# Large-Scale Multi-Domain Recommendation: an Automatic Domain Feature
Extraction and Personalized Integration Framework

Dongbo Xi1,∗, Zhen Chen1,∗, Yuexian Wang1,2, He Cui1,
Chong Peng1, Fuzhen Zhuang3,4, and Peng Yan1 1Meituan
2Shanghai Jiao Tong University, Shanghai 201109, China
3Institute of Artificial Intelligence, Beihang University, Beijing 100191,
China
4SKLSDE, School of Computer Science, Beihang University, Beijing 100191, China
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Feed recommendation is currently the mainstream mode for many real-world
applications (e.g., TikTok, Dianping), in order to increase the overall daily
active users of the application, it is usually necessary to model and predict
user interests in multiple scenarios (domains) within and even outside the
application. Multi-domain learning is a typical solution in this regard. The
usual practice of multi-domain learning is to design shared and specific
modules for each domain separately. While considerable efforts have been made
in this regard, there are still two long-standing challenges: (1) Accurately
depicting the differences among domains using domain features is crucial for
enhancing the performance of each domain. However, manually designing domain
features and models for numerous domains can be a laborious task. (2) Users
typically have limited impressions in only a few domains. Extracting features
automatically from other domains and leveraging them to improve the predictive
capabilities of each domain has consistently posed a challenging problem. In
this paper, we propose an Automatic Domain Feature Extraction and Personalized
Integration (DFEI) framework for the large-scale multi-domain recommendation.
The framework automatically transforms the behavior of each individual user
into an aggregation of all user behaviors within the domain, which serves as
the domain features. Unlike offline feature engineering methods, the extracted
domain features are higher-order representations and directly related to the
target label. Besides, by personalized integration of domain features from
other domains for each user and the innovation in the training mode, the DFEI
framework can yield more accurate conversion identification. Experimental
results on both public and industrial real-world Dianping feed recommendation
datasets, consisting of over 20 domains, clearly demonstrate that the proposed
framework achieves significantly better performance compared with state-of-
the-art baselines. Furthermore, we have released the source code of the
proposed framework at https://github.com/xidongbo/DFEI.

Multi-Domain Learning, Recommender System, Feed Recommendation, CTR Prediction

$*$ Corresponding authors: Dongbo Xi and Zhen Chen.

††conference: ; ; ††isbn: 978-1-4503-XXXX-X/18/06††ccs: Computing
methodologies Multi-task learning††ccs: Information systems Recommender
systems

##  1\. Introduction

In feed recommendation, users can continuously browse items generated by an
endless feed using their mobile phones. Accurate feed recommendation is
crucial for increasing the daily active users of real-world applications.
Especially with the increasing complexity of applications nowadays and the
involvement of more and more feed recommendation scenarios (domains), it has
always been a challenging problem to accurately depict the differences between
domains and utilize the information of domains to improve the recommendation
performance in each domain.

The conventional approaches (Sheng et al., 2021; Ning et al., 2023; Zhou et
al., 2023) for multi-domain learning involve designing shared and specific
modules for each domain separately. However, despite significant efforts in
this area, there are still two long-standing challenges that need to be
addressed. Firstly, manually designing domain features and models for numerous
domains can be a laborious task. This manual process not only consumes a
significant amount of time and effort but also increases the risk of human
error. Therefore, there is a need for an automatic approach that can extract
domain features effectively and efficiently. Secondly, users typically have
limited impressions in only a few domains. When inferring in a new domain, the
domain-related features of users are often missing. This poses a challenging
problem of extracting features automatically from other domains and leveraging
them to improve the predictive capabilities of each domain. Existing methods
struggle to overcome this challenge, and as a result, the performance of
multi-domain recommendation systems remains sub-optimal.

To address these challenges, this paper proposes an Automatic Domain Feature
Extraction and Personalized Integration (DFEI) framework for the large-scale
multi-domain recommendation. The DFEI framework automatically transforms the
behavior of each individual user into an aggregation of all user behaviors
within the domain, which serves as the domain features. In contrast to
traditional offline feature engineering methods, the extracted domain features
are higher-order representations that are directly correlated with the target
label. This approach not only minimizes the manual effort needed for feature
engineering but also improves the performance of the feed recommendation.
Furthermore, the DFEI framework personally integrates domain features from
other domains for each user. This integration, coupled with the innovation in
the training mode, enables the framework to significantly enhance the
recommendation performance.

To evaluate the effectiveness of the proposed framework, extensive experiments
are conducted on both public and industrial real-world Dianping feed
recommendation datasets, consisting of over 20 domains. The experimental
results clearly demonstrate that the DFEI framework outperforms state-of-the-
art baselines, achieving significantly better performance.

To summarize, the contributions of this paper are threefold:

  * •

The DFEI framework achieves automatic domain feature extraction and
personalized integration for the large-scale multi-domain recommendation.

  * •

The experimental results validate the effectiveness of the proposed framework,
highlighting its superiority over existing approaches.

  * •

The release of the source code further promotes collaboration and encourages
future advancements in the field of multi-domain recommendation.

##  2\. Related Work

This section offers a brief overview of some topics related to our study,
including Multi-Task Learning (MTL) and Multi-Domain Learning (MDL).

###  2.1. Multi-Task Learning

Multi-Task Learning (MTL) is a subfield of machine learning in which the model
is trained simultaneously on multiple related tasks. The key idea is to
leverage the commonalities and divergences across tasks to enhance the model’s
performance on each individual task, rather than training separate models for
each task. Share-Bottom (Caruana, 1997) network is the most commonly used
multi-task model which shares the bottom network to discern similarities and
builds multiple task-specific networks at the top to identify differences.
However, the hard parameter sharing approach may suffer from severe
optimization conflicts when tasks are not highly correlated. Cross-Stitch
(Misra et al., 2016) and Sluice Network (Ruder et al., 2019) are proposed to
learn unique combinations of the outputs for each task. MoE (Shazeer et al.,
2017) and MMoE (Ma et al., 2018b) adopt gating networks to combine multiple
experts shared at the bottom to capture different patterns in data, which
makes it more effective when the tasks are loosely related. Based on MMoE, PLE
(Tang et al., 2020) separates expert networks into shared and task-specific to
better tackle with seesaw phenomenon. Unlike preceding approaches, models such
as ESMM (Ma et al., 2018a), ESM2 (Wen et al., 2020), and AITM (Xi et al.,
2021a) primarily concentrate on characterizing the relationship between output
tasks. Concurrently, certain studies (He et al., 2022; Xie et al., 2021; Yang
et al., 2023) have been undertaken to mitigate the issue of negative transfer
from the vantage point of multi-task optimization.

###  2.2. Multi-Domain Learning

Multi-Domain Learning (MDL) has become an increasingly significant research
area due to the prevalence of data collected from various domains in real-
world applications (Dredze et al., 2010; Joshi et al., 2012; Li et al., 2020).
MDL seeks to enable knowledge transfer across different domains to enhance
learning outcomes in all domains simultaneously, distinguishing itself from
Transfer Learning (TL) (Pan and Yang, 2009; Zhuang et al., 2020), which
traditionally involves a unidirectional transfer of knowledge from a source to
a target domain (Li and Tuzhilin, 2019; Hu et al., 2018).

In recent years, Multi-Domain Learning has been applied to recommendation
systems, with a focus on Multi-Domain Recommendation (MDR) (Li et al., 2020;
Chen et al., 2020; Shen et al., 2021; Hao et al., 2021; Sheng et al., 2021;
Chang et al., 2023; Zhou et al., 2023; Tian et al., 2023; Cheng et al., 2023;
Huan et al., 2023; Lan et al., 2023; Xu et al., 2023; Zhao et al., 2023; Wang
et al., 2023; Li et al., 2023; Ning et al., 2023; Zhang et al., 2023; Gao et
al., 2023). Inspired by Multi-task learning, HMoE (Li et al., 2020) takes
advantage of MMoE (Ma et al., 2018b) structure to implicitly model
distinctions and commonalities among different domains. SAML (Chen et al.,
2020) constructs domain-independent and domain-dependent features respectively
through feature transformation and attention mechanism, and proposes a mutual
unit to learn similarities between domains. SAR-Net (Shen et al., 2021)
employs attention modules to transfer users’ interests across domains and uses
a mixture of experts for information extraction. AFT (Hao et al., 2021)
proposes a generative adversarial network framework to capture informative
domain-specific features. STAR (Sheng et al., 2021) constructs a star topology
with shared centered parameters and domain-specific parameters, and trains a
single model to serve all domains. Moreover, some other works start to focus
on performing multiple tasks in multiple domains simultaneously. For instance,
PEPNet (Chang et al., 2023) introduces both parameter personalization and
embedding personalization to accurately capture user interests for multi-task
recommendation in the multi-domain setting. AESM2 (Zou et al., 2022) and HiNet
(Zhou et al., 2023) use a hierarchical MOE to model the commonalities and
differences between different domains and tasks. HiNet pays special attention
to how useful information from one domain can be for another domain.

However, all of the above models do not explicitly and automatically extract
domain features for the large-scale multi-domain recommendation, which is
crucial for enhancing the performance of each domain, and also do not
explicitly consider the information utilization problem when users only have
impressions in a few domains.

Figure 1. The proposed Automatic Domain Feature Extraction and Personalized
Integration (DFEI) framework.

##  3\. Methodology

In this section, we first describe the problem formulation, and then we
present the details of the proposed framework DFEI as shown in Figure 1.

###  3.1. Problem Formulation

As mentioned above, we focus on the large-scale multi-domain recommendation
problem. Given the domain set $\mathcal{D}$, assuming the $d$-th domain sample
is $\bm{x}_{d}$, ($\bm{x}_{d}\in\mathcal{D}_{d},d=1,2,\cdots,D$). The multi-
domain framework needs to predict the conversion probability $\hat{y}_{d}$ of
each domain $d$ based on the input features $\bm{x}_{d}$:

(1) $\hat{y}_{d}=p(y_{d}=1|\bm{x}_{d}).$

###  3.2. Automatic Domain Feature Extraction and Personalized Integration

As shown in Figure 1, given the sample $\bm{x}_{d}$ in domain
$\mathcal{D}_{d}$, we first encode all the features and then embed each of
them to a low dimension dense vector representation, the output of the Shared
Embedding layer is the concatenation of all embedding vectors:

(2) $\bm{e}_{d}=E(\bm{x}_{d}),$

where $E$ denotes the embedding and concatenation operations. By sharing the
same embedding vectors across all domains, several benefits can be achieved.
Firstly, it helps to reduce the number of model parameters for the large-scale
multi-domain recommendation. Secondly, it enhances information sharing between
different domains, thereby improving overall performance.

Next, shared and specific modules are adopted to extract the shared and
specific representation, respectively:

(3) $\bm{s}_{d}=f(\bm{e}_{d}),~{}\bm{u}_{d}=f_{d}(\bm{e}_{d}),$

where $f(\cdot)$ is the Shared Tower, and $f_{d}(\cdot)$ is the
$\mathcal{D}_{d}$ Tower as shown in Figure 1. It should be mentioned that
designing different shared and specific towers is not the focus of this paper
as we aim at designing an Automatic Domain Feature Extraction and Personalized
Integration modules to improve the overall performance for the large-scale
multi-domain recommendation.

DFE: Domain features can accurately depict the differences among domains,
which are crucial for improving the performance of each domain. Although the
domain is static, users within the domain are dynamic, and the domain feature
should be represented by the behavior of users within the domain. In the
training phase, the Automatic Domain Feature Extraction computes the domain
feature $\bm{v}_{d}^{t}$ at training step $t$ as:

(4)
$\bm{v}_{d}^{t}=\alpha*\bm{v}_{d}^{t-1}+(1-\alpha)*pooling(\bm{u}_{d}^{t}),$

where $\bm{v}_{d}^{t-1}$ is output of training step $t-1$,
$pooling(\bm{u}_{d}^{t})$ is the mean-pooling of the samples at training step
$t$ during mini-bath training, $\bm{v}_{d}^{0}=pooling(\bm{u}_{d}^{0})$, and
$\alpha$ defines the decay coefficient. Intuitively, the domain feature can be
understood as the moving averages of all samples throughout the training
process. It transforms the behavior of each individual user into an
aggregation of all user behaviors within the domain. In contrast to
traditional offline feature engineering methods, the extracted domain features
are higher-order representations that are directly associated with the target
label during the training process. In the inference phase, the domain feature
will directly adopt the final trained state, i.e.,
$\bm{v}_{d}=\bm{v}_{d}^{T}$.

Table 1. Basic information of the industrial dataset.

Domains | #A1 | #A2 | #A3 | #A4 | #A5 | #A6 | #A7 | #A8 | #A9 | #A10 | #A11 | #A12 | #A13 | #A14 | #A15 | #A16 | #A17 | #A18 | #A19 | #A20 | #A21
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
#Train | 2,514K | 1,881K | 83K | 18K | 83K | 49K | 328K | 832K | 213K | 51K | 146K | 15K | 292K | 91K | 9K | 20K | 66K | 28,625K | 35K | 2,135K | 127K
#Validation | 313K | 235K | 11K | 2K | 10K | 6K | 41K | 104K | 27K | 6K | 18K | 2K | 37K | 11K | 1K | 2K | 8K | 3,587K | 4K | 266K | 16K
#Test | 314K | 235K | 11K | 2K | 10K | 6K | 41K | 104K | 27K | 6K | 18K | 2K | 37K | 11K | 1K | 3K | 8K | 3,579K | 4K | 267K | 16K
CTR(%) | 1.09 | 1.9 | 2.13 | 3.09 | 3.47 | 3.59 | 3.67 | 3.83 | 4.6 | 5.39 | 5.4 | 5.76 | 5.98 | 6.46 | 6.79 | 6.98 | 7.46 | 7.58 | 7.84 | 7.86 | 9.45

Table 2. Basic information of the public dataset.

Domains | #B1 | #B2 | #B3 | #B4 | #B5 | #B6
---|---|---|---|---|---|---
#Train | 857K | 2,615K | 148K | 326K | 5K | 50K
#Validation | 209K | 671K | 35K | 77K | 2K | 17K
#Test | 1,341K | 4,431K | 219K | 493K | 9K | 117K
CTR(%) | 12.00 | 44.54 | 39.09 | 56.05 | 21.34 | 18.10

DFI: Users usually only have impressions in a few domains, so their domain-
related features in other domains are often missing, which makes it difficult
to directly utilize the users’ domain-related features for other domains. The
Automatic Domain Feature Integration uses domain features
$\bm{v}_{1},\bm{v}_{2},...,\bm{v}_{D}$ to enhance the prediction ability of
each domain:

(5) $\displaystyle\bm{z}_{d}$ $\displaystyle=$
$\displaystyle\sum_{d=1}^{D}w_{d}h_{1}(\bm{v}_{d}),$

where $w_{d}$ is the personalized integration weight which is formulated as:

(6) $\displaystyle
w_{d}=\frac{exp(\hat{w}_{d})}{\sum_{d=1}^{D}exp(\hat{w}_{d})},~{}~{}\hat{w}_{d}=\frac{<h_{2}(\bm{e}_{d}),h_{3}(\bm{v}_{d})>}{\sqrt{k}},$

where $<\cdot~{},\cdot>$ represents the dot product. $h_{1}(\cdot)$,
$h_{2}(\cdot)$, and $h_{3}(\cdot)$ project the input information to one new
unified vector space, which the dimension is $k$. In this paper, we use a
simple MLP (Multi-Layer Perceptron) (Gardner and Dorling, 1998) as
$h_{1}(\cdot)$, $h_{2}(\cdot)$, and $h_{3}(\cdot)$ following previous work (Xi
et al., 2021a). The key idea is similar to previous works (Xi et al., 2020;
Zhu et al., 2020; Xi et al., 2021b) and has been proven to be more effective.
The difference is that query $h_{2}(\bm{e}_{d})$ is personalized for different
samples, while key $h_{3}(\bm{v}_{d})$ and value $h_{1}(\bm{v}_{d})$ represent
domain features, so they are consistent for all samples within the domain.

Finally, the prediction probability of each domain $d$ is:

(7)
$\displaystyle\hat{y}_{d}=sigmoid(MLP(concat([\bm{s}_{d},\bm{u}_{d},\bm{v}_{d},\bm{z}_{d})])).$

###  3.3. Joint Optimization Strategy for MDL

In the joint optimization phase, we need to cyclically feed mini-batch samples
and minimize the cross-entropy loss in each domain:

(8)
$\mathcal{L}_{d}(\theta^{s},\theta_{d}^{u})=-\frac{1}{N}\sum^{N}_{(\bm{x}_{d},y_{d})\in\mathcal{D}_{d}}(\left(y_{d}\log\hat{y}_{d}+(1-y_{d})\log(1-\hat{y}_{d})\right),$

where $N$ is the number of samples, $y_{d}$ is the label of the $d$-th domain
and $\theta^{s}$, $\theta_{d}^{u}$ is the shared and specific (unique)
parameter sets in the MDL framework, respectively. The $\theta^{s}$ contains
the shared parameters in $E(\cdot)$, $f(\cdot)$,
$h_{1}(\cdot),h_{2}(\cdot),h_{3}(\cdot)$ in Equations (2), (3), (5), (6) cross
all domains, and the other parameters is in $\theta_{d}^{u}$. As shown in
Algorithm 1, the learning rate of the shared parameters $\theta^{s}$ is set to
$\frac{1}{D}$ of the $\theta_{d}^{u}$ to balance the learning phase.

The framework is implemented using TensorFlow111https://www.tensorflow.org/
and trained through stochastic gradient descent over shuffled mini-batches
with the Adam (Kingma and Ba, 2015) update rule.

##  4\. Experiments

In this section, we evaluate our proposed framework by conducting extensive
experiments on both industrial and public datasets. The experimental settings
including dataset descriptions, reproducibility information, evaluation
protocol, and baseline methods are introduced first. Then, we move on to our
experimental results and analysis.

Algorithm 1 Joint Optimization Strategy for MDL

0: training data $(\bm{X},\bm{Y})$, batch size $B$, learning rate $\gamma$.

0: The trained parameters $\theta^{s}$ and $\theta_{d}^{u}$.

1: Initializing $\theta^{s}$ and $\theta_{d}^{u}$ randomly.

2: for $epoch\leftarrow 1$ to $EPOCHS$ do

3: for $t\leftarrow 1$ to $STEPS$ do

4: for $d\leftarrow 1$ to $D$ do

5: Calculate gradient $\nabla J(\theta_{d}^{u}),\nabla J(\theta^{s})$ based on
$(\bm{X}_{d},\bm{Y}_{d})$

6: $\theta_{d}^{u}=\theta_{d}^{u}-\gamma\nabla J(\theta_{d}^{u})$

7: $\theta^{s}=\theta^{s}-\frac{1}{D}\gamma\nabla J(\theta^{s})$

8: end for

9: end for

10: end for

###  4.1. Datasets

Two datasets are used to validate our proposed framework. One is the real-
world industrial dataset collected from Dianping222https://www.dianping.com,
and the other is a publicly accessible dataset collected from
Kuaishou333https://www.kuaishou.com. The tasks are all CTR prediction. Tables
1 and 2 show the statistical information of these two datasets.

  * •

Industrial dataset: The industrial dataset is collected from the user log data
of one of the most renowned local life information and trading platforms in
China. We extract a total of 21 scenarios (domains) of different sizes in the
application and denote them as #A1 to #A21 based on their CTR ranking. The
dataset comprises tens of millions of samples, encompassing user behavior,
user attributes, and item attributes. We randomly split the dataset into
training, validation, and test sets in a ratio of 8:1:1 according to the user
id.

  * •

Public dataset: The public dataset is
KuaiRand444https://github.com/chongminggao/KuaiRand dataset (Gao et al.,
2022), which is collected from recommendation logs of the video-sharing mobile
app, Kuaishou. We use the KuaiRand-1K version and split domains according to
the ”tab” field. In order to ensure the stability of evaluation results, we
select the 6 largest domains (tab in [0,1,2,4,5,6]) and denote them as as #B1
to #B6, respectively. According to the official guidelines, we designate the
data from April 8 to 21, 2022, as the training set, the data from April 22 to
May 8, 2022, as the test set, and the data from April 19 to 21, 2022, within
the training set as the validation set.

Table 3. The average AUC performance on the industrial dataset. Underlined
results indicate the best baselines over each domain. “*” indicates that the
improvement of the proposed DFEI is statistically significant compared with
the best baselines at p-value $<$ 0.05 over paired samples t-test, and “**”
indicates that the p-value $<$ 0.01.

Model | #A1 | #A2 | #A3 | #A4 | #A5 | #A6 | #A7 | #A8 | #A9 | #A10 | #A11 | #A12 | #A13 | #A14 | #A15 | #A16 | #A17 | #A18 | #A19 | #A20 | #A21 | avg
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
MLP | 0.6665 | 0.5808 | 0.6980 | 0.5557 | 0.6764 | 0.5699 | 0.5808 | 0.6363 | 0.6132 | 0.6658 | 0.6041 | 0.6331 | 0.5908 | 0.6476 | 0.5677 | 0.6274 | 0.6290 | 0.6715 | 0.6193 | 0.6326 | 0.5975 | 0.6221
MMoE | 0.7137 | 0.6574 | 0.7511 | 0.5487 | 0.7120 | 0.6286 | 0.6103 | 0.6792 | 0.6497 | 0.7058 | 0.6591 | 0.6433 | 0.6177 | 0.6588 | 0.6044 | 0.6419 | 0.6930 | 0.7193 | 0.6468 | 0.6635 | 0.6126 | 0.6579
PLE | 0.7192 | 0.6577 | 0.7571 | 0.5752 | 0.7118 | 0.6219 | 0.6105 | 0.6775 | 0.6554 | 0.7118 | 0.6519 | 0.6495 | 0.6123 | 0.6622 | 0.5941 | 0.6628 | 0.6867 | 0.7170 | 0.6445 | 0.6592 | 0.6148 | 0.6597
AITM | 0.7036 | 0.6541 | 0.7491 | 0.5893 | 0.7296 | 0.6199 | 0.6189 | 0.6877 | 0.6365 | 0.7071 | 0.6591 | 0.6476 | 0.6242 | 0.6797 | 0.6169 | 0.6462 | 0.6828 | 0.7136 | 0.6471 | 0.6704 | 0.6285 | 0.6625
HMoE | 0.7185 | 0.6525 | 0.7515 | 0.5704 | 0.7441 | 0.6266 | 0.6102 | 0.6746 | 0.6548 | 0.7079 | 0.6528 | 0.6040 | 0.6181 | 0.6697 | 0.5966 | 0.6559 | 0.6879 | 0.7106 | 0.6489 | 0.6631 | 0.6177 | 0.6589
STAR | 0.7158 | 0.6340 | 0.7501 | 0.5718 | 0.7579 | 0.6292 | 0.6099 | 0.6708 | 0.6536 | 0.7053 | 0.6543 | 0.6366 | 0.6161 | 0.6669 | 0.6007 | 0.6499 | 0.6775 | 0.7164 | 0.6469 | 0.6580 | 0.6192 | 0.6591
PEPNet | 0.7099 | 0.6585 | 0.7500 | 0.5644 | 0.7443 | 0.6209 | 0.6013 | 0.6807 | 0.6617 | 0.7063 | 0.6585 | 0.6364 | 0.6151 | 0.6628 | 0.6108 | 0.6439 | 0.6942 | 0.7188 | 0.6516 | 0.6623 | 0.6179 | 0.6605
HiNet | 0.7332 | 0.6587 | 0.7409 | 0.5785 | 0.7639 | 0.6363 | 0.6307 | 0.6980 | 0.6721 | 0.6956 | 0.6764 | 0.6576 | 0.6293 | 0.6787 | 0.6091 | 0.6496 | 0.6824 | 0.7213 | 0.6561 | 0.6740 | 0.6356 | 0.6704
DFEI | 0.7316 | 0.6623** | 0.7642* | 0.5866 | 0.7820** | 0.6434* | 0.6383** | 0.7050** | 0.6729 | 0.7137** | 0.6831** | 0.6449 | 0.6294 | 0.6766 | 0.6329 | 0.6558 | 0.6952 | 0.7192 | 0.6535 | 0.6662 | 0.6423** | 0.6761**

Table 4. The summary of the hyper-parameters in all models.

Hyper-parameter | Value
---|---
Optimizer | Adam
Batch size | 512
Learning rate | 1e-3
L2 regularization | 1e-6
Embedding dimension | 16
Dimensions of layers in the MLP Tower and Expert | [128, 64, 32]
Dropout rate in each layers of MLP | [0.1, 0.2, 0.3]
Activation function in MLP | Relu

###  4.2. Reproducibility Information

To ensure fair comparison and evaluate the generalization performance of
different models, we adopt the same set of common hyper-parameters for all
models on two datasets. These hyper-parameters are chosen based on empirical
values and computational efficiency. For both industrial and public datasets,
the output dimension $k$ of the attention in Equation (6) is $32$, the decay
coefficient $\alpha$ in Equation (4) is $0.9$, we only fine-tune the hyper-
parameters of $\alpha$ according to grid search on the validation set.
Besides, the $h_{1}(\cdot)$, $h_{2}(\cdot)$ and $h_{3}(\cdot)$ are all single-
layer MLP with dimension $32$ in Equations (5), (6). For a fair comparison, we
try our best to ensure that the main architecture of different models is
consistent. The summary of different models is shown in Table 4. We do not
perform any dataset-specific tuning, except for the early-stopping on
validation sets for all models, the experiments of all models are conducted
with NVIDIA Tesla V100 GPU with 16G memory.

Table 5. The average AUC performance on the public dataset. Underlined results
indicate the best baselines over each domain. “*” indicates that the
improvement of the proposed DFEI is statistically significant compared with
the best baselines at p-value $<$ 0.05 over paired samples t-test, and “**”
indicates that the p-value $<$ 0.01.

Model | #B1 | #B2 | #B3 | #B4 | #B5 | #B6 | avg
---|---|---|---|---|---|---|---
MLP | 0.6505 | 0.7314 | 0.7658 | 0.7160 | 0.6770 | 0.7235 | 0.7107
MMoE | 0.6983 | 0.7325 | 0.7717 | 0.7203 | 0.6786 | 0.7284 | 0.7225
PLE | 0.6908 | 0.7347 | 0.7718 | 0.7223 | 0.6695 | 0.7207 | 0.7183
AITM | 0.6949 | 0.7370 | 0.7784 | 0.7220 | 0.6800 | 0.7288 | 0.7236
HMoE | 0.6995 | 0.7364 | 0.7801 | 0.7208 | 0.6857 | 0.7214 | 0.7240
STAR | 0.6989 | 0.7370 | 0.7771 | 0.7223 | 0.6831 | 0.7273 | 0.7243
PEPNet | 0.6948 | 0.7354 | 0.7837 | 0.7234 | 0.6853 | 0.7271 | 0.7250
HiNet | 0.7051 | 0.7364 | 0.7842 | 0.7224 | 0.6866 | 0.7190 | 0.7256
DFEI | 0.7103** | 0.7372 | 0.7931** | 0.7195 | 0.6757 | 0.7328* | 0.7281**

###  4.3. Evaluation Protocol

Following existing works (Xi et al., 2021a; Zhou et al., 2023), AUC (the area
under ROC curve), a commonly used metric in ranking tasks, is adopted as the
evaluation metric to compare the performance of the the baselines and the
proposed DFEI framework. To avoid interference caused by experimental
fluctuations, all experiments were run five times with different random seeds.
In the end, we presented the average AUC over the five runs and performed the
paired samples t-test.

Figure 2. Ablation Study of the DFEI modules and the learning rate of the
shared parameters $\theta^{s}$ on the public dataset.

###  4.4. Baseline Methods

To demonstrate the effectiveness of our proposed framework, we compare it with
several state-of-the-art baselines, including a single-task learning model,
multi-task learning models, and multi-domain learning models. Below are more
details about baselines:

  1. i)

Single-task learning model

     * $\bullet$

MLP: This method utilizes a Multi-Layer Perceptron as the backbone and treats
all domains as a single domain.

  2. ii)

Multi-task learning models

     * $\bullet$

MMoE(Ma et al., 2018b): This model adopts a gating mechanism to combine N
expert networks shared at the bottom to capture task-related information.
Here, we consider different domains as distinct tasks and apply domain-
specific towers and gating networks for each domain.

     * $\bullet$

PLE(Tang et al., 2020): This model based on MMoE, separates the experts into
shared and task-specific to alleviate the negative transfer problem.

     * $\bullet$

AITM(Xi et al., 2021a): This model focuses on multi-task learning with
sequential dependence, using a gating unit and an information transfer unit to
adaptively learn what and how much information to transfer for different
conversion stages of different audiences, thereby facilitating effective
knowledge transfer.

  3. iii)

Multi-domain learning models

     * $\bullet$

HMoE(Li et al., 2020): This model is adapted from MMoE, utilizing a scenario-
aware MOE structure to implicitly model distinctions and commonalities among
different domains.

     * $\bullet$

STAR(Sheng et al., 2021): This model constructs a star topology incorporating
both shared and domain-specific parameters, utilizing a single model to cater
to all domains.

     * $\bullet$

PEPNet(Chang et al., 2023): This model introduces both parameter
personalization and embedding personalization to accurately capture users’
interest for multi-task recommendation in the multi-domain setting.

     * $\bullet$

HiNet(Zhou et al., 2023): This model applies a hierarchical information
extraction structure with MOE module to model complex correlations between
different domains and tasks.

###  4.5. Performance Comparison

The comparison of AUC scores among different methods on two datasets is
reported in Tables 3 and 5, where the best score among all methods is marked
in boldface and the best-performed baseline is underlined. From the results,
we have the following observations:

  * •

The simple MLP regards all domains as a single entity, failing to model the
differences among domains. It employs entirely shared parameters for multi-
domain learning, which makes the experimental performance fall short when
compared to multi-task and multi-domain learning methods, particularly in the
case of a large number of domains.

  * •

Compared with MLP, multi-task learning models (MMoE, PLE, AITM) achieve better
results on both datasets. We treat various domains as separate tasks, and this
approach allows multi-task models to exhibit adaptability when dealing with
multi-domain settings. Besides, AITM focuses on multi-task learning with
sequence dependence. When applied to multi-domain settings, it can establish a
sequential relationship among different domains. Experimental results have
indicated that domains ranked later in the sequence show a performance boost
in the AITM model, compared to other multi-task models. Its overall
performance is even better than some multi-domain baselines on the industrial
dataset. However, multi-tasking methods focus more on modeling the
relationships between tasks, thus ignoring domain differences when applied to
multi-domain settings.

  * •

Most multi-domain learning methods have modeled the similarities and
differences among various domains, yielding positive results. In addition,
HiNet applies scenario-aware attentive network to effectively counteract the
phenomenon of negative transfer, which makes it better capture and leverage
the correlations among domains, leading to commendable performance. However,
these models do not explicitly and automatically extract domain features for
the large-scale multi-domain recommendation, and also do not explicitly
address the issue of information utilization when users only have impressions
in a limited number of domains, leading to sub-optimal performance.

  * •

DFEI outperforms all baselines in average AUC score by a significant margin,
showing superior predictive capabilities among domains and proving the
effectiveness of combining automatic domain feature extraction and
personalized integration modules.

Figure 3. Hyper-parameter Study of the decay coefficient $\alpha$ on the
public dataset. The metric is the average AUC of all domains.

Figure 4. The t-SNE visualization of the final domain features
$\bm{v}_{1},...,\bm{v}_{D}$ on the industrial dataset. The red dots represent
CTR less than $5\%$, while the blue stars represent CTR greater than $5\%$ as
shown in Table 1.

###  4.6. Ablation Study

In this subsection, we perform the ablation study of the DFEI module and the
learning rate of the shared parameters $\theta^{s}$ in Algorithm 1 on the
public dataset.

Firstly, considering the impact of DFE and DFI in Figure 1, we remove the DFEI
(DFE + DFI, i.e., w/o DFEI) and DFI (i.e., w/o DFI) respectively and the
results are shown in Figure 2 (a). Secondly, we study the impact of the
learning rate of the shared parameters $\theta^{s}$, and the results are shown
in Figure 2 (b). From Figure 2, we can obtain the following inspiring
observations:

  * •

DFEI outperforms both w/o DFEI and w/o DFI in most cases, which suggests that
the automatic domain feature extraction and personalized integration from
different domains effectively enhance the prediction performance. Except for
domains #B2 and #B5, the AUC score of w/o DFI is usually higher than w/o DFEI.
This suggests that employing automatic domain feature extraction is an
effective strategy for illustrating the differences among domains, thereby
enhancing the performance of DFEI. Some fluctuations occur in the results
among domains, due to the significant disparity in data volumes. We consider
the average AUC across all domains as our primary focus.

  * •

With the learning rate scaling factor $\frac{1}{D}$ on the shared parameters
$\theta^{s}$, the DFEI leads to higher average AUC scores. This is because in
Algorithm 1, each training step will be fed the samples of each domain, thus
updating each domain-specific parameter $\theta^{d}$ once, while updating the
shared parameter $\theta^{s}$ $D$ times. Therefore, using $\frac{1}{D}$ as the
scaling factor will better balance the entire learning process.

From the above results, we could see that the learning rate scaling factor of
the shared parameters and the DFEI module can indeed improve the performance
of multi-domain recommendation.

###  4.7. Hyper-parameter Study

In order to study the impact of the hype-parameter decay coefficient $\alpha$
in Equation (4), we perform the hyper-parameter study on the public dataset.
As shown in Figure 4, we vary the decay coefficient as [0, 0.1, 0.3, 0.5, 0.7,
0.9, 1] and report the average AUC performance on the test set. When $\alpha$
is small, it will forget more historical information and record more of the
latest batch information, resulting in poor performance. As $\alpha$ gradually
increases, more historical information is preserved and accumulated. Finally,
the behavior of each individual user is transformed into an aggregation of all
user behaviors within the domain, and the best performance is achieved when
$\alpha=0.9$.

###  4.8. Visualization Study

In order to determine if the final domain features
($\bm{v}_{1},...,\bm{v}_{D}$ in Equation (4)) have successfully captured
specific information across different domains, we utilize t-SNE (t-distributed
Stochastic Neighbor Embedding (van der Maaten and Hinton, 2008)) to visualize
them in Figure 4. Due to the limited domain number of the public dataset and
the lack of statistical significance, our visualization study is conducted on
the industrial dataset. From the visualization, we have the following
interesting findings: Domains with a CTR less than $5\%$ ($\\#A1-\\#A9$) are
concentrated in the area with x coordinates greater than 0, while domains with
a CTR greater than $5\%$ ($\\#A10-\\#A21$) are mostly concentrated in the area
with x coordinates less than 0. This indicates that the domain features have
learned the CTR information of the domain, and the CTR is directly correlated
with the target label. Furthermore, the domain features are higher-level
feature representations, and the CTR information is merely a dimension
indicating differences between domains.

##  5\. Conclusion

In this paper, we proposed an Automatic Domain Feature Extraction and
Personalized Integration (DFEI) framework for the large-scale multi-domain
recommendation. The proposed framework could automatically extract and
personalize the integration of domain features for each user, which can yield
more accurate conversion identification. Experimental results on large-scale
multi-domain datasets demonstrate significant improvement compared with state-
of-the-art baseline models.

Several directions are available for future research in the area. Firstly, the
DFEI could be applied as a plug-and-play method to other advanced models.
Secondly, combining the MTL and MDL may bring additional gains.

## References

  * (1)
  * Caruana (1997) Rich Caruana. 1997.  Multitask Learning.  _Machine Learning_ 28 (07 1997).  https://doi.org/10.1023/A:1007379606734
  * Chang et al. (2023) Jianxin Chang, Chenbin Zhang, Yiqun Hui, Dewei Leng, Yanan Niu, Yang Song, and Kun Gai. 2023.  PEPNet: Parameter and Embedding Personalized Network for Infusing with Personalized Prior Information. In _Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_ _(KDD ’23)_. 3795–3804.  https://doi.org/10.1145/3580305.3599884
  * Chen et al. (2020) Yuting Chen, Yanshi Wang, Yabo Ni, Anxiang Zeng, and Lanfen Lin. 2020.  Scenario-aware and Mutual-based approach for Multi-scenario Recommendation in E-Commerce.  _2020 International Conference on Data Mining Workshops (ICDMW)_ (2020), 127–135.  https://api.semanticscholar.org/CorpusID:229211225
  * Cheng et al. (2023) Wenzhuo Cheng, Ke Ding, Xin Dong, Yong He, Liang Zhang, and Linjian Mo. 2023.  MI-DPG: Decomposable Parameter Generation Network Based on Mutual Information for Multi-Scenario Recommendation. In _Proceedings of the 32nd ACM International Conference on Information and Knowledge Management_. 3803–3807.
  * Dredze et al. (2010) Mark Dredze, Alex Kulesza, and Koby Crammer. 2010.  Multi-domain learning by confidence-weighted parameter combination.  _Machine Learning_ 79 (2010), 123–149.
  * Gao et al. (2022) Chongming Gao, Shijun Li, Yuan Zhang, Jiawei Chen, Biao Li, Wenqiang Lei, Peng Jiang, and Xiangnan He. 2022.  KuaiRand: An Unbiased Sequential Recommendation Dataset with Randomly Exposed Videos. In _Proceedings of the 31st ACM International Conference on Information and Knowledge Management_ _(CIKM ’22)_. 3953–3957.  https://doi.org/10.1145/3511808.3557624
  * Gao et al. (2023) Jingtong Gao, Bo Chen, Menghui Zhu, Xiangyu Zhao, Xiaopeng Li, Yuhao Wang, Yichao Wang, Huifeng Guo, and Ruiming Tang. 2023.  Scenario-Aware Hierarchical Dynamic Network for Multi-Scenario Recommendation.  _arXiv preprint arXiv:2309.02061_ (2023).
  * Gardner and Dorling (1998) Matt W Gardner and SR Dorling. 1998.  Artificial neural networks (the multilayer perceptron)—a review of applications in the atmospheric sciences.  _Atmospheric environment_ (1998).
  * Hao et al. (2021) Xiaobo Hao, Yudan Liu, Ruobing Xie, Kaikai Ge, Linyao Tang, Xu Zhang, and Leyu Lin. 2021.  Adversarial Feature Translation for Multi-domain Recommendation. In _Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining_ _(KDD ’21)_. 2964–2973.  https://doi.org/10.1145/3447548.3467176
  * He et al. (2022) Yun He, Xue Feng, Cheng Cheng, Geng Ji, Yunsong Guo, and James Caverlee. 2022.  MetaBalance: Improving Multi-Task Recommendations via Adapting Gradient Magnitudes of Auxiliary Tasks. In _Proceedings of the ACM Web Conference 2022_ _(WWW ’22)_. 2205–2215.  https://doi.org/10.1145/3485447.3512093
  * Hu et al. (2018) Guangneng Hu, Yu Zhang, and Qiang Yang. 2018.  CoNet: Collaborative Cross Networks for Cross-Domain Recommendation. In _Proceedings of the 27th ACM International Conference on Information and Knowledge Management_ _(CIKM ’18)_. 667–676.  https://doi.org/10.1145/3269206.3271684
  * Huan et al. (2023) Zhaoxin Huan, Ang Li, Xiaolu Zhang, Xu Min, Jieyu Yang, Yong He, and Jun Zhou. 2023.  SAMD: An Industrial Framework for Heterogeneous Multi-Scenario Recommendation. In _Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_. 4175–4184.
  * Joshi et al. (2012) Mahesh Joshi, Mark Dredze, William Cohen, and Carolyn Rosé. 2012.  Multi-Domain Learning: When Do Domains Matter?  _In Proceedings of the 2012 Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning._
  * Kingma and Ba (2015) Diederik P Kingma and Jimmy Ba. 2015.  Adam: A method for stochastic optimization. In _ICLR_ , Vol. 5.
  * Lan et al. (2023) Zerong Lan, Yingyi Zhang, and Xianneng Li. 2023.  M3REC: A Meta-based Multi-scenario Multi-task Recommendation Framework. In _Proceedings of the 17th ACM Conference on Recommender Systems_. 771–776.
  * Li et al. (2020) Pengcheng Li, Runze Li, Qing Da, An-Xiang Zeng, and Lijun Zhang. 2020.  Improving Multi-Scenario Learning to Rank in E-commerce by Exploiting Task Relationships in the Label Space. In _Proceedings of the 29th ACM International Conference on Information & Knowledge Management_ _(CIKM ’20)_. 2605–2612.  https://doi.org/10.1145/3340531.3412713
  * Li and Tuzhilin (2019) P. Li and Alexander Tuzhilin. 2019.  DDTCDR: Deep Dual Transfer Cross Domain Recommendation.  _Proceedings of the 13th International Conference on Web Search and Data Mining_ (2019).  https://api.semanticscholar.org/CorpusID:204402822
  * Li et al. (2023) Xiaopeng Li, Fan Yan, Xiangyu Zhao, Yichao Wang, Bo Chen, Huifeng Guo, and Ruiming Tang. 2023.  Hamur: Hyper adapter for multi-domain recommendation. In _Proceedings of the 32nd ACM International Conference on Information and Knowledge Management_. 1268–1277.
  * Ma et al. (2018b) Jiaqi Ma, Zhe Zhao, Xinyang Yi, Jilin Chen, Lichan Hong, and Ed H. Chi. 2018b.  Modeling Task Relationships in Multi-task Learning with Multi-gate Mixture-of-Experts. In _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_ _(KDD ’18)_. 1930–1939.  https://doi.org/10.1145/3219819.3220007
  * Ma et al. (2018a) Xiao Ma, Liqin Zhao, Guan Huang, Zhi Wang, Zelin Hu, Xiaoqiang Zhu, and Kun Gai. 2018a.  Entire Space Multi-Task Model: An Effective Approach for Estimating Post-Click Conversion Rate. In _The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval_ _(SIGIR ’18)_. 1137–1140.  https://doi.org/10.1145/3209978.3210104
  * Misra et al. (2016) Ishan Misra, Abhinav Shrivastava, Abhinav Gupta, and Martial Hebert. 2016.  Cross-Stitch Networks for Multi-task Learning. In _2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_. 3994–4003.  https://doi.org/10.1109/CVPR.2016.433
  * Ning et al. (2023) Wentao Ning, Xiao Yan, Weiwen Liu, Reynold Cheng, Rui Zhang, and Bo Tang. 2023.  Multi-domain Recommendation with Embedding Disentangling and Domain Alignment. In _Proceedings of the 32nd ACM International Conference on Information and Knowledge Management_. 1917–1927.
  * Pan and Yang (2009) Sinno Jialin Pan and Qiang Yang. 2009.  A survey on transfer learning.  _IEEE Transactions on knowledge and data engineering_ 22, 10 (2009), 1345–1359.
  * Ruder et al. (2019) Sebastian Ruder, Joachim Bingel, Isabelle Augenstein, and Anders Søgaard. 2019.  Latent Multi-Task Architecture Learning.  _Proceedings of the AAAI Conference on Artificial Intelligence_ 33 (07 2019), 4822–4829.  https://doi.org/10.1609/aaai.v33i01.33014822
  * Shazeer et al. (2017) Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc Le, Geoffrey Hinton, and Jeff Dean. 2017.  Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer.  arXiv:1701.06538 [cs.LG]
  * Shen et al. (2021) Qijie Shen, Wanjie Tao, Jing Zhang, Hong Wen, Zulong Chen, and Quan Lu. 2021.  SAR-Net: A Scenario-Aware Ranking Network for Personalized Fair Recommendation in Hundreds of Travel Scenarios. In _Proceedings of the 30th ACM International Conference on Information & Knowledge Management_ _(CIKM ’21)_. 4094–4103.  https://doi.org/10.1145/3459637.3481948
  * Sheng et al. (2021) Xiang-Rong Sheng, Liqin Zhao, Guorui Zhou, Xinyao Ding, Binding Dai, Qiang Luo, Siran Yang, Jingshan Lv, Chi Zhang, Hongbo Deng, and Xiaoqiang Zhu. 2021.  One Model to Serve All: Star Topology Adaptive Recommender for Multi-Domain CTR Prediction. In _Proceedings of the 30th ACM International Conference on Information & Knowledge Management_ _(CIKM ’21)_. 4104–4113.  https://doi.org/10.1145/3459637.3481941
  * Tang et al. (2020) Hongyan Tang, Junning Liu, Ming Zhao, and Xudong Gong. 2020.  Progressive Layered Extraction (PLE): A Novel Multi-Task Learning (MTL) Model for Personalized Recommendations. In _Proceedings of the 14th ACM Conference on Recommender Systems_ _(RecSys ’20)_. 269–278.  https://doi.org/10.1145/3383313.3412236
  * Tian et al. (2023) Yu Tian, Bofang Li, Si Chen, Xubin Li, Hongbo Deng, Jian Xu, Bo Zheng, Qian Wang, and Chenliang Li. 2023.  Multi-Scenario Ranking with Adaptive Feature Learning. In _Proceedings of the 46th International ACM SIGIR Conference on Research and Development in Information Retrieval_. 517–526.
  * van der Maaten and Hinton (2008) Laurens van der Maaten and Geoffrey Hinton. 2008.  Visualizing Data using t-SNE.  _Journal of Machine Learning Research_ 9, 86 (2008), 2579–2605.
  * Wang et al. (2023) Yuhao Wang, Xiangyu Zhao, Bo Chen, Qidong Liu, Huifeng Guo, Huanshuo Liu, Yichao Wang, Rui Zhang, and Ruiming Tang. 2023.  PLATE: A Prompt-Enhanced Paradigm for Multi-Scenario Recommendations. In _Proceedings of the 46th International ACM SIGIR Conference on Research and Development in Information Retrieval_. 1498–1507.
  * Wen et al. (2020) Hong Wen, Jing Zhang, Yuan Wang, Fuyu Lv, Wentian Bao, Quan Lin, and Keping Yang. 2020.  Entire Space Multi-Task Modeling via Post-Click Behavior Decomposition for Conversion Rate Prediction. In _The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval_ _(SIGIR ’20)_. 2377–2386.  https://doi.org/10.1145/3397271.3401443
  * Xi et al. (2021a) Dongbo Xi, Zhen Chen, Peng Yan, Yinger Zhang, Yongchun Zhu, Fuzhen Zhuang, and Yu Chen. 2021a.  Modeling the Sequential Dependence among Audience Multi-step Conversions with Multi-task Learning in Targeted Display Advertising. In _Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining_ _(KDD ’21)_. 3745–3755.  https://doi.org/10.1145/3447548.3467071
  * Xi et al. (2021b) Dongbo Xi, Bowen Song, Fuzhen Zhuang, Yongchun Zhu, Shuai Chen, Tianyi Zhang, Yuan Qi, and Qing He. 2021b.  Modeling the Field Value Variations and Field Interactions Simultaneously for Fraud Detection. In _AAAI_.
  * Xi et al. (2020) Dongbo Xi, Fuzhen Zhuang, Bowen Song, Yongchun Zhu, Shuai Chen, Dan Hong, Tao Chen, Xi Gu, and Qing He. 2020.  Neural Hierarchical Factorization Machines for User’s Event Sequence Analysis. In _SIGIR_. 1893–1896.
  * Xie et al. (2021) Ruobing Xie, Yanlei Liu, Shaoliang Zhang, Rui Wang, Feng Xia, and Leyu Lin. 2021.  Personalized Approximate Pareto-Efficient Recommendation. In _Proceedings of the Web Conference 2021_ _(WWW ’21)_. 3839–3849.  https://doi.org/10.1145/3442381.3450039
  * Xu et al. (2023) Senrong Xu, Liangyue Li, Yuan Yao, Zulong Chen, Han Wu, Quan Lu, and Hanghang Tong. 2023.  MUSENET: Multi-scenario learning for repeat-aware personalized recommendation. In _Proceedings of the Sixteenth ACM International Conference on Web Search and Data Mining_. 517–525.
  * Yang et al. (2023) Enneng Yang, Junwei Pan, Ximei Wang, Haibin Yu, Li Shen, Xihua Chen, Lei Xiao, Jie Jiang, and Guibing Guo. 2023.  AdaTask: a task-aware adaptive learning rate approach to multi-task learning. In _Proceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence_ _(AAAI’23/IAAI’23/EAAI’23)_. AAAI Press.  https://doi.org/10.1609/aaai.v37i9.26275
  * Zhang et al. (2023) Yingyi Zhang, Xianneng Li, Yahe Yu, Jian Tang, Huanfang Deng, Junya Lu, Yeyin Zhang, Qiancheng Jiang, Yunsen Xian, Liqian Yu, et al. 2023\.  Meta-generator enhanced multi-domain recommendation. In _Companion Proceedings of the ACM Web Conference 2023_. 485–489.
  * Zhao et al. (2023) Pengyu Zhao, Xin Gao, Chunxu Xu, and Liang Chen. 2023.  M5: Multi-Modal Multi-Interest Multi-Scenario Matching for Over-the-Top Recommendation. In _Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_. 5650–5659.
  * Zhou et al. (2023) Jie Zhou, Xia Cao, Wenhao Li, Kun Zhang, Chuan Luo, and Qian Yu. 2023.  HiNet: Novel Multi-Scenario & Multi-Task Learning with Hierarchical Information Extraction.  _2023 IEEE 39th International Conference on Data Engineering (ICDE)_ (2023), 2969–2975.  https://api.semanticscholar.org/CorpusID:257482348
  * Zhu et al. (2020) Yongchun Zhu, Dongbo Xi, Bowen Song, Fuzhen Zhuang, Shuai Chen, Xi Gu, and Qing He. 2020.  Modeling Users’ Behavior Sequences with Hierarchical Explainable Network for Cross-domain Fraud Detection. In _TheWebConf_. 928–938.
  * Zhuang et al. (2020) Fuzhen Zhuang, Zhiyuan Qi, Keyu Duan, Dongbo Xi, Yongchun Zhu, Hengshu Zhu, Hui Xiong, and Qing He. 2020.  A comprehensive survey on transfer learning.  _Proc. IEEE_ 109, 1 (2020), 43–76.
  * Zou et al. (2022) Xinyu Zou, Zhi Hu, Yiming Zhao, Xuchu Dennis Ding, Zhongyi Liu, Chenliang Li, and Aixin Sun. 2022.  Automatic Expert Selection for Multi-Scenario and Multi-Task Search.  _Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval_ (2022).  https://api.semanticscholar.org/CorpusID:249191419


# Higher-order transverse momentum fluctuations in heavy-ion collisions

Somadutta Bhatta  Department of Chemistry, Stony Brook University, Stony
Brook, NY 11794, USA  Chunjian Zhang  Department of Chemistry, Stony Brook
University, Stony Brook, NY 11794, USA  Jiangyong Jia
<EMAIL_ADDRESS>Department of Chemistry, Stony Brook University,
Stony Brook, NY 11794, USA  Physics Department, Brookhaven National
Laboratory, Upton, NY 11976, USA

###### Abstract

In relativistic heavy-ion collisions, the event-by-event mean transverse
momentum fluctuations are sensitive to overlap area and energy density
fluctuations in initial state. We present a framework to calculate
$p_{\mathrm{T}}$ fluctuations up to $4^{\mathrm{th}}$-order using standard and
subevent methods, which is validated using the HIJING model. We observe a
power-law dependence for cumulants of all orders as a function of charged
particle multiplicity $N_{\mathrm{ch}}$, consistent with a simple independent
source picture. The fluctuation in $pp$ collisions is observed to be larger
than for $p+$Pb, Pb+Pb and Xe+Xe collisions at the same $N_{\mathrm{ch}}$ due
to bias in number of contributing sources. The short-range correlations are
greatly suppressed in subevent method in comparison to calculations based on
the standard method. This study provides a baseline for transverse momentum
fluctuations without the presence of final state effects.

###### pacs:

25.75.Gz, 25.75.Ld, 25.75.-1

##  I Introduction

Heavy-ion collisions at ultrarelativistic energies creates a hot and dense QCD
matter whose space-time evolution is well described by relativistic viscous
hydrodynamics Romatschke and Romatschke (2019); Busza  _et al._ (2018).
Fluctuations of the position of participating nucleons lead to large event-by-
event fluctuations in the initial state of the collision. As the system
expands, initial-state fluctuations survive various stages of system evolution
and influence the final-state observables Alver and Roland (2010); Teaney and
Yan (2011); Schenke  _et al._ (2020). Within hydrodynamic model framework, the
average transverse momentum $\mathrm{p_{\mathrm{T}}}$ in each event depends on
the initial energy density and is inversely related to the size of overlap
region area Shuryak (1998); Gardim  _et al._ (2020); Giacalone  _et al._
(2021); Jia (2022a). The variance of the $\mathrm{p_{\mathrm{T}}}$ and the
initial overlap size $R$ is expected to have a simple relation
$\sigma(\mathrm{p_{\mathrm{T}}})/\left\langle\mathrm{p_{\mathrm{T}}}\right\rangle=\sqrt{\big{\langle}(\mathrm{p_{\mathrm{T}}}-\left\langle\mathrm{p_{\mathrm{T}}}\right\rangle)^{2}\big{\rangle}}/\left\langle\mathrm{p_{\mathrm{T}}}\right\rangle\propto\sigma(R)/\left\langle
R\right\rangle$ Bozek and Broniowski (2012). The $\mathrm{p_{\mathrm{T}}}$
fluctuations can be quantified by its cumulants such as mean, variance,
skewness and kurtosis. But so far, only the mean and variance have been
measured experimentally Adams  _et al._ (2005); Abelev  _et al._ (2013, 2014);
Adamczyk  _et al._ (2017); Adam  _et al._ (2019); Aad  _et al._ (2019).
Recently Ref. Giacalone  _et al._ (2021) predicted that the skewness of
$\mathrm{p_{\mathrm{T}}}$ fluctuations should be significantly larger than
expectation from the independent source scenario, and they can be used to
probe nuclear deformation effects Jia (2022a, b). But no experimental
measurement has been carried out for skewness and kurtosis.

The cumulants of $\mathrm{p_{\mathrm{T}}}$ fluctuations are intensive
quantities. Within the independent source picture such as that implemented in
the HIJING model, each collision comprises of superposition of independent
$pp$-like collisions and interaction between the sources are ignored Wang and
Gyulassy (1991). Under these conditions, the $n^{\mathrm{th}}$-order cumulant
is expected to scale as $\propto 1/\mathrm{N_{s}}^{(n-1)}$, where
$\mathrm{N_{s}}$ is the number of sources often taken to be
$N_{\mathrm{part}}$ (number of participating nucleons) or $N_{\mathrm{ch}}$
(charged particle multiplicity) Zhou and Jia (2018); Giacalone  _et al._
(2021); Cody  _et al._ (2021). However, the dynamics of evolution and final-
state interaction can lead to a deviation from this power-law behavior.
Indeed, experimental measurements of $\mathrm{p_{\mathrm{T}}}$ variance in
Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV and Pb+Pb collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV have reported the power to be $\sim 0.81$
instead of the expected value of 1. This clear deviation from the baseline of
independent source picture in the experimental data indicates the presence of
long-range collective correlations and significant final-state effects Abelev
_et al._ (2014); Aaboud  _et al._ (2018); Adam  _et al._ (2019).

The direct calculation of higher order cumulants in heavy-ion collisions are
computationally expensive, we provide a framework based on the standard and
subevent methods to ease this calculationBilandzic  _et al._ (2011); Jia  _et
al._ (2017). We use the HIJING model to extract the baseline in which
$\mathrm{p_{\mathrm{T}}}$ fluctuations arise mostly from superposition of
contributions from independent sources. The contribution from each source
contains both long-range correlations associated with strings and short-range
correlations from jets and resonance decays Wang and Gyulassy (1991). The
short-range correlations can be suppressed using rapidity-separated subevent
methods, which are widely used in in previous flow analyses Aaboud  _et al._
(2018); Jia  _et al._ (2017); Huo  _et al._ (2018); Aaboud  _et al._ (2019);
Sirunyan  _et al._ (2021); Aaboud  _et al._ (2020); Aad  _et al._ (2021). To
test the power-law scaling and its system size dependence, we repeated the
study using Pb+Pb, Xe+Xe $p$+Pb and $pp$ collisions at similar center of mass
energy around 5 TeV.

The paper is organised as follows. Section II provides the formulae for
calculating $\mathrm{p_{\mathrm{T}}}$ cumulants in both standard and subevent
methods and describe the setup for HIJING model. The main results are
presented in Section III. Section IV gives a summary.

##  II Methodology and Model Setup

We first provide the framework for mean-$\mathrm{p_{\mathrm{T}}}$ cumulants
following the approach prescribed in Refs. Di Francesco  _et al._ (2017); Jia
_et al._ (2017). The $n$-particle $\mathrm{p_{\mathrm{T}}}$ correlator in one
event is defined as:

$\displaystyle c_{n}=\frac{\sum_{i_{1}\neq...\neq
i_{n}}w_{i_{1}}...w_{i_{n}}(p_{\rm{T},i_{1}}-\llangle\mathrm{p_{\mathrm{T}}}\rrangle)...(p_{\rm{T},i_{n}}-\llangle\mathrm{p_{\mathrm{T}}}\rrangle)}{\sum_{i_{1}\neq...\neq
i_{n}}w_{i_{1}}...w_{i_{n}}}$ (1)

where $w_{i}$ is the weight for particle $i$. Following Jia  _et al._ (2017)
and denoting $p\equiv\mathrm{p_{\mathrm{T}}}$, this relation can be expanded
algebraically into a simple polynomial function of the following quantities:

$\displaystyle
p_{mk}=\sum_{i}w_{i}^{k}p^{m}_{i}/\sum_{i}w_{i}^{k},\tau_{k}=\frac{\sum_{i}w_{i}^{k+1}}{\left(\sum_{i}w_{i}\right)^{k+1}}$
$\displaystyle\bar{p}_{1k}\equiv
p_{1k}-\llangle\mathrm{p_{\mathrm{T}}}\rrangle$
$\displaystyle\bar{p}_{2k}\equiv
p_{2k}-2p_{1k}\llangle\mathrm{p_{\mathrm{T}}}\rrangle+\llangle\mathrm{p_{\mathrm{T}}}\rrangle^{2}$
$\displaystyle\bar{p}_{3k}\equiv
p_{3k}-3p_{2k}\llangle\mathrm{p_{\mathrm{T}}}\rrangle+3p_{1k}\llangle\mathrm{p_{\mathrm{T}}}\rrangle^{2}-\llangle\mathrm{p_{\mathrm{T}}}\rrangle^{3}$
$\displaystyle\bar{p}_{4k}\equiv
p_{4k}-4p_{3k}\llangle\mathrm{p_{\mathrm{T}}}\rrangle+6p_{2k}\llangle\mathrm{p_{\mathrm{T}}}\rrangle^{2}-4p_{1k}\llangle\mathrm{p_{\mathrm{T}}}\rrangle^{3}+\llangle\mathrm{p_{\mathrm{T}}}\rrangle^{4}$
(2)

Note there $\llangle\mathrm{p_{\mathrm{T}}}\rrangle=\left\langle
p_{11}\right\rangle$ is the mean-$\mathrm{p_{\mathrm{T}}}$ averaged over the
event ensemble.

Using these auxiliary variables, the correlator in Eq. 1 in the standard
method can be expressed as,

$\displaystyle
c_{2}=\frac{\bar{p}_{11}^{2}-\tau_{1}\bar{p}_{22}}{1-\tau_{1}},\quad
c_{3}=\frac{\bar{p}_{11}^{3}-3\tau_{1}\bar{p}_{22}\bar{p}_{11}+2\tau_{2}\bar{p}_{33}}{1-3\tau_{1}+2\tau_{2}},\quad$
$\displaystyle
c_{4}=\frac{\bar{p}_{11}^{4}-6\tau_{1}\bar{p}_{22}\bar{p}^{2}_{11}+3\tau_{1}^{2}\bar{p}^{2}_{22}+8\tau_{2}\bar{p}_{33}\bar{p}_{11}-6\tau_{3}\bar{p}_{44}}{1-6\tau_{1}+3\tau^{2}_{1}+8\tau_{2}-6\tau_{3}}$
(3)

where particles are taken from $|\eta|<2.5$ and only unique particle
combinations in the event are considered.

In the subevent method (2sub), particle combinations are chosen from two
rapidity separated subevents $a$ ($-2.5<\eta<-0.75$) and $c$
($2.5>\eta>0.75$). The gap between the subevents reduce short-range
correlations. The correlators in this case are given by

$\displaystyle
c_{2,\mathrm{2sub}}=(\bar{p}_{11})_{a}(\bar{p}_{11})_{c},\;c_{3,\mathrm{2sub1}}=\frac{(\bar{p}^{2}_{11}-\tau_{1}\bar{p}_{22})_{a}(\bar{p}_{11})_{c}}{1-\tau_{1a}},\;c_{3,\mathrm{2sub2}}=\frac{(\bar{p}^{2}_{11}-\tau_{1}\bar{p}_{22})_{c}(\bar{p}_{11})_{a}}{1-\tau_{1c}},\;c_{4,\mathrm{2sub}}=\frac{(\bar{p}^{2}_{11}-\tau_{1}\bar{p}_{22})_{a}(\bar{p}^{2}_{11}-\tau_{1}\bar{p}_{22})_{c}}{(1-\tau_{1a})(1-\tau_{1c})}$
(4)

where the lettered subscripts in Eq. 4 denote the subevents from which
particles are taken. Note that there are two different ways of calculating
two-subevent $c_{3}$, and the final results is calculated as average, i.e.
$2c_{3,\mathrm{2sub}}=c_{3,\mathrm{2sub1}}+c_{3,\mathrm{2sub2}}$. The validity
of these formulae is confirmed by comparison with results obtained from direct
loop calculations.

The cumulants are then calculated by averaging $c_{n}$ over a given ensemble
of events. In practise, they are calculated over an unit $N_{\mathrm{ch}}$ bin
and are then combined over wider bins. In this paper, we use the following
dimensionless definition for cumulants (also called scaled-cumulants),

$\displaystyle k_{2}=\frac{\left\langle
c_{2}\right\rangle}{\llangle\mathrm{p_{\mathrm{T}}}\rrangle^{2}},\quad
k_{3}=\frac{\left\langle
c_{3}\right\rangle}{\llangle\mathrm{p_{\mathrm{T}}}\rrangle^{3}},\quad
k_{4}=\frac{\left\langle c_{4}\right\rangle-3\left\langle
c_{2}\right\rangle^{2}}{\llangle\mathrm{p_{\mathrm{T}}}\rrangle^{4}},\quad
k_{2,\mathrm{2sub}}=\frac{\left\langle
c_{2,\mathrm{2sub}}\right\rangle}{\llangle\mathrm{p_{\mathrm{T}}}\rrangle_{a}\llangle\mathrm{p_{\mathrm{T}}}\rrangle_{c}},$
$\displaystyle k_{3,\mathrm{2sub1}}=\frac{\left\langle
c_{3,\mathrm{2sub1}}\right\rangle}{\llangle\mathrm{p_{\mathrm{T}}}\rrangle_{a}^{2}\llangle\mathrm{p_{\mathrm{T}}}\rrangle_{c}},\quad
k_{3,\mathrm{2sub2}}=\frac{\left\langle
c_{3,\mathrm{2sub2}}\right\rangle}{\llangle\mathrm{p_{\mathrm{T}}}\rrangle_{a}\llangle\mathrm{p_{\mathrm{T}}}\rrangle_{c}^{2}},\quad
k_{4,\mathrm{2sub}}=\frac{\left\langle
c_{4,\mathrm{2sub}}\right\rangle-2\left\langle
c_{2,\mathrm{2sub}}\right\rangle^{2}-\left\langle
c_{2}\right\rangle_{a}\left\langle
c_{2}\right\rangle_{c}}{\llangle\mathrm{p_{\mathrm{T}}}\rrangle_{a}^{2}\llangle\mathrm{p_{\mathrm{T}}}\rrangle_{c}^{2}}\;.$
(5)

Note that the $\left\langle c_{2}\right\rangle_{a}$ and $\left\langle
c_{2}\right\rangle_{c}$ denote the $\left\langle c_{2}\right\rangle$ obtained
from the standard method in subevent $a$ and $c$, respectively. The final
skewness is taken as
$2k_{3,\mathrm{2sub}}=k_{3,\mathrm{2sub1}}+k_{3,\mathrm{2sub2}}$.

For HIJING simulation, we generate $pp$, $p$+Pb, and Pb+Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV and Xe+Xe collisions at
$\sqrt{s_{\mathrm{NN}}}=5.44$ TeV. The particles are chosen from $|\eta|<2.5$
and two $\mathrm{p_{\mathrm{T}}}$ ranges: $0.2<\mathrm{p_{\mathrm{T}}}<2$ GeV
and $0.2<\mathrm{p_{\mathrm{T}}}<5$ GeV. The lower $\mathrm{p_{\mathrm{T}}}$
range is less sensitive to contribution from short-range correlations. To
facilitate direct comparison with experimental measurements, the calculations
are carried out as a function of $N_{\mathrm{ch}}$ consisting only of charged
hadrons within $|\eta|<2.5$ and $0.5<\mathrm{p_{\mathrm{T}}}<5$ GeV. The
$N_{\mathrm{ch}}$ range thus obtained is consistent with the choice used in
the ATLAS experiment Aaboud  _et al._ (2020).

##  III Results

Figure 1 shows the $\mathrm{p_{\mathrm{T}}}$ cumulants without the
normalization by $\llangle\mathrm{p_{\mathrm{T}}}\rrangle$ in Pb+Pb and Xe+Xe
collisions as a function of $N_{\mathrm{ch}}$. A double-log scale is used to
show clearly the power-law dependence as a function of $N_{\mathrm{ch}}$. The
extracted power law index from a simple fit of the form
$a/(N_{\mathrm{ch}})^{b}$ gives the value $b$ that is very close to $(n-1)$
expected for the $n^{\mathrm{th}}-\mathrm{order}$ cumulant. This confirms that
the $\mathrm{p_{\mathrm{T}}}$ fluctuations in HIJING originate from
superposition of independent sources.

Figure 1:  The $\mathrm{p_{\mathrm{T}}}$ cumulants without normalizing by
$\llangle\mathrm{p_{\mathrm{T}}}\rrangle$ for particles in
$0.2<\mathrm{p_{\mathrm{T}}}<2$ GeV as a function of $N_{\mathrm{ch}}$ in
Pb+Pb and Xe+Xe collisions. The solid lines show the fit of the data to
$a/(N_{\mathrm{ch}})^{b}$.  Figure 2: The variance (left), skewness (middle)
and kurtosis (right) in Pb+Pb collisions using the standard (solid points) and
two-subevent method (open points) for charged particles in
$0.2<\mathrm{p_{\mathrm{T}}}<5.0$ GeV as a function of $N_{\mathrm{ch}}$.

Figure 2 compares $k_{n}$ between the standard and subevent calculations to
estimate the influence of short-range correlations. The $k_{2,\mathrm{2sub}}$
is suppressed by a factor of 3 in comparison to $k_{2}$ from the standard
method, suggesting the scaled variance in HIJING model is dominated by the
short-range correlations. The influence for $k_{3}$ and $k_{4}$ are
significantly smaller. This is consistent with the expectation that the short-
range correlations have smaller impact for higher-order cumulants.

Figure 3: (Left) The variance obtained for the standard method and two-
subevent method with various rapidity separations in Pb+Pb collisions for
$0.2<\mathrm{p_{\mathrm{T}}}<2$ GeV as a function of $N_{\mathrm{ch}}$.
(Right) Ratio of results from two-subevent method to those obtained from the
standard method. Figure 4: The $\mathrm{p_{\mathrm{T}}}$ cumulants compared
between different collision systems as a function of $N_{\mathrm{ch}}$

Figure 3 shows how the scaled variance $k_{2}$ depends on the rapidity
separation between the two subevents. Most of the decrease is obtained between
the subevent method and standard method. Further decrease is observed when one
increase the separation between the two subevents. Beyond the edge gap of 0.5
unit, however, the values of $k_{2}$ do not change further. This residual
signal probably reflects the genuine long-range correlations associated with
strings in the HIJING model.

Figure 4 compares the $\mathrm{p_{\mathrm{T}}}$ cumulants between $pp$,
$p+$Pb, Xe+Xe and Pb+Pb collisions as a function of $N_{\mathrm{ch}}$. In
HIJING, we expect fluctuations to decrease with increasing system size in
accordance with increase of the number of sources. The magnitude of
$\llangle\mathrm{p_{\mathrm{T}}}\rrangle$ shows a clear system size ordering,
whereas for higher order cumulants only $pp$ has a higher magnitude than those
for $p+$Pb, Xe+Xe and Pb+Pb collisions. The latter is expected since the
number of sources in $pp$ is small, and the events with large
$N_{\mathrm{ch}}$ is dominated by fluctuations in the particle production in
each source, i.e. from minijets, instead of fluctuations in the number of
sources.

##  IV Summary

In summary, we provide a framework for calculating the higher-order dynamical
$\mathrm{p_{\mathrm{T}}}$ cumulants up to $4^{\mathrm{th}}$ order using the
standard and subevent methods. In the HIJING model, the higher-order cumulants
are found to follow a simple power-law scaling as a function of charged
particle multiplicity $N_{\mathrm{ch}}$, as expected from a simple
superposition of independent sources. The subevent method is found to
effectively suppress short-range correlations that dominates the variance of
the $\mathrm{p_{\mathrm{T}}}$ fluctuations; such short-range correlations have
smaller influence on skewness and kurtosis of the $\mathrm{p_{\mathrm{T}}}$
fluctuations. The $\mathrm{p_{\mathrm{T}}}$ cumulants are found to follow a
common scaling as a function of $N_{\mathrm{ch}}$, except in the $pp$
collisions where they are influenced by fluctuations of particle production
within each source. Our study provides a useful baseline for
$\mathrm{p_{\mathrm{T}}}$ fluctuations that is based on simple superposition
of independent sources and short-range correlations but without final state
effects.

Acknowledgements: This work is supported by DOE DE-FG02-87ER40331.

## References

  * Romatschke and Romatschke (2019) P. Romatschke and U. Romatschke,  _Relativistic Fluid Dynamics In and Out of Equilibrium_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2019) arXiv:1712.05815 [nucl-th] .
  * Busza  _et al._ (2018) W. Busza, K. Rajagopal, and W. van der Schee, Ann. Rev. Nucl. Part. Sci. 68, 339 (2018), arXiv:1802.04801 [hep-ph] .
  * Alver and Roland (2010) B. Alver and G. Roland, Phys. Rev. C 81, 054905 (2010), [Erratum: Phys.Rev.C 82, 039903 (2010)], arXiv:1003.0194 [nucl-th] .
  * Teaney and Yan (2011) D. Teaney and L. Yan, Phys. Rev. C 83, 064904 (2011), arXiv:1010.1876 [nucl-th] .
  * Schenke  _et al._ (2020) B. Schenke, C. Shen, and D. Teaney, Phys. Rev. C 102, 034905 (2020), arXiv:2004.00690 [nucl-th] .
  * Shuryak (1998) E. V. Shuryak, Phys. Lett. B 423, 9 (1998), arXiv:hep-ph/9704456 .
  * Gardim  _et al._ (2020) F. G. Gardim, G. Giacalone, M. Luzum, and J.-Y. Ollitrault, Nature Phys. 16, 615 (2020), arXiv:1908.09728 [nucl-th] .
  * Giacalone  _et al._ (2021) G. Giacalone, F. G. Gardim, J. Noronha-Hostler, and J.-Y. Ollitrault, Phys. Rev. C 103, 024910 (2021), arXiv:2004.09799 [nucl-th] .
  * Jia (2022a) J. Jia, Phys. Rev. C 105, 044905 (2022a), arXiv:2109.00604 [nucl-th] .
  * Bozek and Broniowski (2012) P. Bozek and W. Broniowski, Phys. Rev. C 85, 044910 (2012), arXiv:1203.1810 [nucl-th] .
  * Adams  _et al._ (2005) J. Adams _et al._ (STAR), Phys. Rev. C 72, 044902 (2005), arXiv:nucl-ex/0504031 .
  * Abelev  _et al._ (2013) B. B. Abelev _et al._ (ALICE), Phys. Lett. B 727, 371 (2013), arXiv:1307.1094 [nucl-ex] .
  * Abelev  _et al._ (2014) B. B. Abelev _et al._ (ALICE), Eur. Phys. J. C 74, 3077 (2014), arXiv:1407.5530 [nucl-ex] .
  * Adamczyk  _et al._ (2017) L. Adamczyk _et al._ (STAR), Phys. Rev. C 96, 044904 (2017), arXiv:1701.07065 [nucl-ex] .
  * Adam  _et al._ (2019) J. Adam _et al._ (STAR), Phys. Rev. C 99, 044918 (2019), arXiv:1901.00837 [nucl-ex] .
  * Aad  _et al._ (2019) G. Aad _et al._ (ATLAS), Eur. Phys. J. C 79, 985 (2019), arXiv:1907.05176 [nucl-ex] .
  * Jia (2022b) J. Jia, Phys. Rev. C 105, 014905 (2022b), arXiv:2106.08768 [nucl-th] .
  * Wang and Gyulassy (1991) X.-N. Wang and M. Gyulassy, Phys. Rev. D 44, 3501 (1991).
  * Zhou and Jia (2018) M. Zhou and J. Jia, Phys. Rev. C 98, 044903 (2018), arXiv:1803.01812 [nucl-th] .
  * Cody  _et al._ (2021) M. Cody, S. Gavin, B. Koch, M. Kocherovsky, Z. Mazloum, and G. Moschelli, (2021), arXiv:2110.04884 [nucl-th] .
  * Aaboud  _et al._ (2018) M. Aaboud _et al._ (ATLAS), Phys. Rev. C 97, 024904 (2018), arXiv:1708.03559 [hep-ex] .
  * Bilandzic  _et al._ (2011) A. Bilandzic, R. Snellings, and S. Voloshin, Phys. Rev. C 83, 044913 (2011).
  * Jia  _et al._ (2017) J. Jia, M. Zhou, and A. Trzupek, Phys. Rev. C 96, 034906 (2017), arXiv:1701.03830 [nucl-th] .
  * Huo  _et al._ (2018) P. Huo, K. Gajdošová, J. Jia, and Y. Zhou, Phys. Lett. B 777, 201 (2018), arXiv:1710.07567 [nucl-ex] .
  * Aaboud  _et al._ (2019) M. Aaboud _et al._ (ATLAS), Phys. Lett. B 789, 444 (2019), arXiv:1807.02012 [nucl-ex] .
  * Sirunyan  _et al._ (2021) A. M. Sirunyan _et al._ (CMS), Phys. Rev. C 103, 014902 (2021), arXiv:1905.09935 [hep-ex] .
  * Aaboud  _et al._ (2020) M. Aaboud _et al._ (ATLAS), JHEP 01, 051 (2020), arXiv:1904.04808 [nucl-ex] .
  * Aad  _et al._ (2021) G. Aad _et al._ (ATLAS), Phys. Rev. Lett. 126, 122301 (2021), arXiv:2001.04201 [nucl-ex] .
  * Di Francesco  _et al._ (2017) P. Di Francesco, M. Guilbaud, M. Luzum, and J.-Y. Ollitrault, Phys. Rev. C 95, 044911 (2017), arXiv:1612.05634 [nucl-th] .