Federated Learning from Only Unlabeled Data with Class-conditional-sharing Clients
Authors: Nan Lu
Zhao Wang
Xiaoxiao Li
Gang Niu
Qi Dou
Masashi Sugiyama
Venue: ICLR 2022 Poster (2022)
Keywords: unsupervised federated learning
unlabeled data
class prior shift

Abstract: Supervised federated learning (FL) enables multiple clients to share the trained model without sharing their labeled data. However, potential clients might even be reluctant to label their own data, which could limit the applicability of FL in practice. In this paper, we show the possibility of unsupervised FL whose model is still a classifier for predicting class labels, if the class-prior probabilities are shifted while the class-conditional distributions are shared among the unlabeled data owned by the clients. We propose federation of unsupervised learning (FedUL), where the unlabeled data are transformed into surrogate labeled data for each of the clients, a modified model is trained by supervised FL, and the wanted model is recovered from the modified model. FedUL is a very general solution to unsupervised FL: it is compatible with many supervised FL methods, and the recovery of the wanted model can be theoretically guaranteed as if the data have been labeled. Experiments on benchmark and real-world datasets demonstrate the effectiveness of FedUL. Code is available at https://github.com/lunanbit/FedUL.

Published as a conference paper at ICLR 2022
FEDERATED LEARNING FROM ONLY UNLABELED DATA WITH CLASS-CONDIIONAA--SAARNG CLIENTS
Nan Lu' Zhao Wang"
Xiaoxiao Li
Gang N Qi Dou Masashi Sugiyama
The University of Tokyo The Chinese University of Hong Kong
The University of British Columbia *RIKEN
(lu@edu. sugie @ k. wu-tokyo. ac. jp, (zwang21@cse. q gidoue) cuhk. edu.h. xiaoxiac. Li@ece. ubo. ca, gang. niu. ml @egmagmmll..... com
ABSTRACT
Supervised federated learning (FL) enables multiple clients to share the trained model without sharing their labeled data. However, potential clients might even be reluctant to label their own data, which could limit the applicability of FL in practice. In this paper, we show the possibility of unsupervised FL whose model is still classifier for predicting class labels, if the class-prior probabilities are shifted while the class-conditional distributions are shared among the unlabeled data owned by the clients. We propose federation of unsupervised learning (FedUL), where the unlabeled data are transformed into surrogate labeled data for each the clients, modified model is trained by supervised FL., and the wanted model is recovered from the modified model. FedUL a very general solution to unsuper- vised FL: it is compatible with many supervised FL methods, and the recovery of the wanted model can be theoretically guaranteed as if the data have been labeled. Experiments on benchmark and real-world datasets demonstrate the effectiveness of FedUL. Code is available at https: / /g1 /gggtttuu thub. com/ lunanbit Sereaaniit/Feeee FFedUL.
INTRODUCTION
Federated learning (FL) has received significant attention from both academic and industrial perspec- tives in that can bring together separate data sources and allow multiple clients to train a central model in collaborative but private manner (McMahan et al., 2017; Kairouz et al., 2019; Yang et al., 2019). So far, the majority of FL researches focused on the supervised setting, requiring collected data at each client to be fully labeled. In practice, this may hinder the applicability of FL since manually labeling large-scale training data can be extremely costly, and sometimes may not even be possible for privacy concerns in for example medical diagnosis (Ng et al., 2021).
To promote the applicability of FL, we are interested in challenging unsupervised FL setting where only unlabeled (U) data are available the clients under the condition described blow. Our goal is still to train a classification model that can predict accurate class labels. It is often observed that the clients collect their own data with different temporal and/or spatial patterns, e.g., a hospital (data center) may store patient data every month particular user (mobile device) may take photos at different places. Therefore, we consider a realistic setting that the data at a client come in the form of separate data sets, and the data distribution of each U set and the number of U sets available at each client may vary. Without any labels, it is unclear whether FL can be performed effectively.
In this paper, we show the possibility: the aforementioned unsupervised FL problem be solved if the class-prion probabilities are shifted while the classccontiiional distributions are shared between the available U sets at the clients, and these class priors are known to the clients. Such learning scenario is conceivable in many real-world problems. For example, the hospital may not release the diagnostic labels of patients due to privacy concerns, but the morbidity rates of diseases (corresponding the class priors) may change over time and can be accessible from public medical reports (Croft et al., 2018); moreover, in many cases it possible to estimate the class priors much more cheaply than to collect ground-truth labels (Quadrianto et al., 2009a; Sugiyama et al., 2022).
"Correspondence Gang Niu <gang.niu.ml mwi.ammmmmmmmmlmmmmmm..m..

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Published as a conference paper at ICLR 2022
Server update
Server
Local update
Server
Q1
f2
Q2
fc
Qc
Surrogate label
Surrogate label
Surrogate label
True label
True label
True label
M
M
M
Client
Client2
ClientC
Client
Client
Client
In the left panel, each client fo (c E (C]) has access to fully labeled data, and a global model f can be trained using a supervised FL scheme, e.g., Federated Averaging (FedAvg) (McMahan al., 2017). the right panel, each client has access to only unlabeled (U) data coming in the form of separate sets, where supervised FL methods cannot directly applied. We propose FedUL treats the indexes the sets surrogate labels, transforms the local U datasets to (surrogate) labeled datasets, and formulates a surrogate supervised FL task. Then we modify each client model be compatible with surrogate task by adding a fixed transition layer Q. the original model output fe. FedUI. can incorporate existing supervised FI. methods its hase method for surrogate training, the wanted model f can retrieved from the surrogate model.
Figure Illustration of standard supervised FL. scheme vs. proposed FedUL scheme.
Based on this finding, we propose the federation of unsupervised learning (FedUL) (see Figure 1). In FedUL, the original clients with U data are transformed to surrogate clients with (surrogate) labeled data and then supervised FL. can be applied, which we call the surrogate task. Here, the difficulty is how to infer the wanted model for the original classification task from the model learned by the surrogate task. We solve this problem by bridging the original and surrogate class-posterior probabilities with certain injective transition functions. This can be implemented by adding specifically designed transition layer to the output of each client model, so that the learned model is guaranteed to be good approximation of the original class- posterior probability.
FedUL has many key advantages. On the one hand, FedUL is a very general and flexible solution: works as wrapper that transforms the original clients to the surrogate ones and therefore is compatible with many supervised FL methods, e.g., FedAvg. On the other hand, FedUL. is computationally efficient and casy-to-implement: since the added transformation layer is fixed and determined by
the class priors, FedUL adds no more burden on hyper-parameter tuning and/or optimization. Our contributions are summarized as follows:
Methodologically, propose novel FedUL method that solves certain unsupervised FL problem, expanding the applicability of FL.
Theoretically, we prove the optimal global model learned by FedUL from only U data converges to the optimal global model learned by supervised FL. from labeled data under mild conditions. Empirically, we demonstrate the effectiveness of FedUL on benchmark and real-world datasets.
FL has been extensively studied in the supervised setting, One of the most popular supervised FL paradigms is FedAvg (McMahan et al., 2017) which aggregates the local updates at the server and transmits the averaged model back to local clients. contrast, FL in the unsupervised setting is less explored. Recent studies have shown the possibility of federated clustering with U data (Ghosh et al., 2020; Dennis et al., 2021). However, these clustering based methods rely on geometric or information-thes -hheorctic assumptions build their learning objectives (Chapelle et al., 2006), and are suboptimal for classification goals. Our work is intrinsically different from them in the sense that show the possibility of federated classification with U data based empirical risk minimization (ERM) (Vapnik, 1998), and therefore the optimal model recovery can be guaranteed.
Related Work:
Leamning from multiple U sets has also been studied in classical centralized learning. Mainstream methods include learning with label proportions (LLP) (Quadrianto et al., 2009b) and U classi- fication (Lu et al., 2021). The former leams multiclass classifier from multiple U sets based
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empirical proportion risk minimization (EPRM) (Yu et al., 2014), while the latter learns binary classifier from multiple sets based on ERM. EPRM is inferior to ERM since its learning is not consistent. Yet, it is still unexplored how to leam multiclass classification model from multiple sets based on ERM. To the best of our knowledge, this work is the first attempt to tackle this problem in the FL setup, which is built upon the binary U classification in centralized learning. A full discussion of related work can be found at Appendix B.
STANDARD FEDERATED LEARNING
We begin by reviewing the standard FL. Let us consider K-class classification problem with the feature space X C Rd and the label space
[K], where d is the input dimension and [K] = (1,.... K}. Let x E X' and y E ) be the input and output random variables following an underlying joint distribution with density p(z, y). which can be identified via the class priors === ply k and the class-conditional densities {p(x
k)} Let f X - TR k be classification model that assigns score to each of the K classes for a given input and then outputs
the predicted label by Upred = argmax (())) where (f(x)) the k-th element of f fax).
In the standard FL setup with clients, for c E [C]. the e-th client has access to a labeled training set D. [(05 y (al y) of sample size ne and learns its local model fo by minimizing the risk Ro(fa) = ((a.)))pe(a..)) (((c(())) v)].
(1)
Here, denotes the expectation, R.K x x - IR, proper loss function, e.g., the softmax cross-entropy loss (f=()(())). log E (xp((f((())))) (ff(x))y where 1(-) is the indicator function. In practice, ERM commonly used compute an approximation of Ro(fo) based on the client's local data De by Re(fe: ne Em= ((f((x)).
Ek 1(y kog exp(((()))))))
ex)((f.(())))
The goal of standard FL (MeMahan al., 2017) that the clients collaboratively train global classification model f that generalizes well with respect to p(z, y), without sharing their local data D.. The problem can be formalized as minimizing the aggregated risk:
R(f) =
c=1 (f)
(2)
Typically, employ server to coordinate the iterative distributed training as follows:
in each global round of training, the server broadcasts its current model to all the clients (fa) each client copies the current server model fc = f. performs local step updates
Se Sa - oy V (.,;
where a is the local step-size, and sends fc - f back to the server;
the server aggregates the updates {fc - to form a new server model using FedAvg:
f f f 0g Σ (fe
Ee ff).
(4)
where og the global step-size.
3 FEDERATED LEARNING: NO LABEL NO CRY
Next, we formulate the problem setting of FL with only training sets, propose our method called federation of unsupervised learning (FedUL), and provide the theoretical analysis of FedUL. All proofs are given in Appendix C.
3.1 PROBLEM FORMULATION
In this paper, we consider a challenging setting: instead of fully labeled training data, each client E [C] observes M. (M, K, Ve) sets of unlabeled samples, Ur = { where 4
Note that we do not require each client has the same number of U sets. We only assume the condition M. K, Ve ensure the full column rank matrix TT, which is essential for FL with U training sets.
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a. Me denotes the collection of the m-th U set for client with sample size n..m- Each U set can be seen as a set of data points drawn from a mixture of the original condition densities:
{
Σ k= Te p( (z <'=>"<((>
Una Pe (x)
y k),
where ,. (v = k) denotes the k-th class prior of the m-th U set at client c. These class priors form full column rank matrix II, MRM with the constraint that each row sums up 1. Note that we do not require any labels to be known, only assume the class priors within each of the involved datasets are available, i.e., the training class priors ..t and the test class prior Te p(y k)).
Our goal is the same as standard FL: the C' clients are interested in collaboratively training a global classification model f that generalizes well with respect to p(x: y), but using only U training sets (5) on each local client.
3.2 PROPOSED METHOD
As discussed, FedAvg is a representative approach for aggregating clients in the FL framework. also serves as the basis for many advanced federated aggregation techniques, e.g., Fedprox (Li et al., 2020), FedNova (Wang et al., 2020), and SCAFFOLD (Karimireddy et al., 2020). However, these methods cannot handle clients without labels. To tackle this problem, we consider the construction of surrogate clients by transforming the local datasets into labeled datasets and modifying the local model such that it is compatible with the transformed data for training.
Local Data Transformation First, we transform the data for the clients. For each client c E [C], let be the index of given U sets, and Pe D) be underlying joint distribution for the random variables x E X' and y E [M), where M = max,c(c) [c] M..² By treating ÿ as a surrogate label, we M.
may transform the original U training sets Ue [ Jm= to labeled training set for client c,
U. {x D. (x.
(6)
where n.
EM=_ m= ne..- Obviously, the surrogate class-conditional density P. T m) corresponds to p. (z) in (5), for m E [Mr]. If 3Me < m I M, we pad P. T = m) = 0. The surrogate class prior "." Py = m) can be estimated by ne.,m/
Mc
Based on the data transformation, we then formulate a surrogate supervised FL task: the C' clients aim to collaboratively train a global classification model X + RM that predicts the surrogate label for the given input æ, without sharing their local data 27.. This task can be easily solved by standard supervised FL methods. More specifically, we minimize
1 J(g) = C E Jelg) E (((gg))()))))(((())))
by employing a server that iteratively broadcasts its model to all local clients {ge )g and aggregates local updates from all clients using e.g., FedAvg. The local and global update procedures exactly follow (3) and (4) in standard FL.
Now natural question arises: we infer our desired model from the surrogate model g? To answer it, we study the relationship between the original and surrogate class- caass-oottrioor probabilities.
Theorem 1. For each client [C1, let Th X AM-1 and n : X + AK-1 be the surrogate and original class-posterior probubility functions, where (T) (m (y mm x) for m E [M), (n(x))) = p(y = k x)) for k E [K), -1 and AK-1 are the M- and K K-dimensional simplexes. Let ग.. [T. JMM and | ..... R be vector forms of the surrogate and original class priors, and IT, R M be the matrix form of mikk defined in (5). Then we have
(x) (n((()); π,, Te, Il))
(8)
where the unnormalized version of vector function Q. is given by
(n(x); Te, I1.) D#. II D (n)x).
Da denotes the diagonal matrix with diagonal terms being vector a and - denotes matrix multiplica-
(Q:).
tion. Qo is normalized by the sum of all entries, i.e., (Qo): = E,(9,))
"To make the prohlem well-defined, for elient e with Mc < M, set = V.. < m M.
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Algorithm Federation of unsupervised learning (FedUL)
Server Input:
initial f. global step-size ag. and global communication round R
M. Client c's Input: local model fa, unlabeled training sets Ue - ( 1m class priors II, and , local stee-size ag, and local updating iterations L
1: We start with initializing clients with Procedure A.
2: For r 1 + It rounds, we run Procedure B and Procedure iteratively.
3: procedure A. CLIENTINIT(C)
4: transform Ue to surrogate labeled dataset Uf according to (6)
5: modify f to 9. Qo(f), where Qu computed according Theorem 1
6: end procedure
7: procedure B. CIJENTUPPATE(())
8:
fc 4 f
Receive updated model from SERVEREXECUTE
9:
9. - Qo(fc) do
10:
for 1
11:
9c + 9. aaj (Je(gc:
b SGD update based on objective (7) The update on g. induces update on fc
12:
fe < fc an VJe(Qc(f..;U))
end for
14: send fo f to SERVEREXECUTE
15: end procedure
16: procedure C. SERVEREXECUTE(")
17:
receive local models' updates {fc- ffe from CLIENTPPDATE
18: f -f0-0g 0ig Ee
D FL aggregation
broadcast f to CLIENTUPDATE
20: end procedure
Note that æ,, Te, and II, are all fixed, Qe is deterministic and can be identified with the knowledge of π, Ll and an estimate of π.. In addition, we note that if M, < M, the last M - M, entries of Q. are padded with zero by construction. We further study the property of Qe in the following lemma.
Lemma 2. For each client c E [C], the transition function Qe : (X AK-1) + (X' - AM-1) defined in Theorem I is an injective function.
In this sense, the transition function Qe naturally bridges the class-posterior probability of our desired task (x) and that of the learnable surrogate task (x)) given only U training sets. This motivates embed the estimation of (7)x) into the estimation of (x( at the client side.
Local Model Modification Second, we modify the model for the clients. Let f(x) be the original global model output that estimates (7))) At each client [C), we implement Qe by adding transition layer following fa and then we have ge( Q (f(()))) Based on the local model modification, (7) can be reformulated follows:
J(f) - C
(o() (. (f. 7)
Therefore, local updates on the surrogate model gr naturally yield updates on the underlying model f. Now we can aggregate the surrogate clients by a standard FL scheme. A detailed algorithm, example, is given in Algorithm 1.
0
using FedAvg
g
The relationship elucidating the supervised and sur- rogate FL. paradigms (for a single client) is visually shown in Figure 2. The blue part illustrates a supervised FL scheme: the global model is f(z) parameterized by 0; the loss is R(f; y) in (2) where labels y are observed. However, we cannot observe y and thus employ surrogate FedUL scheme which is shown in green: the global model is gæx) directly modified from f; the loss is [g: y) in (7) where y surrogate labels.
Remarks
Q g
R(f.y)
(g:
Figure 2: Graphical representa- tion of supervised FL (in blue) and FedUL (in green).
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3.3 THEORETICAL ANALYSIS
In what follows, we provide theoretical analysis on the convergence and optimal model recovery for the proposed method.
Convergence of the Algorithm In supervised FI., convergence analysis has been studied with regularity conditions (see Section 3.2.2 in Kairouz al. (2019) for comprehensive survey), e.g., the bounded gradient dissimilarity (Karimireddy et al., 2020) that associates the client model with the server model for non-IID data distributions." In our analysis, we focus on the behaviour of our surrogate learning objective J with the effect from transformation Qc.
Proposition 3 (Theorem V in Karimireddy et al. (2020)), Assume that Jg) in (7) satisfies (A1)-(A3) in Appendix C.3. Denote g" = argminges J(g), where J(g) = ΣE Ee Je(g) 1 ne E ((g(z)) U.). Let the global step-size 0g 2 1 and local step-size ai I 8(1+ B2) FedUL given by Algorithm is expected to have contracting gradient: with the initialized model g F J(g)) J(g') and constant M 0 1 E. in R. rounds, the learned model g" satisfies
=
3MVF /1/3
3B2F R
[ ( 0O
VRLC +
(R+1)
Note that the proposition statement is w.r.t. the convergence of g learned from surrogate task. By model construction, is built from Qc(f). By Lemma 2, the convergence of trained model to the optimal model g" in the function class G implies the convergence of the retrieved model f to the optimal model f = argmin J(f) in the corresponding function class F (induced by G), where the empirical version of (9).
J(f) = Ee J((Q)()) = 1 Ei ((:)) M/)
Optimal Model Recovery In our context, the model is optimal the sense that the minimizer of the surrogate task yf coincide with the minimizer of the supervised task R(f: y). as if all the data have been labeled. Based on the convergence of surrogate training, we show that FedUL can recover this optimal model.
Theorem 4. Assume that the cross-entropy loss is chosen for f, and the model used for g is very flexible, e.g., deep neural networks, that g" E G where g" argming J(g: v)- Let f = frgmin be the optimal classifier learned with FedUL by Algorithm 1, and f = argmin, R(f:y) be the optimal classifier learned with supervised FL. By assumptions in Proposition 3, we have f" === f-
4 EXPERIMENTS
In this section, we conduct experiments to validate the effectiveness of the proposed FedUL method under various testing scenarios. Compared with the baseline methods, FedUL consistently achieves superior results on both benchmark and real-world datasets. The detailed experimental settings and additional supportive results are presented in Appendix E.
4.1 BENCHMARK EXPERIMENTS
Setup: As proof-of-conceptt we first perform experiments on widely adopted benchmarks MNIST (LeCun et al., 1998) and CIFAR (Krizhevsky et al., 2009). We generate sets of training data according (5) for each client E [C]. The total number of training samples
each client is fixed, and the number of instances contained each set indexed by m is also fixed as M. In order to simulate the real world cases, uniformly select class priors n.m.k from range (..1 and then
regularize them formulate valid IT. E TR. M discussed in Section 3.1.
For model training, we use the cross-entropy loss and Adam (Kingma Ba, 2015) optimizer with learning rate of le-4 and train 100 rounds. If specified, our default setting for local update
'The IID case is equivalent to hatch sampling from the same distribution, where the analysis is less interesting associated with our Fl. setting and omit details here.

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Table 1: Quantitative comparison of method with the baseline methods benchmark datasets under IID and Non-IID setting (mean error (std)). The best method (paired t-test at significance level 5%) is highlighted in boldface.
IID Task with 5 Clients
Dataset
Sets
FedPL.
FedLLP
VeT
FedUL.
FedAvg 10%
10
2.12 (0.10)
19.00 (15.90)
20.53 (5.39)
0.78 (0.06)
MNIST
20
2.98 (0.15)
4.38 (3.95)
4,40 (5.11)
1.12 (0.09)
1.79 (0.09)
40
3.90 (0.22)
10.11 (15.74)
11.96 (9.56)
1.00 (0.29)
10
34.60 (0.30)
77.01 (5.02)
77.56 (8.71)
18.56 (0.04)
CIFARIO
20
42.06 (1.63)
78.74 (6.96)
84.18 (5.95)
19.66 (0.30)
32.85 (1.03)
40
46.20 (0.67)
73.75 (4.78)
78.65 (6.91)
20.31 (0.69)
Non-IID Task with 5 Clients
Dataset
Sets
FedPL.
FedLLPP
FedLLP-VAT
FedUL.
FedAvg 10%
10
15.54 (3.48)
20.52 (15.00)
23.39 (15.34)
2.98 (1.72)
MNIST
20
7.54 (2.77)
6.63 (7.94)
11.76 (6.60)
1.77 (0.49)
3.82 (1.56)
40
5.76 (0.90)
5.53 (2.51)
7.82 (10.73)
1.64 (0.18)
10
70.65 (2.38)
81.12 (5.41)
81.55 (7.09)
42.92 (3.02)
CIFARIO
20
63.53 (1.24)
65.51 (13.16)
70,51 (7.66)
35.24 (0.98)
58.62 (8.63)
40
53.78 (0.13)
65.72 (3.93)
60.23 (5.71)
31.43 (1.72)
epochs (E) is 1 and batch size (B) 128 with 5 clients (C) in our FL system. We use LeNet (LeCun et al., 1998) and ResNet18 (He et al., 2016) as backbone for MNIST and CIFARIO, respectively. Here, we investigate two different FL settings: IID and Non-IID, where the overall label distribution across clients is the same in the IID setting, whereas clients have different label distributions in the non-IID setting. Specially, for Non-IID setting, we impose class-prior shift as follows:
the classes are divided into the majority and minority classes, where the fraction of each minority class is less than 0.08 while the fraction of each majority class is in the range [0.15, 0.25];
the training data for every client are sampled from 2 majority classes and 8 minority classes;
the test data are uniformly sampled from all classes.
Baselines: (1) Federated Pseudo Labeling (FedPL.) (Diao et al., 2021): at each client of FedAvg
system, we choose the label with the largest class prior in a U set as the pseudo label for all the data in this set. During local training, these pseudo labels are boosted by Mixup (Zhang et al., 2017) high-confidence data and low-confidence data, where the confidence is given by model predictions. (2) Federated LLP (FedLLP) (Dulac- Arnold et al., 2019): each client model is updated by minimizing the distance between the ground truth label proportion and the predicted label proportion, and we aggregate them using FedAvg. (3) FedLLP with Virtual Adversarial Training (FedLLP- VAT) (Tsai & Lin, 2020): a consistency term is added to FedLLP as regularization for optimizing client model in FedAvg. (4) FedAvg with 10% labeled data (FedAvg 10%): following Chen al. (2020), also compare with supervised learning based on FedAvg using 10% fully labeled data. The detailed baseline settings are depicted in Appendix D.
The experimental results of our method and baselines on two benchmark datasets under IID and Non-IID settings are reported in Table where the mean error (std) over clients based on 3 independent runs are shown. Given the limited number of total data available in the benchmark datasets, we compare our method with baselines in an FL system with five clients and vary the number of U sets M E {10, 20, 40} at each client. In this regard, each set contains a sufficient number of data samples to train the model.
Results:
Our observations are as follows. First, the proposed FedUL method outperforms other baseline methods by large margin under both IID and Non-IID settings. Second, when varying the number of sets M E {10, 20, 40). FedUL not only achieves lower classification error but also shows more stable performance compared to baselines. Third, as we use FedAvg as the default aggregation strategy for all the methods, the performances of FedUL, FedPL, and FedAvg all degrade under the inter-client NoD-IID setting, as expected. The eπror rates of FedLLP and FedLLP- VAT are high and unstable, which demonstrates that the EPRM based methods are inferior as discussed in Section 1.
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20
20
20
Cassitcatio emor 18)
10
Cinse (%)
Clavviltaatinn (S3)
-.6
12
1.6
15
64
Notie rario
Local updating (c)
(a)
Figure 3: Ablation studies on MNIST under Non- system. (a) Study on the robustness of our method with noisy class priors. (b) Effects of the number of accessible Non-IID U data. (c) Analysis of local updating epochs. (d) Effects of training batch size.
4.2 ABLATION STUDIES
In the following, we present comprehensive investigation on the ablation studies of the proposed FedUL method on MNIST benchmark.
To investigate the robustness of our method, we further apply FedUL in a noisy environment. Specifically, we perturb the class priors n. by a random noise y, and let the method treat the noisy class prior T mm Te ((2e - I)y + +1, where 7 uniform randomly take values in [0, 1] and E 0.2, 0.4, 0.8, 1.6), as the true me m. during the whole learning process. Note that we tailor the noisy * if surpasses the range. We conduct experiments with five clients and ten U sets at each client. The experimental setup is the same before except for the replacement of Th From Figure 3()), we can see that FedUL performs stably even with some challenging noise perturbations (see when noise ratio > 0).
Robustness with Noisy Class Priors:
Effects of the Number of Accessible Non-IID U Data: The purpose FL is utilizing more data to empower model performance. While how unlabeled data, especially unlabeled Non-IID data, can contribute to FL model performance needs investigation. To this end, we analyze the effects of the number of accessible Non-IID U data. We set each client with ne 500 samples Ve E [C] and conduct experiments by changing the amount of Non-IID U data by increasing the number of clients C' {10, 20, 30, 40, 50). As shown in Figure 3(b), the model performance improves by a large margin as the number of accessible Non-IID U data increases by using our proposed FedUL method.
Analysis of Local Updating Epochs: The frequency of aggregation may sometimes influence the model performance in an FL system. We would like to explore if FedUL is stable when the local clients are updated with different local epochs E. Figure 3(c), we explore E {1,2, 4, 8, 16) with fixed B = 128 and C = 5. The results demonstrate that with different E, the performances of FedUL are reasonably stable.
The training batch size is always important factor in machine learning tasks, especially in an FL system. Hence here, we study how batch size affects our model performance. Specifically,
Effects of Batch Size:
train the model using different batch sizes B E {16, 64, 128, 512, xo with fixed E = and C 5. Here B = co stands for loading all the training samples into a single batch. The results in Figure 3(d) indicate that FedUL can effectively work under wide range of batch sizes.
4.3 CASE STUDY ON REAL-WORLD PROBLEM
To demonstrate the feasibility of our proposed FedUL. scheme to the real problem, we test FedUI. on a medical application for the diagnosis of brain disease. the rest of this section, we introduce the problem formulation, data and setting, and then discuss the results.
Real Problem Formulation: For the real-world scenario, we consider a clinical setting for disease classification in C different healthcare systems. Suppose each healthcare system has M hospitals located in different regions, each associated with unique class priors due to population diversities. Medical data not shareable between the healthcare systems for privacy reasons. We assume patient-wise diagnosis information is unavailable, but each hospital provides the prevalence of these patients (i.e., class priors of the disease classes).
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Table 2: Quantitative comparison of our method with the baseline methods for the five healthcare systems (HS) using the real-world intracranial hemorrhage detection dataset. Results of three runs are reported in mean error (std) format. The best methods are highlighted in boldface.
Hospitals
Method
HS
HS 2
HS
HS 4
HS
FedPL
44.69 (4.28)
46.42 (4.35)
46.29 (1.83)
47.55 (2.84)
46.53 (2.40)
12
FedUL (ours)
40.26 (1.39)
40.35 (1.10)
39.31 (1.10)
39.72 (0.67)
39.43 (1.42)
FedPl.
42.71 (2.97)
42.18 (3.00)
42.30 (1.28)
41.79 (1.98)
41,46 (0.53)
24
FedUL (ours)
32.27 (2.71)
30.99 (2.03)
31.29 (1.05)
31.72 (1.23)
31.37 (1.58)
FedPl.
41.44 (3.04)
40.38 (1.38)
41.06 (2.88)
41.44 (1.78)
40.26 (2.35)
48
FedUL (ours)
31.48 (2.25)
30.92 (0.88)
31.16 (0.84)
31.55 (0.58)
30.57 (1.37)
FedAvg 10%
46.79 (1.91)
46.44 (0.78)
47.19 (1.05)
47.29 (1.93)
46.20 (1.61)
(a) Healthy
(b) Epidural
(c) Intraparenchymal
(c) Intraventricular
Subanchhood
(f) Subdural
Figure 4: Example samples of each class in the distributed medical dataset (Flanders et al., 2020).
Dataset and Settings: We use the RSNA Intracranial Hemorrhage Detection dataset (Flanders et al., 2020), which contains large-scale Magnetic Response Image (MRI) data from several hospitals (see Figure 4), to simulate this real-world scenario. There are 106k samples in the dataset belonging to six classes, including five types of intracranial hemorrhage (Epidural, Intraparenchymal. Intraventricular, Subarachnoid, and Subdural) and healthy control type. We model FL system with five healthcare systems (i.e., clients). Each healthcare system contains 21.2k data samples. To show the effect on the number of samples in each U set, we vary the number of hospitals (i.e., sets) M 12, 24, and 48 in each system. Here we generate each -. for the k-th class in m-th U set at c-th client from range [0.1, 0.9] and check that the generated class priors are not all identical. For model training, we use the cross-entropy loss and Adam (Kingma & Ba, 2015) optimizer with learning rate of le 4 and train 100 rounds. We use ResNet18 (He et al., 2016) as backbone for this experiment.
Results and Analysis: As indicated in our proof-of-concept (Table 1), FedPL consistently outper- forms the other two LLP baselines, i.e., FedLLP and FedLLP-VAT. Here we focus on comparing FedUL with FedPL. The supervised baseline FedAvg with 10% fully labeled data is also included for reference. We report the results of each hospital in mean error (std) format over three runs in Table 2. We can see that our proposed FedUL method consistently outperforms FedPL and FedAvg 10% for all the hospitals and under different set number settings. The superiority of FedUL this challenging wal-world FL setting further indicates the effectiveness and robustness of our algorithm. These results are inspiring and bring the hope of deploying FedUL. to the healthcare field, where data are often very costly to be labeled, heterogeneous, and distributed.
5 CONCLUSION
In this paper, we propose FedUL., a novel FL algorithm using only U data. FedUL, works as a wrapper that transforms the unlabeled client data and modifies the client model to form a surrogate supervised FL task, which can be easily solved using existing FL paradigms. Theoretically, we prove that the recovery of the optimal model by using the FedUL method can be guaranteed under mild conditions. Through extensive experiments on benchmark and real-world datasets, we show FedUL expect significantly outperform the baseline methods under different settings. As a wrapper, FedUL can be further improved by incorporating advanced FL aggregation or optimization schemes for the non-IID setting (Wang et al., 2020; Li al., 2020; 2021; Karimireddy et al., 2020).

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ETHICS STATEMENT
Collecting sample-wise labels for training can be costly and may violate data privacy. Considering data in the real world are often distributed at large scale, we provide promising solution to perform multi-class classification on distributed U data in the FL system. Our method FedUL can be widely applied to data-sensitive fields, such as healthcare, politics, and finance, where the sample-wise label is not available, but populational prevalence can be accessed. With FedUL. we hope the future FL can enjoy significant labeling cost reduction and improve its privacy. The real datasets used this work are publicly available with appropriate prepossessing to de-identify patients' identities. We provide proper references to the datasets, libraries, and tools used in our study. This study does not have any legal compliance, conflict of interest, research integrity issues.
REPRODUCIBILITY STATEMENT
We provide the codes to reproduce the main experimental results at https: /ttp///gttt///// com/ lunanbit /eettFFFedllll We further provide experimental details and additional information of the datasets in the supplementary materials.
ACKNOWLEDGMENTS
NL was supported by JST AIP Challenge Program and the Institute for AI and Beyond, UTokyo. ZW and QD were supported by project Shenzhen-HK Collaborative Development Zone. XL was thankful for the support by NVIDIA. GN was supported by JST AIP Acceleration Research Grant Number JPMJCR20U, MS was supported by JST AIP Acceleration Research Grant Number JPMJCR20U3 and the Institute for AI and Beyond, UTokyo.
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Supplementary Material
Roadmap The appendix is organized follows. We summarize notations used in our paper in Section A and give a full discussion of related work in Section B. Our proofs are presented in Section C. We provide detailed experimental settings in Section and supplementary experimental results on the benchmark datasets in Section E.
A NOTATION TABLE
Table A.1: Notations used in the paper.
Notations
Description
input dimension
K
number of classes for the original classification task
index of the class & C [K]
X
input space X C I [K] where [K] == {1....,K) K)
y
label space input x E X' true label y E y test data distribution
y
p
*
k-th class prior n² = p(y= of the set
vector form of 7 where = ( :.... x*]
loss function for classification task risk for the original FL task
f
global model predicting label for inpul number of clients in the FL setup index of the client E [C]
C
fo
e-th client model predicting label y for input æ local data distribution at the e-th client
Pe
na
number training data at
client
De
labeled training set D. = {(x M()'s, at the e-th client
ag
local step size
Og
global step size
global communication round
L
local updating iterations number available sets e-th client
Me
index of the U set m E [Ma]
Dem
number training data in the m-th U set the c-th client
Pe
m-th U data distribution at the e-th client
U
m-th U set at the e-th client 2tth = (x; c.m ne.m
Ji=1
U
all available U sets at the e-th client U4 = [24mh Mr
M
the maximum number of available U sets among all clients M max Ma
Te m.kk
k-th class prior Tm p. (y k)) of the m-th U set at the e-th client matrix form of m,,k where II, R.M. M.
II
surrogate label y E: [M]
Pe
surrogate local data distribution the c-th client
Ur
surrogate labeled training set Ut = { Ts He at the e-th client
m-th surrogate class prior ,. = (T m) at the c-th client vector form of Th where म. = [ , =
n.
surrogate global model predicting surrogate label 7 for input x surrogate class-posterior probability the e-th client where (x)) (T Te
true class-posterior probability at the e-th client where (1(())) h p(y = (
Q.
vector-valued transition function Q. : (X Ak) (X Am) surrogate risk for the surrogate FL task
J
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B RELATED WORK
B.1 PERSONALIZED FEDERATED LEARNING
Considering the heterogeneity of data, clients, and systems, personalization a natural and trending solution to improve FL performance. Many efforts have been made in personalized FL. FedProx(Li al., 2020), FedCurv (Shoham et al., 2019), and FedCL (Yao & Sun, 2020) propose regularization- based methods that include special regularization terms to improve FL model generalizing to various clients. Meta-learning methods, such as MAML (Finn et al., 2017), are also used to learn personalized models for each client (Jiang al., 2019; Fallah et al., 2020; Li et al., 2019). Unlike the methods mentioned above that yield a single generalizable model, multiple FL models can be trained using clustering strategies (Sattler et al., 2020; Ghosh et al., 2020). Different local FL models can also be achieved by changing the last personalized layers (Li et al., 2021; Arivazhagan et al., 2019).
At first glance, FedUL looks similar personalized FL since each client learns "personalized" model 9. which contains a client-specific transition layer Qc. However, our learning setting and method are very different from existing personalized FL approaches, explained below. First, our framework focuses on FL with only unlabeled data. The personalized model 9r only serves as a proxy, in order to compatible with the transformed unlabeled data, and our ultimate goal is to learn f that is not exactly personalized. Specifically, 1) if the inter-clients are IID, FedUL yields the same proxy model Gr and federated model f for all the clients; 2) if the inter-clients are Non-IID,, although ge is different (as Q. is different), the underlying model f (what we want) is still the same. Second, our method differs from the previous personalized FL methods by its surrogate training nature: works wrapper that transforms unlabeled clients to (surrogate) labeled ones and is compatible with many supervised FL methods.
B.2 SEMI-SUPERVISED FEDERATED LEARNING
Another line of research working on utilizing unlabeled data in FL follows the semi-supervised learning framework (Zhu, 2005; Chapelle et al., 2006), which assumes a small amount of labeled data available at the client side (labels-at-client or at the server side (labels-at-server), For example, FedMatch (Jeong et al., 2021) introduces a new inter-client consistency loss that maximizes the agreement between multiple clients and decomposes the model parameters into one for labeled data and the other for unlabeled data for disjoint learning. FedSem (Albaseer et al., 2020) generates pseudo labels for unlabeled data based on the trained FedAvg model with labeled data, and the generated (pscudo) labeled data are further used retrain FedAvg to obtain the global model. Moreover, distillation-base semi-supervised FL (Itahara et al., 2020) considers using shared unlabeled data for biseillation-based message exchanging.
Although the motivation FedUL can be related to semi semi- supervised FL since both work on utilizing unlabeled data in FL, our learning setting is different since we only have unlabeled data at both the client and the server sides, and our method is different since FedUl. follows the empirical risk minimization framework.
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C PROOFS
In this appendix, we prove all theorems.
C.I PROOF OF THEOREM 1
By the Bayes' rule, Vm [M] we have:
(Z))m = Po Po y = m.)
(x)
(e D m)
Mr
De(a, T m)
(a (T
Σ m= (a y (y m)
Me
Pe (x)
Σ Me m==1 P. (x) T
Tee 2k=1 To -
K
p(
k)
Mr T. k== m.sk p( y k)
Re E Te
m. (m(x))e
(10)
Mr Ek= m.k (n(x))
Em=1
n²
By transforming (10) to the matrix form, complete the proof.
C.2 PROOF OF LEMMA 2
Given our data generation process in Section 3.1, we have that M. 2 k and II, E IR M. X is a full column rank matrix. With D7. and D being non-degenerate, we obtain that Qc(n(x)) presents an injective mapping from (z) to 17. (x).
C.3 PROOF OF PROPOSITION 3
The assumptions (A1)-(A3) are listed as follows.
(A1) Bounded gradient dissimilarity: 3 constants G 2 and B 2 1 s.t.
[Vc(g)]11 < G V Vg.
(A2) 3-smooth function satisfies
(O) vJ.( < e| - e' Ve, 0.0'.
(A3) Unbiased stochastic gradient with bounded variance:
[ll - VJ.(g)) < 0², Vc,
where = = E. ((g(z)).
Note that this proposition statement w.r.t. the convergence of model g learned from the surrogate supervised learning task. The proof follows directly from Theorem V in Karimireddy et al. (2020).
C.4 PROOF OF THEOREM 4
When proper loss (Reid & Williamson, 2010) is used for L, the mapping of g"* is the unique minimizer of J(g). We show an example of the a widely used proper loss function: the cross-entropy loss (De Boer et al., 2005) defined
Lee (g))) =
1 m) log(g(z))m. = llog(g (x)ggg())) (x)
m=1
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We that the cross-entropy loss is non-negative by its definition. Therefore, minimizing (g) can be obtained by minimizing the conditional risk ((g))) now optimizing
7) x for every a E X. So we
(g)
Σ
p m x) log(g()))m
s.t. (g(x)) 1.
By using the Lagrange multiplier method (Bertsekas, 1997), have
C
Σ
p - m x) log(g(x))m A- Σ (g(x)) 1).
The derivative of L with respect to g is
ac =
p(u x1 X,
p(u = M x3)
0g
(g(x))1
(g(x))
By setting this derivative to we obtain
(g(x))m m
ply x)), Vm mm=1,.... 1,..., M and Va E X.
Since g AM-1 is the M -idiinnsional simplex, we have Em (g""(()) and ΣM p(T
j x) = Then
M Σ
M Σ
g (z))
plu
= m x)) =
m=1
m=
Therefore we obtain A = and (g (xx) m p m x)) (T(()) Vm = I,..., M and Vo X, which is equivalent g" (x) n7))) Similarly, we obtain
f (a:) ()x))
(11)
Given Proposition and the assumption that the model used for is sufficiently flexible, we obtain that the classification model learned by FedUI. (see Algorithm satisfies g" =g" By our model construction, g is built from (f( Then we have
(12)
where f = argmin, J(f) the model induced by g"* Given Theorem and Lemma 2, we obtain that (xx) = 27) if and only if
(z) 17x).
(13)
By summarizing (11), (12), and (13) obtain f"= ft
L
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D EXPERIMENTAL DETAILS
We illustrate our model architectures and training details of the benchmark and real-world experiments in this section.
D.1 MODEL. ARCHITECTURE
For our benchmark experiment on MNIST, we use six-layer convolutional neural network. The network details are listed in Table D.2. For our benchmark experiment on CIFAR10 and real-world experiment, we use ResNet18 as our backbone. The network details are listed in Table D.3.
Layer
Details
Conv2D(3, 64, 5, 1, 2), BN(64), ReLU, MaxPool2D(2,, 2) Conv21((64, 64, 5, 1, 2), BN(64), ReL.U, MaxPool2D(2,, 2) Conv2D(64, 128, 5, 1, 2), BN((28), ReLU FC(6272, 2048), BN(2048), ReLU
2
3
5
FC(2048, 512), BN(512), ReLU FC(512, 10)
6
Table D.2: Model architecture of the benchmark experiment MNIST. For convolutional layer (Conv2D), we list parameters with sequence of input and output dimension, kernel size, stride and padding. For max pooling layer (MaxPool2D), list kernel and stride. For fully connected layer (FC), we list input and output dimension. For BatchNormalization layer (BN), we list the channel dimension.
D.2 TRAINING DETAILS
Benchmark Experiments: We implement all the methods by PyTorch and conduct all the experi- ments on an NVIDIA TITAN X GPU. Please note for all experiments, we split 20% original data for validation and model selection. During training process, we use Adam optimizer with learning rate le 4 and the cross-entropy loss. We set batch size 128 and training rounds 100. To avoid overfitting, we use L1 regularization. The detailed weight of L1 regularization used in benchmark experiments shown in Table D.4. The detailed number of samples at each client for MNIST and CIFARIO under IID and Non-III) setting is given in Table D.5 and Table D.6, respectively.
We describe details of the baseline methods and corresponding hyper- sppee-paraeeters follows.
FedPL (Diao et al., 2021): this method is based on empirical classification risk minimization. The learning objective
M ffrr + XLmins
R(
1
(14)
C
where
Lix = E (f (zt) v5
(15)
is the "fix" loss. (at v represents for sample x with high confidence pseudo label y² and
Liix Amix E(f (Xmix) v/5 ) + (1 Amix) (f (amixx v15))
(16)
is the "mix" loss, in which
Amix Beta(a, a), Tmix f- Amixa + (1 Amix) 2T
(17)
is the mixup hyper-parameter and (2, 2b represents for sample 2, with low confidence pseudo label y Following the default implementation in Diao et al. (2021), set hyper- parameter a = 0.755. Moreover, we set the weight of "mix" loss 0.3 and the confidence threshold 0.4.
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Layer
Details
Conv2D(3, 64, 2, 3), BN(64), ReLU
Conv2D(64, 64, 3, 1, 1), BN(64), ReLU
Conv2D(644, 64, 3, 1, 1), BN(64)
Conv2D(64, 64, 3, 1), BN(64), ReLU
Conv2D((64, 64, 3, 1, 1). BN(64)
Conv2D(64, 128, 3, 2, 1), BN(128), ReLU
Conv2D(128, 128, 3, 1), BN(64)
Conv2D(64, 128, 1, 2, O), BN(128)
Conv2D(128, 128, 3, 1, I), BN(128), ReLU
10
Conv2D(128, 128, 3, 1), BN(64)
Conv2D(128, 256, 3, 2, 1). BN((28), ReLU
12
Conv2D(256, 256, 3, 1, 1), BN(64)
13
Conv2D(128, 256, 2. O), BN(128)
14
Conv2D(256, 256, 3, 1, 1). BN(128), ReLU
15
Conv2I((56,, 256, 3, 1, 1), BN(64)
16
Conv2D(256, 512, 3, 1), BN(512), ReLU
17
Conv2I((52,, 512, 3, 1, 1), BN(512)
18
Conv2D(256, 512, 2, O), BN(512)
19
Conv2D(512, 512, 1), BN(512), ReLU
Conv2D(512, 512, 3, 1, 1), BN(512)
21
AvgPool2D
22
FC(512, 10)
Table D.3: Model architecture of the benchmark experiment CIFARIO. For convolutional layer (Conv2I)), we list parameters with sequence of input and output dimension, kernel size, stride and padding. For max pooling layer (MaxPool2D), (FC), we list input and output dimension. For BatchNormalization layer (BN), we list the channel dimension.
list kernel and stride. For fully connected layer
Table D.4: Detailed weight of LI regularization used in benchmark experiments.
Dataset
MNIST
CIFAR10
Sets
10
20
30
10
20
30
Weight
le-5
5e-5
2e-6
2e-5
le-5
4e-6
FedLLP (muld et al., 2019): this baseline method is based on empirical proportion risk minimization. The learning objective is the aggregated proportion risk, i.e.,
Ma dprop
R(
(TT m
mm)
(18)
C
m
where The and 7. are the true and predicted label proportions for the m-th U ue at the e-th client, 7m M. Exrum f(z). For each client c, dprop is defined as
dprop (
π
log Am
RedLLP-VAT (Tsai & Lin, 2020): this method is based on empirical proportion risk mini- mization, integrated with consistency regularization term. The learning objective is the
18

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Published as a conference paper at ICLR 2022
Table D.5: Detailed number of samples each client for MNIST benchmark
Class
2
3
4
6
7
8
9
Client1
920
1098
970
972
945
893
898
1007
971
926
Client2
920
1098
970
972
945
898
1007
971
926
Client3
920
1098
970
972
945
893
898
1007
971
926
IID
Clicnt4
920
1098
970
972
945
893
898
1007
971
926
Client5
920
1098
970
972
945
893
898
1007
971
926
Client1
4498
4646
56
56
54
57
54
60
60
54
Client2
$7
54
4425
4737
49
50
49
58
57
58
Client3
64
61
57
61
4316
4217
54
53
58
57
Non-III Client4
58
55
35
50
4572
4576
60
57
Client5
57
58
61
0
60
19
54
4418
4576
Table D.6: Detailed number of samples at each client for CIFARIO benchmark
Class
airplane
automobile
bird
deer
dog
frog
horse
ship
truck
Clientl
755
788
794
796
791
842
827
788
820
799
Client2 755
788
794
796
791
842
827
788
820
799
Client3
755
788
794
796
791
842
827
788
820
799
Client4
755
788
794
796
791
842
827
788
820
799
Client5
755
788
794
796
791
842
827
788
820
799
Clientl
3748
3872
46
45
47
45
50
50
45
Client2
48
45
3688
3938
41
41
41
48
48
48
Client3
51
3597
3887
45
53
48
Non-IID
Client4
44
18
46
0
42
0
3810
3813
50
48
Client5
47
0
50
45
3788
3790
aggregated proportion risk with consistency regularization, i.e.,
R(f)= - 2
Σ Σ
C
Me
+ all (1).
dgrop
(m"",
(19)
m==1
where The and 7m are the true and predicted label proportions for the m-th set 4 at the e-th client, and dprop is distance function.
looms (f) deans (f(x),
(20)
is the consistency loss, where deons is distance function, and ã is perturbed input from the original æ. Following the default implementation in their paper (Tsai Lin, 2020), we set the hyper- # = 6.0 for VeT on CIFAR1O benchmark experiment. For MNIST benchmark, we set the hyper-parameter, the weight of consistency loss a 5e 4 and the perturbation weight 1e-
the weight of consistency loss 0.05 and the perturbation weight
Real-world Experiments: We perform case study on the clinical task intracranial hemorrhage diagnosis. We use the RSNA Intracranial Hemorrhage Detection dataset (Flanders et al., 2020) (Figure 4), which contains large-scale Magnetic Response Image (MRI) data from several hospi- There are 106k samples in the dataset belonging to six classes, including five types of intracranial hemorrhage (Epidural, Intraparenchymal, Intraventricular, Subarachnoid, and Subdural) and healthy control type. In our experiments, we simulate a real-world scenario with five healthcare systems (a.k.a. clients) and assume the number of hospitals (a.k.a sets) each healthcare systems varies M = 12, 24, and 48. At first, randomly split the whole dataset for five healthcare systems equally, so each healthcare systems is assigned 21.2k samples. Then we generate ा. for m-th set in c-th client from range 1, 0.9] for all the hospitals with U data only. We check that the generated class priors are not all identical. Finally, we load samples for each the hospitals following the generated n. Detailed dataset statistics is given in Table D.7.
During training process, we use Adam optimizer with learning rate le 4 and cross-entropy loss. We set batch size to 128 and training rounds 100. To avoid overfitting, use L1 regularization. We
19

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Published as a conference paper at ICLR 2022
Table D.7:
Detailed number samples in each healthcare system (HS) for real-world RSNA
Intracranial Hemorrhage Detection dataset (Flanders et al., 2020).
Type
Healthy
Epidural
Intraparenchymal
Intraventricular
Subarachnoid
Subdural
HS 1
3726
221
2023
1198
2027
4005
HS 2
3756
226
1923
1190
2063
4042
HS 3
3804
177
1932
1266
2040
3981
HS 4
3805
224
1939
1187
2074
3971
HS
3711
1929
1299
2080
3983
set the weight of LI regularization as 2e - 6, 8e 7 and 4e 7 for 10, 20 and 40 sets, respectively. In real-world experiments, we use the same syper-parameters as benchmark experiments for all methods except FedILP-VAIT For FedLLPP-AA,, we set the weight of consistency loss 0.05 and the perturbation weight μ 6.0.
Training Efficiency: We further evaluate the training efficiency of our method with comparison to baseline methods. The detailed one epoch training time (mean) and forward FLOOPs are given in Table D.8. The computation time is evaluated see that our method slightly increase the forward FLOPs and can achieve higher training efficiency compared with FedPL and FedLLP-VAT.
PC with Nvidia TITAN X GPU. We
Table D.8: Quantitative comparison of our method with the baseline methods training epoch time and forward flops on benchmark and real-world datasets under Non-IID and IID setting.
Non-IID Task with Clients
Task with 5 Clients
Dataset
Method
Computation time (s)
Fl.OPs (G)
Computation time (8)
FL.OPs (G)
FedPL
3.29
4.78
FedLLP
1.67
0.048125312
2.45
0.048125312
MNIST
FedLLP VAT
2.10
2.87
FedUL
1.80
0.048125422
2.35
0.048125422
FedPL
3.37
5.70
FedLLP
1.95
0.141595648
2.44
0.141595648
CIFARIO
FedLL,P. VAT
3.43
4.86
FedUL
2.63
0.141595758
3,80
0.141595758
FedPL.
32.47
6.937938944
Real-world
FedUl.
32.34
6.937939028
20

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Published as a conference paper at ICLR 2022
E MORE EXPERIMENTAL RESULTS ON THE BENCHMARK DATASETS
E.1 DETAILED STATISTICS OF FIGURE 3(A)
We show detailed statistics of Figure 3(a) in Table E.9. With different noisy class priors, even with challenging 0.8 and = 1.6, our method still achieves low error rate, which verify the robustness and effectiveness of our method.
Table E.9: Inaccurate class priors experiments with different noise ratio on MNIST under Non-III setting with 5 clients and 10 sets for every client (mean error (std)).
(Clean)
0.2
c 0.4
0.8
1.6
2.98 (1.72) 3.02 (1.30) 3.10 (0.96) 3.02 (1.36) 3.49 (1.74)
E.2 DETAILED STATISTICS OF FIGURE 3(B)
We show detailed statistics of Figure 3(b) in Table E.10. With more Non-IID data, our method performs better, which validates that more unlabeled Non-IID data can be helpful for FL..
Table E.10: Experiments on accessible data amount under the Non- -on-IID setting (mean error (std)).
5k
10k
15k
20k
25k
3.56 (0.09) 2.62 (0.07) 1.97 (0.38)
1.75 (0.11) 1.72 (0.13)
E.3 DIFFERENT COMBINATIONS OF AND
We compare method with different combinations of E and B on benchmark dataset MNIST in the Non-IID setting. The classification error rate (mean and std) is consistently small as presented (cf. Table E.11), which indicates our method can work under wide range combinations of E and B.
Table E.11: Experiments using different combinations of training batch size B and local update epoch E on benchmark dataset MNIST in the Non-IID setting (mean error (std)).
E
4
16
00
128
00
00
7.99
2,93
3.69
2.54
4.16
2.81
2.92
3.51
2,88
Error
(9.83)
(1.20)
(0.88)
(1.09)
(3.44)
(0.37)
(1.12)
(1.88)
(1.06)
E.4 BENCHMARK EXPERIMENTS ON CLIENTS WITH DIFFERENT SETS
Here we conduct experiments on benchmark datasets under IID) and Non-IID setting with 5 clients and different for each client (10, 20, 30, 40, 50). The experimental results shown in Table E. which further indicates the effectiveness of our method even under this challenging setting.
E.5 BENCHMARK EXPERIMENTS WITH MORE CLIENTS
To study the influence of the number of clients, here we further conduct benchmark experiments on both MNIST and CIFARIO with 10 clients. The detailed results shown in Table E. The observations are similar to Table
21

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Published as a conference paper at ICLR 2022
Table E. 12: Quantitative comparison of our method with the baseline methods on benchmark datasets under Non-IID and IID setting (mean error (std)) with different sets for each client. The best method (paired t-test at significance level 5%) is highlighted in boldface.
IID Task with 5 Clients
Dataset
FedPL.
FedLLP
FedL.P- VAT
FedUL
FedAvg 10%
MNIST
3.74 (0.78)
5.22 (5.38)
2.24 (0.05)
1.19 (0.11)
2.21 (0.07)
CIFARIO
45.49 (2.74)
84.16 (0.71)
86.01 (2.54)
31.17 (2.80)
42.27 (1.06)
Non-IID Task with 5 Clients
Dataset
FedPL.
FedLLP
FedLLP-VAT
FedUL
FedAvg 10%
MNIST
12.03 (2.89)
4.26 (3.20)
3.88 (2.89)
2.31 (1.14)
3.42 (0.20)
CIFARIO
59.06 (3.73)
62.44 (16.99)
59.72 (18.73)
35.46 (1.40)
53.13 (0.22)
Table E.13: Quantitative comparison of our method with the baseline methods on benchmark datasets under Non-IID and IID setting (mean error (std)). The best method (paired t-test significance level 5%) is highlighted in boldface.
IID Task with 10 Clients
Dataset
Sets
FedPI.
FedL.I.P
PATT
FedUl.
FedAvg 10%
10
2.19 (0.04)
10.11 (7.12)
25.83 (22.64)
0.74 (0.14)
MNIST
20
2.55 (0.30)
24.23 (17.52)
28.40 (23.20)
1.05 (0.05)
1.58 (0.23)
40
3.13 (0.25)
32.53 (16.93)
34.46 (15.46)
1.47 (0.11)
10
40.47 (0.58)
76.23 (1.47)
76.25 (1.65)
20.40 (0.63)
CIFARIO
20
46.44 (0.71)
78.10 (3.58)
76.26 (3.83)
21.61 (0.23)
33.57 (1.03)
40
51.58 (1.18)
79.10 (2.34)
80.44 (1.60)
22.71 (0.17)
Non-III Task with 10 Clients
Dataset
Sets
FedPL.
FedL.IPP
FedI.LP-VAT
FedUL.
FedAvg 10%
10
12.24 (1.36)
20.81 (14.72)
17.26 (10.39)
1.91 (0.16)
MNIST
20
5.73 (0.89)
17.67 (11.48)
18.45 (4.30)
1.45 (0.27)
3.03 (0.24)
40
4.16 (0.56)
14.98 (10.11)
13.62 (5.47)
1.27 (0.12)
10
68.53 (1.14)
84.11 (2.63)
82.17 (2.58)
39.20 (0.79)
CIFARIO
20
61.49 (1.42)
71.69 (7.59)
72.34 (7.39)
34.12 (0.57)
52.48 (1.55)
40
55.89 (2.19)
58.21 (4.95)
59.42 (9.59)
32.21 (1.02)
22


Reviews:
[{"review_id": "mNaAsKpCjsP", "reviewer": "Reviewer_4sTZ", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "# Pros\n\n-The proposed method is novel and interesting. By using the surrogate loss, the method provides a framework to enable unsupervised clients (without any labels) to supervised clients (with labels in particular form); and learning is performed via the surrogate supervised learning loss. With such a framework, many existing supervised federated learning methods can also be applied in this client-without-label setting. I think this framework provides a novel perspective for solving unsupervised federated learning, instead of following the unsupervised learning approach such as clustering.\n\n-It is also nice that the method can be easily implemented by adding a transition layer to the existing models, neither changes on the optimization process nor introduces additional hyperparameters to be tuned.\n\n-The theoretical analysis on the optimal model recovery is also a very useful result providing both conditions for unsupervised surrogate task to match the supervised task optimal; and the interpretations for these surrogates. Its convergence directly relies on the chosen federated aggregation approach, which has quite a flexibility on advanced methods with guarantees.\n\n\n# Cons\n\n-The discussions in the related work section may not be comprehensive. The work can be quite related to personalized federated learning in the sense that each client learns a personalized model (g1,...,gc), see:\nTowards Personalized Federated Learning, Tan et al., 2021.\nDiscussing relations and differences with this line of research seems necessary.\n\n-Another line of research working on utilizing unlabeled data in federated learning follows the semi-supervised learning framework, see\nA Survey towards Federated Semi-supervised Learning, Jin et al., 2020.\nAlthough they assume some labeled data are available, which is not exactly the unsupervised federated learning setting, adding discussions with them in the related work may help position the paper well.\n\n-In this paper, the authors do assume certain conditions on the unlabeled data, e.g. the assumptions on the class prior knowledge on each unlabeled dataset. Can the authors give more examples on how this information can be obtained in the real world?\n\n# Minor comments\n\n-In algorithm 1, it seems that only the model f is trained. In my understanding, equation 7 is the surrogate loss and model g is what is being trained. If it is correct, the current algorithm may be a bit misleading.\n\n-In table 1, the performances of FedPL and FedUL are reasonable when the set number changes. However FedLLP and FEdLLP-VAT are not very stable. Could the authors explain more on this phenomenon?\n\n-In the experiments, it seems that the experiments are conducted only on 5 clients, could the authors show more results with more clients?\n\n", "summary_of_the_paper": "This work presents a novel federated learning scheme to address the problem of learning from only unlabeled data. The main idea is clean and interesting, which constructs a global (server) model by aggregating the surrogate clients’ tasks from observing only unlabeled data for the classification tasks. The unlabeled data are transformed at each client to make them compatible with supervised federated learning; consequently, the learned models are transformed accordingly. Existing federated aggregation techniques, e.g. FedAvg, are applied on these transformed clients (where they call surrogate clients). Theoretical results are provided for learning the optimal model and experiments on benchmark and real data show superior performance compared to baselines.\n", "summary_of_the_review": "Overall, this paper is novel and interesting. However, some points should be improved in the revision.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "aMaMvksKXVS", "reviewer": "Reviewer_U3W9", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Overall there are some good theoretical results in the paper with a thoroughly analyzed experiment section. I felt that Section 2 is not really needed in the paper as explaining the context of supervised federated learning is not relevent to the contributions in this paper. This could probably be moved to the appendix and then some of the theoretical derivations could be moved to the main body.\n\nFrom the experiments, it was not exactly clear to me which class prior was used. It says \"we randomly sampled class priors from range [0.1, 0.9] and then regularized them to formulate a valid \\Pi_c as discussed in Section 3.1.\". But it does not appears to explain what distribution the class prior is. Could this be explained.\n\nI felt that U shouldn't be used a shorthand for unsupervised. But I guess this is up to the authors.", "summary_of_the_paper": "The authors are proposing an approach for unsupervised Federated Learning. The authors propose to use Emperical risk minimizing for unsupervised learning. The authors assign labels to each of the class and uses a prior for the classes so that each of the client can learn without any labels.\n\nThe authors have shown theoretical properties for their proposed solution. Theorem 1 showing that there is a map that that exist to transform the data to the classes, Lemma 2 shows that there is a transition function for each of the clients. The authors have also shown convergences of the algorithm by deriving the upper bound.\n", "summary_of_the_review": "The paper has provided some novel contribution in terms of the theoretical properties for unsupervised learning. I felt that the paper could have been better written and more experiments could have been conducted to make their results more convincing. But nevertheless, I do think that the results that the authors have presented are novel enough for a publication at ICLR.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "a5I4RAe0bZr", "reviewer": "Reviewer_SBTk", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The proposed method allows different clients to have different unlabeled data distributions. There are two conditions required for learning from such data. First, all unlabeled data distributions must share the same set of class-conditional distributions. Second, the ratios of classes must be given, even though the data are unlabeled. Then, with the additional information about the class ratios, each client could align with the supervised learning counterpart itself, and the server would not be affected by the fact that the training data are all unlabeled.\n\nIt is worth mentioning that unsupervised federated learning is still based on empirical risk minimization (the loss function is new and the regularization is optional). This means that it goes along different lines from semi-supervised federated learning based on semi-supervised regularizations (the loss function is the same as supervised federated learning and the regularization is mandatory).\n\nPositive points:\n\n1. The idea is novel. The underlying task is still multi-class classification rather than clustering. As far as I know, the problem has not been studied yet, though there is already semi-supervised federated learning.\n\n2. The idea is general. The paper proposed a new loss function, and thus it does not affect our choices of the deep network and optimizer used for training. Moreover, the new loss itself is fine with both image data and text data, which is more general than many semi-supervised techniques that can only be used with image data.\n\n3. The idea is motivated. Since all clients have to prepare training data by themselves and no client would like to share its data with any other party, it is natural that the clients do not want to label all the unlabeled data due to the money and/or time concerns.\n\nNegative points:\n\n1. The assumption that \"all unlabeled data distributions must share the same set of class-conditional distributions\" sounds strong to me. How can the authors extend the current method so that this assumption is removed or reduced to a milder assumption?\n\n2. The new loss function leads to some additional operations in forward and backward passes. However, in section 4 experiments, there is no experimental result about the computational efficiency. I am working on the practical side of deep learning and I think the issue of computational efficiency is very important. In tables 1 and 2, the proposed method worked much better than the baseline methods, but might the proposed method be much slower than the baseline methods too?", "summary_of_the_paper": "The paper proposed federated learning in an unsupervised manner with some nice theoretical guarantees and nice experimental results, where the unlabeled data distributions must satisfy some non-trivial assumptions.", "summary_of_the_review": "This is an overall well-executed paper. Besides the above comments, the authors need clarify on a few raised questions.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "AC_7E1WEI4c", "reviewer": "Reviewer_mco5", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strong:\n- relevant topic with clear motivation and useful practical examples\n- sufficient experiments\n- outlined theoretical derivations + code\n\nWeakness:\n- the title is a bit misleading ... because the approach does not cover\n  a general framework for unsupervised learning -> maybe the title should be changed", "summary_of_the_paper": "The paper suggests a new strategy for pseudo-labeling\nbased on cluster structures in FedLearn. Theoretical \nderivations, abl.-studies and experiments are provided\nshowing the effectivness of the approach.", "summary_of_the_review": "Unsupervised FedLearn\n\ncomments:\n- not so much to mention but only some 'concerns'\n-  Unsupervised learning is really different to supervised learning.\n   In your case you consider local cluster assignments as labels and use\n   subsequently a supervised learning scheme to generate predictions.\n   This is not really unsupervised because in general SL methods focus \n   on margin maximization of the class boundaries ... which is not the same\n   as approximating a cluster distribution (so I would say your title is a bit\n   wrong). Maybe better: Federated learning with surrogate labeling?\n-  Further what happens if you have no uni-modal data as shown in Fig 1?\n   In fact your # of labelings may change during the optimization or you\n   artificially colaps different distribution into single once if the #of cluster\n   is not appropriate\n- 'Without any labels, FL becomes significantly harder than before, since it is unclear how to compute\n   the local gradients at each client and how to aggregate them for updating the global model  '\n   --> well there are unsupervised learning approaches which are gradient based\n   ... you only need a differentiable cost function!\n-  if the labeling in the experiment (for labeled data) follows the data distribution\n   I do not expect a substantial problem ... but if the data are non-unimodal and the\n   #of clusters does not fit it may become a challenge to get reliable models\n- theory is likely correct and code is provided so reproducibility should be given\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper demonstrates a case of federated learning with unlabelled but systematically partitioned data between clients. A title along terms like "FL with unlabelled data" would be much better - the considered setting here is not fully unsupervised but relies on the key assumption that while not the labels, at lease the precise label frequencies have to be known on each client, which is a strong assumption (also iid up to the class shift). Semi-supervised FL approaches should also be discusses.

Overall, reviewers all agreed that the paper is interesting, well-motivated and deserves acceptance.
We hope the authors will incorporate the open points as mentioned by the reviewers.
Decision: Accept (Poster)