FedMM: A Communication Efficient Solver for Federated
Adversarial Domain Adaptation
Yan Shen*

University at Buffalo
Buffalo, New York, USA
yshen22@buffalo.edu

Zhanghexuan Ji

University at Buffalo
Buffalo, New York, USAzhanghex@buffalo.edu

Jian Du*

TikTok Inc.
San Jose, California, USA
dujianeee@gmail.com

Han Zhao

University of Illinois Urbana
Champaign
Urbana, Illinois, USA
hanzhao@illinois.edu

Chunwei Ma

University at Buffalo
Buffalo, New York, USA
chunweim@buffalo.edu

Mingchen Gao

University at Buffalo
Buffalo, New York, USA
mgao8@buffalo.edu

ABSTRACT

1

Federated adversary domain adaptation is a unique distributed minimax training task due to the heterogeneous data among different
local clients, where each client only sees a subset of the data thatmerely belongs to either the source or target domain. Despite theextensive research in distributed minimax optimization, existing
communication efficient solvers that exploit multiple steps of thelocal update are still not able to generate satisfactory solutions
for federated adversarial domain adaptation because of the gradient divergence issue among clients. To tackle this problem, we
propose a distributed minimax optimizer, referred to as FedMM,
by introducing dual variables to bridge the gradient gap amongclients. This algorithm is effective even in the extreme case where
each client has different label classes and some clients only have
unlabeled data. We prove that FedMM admits benign convergence
to a stationary point under domain-shifted unlabeled data. On avariety of benchmark datasets, extensive experiments show that
FedMM consistently achieves both better communication savingsand significant accuracy improvements over existing federated optimizers based on the stochastic gradient descent ascent (SGDA)
algorithm. When training from scratch, for example, it outperforms
other SGDA based federated average methods by around 20% in
accuracy over the same communication rounds; and it consistently
outperforms when training from pre-trained models.

Federated Learning (FL) is gaining popularity because it enables
multiple clients to train machine learning models without directly
sharing the potentially sensitive data with other clients [11, 13].
A typical FL training pipeline involves exchanging local model
parameters with a centralized server to update the global model parameter, and its communication overhead has been, in many cases,
identified as the bottleneck [3, 20] of the training pipeline. Moreover, domain shift often exists between clients’ data [23], which
is another inherent characteristic of FL training, where the data
are sampled from different parts of the sample space on different
clients. Because of the aforementioned unique features, FL training
needs efficient optimizers that converge over heterogeneous data
among clients with fewer communication rounds.
For data with distributional shifts, one of the most challenging
settings is that each local client only has access to a subset ofthe label classes in order to train the global/common model. In
this situation, the global model’s accuracy suffers considerably
as a result of the gradient/model drift [20]. In the literature ofdomain adaptation, this problem is also known as label shift [29,38]. Under the setting of FL, however, label shift is ubiquitous
due to the imbalance between clients’ label distributions, withthe extreme case being that individual clients have disjoint labelsor only unlabeled data. While domain adversarial training is aclassic technique in centralized settings [7, 31, 43], the distributed
nature of FL makes the direct application of this line of approaches
particularly challenging.
One method is to use the stochastic gradient descent ascent(SGDA) method [15] directly as if the data are homogeneous andcentralized, where data are aggregated together to find the saddle
point solutions [10, 16] of a minimax problem. However, becauseof the potential domain shifts among clients in FL settings, a single
client cannot access an unbiased sample of the gradient from theglobal objective function. A natural solution would be to averageeach client’s gradients, which exactly corresponds to the FedSGDA
approach in [22]. Its training efficiency, on the other hand, is low
due to the requirement of large communication rounds between
the server and clients. Without considering the issue of domain
shift, there are several works on communication-efficient FL algorithms. A large spectrum of these algorithms are variations of theclassic FedAvg [20]. Following this pipeline, a natural extension of

KEYWORDS
Federated Learning; Domain Adaptation; Adversarial Learning
ACM Reference Format:
Yan Shen*, Jian Du*, Han Zhao, Zhanghexuan Ji, Chunwei Ma, and Mingchen
Gao. 2023. FedMM: A Communication Efficient Solver for Federated Adversarial Domain Adaptation. In Proc. of the 22nd International Conference on
Autonomous Agents and Multiagent Systems (AAMAS 2023), London, United
Kingdom, May 29 – June 2, 2023, IFAAMAS, 9 pages.

*The first two authors contributed equally to this work.
Proc. of the 22nd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2023), A. Ricci, W. Yeoh, N. Agmon, B. An (eds.), May 29 – June 2, 2023,
London, United Kingdom. © 2023 International Foundation for Autonomous Agentsand Multiagent Systems (www.ifaamas.org). All rights reserved.

INTRODUCTION

Figure 1: FedAvgSGDA for CDAN with 2 clients. The ratios forsource and target data allocated to client 1 are 𝑝 and (1 − 𝑝 ). The
remaining data pertains to client 2. It shows that the performanceof FedAvgSGDA degrades rapidly as the data distribution becomes
imbalanced, which motivates our FedMM algorithm.

FedSGDA is FedAvgSGDA shown in Algorithm 3 in the appendix.
However, if the data are non-i.i.d among clients, especially in thecase of imbalanced label distributions [42], the performance of FedAvg would be significantly lower than that when all the data weretrained on a single client. As empirically demonstrated in Fig. 1, we
can see that the performance of FedAvgSGDA degrades rapidly as
the data distributions become more and more imbalanced.

Figure 2: A federated adversarial domain adaptation model. Only
the source risk of the client’s local source data (if any) and the domain risk of the client’s source/target data are accessible locally. The
source data labels are non-i.i.d.

FedMM. In light of the above challenges, to optimize a federated
minimax objective, we formulate this distributed saddle point optimization as a Federated MiniMax (FedMM) optimization on a sum of
non-identical distributions. In particular, we use an augmented Lagrange function to enforce the global model consensus constraints.
In each client’s local optimization oracle, FedMM deconstructs theglobal sum by solving the augmented Lagrange of each function
individually in a finite number of steps. The collection of Lagrange
dual variables will then locally compensate for the client-to-clientmodel divergence caused by domain shift. To summarize, our main
contributions are listed as follows.
Contributions:(i) We present, FedMM, a stochastic federated optimizer tailoredfor federated minimax optimizations with non-separable minimization and maximization variables, as well as clients with imbalancedlabel class distributions. FedMM is effective in the extreme casewhere each client has disjoint classes of labels or unlabeled data.
(ii) Under the generic federated saddle point optimization problem with a nonconvex-concave global objective function assumption, we prove that FedMM converges asymptotically to a stationary
point for the nonconvex-strongly-concave setting under local update residual errors and distribution shifts(iii) Empirically, we show that FedMM consistently achieves
either significant communication savings or accuracy improvements over the federated SGDA method on a variety of benchmark
datasets with varying adversarial domain adaptation networks. For
example, when training from scratch, it outperforms other SGDA
based federated average methods by around 20% in accuracy overthe same communication rounds.

2 PRELIMINARIES
2.1 Adversarial Domain Adaptation
Domain adaptation refers to the process of transferring knowledge
from a labeled source domain to an unlabeled target domain [2, 42].
Let P and Q be the source and target distributions, respectively.
In a general formulation, the upper bound of the target prediction
error is given by Ben-David et al. [2]err Q (𝜁 ) ≤ err P (𝜁 ) +𝑑 H (P, Q) + min {err P (𝜁 ∗ ) + err Q (𝜁 ∗ )}, (1)𝜁∗∈F

where err Q (𝜁 ) denotes the population loss of 𝜁 under the target
distribution Q, i.e., err Q (𝜁 ) ≜ E (x𝑖 ,y𝑖 )∼Q [ℓ (𝜁 (x𝑖 ), y𝑖 )], and we
use the parallel notation err P (𝜁 ) for the source domain error. Besides, 𝑑 H (P, Q) is a discrepancy-based distance, known as the
H -divergence, and min𝜁 ∗ ∈ F {err P (𝜁 ∗ ) + err Q (𝜁 ∗ )} is the optimal
joint error, i.e., the sum of source and target domain’s population
loss of 𝜁 in a hypothesis class F . For the unsupervised domain
adaptation problem, it has been proven that minimizing the upperbound, which is the r.h.s in (1), leads to an architecture consistingof a feature extractor parameterized by 𝜔, i.e., 𝜁𝜔1 , a label predictor,
parameterized also by 𝜔 i.e., 𝜁𝜔2 ( 𝜁𝜔 ≜ 𝜁𝜔2 ◦ 𝜁𝜔1 ), 1 and a domainclassifier parameterized by 𝜓 , i.e., ℎ𝜓 , as shown in Fig 2 [6, 43]. The
feature extractor generates the domain-independent feature representations, which are then fed into the domain classifier and label
predictor. The domain classifier then tries to determine whether
the extracted features belong to the source or target domain. Meanwhile, the label predictor predicts instance labels based on theextracted features of the labeled source-domain instances.
Minimizing the upper bound in (1) encourages the extracted
features to be both discriminative and invariant to changes between
the source and target domains. The upper bound minimization
corresponding to a saddle point over the parameter space of 𝜔 and𝜓 has been demonstrated using 𝜔b ≜ arg min𝜔 𝐿1 (𝜔) − 𝜈𝐿2 (𝜔, 𝜓b)band 𝜓 ≜ arg min𝜓 𝐿2 (b𝜔 ,𝜓 ) with an equivalent minimax form as
min max 𝐹 = min max 𝐿1 (𝜔) − 𝜈𝐿2 (𝜔,𝜓 ) .
𝜔

𝜓

𝜔

(2)

𝜓

In the majority of adversarial domain adaptation problems, 𝐿1 (𝜔) ≜E (x𝑖 ,y𝑖 )∼Q [ℓ (𝜁𝜔 (x𝑖 ), y𝑖 )] is the supervised learning loss on 𝜁 ,𝐿2 (𝜔,𝜓 )1 The parameters of 𝜁 1 and 𝜁 1 are not the same. In this case, we abuse the notation tosimplify the expression.

≜ E (x𝑖 )∼Q 𝐷𝑄 (ℎ𝜓 𝜁𝜔′ (x𝑖 ) − E (x𝑖 )∼P 𝐷 𝑃 (ℎ𝜓 𝜁𝜔′ (x𝑖 ) is the domain classification loss, and 𝜈 is the trade-off coefficient between
𝐿1 (𝜔) and 𝐿2 (𝜔,𝜓 ). With the commonly used cross-entropy loss
for 𝐿2 , we have 𝐷𝑄 (𝑥) ≜ 1 − log(𝑥) and 𝐷 𝑃 (𝑥) ≜ log(1 − 𝑥). Besides, 𝜁𝜔′ is the feature and ℎ𝜓 (·) : R𝐷 → [0, 1] is the probabilistic
prediction of the domain label. In general, 𝜁𝜔′ and ℎ𝜓 (·) include, but
is not limited to, the following cases: (i) Domain-Adversarial Neural
Networks (DANN) [6]: In DANN, the input of ℎ𝜓 (·) is designed

simply to be the domain invariant feature 𝜁𝜔1 (x𝑖 ), i.e., ℎ𝜓 𝜁𝜔1 (x𝑖 ) .
(ii) Margin Disparity Discrepancy (MDD) [41]: In MDD, the input ofℎ𝜓 (·) is the concatenation of 𝜁𝜔1 and arg max𝑐 𝜁𝜔 (x𝑖 ; 𝑐) with 𝑐 theclass type i.e., ℎ𝜓 𝜁𝜔1 (x𝑖 ) , arg max𝑐 𝜁𝜔 (x𝑖 ; 𝑐) . (iii) Conditional
Domain Adaptation Network (CDAN) [17]: In CDAN, the inputof ℎ𝜓 is from the cross-product space of 𝜁𝜔1 (x𝑖 ) and 𝜁𝜔 (x𝑖 ), i.e.,ℎ𝜓 𝜁𝜔1 (x𝑖 ) ⊗ 𝜁𝜔 (x𝑖 ) .
Our FedMM is a generic federated adversarial domain adaptation framework in which each client is equipped with ℎ𝜓 and 𝜁𝜔
depending on the availability of source data, target data, or both.
The objective function in an adversarial domain adaptation problem is determined by whether the data is from the source domainor the target domain, i.e.,ℓ (𝜁𝜔 (x𝑖 ) , y𝑖 ) + 𝜈 log(1−



(𝑖 )
𝐹𝑖 𝜔,𝜓 ; 𝜉 𝑗≜ℎ𝜓 𝜁𝜔′ (x𝑖 ) ),if 𝜉𝑖 ∈ P, 𝜈 log(ℎ𝜓 𝜁 ′ (x𝑖 ) ),if
𝜉𝑖 ∈ Q.
𝜔







2.2

(3)

Federated Learning under Domain Shifts𝑁

min max 𝑓 (𝜔,𝜓 ) ≜ min max
𝜔

1
𝛼𝑖
𝜓 𝑁
𝑖=1

= min max
𝜔

𝜔

𝜓
𝑁
∑︁

∑︁



(𝑖 )
𝐹𝑖 𝜔,𝜓 ; 𝜉 𝑗 ,3

FEDMM ALGORITHM

In this section, we look at the federated minimax problem by reformulating the centralized problem in (4) into the federated saddlepoint optimization problem with consensus constraints given by

(4)

(𝑖 )

𝜉 𝑗 ∈ D𝑖where 𝑁 is the number of clients, 𝑓𝑖 (𝜔,𝜓 ) is the loss functionat
the 𝑖-th client, 𝛼𝑖 is the weight coefficient, and 𝐹𝑖 𝜔,𝜓 ; 𝜉 𝑗 is the𝜔 0 ,𝜔𝑖 𝜓 0 ,𝜓𝑖

𝑓 (𝜔,𝜓 ) =𝜓𝑖 = 𝜓 0 ,(5)∀𝑖 ∈ [𝑁 ].

The corresponding augmented Lagrangian form for each client isdefined as
L𝑖 (𝜔 0, 𝜔𝑖 , 𝜆𝑖 ,𝜓 0,𝜓𝑖 , 𝛽𝑖 ) ≜ 𝑓𝑖 (𝜔𝑖 ,𝜓𝑖 ) + ⟨𝜆𝑖 , 𝜔𝑖 − 𝜔 0 ⟩
(6)
𝜇1
𝜇2
+ ∥𝜔𝑖 − 𝜔 0 ∥ 22 − ⟨𝛽𝑖 ,𝜓𝑖 − 𝜓 0 ⟩ − ∥𝜓𝑖 − 𝜓 0 ∥ 22, 𝜇1, 𝜇2 > 0.
2
2
The centralized optimization problem in (4) is then transformed
into a saddle-point minimax optimization of augmented Lagrangianfunctions over all primal-dual pairs, i.e., {𝜔𝑖 , 𝜔 0, 𝜆𝑖 ,𝜓𝑖 ,𝜓 0, 𝛽𝑖 } forall clients 𝑖 ∈ [𝑁 ]:


𝑁
𝑁
𝑁
𝑁
min max L {𝜔𝑖 }𝑖=0
, {𝜓𝑖 }𝑖=0
, {𝜆𝑖 }𝑖=1
, {𝛽𝑖 }𝑖=1
𝜔 0 ,𝜔𝑖 𝜓 0 ,𝜓𝑖

1 ∑︁L𝑖 (𝜔 0, 𝜔𝑖 ,𝜓 0,𝜓𝑖 , 𝜆𝑖 , 𝛽𝑖 ).
𝜔 0 ,𝜔𝑖 𝜓 0 ,𝜓𝑖 𝑁
𝑖=1
𝑁

(7)

≜ min max

By fixing the global consensus variables {𝜔 0,𝜓 0 }, the above problem
is separable w.r.t local pairs {𝜔𝑖 ,𝜓𝑖 , 𝜆𝑖 , 𝛽𝑖 } for all 𝑖 ∈ [𝑁 ]. And thedecomposed task could be independently updated on local clients
periodically without global communication. The only problem left
is to align the update of global consensus 𝜔 0,𝜓 0 and local updates
𝜔𝑖 ,𝜓𝑖 for all 𝑖 ∈ [𝑁 ]. Next, we demonstrate how to achieve distributed local updates and align local updates with global consensus.
By substituting (6) into (7), we obtain the augmented Lagrangianfunctions over all primal-dual parameters:


𝑁
𝑁
𝑁
𝑁
min max L {𝜔𝑖 }𝑖=0
, {𝜓𝑖 }𝑖=0
, {𝜆}𝑖=1, {𝛽}𝑖=1
𝜔𝑖(𝑖 )loss function w.r.t the data point 𝜉 𝑗 ≜ {x 𝑗 , y 𝑗 } with specific form
determined by whether the data is from the source domain or thetarget domain.
This novel problem structure introduces several unique challenges in federated adversarial domain adaptation that do not exist
in existing adversarial domain adaptation problems or the FL literature: (i) Clients are restricted to compute the minimax optimization1 ∑︁𝑓𝑖 (𝜔𝑖 ,𝜓𝑖 )𝑁 𝑖=1
𝑁

min maxs.t. 𝜔𝑖 = 𝜔 0,

We focus on the cross-silo FL adversarial domain adaptation problem, in which the training dataset is distributed across silos in a
multi-organizational context, such as in healthcare, banking, finance and so on, where institutions hold users’ data but cannot
share it directly with other institutions for collaborative learning.
As a result, federated adversarial domain adaptation addresses the
problem by training a model among clients from a labeled source
domain to an unlabeled target domain. A centralized server coordinates between the clients to solve the learning task. To express thefederated adversarial domain adaptation objective, we convert thejoint learning objective of (2) into the form of a centralized averageof all the clients’ objective functions, as given by1 ∑︁
𝑓𝑖 (𝜔,𝜓 )𝜓 𝑁
𝑖=1

in a distributed manner rather than the centralized minimax optimization. (ii) To train a centralized model, both the set of feature
extractor variables 𝜔 and domain classifier variables 𝜓 are nonseparable cross clients.
(iii) The marginal label distributions are class-imbalanced acrossclients due to the imbalanced distribution of source domain data
and target domain data. In extreme cases, each client may only haveaccess to data from the target domain or the source domain, but
not both.
To address the above unique challenges in federated domain
adaptation, we propose FedMM, a general algorithm that works forminimax optimization under FL. FedMM is designed for imbalancedlabel classes among clients in federated minimax training, a uniqueproblem in domain adaptation.

𝜓𝑖

𝑁
1 ∑︁


min max 𝑓𝑖 (𝜔𝑖 ,𝜓𝑖 ) + ⟨𝜆𝑖 , 𝜔𝑖 − 𝜔 0 ⟩(8)
𝑁 𝑖=1 𝜔𝑖 𝜓𝑖
𝜇2
𝜇1+ ∥𝜔𝑖 − 𝜔 0 ∥ 22 − ⟨𝛽𝑖 ,𝜓𝑖 − 𝜓 0 ⟩ − ∥𝜓𝑖 − 𝜓 0 ∥ 22 ,22
where 𝜇1 and 𝜇 2 are the penalty parameters. The minimax optimization w.r.t the global consensus variable 𝜔 0 and 𝜓 0 is given
=

by:
1
𝑓𝑖 (𝜔𝑖 ,𝜓𝑖 ) + ⟨𝜆𝑖 , 𝜔𝑖 − 𝜔 0 ⟩
𝑁 𝑖=1

𝜇1
𝜇2
+ ∥𝜔𝑖 − 𝜔 0 ∥ 22 − ⟨𝛽𝑖 ,𝜓𝑖 − 𝜓 0 ⟩ − ∥𝜓𝑖 − 𝜓 0 ∥ 22
2
2

𝑁 
1 ∑︁
1
=
𝜔𝑖 + 𝜆𝑖 ,
𝑁 𝑖=1
𝜇1

𝜔
b0 = arg min{𝜔 0,𝜓 0 } in (16). After one round of global communication. The
global coordinated value of {𝜔 0𝑡 +1,𝜓 0𝑡 +1 } is then broadcasted backto each client, triggering next-round updates. The summarized
diagram and algorithm of FedMM is shown in Fig. 3 and Algorithm 1

𝑁 
∑︁

𝜔0

Algorithm 1 FedMM Algorithm(9)

where the closed-form solution is due to the quadratic optimization.
Similarly, we obtain

𝑁 
1 ∑︁
1
𝜓b0 =
𝜓𝑖 + 𝛽𝑖 .
(10)
𝑁 𝑖=1
𝜇2
Eqn. (9) and (10) provide guidance for local update alignment with
global consensus. More specifically, in each round, we optimize
each client’s individual 𝜔𝑖 and 𝜓𝑖 , by fixing the global consensus
constraints (𝜔 0 and 𝜓 0 ) and dual parameters (𝜆𝑖 and 𝛽𝑖 ). Taking
the (𝑡 + 1)-th round update as an example. Client 𝑖 receives theglobal parameters {𝜔 0𝑡 ,𝜓 0𝑡 } from the server and sets local parameters 𝜔
b𝑖0 = 𝜔 0𝑡 , 𝜓b𝑖0 = 𝜓 0𝑡 . 2 Then, the local saddle-point optimizationof (8) w.r.t {𝜔𝑖 ,𝜓𝑖 } is updated by multiple-step SGDA to reduce thecommunication rounds between a client and the server:𝜔b𝑖𝑚+1 = 𝜔b𝑖𝑚 − 𝜂 1 ∇𝜔𝑖 L𝑖 (b𝜔𝑖𝑚 , 𝜓b𝑖𝑚 )hi(11)= 𝜔𝑖𝑚 − 𝜂 1 ∇𝜔𝑖 𝑓𝑖 (b𝜔𝑖𝑚 , 𝜓b𝑖𝑚 ) + 𝜇1 (b𝜔𝑖𝑚 − 𝜔 0𝑡 ) + 𝜆𝑖𝑡𝜔𝑖𝑚 , 𝜓b𝑖𝑚 )𝜓b𝑖𝑚+1 = 𝜓b𝑖𝑚 + 𝜂 2 ∇𝜓𝑖 L𝑖 (bhi𝜔𝑖𝑚 , 𝜓b𝑖𝑚 ) − 𝜇2 (𝜓b𝑖𝑚 − 𝜓 0𝑡 ) − 𝛽𝑖𝑡 ,= 𝜓b𝑖𝑚 + 𝜂 2 ∇𝜓𝑖 𝑓𝑖 (b

(12)

𝑁 ,𝑇
Require: Initialize 𝜔 00,𝜓 00, 𝜇1, 𝜇2, 𝜂 1, 𝜂 2, 𝜂 3, {𝑀𝑖 }𝑖=0
1: for 𝑡 = 0, . . . ,𝑇 − 1 do
2:for each client 𝑖 ∈ [𝑁 ] in parallel do3:𝜔b𝑖0 = 𝜔 0𝑡 ;
𝜓b𝑖0 = 𝜓 0𝑡
4:
# Local Update:5:
for 𝑚 = 0, . . . , 𝑀𝑖 − 1 do
6:
# Stochastic Gradient Descent:
7:
8:
9:
10:
11:
12:
13:
14:
15:

𝜔
b𝑖𝑚+1 = 𝜔b𝑖𝑚 − 𝜂 1 [ ∇𝜔𝑖 𝑓𝑖 (𝜔b𝑖𝑚 , 𝜓b𝑖𝑚 ) + 𝜇 1 (𝜔b𝑖𝑚 − 𝜔 0𝑡 ) + 𝜆𝑖𝑡 ]# Stochastic Gradient Ascent𝜓b𝑖𝑚+1 = 𝜓b𝑖𝑚 + 𝜂 2 [ ∇𝜓𝑖 𝑓𝑖 (𝜔b𝑖𝑚 , 𝜓b𝑖𝑚 ) − 𝜇2 (𝜓b𝑖𝑚 − 𝜓 0𝑡 ) − 𝛽𝑖𝑡 ]end for𝜔𝑖𝑡 +1 = 𝜔b𝑖𝑀𝑖 ; 𝜓𝑖𝑡 +1 = 𝜓b𝑖𝑀𝑖# Dual Descent:𝜆𝑖𝑡 +1 = 𝜆𝑖𝑡 + 𝜇1 (𝜔𝑖𝑡 +1 − 𝜔 0𝑡 )# Dual Ascent:𝛽𝑖𝑡 +1 = 𝛽𝑖𝑡 + 𝜇 2 (𝜓𝑖𝑡 +1 − 𝜓 0𝑡 )𝜂𝑡𝜂𝑡

𝜔𝑖𝑡 + = 𝜔𝑖𝑡 +1 + 𝜇31 𝜆𝑖𝑡 +1 ; 𝜓𝑖𝑡 + = 𝜓𝑖𝑡 +1 + 𝜇32 𝛽𝑖𝑡 +1
17:
end for18:# Global Update:Í𝑁 𝑡 +
Í𝑁 𝑡 +
19:
𝜔 0𝑡 +1 = 𝑁1 𝑖=1
𝜔𝑖 ;𝜓 0𝑡 +1 = 𝑁1 𝑖=1𝜓𝑖
20: end for

16:

where 𝑚 ∈ [𝑀𝑖 ]. We denote 𝜔𝑖𝑡 +1 = 𝜔b𝑖𝑀𝑖 and 𝜓𝑖𝑡 +1 = 𝜓b𝑖𝑀𝑖 for the
results of 𝑀𝑖 -th step local update. The dual parameters are thenupdated using SGDA by
𝜆𝑖𝑡 +1 = 𝜆𝑖𝑡 + 𝜇 1 (𝜔𝑖𝑡 +1 − 𝜔 0𝑡 ),(13)

𝛽𝑖𝑡 +1 = 𝛽𝑖𝑡 + 𝜇2 (𝜓𝑖𝑡 +1 − 𝜓 0𝑡 ).

(14)

To align with the global consensus constraint obtained in (9) and(10), we set
𝜂𝑡
𝜂𝑡
(15)
𝜔𝑖𝑡 + = 𝜔𝑖𝑡 +1 + 3 𝜆𝑖𝑡 +1 ; 𝜓𝑖𝑡 + = 𝜓𝑖𝑡 +1 + 3 𝛽𝑖𝑡 +1 .
𝜇1𝜇2Note that different from vanilla augmented Lagrangian, we introduce the decay factor 𝜂 𝑡3 < 1, which helps FedMM converge withsmaller local steps. Therefore, the global consensus constraint issatisfied by the global update at the server with𝑁𝑁
1 ∑︁ 𝑡 +1 ∑︁ 𝑡 +(16)
𝜔 0𝑡 +1 =𝜔𝑖 , and 𝜓 0𝑡 +1 =
𝜓 .
𝑁 𝑖=1
𝑁 𝑖=1 𝑖
We can now summarize one round of the FedMM algorithm, which
consists of three major steps: (i) Parallel saddle-point optimizationon all local augmented Lagrangian function L𝑖 ’s. One optimization
oracle example is based on SGDA, as shown in (11) and (12). (ii) Local
gradient descent and ascent updates on dual variable ({𝛽𝑖 , 𝜆𝑖 }) as
shown in (14). (iii) Aggregation to update global consensus variables
2We use { 𝜔b𝑖 , 𝜓b𝑖 } to denote the local iterative updates for {𝜔𝑖 ,𝜓𝑖 } to differentiate

symbols of local updates and global updates.

Figure 3: The FedMM algorithm addresses the federated adversarial domain adaptation as shown in the flowchart. Eachsource and target client has unique local minimax objectives
due to domain distribution differences. Clients conduct local
optimization, upload parameters to the server, and receive averaged parameter updates in parallel, completing one-round
updates.

data from distinct domains. We will focus on the 𝑝 = 1 case for theremainder of the experiment to test the effectiveness of FedMM.
Unlike the traditional augmented Lagrangian method, FedMM
introduces 𝜂 3 < 1 for the FL setting to reduce the need for large
local update steps 𝑀𝑖 for convergence. When 𝜂 3 = 1, as shown in
Fig.6, large local steps with 𝑀𝑖 > 50 are required. With appropriate𝜂 3 , one can reduce 𝑀𝑖 from 50 to 25 with negligible performanceloss for all three adversarial domain adaptation models. Note thatif 𝜂 3 is less than the feasible range, the outcome will be suboptimal.

4.2
Figure 4: FedMM is robust to label imbalance. Experiments withthe same setting as that in Fig. 1.

4

EXPERIMENTS

Our experiments have three main goals. (1) We create a federated domain adaptation benchmark that is communication-efficientby combining three representative domain adaptation methods:
DANN [6], MDD [41], and CDAN [17]. (2) We demonstrate thestrength of our proposed FedMM algorithm on the Federated domain adaptation benchmark, showing how our proposed FedMM
improves model generalization while significantly reducing communication rounds. (3) We investigate the contributions of various
components of our algorithm, such as multi-step local updates,
proximal terms in local objectives, dual variables, and the choiceof 𝜂 3 . Our experiments are primarily concerned with the training
communication overhead and test accuracy on the label-free target
data set. Appendix B contains detailed descriptions of the dataset
used in this experiment.
Experiment Setup. On MNISTM, we use a three-layer convolutional network as the invariant feature extractor, and the networkmodels are trained from random initialization on server. On Office31, we use the pre-trained MobileNetV2 [28] on ImageNet [27] as
the feature extractor. For a fair comparison, the pre-trained MobileNetV2 is downloaded from the pre-trained one by [28]. Boththe task classifier and the domain classifier are two-layer fullyconnected neural networks. The domain classifier’s parameterare trained from random initialization in all settings. The hyperparameter settings are provided in Appendix C and our code isavailable atgithub.com/fedmm/FedMM.

4.1

Ablation Experiment on FedMM

We first investigate how a change in label imbalance affects modeltraining performance. Consider the case of two clients and set theratios for source and target data assigned to client 1 as 𝑝 and 1 − 𝑝,respectively. The commonly used adversarial domain adaptation
models including DANN [6], CDAN [41] , and MDD [17] are tested
separately as the local model. The label imbalance degree varies
from 𝑝 = 0.5 to 𝑝 = 1. FedMM is robust to variations in label
distribution, as shown in Fig. 4, and performs well even in theworst-case scenario, in which the source domain data and target
domain data are allocated to different clients separately, i.e., 𝑝 = 1.
In practice, 𝑝 = 1 occurs frequently because different silos contain

FedMM is the State-of-the-Art of Federated
Domain Adaptation

With extensive experiments, we show that FedMM achieves SOTA
performance in terms of accuracy and communication overhead.
We include the following two kinds of baselines:(i) Recent work on distributed minimax optimization including [4,26, 30, 35] with extensive studies for different adversarial domain
adaptation tasks shown in Table 1. As MDD is the SOTA networkof centralized domain adaptation in current stage, we only compare
these works on federated domain adaptation with MDD loss.
(ii)Peng et al.[22] proposed FedSGDA to extend FedSGD to SGDA
for federated domain adaptation, where the single step per communication round in SGDA resulted in massive communication
overhead as observed in experiments. Though, to the best of ourknowledge, there is no communication-efficient federated domain
adaptation algorithm. To reduce communication overhead, we inspire from the existing multi-step federated minimization optimizers like FedAvg [20], and FedProx [14], which leads to FedAvgSGDAand FedProxSGDA in Algorithm 3 as described in Appendix A.
Performance comparison: FedMM has far fewer global communication rounds than FedSGDA, as illustrated in Fig.6. FedMM
saves more than 90% of the rounds required to achieve the samelevel of test accuracy on the target domain as FedSGDA in all three
categories of representative domain adaptation networks due tothe deliberately designed FedMM’s local multi-step minimax optimization at each client.
Moreover, FedMM not only reduces communication overhead
but also ensures accuracy. Gradient drift causes severe performancedegradation in multi-step local SGDA on existing baselines. In Fig. 7,
we compare the convergence property of our proposed FedMM
to FedAvgSGDA and FedProxSGDA with 𝑀𝑖 = 20 for different
source/target client settings. While both the FedAvgSGDA and
FedProxSGDA algorithms converge in a communication-saving
manner at the expense of severe performance decay, FedMM consistently outperforms them by more than 20% in terms of test accuracy
for training from scratch with all the three domain adaptation networks. The results clearly show how that FedMM addresses the
problem of gradient drifts in multi-steps local minimax, which havenot been observed in any previous FL problems other than federated
adversarial domain adaptation.
We compare FedMM to all of the above baselines on commonlyused domain adaptation tasks on Office-31, including 𝐴 → 𝑊 , etc.
six tasks with pretrained model, i.e., MobileNetV2 [28] on ImageNet[27]. When compared to the training from scratch case in Fig. 7, it isexpected that FedMM’s performance improvement will be reduced.

(a) DANN(b) MDD(c) CDAN

Figure 5: Accuracy heatmap using various pairs of 𝜂 3 and 𝑀𝑖 . The blank sections represent the settings from which FedMM deviates.

(a) DANN(b) MDD(c) CDAN

Figure 6: Comparisons of convergence for the proposed FedMM with FedSGDA [22] on different number of local steps of 𝑀𝑖 = 20 and 𝑀𝑖 = 25.
The comparison is based on different adversarial domain networks, i.e., DANN, MDD, and CDAN. Models are trained from scratch on
MNISTM.
Because the feature extractor parameters in these pre-trained models have approached optimal values. FedMM achieves SOTA withsignificant accuracy improvement while save much communication overhead over single step local update method. Interestingly,
even though other multi-step local update distributed minimax
optimization methods including [4, 26, 30] save the communication overhead, they do not have a evident performance gain over
FedAvgSGDA.

5

CONVERGENCE ANALYSIS

For a theoretical analysis, finding a global saddle point, i.e.,
min𝑥 max𝑦 𝑓 (𝑥, 𝑦), in general is intractable [16]. One approach is
to equivalently
reformulate the problem bymin𝑥 Φ(𝑥) := max𝑦 ∈ Y 𝑓 (𝑥, 𝑦) , and define an optimality notion
for the local surrogate of global optimum of Φ. A series of theoreticalanalyses on the stationary point convergence condition of Φ with
first-order algorithm were carried out to extend the convex-concave
assumption to assumptions of nonconvex-strongly-concave [18, 24,34], nonconvex-concave [16, 21], and nonconvex-nonconcave [10].
Convergence analysis for a federated optimizer, such as FedMM thatinvolves bounding client’s drift from global parameter via primaldual method, on the other hand, is more challenging. We establish
our main convergence results and show that FedMM converges
asymptotically to the stationary point for the nonconvex-stronglyconcave case.
Let 𝜓 ★ (𝜔) ≜ arg max𝜓 𝑓 (𝜔,𝜓 ) be the optimal value of 𝜓 for the
global objective function 𝑓 w.r.t 𝜔. Then (4) can be reformulated asmin𝜔 𝑓 (𝜔,𝜓 ) = min𝜔 𝑁1𝑖 Φ𝑖 (𝜔) with

Í1 ∑︁Φ𝑖 (𝜔).
𝑁 𝑖=1
𝑁

Φ𝑖 (𝜔) ≜ 𝑓𝑖 (𝜔,𝜓 (𝜔)),★

Φ(𝜔) ≜(17)

In thisway, we equivalently reformulate the problem asmin𝜔 Φ(𝜔) = max𝜙 𝑓 (𝜔, 𝜙) . To ease the presentation, we further
define the augmented Lagrange function of Φ𝑖 by
𝜇1 𝑡
2
L𝑖Φ (𝜔𝑖𝑡 , 𝜔 0𝑡 , 𝜆𝑖𝑡 ) = Φ𝑖 (𝜔𝑖𝑡 ) + ⟨𝜆𝑖𝑡 , 𝜔𝑖𝑡 − 𝜔 0𝑡 ⟩ +𝜔𝑖 − 𝜔 0𝑡 . (18)2

5.1

Assumptions

Note that we concentrate on the convergence analysis of the federated nonconvex-strongly-concave case, which is difficult even in acentralized setting and has recently received increased attention in
the literature [10, 16, 19]. Thus, we set the standard assumptions
by following the minimax optimization literature [10, 15, 16, 19] to
impose customary conditions on the gradients of local functions.
Assumption 1. (Lipschitz continuous gradients) For all 𝑖 ∈ [𝑁 ],there exists positive constants 𝐿11 , 𝐿12 , 𝐿21 , and 𝐿22 such that for any
𝜔, 𝜔 ′ ∈ R𝑑1 , and 𝜓,𝜓 ′ ∈ R𝑑2 , we have∇𝜔𝑖 𝑓𝑖 (𝜔,𝜓 ) − ∇𝜔𝑖 𝑓𝑖 (𝜔 ′,𝜓 ) ≤ 𝐿11 𝜔 − 𝜔 ′∇𝜔𝑖 𝑓𝑖 (𝜔,𝜓 ) − ∇𝜔𝑖 𝑓𝑖 (𝜔,𝜓 ′ ) ≤ 𝐿12 𝜓 − 𝜓 ′ ,∇𝜓𝑖 𝑓𝑖 (𝜔,𝜓 ) − ∇𝜓𝑖 𝑓𝑖 (𝜔 ′,𝜓 ) ≤ 𝐿21 𝜔 − 𝜔 ′∇𝜓𝑖 𝑓𝑖 (𝜔,𝜓 ) − ∇𝜓𝑖 𝑓𝑖 (𝜔,𝜓 ′ ) ≤ 𝐿22 𝜓 − 𝜓 ′ .

(a) DANN: 1-source/1-target clients(b) MDD: 1-source/1-target clients(c) CDAN: 1-source/1-target clients(d) DANN: 1-source/2-target clients(e) MDD: 1-source/2-target clients(f) CDAN: 1-source/2-target clients(g) DANN: 2-source/1-target clients(h) MDD: 2-source/1-target clients(i) CDAN: 2-source/1-target clients

Figure 7: Comparisons of convergence for the proposed FedMM with FedAvgSGDA and FedProxSGDA. In the legend, we use
FedAvg+ and FedProx+ to denote FedAvgSGDA and FedProxSGDA, respectively, for simplification. Models are trained fromscratch on MNISTM with different number of source and target clients
Assumption 2. (Strongly concave 𝑓𝑖 (·,𝜓𝑖 )) For all 𝑖 ∈ [𝑁 ], 𝑓𝑖 (𝜔,𝜓 )are strongly concave on 𝜓 with constant 𝐵 > 0 such that for any𝜔 ∈ R𝑑1 , and 𝜓,𝜓 ′ ∈ R𝑑2 , we have∇𝜓 𝑓𝑖 (𝜔,𝜓 ) − ∇𝜓 𝑓𝑖 (𝜔,𝜓 ′ ),𝜓 − 𝜓 ′ ≤ −𝐵 𝜓 − 𝜓 ′2.

Assumption 3. The 𝜅-Lipschitz continuity of 𝜓 ★ (𝜔), i.e.,𝜓 ★ 𝜔𝑖𝑡 −1 − 𝜓 ★ 𝜔𝑖𝑡 ≤ 𝜅 𝜔𝑖𝑡 −1 − 𝜔𝑖𝑡 , ∀𝑡 ∈ [𝑇 ].

(19)

Theorem 1. (Convergence on Φ(𝜔)) With Assumption 1, 2, 20 and
4 holds. Then there exist positive constants 𝐸 1 , 𝐸 2 , and 𝐸 3 , which are
independent of 𝑇 , such that after 𝑇 rounds of global updates, the upperbound for the accumulate descent of Φ(𝜔 0𝑡 ) is given byΦ(𝜔 00 ) − Φ(𝜔𝑇0 ) ≤ −𝐸 1

Assumption 4. (Sufficient local training) For all 𝑖 ∈ [𝑁 ], after𝑀𝑖 -step update, the gradients w.r.t. 𝜔𝑖 and 𝜓𝑖 are finite and denotedby∇L𝑖 (𝜔𝑖𝑡 ,𝜓𝑖𝑡 ) ≤ 𝜖 ∀𝑡 ∈ [𝑇 ].
(21)∇Φ 𝜔 0𝑡

 2+ 𝐸 3𝑇 𝜖

𝑡 =1

(20)

Next, we make the following assumptions that 𝑀𝑖 in FedMM ischosen that local objective are sufficiently trained that 𝜔𝑖𝑡 ,𝜓𝑖𝑡 is 𝜖stationary on L𝑖 .

𝑇
∑︁

+𝐸 2

𝑁
∑︁
𝑖=1

𝜓𝑖0 − 𝜓 ★ (𝜔𝑖0 )

2

+ 𝐸3

∑︁

𝜔𝑖0 − 𝜔 0𝑗

(22)

2

.

𝑖≠𝑗

 
In particular, this implies lim sup𝑡 →∞ ∇Φ 𝜔 0𝑡 = 𝑂 (𝜖) with alocal residue gradient error bound, i.e., ∇L𝑖 (𝜔𝑖𝑡 ,𝜓𝑖𝑡 )

2

≤ 𝜖..

Remark Because the l.h.s. of (22) admits a lower bound, so is  2Íthe r.h.s. As a result, lim sup𝑇 →∞ 𝑇𝑡=1 ∇Φ 𝜔 0𝑡
must converge,

Office-31 Domain Adaptation Tasks
𝐴 →𝑊
Optimizer

𝐷 →𝑊

𝑊 →𝐷

𝐴→𝐷

𝐷 →𝐴

𝑊 →𝐴

Network

CR↓

ACC↑

CR↓

ACC↑

CR↓

ACC↑

CR↓

ACC↑

CR↓

ACC↑

CR↓

ACC↑

FedAvgSGDA

DANN
CDAN
MDD

10
17
31

60.1
62.9
73.2

13
13
27

86.1
86.8
93.6

8
7
11

93.6
94.2
97.8

7
21
22

63.5
65.1
72.1

24
34
31

33.7
40.3
47.9

18
14
18

40.5
45.5
51.7

FedSGDA†

DANN
CDAN
MDD

59
78
255

60.3
55.3
76.4

40
49
188

84.9
83.4
94.7

29
16
92

93.7
94.0
98.3

56
13
400

65.3
67.7
75.3

48
95
300

36.9
47.1
49.2

88
85
321

40.3
43.3
52.7

[4]

MDD

26

72.9

32

92.5

21

96.3

25

71.9

30

47.2

20

51.0

[26]

MDD

35

71.5

30

93.0

15

95.1

25

73.0

35

48.8

22

51.9

[30]

MDD

−

−

29

82.3

17

78.9

−

−

−

−

−

−

[35]†

MDD

150

77.0

121

95.0

62

98.0

220

74.0

210

51.2

253

50.7

FedMM

DANN
CDAN
MDD

13
29
23

65.5
64.7
79.7

15
9
18

89.6
93.4
95.9

14
10
19

96.7
94.0
98.5

7.5
32
22

67.8
66.9
78.8

39
17
19

44.3
51.4
60.3

25
13
15

48.7
59.6
55.5

Table 1: Comparison of communication round (CR) (×100) and accuracy (acc) for different optimizers on Office-31 data sets fordifferent domain adaptation tasks (𝐴 → 𝑊 ,. . . , 𝑊 → 𝐴). The Optimizer’s name labeled with † denotes the method with only
one step local update in each CR, and all others involve multiple local updates. We uniformly use a local steps of 𝑀𝑖 = 10 inmulti-step local update algorithms. The symbol - denotes the optimizer fails to converge.

 which implies that Φ 𝜔 0𝑡 converges to a 𝜖-stationary point. Morespecifically, dividing both sides of (22) by 𝑇 and taking lim sup𝑇 →∞ ,Í𝑇∥ ∇Φ (𝜔 𝑡 )∥ 2
we obtain lim sup𝑇 →∞ 𝑡 =1 𝑇 0≤ 𝐸𝐸31𝜖 , which implies that
  2
  2
Í𝑇
𝑡
and
for
sufficiently
large
𝑡,
∇Φ
𝜔
=
𝑂
(𝑇
𝜖)
∇Φ
𝜔 0𝑡
=
𝑡 =1
0
𝑂 (𝜖). In the special case of 𝜖 = 0, i.e., strict optimality is obtainedat each local client, this result shows that the limiting point is astationary point.

6

RELATED WORK

Distributed minimization: Since the invention of FedSGD and
its communication efficient version FedAvg [20], several work havebeen developed to address the suboptimality of FedAvg over noni.i.d data, including FedProx [14], FedPD [40], SCAFFOLD [12],FedNova [32], dynamic gradient aggregation [5], and FedDyn [1].
These works aim to minimize a sum of non-identical functions,where each function can only be accessed locally. Auto-FedAvg [33]adjusted weights at the aggregation during training. These resultscannot be directly applied to federated saddle point optimization
problems, such as the federated adversarial domain adaptation,
which seeks a federated minimax optimization. Similarly, off-theshelf distributed augmented Lagrangian minimization methods
and convergence analyses in Jakovetić et al. [8, 9], Yue et al. [37]fails to address the unique challenges in FL minimax optimization
including the domain adaptation problem.
Distributed minimax: Several works [4, 25, 26] have improved
communication efficiency in FedAvg-based minimax optimization,
including the federated GAN. However, FedAvgSGDA is sensitive todata imbalance in our federated domain adaptation problem, unlikein the federated GAN where the binary classification function works
well since there is no label-imbalanced problem. Note that the FLRA
algorithm in Reisizadeh et al. [26] also corresponds to FedAvgSGDA,
and its convergence analysis cannot be directly adapted to our
setting because each local client in our study is optimized on theaugmented Lagrangian local function rather than the pure local risk
minimization objective. There are also works on federated domain
adaptation with the specific assumption that a centralized labeleddataset is available on the server [36]. Other works, including [30,35, 39], look into fully distributed minimax optimization without acentralized server. Note that the work in Zhang et al. [39] assumes
only one participating client each time which is not realistic underFL settings.

7

CONCLUSIONS

We have proposed FedMM for federated adversarial domain adaptation. FedMM is designed specifically for federated minimax optimizations with non-separable minimization and maximization
variables, as well as clients with uneven label class distributions.
We have symptomatically performed a theoretical analysis on theconvergence property of our proposed FedMM. Experiments showthat FedMM outperforms state-of-the-art algorithms in terms ofcommunication rounds and test accuracy on various benchmark
datasets.

8

ACKNOWLEDGEMENT

This research was supported in part by NSF through grant IIS1910492. Han Zhao would like to thank the support from a Facebook
research award and Amazon AWS Cloud Credits. Jian Du’s work
was conducted during his time at Ant Group.

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