ecReferences

Mao et al.

Adjustable Robust Optimization via Fourier-Motzkin Elimination

Early Detection of Misinformation for Infodemic Management: A Domain
Adaptation Approach

Minjia Mao University of Delaware

Xiaohang Zhao∗ Shanghai University of Finance and Economics

Xiao Fang∗ University of Delaware

∗Corresponding author

An infodemic refers to an enormous amount of true information and
misinformation disseminated during a disease outbreak. Detecting
misinformation at the early stage of an infodemic is key to manage it and
reduce its harm to public health. An early stage infodemic is characterized by
a large volume of unlabeled information concerning a disease. As a result,
conventional misinformation detection methods are not suitable for this
misinformation detection task because they rely on labeled information in the
infodemic domain to train their models. To address the limitation of
conventional methods, state-of-the-art methods learn their models using
labeled information in other domains to detect misinformation in the infodemic
domain. The efficacy of these methods depends on their ability to mitigate
both covariate shift and concept shift between the infodemic domain and the
domains from which they leverage labeled information. These methods focus on
mitigating covariate shift but overlook concept shift, rendering them less
effective for the task. In response, we theoretically show the necessity of
tackling both covariate shift and concept shift as well as how to
operationalize each of them. Built on the theoretical analysis, we develop a
novel misinformation detection method that addresses both covariate shift and
concept shift. Using two real-world datasets, we conduct extensive empirical
evaluations to demonstrate the superior performance of our method over state-
of-the-art misinformation detection methods as well as prevalent domain
adaptation methods that can be tailored to solve the misinformation detection
task.

misinformation detection, infodemic management, domain adaptation, covariate
shift, concept shift, deep learning, contrastive learning, transfer learning

##  1 Introduction

An infodemic refers to an overwhelming volume of true and false information
spread during a disease outbreak (Van Der Linden 2022). Misinformation, or
false information, in an infodemic misleads the public about the disease and
causes significant harm to public health (Imhoff and Lamberty 2020, Freeman et
al. 2022, Van Der Linden 2022). For example, the misinformation that the Ebola
virus is intentionally created by the Congolese government to eliminate people
in the city of Beni has led to attacks on Ebola clinics by local residents,
hindering timely treatments for Ebola.111See
https://news.un.org/en/story/2019/03/1034381 (last accessed on May 28, 2024)
More recently, during the outbreak of the coronavirus disease (COVID-19), a
significant amount of misinformation has been diffused on social media.
Examples include the misinformation that baking soda can cure
coronavirus,222See https://apnews.com/article/archive-fact-checking-8736262219
(last accessed on May 28, 2024) the false claim that injecting disinfectant
can prevent the virus (Borah et al. 2022), and the incorrect attribution of
the outbreak in Italy to Middle East illegal immigrants.333See
https://time.com/5789666/italy-coronavirus-far-right-salvini/ (last accessed
on May 28, 2024) Such widespread of misinformation about COVID-19 has caused
social panic, led people to dismiss health guidance, and weakened their
confidence in vaccines, ultimately undermining pandemic response efforts
(Bursztyn et al. 2020). Therefore, it is imperative to manage an infodemic to
mitigate its adverse impact on public health (Freeman et al. 2022, Hwang and
Lee 2024).

According to the World Health Organization (WHO), infodemic management is the
use of risk- and evidence-based approaches to manage an infodemic and reduce
its harm to public health.444See https://www.who.int/health-
topics/infodemic#tab=tab_1 (last accessed on May 28, 2024) The key to
effective infodemic management is the detection of misinformation at the early
stage of an infodemic (Buchanan 2020, Van Der Linden 2022). Early
identification of misinformation discourages people from sharing it and
prevents it from reaching a much larger population, thereby mitigating its
potential harm to public health (Buchanan 2020). Moreover, the likelihood that
a person trusts misinformation increases as the person is exposed longer to
the misinformation (Zajonc 2001, Moravec et al. 2019). Early identification of
misinformation reduces individuals’ exposure time to misinformation, thereby
lowering their likelihoods of trusting it. This, in turn, increases their
chances of adhering to health guidance, ultimately benefiting public health as
a whole.

An early stage infodemic features two salient characteristics. First, an
emerging health event triggers a huge amount of true and false information
spreading on various media platforms in a short time period. For example,
during the early stage of the COVID-19 infodemic, a significant volume of true
and false information concerning COVID-19 is diffused on social media (Yue et
al. 2022), and the WHO declares a worldwide infodemic (Van Der Linden 2022).
Second, it requires expert knowledge to distinguish between true information
and misinformation in an infodemic. Moreover, during the early stage of a
disease outbreak and its ensuing infodemic, even experts have no or limited
knowledge about the disease, making it even more difficult to discern true
information from misinformation. For example, during the early stage of the
Ebola outbreak, scientists lack knowledge about the disease and its treatment
(Adebimpe et al. 2015). Consequently, it is common that information
disseminated at the early stage of an infodemic is unlabeled. As there is no
labeled information (i.e., true or false), it is impossible to learn a
misinformation detection model using conventional misinformation detection
methods such as Abbasi et al. (2010), Nan et al. (2021), and Wei et al.
(2022), which require labeled information in their training data.

A viable solution is to leverage labeled information in other domains to build
a misinformation detection model for the infodemic domain. Indeed, popular
fact-checking websites like PolitiFact555See https://www.politifact.com/ (last
accessed on May 28, 2024) and Gossipcop666See
https://en.wikipedia.org/wiki/Gossip_Cop (last accessed on May 28, 2024)
contain a wealth of verified true and false information in domains such as
politics and entertainment. Moreover, misinformation across domains shares
some common characteristics. For example, compared to true information,
misinformation spreads faster, farther, and more widely (Oh et al. 2013,
Vosoughi et al. 2018). As another example, fake news, an important type of
misinformation, often features emotional headlines and demonstrates
inconsistency between their headlines and contents (Siering et al. 2016).
Because of these common characteristics shared across domains, it is possible
to learn a misinformation detection model using labeled information in other
domains and transfer it to the infodemic domain. However, information in each
domain also has its unique characteristics. Understandably, the choice of
words and the organization of words to covey sentiments and semantics differ
across domains. For example, less than 20% of words used in politics news
overlaps with those found in entertainment news (Shu et al. 2020). Because
each domain possesses unique characteristics, a misinformation detection model
learned with labeled information in other domains usually performs poorly in
the infodemic domain.

Domain adaptation is a widely employed technique to enhance the performance of
a model trained in one domain but applied to another (Ben-David et al. 2010).
It aims to mitigate disparities between data in different domains such that a
model learned using data in one domain performs well in another domain (Ben-
David et al. 2010). Concretely, data in a domain is represented as a joint
distribution $p(\textbf{x},y),\textbf{x}\in\mathcal{X},y\in\mathcal{Y}$, where
$\mathcal{X}$ denotes the feature space and $\mathcal{Y}$ is the label space
(Liu et al. 2022). In the context of misinformation detection, x might be
explicit features, such as the number of words in a piece of information, or
latent features, such as the embedding of a piece of information (Zhou and
Zafarani 2020). Label $y$ denotes whether a piece of information is true or
false. The join distribution $p(\textbf{x},y)$ can be expressed as the product
of $p(\textbf{x})$ and $p(y|\textbf{x})$, i.e.,
$p(\textbf{x},y)=p(\textbf{x})p(y|\textbf{x})$. Accordingly, to effectively
alleviate disparities between data in different domains, it is essential to
reduce both their differences in terms of $p(\textbf{x})$ and their
differences in terms of $p(y|\textbf{x})$, where the former type of
differences is referred to as covariate shift and the latter type of
differences is known as concept shift (Liu et al. 2022). Existing domain
adaptation methods focus on mitigating covariate shift between domains (Ganin
et al. 2016, Long et al. 2017, Zhu et al. 2019, Peng et al. 2019). By applying
these domain adaptation methods, Huang et al. (2021), Li et al. (2021), Yue et
al. (2022) develop state-of-the-art misinformation detection methods, which
learn a model from labeled information in one or more source domains to
identify misinformation in a target domain with no labeled information.
Although these misinformation detection methods are applicable to our problem,
their efficacy is limited because they fail to account for concept shift.

In response to the research gaps, this study contributes to the literature
with a novel misinformation detection method that tackles both covariate shift
and concept shift. Our method design is rooted in our theoretical analysis,
which shows the necessity of addressing both covariate shift and concept shift
for effective domain adaptation as well as how to operationalize each of them.
Guided by the theoretical analysis, we develop two modules in our method: one
for assessing and reducing covariate shift and the other for evaluating and
mitigating concept shift. Therefore, the primary methodological difference
between our method and state-of-the-art domain adaptation-based misinformation
detection methods (e.g., Yue et al. (2022)) is the addressing of concept shift
by our method. It also differs from conventional misinformation detection
methods such as Wei et al. (2022). Specifically, our method is a cross-domain
method that learns a model from labeled information in other domains to
classify unlabeled information in the infodemic domain. In contrast,
conventional misinformation detection methods are single-domain methods that
learn a model from labeled information in a domain (e.g., infodemic domain) to
classify unlabeled information in the same domain.

##  2 Related Work

Existing methods for misinformation detection can be classified into three
categories: single-domain, multi-domain, and cross-domain approaches. In this
section, we review each category of these methods and highlight the
methodological novelty of our study.

###  2.1 Single and Multi-domain Misinformation Detection Methods

Single-domain misinformation detection methods train a classification model
using labeled information from a specific domain (e.g., true and fake news
from the domain of politics), and then employ the model to classify unlabelled
information in the same domain (e.g., unlabelled political news). In this
category, knowledge-based methods classify unlabelled information by comparing
the content of the information (e.g., textual content of news) against known
facts (Ciampaglia et al. 2015). More specifically, these methods retrieve
pertinent facts from authoritative websites like Wikipedia, represent
extracted facts as a knowledge graph, and employ the knowledge graph to
automatically detect misinformation. Additionally, writing styles identified
from the content of information, such as readability, sensationalism,
informality, and subjectivity, have been utilized to detect misinformation
(Zhou and Zafarani 2020). Recent methods consider the propagation of
information on social media, in addition to the content of information
(Papanastasiou 2020). Some methods employ propagation patterns to detect
misinformation because misinformation spreads faster, farther, and more widely
compared to true information (Vosoughi et al. 2018). Others harness the
collective intelligence of crowds, such as likes, shares, and comments from
social media users (Atanasov et al. 2017, Pennycook and Rand 2021). For
example, Wei et al. (2022) employ both the content of news articles and social
media users’ comments on these news articles to detect fake news.

Multi-domain misinformation detection methods learn a classification model
from labeled information in various domains (e.g., true and fake news in the
politics, health, and sports domains), and then employ the model to classify
unlabelled information in each of these domains (Nan et al. 2021, Zhu et al.
2023). The rationale underlying these methods is to leverage labeled
information in multiple domains to enhance misinformation detection in each
individual domain. However, because of variations in information content among
different domains, a model learned with labeled information in these domains
exhibits significant performance differences when applied to each of these
domains (Zhu et al. 2023). That is, the effectiveness of the learned model in
identifying misinformation in one domain could be significantly inferior to
its effectiveness in detecting misinformation in another domain. To tackle
this problem, several methods incorporate domain-specific characteristics into
the model training. For example, Nan et al. (2021) propose to embed different
domains as different vectors and integrate these vectors into the learning of
a multi-domain misinformation detection model. Zhu et al. (2023) introduce
domain adapters tailored to each specific domain; representations of news
articles in each domain are then adjusted by their corresponding domain
adapter to train a multi-domain model.

###  2.2 Cross-domain Misinformation Detection Methods

Cross-domain misinformation detection methods leverage labeled information in
one or more source domains (e.g., true and fake news in the politics and
sports domains) to classify unlabelled information in a target domain (e.g.,
unlabelled news in the health domain). Unlike multi-domain methods, which
identify misinformation in various domains, cross-domain methods focus on
detecting misinformation in one target domain. Depending on whether training
data contain labeled or unlabeled information in a target domain, cross-domain
methods can be further categorized into two subgroups.

One subgroup of cross-domain methods learns a model from training data
consisting of labeled information in source domains as well as labeled
information in a target domain to classify unlabeled information in the target
domain (Mosallanezhad et al. 2022, Nan et al. 2022). These methods essentially
utilize labeled source domain information as out-of-distribution data to
improve the performance of misinformation detection in the target domain
(Zhang et al. 2023). For example, Nan et al. (2022) pretrain a misinformation
detection model using labeled information in source and target domains. Next,
a language model is learned from labeled target domain information. Each piece
of source domain information is then assigned a transferability score based on
the degree to which the language model can predict its content. Finally, the
pretrained model is fine-tuned using labeled source domain information
weighted by their corresponding transferability scores as well as labeled
target domain information.

The other subgroup learns a model from training data consisting of labeled
information in source domains and unlabeled information in a target domain.
Since there is no labeled target domain information, this subgroup of methods
aims to transfer a model learned from labeled source domain information to the
target domain. To this end, domain adaptation is a suitable and widely adopted
technique. In the following, we review domain adaptation methods and their
applications in cross-domain misinformation detection. In the literature of
domain adaption, a domain consists of a feature space $\mathcal{X}$, a label
space $\mathcal{Y}$, and a probability distribution
$p(\textbf{x},y),\textbf{x}\in\mathcal{X}$ and $y\in\mathcal{Y}$ (Liu et al.
2022). As the probability distribution in a source domain usually differs from
that in a target domain, a model learned from source domain data often
performs poorly in the target domain (Ben-David et al. 2010). To tackle this
issue, it is necessary to reduce the disparity between source and target
domains in terms of $p(\textbf{x},y)$. Existing domain adaptation methods
concentrate on mitigating the difference in feature distribution (i.e.,
$p(\textbf{x})$) between source and target domains, known as covariate shift
(Kouw and Loog 2018).777In the field of data stream mining, concept shift or
concept drift refers to the phenomenon that data samples with same features
could have different labels at different timestamps (Agrahari and Singh 2022,
Roychowdhury et al. 2023). Given a model trained with previous data samples
and a continuous influx of new data samples, the objective of data stream
mining methods is to determine when and how to retrain the model (Vorburger
and Bernstein 2006, Fang et al. 2013, Agrahari and Singh 2022, Roychowdhury et
al. 2023). These methods are not applicable to our problem because they
require labeled data samples at all timestamps while instances of target
domain information are unlabeled in our problem. Moreover, these methods aim
to decide when and how to retrain a model in a dynamic environment, whereas
the objective of our problem is to classify unlabeled target domain instances.
In this vein, Ganin et al. (2016) propose a domain adversarial network to
learn domain-invariant features from data in source and target domains. Saito
et al. (2018) devise a mini-max mechanism to align feature distributions
between source and target domains. In the maximum stage, two classifiers are
learned from source samples such that the discrepancy between their
classifications of target samples is maximized. In the minimum stage, features
of target samples are adjusted to minimize such discrepancy. Rangwani et al.
(2022) develop a domain adaption method that incorporates a smoothing
mechanism into the method proposed by Ganin et al. (2016). Rostami and
Galstyan (2023) construct a pseudo-dataset using a model learned from labeled
source domain data and then mitigate feature distribution difference between
the pseudo-dataset and target domain data.888Although the title of the paper
contains the phrase of concept shift, it actually mitigates covariate shift.
This is evident in the objective function of the proposed algorithm, i.e.,
Equation (3) in (Rostami and Galstyan 2023). According to the equation, the
algorithm aims to mitigate the difference between the feature distribution
$p_{T}(\textbf{X}_{T})$ of target domain data and the feature distribution
$\hat{p}_{J}(\textbf{Z}_{p})$ of the pseudo-dataset. While the methods
reviewed above implicitly minimize covariate shift between source and target
domains, there are methods explicitly measure and minimize covariate shift.
For example, Long et al. (2015) utilize a metric known as the multi-kernel
Maximum Mean Discrepancy (MMD) to quantify covariate shift, integrate this
metric into a loss function, and minimize covariate shift by minimizing the
loss function. In a follow-up study, Long et al. (2017) introduce another
metric , known as the joint MMD, to measure covariate shift. In addition, Chen
et al. (2022) employ the nuclear-norm Wasserstein discrepancy to evaluate
covariate shift between source and target domains. Recent studies have
employed domain adaptation techniques to solve the cross-domain misinformation
detection problem with unlabeled target domain information. For example, Li et
al. (2021) extend the work of Ganin et al. (2016) to detect misinformation in
a target domain by leveraging labeled information in multiple source domains
whereas Ng et al. (2023) apply the domain adversarial network developed by
(Ganin et al. 2016) to identify fake news and reviews. Huang et al. (2021)
utilize the MMD metric to measure covariate shift between source domain
information and target domain information, and mitigate covariate shift by
minimizing the metric. Yue et al. (2022) develop a variant of the MMD metric
to measure covariate shift between source and target domain information.

Table 1: Comparison Between Our Method and Existing Misinformation Detection Methods.  |  Applicable to our problem |  Addressing covariate shift |  Addressing concept shift
---|---|---|---
Single- and multi-domain misinformation detection methods, e.g., Wei et al. (2022), Zhu et al. (2023) |  No |  No |  No
Cross-domain misinformation detection methods with labeled target domain information, e.g., Mosallanezhad et al. (2022), Nan et al. (2022) |  No |  Yes |  No
Cross-domain misinformation detection methods with unlabeled target domain information, e.g., Li et al. (2021), Yue et al. (2022) |  Yes |  Yes |  No
Our method |  Yes |  Yes |  Yes

###  2.3 Key Novelty of Our Study

Our literature review suggests the following research gaps, as summarized in
Table 1. First, existing single and multi-domain misinformation detection
methods as well as cross-domain methods with labeled target domain information
are not applicable to our problem. These methods require labeled target domain
information (i.e., the infodemic domain in our study) in their training
datasets, while our problem involves completely unlabeled information in the
infodemic domain. Second, although existing cross-domain methods with
unlabeled target domain information are applicable to our problem, they are
less effective in solving the problem because they fail to address concept
shift. In fact, none of the existing misinformation detection methods is
capable of tackling concept shift. However, to ensure the efficacy of a model
learned using labeled source domain information in classifying unlabeled
target domain information, it is essential to mitigate both covariate shift
and concept shift between source and target domains (Liu et al. 2022). To this
end, we propose a misinformation detection method that tackles both covariate
shift and concept shift, in contrast to existing methods that focus on
covariate shift solely. Therefore, the key methodological novelty of our
method lies in its addressing of concept shift. To implement this
methodological novelty, our study features two innovations. First, our
theoretical analysis of domain adaptation not only underscores the importance
of addressing both covariate shift and concept shift but also shows how to
operationalize each of them. Second, informed by our theoretical analysis, we
introduce a novel misinformation detection method, two modules of which are
respectively designed to tackle covariate shift and concept shift.

##  3 Problem Formulation

We consider a misinformation detection problem at the early stage of an
infodemic, when all instances of information in this domain are unlabeled.
Concretely, let $\mathcal{D}_{T}$ be a dataset of $N_{T}$ pieces of unlabeled
information in the infodemic domain. The subscript $T$ signifies target
domain, which is the infodemic domain in this study. As an example, a piece of
unlabeled information might encompass the textual content of a news article
about COVID-19. Let $\textbf{x}_{i}^{T}$ denote a set of features extracted
from a piece of unlabeled information in $\mathcal{D}_{T}$,
$i=1,2,\dots,N_{T}$. We aim to predict the label (true or false) for each
piece of information in $\mathcal{D}_{T}$.

To accomplish this objective, we are given $k$ datasets of labeled
information, i.e.,
$\mathcal{D}_{S_{1}},\mathcal{D}_{S_{2}},\dots,\mathcal{D}_{S_{k}}$. The
subscript $S$ refers to source domain, which provides labeled information to
facilitate the label prediction in the target domain. Concretely,
$\mathcal{D}_{S_{j}}$ consists of $N_{S_{j}}$ pieces of labeled information,
$j=1,2,\dots,k$. Considering an example of $S_{j}$ being the politics domain,
a piece of information in this domain could comprise the textual content of a
political news article. Each piece of labeled information in
$\mathcal{D}_{S_{j}}$ is represented as
$(\textbf{x}_{i}^{S_{j}},y_{i}^{S_{j}})$, where $\textbf{x}_{i}^{S_{j}}$
denotes features extracted from the information, $y_{i}^{S_{j}}$ is the label
of the information, and $i=1,2,\dots,N_{S_{j}}$. Label $y_{i}^{S_{j}}=0$
indicates that the information is true and $y_{i}^{S_{j}}=1$ shows that the
information is false. We now formally define the problem of early detection of
misinformation for infodemic management.

Definition 1 (Early Detection of Misinformation Problem (EDM)). Given a
dataset $\mathcal{D}_{T}$ of $N_{T}$ pieces of unlabeled information in the
infodemic domain and $k$ datasets of labeled information in various source
domains, $\mathcal{D}_{S_{1}},\mathcal{D}_{S_{2}},\dots,\mathcal{D}_{S_{k}}$,
where $\mathcal{D}_{S_{j}}$ consists of $N_{S_{j}}$ pieces of labeled
information, $j=1,2,\dots,k$, the objective of the problem is to learn a model
from the data to classify each piece of information in $\mathcal{D}_{T}$ as
true or false.

##  4 Method

In this section, we theoretically analyze the performance of a model learned
using labeled source domain instances but applied to classify unlabeled target
domain instances. Guided by the insights from our theoretical analysis, we
then propose a novel method to solve the EDM problem.

###  4.1 Theoretical Analysis

We start with the setting of one target domain $T$ and one source domain $S$.
Let $\mathcal{D}_{T}$ be a dataset of $N_{T}$ unlabelled instances in $T$ and
$\mathcal{D}_{S}$ be a dataset of $N_{S}$ labeled instances in $S$. In the
context of misinformation detection, an instance is a piece of information. We
denote $f_{T}$ as the true labeling function in the target domain that maps an
instance with features x into the probability of the instance being 1, i.e.,
$f_{T}(\textbf{x})=p_{T}(y=1|\textbf{x})$. Similarly, $f_{S}$ is the true
labeling function in the source domain and
$f_{S}(\textbf{x})=p_{S}(y=1|\textbf{x})$. Let $h$ be a hypothesis learned
from labeled instances in $\mathcal{D}_{S}$ and
$h(\textbf{x})=p_{h}(y=1|\textbf{x})$. Following Ben-David et al. (2010), we
define the source error $\epsilon_{S}(h)$ of hypothesis $h$ as the expected
difference between $h$ and the true labeling function $f_{S}$ in the source
domain

$\epsilon_{S}(h)=E_{\textbf{x}\sim\mathbb{D}_{S}}[|h(\textbf{x})-f_{S}(\textbf{x})|],$
(1)

where $\mathbb{D}_{S}$ represents the feature distribution in the source
domain. In a similar manner, we define the target error $\epsilon_{T}(h)$ of
hypothesis $h$ as the expected difference between $h$ and the true labelling
function $f_{T}$ in the target domain

$\epsilon_{T}(h)=E_{\textbf{x}\sim\mathbb{D}_{T}}[|h(\textbf{x})-f_{T}(\textbf{x})|],$
(2)

where $\mathbb{D}_{T}$ denotes the feature distribution in the target domain.
Recall that domain adaptation aims to learn $h$ from labeled instances in
$\mathcal{D}_{S}$ to classify each unlabelled instance in $\mathcal{D}_{T}$.
Thus, the objective of domain adaptation is to minimize the target error
$\epsilon_{T}(h)$. To achieve this objective, we show a bound of
$\epsilon_{T}(h)$ in the following theorem.

Theorem 1  For a source domain instance with features $\textbf{x}_{i}^{S}$,
let $c(\textbf{x}_{i}^{S})$ be the features of its nearest target domain
instance, where $i=1,2,\dots,N_{S}$ and the distance between a pair of
instances are measured over their feature space. If $f_{T}$ is $L$-Lipschitz
continuous,999The L-Lipschitz continuous assumption is commonly used in
theoretical analyses of machine learning algorithms (e.g., Arjovsky et al.
2017, Asadi et al. 2018, Kim et al. 2021).  then for any $\eta\in(0,1)$, with
probability at least $(1-\eta)^{2}$,

$\displaystyle\epsilon_{T}(h)\leq\epsilon_{S}(h)+\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|f_{S}(\textbf{x}_{i}^{S})-f_{T}(c(\textbf{x}_{i}^{S}))|+\frac{L}{N_{S}}\sum_{i=1}^{N_{S}}||\textbf{x}_{i}^{S}-c(\textbf{x}_{i}^{S})||+C_{1},$
(3)

where $L$ is the Lipschitz constraint constant, $C_{1}$ is a constant,
$\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$ denotes the empirical
$\mathcal{H}$-divergence between distributions $\mathbb{D}_{S}$ and
$\mathbb{D}_{T}$, estimated using features of source domain instances in
$\mathcal{D}_{S}$ and features of target domain instances in
$\mathcal{D}_{T}$, and $||\textbf{x}_{i}^{S}-c(\textbf{x}_{i}^{S})||$
represents the distance between $\textbf{x}_{i}^{S}$ and
$c(\textbf{x}_{i}^{S})$.

Proof. See Appendix B.

Theorem 4.1 establishes an upper bound of $\epsilon_{T}(h)$, which can be
empirically computed using datasets $\mathcal{D}_{S}$ and $\mathcal{D}_{T}$
drawn from distributions $\mathbb{D}_{S}$ and $\mathbb{D}_{T}$, respectively.
By Theorem 4.1, to minimize $\epsilon_{T}(h)$, we need to minimize the first
three terms of its upper bound specified in Inequality (3). This is because
the fourth term of the upper bound is inherently minimized, given that
$c(\textbf{x}_{i}^{S})$ is the closest to $\textbf{x}_{i}^{S}$, and the last
term $C_{1}$ is a constant.

The first term $\epsilon_{S}(h)$ can be minimized by learning a classification
model from labeled source domain instances. The second term
$\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$ is the empirical
estimation of the $\mathcal{H}$-divergence between the source domain feature
distribution $\mathbb{D}_{S}$ and the target domain feature distribution
$\mathbb{D}_{T}$. $\mathcal{H}$-divergence has been extensively utilized in
domain adaptation studies to measure distances between distributions. Please
refer to Appendix A for a description of $\mathcal{H}$-divergence and its
empirical estimation $\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$.
Fundamentally, minimizing the $\mathcal{H}$-divergence between source and
target domain feature distributions is to minimize covariate shift between
source and target domains. Therefore, existing domain adaptation methods for
mitigating covariate shift, such as Ganin et al. (2016) and Zhu et al. (2019),
can be applied to minimize
$\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$.

For the third term
$\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|f_{S}(\textbf{x}_{i}^{S})-f_{T}(c(\textbf{x}_{i}^{S}))|$,
recall that $f_{S}(\textbf{x}_{i}^{S})=p_{S}(y=1|\textbf{x}_{i}^{S})$ and
$f_{T}(c(\textbf{x}_{i}^{S}))=p_{T}(y=1|c(\textbf{x}_{i}^{S}))$. Hence, to
minimize the third term, we need to minimize the summation of differences
between $p_{S}(y=1|\textbf{x}_{i}^{S})$ and
$p_{T}(y=1|c(\textbf{x}_{i}^{S}))$, summed over all source domain instances
and their corresponding nearest target domain instances. Therefore, minimizing
the third term is essentially to minimize concept shift between source and
target domains.

Next, we consider the setting of one target domain $T$ and multiple source
domains $S_{1},S_{2},\dots,S_{k}$. Similar to the notations used in the
setting of one source domain, we denote $\mathcal{D}_{T}$ as a dataset of
$N_{T}$ unlabelled instances in $T$ and $\mathcal{D}_{S_{j}}$ as a dataset of
$N_{S_{j}}$ labeled instances in $S_{j}$, $j=1,2,\dots,k$. Let $h$ be a
hypothesis learned from labeled instances in
$\mathcal{D}_{S_{1}},\mathcal{D}_{S_{2}},\dots,\mathcal{D}_{S_{k}}$. We denote
$f_{T}$ and $f_{S_{j}}$ as the true labeling functions in $T$ and $S_{j}$,
respectively, $j=1,2,\dots,k$. For a hypothesis $h$, $\epsilon_{T}(h)$ and
$\epsilon_{S_{j}}(h)$ represent its errors in $T$ and $S_{j}$, respectively,
$j=1,2,\dots,k$. Built on Theorem 4.1, the following proposition gives an
upper bound of $\epsilon_{T}(h)$ in the setting of multiple source domains.

Proposition 1 For a source domain instance with features
$\textbf{x}_{i}^{S_{j}}$ in $\mathcal{D}_{S_{j}}$, let
$c(\textbf{x}_{i}^{S_{j}})$ be the features of its nearest instance in
$\mathcal{D}_{T}$, where $j=1,2,\dots,k$ and $i=1,2,\dots,N_{S_{j}}$. If
$f_{T}$ is $L$-Lipschitz continuous, then for any $\eta\in(0,1)$, with
probability at least $(1-\eta)^{2k}$,

$\displaystyle\epsilon_{T}(h)\leq\frac{1}{k}\sum_{j=1}^{k}\\{\epsilon_{S_{j}}(h)+$
$\displaystyle\hat{d}_{\mathcal{H}}(\mathcal{D}_{S_{j}},\mathcal{D}_{T})+\frac{1}{N_{S_{j}}}\sum_{i=1}^{N_{S_{j}}}|f_{S_{j}}(\textbf{x}_{i}^{S_{j}})-f_{T}(c(\textbf{x}_{i}^{S_{j}}))|+\frac{L}{N_{S_{j}}}\sum_{i=1}^{N_{S_{j}}}||\textbf{x}_{i}^{S_{j}}-c(\textbf{x}_{i}^{S_{j}})||+C_{j}\\},$
(4)

where $L$ is the Lipschitz constraint constant, $C_{j}$ is a constant, and
$\hat{d}_{\mathcal{H}}(\mathcal{D}_{S_{j}},\mathcal{D}_{T})$ denotes the
empirical $\mathcal{H}$-divergence between the feature distribution in the
source domain $S_{j}$ and that in the target domain $T$.

Proof. See Appendix B.

Proposition 4.1 offers useful insights for solving the EDM problem, which aims
to learn a model from labeled information in multiple source domains to
classify unlabeled information in the target infodemic domain with minimum
error (i.e., $\epsilon_{T}(h)$). Informed by the proposition, we design a
method that minimizes $\epsilon_{T}(h)$ through its three modules, each
dedicated to reducing one of the first three terms in Inequality (4).
Specifically, to reduce $\sum_{j=1}^{k}\epsilon_{S_{j}}(h)$, we implement a
classification module, which is trained to accurately predict the label for
each piece of source domain information. A covariate alignment module is
developed to reduce
$\sum_{j=1}^{k}\hat{d}_{\mathcal{H}}(\mathcal{D}_{S_{j}},\mathcal{D}_{T})$,
thus diminishing covariate shift between source domains and the target domain.
Finally, we propose a concept alignment module to mitigate
$\sum_{j=1}^{k}\sum_{i=1}^{N_{S_{j}}}|f_{S_{j}}(\textbf{x}_{i}^{S_{j}})-f_{T}(c(\textbf{x}_{i}^{S_{j}}))|$,
thereby reducing concept shift between source domains and the target domain.
The concept alignment module constitutes the main methodological novelty of
our method, and we designate our method as Domain Adaptation with Concept
Alignment (DACA).

Overall Architecture of Domain Adaptation with Concept Alignment (DACA) Method
Better view in color.

###  4.2 Method Overview

Grounded in our theoretical analysis, the DACA method adeptly alleviates
disparities between source and target domains through its covariate alignment
and concept alignment modules. As a result, the model learned from labeled
source domain information by its classification module can effectively
classify unlabeled infodemic domain information. Figure 4.1 illustrates the
overall architecture of the DACA method. As shown, the inputs to the method
consist of $k$ datasets of labeled information in various source domains,
$\mathcal{D}_{S_{1}},\mathcal{D}_{S_{2}},\dots,\mathcal{D}_{S_{k}}$, and a
dataset $\mathcal{D}_{T}$ of unlabeled information in the target infodemic
domain. A content embedding function extracts features from the content of
each piece of input information. Various content embedding functions (e.g.,
Kenton and Toutanova 2019, Zhu et al. 2023) could be used for this purpose,
and we describe the content embedding function employed in this study in
Section 5.2. Let
$X^{S_{j}}=\\{\textbf{x}_{i}^{S_{j}}|i=1,2,\dots,N_{S_{j}}\\}$ and
$X^{T}=\\{\textbf{x}_{i}^{T}|i=1,2,\dots,N_{T}\\}$ denote the set of features
extracted from input information in source domain $S_{j}$ and target domain
$T$, respectively, $j=1,2,\dots,k$. These features serve as the inputs to the
three modules of the DACA method.

The classification module trains a label classifier to accurately predict the
label of a piece of information as either true (0) or false (1). Concretely,
the label classifier takes the features x of each piece of information as
input and predicts its probability of being 1, i.e.,
$p(\hat{y}=1|\textbf{x})$. Given that only source domain information is
labeled, the label classifier is trained using labelled source domain
information. The covariate alignment module trains a domain classifier to
predict whether a piece of information is from a source domain or the target
domain. As the objective of this module is to alleviate the feature
distribution difference between source domains and the target domain, the
domain classifier is trained to be perplexed in distinguishing between source
domain information and target domain information (Ganin et al. 2016).

Proposition 4.1 sheds light on the design of the concept alignment module, the
main innovation of our method. According to the proposition, for each piece of
information in a source domain, we need to find its nearest target domain
information. Therefore, it is imperative to have a function that measures the
similarity between two pieces of information over their feature space. To this
end, we introduce a contrastive learning sub-module that learns a similarity
function using labeled source domain information. Utilizing the similarity
function, a concept alignment sub-module identifies the nearest target domain
information for each piece of source domain information. By Proposition 4.1,
this sub-module then mitigates concept shift between source domains and the
target domain by minimizing
$\sum_{j=1}^{k}\sum_{i=1}^{N_{S_{j}}}|f_{S_{j}}(\textbf{x}_{i}^{S_{j}})-f_{T}(c(\textbf{x}_{i}^{S_{j}}))|$,
where $f_{S_{j}}(\textbf{x}_{i}^{S_{j}})$ and
$f_{T}(c(\textbf{x}_{i}^{S_{j}}))$ are estimated using the label classifier in
the classification module.

###  4.3 Classification Module and Covariate Alignment Module

As depicted in Figure 4.1, the classification module and the covariate
alignment module share a feature transformation layer. This layer, denoted by
$F_{h}$, takes each feature vector x in $X^{S_{1}}$, $X^{S_{2}}$, …,
$X^{S_{k}}$, $X^{T}$ as input and outputs the transformed vector h. Formally,
we have

$\displaystyle\textbf{h}=F_{h}(\textbf{x})=\text{MLP}_{1}(\textbf{x};\theta_{h}),$
(5)

where $\text{MLP}_{1}$ is a multi-layer perceptron (MLP) with trainable
parameters $\theta_{h}$.

The classification module features an MLP-based label classifier, denoted by
$F_{y}$, which takes the transformed feature vector h as input and outputs the
probability of being labeled as 1 (i.e., false) for the piece of information
characterized by feature vector x. Concretely, we have

$\displaystyle
p(\hat{y}=1|\textbf{x})=F_{y}(\textbf{h})=\text{Sigmoid}\big{(}\text{MLP}_{2}(\textbf{h};\theta_{y})\big{)},$
(6)

where $\hat{y}$ is the predicted label for the piece of information,
$\text{MLP}_{2}$ is an MLP with trainable parameters $\theta_{y}$, and
Sigmoid() denotes the sigmoid function. The classification module is trained
by minimizing the total cross-entropy loss summed over all labeled source
domain information:

$\displaystyle\mathcal{L}_{y}(\theta_{h},\theta_{y})$
$\displaystyle=-\sum_{\textbf{x}\in
X^{S}}\bigg{[}y\log\big{(}p(\hat{y}=1|\textbf{x})\big{)}+(1-y)\log\big{(}1-p(\hat{y}=1|\textbf{x})\big{)}\bigg{]}$
(7) $\displaystyle=-\sum_{\textbf{x}\in X^{S}}\bigg{[}y\log
F_{y}(F_{h}(\textbf{x}))+(1-y)\log\big{(}1-F_{y}(F_{h}(\textbf{x}))\big{)}\bigg{]},$

where $X^{S}=\cup_{j=1}^{k}{X}^{S_{j}}$ and $y$ is the true label of x.
Minimizing $\mathcal{L}_{y}$ trains the classification module to score high
probabilities for true labels, thereby inducing a model with low source error
(i.e., $\sum_{j=1}^{k}\epsilon_{S_{j}}(h)$ in Proposition 4.1).

The covariate alignment module aims to mitigate covariate shift between source
domains and the target domain, i.e., reducing
$\sum_{j=1}^{k}\hat{d}_{\mathcal{H}}(\mathcal{D}_{S_{j}},\mathcal{D}_{T})$ in
Proposition 4.1. To this end, we adapt the domain adversarial method developed
by Ganin et al. (2016) to implement this module. Let $d$ denote the true
domain for a piece of information characterized by feature vector x, where
$d=0$ if it belongs to a source domain and $d=1$ if it comes from the target
domain. The module employs a domain classifier, denoted by $F_{d}$, that takes
the transformed feature vector h as input and outputs its probability of
belonging to the target domain. Mathematically, we have

$\displaystyle
p(\hat{d}=1|\textbf{x})=F_{d}(\textbf{h})=\text{Sigmoid}\big{(}\text{MLP}_{3}(\textbf{h};\theta_{d})\big{)},$
(8)

where $\hat{d}$ is the predicted domain (i.e., source or target) for x, and
$\text{MLP}_{3}$ is an MLP with trainable parameters $\theta_{d}$. On the one
hand, $F_{d}$ is trained to accurately predict the domain for each piece of
information. On the other hand, to reduce covariate shift, the vector h
produced by the feature transformation layer $F_{h}$ should be domain-
invariant, i.e., when presented to $F_{d}$, the classifier cannot tell whether
it is from a source domain or the target domain (Li et al. 2021). With this
reasoning in mind, the learning of domain-invariant features can be formulated
as a min-max game defined as follows(Ganin et al. 2016, Li et al. 2021):

$\displaystyle\mathop{\min}_{\theta_{h}}\mathop{\max}_{\theta_{d}}\
\sum_{\textbf{x}\in
X}\bigg{[}d\log\big{(}p(\hat{d}=1|\textbf{x})\big{)}+(1-d)\log\big{(}1-p(\hat{d}=1|\textbf{x})\big{)}\bigg{]}$
(9) $\displaystyle=\mathop{\min}_{\theta_{h}}\mathop{\max}_{\theta_{d}}\
\sum_{\textbf{x}\in X}\bigg{[}d\log
F_{d}(F_{h}(\textbf{x}))+(1-d)\log\big{(}1-F_{d}(F_{h}(\textbf{x}))\big{)}\bigg{]},$

where $X=(\cup_{j=1}^{k}{X}^{S_{j}})\cup X^{T}$ and $d$ is the true domain
(i.e., source or target) that x belongs to. The maximization objective in
Equation 9 promotes a more accurate domain classifier, while the minimization
objective strengthens the domain invariance of the transformed feature vector
h by making hard even for the most accurate domain classifier to discern its
domain identity. Following Ganin et al. (2016), the two adversarial objectives
in Equation 9 can be unified using a gradient reversal layer $R(\textbf{z})$
defined as:

$R(\textbf{z})=\textbf{z};\frac{dR(\textbf{z})}{d\textbf{z}}=-\textbf{I},$
(10)

where I is an identity matrix. According to Equation 10, the gradient reversal
layer makes no change to its input but flips the sign of the gradient passed
through it. Equipped with the gradient reversal layer, solving the min-max
problem in Equation 9 is equivalent to minimize the following loss (Ganin et
al. 2016):

$\displaystyle\mathcal{L}_{d}(\theta_{h},\theta_{d})=-\sum_{\textbf{x}\in
X}\bigg{[}d\log
F_{d}(R(F_{h}(\textbf{x})))+(1-d)\log\big{(}1-F_{d}(R(F_{h}(\textbf{x})))\big{)}\bigg{]},$
(11)

where we replace $F_{h}(\textbf{x})$ in Equation 9 with
$R(F_{h}(\textbf{x}))$. To minimize the loss defined in Equation 11,
parameters $\theta_{d}$ of the domain classifier $F_{d}$ are updated with
gradient descent, leading to a more accurate domain classifier. In contrast,
because of the gradient reversal layer, parameters $\theta_{h}$ of the feature
transformation layer $F_{h}$ are adjusted with gradient upscent, resulting in
more domain-invariant h (Ganin et al. 2016).

###  4.4 Concept Alignment Module

The key novelty of our method is the proposed concept alignment module, which
mitigates concept shift between source domains and the target domain. By
Proposition 4.1, to mitigate concept shift, we need to learn a function that
measures the similarity between two pieces of information. To this end, we
propose a contrastive learning sub-module. Specifically, for two pieces of
information respectively characterized by feature vectors $\textbf{x}_{i}$ and
$\textbf{x}_{j}$, we define the similarity between them as

$F_{s}(\textbf{x}_{i},\textbf{x}_{j})=\frac{\text{MLP}_{4}(\textbf{x}_{i};\theta_{s})\cdot\text{MLP}_{4}(\textbf{x}_{j};\theta_{s})}{||\text{MLP}_{4}(\textbf{x}_{i};\theta_{s})||\hskip
1.42262pt||\text{MLP}_{4}(\textbf{x}_{j};\theta_{s})||},$ (12)

where $F_{s}$ denotes the similarity function, $\text{MLP}_{4}$ is an MLP with
learnable parameters $\theta_{s}$ that transforms a feature vector (e.g.,
$\textbf{x}_{i}$) into a different representation space, notation
$\mathbb{\cdot}$ denotes the dot product of two vectors, and $||\hskip
2.84544pt||$ represents the $L^{2}$ norm of a vector.

According to Equation 12, learning the similarity function $F_{s}$ is
essentially learning the parameters $\theta_{s}$ of $\text{MLP}_{4}$. The
rationale underlying the learning is that information with identical labels
exhibit greater similarity than information with distinct labels in their
representation space (Frosst et al. 2019). Therefore, we design the
contrastive learning sub-module that learns the parameters $\theta_{s}$ via
the objective of maximizing the similarity between identically labeled
information in the transformed representation space while minimizing the
similarity between distinctly labeled information. More specifically, for each
piece of source domain information characterized by feature vector x, we
randomly select another piece of source domain information, bearing the same
label, as its positive peer. We denote the feature vector of the positive peer
as $\textbf{x}^{+}$. In addition, we randomly select $m$ other pieces of
source domain information, each having the opposite label of the focal
information characterized by x, to serve as its negative peers. We denote the
feature vectors of these negative peers as $\textbf{x}_{l}^{-}$,
$l=1,2,\dots,m$. The objective is thus to maximize the similarity
$F_{s}(\textbf{x},\textbf{x}^{+})$ between x and $\textbf{x}^{+}$, while
minimizing the similarity $F_{s}(\textbf{x},\textbf{x}_{l}^{-})$ between x and
$\textbf{x}_{l}^{-}$, $l=1,2,\dots,m$. Accordingly, the contrastive learning
sub-module learns the similarity function by minimizing the following loss
summed over all labelled source domain information:

$\mathcal{L}_{s}(\theta_{s})=-\sum_{\textbf{x}\in
X^{S}}\log\frac{\exp(F_{s}(\textbf{x},\textbf{x}^{+})/\tau)}{\exp(F_{s}(\textbf{x},\textbf{x}^{+})/\tau)+\Sigma_{l=1}^{m}\exp(F_{s}(\textbf{x},\textbf{x}_{l}^{-})/\tau)},$
(13)

where $X^{S}=\cup_{j=1}^{k}{X}^{S_{j}}$, $F_{s}$ is given in Equation 12, and
$0<\tau<1$ is a hyperparameter.101010 Hyperparameter $\tau$ adjusts the
emphasis on the two objectives (Frosst et al. 2019). Lower $\tau$ enables the
learning of the similarity function to place more emphasis on maximizing
$F_{s}(\textbf{x},\textbf{x}^{+})$ while higher $\tau$ shifts the learning to
emphasize more on minimizing $F_{s}(\textbf{x},\textbf{x}_{l}^{-})$. By
minimizing $\mathcal{L}_{s}$ w.r.t. $\theta_{s}$, this sub-module learns
$F_{s}$ that measures the similarity between two pieces of information in
their transformed representation space such that identically labeled
information are close to each other and distinctly labeled information are
distant from each other.

Applying the similarity function $F_{s}$, for each piece of source domain
information characterized by feature vector $\textbf{x}_{i}^{S_{j}}$, the
concept alignment sub-module identifies its most similar target domain
information characterized by feature vector $c(\textbf{x}_{i}^{S_{j}})$,
$j=1,2,\dots,k$ and $i=1,2,\dots,N_{S_{j}}$. It then mitigates concept shift
between source domains and the target domain by minimizing
$\sum_{j=1}^{k}\sum_{i=1}^{N_{S_{j}}}|f_{S_{j}}(\textbf{x}_{i}^{S_{j}})-f_{T}(c(\textbf{x}_{i}^{S_{j}}))|$
in Proposition 4.1. To this end, $f_{S_{j}}(\textbf{x}_{i}^{S_{j}})$ and
$f_{T}(c(\textbf{x}_{i}^{S_{j}}))$ are estimated using the label classifier
described in Section 4.3. Specifically, we have

$\hat{f}_{S_{j}}(\textbf{x}_{i}^{S_{j}})=p(\hat{y}=1|\textbf{x}_{i}^{S_{j}})=F_{y}(F_{h}(\textbf{x}_{i}^{S_{j}})),$

where $\hat{f}_{S_{j}}$ is the estimation of $f_{S_{j}}$, and $F_{h}$ and
$F_{y}$ are defined by Equations 5 and 6, respectively. Similarly,
$f_{T}(c(\textbf{x}_{i}^{S_{j}}))$ is estimated by

$\hat{f}_{T}(c(\textbf{x}_{i}^{S_{j}}))=p(\hat{y}=1|c(\textbf{x}_{i}^{S_{j}}))=F_{y}(F_{h}(c(\textbf{x}_{i}^{S_{j}}))).$

With the estimations of $f_{S_{j}}$ and $f_{T}$, the concept alignment sub-
module reduces concept shift by minimizing
$\sum_{j=1}^{k}\sum_{i=1}^{N_{S_{j}}}|F_{y}(F_{h}(\textbf{x}_{i}^{S_{j}}))-F_{y}(F_{h}(c(\textbf{x}_{i}^{S_{j}})))|$.
Accordingly, this sub-module is trained by minimizing the following loss:

$\mathcal{L}_{c}(\theta_{h},\theta_{y})=\sum_{\textbf{x}\in
X^{S}}\Big{[}\big{(}F_{y}(F_{h}(\textbf{x}))-F_{y}(F_{h}(c(\textbf{x})))\big{)}^{2}-\big{(}F_{y}(F_{h}(\textbf{x}))-F_{y}(F_{h}(d(\textbf{x})))\big{)}^{2}\Big{]}.$
(14)

where $X^{S}=\cup_{j=1}^{k}{X}^{S_{j}}$. For each piece of source domain
information with feature vector $\textbf{x}\in X^{S}$, $c(\textbf{x})$ and
$d(\textbf{x})$ respectively denote the feature vectors characterizing its
most similar and most dissimilar target domain information, both of which are
identified using the similarity function $F_{s}$. Minimizing the first term in
$\mathcal{L}_{c}$ essentially minimizes
$\sum_{j=1}^{k}\sum_{i=1}^{N_{S_{j}}}|F_{y}(F_{h}(\textbf{x}_{i}^{S_{j}}))-F_{y}(F_{h}(c(\textbf{x}_{i}^{S_{j}})))|$.
Consequently, it enforces the desired property that similar source and target
domain information have similar label predictions (Liu et al. 2022), thus
reducing concept shift between source domains and the target domain.
Minimizing the second term
$-\big{(}F_{y}(F_{h}(\textbf{x}))-F_{y}(F_{h}(d(\textbf{x})))\big{)}^{2}$
maximizes the label prediction difference between $F_{y}(F_{h}(\textbf{x}))$
and $F_{y}(F_{h}(d(\textbf{x})))$. As a result, label predictions diverge for
dissimilar source and target domain information, which further mitigates
concept shift.

###  4.5 The DACA Method

The DACA method is trained to minimize the combined losses of its three
modules:

$\displaystyle\mathcal{L}_{\text{DACA}}(\theta_{h},\theta_{y},\theta_{d},\theta_{s})=\mathcal{L}_{y}(\theta_{h},\theta_{y})+\mathcal{L}_{d}(\theta_{h},\theta_{d})+\mathcal{L}_{s}(\theta_{s})+\mathcal{L}_{c}(\theta_{h},\theta_{y})$
(15)

where $\mathcal{L}_{y}$, $\mathcal{L}_{d}$, $\mathcal{L}_{s}$, and
$\mathcal{L}_{c}$ are defined by Equations 7, 11, 13, and 14, respectively.
There are two practical considerations when training DACA. First, the DACA
method is trained in a two-stage manner. In the first stage (or the warmup
stage), the method is trained by minimizing
$\mathcal{L}_{\text{DACA}}-\mathcal{L}_{c}$. The rationale is that, to
accurately estimate the concept alignment loss $\mathcal{L}_{c}$, DACA needs
to learn a reliable similarity function first. In the second stage, DACA is
trained by minimizing $\mathcal{L}_{\text{DACA}}$ as defined by Equation 15.
Second, to speed up the training of the DACA method, it is beneficial to
mitigate concept shift between each pair of source domains. This is
accomplished by minimizing a loss function for each pair of source domains.
Concretely, for a pair of source domains $S_{j}$ and $S_{n}$, the loss
function has the same form as Equation 14 but with $S_{j}$ treated as the
source domain in the equation and $S_{n}$ regarded as the target domain. Once
trained, DACA predicts the probability that a piece of target domain
information characterized by feature vector $\textbf{x}_{T}$ is false as
$F_{y}(F_{h}(\textbf{x}_{T}))$, where $F_{h}$ and $F_{y}$ are defined by
Equations 5 and 6, respectively.

##  5 Empirical Evaluation

###  5.1 Data

We evaluated the performance of our proposed DACA method using the publicly
available datasets of English news, which have been widely employed to assess
the performance of misinformation detection methods (Nan et al. 2021,
Mosallanezhad et al. 2022, Zhu et al. 2023). One is the MM-COVID dataset,
which contains 4,750 pieces of true news (i.e., true information) and 1,317
pieces of fake news (i.e., misinformation) on COVID-19 as well as user
comments on these news (Li et al. 2020). Specifically, 8% of COVID news are
accompanied by user comments. In our evaluation, we treated the COVID domain
as the infodemic or target domain. For source domains, we utilized the
FakeNewsNet dataset, which consists of true and fake news alongside their
associated comments from the domains of entertainment and politics (Shu et al.
2020). Specifically, there are 16,804 pieces of true news and 5,067 pieces of
fake news from the entertainment domain, and 1,583 instances of true news and
1,287 instances of fake news from the politics domain. User comments accompany
27% of entertainment news and 59% of politics news. Table 2 reports the
summary statistics of the datasets used in our evaluation. Examples of true
and fake news from the source and target domains are given in Figure 1.

Table 2: Summary Statistics of Evaluation Datasets.  Domain | Number of true news | Number of fake news
---|---|---
COVID (Infodemic / Target) | 4,750 | 1,317
Entertainment (Source) | 16,804 | 5,067
Politics (Source) | 1,583 | 1,287
Figure 1: Examples of True and Fake News in Evaluation Datasets

###  5.2 Evaluation Procedure and Benchmark Methods

We considered two scenarios of early detection of misinformation in our
evaluation. During the early stage of an infodemic, there is little
understanding of the disease that causes the infodemic. Consequently, even
experts encounter difficulty distinguishing true news from fake news in the
infodemic domain, leaving all news in this domain unlabeled. Hence, the first
evaluation scenario entails each method (ours or benchmark) utilizing labeled
news in the entertainment and politics domains to predict the label for each
piece of news in the COVID domain. Accordingly, the inputs to each method
encompass labeled news in the entertainment and politics domains, along with
user comments on these news, as well as unlabeled news in the COVID domain and
their associated comments. In the second evaluation scenario, user comments
accompanying COVID news are dropped from the inputs. As a result, the inputs
in this scenario consist of labeled news in the entertainment and politics
domains, alongside user comments on these news, and unlabeled news in the
COVID domain. This scenario emulates the onset of an infodemic, characterized
by a large volume of information, yet with very few or no comments
accompanying these information.

In each evaluation scenario, we conducted 25 experiments for every method
(ours or benchmark) and measured its performance using the metrics recall and
F1-score. To curb the spread of fake news and minimize their societal impact,
it is crucial to identify as many instances of fake news as possible (Zhu et
al. 2023). To this end, recall is an important metric as it measures the
effectiveness of a method in identifying fake news. In addition, F1-score
evaluates a method’s performance in both identifying fake news and avoiding
predicting true news as fake. Concretely, let P be the number of fake news, TP
be the number of fake news that are predicted as fake, and FP be the number of
true news that are predicted as fake. Recall is defined as
$\text{TP}/\text{P}$, and F1-score is computed as
$2\text{TP}/(\text{TP}+\text{FP}+\text{P})$.

As reviewed in Section 2 and summarized in Table 1, among all existing
misinformation detection methods, only cross-domain misinformation detection
methods with unlabeled target domain information can solve the EDM problem
investigated in this paper. Therefore, we benchmarked our method against
state-of-the-art methods in this category. Specifically, one benchmark is the
MMD method, which minimizes covariate shift between source and target domain
information, measured using the maximum mean discrepancy (MMD) metric (Huang
et al. 2021). Another benchmark is the contrastive adaptation network for
misinformation detection (CANMD) method, which assesses covariate shift using
a variant of the MMD metric (Yue et al. 2022). In addition, we also compared
our method against the domain adversarial neural network (DANN) method; it
minimizes covariate shift by learning domain-invariant features from source
and target domain information (Li et al. 2021). Moreover, general-purpose
domain adaptation methods that were not originally designed for misinformation
detection can also be adapted to solve our problem. As analyzed in Section
2.2, there are two categories of domain adaptation methods. One category
explicitly measures and minimizes covariate shift while the other category
implicitly minimizes covariate shift by learning domain-invariant features.
Accordingly, we benchmarked against representative methods in each category.
For the category that explicitly measures and minimizes covariate shift, we
considered the multi-kernel MMD (MK-MMD) method (Long et al. 2015), the joint
adaptation network (JAN) method (Long et al. 2017), and the discriminator-free
adversarial learning network (DALN) method (Chen et al. 2022) as our
baselines. MK-MMD and JAN (Long et al. 2015, 2017) are commonly used domain
adaption methods whereas DALN (Chen et al. 2022) is a state-of-the-art method
in this category. Representative methods in the other category include the
maximum classifier discrepancy (MCD) method (Saito et al. 2018) and the smooth
domain adversarial training (SDAT) method (Rangwani et al. 2022). MCD is
widely used for domain adaptation and SDAT is a state-of-the-art implicit
domain adaptation method. Table 3 lists all methods compared in our
evaluation.

Table 3: Methods Compared in Our Evaluation.  Method |  Notes
---|---
DACA |  Our method
MMD |  Cross-domain misinformation detection method based on the MMD metric (Huang et al. 2021)
CANMD |  Cross-domain misinformation detection method based on a variant of the MMD metric (Yue et al. 2022)
DANN |  Cross-domain misinformation detection method based on the learning of domain-invariant features (Li et al. 2021)
MK-MMD |  Domain adaptation method based on the multi-kernel MMD metric (Long et al. 2015)
JAN |  Domain adaptation method based on the joint MMD metric (Long et al. 2017)
DALN |  Domain adaptation method using the Nuclear-norm Wasserstein Discrepancy (Chen et al. 2022)
MCD |  Domain adaptation method utilizing a mini-max mechanism to align feature distributions between source and target domains (Saito et al. 2018)
SDAT |  Domain adaptation method employing a smoothing mechanism to learn domain-invariant features (Rangwani et al. 2022)

In each evaluation scenario, all compared methods took identical inputs and
employed the same content embedding method to represent the textual contents
of these inputs. Specifically, English words in the inputs were embedded using
the pretrained Roberta model (Liu et al. 2019). Next, the content embedding
method proposed by Zhu et al. (2023) was utilized to embed the textual
contents of the inputs. This method was developed based on BERT (Kenton and
Toutanova 2019) and extracted three types of features from each instance of
textual content (e.g., a news article): (1) semantic features, such as word
usage patterns, (2) style features, such as writing styles, and (3) emotional
features, such as emotional signals expressed in a news article. It then
aggregated these features into a single feature vector through an attention
mechanism. We trained our DACA method using the Adam optimizer with a learning
rate of $0.0001$ (Kingma and Ba 2015). The hyperparamters in Equation 13 of
our method were set as follows: the number of instances of source domain
information $m$ was set to 3, and the hyperparameter $\tau$ was set to 0.5.
Implementation details of the benchmark methods are given in Appendix C.

###  5.3 Evaluation Results

Table 4 presents the average recall and F1-score for each method in evaluation
scenario 1, along with standard deviations (in parentheses), across 25
experiments. As reported, our DACA method significantly outperforms each
benchmark method in both recall and F1-score. In particular, our method
achieves an average recall of 0.745, indicating that, on average, it correctly
identifies 74.5% of fake news in the COVID domain. Such performance is
attained without any labeled COVID news in the training data, highlighting the
efficacy of our method in transferring a model learned from labeled news in
the entertainment and politics domains to predict the labels of news in the
COVID domain. Moreover, our method respectively outperforms three state-of-
the-art misinformation detection methods—MMD, CANMD, and DANN—by 5.82%, 6.13%,
and 13.22% in recall, and by 6.05%, 8.77%, and 10.62% in F1-score.
Additionally, the performance advantages of our method over representative
domain adaptation methods—MK-MMD, JAN, DALN, MCD, and SDAT—range from 4.20% to
11.36% in recall and from 4.66% to 18.26% in F1-score. Since the key
methodological difference between our method and the benchmark methods lies in
the mitigation of concept shift by its concept alignment module, the
performance improvements achieved by our method can be largely attributed to
this module. Given the huge volume of information generated at the early stage
of an infodemic, such performance advantages achieved by our method could
result in substantially more instances of misinformation being identified by
our method, in comparison to the benchmarks. As a result, a greater volume of
misinformation could be prevented from dissemination, thereby significantly
benefiting public health and society at large (Buchanan 2020, Van Der Linden
2022).

Table 4: Performance Comparison between Our Method and Benchmark Methods
(Evaluation Scenario 1).

Method |  Recall |  Improvement by DACA |  F1-score |  Improvement by DACA
---|---|---|---|---
DACA |  0.745 |  |  0.719 |
(0.049) |  |  (0.034) |
MMD |  0.704∗∗ |  5.82% |  0.678∗∗ |  6.05%
(0.044) |  |  (0.049) |
CANMD |  0.702∗∗ |  6.13% |  0.661∗∗ |  8.77%
(0.026) |  |  (0.030) |
DANN |  0.658∗∗ |  13.22% |  0.650∗∗ |  10.62%
(0.053) |  |  (0.047) |
MK-MMD |  0.715∗ |  4.20% |  0.687∗∗ |  4.66%
(0.038) |  |  (0.041) |
JAN |  0.710∗∗ |  4.93% |  0.686∗∗ |  4.81%
(0.030) |  |  (0.034) |
DALN |  0.669∗∗ |  11.36% |  0.660∗∗ |  7.47%
(0.039) |  |  (0.052) |
MCD |  0.671∗∗ |  11.03% |  0.608∗∗ |  18.26%
(0.020) |  |  (0.049) |
SDAT |  0.712∗∗ |  4.63% |  0.682∗∗ |  5.43%
(0.002) |  |  (0.015) |

  * *

Note: Significance levels are denoted by * and ** for 0.05 and 0.01,
respectively.

Table 5 reports the average recall and F1-score for each method in evaluation
scenario 2, with standard deviations given in parentheses. Again, our method
significantly outperforms each benchmark method in both evaluation metrics.
Specifically, it surpasses three state-of-the-art misinformation detection
methods with recall improvements ranging from 6.15% to 17.00% and F1-score
enhancements ranging from 5.47% to 13.92%. Moreover, in comparison to five
representative domain adaptation methods, our method boosts recall by a range
of 4.73% to 12.74% and increases F1-score by a range of 4.13% to
13.47%.121212For each method, its performance change in two evaluation
scenarios is small because only 8% of COVID news is accompanied by user
comments. Further, different methods respond differently to noises and signals
in user comments. As a result, dropping user comments associated with COVID
news leads to decreased performance for some methods and improved performance
for others. As a robustness check, we conducted another empirical evaluation
using a public dataset of Chinese news. Evaluation results reported in
Appendix D further support the superior performance of our method over the
benchmarks.

Table 5: Performance Comparison between Our Method and Benchmark Methods
(Evaluation Scenario 2).

Method |  Recall |  Improvement by DACA |  F1-score |  Improvement by DACA
---|---|---|---|---
DACA |  0.757 |  |  0.720 |
(0.019) |  |  (0.021) |
MMD |  0.713∗∗ |  6.15% |  0.682∗∗ |  5.47%
(0.048) |  |  (0.046) |
CANMD |  0.701∗∗ |  7.89% |  0.657∗∗ |  9.59%
(0.031) |  |  (0.033) |
DANN |  0.647∗∗ |  17.00% |  0.632∗∗ |  13.92%
(0.038) |  |  (0.039) |
MK-MMD |  0.722∗∗ |  4.73% |  0.691∗∗ |  4.13%
(0.031) |  |  (0.033) |
JAN |  0.710∗∗ |  6.63% |  0.680∗∗ |  5.79%
(0.026) |  |  (0.036) |
DALN |  0.671∗∗ |  12.74% |  0.661∗∗ |  8.87%
(0.040) |  |  (0.043) |
MCD |  0.679∗∗ |  11.50% |  0.633∗∗ |  13.47%
(0.021) |  |  (0.025) |
SDAT |  0.695∗∗ |  8.80% |  0.665∗∗ |  8.20%
(0.025) |  |  (0.035) |

  * *

Note: Significance levels are denoted by * and ** for 0.05 and 0.01,
respectively.

###  5.4 Performance Analysis

Having demonstrated the superior performance of our method over the
benchmarks, it is intriguing to delve deeper and analyze the factors
contributing to its performance advantages. To this end, we conducted ablation
studies to investigate the contribution of each novel design artifact of our
DACA method to its performance. In particular, we focused on the key novelty
of our method – the concept alignment module. As elaborated in Section 4.4, it
consists of two sub-modules: the contrastive learning sub-module and the
concept alignment sub-module. The former represents instances of information
in a transformed space, whereas the latter computes distances between these
instances in the transformed space and mitigates concept shift. To evaluate
the overall contribution of the concept alignment module, we removed this
module from the DACA method and designated the resulting method as W/o_CL_CA
(Without both the Contrastive Learning and the Concept Alignment sub-modules).
The performance difference between DACA and W/o_CL_CA reveals the the overall
contribution of the concept alignment module to the performance of our method.
To further investigate the functioning mechanism of the concept alignment
module, we dropped its contrastive learning sub-module but kept its concept
alignment sub-module. We named the resulting method W/o_CL (Without the
Contrastive Learning sub-module). The performance difference between DACA and
W/o_CL uncovers the role of the contrastive learning sub-module in concept
alignment.

Table 6: Ablation Study of DACA (Evaluation Scenario 1).

Method |  Recall |  Improvement by DACA |  F1-score |  Improvement by DACA
---|---|---|---|---
DACA |  0.745 |  |  0.719 |
(0.049) |  |  (0.034) |
W/o_CL_CA |  0.658∗∗ |  13.22% |  0.650∗∗ |  10.62%
(0.053) |  |  (0.047) |
W/o_CL |  0.663∗∗ |  12.37% |  0.648∗∗ |  10.96%
(0.047) |  |  (0.052) |

  * *

Note: Significance level is denoted by ** for 0.01.

Table 7: Ablation Study of DACA (Evaluation Scenario 2).

Method |  Recall |  Improvement by DACA |  F1-score |  Improvement by DACA
---|---|---|---|---
DACA |  0.757 |  |  0.720 |
(0.019) |  |  (0.021) |
W/o_CL_CA |  0.647∗∗ |  17.00% |  0.632∗∗ |  13.92%
(0.038) |  |  (0.039) |
W/o_CL |  0.663∗∗ |  14.18% |  0.638∗∗ |  12.85%
(0.034) |  |  (0.025) |

  * *

Note: Significance level is denoted by ** for 0.01.

Table 6 presents the average performance of DACA, W/o_CL_CA, and W/o_CL in
evaluation scenario 1, along with their respective standard deviations (in
parentheses), across 25 experiments. As reported, the performance of W/o_CL_CA
is significantly inferior to that of DACA, due to the removal of the concept
alignment module. More specifically, W/o_CL_CA trails DACA by 13.22% in recall
and 10.62% in F1-score, which collectively demonstrate the contribution of the
concept alignment module to the performance of DACA. Furthermore, W/o_CL
performs significantly worse than DACA but comparable to W/o_CL_CA. It is
noted that W/o_CL has the concept alignment sub-module while omitting the
contrastive learning sub-module. The comparison results thus show that, to
realize the benefits of the concept alignment sub-module, it must be coupled
with the contrastive learning sub-module. This empirical finding is in line
with the theoretical analysis in Section 4.1, which defines concept shift over
source domain instances and their corresponding nearest target domain
instances. Therefore, to mitigate concept shift, we need a suitable similarity
function that measures similarities between instances in an appropriate
representation space. And the similarity function is learned by the
contrastive learning sub-module. Table 7 reports the results of the ablation
study in evaluation scenario 2, consistent with our findings from the ablation
study in evaluation scenario 1. Overall, the ablation studies in both
scenarios demonstrate the significant contribution of its concept alignment
module to the performance of our method. Moreover, the ablation studies show
that it is appropriate to design this module with two sub-modules: contrastive
learning and concept alignment.

##  6 Conclusions

###  6.1 Summary and Contributions

To contain harmful effects of an infodemic on public health, it is crucial to
detect misinformation at the early stage of the infodemic. An early stage
infodemic is characterized by a large volume of unlabeled information spread
across various media platforms. Consequently, conventional misinformation
detection methods are not suitable for this misinformation detection task
because they rely on labeled information in the infodemic domain to train
their models. State-of-the-art misinformation detection methods learn their
models with labeled information in other domains to detect misinformation in
the infodemic domain, thereby applicable to the task. The efficacy of these
methods depends on their ability to mitigate both covariate shift and concept
shift between the infodemic domain and the domains from which they leverage
labeled information. These methods focus on mitigating covariate shift but
overlook concept shift, making them less effective for the task. In response,
we propose a novel misinformation detection method that addresses both
covariate shift and concept shift. Through extensive empirical evaluations
with two widely used datasets, we demonstrate the superior performance of our
method over state-of-the-art misinformation detection methods as well as
prevalent domain adaptation methods that can be tailored to solve the
misinformation detection task.

Our study makes the following contributions to the extant literature. First,
our study belongs to the area of computational design science research in the
Information Systems (IS) field (Rai et al. 2017, Padmanabhan et al. 2022).
This area of research develops computational algorithms and methods to solve
business and societal problems and aims at making methodological contributions
to the literature, e.g., Abbasi et al. (2010), Li et al. (2017), Zhao et al.
(2023). In particular, the methodological contribution of our study lies in
its addressing of concept shift, in addition to covariate shift. More
specifically, we theoretically show the importance of addressing concept shift
and how to operationalize it. Built on the theoretical analysis, we develop a
novel concept alignment module to mitigate concept shift, as described in
Section 4.4. Second, given its significant social and economic impact,
misinformation detection and management has attracted attention from IS
scholars, e.g., Moravec et al. (2019), Wei et al. (2022), Hwang and Lee
(2024). Our study adds to this stream of IS research with a novel method that
is effective in detecting misinformation at the early stage of an infodemic.

###  6.2 Implications for Infodemic Management and Future Work

Every epidemic is accompanied by an infodemic, a phenomenon known since the
Middle Ages (Zarocostas 2020). The wide dissemination of misinformation during
an infodemic misleads people to dismiss health guidance and pursue
unscientific treatments, resulting in substantial harm to public health and
significant social and economic consequences (Bursztyn et al. 2020, Romer and
Jamieson 2020). Furthermore, the pervasive reach of the Internet and social
media platforms accelerates the spread of misinformation and amplifies its
harmful impacts on public health and society (Zarocostas 2020). Detecting
misinformation at the early stage of an infodemic discourages people from
believing and sharing it, thereby preventing it from going viral (Buchanan
2020). Hence, early detection of misinformation is crucial for managing an
infodemic and mitigating its adverse effects. Accordingly, a direct
implication of our study is to provide an effective early misinformation
detection method for infodemic management. Our method effectively overcomes
obstacles to early detection of misinformation. First, vast amount of
information spread during the early stage of an infodemic makes manual
identification of misinformation impractical. In response, our deep learning-
based method automatically learns to differentiate misinformation from true
information. Second, there is no labeled information at the early stage of an
infodemic, rendering traditional misinformation detection methods
inapplicable. Accordingly, our method leverages labeled information in other
domains to detect misinformation in the infodemic domain.

Our study empowers infodemic management in several ways. Operators of media
platforms can employ our method to effectively detect misinformation in the
infodemic domain that spreads on their platforms. Subsequently, they can flag
and debunk identified misinformation, which helps curb the diffusion of
misinformation and alleviate its negative impacts (Pennycook et al. 2020, Wei
et al. 2022). Additionally, flagged misinformation is less likely to be
clicked, thereby reducing revenues for those who monetize misinformation and
diminishing their incentives to create more misinformation (Pennycook and Rand
2021). Moreover, platform operators can trace misinformation identified by our
method back to its producers. They could then restrict those who frequently
produce misinformation from publishing on their platforms. In this regard,
platforms are encouraged to set up guidelines to regulate their content
producers (Hartley and Vu 2020). Platform operators can also trace
misinformation detected by our method to its spreaders. Understandably,
blocking individuals who disseminate a large amount of misinformation is
important for combating misinformation on their platforms.

Furthermore, our study sheds light on the value of cross-domain data sharing
for infodemic management. At the early stage of an infodemic, even experts
have difficulty in distinguishing between misinformation and true information.
Moreover, recruiting experts to verify and label information is costly (Kim et
al. 2018). Cross-domain data sharing enables us to utilize labeled information
in other domains to detect misinformation in the infodemic domain. Hence, it
is a viable and cost-effective approach to misinformation detection and
infodemic management. To implement this approach, we need to address a
challenge. Information from different domains exhibits different marginal and
conditional distributions, known as covariate shift and concept shift,
respectively. Thus, the challenge is how to mitigate covariate shift and
concept shift between the infodemic domain and source domains that provide
labeled information. Our proposed DACA method is effective in tackling this
challenge. To facilitate cross-domain data sharing for infodemic management,
media platforms are recommended to establish data exchange mechanisms to
ensure that participating platforms are properly incentivized and labeled
information is securely shared.

Our study has limitations and can be extended in several directions. First,
our method employs information content to detect misinformation. Future work
could extend our method by incorporating information propagation patterns. To
this end, graphical neural networks are useful tools for modeling such
patterns. Second, our empirical evaluations are based on fake news datasets,
which have been widely utilized to assess the performance of misinformation
detection methods in the literature. It is also interesting to evaluate the
performance of our method in detecting other types of misinformation such as
hyperpartisan news and yellow journalism (Pennycook and Rand 2021). Third, our
method could be generalized to solve other domain adaptation problems, in
addition to the misinformation detection problem tackled in this study. Hence,
another area worthy of future investigation is how to generalize our method to
solve the problem of classifying unlabeled target domain instances by
leveraging labeled source domain instances.

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##  Appendix A $\mathcal{H}$-divergence

$\mathcal{H}$-divergence has been widely utilized to measure distances between
distributions over a feature space.

Definition A.1. ($\mathcal{H}$-divergence) ben2010theory2. Given a feature
space $\mathcal{X}$, let $\mathcal{H}$ denote a hypothesis space defined on
$\mathcal{X}$ and $\mathbb{D}_{S}$ and $\mathbb{D}_{T}$ be two probability
distributions over $\mathcal{X}$. For a hypothesis $h\in\mathcal{H}$, let
$I(h)$ be the set such that $\textbf{x}\in I(h)$ if and only if
$h(\textbf{x})>\tau$, where $\tau$ is the classification threshold and an
instance with features x is classified as 1 if $h(\textbf{x})>\tau$ and 0
otherwise. The $\mathcal{H}$-divergence between $\mathbb{D}_{S}$ and
$\mathbb{D}_{T}$ is defined as

$d_{\mathcal{H}}(\mathbb{D}_{S},\mathbb{D}_{T})=2\sup_{h\in\mathcal{H}}|Pr_{\mathbb{D}_{S}}[I(h)]-Pr_{\mathbb{D}_{T}}[I(h)]|,$
(A1)

where $Pr$ denotes probability.131313ben2010theory2 consider a hypothesis $h$
that produces a discrete output, i.e., 0 or 1, while we consider a hypothesis
$h$ that predicts a continuous output between 0 and 1.

The $\mathcal{H}$-divergence between two distributions can be empirically
estimated using two sample datasets drawn from these distributions,
respectively ben2010theory2. Given a dataset of $N_{S}$ source domain
instances $\mathcal{D}_{S}$ and a dataset of $N_{T}$ target domain instances
$\mathcal{D}_{T}$, features of which are from distributions $\mathbb{D}_{S}$
and $\mathbb{D}_{T}$ respectively, the $\mathcal{H}$-divergence between
$\mathbb{D}_{S}$ and $\mathbb{D}_{T}$ can be estimated using
$\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$ ben2010theory2:

$\hat{d}_{\mathcal{H}}\left(\mathcal{D}_{S},\mathcal{D}_{T}\right)=2\left(1-\min_{h\in\mathcal{H}}\left[\frac{1}{N_{S}}\sum_{\mathbf{x}:h(\mathbf{x})\leq\tau}I[\mathbf{x}\in\mathcal{D}_{S}]+\frac{1}{N_{T}}\sum_{\mathbf{x}:h(\mathbf{x})>\tau}I\left[\mathbf{x}\in\mathcal{D}_{T}\right]\right]\right),$
(A2)

where indicator function $I[\mathbf{x}\in\mathcal{D}_{S}]$ is 1 if an instance
with feature x belongs to $\mathcal{D}_{S}$ and $\mathcal{H}$ is a symmetric
hypothesis space (i.e., for each hypothesis $h\in\mathcal{H}$, its inverse
$1-h$ is also in $\mathcal{H}$).

##  Appendix B Proof of Theorem and Proposition

Let
$\epsilon_{S}(g_{1},g_{2})=E_{\textbf{x}\sim\mathbb{D}_{S}}[|g_{1}(\textbf{x})-g_{2}(\textbf{x})|]$
be the expected difference between $g_{1}$ and $g_{2}$ in the source domain,
where $g_{1}$ and $g_{2}$ are two labeling functions. Similarly,
$\epsilon_{T}(g_{1},g_{2})=E_{\textbf{x}\sim\mathbb{D}_{T}}[|g_{1}(\textbf{x})-g_{2}(\textbf{x})|]$
denotes the expected difference between $g_{1}$ and $g_{2}$ in the target
domain. To prove Theorem 4.1, we first show an upper bound of
$\epsilon_{S}\left(f_{S},f_{T}\right)$:

Lemma 1 Let $f_{S}$ and $f_{T}$ respectively denote the true source and target
domain labeling functions, $\mathbb{D}_{S}$ be the source domain feature
distribution, and $\mathcal{D}_{S}$ be a dataset with $N_{S}$ samples drawn
from $\mathbb{D}_{S}$, for any $\eta\in(0,1)$, the following inequality holds
with probability at least $1-\eta$:

$\displaystyle\epsilon_{S}\left(f_{S},f_{T}\right)=E_{\textbf{x}\sim\mathbb{D}_{S}}[|f_{S}(\textbf{x})-f_{T}(\textbf{x})|]$
$\displaystyle\leq\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|f_{S}(\textbf{x}_{i}^{S})-f_{T}(\textbf{x}_{i}^{S})|+\sqrt{-\frac{\ln(\frac{\eta}{2})}{2N_{S}}},$
(A3)

where $\textbf{x}_{i}^{S}$ is the $i$-th sample drawn i.i.d. from
$\mathbb{D}_{S}$.

Proof. This proof relies on McDiarmid’s inequality doob1940regularity2. For
notation convenience, define $\textbf{z}=f_{S}(\textbf{x})-f_{T}(\textbf{x})$
where $\textbf{x}\sim\mathbb{D}_{S}$. We then have
$\textbf{z}_{i}=f_{S}(\textbf{x}_{i}^{S})-f_{T}(\textbf{x}_{i}^{S}),i=1,2,\cdots,N_{s}$.
Further define
$g(\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}})=\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|\textbf{z}_{i}|$.
Function $g$ satisfies the property that

$\displaystyle|g(\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}})-g(\textbf{z}_{1},\dots,\textbf{z}_{k-1},\textbf{z}_{k}^{\prime},\textbf{z}_{k+1},\dots,\textbf{z}_{N_{S}})|\leq\frac{1}{N_{S}},$
(A4)

which means that by changing any $k$-th coordinate, the function will change
at most $\frac{1}{N_{S}}$. This is because
$|\textbf{z}_{k}|=|f_{S}(\textbf{x}_{k}^{S})-f_{T}(\textbf{x}_{k}^{S})|\in[0,1]$.

Given that $\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}}$ are i.i.d.
random samples, for any $\epsilon>0$, by McDiarmid’s inequality

$\displaystyle
p(|g(\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}})-E[g(\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}})]|\geq\epsilon)$
$\displaystyle\leq
2\exp(\frac{-2\epsilon^{2}}{\sum_{k=1}^{N_{S}}\frac{1}{N_{S}^{2}}})$ (A5)
$\displaystyle=2\exp(-2N_{S}\epsilon^{2}).$

Equivalently,

$\displaystyle
p(|g(\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}})-E[g(\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}})]|\leq\epsilon)$
$\displaystyle\geq 1-2\exp(-2N_{S}\epsilon^{2}).$ (A6)

By replacing $\epsilon$ with $\sqrt{-\frac{\ln(\frac{\eta}{2})}{2N_{S}}}$
where $\eta\in(0,1)$, we have

$\displaystyle
p\Bigg{(}|g(\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}})-E[g(\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}})]|\leq\sqrt{-\frac{\ln(\frac{\eta}{2})}{2N_{S}}}\Bigg{)}$
$\displaystyle\geq 1-\eta,$ (A7)

Note that
$E[g(\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}})]=E_{\textbf{x}\sim\mathbb{D}_{S}}[|f_{S}(\textbf{x})-f_{T}(\textbf{x})|]$
by construction, while
$g(\textbf{z}_{1},\textbf{z}_{2},\dots,\textbf{z}_{N_{S}})=\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|f_{S}(\textbf{x}_{i}^{S})-f_{T}(\textbf{x}_{i}^{S})|$
by definition. Based on the inequality $a-b\leq|b-a|$, Inequality A7 means
that Inequality A3 must hold with probability at least $1-\eta$, which
concludes our proof of Lemma 1.

Next, we show, in Lemma 2, that the population $\mathcal{H}$-divergence is
bounded by the empirical $\mathcal{H}$-divergence.

Lemma 2  Let $\mathcal{H}$ be a hypothesis space with VC dimension $d$.
$\mathcal{D}_{S}$ and $\mathcal{D}_{T}$ are datasets of instances drawn from
distributions $\mathbb{D}_{S}$ and $\mathbb{D}_{T}$, respectively. Let
$d_{\mathcal{H}}(\mathbb{D}_{S},\mathbb{D}_{T})$ denote the population
$\mathcal{H}$-divergence between distributions $\mathbb{D}_{S}$ and
$\mathbb{D}_{T}$, and $\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$
be its empirical estimation from datasets $\mathcal{D}_{S}$ and
$\mathcal{D}_{T}$. For any $\eta\in(0,1)$, with probability at least $1-\eta$,

$d_{\mathcal{H}}\left(\mathbb{D}_{S},\mathbb{D}_{T}\right)\leq\hat{d}_{\mathcal{H}}\left(\mathcal{D}_{S},\mathcal{D}_{T}\right)+4\max\\{\sqrt{\frac{d\ln(2N_{S})+\ln(\frac{2}{\eta})}{N_{S}}},\sqrt{\frac{d\ln(2N_{T})+\ln(\frac{2}{\eta})}{N_{T}}}\\},$
(A8)

where $N_{S}$ is the number of instances in $\mathcal{D}_{S}$ and $N_{T}$ is
the number of instances in $\mathcal{D}_{T}$.

Proof. According to kifer2004detecting2, we have

$\displaystyle
P\left[\left|d_{\mathcal{H}}\left(\mathbb{D}_{S},\mathbb{D}_{T}\right)-\hat{d}_{\mathcal{H}}\left(\mathcal{D}_{S},\mathcal{D}_{T}\right)\right|>\epsilon\right]\leq\left(2N_{S}\right)^{d}e^{-N_{S}\epsilon^{2}/16}+\left(2N_{T}\right)^{d}e^{-N_{T}\epsilon^{2}/16}.$
(A9)

By replacing
$\left(2N_{S}\right)^{d}e^{-N_{S}\epsilon^{2}/16}+\left(2N_{T}\right)^{d}e^{-N_{T}\epsilon^{2}/16}$
with $\eta$, we get

$\eta\leq
2\max\\{\left(2N_{S}\right)^{d}e^{-N_{S}\epsilon^{2}/16},\left(2N_{T}\right)^{d}e^{-N_{T}\epsilon^{2}/16}\\},$
(A10)

which means that $\eta\leq 2\left(2N_{S}\right)^{d}e^{-N_{S}\epsilon^{2}/16}$
as well as $\eta\leq 2\left(2N_{T}\right)^{d}e^{-N_{S}\epsilon^{2}/16}$. In
other words, we have $\epsilon\leq 4\sqrt{(d\ln(2N_{S})+\ln(2/\eta))/N_{S}}$
as well as $\epsilon\leq 4\sqrt{(d\ln(2N_{T})+\ln(2/\eta))/N_{T}}$. Because
increasing $\epsilon$ will decrease the probability
$P\left[\left|d_{\mathcal{H}}\left(\mathbb{D}_{S},\mathbb{D}_{T}\right)-\hat{d}_{\mathcal{H}}\left(\mathcal{D}_{S},\mathcal{D}_{T}\right)\right|>\epsilon\right]$,
for

$\epsilon_{\max}=4\max\\{\sqrt{\frac{d\ln(2N_{S})+\ln(\frac{2}{\eta})}{N_{S}}},\sqrt{\frac{d\ln(2N_{T})+\ln(\frac{2}{\eta})}{N_{T}}}\\},$
(A11)

we have

$\displaystyle\eta$ $\displaystyle\geq
P\left[\left|d_{\mathcal{H}}\left(\mathbb{D}_{S},\mathbb{D}_{T}\right)-\hat{d}_{\mathcal{H}}\left(\mathcal{D}_{S},\mathcal{D}_{T}\right)\right|>\epsilon\right]$
(A12) $\displaystyle\geq
P\left[\left|d_{\mathcal{H}}\left(\mathbb{D}_{S},\mathbb{D}_{T}\right)-\hat{d}_{\mathcal{H}}\left(\mathcal{D}_{S},\mathcal{D}_{T}\right)\right|>\epsilon_{\max}\right]$

which concludes the proof of Lemma 2. If $N_{S}=N_{T}$, Lemma 2 will reduce to
Lemma 1 in ben2010theory2.

Definition A.2. (The ideal hypothesis). The ideal hypothesis $h^{*}$ is
defined as the hypothesis that minimizes the summation of the expected
differences between $h\in\mathcal{H}$ and the target domain labeling function
$f_{T}$ in both source and target domains

$h^{*}=\argmin_{h\in\mathcal{H}}\epsilon_{S}(h,f_{T})+\epsilon_{T}(h,f_{T}),$
(A13)

where
$\epsilon_{S}(h,f_{T})=E_{\textbf{x}\sim\mathbb{D}_{S}}[|h(\textbf{x})-f_{T}(\textbf{x})|]$
and
$\epsilon_{T}(h,f_{T})=E_{\textbf{x}\sim\mathbb{D}_{T}}[|h(\textbf{x})-f_{T}(\textbf{x})|]$.

The error incurred by the ideal hypothesis is
$\lambda=\epsilon_{S}(h^{*},f_{T})+\epsilon_{T}(h^{*},f_{T})$. Now, we provide
the proof of Theorem 4.1 utilizing the lemmas established above.

Theorem 1  For a source domain instance with features $\textbf{x}_{i}^{S}$,
let $c(\textbf{x}_{i}^{S})$ be the features of its nearest target domain
instance, where $i=1,2,\dots,N_{S}$ and the distance between a pair of
instances are measured over their feature space. If $f_{T}$ is $L$-Lipschitz
continuous, then for any $\eta\in(0,1)$, with probability at least
$(1-\eta)^{2}$,

$\displaystyle\epsilon_{T}(h)\leq\epsilon_{S}(h)+\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|f_{S}(\textbf{x}_{i}^{S})-f_{T}(c(\textbf{x}_{i}^{S}))|+\frac{L}{N_{S}}\sum_{i=1}^{N_{S}}||\textbf{x}_{i}^{S}-c(\textbf{x}_{i}^{S})||+C_{1},$
(A14)

where $L$ is the Lipschitz constraint constant,
$\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})$ denotes the empirical
$\mathcal{H}$-divergence between distributions $\mathbb{D}_{S}$ and
$\mathbb{D}_{T}$, $||\textbf{x}_{i}^{S}-c(\textbf{x}_{i}^{S})||$ represents
the distance between $\textbf{x}_{i}^{S}$ and $c(\textbf{x}_{i}^{S})$, and
$C_{1}=\lambda+\sqrt{-\frac{\ln(\frac{\eta}{2})}{2N_{S}}}+2\max\\{\sqrt{\frac{d\ln(2N_{S})+\ln(\frac{2}{\eta})}{N_{S}}},\sqrt{\frac{d\ln(2N_{T})+\ln(\frac{2}{\eta})}{N_{T}}}\\}$.

Proof. With probability at least $(1-\eta)^{2}$,

$\displaystyle\epsilon_{T}(h)$
$\displaystyle\leq\epsilon_{T}\left(h^{*}\right)+\epsilon_{S}(h)+\epsilon_{S}\left(h^{*}\right)+\frac{1}{2}d_{\mathcal{H}}\left(\mathbb{D}_{S},\mathbb{D}_{T}\right)$
(A15)
$\displaystyle\leq\epsilon_{T}\left(h^{*}\right)+\epsilon_{S}(h)+\epsilon_{S}\left(h^{*},f_{T}\right)+\epsilon_{S}\left(f_{S},f_{T}\right)+\frac{1}{2}d_{\mathcal{H}}\left(\mathbb{D}_{S},\mathbb{D}_{T}\right)$
$\displaystyle=\epsilon_{S}(h)+\frac{1}{2}d_{\mathcal{H}}\left(\mathbb{D}_{S},\mathbb{D}_{T}\right)+\epsilon_{S}\left(f_{S},f_{T}\right)+\lambda$
$\displaystyle\leq\epsilon_{S}(h)+\frac{1}{2}\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|f_{S}(\textbf{x}_{i}^{S})-f_{T}(\textbf{x}_{i}^{S})|+C_{1}$
$\displaystyle=\epsilon_{S}(h)+\frac{1}{2}\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|f_{S}(\textbf{x}_{i}^{S})+f_{T}(c(\textbf{x}_{i}^{S}))-f_{T}(c(\textbf{x}_{i}^{S}))-f_{T}(\textbf{x}_{i}^{S})|+C_{1}$
$\displaystyle\leq\epsilon_{S}(h)+\frac{1}{2}\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|f_{S}(\textbf{x}_{i}^{S})-f_{T}(c(\textbf{x}_{i}^{S}))|+\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|f_{T}(c(\textbf{x}_{i}^{S}))-f_{T}(\textbf{x}_{i}^{S})|+C_{1}$
$\displaystyle\leq\epsilon_{S}(h)+\frac{1}{2}\hat{d}_{\mathcal{H}}(\mathcal{D}_{S},\mathcal{D}_{T})+\frac{1}{N_{S}}\sum_{i=1}^{N_{S}}|f_{S}(\textbf{x}_{i}^{S})-f_{T}(c(\textbf{x}_{i}^{S}))|+\frac{L}{N_{S}}\sum_{i=1}^{N_{S}}||\textbf{x}_{i}^{S}-c(\textbf{x}_{i}^{S})||+C_{1}.$

The first step is based on the proof of Theorem 2 in ben2010theory2. The
second step is followed by triangle inequality. The third step utilizes the
definition $\lambda=\epsilon_{S}(h^{*},f_{T})+\epsilon_{T}(h^{*},f_{T})$ and
the definition $\epsilon_{T}\left(h^{*}\right)=\epsilon_{T}(h^{*},f_{T})$. The
fourth step employs Lemma 1 and Lemma 2 to respectively expand the terms
$\epsilon_{S}\left(f_{S},f_{T}\right)$ and
$d_{\mathcal{H}}(\mathbb{D}_{S},\mathbb{D}_{T})$ while collecting the
constants into $C_{1}$. Because the inequalities given by Lemma 1 and Lemma 2
hold both with at least probability $1-\eta$, the inequality at the second
step holds with at least probability $(1-\eta)^{2}$ under the independence
assumption. The fifth step adds and subtracts $f_{T}(c(\textbf{x}_{i}^{S}))$.
The sixth step utilizes triangle inequality. The last step is from the
assumption of $L$-Lipschitz continuous for $f_{T}$.

By extending Theorem 1 to the case of multiple source domains, we have the
following proposition.

Proposition 1  For a source domain instance with features
$\textbf{x}_{i}^{S_{j}}$ in $\mathcal{D}_{S_{j}}$, let
$c(\textbf{x}_{i}^{S_{j}})$ be the features of its nearest instance in
$\mathcal{D}_{T}$, where $j=1,2,\dots,k$ and $i=1,2,\dots,N_{S_{j}}$. If
$f_{T}$ is $L$-Lipschitz continuous, then for any $\eta\in(0,1)$, with
probability at least $(1-\eta)^{2k}$,

$\epsilon_{T}(h)\leq\frac{1}{k}\sum_{j=1}^{k}\\{\epsilon_{S_{j}}(h)+\frac{1}{2}\hat{d}_{\mathcal{H}}(\mathcal{D}_{S_{j}},\mathcal{D}_{T})+\frac{1}{N_{S_{j}}}\sum_{i=1}^{N_{S_{j}}}|f_{S_{j}}(\textbf{x}_{i}^{S_{j}})-f_{T}(\textbf{x}_{i}^{{S_{j}},T})|+\frac{L}{N_{S_{j}}}\sum_{i=1}^{N_{S_{j}}}||\textbf{x}_{i}^{S_{j}}-\textbf{x}_{i}^{{S_{j}},T}||+C_{j}\\},$
(A16)

where
$C_{j}=\lambda_{j}+\sqrt{-\frac{\ln(\frac{\eta}{2})}{2N_{S_{j}}}}+2\max\\{\sqrt{\frac{d\ln(2N_{S_{j}})+\ln(\frac{2}{\eta})}{N_{S_{j}}}},\sqrt{\frac{d\ln(2N_{T})+\ln(\frac{2}{\eta})}{N_{T}}}\\}$,
$\lambda_{j}$ is the error incurred by the ideal hypothesis
$\lambda_{j}=\epsilon_{S_{j}}(h^{*},f_{T})+\epsilon_{T}(h^{*},f_{T})$, and $L$
is the constant for the Lipschitz condition.

Proof. By Theorem 1, for any source domain $j=1,2,\cdots,k$, the following
inequality holds with probability at least $(1-\eta)^{2}$ for any
$\eta\in(0,1)$,

$\displaystyle\epsilon_{T}(h)\leq\epsilon_{S_{j}}(h)+\hat{d}_{\mathcal{H}}(\mathcal{D}_{S_{j}},\mathcal{D}_{T})+\frac{1}{N_{S_{j}}}\sum_{i=1}^{N_{S_{j}}}|f_{S_{j}}(\textbf{x}_{i}^{S_{j}})-f_{T}(c(\textbf{x}_{i}^{S_{j}}))|+\frac{L}{N_{S_{j}}}\sum_{i=1}^{N_{S_{j}}}||\textbf{x}_{i}^{S_{j}}-c(\textbf{x}_{i}^{S_{j}})||+C_{j}.$
(A17)

Proposition 1 follows immediately by summing on each side of inequality (A17)
over all source domains.

##  Appendix C Implementation Details of Benchmark Methods

MMD.  Maximum Mean Discrepancy (MMD) is a metric to measure the discrepancy
between distributions in a reproducing kernel Hilbert space (RKHS)
borgwardt2006integrating2. Specifically, a kernel function $k$ is associated
with a feature map $\phi$, i.e.,
$k(\textbf{x}_{S},\textbf{x}_{T})=<\phi(\textbf{x}_{S}),\phi(\textbf{x}_{T})>$,
where $\textbf{x}_{S}$ and $\textbf{x}_{T}$ are source and target
distributions. Let $\mathcal{H}$ be an RKHS, and MMD is defined as the
discrepancy between the mean embeddings of source and target distributions in
$\mathcal{H}$:

$\text{MMD}=d(p_{S},p_{T})^{2}=||\frac{1}{N_{S}}\phi(\textbf{x}_{S})-\frac{1}{N_{T}}\phi(\textbf{x}_{T})||_{\mathcal{H}}^{2}.$
(A18)

As reviewed in Section 2.2, MMD is an explicit metric that measures the
discrepancy between source and target domains in terms of their feature
distributions. In our implementation, we utilized the Gaussian kernel
$k(\textbf{x}_{S},\textbf{x}_{T})=e^{||\textbf{x}_{S}-\textbf{x}_{T}||/\gamma}$
and set $\gamma=0.5$.

CANMD.  Contrastive Adaptation Network for Early Misinformation Detection
(CANMD) is a domain adaptation method devised for misinformation detection
yue2022contrastive2. This method assigns pseudo labels to target domain
instances. Let $D_{ij}$ denote the multi-kernel MMD discrepancy between the
covariate distribution of source domain instances with label $i$ and that of
target domain instances with pseudo label $j$, where $i,j\in\\{0,1\\}$. To
minimize the divergence between source and target domains, the loss is defined
as

$\mathcal{L}=D_{00}+D_{11}-\frac{1}{2}(D_{01}+D_{10}).$ (A19)

Following yue2022contrastive2, we implemented the multi-kernel MMD using 5
Gaussian kernels with
$k(\textbf{x}_{S},\textbf{x}_{T})=e^{||\textbf{x}_{S}-\textbf{x}_{T}||/\gamma}$
and $\gamma=2^{k},$ where the value of $k$ for each Gaussian kernel was set to
-3, -2, -1, 0, and 1, respectively. The Gaussian kernels were combined with
equal weights.

DANN.  The Domain Adversarial Neural Networks (DANN) method has two
classifiers: a label classifier and a domain classifier
ganin2016domain2,li2021multi2. The former is learned to classify each piece of
source or target domain information as true or false. The latter is used to
learn domain-invariant features. The hyperparameter $\lambda$ of DANN controls
the relative weight between its domain and label classifiers. In our
evaluation, we performed a grid search for the value of $\lambda$ from 1 to
10, with an interval of 1. We found that $\lambda=5$ yielded the best
performance for DANN and chose $\lambda=5$ accordingly.

MK-MMD.  Multi-kernel MMD (MK-MMD) is an extension of single-kernel MMD
long2015learning2. Instead of using a single kernel function, MK-MMD employs a
convex combination of multiple kernel functions. In our implementation, we
used Gaussian kernels
$k(\textbf{x}_{S},\textbf{x}_{T})=e^{||\textbf{x}_{S}-\textbf{x}_{T}||/\gamma}$
and $\gamma=2^{k},$ where the value of $k$ for each Gaussian kernel was set to
-3, -2, -1, 0, and 1, respectively. The Gaussian kernels were combined with
equal weights.

JAN.  Joint Adaptation Networks (JAN) measures the discrepancy between source
and target domains at each layer of its classifier using MK-MMD long2017deep2.
It then combines these discrepancies and minimizes the aggregated discrepancy.
The implementation of MK-MMD in JAN is the same as the implementation of MK-
MMD described above.

DALN.  Discriminator-free Adversarial Learning Network (DALN) utilizes the
Nuclear-norm Wasserstein Discrepancy (NWD) to measure the covariate shift
chen2022reusing2. To compute NWD, chen2022reusing2 devise a domain classifier
that predicts the probability of an instance belonging to the source domain.
NWD is then defined as the difference between the average probability of
source domain instances and that of target domain instances. The DALN method
minimizes the covariate shift by minimizing NWD. In addition, the DALN method
has a label classifier that predicts the label (true or false) for each
instance. The loss of the method is a summation of NWD and the label
classification loss.

MCD.  The Maximum Classifier Discrepancy (MCD) method consists of three steps
saito2018maximum2. In the first step, it trains two classifiers using labeled
source domain instances. Next, it maximizes the discrepancy between two
classifiers when applied to target domain instances. In the third step, it
adjusts its feature extractor to minimize such discrepancy. MCD can be viewed
as a generative adversarial network (GAN), with the feature extractor serving
as the generator and the two classifiers functioning as the discriminator.

SDAT.  Smooth Domain Adversarial Training (SDAT) is an implicit domain
adaptation method based on DANN rangwani2022closer2. Inspired by the
observation that the label classifier of DANN often falls into sharp minima,
rangwani2022closer2 devise a smooth training loss based on the Sharpness Aware
Minimization (SAM) proposed by foret2020sharpness2. In our implementation, we
set the neighborhood size $\rho$ in SAM to 0.05 by following
foret2020sharpness2.

##  Appendix D Additional Empircal Evaluation

Table A1: Summary Statistics of Chinese News Dataset.  Domain | Number of true news | Number of fake news
---|---|---
COVID (Infodemic / Target) | 201 | 124
Entertainment (Source) | 1000 | 440
Politics (Source) | 306 | 546
Table A2: Performance Comparison between Our Method and Benchmark Methods
using Chinese News Dataset (Evaluation Scenario 1).

Method |  Recall |  Improvement by DACA |  F1-score |  Improvement by DACA
---|---|---|---|---
DACA |  0.836 |  |  0.824 |
(0.004) |  |  (0.010) |
MMD |  0.804∗∗ |  3.88% |  0.797∗∗ |  3.33%
(0.006) |  |  (0.001) |
CANMD |  0.792∗∗ |  5.52% |  0.789∗∗ |  8.77%
(0.013) |  |  (0.012) |
DANN |  0.713∗∗ |  17.25% |  0.709∗∗ |  16.20%
(0.002) |  |  (0.001) |
MK-MMD |  0.786∗∗ |  6.33% |  0.780∗∗ |  5.64%
(0.006) |  |  (0.009) |
JAN |  0.723∗∗ |  15.49% |  0.718∗∗ |  14.70%
(0.014) |  |  (0.023) |
DALN |  0.764∗∗ |  9.45% |  0.763∗∗ |  7.95%
(0.013) |  |  (0.002) |
MCD |  0.667∗∗ |  25.33% |  0.658∗∗ |  25.12%
(0.074) |  |  (0.082) |
SDAT |  0.777∗∗ |  7.52% |  0.772∗∗ |  6.66%
(0.002) |  |  (0.008) |

  * *

Note: Significance levels are denoted by * and ** for 0.05 and 0.01,
respectively.

Table A3: Performance Comparison between Our Method and Benchmark Methods
using Chinese News Dataset (Evaluation Scenario 2).

Method |  Recall |  Improvement by DACA |  F1-score |  Improvement by DACA
---|---|---|---|---
DACA |  0.834 |  |  0.818 |
(0.030) |  |  (0.029) |
MMD |  0.793∗∗ |  5.10% |  0.786∗∗ |  4.12%
(0.024) |  |  (0.024) |
CANMD |  0.781∗∗ |  6.74% |  0.776∗∗ |  5.45%
(0.036) |  |  (0.035) |
DANN |  0.686∗∗ |  21.48% |  0.671∗∗ |  22.02%
(0.024) |  |  (0.040) |
MK-MMD |  0.769∗∗ |  8.47% |  0.758∗∗ |  7.94%
(0.019) |  |  (0.019) |
JAN |  0.722∗∗ |  15.42% |  0.708∗∗ |  15.55%
(0.036) |  |  (0.035) |
DALN |  0.706∗∗ |  18.09% |  0.710∗∗ |  15.28%
(0.040) |  |  (0.040) |
MCD |  0.643∗∗ |  29.74% |  0.642∗∗ |  27.50%
(0.004) |  |  (0.006) |
SDAT |  0.744∗∗ |  12.03% |  0.741∗∗ |  10.44%
(0.059) |  |  (0.057) |

  * *

Note: Significance levels are denoted by * and ** for 0.05 and 0.01,
respectively.

We conducted an additional empirical evaluation using a public dataset of
Chinese news nan2021mdfend2. The dataset contains true and fake news in nine
different domains. We employed news in three of these domains: entertainment,
politics, and health, consistent with the setting of the empirical study with
English news datasets. Not all news in the health domain are related to
infodemic. We specifically chose news articles from this domain that contain
words associated with the COVID pandemic, such as COVID, COVID test, or
pandemic. Table A1 lists the number of true and fake news in each domain. User
comments accompany 53.22% of COVID news, 84.27% of politics news, and 94.64%
of entertainment news.

We followed the same evaluation procedure as described in Section 5.2, except
that Chinese words in the dataset were embedded using the pretrained BERT-wwm
model cui2021pre2. Table A2 compares the performance between our method and
each benchmark method in evaluation scenario 1. As reported, our method
significantly outperforms each benchmark, improving recall by a range of 3.88%
to 25.33% and increasing F1-score by a range of 3.33% to 25.12%. Table A3
presents the evaluation results of each method in evaluation scenario 2.
Again, our method attains significantly better performance than each benchmark
method in both recall and F1-score. The additional empirical analysis further
confirms the effectiveness of our method in leveraging labeled information in
other domains to detect misinformation in the infodemic domain.

informs2014 reference_ec