obeypunctuation=true]Hefei University of Technology, Hefei, China

obeypunctuation=true]Hefei University of Technology, Hefei, China

obeypunctuation=true]Hefei University of Technology, Hefei, China

obeypunctuation=true]Hefei University of Technology, Hefei, China

obeypunctuation=true]University of Science and Technology of China, Hefei,
China

obeypunctuation=true]Ant Group, Hangzhou, China

obeypunctuation=true]Ant Group, Hangzhou, China

obeypunctuation=true]Hefei University of Technology, Hefei, China
obeypunctuation=true]Institute of Artificial Intelligence, Hefei Comprehensive
National Science Center, Hefei, China

# Improving Recommendation Fairness via Data Augmentation

Lei Chen  [<EMAIL_ADDRESS>,  Le Wu  [<EMAIL_ADDRESS>,  Kun
Zhang  [<EMAIL_ADDRESS>,  Richang Hong  [<EMAIL_ADDRESS>,
Defu Lian  [<EMAIL_ADDRESS>,  Zhiqiang Zhang  [
<EMAIL_ADDRESS>,  Jun Zhou  [<EMAIL_ADDRESS>and  Meng Wang
[  [<EMAIL_ADDRESS>
(2023)

###### Abstract.

Collaborative filtering based recommendation learns users’ preferences from
all users’ historical behavior data, and has been popular to facilitate
decision making. Recently, the fairness issue of recommendation has become
more and more essential. A recommender system is considered unfair when it
does not perform equally well for different user groups according to users’
sensitive attributes (e.g., gender, race). Plenty of methods have been
proposed to alleviate unfairness by optimizing a predefined fairness goal or
changing the distribution of unbalanced training data. However, they either
suffered from the specific fairness optimization metrics or relied on
redesigning the current recommendation architecture. In this paper, we study
how to improve recommendation fairness from the data augmentation perspective.
The recommendation model amplifies the inherent unfairness of imbalanced
training data. We augment imbalanced training data towards balanced data
distribution to improve fairness. Given each real original user-item
interaction record, we propose the following hypotheses for augmenting the
training data: each user in one group has a similar item preference (click or
non-click) as the item preference of any user in the remaining group. With
these hypotheses, we generate “fake” interaction behaviors to complement the
original training data. After that, we design a bi-level optimization target,
with the inner optimization generates better fake data to augment training
data with our hypotheses, and the outer one updates the recommendation model
parameters based on the augmented training data. The proposed framework is
generally applicable to any embedding-based recommendation, and does not need
to pre-define a fairness metric. Extensive experiments on two real-world
datasets clearly demonstrate the superiority of our proposed framework. We
publish the source code at https://github.com/newlei/FDA.

user modeling, fairness, data augmentation, fair recommendation

††copyright: acmcopyright††journalyear: 2023††copyright:
acmlicensed††conference: Proceedings of the ACM Web Conference 2023; May 1–5,
2023; Austin, TX, USA††booktitle: Proceedings of the ACM Web Conference 2023
(WWW ’23), May 1–5, 2023, Austin, TX, USA††price: 15.00††doi:
10.1145/3543507.3583341††isbn: 978-1-4503-9416-1/23/04††ccs: Information
systems Collaborative filtering††ccs: Human-centered computing User models

##  1\. Introduction

Recommender systems automatically help users to find items they may like, and
have been widely deployed in our daily life for decision-making (Koren et al.,
2009; Covington et al., 2016; Lei et al., 2020). Given the original user-item
historical behavior data, most recommendation models design sophisticated
techniques to learn user and item embeddings, and try to accurately predict
users’ unknown preferences to items (Mnih and Salakhutdinov, 2008; Lei et al.,
2020; He et al., 2020). Recently, researchers argued that simply optimizing
recommendation accuracy lead to unfairness issues. Researchers have found that
current recommender systems show apparent demographic performance bias of
different demographic groups (Ekstrand et al., 2018a, b). Career recommender
systems tend to favour male candidates compared to females even though they
are equally qualified (Lambrecht and Tucker, 2019). Besides, recommendation
accuracy performance shows significant differences between advantaged user
group and disadvantaged user group (Li et al., 2021; Melchiorre et al., 2021).

Since the unfairness phenomenon has been ubiquitous in recommender systems and
user-centric AI applications, how to define fairness measures and improve
fairness to benefit all users is a trending research topic (Pedreshi et al.,
2008; Hardt et al., 2016a). Among all fairness metrics, group based fairness
is widely accepted and adopted, which argues that the prediction performance
does not discriminate particular user groups based on users’ sensitive
attributes (e.g., gender, race) (Dwork et al., 2012; Geyik et al., 2019).
Given the basic ideas of group fairness, researchers proposed various group
fairness measures and debiasing models to improve fairness, such as fairness
regularization based models (Yao and Huang, 2017; Beutel et al., 2019),
sensitive attribute disentangle models (Zhu et al., 2018), and adversarial
techniques to remove sensitive attributes (Wu et al., 2021). E.g., researchers
proposed different recommendation fairness regularization terms and added
these terms in the loss function for collaborative filtering (Yao and Huang,
2017). In summary, these works alleviate unfairness in the modeling process
with the fixed training data and achieve better fairness results.

Table 1. An illustration of unfairness of recommendation accuracy performance on MovieLens dataset. We divide users into two subgroups based on the gender attribute. For each group, we calculate the distribution of clicked items of the corresponding group based on the training data, as well as the hit items on Top-K recommendation from two widely used recommendation models. Then, we measure the distribution differences of the two groups with JS divergence. We consider two typically recommendation models (BPR (Rendle et al., 2009) and GCCF (Lei et al., 2020)). We consider the corrected hit from Top-K ranking list, denoted as “Top-20-Hit” and “Top-50-Hit”. The recommendation accuracy performance has larger JS divergence compared to the training data, showing recommendation models exacerbate unfairness from training data. BPR (Rendle et al., 2009) | JS divergence | GCCF (Lei et al., 2020) | JS divergence
---|---|---|---
Training data | 0.1303 | Training data | 0.1303
Top-20-Hit | 0.4842 | Top-20-Hit | 0.4879
Top-50-Hit | 0.4349 | Top-50-Hit | 0.4229

In fact, researchers agree that unfairness in machine learning mainly comes
from two processes. Firstly, the collected historical data shows imbalanced
distribution among different user groups or reflects the real-world
discrimination (Burke et al., 2018; Iosifidis et al., 2019). After that,
algorithms that learn the typical patterns amplify imbalance or bias inherited
from training data and hurt the minority group (Rastegarpanah et al., 2019; Wu
et al., 2021). To show whether current recommendation algorithms amplify
unfairness, let us take the MovieLens dataset as an example (detailed data
description is shown in Section 5). We divide users into two subgroups based
on the sensitive attribute _gender_. Given the training data, for each
subgroup, we calculate the distribution over all items based on clicked items
of users in this group. Then, we measure the distribution differences of the
two sub user groups with Jensen-Shannon (JS) divergence (Menéndez et al.,
1997). After employing recommendation algorithms for Top-K recommendation, we
measure the distribution difference of hit items of the two user groups. As
can be seen from Table 1, the training data shows the preference distributions
of the two groups are different. Both of the two recommendation models (i.e.,
BPR (Rendle et al., 2009) and GCCF (Lei et al., 2020)) show larger group
divergences of recommendation accuracy results compared to the divergence
value of the training data. As a result, these two groups receive different
benefits from the recommendation algorithms. Since all recommendation models
rely on the training data for model optimization, comparing with the huge
works on model-level fairness research, how to consider and improve
recommendation fairness from the data perspective is equally important.

When considering fairness from data perspective, some previous works proposed
data resampling or data reweighting techniques to change data distribution
(Rastegarpanah et al., 2019; Iosifidis et al., 2019). As a particular group of
users (e.g., females) are underrepresented in the training data, BN-SLIM is
designed to resample a balanced neighborhood set of males and females for
neighborhood-based recommendation (Burke et al., 2018). Besides, given a
specific fairness measure in recommendation, researchers made attempts to add
virtual users via a unified optimization framework (Rastegarpanah et al.,
2019). These previous works show the possibility of improving fairness from
the data perspective. However, they either suffered from applying to specific
recommendation models or relied on a specific predefined fairness metric,
limiting the generality to transferring to current RS architecture.

In this paper, we study the problem of designing a model-agnostic framework to
improve recommendation fairness from data augmentation perspective. Given the
original training data of user-item implicit feedback, we argue that the
augmented training data are better balanced among different user groups, such
that RS algorithms could better learn preferences of different user groups and
avoid neglecting preferences of the minority groups. As a result, we argue the
augmented data should satisfy the following hypotheses: for any user’s two
kinds of preference (a click record or a non-click record) in one group, there
is another user in the remaining group (users with opposite sensitive
attribute value) that has a similar item preference. With these hypotheses, we
propose a general framework of Fairness-aware Data Augmentation (FDA ) to
generate “fake” data that complement the original training data. We design a
bi-level optimization function with the inner and outer loop to optimize FDA .
The proposed FDA is model-agnostic and can be easily integrated to any
embedding-based recommendation. Besides, FDA does not rely on any predefined
specific fairness metrics. Finally, extensive experiments on two real-world
datasets clearly demonstrate the superiority of our proposed framework.

##  2\. Related Work

Fairness Discovery and Measures. As machine learning technologies have become
a vital part of our daily lives with a high social impact, fairness issues are
concerned and raised (Jobin et al., 2019; Pedreshi et al., 2008). Fairness
refers to not discriminating against individuals or user groups based on
sensitive user attributes (Mehrabi et al., 2021; Gajane and Pechenizkiy,
2017). One increasing requirement is how to define and measure fairness.
Current fairness metrics can be categorized into individual fairness and group
fairness. Individual fairness refers to producing similar predictions for
similar individuals, and group fairness argues not discriminating a particular
user group based on the sensitive attribute (Binns, 2020; Biega et al., 2018;
Zafar et al., 2017; Wu et al., 2021). Among all group fairness metrics,
Demographic Parity ($DP$) and Equality of Opportunity ($EO$) are widely
accepted (Hardt et al., 2016b; Caton and Haas, 2020). DP requires that each
particular demographic group has the same proportion of receiving a particular
outcome (Gajane and Pechenizkiy, 2017; Zemel et al., 2013). According to
specific tasks, researchers have proposed specific demographic parity measures
(Kamishima et al., 2012; Calders et al., 2009; Zemel et al., 2013). A notable
disadvantage of demographic parity lies in ignoring the natural differences
across groups. To this end, EO is proposed for an equal proportion of
receiving a particular outcome conditioned on the real outcome (Hardt et al.,
2016b; Berk et al., 2021; Pleiss et al., 2017). In other words, a model is
fair if the predicted results and sensitive attributes are independently
conditioned on the real outcome (Hardt et al., 2016b).

Fairness aware Models. Building on the mathematical fairness metrics, some
researchers design task-oriented model structures to meet fairness
requirements. Current model based fairness approaches can be classified into
regularization based approaches (Aghaei et al., 2019; Feldman et al., 2015;
Goel et al., 2018; Huang and Vishnoi, 2019), causal based approaches (Singh
and Joachims, 2018; Kilbertus et al., 2017; Kusner et al., 2017), adversarial
learning based approaches (Beutel et al., 2017; Zhang et al., 2018; Wu et al.,
2021), and so on. These methods ensure that the outcome of models can meet
fairness requirements, and the modification of model structures heavily relies
on specific fairness definitions. E.g., regularization-based approaches add
one fairness constraint to achieve a specific fairness at a time. Researchers
need to design various constraints for achieving different fairness
requirements (Feldman et al., 2015; Zehlike et al., 2017; Beutel et al.,
2019).

Different from alleviating unfairness in the modeling process, some
researchers attempted to solve fairness problems from the data perspective.
Early works tried to directly remove the sensitive attribute from the training
data (Kamiran and Calders, 2012; Calders and Verwer, 2010). Some researchers
argue that unfairness comes from the imbalanced training data of the protected
and unprotected groups, and employ bagging and balance groups in each bag to
build stratified training samples (Iosifidis et al., 2019). Besides,
perturbation approaches change the training data distribution with some prior
assumptions of sensitive attributes, input features and labels. After that, a
perturbed distribution of disadvantaged groups is used to mitigate performance
disparities (Wang et al., 2019; Gordaliza et al., 2019; Jiang and Nachum,
2020). Most of these data modification approaches are designed for
classification tasks with abundant data samples, in which each data sample is
independent. In CF based recommender systems, users and items are correlated
with sparse interaction data. Therefore, current approaches of modifying data
distribution in the classification task could not be easily adapted for the
recommendation task with limited observed user behavior data. Recently, the
authors (Rastegarpanah et al., 2019) propose two metrics that capture the
polarization and unfairness in recommendation. The framework needs to take one
of the proposed metrics as the optimization direction, in order to generate
corresponding antidote new user profiles for the selected metric. By proposing
the concept of a balanced neighborhood, the authors (Burke et al., 2018)
borrow the idea of data sampling and design corresponding regularization to
control neighbor distribution of each user. The regularization is applied to
the sparse linear method (SLIM) to improve the outcome fairness (Burke et al.,
2018). The above two models explore the possibility of improving
recommendation fairness by changing the training data distribution. However,
they either need to define/introduce specific fairness metrics or are only
suitable for a particular kind of recommendation model with well-designed
heuristics. Therefore, the problem of how to design a general fairness
framework from the data perspective that is suitable for different
recommendation backbones and multiple fairness metrics is still under
explored.

##  3\. Preliminary

In a recommender system, there are two sets of entities: a user set $U$
($|U|=M$) and an item set $V$ ($|V|=N$), we denote the user-item interaction
matrix as $R=[r_{uv}]_{M\times N}$. We consider the common implicit feedback
scenario. If user $u$ has clicked item $v$, then $r_{uv}=1$ indicates user $u$
likes the item $v$, otherwise $r_{uv}=0$. For each user $u$, her clicked item
set is denoted as $R_{u}=\\{v:r_{uv}=1\\}$.

As most modern recommender systems are built on embedding based architecture,
we focus on embedding based recommendation models. Generally speaking, there
are two key components for recommendation. First, a recommendation model Rec
employs an encoder Enc to project users and items into the embedding space,
formulated as
$\mathbf{E}=Enc(U,V)=[\mathbf{e}_{1},..,\mathbf{e}_{u},...,\mathbf{e}_{v},...,\mathbf{e}_{M+N}]$,
where $\mathbf{e}_{u}$ is user $u$’s embedding, $\mathbf{e}_{v}$ is item $v$’s
embedding. Then, the predicted preference $\hat{r}_{uv}$ can be calculated
with $\hat{r}_{uv}=\mathbf{e}_{u}^{T}\mathbf{e}_{v}$. Learning high-quality
user and item embeddings has become the key of modern recommender systems.
There are two typical classes of embedding approaches: the classical matrix
factorization models (Mnih and Salakhutdinov, 2008; Rendle et al., 2009) and
neural graph-based models (Lei et al., 2020; He et al., 2020).

Given a binary sensitive attribute $a\in\\{0,1\\}$, $a_{u}$ denotes user $u$’s
attribute value. We divide the user set into two subsets: $G_{0}$ and $G_{1}$.
If $a_{u}=0$, then $u\in G_{0}$, otherwise $u\in G_{1}$. Please note that as
we do not focus on any specific recommendation models, we assume the embedding
based recommendation models are available, such as matrix factorization models
(Mnih and Salakhutdinov, 2008) or neural graph-based models (Lei et al.,
2020). Our goal is to improve recommendation fairness with relatively high
accuracy from the data augmentation perspective.

Figure 1. The overall architecture of the proposed FDA framework. By not changing the recommendation model, our key idea is to improve data balance between different user groups via data augmentation. For each real data $(u_{0},i,r)$ of user $u_{0}$ ($u_{0}\in G_{0}$), item $i$, and the implicit feedback value $r$ of this user-item pair from the training data, we hope there is a user in the remaining group $G_{1}$ that shows the same preference value $r$ to a similar item $\bar{i}$. As the detailed implicit feedback value $r$ can be positive or unknown, we turn the above idea into two hypotheses. In the upper part of the figure, according to Hypothesis 1, we generate two kinds of fake positive examples ($\bar{r}_{u_{0}\bar{i}_{1}}$ and $\bar{r}_{u_{1}\bar{i}_{0}}$). Correspondingly, based on the generated fake data, we can obtain fake records ($<u_{0},\bar{i}_{1},j_{0}>,<u_{1},\bar{i}_{0},j_{1}>$). As shown in the lower part of the figure, according to Hypothesis 2, we generate two kinds of fake negative data ($\bar{r}_{u_{0}\bar{j}_{1}}$ and $\bar{r}_{u_{1}\bar{j}_{0}}$), and obtain fake records ($<u_{0},{i}_{0},\bar{j}_{1}>,<u_{1},{i}_{1},\bar{j}_{0}>$).  Table 2. Mathematical Notations Notations | Description
---|---
U, V | userset $|U|=M$, itemset $|V|=N$
$a\in\\{0,1\\}$ | a binary sensitive attribute
$G_{0},G_{1}$ | user group $\begin{cases}u\in G_{0}&\text{ if }a_{u}=0\\\ u\in G_{1}&\text{ if }a_{u}=1\end{cases}$
$u_{0},u_{1}$ | users, $u_{0}\in G_{0}$, $u_{1}\in G_{1}$
$i_{0},i_{1},j_{0},j_{1}$ | real items
$\bar{i}_{0},\bar{i}_{1},\bar{j}_{0},\bar{j}_{1}$ | fake items
$r_{ui}$,$r_{uj}$ | real positive data, real negative data
$\bar{r}_{ui}$,$\bar{r}_{uj}$ | fake positive data, fake negative data

##  4\. The Proposed Framework

In this section, we first introduce two hypotheses in our proposed FDA
framework. Then, we show how to optimize two hypotheses given a recommendation
model.

###  4.1. Hypotheses for Generating Fake Data

We argue that the augmented data should be balanced between two user groups,
such that RS could learn latent preferences of different groups without
neglecting the minority group. Since users have two kinds of behaviors, i.e.,
click (positive behavior) and non-click (negative behavior), we corresponding
propose two hypotheses to augment data towards balanced distribution. In other
words, for each behavior data $(u_{0},i,r)$ of user $u_{0}$ ($u_{0}\in
G_{0}$), item $i$, and the implicit feedback value $r$, we hope there is a
user in the remaining group $G_{1}$ that shows the same preference value $r$to
a similar item. Under this assumption, we can improve data balance of
different user groups by generating fake behavior data that complements the
training data. Specifically, the first hypothesis focuses on the positive
behavior among two groups ($G_{0}$ and $G_{1}$).

Hypothesis 1.  Assume that there is user $u_{0}\in G_{0}$, and item $i_{0}$ is
one of user $u_{0}$’s clicked items $i_{0}\in R_{u_{0}}$. There should exist
user $u_{1}\in G_{1}$ that also clicks a similar item
$\bar{i}_{0}\simeq{i}_{0}$. Similarly, if user $u_{1}\in G_{1}$ has clicked
item $i_{1}\in R_{u_{1}}$, user $u_{0}\in G_{0}$ should also click a similar
item $\bar{i}_{1}\simeq{i}_{1}$. This hypothesis can be formulated as follows:

(1) $\displaystyle\forall u_{0}\in G_{0},u_{1}\in
G_{1},if~{}r_{u_{0}i_{0}}=1,then~{}\bar{r}_{u_{1}\bar{i}_{0}}=1;where~{}i_{0}\simeq\bar{i}_{0}.$
(2) $\displaystyle\forall u_{1}\in G_{1},u_{0}\in
G_{0},if~{}r_{u_{1}i_{1}}=1,then~{}\bar{r}_{u_{0}\bar{i}_{1}}=1;where~{}i_{1}\simeq\bar{i}_{1}.$

In the above equations, $\bar{r}_{u_{1}\bar{i}_{0}}$ and
$\bar{r}_{u_{0}\bar{i}_{1}}$ are fake interactions data that does not appear
in the training data.

The second hypothesis focuses on the negative behavior among two groups
($G_{0}$ and $G_{1}$):

Hypothesis 2.  If user $u_{0}\in G_{0}$ does not click item $j_{0}\in
V-R_{u_{0}}$, there should also exist user $u_{1}\in G_{1}$ that does not
click similar item $\bar{j}_{0}\simeq{j}_{0}$. Correspondingly, if user
$u_{1}\in G_{1}$ does not click item $j_{1}\in R\backslash R_{u_{1}}$, user
$u_{0}\in G_{0}$ should not click similar item $\bar{j}_{1}\simeq{j}_{1}$.
This hypothesis can be formulated as follows:

(3) $\displaystyle\forall u_{0}\in G_{0},u_{1}\in
G_{1},if~{}r_{u_{0}j_{0}}=0,then~{}\bar{r}_{u_{1}\bar{j}_{0}}=0;where~{}j_{0}\simeq\bar{j}_{0}.$
(4) $\displaystyle\forall u_{1}\in G_{1},u_{0}\in
G_{0},if~{}r_{u_{1}j_{1}}=0,then~{}\bar{r}_{u_{0}\bar{j}_{1}}=0;where~{}j_{1}\simeq\bar{j}_{1}.$

In the above equations, $\bar{r}_{u_{1}\bar{j}_{0}}$ and
$\bar{r}_{u_{0}\bar{j}_{1}}$ are fake interactions.

By employing Hypothesis 1& 2, we can adjust the training data to make sure
that the augmented training data are balanced for different groups.

###  4.2. Optimization For Fake Data

After generating fake data, we focus on augmenting the original training data
for implicit feedback based recommendation. In implicit feedback based
recommendation, pairwise learning has been widely used (Rendle et al., 2009;
Lei et al., 2020; He et al., 2020). For each user $u$, if item $i$ is clicked
by user $u$, and item $j$ is not clicked by user $u$, then this clicked item
$i$ is more relevant than a non-clicked item $j$, which can be formulated as
$i>_{u}j$. As a result, the clicked item $i$ should be assigned higher
prediction value compared to the predicted preference of any non-clicked item
$j$:

(5) $\displaystyle\centering\forall i\in R_{u},j\in
V-R_{u}:\quad\quad\@add@centering$ $\displaystyle\hat{r}_{ui}>\hat{r}_{uj}$
(6) $\displaystyle i.e.,\quad\quad$
$\displaystyle\mathbf{e}_{u}^{T}\mathbf{e}_{i}>\mathbf{e}_{u}^{T}\mathbf{e}_{j}.$

Based on users’ two sensitive attribute values and two kinds of implicit
feedback behaviors, we can obtain four types of interactions (i.e., positive
interactions for users from two groups: $r_{u_{0}i_{0}}$ and $r_{u_{1}i_{1}}$,
negative interactions for users from two groups: $r_{u_{0}j_{0}}$ and
$r_{u_{1}j_{1}}$.) to support Hypothesis 1 and Hypothesis 2. The original
training data contains two types of interactions as:

(7) $\displaystyle D_{real}=\\{<u_{0},i_{0},j_{0}>,<u_{1},i_{1},j_{1}>\\},$

where user $u_{0}$ belongs to user group $G_{0}$. Item $i_{0}$ is a positive
feedback of user $u_{0}$, and item $j_{0}$ is a negative feedback of user
$u_{0}$. Similarly, for user $u_{1}\in G_{1}$, item $i_{1}$ is a positive
feedback and item $j_{1}$ is a negative feedback. Next, we introduce how to
construct the remaining two types of fake interactions and optimize them.

Optimization of Hypothesis 1. Hypothesis 1 focuses on clicked items and
encourages the positive behavior distribution of two groups are balanced. For
each $D_{real}$, we can generate the corresponding fake data $D_{fake1}$
according to Hypothesis 1 as:

(8) $\displaystyle D_{real}=\\{<u_{0},i_{0},j_{0}>,<u_{1},i_{1},j_{1}>\\},$
(9) $\displaystyle
D_{fake1}=\\{<u_{1},\bar{i}_{0},j_{1}>,<u_{0},\bar{i}_{1},j_{0}>\\},$

where $i_{0}\simeq\bar{i}_{0}$ and $i_{1}\simeq\bar{i}_{1}$. $D_{fake1}$
denotes fake positive interaction data. Note that, given any two real positive
behavior data (${r}_{u_{0}{i}_{0}}$ and $\bar{r}_{u_{1}{i}_{1}}$), $D_{fake1}$
also contains two corresponding fake positive behavior data
($\bar{r}_{u_{1}\bar{i}_{0}}$ and $\bar{r}_{u_{0}\bar{i}_{1}}$). Therefore, we
have the following expressions on $D_{fake1}$:

(10) $\displaystyle\bar{i}_{0}>_{u_{1}}j_{1}~{}$ $\displaystyle
and~{}~{}\bar{i}_{1}>_{u_{0}}j_{0}.$

Similar to Eq.(5), we can formulate Eq.(10) with the following optimization
goal:

(11) $\displaystyle\hat{r}_{u_{1}\bar{i}_{0}}>\hat{r}_{u_{1}j_{1}}~{}$
$\displaystyle and~{}\hat{r}_{u_{0}\bar{i}_{1}}>\hat{r}_{u_{0}j_{0}},$

which can also be calculated as follows:

(12)
$\displaystyle\mathbf{e}_{u_{1}}^{T}\bar{\mathbf{e}}_{i_{0}}>\mathbf{e}_{u_{1}}^{T}\mathbf{e}_{j_{1}}~{}and~{}\mathbf{e}_{u_{0}}^{T}\bar{\mathbf{e}}_{i_{1}}>\mathbf{e}_{u_{0}}^{T}{\mathbf{e}}_{j_{0}}.$

Optimization of Hypothesis 2. Hypothesis 2 focuses on non-clicked items and
encourages the non-click behavior of two user groups to be balanced. For each
triplet from $D_{real}$, we can generate the corresponding fake behavior data
$D_{fake2}$ according to Hypothesis 2:

(13) $\displaystyle D_{real}=\\{<u_{0},i_{0},j_{0}>,<u_{1},i_{1},j_{1}>\\},$
(14) $\displaystyle
D_{fake2}=\\{<u_{1},{i}_{1},\bar{j}_{0}>,<u_{0},{i}_{0},\bar{j}_{1}>\\},$

where $j_{0}\simeq\bar{j}_{0}$ and $j_{1}\simeq\bar{j}_{1}$. Therefore, we
have the following goal on $D_{fake2}$:

(15) $\displaystyle{i}_{1}>_{u1}\bar{j}_{0}~{}$ $\displaystyle
and~{}{i}_{0}>_{u0}\bar{j}_{1},$

Similarly, we can turn the above goal into optimization functions as:

(16) $\displaystyle\hat{r}_{u_{1}{i}_{1}}>\hat{r}_{u_{1}\bar{j}_{0}}~{}$
$\displaystyle and~{}\hat{r}_{u_{0}{i}_{0}}>\hat{r}_{u_{0}\bar{j}_{1}},$

which can be calculated as follows:

(17)
$\displaystyle\mathbf{e}_{u_{1}}^{T}{\mathbf{e}}_{i_{1}}>\mathbf{e}_{u_{1}}^{T}\bar{\mathbf{e}}_{j_{0}}~{}and~{}\mathbf{e}_{u_{0}}^{T}{\mathbf{e}}_{i_{0}}>\mathbf{e}_{u_{0}}^{T}\bar{\mathbf{e}}_{j_{1}}.$

We construct corresponding fake data for each hypothesis, then we integrate
all fake data from $D_{fake1}$ and $D_{fake2}$:

(18) $\displaystyle D_{fake}$ $\displaystyle=\\{D_{fake1},D_{fake2}\\}$
$\displaystyle=\\{<u_{1},\bar{i}_{0},j_{1}>,<u_{0},\bar{i}_{1},j_{0}>,<u_{1},{i}_{1},\bar{j}_{0}>,<u_{0},{i}_{0},\bar{j}_{1}>\\}.$

With the implicit feedback, Bayesian Personalized Ranking (BPR) is widely used
for learning the pairwise based optimization function (Mnih and Salakhutdinov,
2008; Rendle et al., 2009). We also adopt BPR loss to optimize the fake data
generation process:

(19) $\displaystyle\min\mathcal{L}_{fake}=-$ $\displaystyle\sum_{<u,i,j>\in
D_{fake}}ln(\sigma(\hat{r}_{ui}-\hat{r}_{uj}))$ $\displaystyle=$
$\displaystyle-\sum_{<u_{1},\bar{i}_{0},j_{1}>\in
D_{fake1}}ln(\sigma(\hat{r}_{u_{1}\bar{i}_{0}}-\hat{r}_{u_{1}j_{1}}))$
$\displaystyle-\sum_{<u_{0},\bar{i}_{1},j_{0}>\in
D_{fake1}}ln(\sigma(\hat{r}_{u_{0}\bar{i}_{1}}-\hat{r}_{u_{0}j_{0}}))$
$\displaystyle-\sum_{<u_{1},{i}_{1},\bar{j}_{0}>\in
D_{fake2}}ln(\sigma(\hat{r}_{u_{1}{i}_{1}}-\hat{r}_{u_{1}\bar{j}_{0}}))$
$\displaystyle-\sum_{<u_{0},{i}_{0},\bar{j}_{1}>\in
D_{fake2}}ln(\sigma(\hat{r}_{u_{0}{i}_{0}}-\hat{r}_{u_{0}\bar{j}_{1}})).$

The key challenge in Eq.(19) lies in estimating the fake data $D_{fake}$. A
direct approach is to define a similarity function among items, and then find
similar items within a predefined threshold. However, with sparse user-item
click behavior data, directly computing item similarities from user-item
interaction matrix is not only time-consuming, but also not accurate.

We propose to find similar items ($\bar{i}$ and $\bar{j}$) from continuous
embedding space. An intuitive idea is to utilize the well-trained embeddings
from the recommendation model. Note that, recommender systems transform
user/item ID to continuous embedding space: $\mathbf{E}=Enc(U,V)$. We
therefore define and find similar items ($\bar{i}$ and $\bar{j}$) based on
continuous embedding space $\mathbf{E}$. In order to satisfy the similarity
requirement, first of all, the similar items need to lie within the original
embedding distribution. Otherwise, it will seriously affect the recommendation
accuracy. Inspired by adversarial examples and poisoning attacks (Biggio et
al., 2012), we employ a non-random perturbation $\delta$ to generate similar
items in continuous embedding $\mathbf{E}$.

For each item $v$, we add small random noise $\delta_{v}$ to the item original
embedding $\mathbf{e}_{v}$, then we can construct the corresponding similar
item embedding $\bar{\mathbf{e}}_{v}$. The similar item can be formulated as
follows:

(20) $\displaystyle\bar{\mathbf{e}}_{v}=\mathbf{e}_{v}+\delta_{v},\quad\quad
where\left\|\delta_{v}\right\|\leq\epsilon$

The noise $\delta_{v}$ is bounded in a small range $\epsilon$, and the
operator $\leq$ enforces the constraint for each dimension of $\delta_{v}$.
Since $\delta_{v}$ is a small “unseen” noise, it is natural that
$\bar{\mathbf{e}}_{v}$ is similar to $\mathbf{e}_{v}$ and
$\bar{\mathbf{e}}_{v}$ also lies within the original embedding distribution.
This method can meet the requirements of Hypothesis 1 and Hypothesis 2. By
combining the similar item requirement in Eq.(20) and the optimization of the
fake data in Eq.(19), the loss function on fake data can be changed into the
embedding form as:

$\displaystyle\min_{\Theta}\mathcal{L}_{fake}=$
$\displaystyle-\sum_{D_{fake}}ln(\sigma(\mathbf{e}_{u_{0}}^{T}(\mathbf{e}_{i_{1}}+\delta_{i_{1}})-\mathbf{e}_{u_{0}}^{T}\mathbf{e}_{j_{0}}))$
$\displaystyle-\sum_{D_{fake}}ln(\sigma(\mathbf{e}_{u_{1}}^{T}(\mathbf{e}_{i_{0}}+\delta_{i_{0}})-\mathbf{e}_{u_{1}}^{T}\mathbf{e}_{j_{1}}))$
$\displaystyle-\sum_{D_{fake}}ln(\sigma(\mathbf{e}_{u_{0}}^{T}\mathbf{e}_{i_{0}}-\mathbf{e}_{u_{0}}^{T}(\mathbf{e}_{j_{1}}+\delta_{j_{1}})))$
(21)
$\displaystyle-\sum_{D_{fake}}ln(\sigma(\mathbf{e}_{u_{1}}^{T}\mathbf{e}_{i_{1}}-\mathbf{e}_{u_{1}}^{T}(\mathbf{e}_{j_{0}}+\delta_{j_{0}}))),$

where
$\delta_{i_{0}},\delta_{i_{1}},\delta_{j_{0}},\delta_{j_{1}}$respectively
denote the small noises adding to corresponding item embeddings, and $\Theta$
denotes the combination of all small noises in the fake data. With fixed
embeddings $\mathbf{E}$ from any recommender model, we optimize the $\Theta$
to generate fake data.

###  4.3. The Overall Bi-Level Optimization

After generating fake training data, we intend to integrate these fake data
with original training data as augmented data. We could not use all the fake
data for data augmentation as too many fake data records would dramatically
modify the original data distribution, leading to decrease accuracy. To better
trade-off, we develop a random mask operation to inject fake data. The mask
operation can be formulated as follows:

(24) $\displaystyle{m}_{v}$
$\displaystyle=\left\\{\begin{array}[]{ll}1,&selected\\\ 0,&\text{ else
}\end{array}\right.\text{and}\sum_{v=1}^{N}{m}_{v}\leq Max_{mask},$

where $Max_{mask}$ denotes the maximum number of selected fake data in each
update. Obviously, $Max_{mask}$ should be less than the number of items $N$.
Since only part of the fake data is selected for training, the impact on
recommendation accuracy can be controlled.

After fake data records are selected by the mask operation, we combine the
selected fake data $D_{fake}$ and original training data $D_{real}$ to
construct the augmented data $D_{aug}$. Next, we retrain the recommendation
model on the augmented data. This training process of a recommender model is
the same as the recommender model except that the data input is augmented. In
short, the overall training process of FDA involves two iterative steps:
generating fake data based on the previous recommendation embeddings and
training recommendation models given updated fake data. The two iterative
steps can be combined by solving the following bi-level optimization problem:

$\displaystyle\min_{\mathbf{E}}\min_{\Theta}\mathcal{L}=\sum_{D_{aug}}$
$\displaystyle-
ln(\sigma([m_{i_{0}}*\bar{r}_{u_{1}\bar{i}_{0}}+(1-m_{i_{0}})*\hat{r}_{u_{1}i_{1}}]-\hat{r}_{u_{1}j_{1}}))$
$\displaystyle-
ln(\sigma([m_{i_{1}}*\bar{r}_{u_{0}\bar{i}_{1}}+(1-m_{i_{1}})*\hat{r}_{u_{0}{i}_{0}}]-\hat{r}_{u_{0}j_{0}}))$
$\displaystyle-
ln(\sigma(\hat{r}_{u_{1}{i}_{1}}-[m_{j_{0}}*\bar{r}_{u_{1}\bar{j}_{0}}+(1-m_{j_{0}})*\hat{r}_{u_{1}{j}_{1}}]))$
(25) $\displaystyle-
ln(\sigma(\hat{r}_{u_{0}{i}_{0}}-[m_{j_{1}}*\bar{r}_{u_{0}\bar{j}_{1}}+(1-m_{j_{1}})*\hat{r}_{u_{0}{j}_{0}}])),$

where $D_{aug}=D_{real}\cup D_{fake}$ is the augmented data. From this
formulation, we can observe that our proposed FDA involves two levels of
optimization.

###  4.4. Discussion

Model Analysis. Given the above bi-level optimization process of FDA , we now
analyze why FDA can achieve a better balance between recommendation accuracy
and fairness without changing the recommendation architecture (i.e., the outer
loop that updates recommendation parameters). In the inner minimization loop,
our key idea is to encourage for each triple behavior of the user, there is a
user in the remaining group that shows the same preference value to a similar
item. As users have two kinds of preferences (i.e., click and non-click), the
key idea turns to two hypotheses for generating fake data. Therefore, the
augmented data that contains both the training data and the fake data are more
balanced compared to the original training data. For the original training
data, the recommendation results are unfairer as recommendation results would
amplify unbalance inherited from the input data. With more balanced data, the
recommender models can output better fairness metrics even though the
recommendation part does not model fairness. Besides, as we constrain the fake
data generated by adding a small random noise, and the fake data is controlled
by a limited ratio with random mask operation, the recommendation accuracy can
also be guaranteed. Compared to other data balance or data augmentation based
fairness enhanced recommendation models, FDA shows the advantage of generally
applicable to embedding based recommendation backbones and does not need to
predefine a specific group fairness metric.

Extension to Multiple Sensitive Values. In FDA , similar as many fairness
aware approaches, we start with the binary sensitive attribute values (Burke
et al., 2018; Yao and Huang, 2017; Hardt et al., 2016b). When dealing with a
sensitive attribute with K ($K>2$) values, a naive extension to multiple
sensitive attribute values is to encourage that: for each user’s one kind of
behavior to an item in one group, we encourage that there are other users in
each remaining group that show the same behavior to a similar item. Therefore,
for each hypothesis, we can generate $K-1$ fake data, and use the bi-level
optimization with both original and the selected fake data with mask
operation.

##  5\. Experiments

###  5.1. Experimental Setup

Datasets. We conduct experiments on two publicly available datasets: MovieLens
(Harper and Konstan, 2015) and LastFM (Celma Herrada et al., 2009; Wu et al.,
2021). For MovieLens 111https://grouplens.org/datasets/movielens/, we adopt
the same data splitting strategy as previous works for fair recommendation (Wu
et al., 2021; Bose and Hamilton, 2019). We treat the items that a user’s
rating is larger than 3 as positive feedback. Moreover, we randomly select
$80\%$ of records for training and the remaining $20\%$ records for the test.
LastFM is a large music recommendation dataset
222http://ocelma.net/MusicRecommendationDataset/lastfm-360K.html. We treat the
items that a user plays as the positive feedback. To ensure the quality of the
dataset, we use the 10-core setting to ensure that users (items) have at least
10 interaction records. We split the historical records into training,
validation, and test parts with the ratio of 7:1:2. Besides the user-item
interaction records, these two datasets also have the user profile data,
including gender (two classes) for users. Similar as previous works, we treat
gender as the sensitive attribute and divide users into two subgroups. The
statistics of these two datasets are shown in Table 3.

Table 3. Statistics of the two datasets.

Datasets | Users | Items | Traning Records | Density
---|---|---|---|---
MovieLens | 6,040 | 3,952 | 513,112 | 2.150%
LastFM | 139,371 | 60,081 | 4,017,311 | 0.048%

Table 4. Recommendation accuracy and fairness performance on MovieLens with
varying Top-K values. We compare all fairness-aware models, the best results
are presented in bold font and the second best results are presented in
underline. The performance improvement of our model against the best baseline
is significant under the paired-t test.

Model | K=10 | K=20 | K=30 | K=40 | K=50
---|---|---|---|---|---
HR$\uparrow$ | NDCG$\uparrow$ | DP$\downarrow$ | EO$\downarrow$ | HR$\uparrow$ | NDCG$\uparrow$ | DP$\downarrow$ | EO$\downarrow$ | HR$\uparrow$ | NDCG$\uparrow$ | DP$\downarrow$ | EO$\downarrow$ | HR$\uparrow$ | NDCG$\uparrow$ | DP$\downarrow$ | EO$\downarrow$ | HR$\uparrow$ | NDCG$\uparrow$ | DP$\downarrow$ | EO$\downarrow$
BPR | 0.2478 | 0.2492 | 0.6541 | 0.7158 | 0.2770 | 0.2498 | 0.6198 | 0.6868 | 0.3147 | 0.2600 | 0.6088 | 0.6803 | 0.3519 | 0.2720 | 0.5960 | 0.6746 | 0.3849 | 0.2830 | 0.5861 | 0.6575
GCCF | 0.2607 | 0.2602 | 0.6391 | 0.7025 | 0.2913 | 0.2617 | 0.6182 | 0.6869 | 0.3301 | 0.2722 | 0.6011 | 0.6842 | 0.3708 | 0.2849 | 0.5856 | 0.6611 | 0.4059 | 0.2967 | 0.5740 | 0.6462
BPR_DP | 0.2209 | 0.2241 | 0.6524 | 0.7039 | 0.2446 | 0.2229 | 0.6063 | 0.6802 | 0.2798 | 0.2324 | 0.6132 | 0.6662 | 0.3145 | 0.2433 | 0.5939 | 0.6672 | 0.3427 | 0.2530 | 0.5921 | 0.6565
BPR_EO | 0.2259 | 0.2225 | 0.6510 | 0.7053 | 0.2507 | 0.2279 | 0.6194 | 0.6712 | 0.2862 | 0.2372 | 0.6162 | 0.6740 | 0.3204 | 0.2480 | 0.5976 | 0.6631 | 0.3529 | 0.2589 | 0.5950 | 0.6608
GCCF_DP | 0.2416 | 0.2420 | 0.6532 | 0.7155 | 0.2691 | 0.2434 | 0.6164 | 0.6865 | 0.3063 | 0.2535 | 0.6053 | 0.6767 | 0.3426 | 0.2650 | 0.5951 | 0.6665 | 0.3761 | 0.2763 | 0.5869 | 0.6607
GCCF_EO | 0.2407 | _0.2428_ | 0.6479 | 0.7060 | 0.2698 | _0.2437_ | 0.6211 | 0.6784 | 0.3068 | 0.2537 | 0.6066 | 0.6773 | 0.3428 | 0.2652 | 0.5980 | 0.6713 | 0.3748 | 0.2759 | 0.5844 | 0.6579
FairUser | 0.2306 | 0.2262 | 0.6502 | 0.7375 | 0.2656 | 0.2318 | 0.6167 | 0.7162 | 0.3088 | 0.2448 | 0.5986 | 0.6927 | 0.3494 | 0.2582 | 0.5864 | 0.6826 | 0.3847 | 0.2705 | 0.5761 | 0.6680
BN-SLIM | 0.2305 | 0.2287 | 0.6502 | 0.7349 | 0.2671 | 0.2334 | 0.6078 | 0.7061 | 0.3096 | 0.2457 | 0.5906 | 0.6958 | 0.3500 | 0.2589 | 0.5743 | 0.6816 | 0.3868 | 0.2716 | 0.5661 | 0.6747
FDA_BPR | 0.2307 | 0.2226 | 0.6132 | 0.6969 | 0.2716 | 0.2321 | 0.5730 | 0.6562 | 0.3157 | 0.2460 | 0.5635 | 0.6488 | 0.3566 | 0.2598 | 0.5537 | 0.6358 | 0.3941 | 0.2729 | 0.5450 | 0.6224
FDA_NCF | _0.2401_ | 0.2322 | _0.6093_ | _0.6946_ | _0.2800_ | 0.2367 | _0.5763_ | 0.6644 | _0.3208_ | _0.2544_ | _0.5681_ | 0.6464 | _0.3601_ | _0.2711_ | _0.5554_ | _0.6306_ | _0.3972_ | _0.2824_ | _0.5445_ | _0.6205_
FDA_GCCF | 0.2476 | 0.2430 | 0.6036 | 0.6773 | 0.2857 | 0.2487 | 0.5786 | _0.6593_ | 0.3290 | 0.2614 | 0.5682 | 0.6462 | 0.3709 | 0.2748 | 0.5510 | 0.6231 | 0.4075 | 0.2873 | 0.5438 | 0.6130

Table 5. Recommendation accuracy and fairness performance on LastFM with
varying Top-K values. The performance improvement of our model against the
best baseline is significant under the paired-t test.

Model | K=10 | K=20 | K=30 | K=40 | K=50
---|---|---|---|---|---
HR$\uparrow$ | NDCG$\uparrow$ | DP$\downarrow$ | EO$\downarrow$ | HR$\uparrow$ | NDCG$\uparrow$ | DP$\downarrow$ | EO$\downarrow$ | HR$\uparrow$ | NDCG$\uparrow$ | DP$\downarrow$ | EO$\downarrow$ | HR$\uparrow$ | NDCG$\uparrow$ | DP$\downarrow$ | EO$\downarrow$ | HR$\uparrow$ | NDCG$\uparrow$ | DP$\downarrow$ | EO$\downarrow$
BPR | 0.1323 | 0.1291 | 0.6372 | 0.6636 | 0.1991 | 0.1602 | 0.6235 | 0.6450 | 0.2480 | 0.1794 | 0.6178 | 0.6410 | 0.2869 | 0.1933 | 0.6126 | 0.6361 | 0.3195 | 0.2042 | 0.6109 | 0.6348
GCCF | 0.1361 | 0.1324 | 0.6321 | 0.6580 | 0.2038 | 0.1640 | 0.6220 | 0.6477 | 0.2533 | 0.1834 | 0.6143 | 0.6419 | 0.2929 | 0.1976 | 0.6105 | 0.6377 | 0.3259 | 0.2087 | 0.6065 | 0.6307
BPR_DP | 0.1272 | 0.1231 | 0.6064 | 0.6451 | 0.1927 | 0.1535 | 0.5956 | 0.6318 | 0.2407 | 0.1724 | 0.5899 | 0.6251 | 0.2789 | 0.1861 | 0.5839 | 0.6215 | 0.3108 | 0.1968 | 0.5821 | 0.6192
BPR_EO | 0.1292 | 0.1247 | 0.5933 | 0.6440 | 0.1958 | 0.1557 | 0.5738 | 0.6271 | 0.2444 | 0.1748 | 0.5637 | 0.6218 | 0.2830 | 0.1886 | 0.5577 | 0.6193 | 0.3155 | 0.1995 | 0.5541 | 0.6150
GCCF_DP | 0.1281 | 0.1234 | 0.6285 | 0.6636 | 0.1950 | 0.1544 | 0.6108 | 0.6473 | 0.2442 | 0.1738 | 0.6046 | 0.6400 | 0.2832 | 0.1877 | 0.5986 | 0.6355 | 0.3163 | 0.1988 | 0.5977 | 0.6377
GCCF_EO | 0.1295 | 0.1245 | 0.5912 | 0.6522 | 0.1969 | 0.1558 | 0.5681 | 0.6308 | 0.2461 | 0.1752 | 0.5530 | 0.6160 | 0.2855 | 0.1892 | 0.5471 | 0.6135 | 0.3176 | 0.2003 | _0.5427_ | 0.6125
BN-SLIM | 0.0986 | 0.0943 | 0.6183 | 0.6630 | 0.1546 | 0.1204 | 0.6017 | 0.6557 | 0.1970 | 0.1371 | 0.5962 | 0.6517 | 0.2318 | 0.1496 | 0.5911 | 0.6467 | 0.2620 | 0.1597 | 0.5847 | 0.6411
FDA_BPR | _0.1301_ | _0.1268_ | 0.5604 | 0.5937 | 0.1965 | _0.1577_ | 0.5535 | 0.5825 | 0.2455 | _0.1770_ | 0.5524 | 0.5788 | 0.2848 | 0.1911 | 0.5479 | 0.5761 | 0.3180 | 0.2022 | 0.5457 | 0.5707
FDA_NCF | 0.1300 | 0.1248 | _0.5599_ | _0.5931_ | _0.1970_ | 0.1575 | _0.5505_ | _0.5801_ | _0.2464_ | _0.1770_ | _0.5470_ | _0.5748_ | _0.2856_ | _0.1913_ | _0.5449_ | _0.5720_ | _0.3194_ | _0.2025_ | 0.5432 | _0.5700_
FDA_GCCF | 0.1304 | 0.1268 | 0.5477 | 0.5789 | 0.1976 | 0.1580 | 0.5450 | 0.5693 | 0.2470 | 0.1774 | 0.5412 | 0.5688 | 0.2868 | 0.1917 | 0.5417 | 0.5676 | 0.3200 | 0.2029 | 0.5403 | 0.5641

Evaluation Metrics. Since we focus on the trade-off between fairness and
recommendation accuracy, we need to evaluate two aspects and report the trade-
off results. On the one hand, we employ two widely used ranking metrics for
recommendation accuracy: HR (Gunawardana and Shani, 2009) and NDCG (Järvelin
and Kekäläinen, 2017) to evaluate the Top-K recommendation. Larger values of
HR and NDCG mean better recommendation accuracy performance. On the other
hand, we adopt two group fairness metrics: _Demographic Parity (DP)_ (Zemel et
al., 2013) and _Equality of Opportunity (EO)_ (Hardt et al., 2016b) to
evaluate fairness.  _DP_ and _EO_ evaluate group fairness from different
aspects. For both fairness metrics, the smaller values mean better fairness
results.

The following equation calculates _DP_ measure:

$\displaystyle DP=1/N\sum_{v\in V}\frac{|\sum_{u\in G_{0}}\mathds{1}_{v\in
TopK_{u}}-\sum_{u\in G_{1}}\mathds{1}_{v\in TopK_{u}}|}{\sum_{u\in
G_{0}}\mathds{1}_{v\in TopK_{u}}+\sum_{u\in G_{1}}\mathds{1}_{v\in
TopK_{u}}},$

where $G_{0}$ denotes user group with sensitive attribute $a_{u}=0$, and
$G_{1}$ denotes user group with sensitive attribute $a_{u}=1$. $TopK_{u}$ is
Top-K ranked items for user $u$.

Please note that as _DP_ forcefully requires similarly predicted results
across different groups, it naturally neglects the natural preference
differences of user clicked patterns (Yao and Huang, 2017). EO is proposed to
solve the limitation of _DP_ (Hardt et al., 2016a; Caton and Haas, 2020).
Specifically, it requires similar predicted results across different groups
conditioned on user real preferences. We calculate EO as follows:

$\displaystyle EO=1/N\sum_{v\in V}\frac{|\sum_{u\in G_{0}}\mathds{1}_{v\in
TopK_{u}\&v\in Test_{u}}-\sum_{u\in G_{1}}\mathds{1}_{v\in TopK_{u}\&v\in
Test_{u}}|}{\sum_{u\in G_{0}}\mathds{1}_{v\in TopK_{u}\&v\in
Test_{u}}+\sum_{u\in G_{1}}\mathds{1}_{v\in TopK_{u}\&v\in Test_{u}}},$

where $Test_{u}$ is items that user $u$ clicks on the test data. Because not
all predicted results have corresponding ground true labels, we calculate _EO_
metric only on the testing data. A smaller EO means there is less unfairness,
as the two groups receive similar recommendation accuracy.

Baselines. The baseline models can be divided into two categories:
recommendation based models _BPR_ (Rendle et al., 2009) and  _GCCF_ (Lei et
al., 2020), and the fairness-oriented models _BN-SLIM_ (Burke et al., 2018),
data augmentation model (_FairUser_) (Rastegarpanah et al., 2019) and the
fairness regularization based model (Yao and Huang, 2017). _FairUser_ adds
virtual users to achieve fairness. As its optimization process is very time-
consuming and needs to compute the full user-item matrix at each iteration, we
only test FairUser on MovieLens dataset. Similar as the fairness
regularization based model (Yao and Huang, 2017), we add different group
fairness metrics (_DP_ and _EO_) as regularization terms into recommendation
based models. E.g., BPR__EO_ denotes treating _EO_ as the regularization term
for the base recommendation model of BPR.

Our proposed framework _FDA_ can be applied on different recommendation
backbones. We select BPR (Rendle et al., 2009), NCF (He et al., 2017) and GCCF
(Lei et al., 2020) as recommendation backbones show state-of-the-art
performance. We use FDA_BPR, FDA_NCF and FDA_GCCF to denote the variants of
our proposed framework with different recommendation backbones.

Parameter Setting. Our implementations are based on Pytorch-GPU 1.6.0. The
embedding size is set as 64 for FDA. All the parameters are differentiable in
the objective function, and we use the Adam optimizer to optimize the model.
In Eq.(4.3), the initial learning rate of Adam is 0.001 for the outer
minimization and the inner minimization.

Table 6. Ablation study of the two modules of FDA: Hypothesis 1 and Hypothesis 2 . Model | Hypothesis 1 | Hypothesis 2 | MovieLens | LastFM
---|---|---|---|---
HR@20 | NDCG@20 | DP@20 | EO@20 | HR@20 | NDCG@20 | DP@20 | EO@20
FDA_BPR | ✓ | ✕ | 0.2706 | 0.2462 | 0.6099 | _0.6564_ | 0.1971 | 0.1613 | 0.6017 | 0.6202
FDA_BPR | ✕ | ✓ | 0.2721 | _0.2360_ | _0.5841_ | 0.6682 | 0.1951 | 0.1552 | _0.5585_ | _0.6016_
FDA_BPR | ✓ | ✓ | _0.2716_ | 0.2321 | 0.5731 | 0.6562 | _0.1965_ | _0.1573_ | 0.5392 | 0.5656
FDA_GCCF | ✓ | ✕ | 0.2650 | 0.2397 | 0.6078 | _0.6610_ | 0.1850 | _0.1511_ | 0.5905 | 0.6021
FDA_GCCF | ✕ | ✓ | 0.2876 | _0.2463_ | _0.5824_ | 0.6696 | _0.1906_ | 0.1499 | _0.5647_ | _0.5991_
FDA_GCCF | ✓ | ✓ | _0.2857_ | 0.2487 | 0.5786 | 0.6593 | 0.1976 | 0.1580 | 0.5450 | 0.5693

###  5.2. Overall Performance on Two Datasets

Table 4 and 5 report the overall results on two datasets. We have several
observations from these two tables. FDA achieves the best performance from the
two aspects: 1) improving  _EO_ and  _DP_ concurrently; 2) the trade-off
between recommender accuracy and fairness.

First, when comparing each fairness metric, our proposed FDA outperforms other
models on both DP and EO group fairness metrics. FairUser and BN-SLIM show
worse performance than BPR (not considering fairness) on EO metric. BPR_EO and
GCCF_EO can improve EO and DP concurrently. But, BPR_EO and GCCF_EO is worse
than FDA. DP and EO reflect the group fairness from two aspects. FDA achieves
good results of multiple group metrics on all datasets. Thus, FDA can well
alleviate the unfairness problem and achieve better fairness performance.
Second, when comparing the trade-off between recommender accuracy and
fairness, all fairness-aware baselines perform worse on recommendation
accuracy performance than FDA. In other words, all fairness-aware baselines
cause a larger decrease in recommender accuracy. Therefore, while achieving
better fairness performance, FDA has the least damage to accuracy on different
datasets. We conclude that FDA can reach the best balance between accuracy and
fairness. Third, no matter the base backbone model is BPR, NCF or GCCF, FDA
can improve fairness and show better recommendation performance. FDA_BPR shows
worse performance than FDA_GCCF. This is due to the fact that the base model
(i.e., BPR) in FDA_BPR does not perform as well as the base graph embedding
model GCCF. On the whole, for different recommendation backbone models and
different experimental settings, FDA can effectively balance accuracy and
fairness. This demonstrates the flexibility and effectiveness of FDA. Fourth,
we observe the improvements of fairness on MovieLens are not so obvious as
results on LastFM. In other words, eliminating unfairness on LastFM is easier
than that on MovieLens. We guess a possible reason is that the sparsity and
number of users are different on these two datasets.

###  5.3. Model Analyses

Effects of fake data numbers. The setting of ${Max}_{mask}$ plays an important
role to control the maximum number of fake data. We conduct experiments on
different ${Max}_{mask}$, as shown in Figure 2. Specifically, we show
different ratios of maximum fake data and the number of all items
($Max_{mask}/N$). Since the fake data is similar to the real data, the fake
data does not seriously affect the recommendation accuracy. When the ratio
increases from 0.1 to 0.6, the recommended performance of FDA_BPR and FDA_GCCF
does not decrease nearly. Among all ratios, we can find that FDA achieves a
better balance on the fairness and recommender accuracy when the ratio equals
0.3 for FDA_BPR and 0.4 for FDA_GCCF on LastFM. The trend is similar on
MovieLens. Due to the page length limit, the results on MovieLens are not
reported.

Figure 2. Performance under different ratios on LastFM.

Measuring the distribution differences of recommendation results.  As shown in
Table 1, we employ the Jensen-Shannon (JS) divergence to measure differences
of two user groups (i.e., $G_{0}$ and $G_{1}$) on the larger LastFM. We pay
attention to Top-20 and Top-50 ranked items, denoted as “Top-20” and “Top-50”.
We also consider the corrected Top-20 and Top-50 ranked items, denoted as
“Top-20-Hit” and “Top-50-Hit” to measure the recommendation accuracy
differences between the two groups. As shown in Figure 3, FDA can also improve
fairness performance compared to its original recommendation backbone. Also,
we observe that JS divergence of “Top-20” and “Top-50” is smaller than that of
“Top-20-Hit” and “Top-50-Hit”. This is reasonable as “Top-K” divergence
measures ranking differences between the two groups and does not take
recommendation accuracy into consideration. In contrast, “Top-K-Hit” measures
the recommendation accuracy differences between the two groups. Nevertheless,
FDA can improve these two metrics.

Figure 3. The distribution differences of recommendation results with two
recommendation backbones on LastFM.

Ablation study. In this part, we investigate the effectiveness of each
hypothesis: Hypothesis 1 and Hypothesis 2 of our proposed FDA framework. The
results are illustrated in Table 6. From this table, we can obtain the
following observations. First, each single hypothesis (Hypothesis 1 or
Hypothesis 2) can help the model achieve comparable performance, indicating
the usefulness of our proposed hypothesis. Second, compared with the
performance of models with a single hypothesis, models with both of them (the
entire FDA framework) have better performance, demonstrating the necessity of
both hypotheses. As these two hypotheses consider different kinds of users’
click or non-click behavior, combining them together reaches the best
performance.

##  6\. Conclusion

In this paper, we studied the recommendation fairness issue from data
augmentation perspective. Given the original training data, we proposed a FDA
framework to generate fake user behavior data, in order to improve
recommendation fairness. Specifically, given the overall idea of balanced
data, we proposed two hypotheses to guide the generation of fake data. After
that, we designed a bi-level optimization target, in which the inner
optimization generates better fake data and the outer optimization finds
recommendation parameters given the augmented data that comprises both the
original training data and the generated fake data. Please note that, FDA can
be applied to any embedding based recommendation backbones, and does not rely
on any specific fairness metrics. Extensive experiments on two real-world
datasets clearly showed FDA is effective to balance recommendation accuracy
and fairness under different recommendation backbones.

##  7\. Acknowledgements

This work is supported in part by grants from the National Key Research and
Development Program of China (Grant No. 2021ZD0111802), the National Natural
Science Foundation of China (Grant No. 72188101, 61932009, U1936219,
U22A2094), Major Project of Anhui Province (Grant No. 202203a05020011), and
the CCF-AFSG Research Fund (Grant No. CCF-AFSG RF20210006).

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# Bott-Chern cohomology and the Hartogs extension theorem for pluriharmonic
functions

Xieping Wang  School of Mathematical Sciences, University of Science and
Technology of China, Hefei 230026, Anhui, People’s Republic of China
<EMAIL_ADDRESS>To Li and Rui’an

###### Abstract.

Let $X$ be a cohomologically $(n-1)$-complete complex manifold of dimension
$n\geq 2$. We prove a vanishing result for the Bott-Chern cohomology group of
type $(1,1)$ with compact support in $X$, which combined with the well-known
technique of Ehrenpreis implies a Hartogs type extension theorem for
pluriharmonic functions on $X$.

###### Key words and phrases:

Bott-Chern cohomology, Hartogs’ extension theorem, pluriharmonic functions

###### 2020 Mathematics Subject Classification:

32D15, 32D20, 31C10, 32L20

The author was partially supported by NSFC grant 12001513, and NSF of Anhui
Province grants 2008085QA18 and 2108085QA04.

##  1\. Introduction

As is well known, the Hartogs extension theorem for holomorphic functions is
one of the striking results in complex analysis that distinguish the higher-
dimensional case from the one-dimensional case. In its general form, the
theorem states:

######  Theorem A.

Let $X$ be a complex manifold of dimension $n\geq 2$ which is $(n-1)$-complete
in the sense of Andreotti-Grauert. Suppose $\Omega$ is a domain in $X$ and $K$
is a compact subset of $\Omega$ such that $\Omega\\!\setminus\\!K$ is
connected. Then every holomorphic function on $\Omega\\!\setminus\\!K$ has a
unique holomorphic extension to $\Omega$.

The theorem was proved by Andreotti-Hill [2] in 1972 and generalized to
$(n-1)$-complete normal complex spaces by Merker-Porten [12] in 2009; see [5]
or [14] for a simple alternative proof. Colţoiu-Ruppenthal [5] also dealt with
the more general cohomologically $(n-1)$-complete case. We refer the reader to
these works and the references therein for a more detailed account of Hartogs’
extension theorem, and to [16] for recent developments on related topics. It
should also be mentioned that there is a metric analogue of Hartogs’ theorem
under appropriate geometric conditions; see [9] for details.

On the other hand, Chen [4] recently proved the following very interesting
result.

######  Theorem B.

Let $\Omega$ be a domain in ${\mathbb{C}}^{n}$, $n\geq 2$, and let $K$ be a
compact subset of $\Omega$ such that $\Omega\\!\setminus\\!K$ is connected.
Then every pluriharmonic function on $\Omega\\!\setminus\\!K$ has a unique
pluriharmonic extension to $\Omega$.

Inspired by this result and the work mentioned above, it seems natural to ask
the following

######  Question .

Does Theorem B still hold when $\mathbb{C}^{n}$ is replaced by a Stein (or
more generally, cohomologically $(n-1)$-complete) manifold of dimension $n\geq
2$?

Now recall that the Bott-Chern cohomology group of type $(1,1)$ with compact
support in a complex manifold $X$ is defined as

$H^{1,\,1}_{{\rm BC},\,c}(X)=\frac{{\rm
Ker}\,d\cap\mathcal{D}^{1,\,1}(X)}{\partial\bar{\partial}\,\mathcal{D}(X)},$

where $\mathcal{D}(X)$ (resp. $\mathcal{D}^{1,\,1}(X)$) denotes the set of all
compactly supported smooth functions (resp. $(1,1)$-forms) on $X$. An
equivalent formulation of Theorem B is $H^{1,\,1}_{{\rm
BC},\,c}({\mathbb{C}}^{n})=0$ for $n\geq 2$. Chen proved this cohomology-
vanishing result by directly solving the $\partial\bar{\partial}$-equation,
which depends heavily on an elegant trick of Lelong [11] and the Sobolev
inequality (see [4, Section 3] for details). According to the celebrated work
of Mok-Siu-Yau [13] and Croke [6], these two key ingredients apply only to
complete Kähler manifolds of nonnegative holomorphic bisectional
curvature.111Despite this, Chen’s method is attractive partly because it
yields the desired solution in an efficient way, with a precise gradient
estimate. For the question posed earlier, these seem to constitute only a
rather restrictive class of complex manifolds, since by virtue of the well-
known embedding theorem and the curvature-decreasing property for Kähler
submanifolds, Stein manifolds–the objects of interest here–are usually those
of nonpositive holomorphic bisectional curvature. In such a situation, to
answer the previous question we instead use a (qualitative) cohomological
argument and prove the following

######  Theorem 1.1.

Let $X$ be a cohomologically $(n-1)$-complete complex manifold of dimension
$n\geq 2$. Suppose $\Omega$ is a domain in $X$ and $K$ is a compact subset of
$\Omega$ such that $\Omega\\!\setminus\\!K$ is connected. Then every
pluriharmonic function on $\Omega\\!\setminus\\!K$ has a unique pluriharmonic
extension to $\Omega$.

By adapting Ehrenpreis’ technique for the $\bar{\partial}$-operator (see [8,
10]) to the case of the $\partial\bar{\partial}$-operator, we reduce Theorem
1.1 to the following cohomology-vanishing result.

######  Theorem 1.2.

Let $X$ be a cohomologically $(n-1)$-complete complex manifold of dimension
$n\geq 2$. Then

$H^{1,\,1}_{{\rm BC},\,c}(X)=0.$

This follows easily from the Serre duality [15], as we will show in Sect. 2.
If $X$ is Stein, the result is previously known; see [3, Teorema 2.4].

We now conclude the introduction with some remarks on Theorem 1.1. Using the
unique continuation property for holomorphic forms, one can easily show that
Theorem 1.1 implies the Hartogs extension theorem for holomorphic functions.
And as essentially pointed out in [12], the latter does not hold in general if
the cohomological $(n-1)$-completeness assumption of $X$ is weakened to
(cohomological) $n$-completeness, which is equivalent to requiring that $X$
have no compact connected components according to the Serre duality and a
result of Greene-Wu (see [7, Chapter IX, Theorem 3.5]). These show that the
cohomological completeness condition imposed on $X$ in Theorem 1.1 is sharp.

Acknowledgements. The author is grateful to the anonymous referees for their
careful reading of the paper and helpful comments.

##  2\. Proof of the results

We begin by recalling the notion of cohomological $q$-completeness, where $q$
is a positive integer.

######  Definition .

A complex manifold $X$ is called cohomologically $q$-complete if its $k$-th
cohomology group with coefficients in $\mathcal{F}$, $H^{k}(X,\,\mathcal{F})$,
vanishes for every coherent analytic sheaf $\mathcal{F}$ over $X$ and integer
$k\geq q$.

Recall also that an $n$-dimensional complex manifold is called $q$-complete in
the sense of Andreotti-Grauert [1] if it possesses a smooth exhaustion
function whose Levi form has at least $n-q+1$ positive eigenvalues at every
point. According to Grauert’s solution of the Levi problem (see, e.g., [10]),
$1$-complete complex manifolds are precisely Stein manifolds. In general, by
an important result of Andreotti-Grauert [1] we know that $q$-complete complex
manifolds are necessarily cohomologically $q$-complete, but the converse
remains unclear except for the cases $q=1$ and $q\geq{\rm dim}\,X$.

We are now ready to prove the results of this paper, starting with Theorem
1.2.

###### Proof of Theorem 1.2.

Without loss of generality, we may assume that $X$ is connected. Let
$\Omega^{p}_{X}$ denote the sheaf of germs of holomorphic $p$-forms on $X$,
$p=0,\ldots,n$. It is well-known that $\Omega^{p}_{X}$ is coherent. Since
$n\geq 2$, the cohomological $(n-1)$-completeness of $X$ implies

$H^{q}(X,\,\Omega^{p}_{X})=0\quad{\rm for}\ \,\,p,\,q=n-1,\,n.$

Then by Serre’s criterion [15, Proposition 6], we can invoke the Serre duality
[15, Théorème 2] to get

$H^{1}_{c}(X,\,\Omega^{p}_{X})=0\quad{\rm for}\ \,\,p=0,\,1.$

Equivalently, we have the corresponding vanishing result for the Dolbeault
cohomology groups of $X$ with compact support:

$H^{0,\,1}_{\bar{\partial},\,c}(X)=0\quad{\rm and}\quad
H^{1,\,1}_{\bar{\partial},\,c}(X)=0,$

in view of the Dolbeault isomorphism theorem (with compact support).
Furthermore, using $H^{n}(X,\,\Omega^{n}_{X})=0$ and the Serre duality again
we obtain

$H^{0}_{c}(X,\,\Omega^{0}_{X})=0.$

This means that $X$ is noncompact.

We now show that what we have obtained in turn implies

$H^{1,\,1}_{{\rm BC},\,c}(X)=0.$

To see this, let $\mathcal{D}^{p,\,q}(X)$ denote the set of all compactly
supported smooth $(p,q)$-forms on $X$. Then for every $d$-closed form
$\alpha^{1,\,1}\in\mathcal{D}^{1,\,1}(X)$, the vanishing of
$H^{1,\,1}_{\bar{\partial},\,c}(X)$ implies that there exists a form
$\alpha^{1,\,0}\in\mathcal{D}^{1,\,0}(X)$ such that
$\bar{\partial}\alpha^{1,\,0}=\alpha^{1,\,1}$. Now

$\bar{\partial}\partial\alpha^{1,\,0}=-\partial\bar{\partial}\alpha^{1,\,0}=-\partial\alpha^{1,\,1}=0.$

It follows that $\partial\alpha^{1,\,0}$ is a compactly supported holomorphic
$2$-form on the noncompact manifold $X$, hence vanishes identically. Since
$H^{0,\,1}_{\bar{\partial},\,c}(X)=0$, then there exists a function
$\alpha\in\mathcal{D}(X)$ such that $\partial\alpha=-\alpha^{1,\,0}$.
Consequently

$\partial\bar{\partial}\alpha=\bar{\partial}\alpha^{1,\,0}=\alpha^{1,\,1},$

and we are done. ∎

###### Proof of Theorem 1.1.

Choose a cut-off function $\chi\in\mathcal{D}(\Omega)$ such that $\chi=1$ on a
neighborhood of $K$. Given a pluriharmonic function $f$ on
$\Omega\\!\setminus\\!K$, we define

$v:=\partial\bar{\partial}((1-\chi)f).$

Then $v$ is a smooth $d$-closed $(1,1)$-form on $X$ satisfying

${\rm supp}\,v\subset{\rm supp}\,\chi\subset\Omega.$

By Theorem 1.2, there exists a function $u\in\mathcal{D}(X)$ such that
$\partial\bar{\partial}u=v$.

Now set

$F:=(1-\chi)f-u,$

which is pluriharmonic on $\Omega$. We claim that
$\left.F\right|_{\Omega\setminus K}=f$. For this we may assume that $X$ itself
is connected, since otherwise $\Omega$ would be contained in a single
connected component of $X$ and we only need to replace $X$ with this connected
component. Note that $u$ is pluriharmonic (and hence real-analytic) on
$X\\!\setminus\\!{\rm supp}\,\chi$, which intersects $X\\!\setminus\\!{\rm
supp}\,u$ since $X$ is noncompact (as we pointed out in the proof of Theorem
1.2). Then by the unique continuation property, $u$ vanishes identically on
some connected component, say $U$, of $X\\!\setminus\\!{\rm supp}\,\chi$. On
the other hand, the connectedness of $X$ implies that $U$ has a nonempty
boundary in $X$, which is necessarily contained in ${\rm
supp}\,\chi\subset\Omega$ (by the fact that $U$ is a connected component of
$X\\!\setminus\\!{\rm supp}\,\chi$). We then conclude that

$U\cap(\Omega\\!\setminus\\!{\rm supp}\,\chi)=U\cap\Omega\neq\emptyset$

and

$\left.F\right|_{U\cap(\Omega\setminus{\rm
supp}\,\chi)}=\left.f\right|_{U\cap(\Omega\setminus{\rm supp}\,\chi)}.$

Consequently, $\left.F\right|_{\Omega\setminus K}=f$ by the connectedness of
$\Omega\\!\setminus\\!K$. ∎

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