-1	Synchronize with PS
	4:		

θ 6: for t = T w + 1, T w + 2, • • • , T do 7: for k = 1, 2, • • • , K do 8: θ k,t , φ k,t ← θ 0,t ,φ 0,t Synchronize with PS 9: θ k,t , φ k,t ← Client Update(θ k,t , φ k,t , C) 10: 
We use the terms "self-supervised learning" and "self-supervised representation learning" interchangeably in this work. 

2 For simplicity, we use FL to denote the federated learning task with standard supervised loss in this paper, unless stated otherwise. 

In fact, the encoder here consists of a feature extractor and a projector. The feature extractor is the model of interest ( ResNet18 [?]). The projector is a multi-layer perceptron (MLP) for dimensionality reduction. The predictor is another MLP. After the unsupervised pre-training, only the feature extractor is kept, both the projector and the predictor are discarded. The interested reader is referred to [?] for further details. 

The experiments with more advanced architectures are considered out of the scope of this work. 

A. Data Preparation

We use the public dataset ChestX-ray14 3 [?] provided by the National Institute of Health (NIH). To facilitate reproducibility , we use the ::: 
and ::::::: preserve :::: the ::::::: non-IID ::::: data :::::::::: distribution :: of :: 
the :::::::: original ::::::: dataset, ::: we :::: use :::: the : default data partitions of NIH. We utilize the original label distributions to reflect the label scarcity and class imbalance of non-IID datasets, as illustrated in Fig. 3. We use the first batch as the test set T , which contains 4999 CXRs. We use the next four batches to create four data nodes and each data node contains 10000 CXRs ::::::: partitions ::: of :::: NIH. 3 https://nihcc.app.box.com/v/ChestXray-NIHCC/ 

B. Experimental Setup

Given the available computing resources, we set C = K = 4 :::::: K = 4, :: i.e. :::::: there :::: are ::::: four ::::::: clients :: in our experiments. We first create four partially labeled data nodes following Sec. ??. We make two sets of thoracic diseases for two sets of experiments. The first set contains ::: use ::: the :::: four :::::: batches :::::: (batch :: 2 : -:::::: batch :: 5) ::: of :::::::::::: ChestX-ray14 :: to :::::: create :::: four ::::: clients :: (i.e. :: K :: = ::: 4) ::::: where ::::: each ::::: client ::::::: contains :::::: 10000 :::::: CXRs. To :::::::: simulate ::: the ::::: label ::::::: scarcity :::: and :::: class :::::::::: imbalance ::::: across :::::: clients, :::: we ::::::::: synthesize :::: the ::::::::: partially ::::::: labeled :::::::: datasets ::: by :::::::: As the class distributions are extremely imbalanced, we consider two evaluation metrics in this study following [?], [?] . The first one is the F1-score, which is the harmonic mean of precision and recall. Without any prior knowledge, we set the threshold as 0.5 for all methods for a fair comparison. Note, in practice, the thresholds for different classes could be selected empirically, which is beyond the scope of this study. The second one is ::: We :::::: follow ::::::::: [?], [?] ::: 
and :::::: choose area under receiver operating characteristic (AUROC) : 
as ::: the :::::::: evaluation :::::: metric :: in ::: this ::::: work. Note, AUROC does not specify the threshold, :::::: unlike :::::::: precision, :::::: recall, :: or :::::::: F1-score, : and is thus preferred in our quantitative comparison. We report the mean over three runs with different random seeds. We select the best performance in each run based on the highest average AUROC. 

C. Baselines

The choice of baseline methods gives consideration to two aspects. First, we want to provide an empirical understanding of FPSL. Second, we want to examine the performance of the seminal methods from existing learning paradigms such as SL, semi-SL, and self -SL, when addressing FPSL. 
We first compare FedPSL against three :: six : robust FPSL baselines, denoted as the first group (G1). 
• FedAVG is an adaptation of FedAVG [?] to FPSL. 
To differentiate from FedAVG, we use FedAVG to denote the method. FedAVG is a seminal method that has robust performance in FL. In the data nodes, we only update the weights given the partial labels in the back-propagation ::::::::::::: backpropagation. Note, this is equivalent to standard SLas we only minimize Eq. . In the PS, only the shared weights are aggregated and synchronized. FedSSP : . The second group (G2) includes four non-FPSL baselines, which are four methods with ::::::: includes ::::: three :::::::::: centralized :::::::: baselines, ::::: where :::: the corresponding constraints in Sec. ?? :: IV :: are : relaxed (i.e. centralized training is feasible or full labels are available). We include these four ::: two :::::::: baseline :::: PSL : methods to provide an empirical understanding of the negative impact of FPSL ::: 
and ::: and ::::::::: supervised ::::::: Oracle :::: with ::: full :::::: labels. 
• M1 is standard SL with centralized data and partial labels, where all partially labeled datasets are trained in a centralized setting. This is the centralized training counterpart of FedAVG :::: IML ::::: [?] : is :::: the :::::::: simplest :::: PSL ::::::: method, :::::: which ::::::: simply :::::: ignores :::::::: missing :::::: labels, i.e. :::: only ::::::::::::: 
backpropagating :::: the ::::::::: gradients ::::::::::::: corresponding ::: to :::: the ::::: partial ::::: labels. • M2 is FixMatch with centralized data and partial labels ::::::: FixMatch ::: is : a :::::::::: centralized :::: semi ::: -SL :::::::: method, ::::: which : is ::::::: adapted :: to :::: PSL ::: for ::: the ::::: same :::::: reason :: of :::::: FedCR. We use the same set of hyperparameters of FedCR. This is ::: also the centralized training counterpart of FedCR. • M3 is standard SL with decentralized data and full labels :::::::::: MixUp-PME ::::: [?] : is ::::::::: centralized :::: PSL ::::::: method ::::: based :: 
on ::::: data :::::::::::: augmentation :::: and ::::::: pseudo :::::::: labeling. We use Oracle is standard SL with centralized data and full labels. This should be the best performance the model of interest can achieve under standard SL, which is also considered as the upper bound performance for centralized data. 224 × 224 image, and µ and σ are the mean and standard deviation of the pixels of x. In the testing process, each CXR is ::: also : resized to 224 × 224, followed by instance normalization. We use the same data augmentation policy proposed in [?] for all methods in the training process. 
2) Network Architecture: All baseline methods use a ResNet18 [?] ::::::::::: DenseNet121 ::::: [?] as the encoder f θ . We choose ResNet18 as ::::::::::: DenseNet121 ::::::::: following ::::: [?] ::: 
and : it is a commonly adopted model in FL [?] for a lightweight experimental setup 7 . Each of the federated methods has K + 1 ResNet18s ::::::::::: 
DenseNet121s : for K nodes ::::: clients :: and the PS, while each of the non-federated methods has one ResNet18 ::::::::::: 
DenseNet121. All models are implemented in Py-Torch on an NVIDIA Tesla V100. For the unsupervised pretraining stage in Sec. V-A, the projector is a 3-layer MLP and the predictor is a 2-layer MLP. The feature dimension of the projector is 512 and the hidden dimension of the predictor is 128. Each fully-connected layer in the MLPs is followed by batch normalization [?] and ReLU [?], except the last layer. In the second stage, each prediction head is a fully-connected layer with hidden dimension 512. ::: we :::::: simply ::: use :::::::: SimSiam :: as : it :::::: allows ::: for :: a ::::::::: lightweight :::::::::::::: implementation. : 
3) Training: For a fair comparison, all networks are initialized with the same random seeds. Note, to provide a comprehensive understanding of the problem, we do not use any additional labeled or unlabeled datasets in the training, as these could lead to impartial conclusions ( domain shift should be taken into consideration if the pre-trained weights come from a different domain). We train all methods for 300 epochs. For methods including unsupervised pre-training, it consists of