# RecGURU: Adversarial Learning of Generalized User Representations for Cross-
Domain Recommendation

Chenglin Li  University of AlbertaEdmonton, ABCanada ,  Mingjun Zhao
University of AlbertaEdmonton, ABCanada ,  Huanming Zhang  Platform and
Content Group, TencentShenzhenChina ,  Chenyun Yu  Sun Yat-sen
UniversityShenzhenChina ,  Lei Cheng  Platform and Content Group,
TencentShenzhenChina ,  Guoqiang Shu  Platform and Content Group,
TencentShenzhenChina ,  Beibei Kong  Platform and Content Group,
TencentShenzhenChina and  Di Niu  University of AlbertaEdmonton, ABCanada

(2022)

###### Abstract.

Cross-domain recommendation can help alleviate the data sparsity issue in
traditional sequential recommender systems. In this paper, we propose the
RecGURU algorithm framework to generate a Generalized User Representation
(GUR) incorporating user information across domains in sequential
recommendation, even when there is minimum or no common users in the two
domains. We propose a self-attentive autoencoder to derive latent user
representations, and a domain discriminator, which aims to predict the origin
domain of a generated latent representation. We propose a novel adversarial
learning method to train the two modules to unify user embeddings generated
from different domains into a single global GUR for each user. The learned GUR
captures the overall preferences and characteristics of a user and thus can be
used to augment the behavior data and improve recommendations in any single
domain in which the user is involved. Extensive experiments have been
conducted on two public cross-domain recommendation datasets as well as a
large dataset collected from real-world applications. The results demonstrate
that RecGURU boosts performance and outperforms various state-of-the-art
sequential recommendation and cross-domain recommendation methods. The
collected data will be released to facilitate future research.

Cross-Domain Recommendation, Sequential Recommendation, Learning
Representation, Autoencoder, Adversarial Learning

††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the
Fifteenth ACM International Conference on Web Search and Data Mining; February
21–25, 2022; Tempe, AZ, USA††booktitle: Proceedings of the Fifteenth ACM
International Conference on Web Search and Data Mining (WSDM ’22), February
21–25, 2022, Tempe, AZ, USA††price: 15.00††doi: 10.1145/3488560.3498388††isbn:
978-1-4503-9132-0/22/02††ccs: Information systems Recommender systems††ccs:
Computing methodologies Learning latent representations

##  1\. Introduction

Sequential recommendation has achieved great success in modeling the
preferences and intentions of online users by utilizing both their recent
actions and long-term behavior history. Existing methods often leverage
recurrent neural networks (RNNs) and attention mechanisms to model the
interests of users based on sequential data (Sun et al., 2019; Ying et al.,
2018; Hidasi and Karatzoglou, 2018; Chen et al., 2018). However, most studies
to date on sequential recommendations have been focusing on a single domain,
where data sparsity issues prevail—chances are that most users may only have a
short behavior history in the domain of interest.

Cross-domain recommendation has been proposed to alleviate the data sparsity
issue by leveraging user behavior in multiple domains to help recommendation
in the target domain and has attracted much attention in both academia and
industry. Recent work on cross-domain recommendation focuses on the transfer
learning of user and item information from diverse perspectives. For example,
mapping functions have been proposed to map user representations from one
domain to another, by learning from the behavior of users that appear in both
domains (Man et al., 2017).

Nevertheless, a common limitation of existing cross-domain recommendation
methods is that they perform transfer learning primarily based on data of the
overlapped users, and fail to function well when there are few or even no
overlapped users in two domains (Man et al., 2017). However, in many real-
world applications, there is often not a sufficient number of overlapped
users.

In this paper, we propose RecGURU, which consists of two parts: the
Generalized User Representation Unit (GURU) to obtain a single Generalized
User Representation (GUR) for each user, and the Cross-Domain Sequential
Recommendation (CDSRec) unit to achieve cross-domain collaboration in
sequential recommendation task. Instead of mapping user embeddings from one
domain to another, we propose to generalize a user’s embedding, i.e., its GUR,
to incorporate information from both domains through an adversarial learning
framework. Once a GUR is obtained for each user, we integrate the extracted
GUR into the CDSRec unit using attention mechanisms to boost recommendation
performance. Specifically, we make the following contributions:

First, in the GURU module, an autoencoder is proposed to generate informative
user embeddings in each domain, which are to be unified later into a
generalized embedding, the GUR, with adversarial learning. The autoencoder
consists of a self-attentive encoder with model weights shared across both the
source and target domains to produce latent user embeddings and two decoders
to reconstruct behavior sequences of users in the source and target domains,
respectively. We train the autoencoder through behavior sequence
reconstructions to generate meaningful preliminary embeddings for users with
unsupervised self-learning.

The GURU module further performs adversarial training to unify domain-
dependent user embeddings into a single global (GUR) for each user, which
domain-independent. Specifically, the encoder part of the proposed autoencoder
serves as a generator to produce user embeddings, while a discriminator is
trained to identify the origin domain of a generated embedding for a user
randomly sampled from either the source or the target domain. The encoder and
discriminator are trained alternately with adversarial objectives until the
discriminator can not distinguish which domain a given user embedding comes
from. This is when the user embeddings in the two domains become statistically
indistinguishable and GURs are supposed to be generalized global user
embeddings, incorporating information from both domains. This method does not
rely on common users present in both target and source domains, therefore
eliminating the dependency on overlapped users as required by prior art.

Furthermore, we introduce an effective and stable training procedure for
RecGURU, consisting of three phases. We first pre-train the autoencoder to
substantially reduce the reconstruction loss which boost-starts the subsequent
adversarial learning. In the adversarial learning phase, the reconstruction
loss is further jointly optimized in a multi-task fashion which prevents the
encoder from generating wild representations, stabilizing adversarial
learning. In the meantime, RecGURU can still leverage overlapped users as
prior work does, by introducing an $l_{2}$ penalty in the optimization
procedure to explicitly force each common user to have the same shared
embedding in different domains. Finally, the CDSRec module, which incorporates
the GUR with attention mechanisms, is fine-tuned in the target domain with the
next-item recommendation task to boost the recommendation performance.

Through extensive experiments, we show that RecGURU has achieved improvement
on sequential recommendation, compared to several state-of-the-art single-
domain and cross-domain recommendation methods. Specifically, we outperform
all the baselines on various metrics by a large margin on the Amazon datasets
including ”Sport”, ”Clothing”, ”Movie”, and ”Book”. Additionally, we have
collected a large cross-domain recommendation dataset with two domains, i.e.
“Wesee” and “Tencent Video”, from two real-world applications which provide
video streams to millions of users. Ablation studies on the collected datasets
with various portions of overlapped users are conducted to show the
effectiveness of each proposed sub-module as well as the robustness of our
method. The collected datasets will be made public to facilitate future
research in the field of cross-domain and sequential recommendation.

##  2\. Related work

Sequential Recommendation. Early studies adopt Markov Chains (MCs) to capture
sequential patterns from users’ historical interactions (Rendle et al., 2010;
He et al., 2017a). Recently, researchers have been putting effort to adapt the
Recurrent Neural networks (RNN) and attention mechanism to solve sequential
recommendation problems. GRU4Rec (Hidasi et al., 2015) and its improved
version GRU4Rec+ (Hidasi and Karatzoglou, 2018) leverage Gated Recurrent Unit
(GRU) with BPR loss to model a user’s sequential behaviors. In SAS (Kang and
McAuley, 2018), unidirectional self-attention is adopted here to encode a
user’s historical behavior. Bert4Rec (Sun et al., 2019) follows the idea of
BERT and trained a bidirectional self-attention model. SHAN (Ying et al.,
2018) adopts a hierarchical attention framework where two attention networks
are used to model user’s long- and short-term preferences.

Cross-domain Recommendation. Cross-domain recommendation alleviates data
sparsity issues posed in single domain recommendation via auxiliary
information from other domains. CoNet (Hu et al., 2018) transfers and combines
knowledge across different domains through cross-connections between feed-
forward neural networks. However, these methods only focus on overlapped
users. CATN (Zhao et al., 2020) solves the cold-start problem via aspect
transfer which requires side information of both users and items. (Krishnan et
al., 2020) leverages meta-transfer learning to address the sparsity problem,
but it requires multiple contextual information such as the user’s average
spend.

To transfer knowledge from the source domain to the target domain, EMCDR (Man
et al., 2017) learns a mapping function on overlapped users which maps user
preferences across domains. DCDCSR (Zhu et al., 2018) maps the latent factors
in the target domain to fit the benchmark factors which combines the features
in both the target and source domains. To reduce the dependency on overlapped
users, SSCDR (Kang et al., 2019) adopts a semi-supervised strategy. DDTCDR (Li
and Tuzhilin, 2020) and its improved version DOML(Li and Tuzhilin, 2021) adopt
dual metric learning (DML).

Generative Adversarial Network (GAN) (Goodfellow et al., 2014) is gaining
popularity in cross-domain recommendation(Perera and Zimmermann, 2019; Wang et
al., 2019; Yan et al., 2020; Li et al., 2020; Wang et al., 2019). CnGAN
(Perera and Zimmermann, 2019) introduces the use of the GAN to learn a better
mapping function of user representations from the source domain to the target
domain. Since the discriminator in CnGAN is trained to distinguish between
real and synthetically mapped pairs, where the real mapped pairs only come
from overlapped users. Thus, CnGAN is critically dependent on the quantity and
quality of overlapped users, which usually cannot be guaranteed in reality.
Additionally, the CnGAN tries to stabilize the training of GAN by introducing
more synthetically mapped pairs. However, this makes the input data to the
discriminator unbalanced, thus may fail to train a good mapping function.
target domain. Yan et al. (Yan et al., 2020) adopt adversarial samples in the
training process to improve the generalization ability of the cross-domain
recommender system. ATLRec (Li et al., 2020) transfers shareable features
across domains, however, it focuses on overlapped users which show up in both
the target and source domains.

Cross-domain Sequential Recommendation. $\pi$-Net (Ma et al., 2019) is able to
generate recommendations for both domains through a cross-domain transfer
unit. However, it requires synchronously shared timestamps and can not be
applied to non-overlapped users. Zhuang et al. (Zhuang et al., 2020) transfers
users’ novelty-seeking properties learned from the sequential data in the
source domain to the target domain. Yuan et al. (Yuan et al., 2020) proposed a
framework that is able to fine-tune a large pre-trained user embedding network
to adapt to downstream tasks in the target domain. However, these studies have
different problem settings from our work. And they can only be applied to
overlapped users, but our method can handle both overlapped and non-overlapped
users.

##  3\. Method

Figure 1. Model structure of the proposed RecGURU. User behavior sequences in
both domain $A$ and $B$ are fed into the GURU encoder to generate generalized
latent user representations, i.e. $\bm{h}_{a}^{i}$ and $\bm{h}_{b}^{i}$. Then,
the generated GUR is fed into the CDSRec model for the next-item
recommendation in each individual domain.

A figure shows the overall structure of the General User Representation
module.

###  3.1. Problem Definition

We first formulate the implicit feedback-based sequential recommendation
problem in a single domain. Let $U=\\{u_{1},\cdots,u_{|U|}\\}$ and
$V=\\{v_{1},\cdots,v_{|V|}\\}$ denote the sets of users and items,
respectively, where $|U|$ and $|V|$ are the total number of users and items.
For each user $u_{i},\forall i\in\\{1,\cdots,|U|\\}$, its interactions with
items, in chronological order, are denoted as
$s^{i}=(v_{1}^{i},\cdots,v_{|s^{i}|}^{i})$, where $|s^{i}|$ is the length of
behavior sequence $s^{i}$. Formally speaking, given a user $u_{i}$, sequential
recommendation aims to predict the next item that the user is most likely to
interact with at the next time step $|s^{i}|+1$ based on his or her past
behavior sequence $s^{i}$, which can be formalized as modeling the probability
over all items:

(1) $p(v_{|s^{i}|+1}^{i}=v|s^{i}).$

In cross-domain scenarios, in order to improve the recommendation performance,
user information from other domains is also taken into account. Specifically,
for two domains $A$ and $B$, given an overlapped user who appears in both
domains, $u_{i}\in(U_{A}\cap U_{B})$, cross-domain sequential recommendation
tries to improve the recommendation accuracy of the next item in one domain by
integrating user information from both domains. For example, for next-item
recommendation in domain $A$, it can be formulated as modeling the probability
over all possible candidates given the behavior sequences $s_{a}^{i}$ and
$s_{b}^{i}$ in domains $A$ and $B$:

(2) $p(v_{|s_{A}^{i}|+1}^{i}=v|s_{A}^{i},s_{B}^{i}).$

For non-overlapped users, due to a lack of their behavior in both domains, we
can only exploit the implicit information from the other domain to enhance the
recommendation performance in the target domain. Therefore, Equation (2) is
reformulated as

(3) $p(v_{|s_{A}^{i}|+1}^{i}=v|s_{A}^{i},\textit{info}(S_{B})),$

where $S_{B}=\\{s_{B}^{j}\\},\forall u_{j}\in U_{B}$ denotes the collection of
behavior sequences for all users in domain $B$ and $\textit{info}(\cdot)$
represent a model which is used to extract any kinds of useful information
from $S_{B}$ to assist recommendation in domain $A$.

###  3.2. Overview of Proposed Method

Inspired by previous studies in single domain sequential recommendations which
try to combine the long-term (static) and short-term (dynamic) preferences of
users for next-item recommendation (wan, 2015; Xu et al., 2019; Ying et al.,
2018), we propose a novel RecGURU framework. It consists of two parts: a
generalized user representation unit (GURU) which learns the generalized user
representation across domains and a cross-domain sequential recommendation
(CDSRec) unit which takes the GUR as input to achieve cross-domain
collaboration for the next-item recommendation task. Here, the GUR represents
the static preference contains the user information in both domains.

The overall architecture of the proposed model is shown in Figure 1. The GURU
takes the long-term behavior sequence as input and generates generalized
representations for all users from both domains. Then, the short-term user
behavior together with the extracted general representation is fed into the
CDSRec module to enhance sequential recommendation in each individual domain.
To construct a GUR, we propose an adversarial training framework that involves
an autoencoder for generating informative user representations and a domain
discriminator to unify the representations across different domains.
Furthermore, we apply an additional $l_{2}$ regularizer on overlapped users to
further ensure each overlapped user has similar representations in both
domains.

Note that, motivated by the idea of the cross-lingual language models (Lample
and Conneau, 2019), we adopt a shared encoder across domains to avoid over-
parameterization and achieve further information sharing via parameter-sharing
across domains.

Figure 1 gives a further example of how input sequences are processed to
produce GURs and next-item recommendations. Specifically, the behavior
sequences $(v_{1}^{i},v_{2}^{i},v_{3}^{i},v_{4}^{i},[eos])$ and
$(v_{1}^{j},v_{2}^{j},v_{3}^{j},[eos])$ of user $u_{i}\in U_{A}$ and $u_{j}\in
U_{B}$, where $[eos]$ denotes the end of sequence token, are fed into two
individual embedding layers to produce the embedding vectors
$(\bm{e}_{1}^{i},\bm{e}_{2}^{i},\bm{e}_{3}^{i},\bm{e}_{4}^{i},\bm{e}_{a}^{i})$
and $(\bm{e}_{1}^{j},\bm{e}_{2}^{j},\bm{e}_{3}^{j},\bm{e}_{b}^{j})$ containing
both item information and sequence position information. Then, the GURU
encoder takes the embedded sequences as inputs and generates the latent user
representations $h_{a}^{i}$ and $h_{b}^{j}$ of users $u_{i}$ and $u_{j}$. To
make sure the user representation is meaningful and informative, decoders are
applied to reconstruct the original input sequences, such that, the encoder
and decoder in each domain form an autoencoder framework. Furthermore, in
order to get generalized representations which combines information from both
the source and target domains, the domain discriminator is applied to
$h_{a}^{i}$ and $h_{b}^{j}$ to pull the distributions of user representation
in the two domains close to each other. Hereafter, in each domain, the
sequential recommendation model combines the GUR and the short-term user
behaviors to generate the next-item recommendation, for example, the predicted
item embeddings $\hat{\bm{I}}_{4}^{j}$ at the 4th timestamp of user $u_{j}$ in
domain B.

###  3.3. Generalized User Representations

As shown in Figure 1, in the GURU module, user representations are learned in
each individual domain, and information from the other domain is incorporated
to make the representation “general” through regularizer achieved by domain
discriminator on latent user representation.

User Representations in Single Domain. In each individual domain, we use an
autoencoder to learn the latent user representation that is capable of
reconstructing the original input sequence of the user. As compared to
extracting user representations from the next-item prediction task, the
autoencoder can produce meaningful representations through the reconstruction
task to boost performance. The autoencoder in our framework consists of an
embedding module, an encoder module, and a decoder module.

Formally, to extract a representation for any given user $u_{i}$ with its
behavior sequence $s^{i}=(v_{1}^{i},\cdots,v_{t}^{i},\cdots,v_{|s^{i}|}^{i})$,
we apply the following procedures:

(4)
$\displaystyle\bm{e}_{1}^{i},\cdots,\bm{e}_{t}^{i},\cdots,\bm{e}_{|s^{i}|}^{i},\bm{e}^{i}$
$\displaystyle=Embed(v_{1}^{i},\cdots,v_{t}^{i},\cdots,v_{|s^{i}|}^{i},[eos]),$
$\displaystyle\bm{h}_{1}^{i},\cdots,\bm{h}_{t}^{i},\cdots,\bm{h}_{|s^{i}|}^{i},\bm{h}^{i}$
$\displaystyle=Encoder(\bm{e}_{1}^{i},\cdots,\bm{e}_{t}^{i},\cdots,\bm{e}_{|s^{i}|}^{i},\bm{e}^{i}),$
$\displaystyle v_{1}^{i},\cdots,v_{t}^{i},\cdots,v_{|s^{i}|}^{i}$
$\displaystyle=Decoder(\bm{h^{i}}),$

where $\bm{h}_{t}^{i}$ is the latent representation of item $v_{t}^{i}$ at
position $t$, and $\bm{e}_{t}^{i}$ represents the sum of item embedding and
positional embedding. $\bm{e^{i}}$ and $\bm{h^{i}}$ are the embedding and
latent representation of the $[eos]$ token. We use bold font to indicate
vector variables.

The input sequence of items and the $[eos]$ token are converted into real-
valued vectors through the Embed module, which consists of an item embedding
layer and a positional embedding layer to incorporate both the item
information and the sequential information of behavior sequences. To this end,
we create two trainable embedding matrices
$\bf{I}\subset\mathbb{R}^{(|V|+1)\times d}$ and
$\bf{P}\subset\mathbb{R}^{(N+1)\times d}$ for item and positional embeddings,
respectively, where $d$ represents the number of dimensions in the latent
space and $N+1$ is the maximum length of input sequences including the $[eos]$
token. By summing up the output of item embeddings and positional embeddings
with point-wise summation, we derive the embedding representations for all
items and the $[eos]$ token, denoted as
$(\bm{e}_{1}^{i},\cdots,\bm{e}_{t}^{i},\cdots,\bm{e}_{|s^{i}|}^{i},\bm{e}^{i})$
shown in Equation (4).

After the derivation of embeddings, we adopt the autoencoder to obtain the
fixed-length representations of the behavior sequences of users. Specifically,
we adopt an encoder with a structure similar to _Transformer_ (Vaswani et al.,
2017) which is composed of a stack of identical transformer layers. Each
transformer layer is composed of a multi-head, bidirectional self-attention
layer, and a position-wise fully connected feed-forward layer. Shown in
Equation (4), the Encoder outputs a list of latent vectors
$(\bm{h}_{1}^{i},\cdots,\bm{h}_{|s^{i}|}^{i},\bm{h}^{i})$ for all input items
in the input sequence. Then, the latent vector of the $[eos]$ token
$\bm{h}^{i}$ is treated as the representation of the whole behavior sequence,
i.e. user representation of $u_{i}$, and is further fed into the decoder.

We adopt a decoder that also consists of multiple transformer layers. In
addition to the self-attention and feed-forward sub-layers in each transformer
layer, another multi-head attention layer over the user representation
$\bm{h^{i}}$ is inserted right after the self-attention layer in each decoder
layer. Additionally, the masking technique is applied to ensure that the
predictions depend only on previous behavior sequences. Formally, shown in
Equation (4), our Decoder takes the latent user representation $\bm{h}^{i}$ as
input and reconstructs the original input sequence in an auto-regressive
manner (Vaswani et al., 2017).

Illustrated in Figure 1, different users have different lengths of behavior
sequences, thus, we transform the input sequences of all users into fixed-
length sequences with a length of $N+1$, including the $[eos]$ token.
Specifically, for users with sequences longer than $N$ (except for the $[eos]$
token), we consider the most recent $N$ items, for users with sequences
shorter than $N$ we add $[pad]$ to the left of its original sequence until it
has a total length of $N$.

Following the convention of autoencoder, by optimizing the reconstruction
error between the input samples and the reconstructed samples, our model can
learn the most important attributes of the input behavior sequence. In this
paper, the autoencoder structure is applied to sequence samples, thus we
formulate the reconstruction loss for input sequence $s^{i}$ as follows:

(5) $\displaystyle\mathcal{L}_{rec}(s^{i},\hat{s}^{i})$ $\displaystyle=-\log
p_{\theta}(\hat{s}^{i}|s^{i})$ $\displaystyle=-\sum_{t=1}^{N}\log
p_{\theta}(\hat{v}_{t}^{i}|\hat{v}_{<t}^{i},\bm{h}^{i}),$

where $s^{i}$ and $\hat{s}^{i}$ are the input and reconstruction output of
autoencoder, and $\bm{h}^{i}=Encoder(Embed(s^{i}))$ denotes the latent user
representation of user $u_{i}$ as is shown in Equation (4). Moreover, at any
position $t$, if its original item is the [pad] token, we simply ignore the
reconstruction loss at this position.

Generalizing User Representation Across Domains. We propose to incorporate
information from other domains into the user representation learned in a
single domain to achieve cross-domain collaboration. Compared to the single-
domain sequential recommendation, the representations regularized by the
distributional constraints are more general and contain extra information from
other domains. A commonly used technique for autoencoder to integrate prior
knowledge is to add extra penalty terms onto the reconstruction objective
function:

(6) $\mathcal{L}_{rec}(s^{i},\hat{s}^{i})+\Omega(\bm{h}^{i}),$

where $\Omega(\cdot)$ denotes the penalty function. For examples, sparse
autoencoder (SAE) (Makhzani and Frey, 2013) encourages sparsity of latent
vectors by adding an $l_{1}$ or KL divergence penalty, variational autoencoder
(VAE) (Kingma and Welling, 2013) assume that the latent representation $h^{i}$
follows a Multidimensional Gaussian distribution.

Similarly, to integrate knowledge from both domains, a Kullback–Leibler (KL)
divergence penalty can be added to the reconstruction loss, where the KL
divergence measures the distance between the distributions of the learned user
representations in both domains, leading to the following reconstruction
objectives in both domains:

(7)
$\displaystyle\mathcal{L}_{rec}(s_{A},\hat{s}_{A})+\textit{KL}(\rho^{A}||\rho^{B}),$
$\displaystyle\mathcal{L}_{rec}(s_{B},\hat{s}_{B})+\textit{KL}(\rho^{B}||\rho^{A}),$

where $\rho^{A}$ and $\rho^{B}$ denote the learned distributions of latent
user representations in domain $A$ and domain $B$, respectively. However,
there is no closed-form expression for the distributions of latent user
representations. Furthermore, the latent distributions are characterized by
the parameters of the autoencoder, which is being constantly updated along the
training process.

Instead of directly estimating the ground-truth distributions, which is
extremely difficult, we propose to bypass this challenge by adopting an
adversarial training strategy to implicitly minimize the KL divergence between
the two learned distributions $\rho^{A}$ and $\rho^{B}$. As illustrated in
Figure 1, the encoders generate latent representations
$\bm{h}_{a}^{i}\sim\rho^{a}$ and $\bm{h}_{b}^{j}\sim\rho^{b}$, where the
distributions are characterized by the parameters of encoders in domain $A$
and $B$, respectively. A discriminator is then built for a binary
classification task, where the input is a single latent representation either
from domain $A$ or domain $B$, while the output is the prediction on which
domain the input representation originates from. We adopt the neg log-
likelihood loss as the objective function for the adversarial optimization
process, denoted as:

(8) $\displaystyle\mathcal{L}_{dis}(h^{i})$ $\displaystyle=-y\cdot
log\sigma(f(h^{i}))-(1-y)\cdot log(1-\sigma(f(h^{i}))),$ $\displaystyle y$
$\displaystyle=\mathbb{I}(u^{i}\in U^{a})$

where $\mathbb{I}(\cdot)$ is the indicator function which equals to $1$ when
the condition $u^{i}\in U^{a}$ is true, otherwise, $y=0$. $\sigma(\cdot)$ is
the sigmoid function and the logits value $f(h^{i})$ is calculated by the
domain discriminator $f(\cdot)$.

Following the conventional optimization scheme of GAN, we alternately update
the domain discriminator and the encoder until the model reaches the point
where the discriminator is unable to tell whether a given user representation
is from domain $A$ or $B$. At this point, the distributions of user
representations $\rho^{A}$ and $\rho^{B}$ are supposed to be close to each
other, i.e., which contributes to reducing the KL divergence
$KL(\rho^{A}||\rho^{B})$ between the two distributions. Therefore, we have
achieved information sharing through the distributional constraint on latent
user representations without relying on the information of overlapped users.

In addition to implicitly unifying the user representations with all users, we
also propose an $l_{2}$ penalty to enforce the explicit representation sharing
over domains for each overlapped user. That is, the representations of the
same user should be the same or close across domains. Illustrated in Figure 1,
for an overlapped user, i.e. $u_{i}=u_{j}$, we add an $l_{2}$ regularizer on
its latent representations in domain $A$ and $B$:

(9) $\mathcal{L}_{l_{2}}=||\bm{h}_{a}^{i}-\bm{h}_{b}^{j}||_{2},i=j.$

By adding the $l_{2}$ regularizer to the loss function for overlapped users,
representations of the same user in different domains are pushed to be equal
or close to each other, exploiting the information of overlapped users in an
explicit way.

Input: User sets $U^{A}$, $U^{B}$, item sets $V^{A}$, $V^{B}$;

Behavior sets: $S_{A}=\\{s_{A}^{i}\\},\forall u_{i}\in U^{A}$,
$S_{B}=\\{s_{B}^{j}\\},\forall u_{j}\in U^{B}$;

Overlapped users $U^{o}=U^{A}\cap U^{B}$.

Hyper-parameter: CRITIC_ITERS.

Initialization:

Initiate the item and positional embedding matrices:
$\bf{I}^{A}\subset\mathbb{R}^{(|V^{A}|+1)\times d}$,
$\bf{P}^{A}\subset\mathbb{R}^{(N+1)\times d}$ and
$\bf{I}^{B}\subset\mathbb{R}^{(|V^{B}|+1)\times d}$,
$\bf{P}^{A}\subset\mathbb{R}^{(N+1)\times d}$.

Initiate the GURU, CDSRec and the domain Discriminator.

Pre-training Phase:

Train autoencoders according to (5) on $S_{A}$ and $S_{B}$.

Multi-task Adversarial Training Phase:

while  _not converged_ do

for  _ $i\leftarrow 0$ to CRITIC_ITERS _ do

Sample two batches of users $\bar{U}^{A}\sim U^{A}$, $\bar{U}^{B}\sim U^{B}$.

Optimize the discriminator loss (8).

Sample two batches of users $\bar{U}^{A}\sim U^{A}$, $\bar{U}^{B}\sim U^{B}$
and one batch of overlapped users $\bar{U}^{o}\sim U^{o}$.

Fix the parameters of the domain Discriminator.

Train the model according to the loss defined in (13).

Fine-tune:

Fine-tune the _CDSRec_ model according to the BPR loss defined in (12) in each
domain.

Algorithm 1 RecGURU Training Algorithm

###  3.4. Cross-domain Sequential Recommendation

The generalized user representation extracted by the proposed GURU module
represents the overall preference of users in different domains, which is
beneficial to the recommendation task in a specific domain.

For the next-item recommendation task in each domain, a sequential recommender
is built on the derived GURs and it is composed of multiple unidirectional
attention layers and feed-forward layers. Specifically, the GURs are passed
into the model through multi-head attention mechanisms.

Formally, for a given user $u_{i}$ with sequence
$s^{i}=(v_{1}^{i},\cdots,v_{|s^{i}|}^{i})$, the sequential recommender takes
its generalized user representation $\bm{h}^{i}$ and short-term behaviors
$(v_{|s^{i}|-m}^{i},\cdots,v_{|s^{i}|}^{i})$ as the input and outputs the
current preference vector of the user $\bm{q}^{i,|s^{i}|}$ at time step
$|s^{i}|$ in the latent space, given by

(10) $\displaystyle\bm{h}^{i}$
$\displaystyle=Encoder(Embed(v_{1}^{i},\cdots,v_{|s^{i}|}^{i},[eos])),$
$\displaystyle\bm{q}^{i,|s^{i}|}$
$\displaystyle=CDSRec(\bm{h}^{i},(v_{|s^{i}|-m}^{i},\cdots,v_{|s^{i}|}^{i})),$

where $m$ denotes the length of the short-term behavior sequence and the GUR
is extracted from the long-term behavior sequence
$(v_{1}^{i},\cdots,v_{|s^{i}|}^{i})$ of user $u_{i}$. _CDSRec_ denotes the
cross-domain sequential recommender.

The preference scores of the user to all the candidate items are then computed
as the inner product between its current preference $\bm{q}^{i,|s^{i}|}$ and
the item embeddings of all candidates, denoted as

(11) $r^{i,|s^{i}|,v}=\bm{q}^{i,|s^{i}|}\bm{I}_{v},\forall v\in V^{c},\forall
u_{i}\in U,$

where $\bm{I}_{v}$ is the item embedding of $v$ and the candidate set
$V^{c}\subset V$ is a subset of the the entire item set $V$. Candidate items
are then ranked and recommended according to the calculated preference scores.

We adopt a Bayesian Personalized Ranking (BPR) loss (Rendle et al., 2012) to
train the recommendation model. For a given user $u_{i}$ from domain $A$, we
calculate the loss of an item recommendation at time step $t$ as

(12)
$\mathcal{L}_{bpr}^{t}=-\log\sigma(\bm{q}_{A}^{i,t}\bm{I}_{v})-\log(1-\sigma(\frac{1}{|N_{s}|}\sum_{v^{\prime}\in
N_{s}}{}\bm{q}_{A}^{i,t}\bm{I}_{v^{\prime}})),$

where $v$ and $\bm{I}_{v}$ are the target item and its corresponding item
embedding, $N_{s}$ is the set of negative item samples and $\sigma(\cdot)$
represents the sigmoid function.

###  3.5. Training Strategy

We propose a three-phase training algorithm shown in Algorithm 1 to optimize
the proposed model.

In the first phase, we pre-train the autoencoders in each domain individually
with the reconstruction task. Through the pre-training process, the
reconstruction loss is largely reduced, producing a boost-start for the
following adversarial training. In the second phase, following the training
process of GAN in (Gulrajani et al., 2017), at each iteration, we first
optimize the discriminator loss $\mathcal{L}_{dis}$ for CRITIC_ITERS steps
(which equals to 5 in our implementation). Then, the reconstruction task,
negative discriminator loss, and $l_{2}$ loss on overlapped users are jointly
optimized in a multi-task fashion by minimizing the loss:

(13) $\mathcal{L}=\mathcal{L}_{rec}-\mathcal{L}_{dis}+\mathcal{L}_{l_{2}}.$

With the reconstruction task, we prevent encoders from generating wild
representation stabilizing the adversarial learning. Following the common
practice in GAN training (Gulrajani et al., 2017), we also adopt the gradient
penalty term in the critic optimization step. Finally, we fine-tune the
_CDSRec_ model in each individual domain with the next-item recommendation
task.

##  4\. Experiments

In this section, we conduct extensive experiments on two public and one
collected cross-domain sequential recommendation scenarios. Comparison with
the state-of-the-art single domain and cross-domain baselines shows the
effectiveness of our proposed method. Furthermore, ablation tests demonstrate
the impact of each proposed sub-modules on the recommendation result and the
robustness of our model under scenarios of a different portion of overlapped
users.

###  4.1. Dataset and Experiment Setup

Table 1. Statistics for the three cross-domain scenarios.  Domain | #Users | #Items | Avg. Seqlen. | #Overlap.
---|---|---|---|---
Sport | 9,024 | 11,835 | 6.62 | 1,062
Cloth | 46,810 | 42,139 | 7.51
Movie | 4,261 | 5,536 | 8.307 | 584
Book | 42,940 | 51,366 | 9.490
Wesee | 1,952,403 | 335,648 | 18.196 | 1,692,893
Tencent Video | 2,183,927 | 1,455,595 | 28.53

Four publicly available Amazon datasets (Ni et al., 2019) and two collected
datasets are used to form three cross-domain sequential recommendation
datasets: “Sports-Clothing”, “Movie-Book”, and the collected dataset. On
Amazon datasets, only recent positive reviews, posted after October 1, 2017,
with a rating score higher than 2, are selected. We collect data from two
popular video applications, “Wesee” and “Tencent Video”, both with over
billions of daily active users. Watching histories for three consecutive
days,from June 26, 2020, to June 28, 2020, of these two applications are
collected in this dataset. The detailed breakdown of the three cross-domain
sequential recommendation datasets is shown in Table 1. Note that, the
collected dataset mostly consists of overlapped users, thus, we can manually
adjust the portion of overlapped users in a wide range to test the robustness
of our method.

Following common practices in sequential recommendation (Sun et al., 2019),
for a given user the second to last item in the behavior sequence is selected
as the validation item, and the last item is used for testing, while the
remaining items are used for training.

Table 2. Comparison between the proposed method with single-domain and cross-domain baselines on “Sport-Cloth” and “Movie-Book” scenarios. “AutoRec” and “AutoEM” are two variants of our proposed method. All values are in percentage. Datasets | Metric | Single-domain algorithms | Cross-domain algorithms
---|---|---|---
POP | BPRMF | SAS | Bert4Rec | AutoRec | CMF | MFEM | CnGAN | DOML | AutoEM | RecGURU
Sport | HR@5 | 3.63 | 16.17 | 17.51 | 18.04 | 18.82 | 17.04 | 15.87 | 15.88 | 15.26 | 17.63 | 20.78
HR@10 | 5.70 | 19.62 | 22.09 | 22.46 | 23.37 | 20.55 | 19.44 | 19.41 | 20.29 | 22.57 | 24.88
NDCG@5 | 3.03 | 12.43 | 14.47 | 14.57 | 14.93 | 13.37 | 12.68 | 12.89 | 11.66 | 14.06 | 16.56
NDCG@10 | 3.70 | 13.73 | 15.94 | 16.01 | 16.36 | 14.45 | 13.77 | 14.05 | 13.17 | 15.61 | 17.87
Cloth | HR@5 | 1.48 | 21.03 | 23.71 | 22.68 | 23.89 | 21.53 | 20.56 | 20.10 | 22.70 | 24.27 | 25.68
HR@10 | 2.10 | 23.98 | 28.40 | 26.57 | 27.67 | 24.33 | 23.41 | 23.29 | 28.36 | 28.71 | 29.16
NDCG@5 | 0.08 | 17.89 | 19.81 | 19.95 | 20.51 | 18.44 | 17.51 | 16.72 | 17.47 | 20.44 | 22.44
NDCG@10 | 0.96 | 18.86 | 21.29 | 21.20 | 21.70 | 19.30 | 18.40 | 17.70 | 19.29 | 21.83 | 23.52
Movie | HR@5 | 2.07 | 12.80 | 12.18 | 13.78 | 14.21 | 12.34 | 14.70 | 14.03 | 14.41 | 12.93 | 18.97
HR@10 | 5.37 | 17.28 | 17.20 | 19.39 | 19.51 | 15.46 | 19.52 | 18.49 | 19.78 | 18.94 | 24.24
NDCG@5 | 1.26 | 9.36 | 8.76 | 9.79 | 10.15 | 8.36 | 10.80 | 10.18 | 9.75 | 8.67 | 14.59
NDCG@10 | 2.30 | 10.85 | 10.38 | 11.55 | 11.83 | 8.75 | 12.31 | 11.57 | 11.94 | 10.61 | 16.17
Book | HR@5 | 2.69 | 22.37 | 27.15 | 27.86 | 28.10 | 22.70 | 23.17 | 23.24 | 20.67 | 28.92 | 28.15
HR@10 | 5.31 | 31.28 | 36.85 | 37.58 | 38.20 | 31.56 | 32.33 | 32.33 | 30.86 | 38.63 | 38.35
NDCG@5 | 1.79 | 16.32 | 19.50 | 19.93 | 20.09 | 16.38 | 16.63 | 16.63 | 14.18 | 20.89 | 20.07
NDCG@10 | 2.62 | 19.09 | 22.60 | 23.03 | 23.33 | 19.19 | 19.36 | 19.36 | 17.14 | 23.99 | 23.06

Hit Ratio (HR) and Normalized Discounted Cumulative Gain (NDCG) (Järvelin and
Kekäläinen, 2002) are adopted to evaluate the performance of all methods. We
follow the strategy used in (He et al., 2017b) to reduce the heavy computation
cost. Specifically, for users from public datasets, we sample 200 negative
items in the item list with respect to their frequencies, which together with
the ground-truth item, form the candidates for recommendation. On collected
datasets, the size of the candidate set becomes 20,000. HR and NDCG with
$k=5,10,20$ are reported. The following single-domain and cross-domain
baseline algorithms are evaluated.

  * •

POP: All items are recommended according to their popularities.

  * •

BPRMF (Rendle et al., 2012): It optimizes the matrix factorization with the
BPR loss.

  * •

SAS (Kang and McAuley, 2018): It adopts unidirectional self-attention to model
user behaviors.

  * •

Bert4Rec (Sun et al., 2019): It incorporates the idea of Bert (Devlin et al.,
2018) to the next item recommendation task.

  * •

AutoRec: It is the single-domain version of proposed model that adopts
autoencoder to generate static user representation which is used for the
sequential recommendation.

  * •

CMF (Singh and Gordon, 2008): It simultaneously factors interaction matrices
in both target and source domains.

  * •

MFEM: It learns a mapping function on overlapped users. Thus, for non-
overlapped users, we can get their embeddings in the source domain through the
trained mapping function as well as the cross-domain recommendation through
Equation (2). Here, the user embeddings are learned with the BPRMF model.

  * •

CnGAN (Perera and Zimmermann, 2019): It adopts adversarial learning to learn a
better mapping function that maps user embeddings from the source domain to
the target domain or vice versa.

  * •

DOML (Li and Tuzhilin, 2021): It adopts dual metrics learning in cross-domain
recommendation when there are few overlapped users.

  * •

AutoEM: As a variation of proposed method, it also learns a mapping function
on overlapped users to get the embeddings of non-overlapped users. However,
here, the item and user embeddings are learned by the proposed AutoRec model.

###  4.2. Implementation Details

We implement the models using PyTorch with python 3.6 and train our framework
on Tesla P40 GPUs with a memory size of 22.38 GiB and a 1.53 GHz memory clock
rate.

On Amazon datasets, 3 transformer layers are adopted, whereas 6 transformer
layers are used on the collected datasets in the encoder and decoder module.
We use 2 attention heads for all the attention layers throughout the model.
And the dimension of all feed-forward layers is set to 512. For adversarial
training, we build the domain discriminator with four fully connected layers
with a hidden size of 128. Furthermore, we adopt the improved W-GAN (Gulrajani
et al., 2017) framework to alternately optimize the domain discriminator and
the GURU model through the discriminator loss $\mathcal{L}_{dis}$. For
simplification, we adopt the same length of sequence for both autoencoder and
recommendation tasks, which is 100 on all datasets. The number of
dimensionality of user embedding is 64 on both public and collected datasets.
Multiple Adam optimizers (Kingma and Ba, 2014) are used to update different
modules of the proposed RecGURU framework.

For all the transfer learning-based cross-domain baselines, we first
concatenate the user embeddings from the source and target domains. Then, a
fully connected layer is applied to get the cross-domain user embedding which
is further used for next item recommendation. More details are given in the
supplementary.

###  4.3. Experimental Results and Analysis

Table 3. Ablation studies on customized collected datasets with the portion of overlapped users ranging from 10% to 75%. All values are in percentage. Variants methods | SeqRec | +Auto | +GURU
---|---|---|---
Dataset | Metric | 10% | 30% | 50% | 75% | 10% | 30% | 50% | 75% | 10% | 30% | 50% | 75%
Wesee | HR@5 | 14.58 | 15.19 | 15.19 | 15.59 | 15.41 | 15.49 | 15.52 | 16.12 | 16.13 | 17.11 | 17.15 | 16.59
HR@10 | 21.72 | 22.47 | 22.39 | 23.08 | 22.71 | 22.84 | 23.08 | 23.9 | 23.61 | 24.87 | 24.87 | 24.34
NDCG@5 | 9.67 | 10.1 | 10.05 | 10.31 | 10.36 | 10.55 | 10.37 | 10.89 | 10.91 | 11.64 | 11.66 | 11.31
NDCG@10 | 11.94 | 12.43 | 12.34 | 12.68 | 12.71 | 12.91 | 12.8 | 13.39 | 13.34 | 14.14 | 14.11 | 13.79
Tencent Video | HR@5 | 25.33 | 25.95 | 27.14 | 26.71 | 28.48 | 29.21 | 31.99 | 31.60 | 28.26 | 29.14 | 32.18 | 31.23
HR@10 | 32.56 | 33.88 | 35.89 | 34.92 | 35.8 | 36.41 | 39.76 | 39.76 | 36.11 | 37.12 | 38.49 | 39.14
NDCG@5 | 18.62 | 18.66 | 19.09 | 18.91 | 21.49 | 21.7 | 23.91 | 24.33 | 20.78 | 21.41 | 23.12 | 23.39
NDCG@10 | 20.95 | 21.16 | 21.91 | 21.55 | 23.87 | 24.01 | 26.4 | 26.78 | 23.32 | 23.98 | 25.51 | 25.95

Training Loss.  Figure 2 shows the training losses on the “Movie-Book”
dataset. Specifically, the Wasserstein distance and the critic loss, i.e.
discriminator loss, are given in Figure 2(a). Both processes converge around a
thousand adversarial iterations. In the beginning, the discriminator is good
at distinguishing representations from different domains with an increasing
Wasserstein distance. After more adversarial training iterations, the
Wasserstein distance is reduced and converged which in line with the training
process of standard GANs (Gulrajani et al., 2017). The reconstruction tasks in
both target and source domains converge after 400 iterations as is shown in
Figure 2(b). The recommendation task in the target domain converges after only
100 steps of fine-tune iterations.

(a) losses of the adversarial training

(b) Reconstruction and BPR losses

Figure 2. Training losses on the “Movie-Book” dataset.

loss

Main Results. Table 2 summarizes the performances of all baselines and the
proposed methods on the two public cross-domain scenarios. AutoRec, the single
domain version of our solution, outperforms SAS and Bert4rec on single domain
sequential recommendation tasks on most of the public datasets. The success of
AutoRec can be attributed to the adoption of user representations extracted
from the history user behavior through an autoencoder.

The proposed RecGURU outperforms all other baselines with an improvement on
HR@10 and NDCG@10 of $6.5\%$, $9.2\%$, and $22.5\%$, $31.3\%$ on the “Sports”
and “Movie” datasets, respectively, compared to the best given by all the
other baselines. We can also outperform the state-of-the-art single-domain and
cross-domain methods on “Cloth” and “Book” datasets in most cases but with a
relatively smaller margin compared to the results reported on the “Sport” and
“Movie” datasets. This is reasonable, as is shown in Table 1, the “Sport” and
“Movie” datasets are much smaller and with more sparsity than the “Cloth” and
“Book” datasets. Obviously, when the data in the source domain is highly
sparse, we can only get a limited amount of information from the source domain
to help with recommendations in the target domain. Therefore, AutoRec, AutoEM,
and RecGURU achieve close performance on the “Book” dataset. However, overall
RecGURU outperforms other baselines in most cases. The superiority of RecGURU
over AutoRec and AutoEM can be attributed to the generalization of user
representations achieved through the adversarial training of the domain
discriminator and the autoencoder. Furthermore, compared with the improvements
RecGURU has achieved over single domain sequential recommendation baselines,
cross-domain methods such as MFEM, CnGAN, and DOML achieved limited
improvements over BPRMF. This can be attributed to the fact that these
baselines either rely on overlapped users (MFEM, CnGAN) or need side
information for more general user and item embeddings (DOML).

###  4.4. Ablation Study

Variants of the proposed method are evaluated on the collected datasets for
ablation studies to show the impact of each proposed sub-modules. We
incrementally accommodate different modules into the single-domain sequential
recommendation model SeqRec, until we incorporate all the proposed sub-modules
and features. Specifically, the following models are evaluated:

  * •

SeqRec: Sequential recommendation model without autoencoder.

  * •

+Auto: The AutoRec model introduced in Sec. 4.1.

  * •

+GURU: The proposed RecGURU model.

Furthermore, to test the robustness of the proposed method under a variety of
overlapping rates, we manually change the number of overlapped users to form
four new datasets with an overlapping rate of 10%, 30%, 50%, and 75%.

Table 3 summarizes the results of ablation tests of all the introduced
variants on all datasets with various overlapping rates. On the “Wesee”
dataset, each time we add a new sub-module or feature incrementally on top of
the previous model, we can observe improvement in the overall recommendation
performance, which illustrates the effectiveness of autoencoder and GURU
modules. Moreover, this phenomenon appears on all the four datasets with
overlapping ranging from 10% to 75% which shows the robustness of our method
to user overlapping rate. Similar to the “Book” dataset, on the “Tencent
Video” dataset, the single-domain version of our proposed method, AutoRec, is
slightly better than its cross-domain version, which is also due to the
sparsity issue in the source domain, i.e. “Wesee”, as we have explained
before.

##  5\. Conclusion

In this paper, we propose RecGURU, a novel cross-domain sequential
recommendation framework based on Generalized User Representations (GURs).
Different from previous work which aims to transfer knowledge across domains,
in the RecGURU system, we propose the GURU module that is capable of
extracting a generalized user representation unified over different domains
via adversarial learning. Specifically, an autoencoder is adopted to generate
user representations in each individual domain. Cross-domain generalization of
user representations is achieved by adversarially training a discriminator and
the encoder until the domain-dependent embeddings are statistically
indistinguishable among different domains. We further propose various schemes
to stabilize and boost the learning effectiveness of RecGURU. Experimental
results on both publicly available datasets and collected datasets show the
effectiveness of the proposed method.

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##  6\. SUPPLEMENTARY

###  6.1. Model Implementation details

As mentioned in the paper, the proposed RecGURU is composed of two parts: the
generalized user representation unit (GURU) and the cross-domain sequential
recommendation (CDSRec) module. Here we introduce the detailed implementation
of these two modules.

####  6.1.1. GURU

.The generalized user representation unit (GURU) consists of one shared
encoder and two decoders which form two Autoencoder frameworks in both the
target and source domains. Furthermore, a domain discriminator is adopted to
unify user representations from both domains in an adversarial learning
scheme.

Encoder: As we have mentioned in the paper, the encoder is composed of several
transformer layers where each transformer layer consists of one multi-headed
self-attention layer and a fully connected feedforward layer. Specific hyper-
parameters of the encoder are:

  * •

Number of layers: The number of transformer layers used to build the encoder.
We have grid searched the number of transform layers in the range of
$[2,3,4,5,6]$ on all public datasets and finally decided to adopt $3$ layers
on all public datasets, i.e. the “Movie”, “Book”, “Sport” and “Clothing”
datasets. On the collected datasets, i.e. “Wesee” and “Tencent Video”, 6
layers of transformer layers are adopted to build the encoder.

  * •

Number of heads: Number of heads for all multi-head attention layers within
each transformer layer, which equals 2 in our model.

  * •

Dimension of embedding: The dimensionality of the latent space for user
representation and item embedding, which is set to be 64. To have a fair
comparison between different methods, the number of dimensions of the item and
user embeddings are all set to be 64 for all experiments that appeared in this
paper.

  * •

Dimension of feedforward layer: We adopt feed-forward layers of size 512 for
all transformer layers.

Decoder: we use the same hyperparameters as the encoder to build decoders in
each domain (datasets), the only difference is that for each transformer layer
in the decoder, we insert another attention layer to incorporate latent user
representation generated by the encoder.

Domain Discriminator: The domain discriminator, as a binary classification
model, takes a user representation, from either the source domain or the
target domain, as input and predicts the origin domain that the user comes
from. In this paper, the domain discriminator is composed of several fully
connected neural layers. Specifically, the hyper-parameters of the domain
discriminator are defined as:

  * •

Number of layers: Four fully connected layers are used in our domain
discriminator.

  * •

Middle dimension: The dimensionality of the middle layers in our domain
discriminator, which is set to be double the size of the latent vector, i.e.
128.

####  6.1.2. CDSRec

We adopt the same model structure as the decoder for the cross-domain
sequential recommendation. However, the input of the CDSRec is the short-time
activity of a user which aims to model users’ dynamic preference in the target
domain. The long-term interest and cross information are provided by the GUR
(generalized user representation) given by the GURU module. Through attention
layers, we are able to combine the GUR with the user’s short-term preference
to provide a more precise recommendation in the target domain.

####  6.1.3. Model Training

The training procedure of the proposed RecGURU has been introduced in the
paper (Sec. 3.5), here we provide some important hyper-parameters and related
to model training for reproducibility. Specifically, the following aspects are
discussed here:

Optimizer: The ADAM optimizer is used in our work to train all the modules. We
adopt different ADAM optimizers with different learning rates at each training
phase of the RecGURU Training Algorithm. Specifically, at the pre-training
phase, we use a wrapped ADAM optimizer with a starting learning rate of $1.0$,
$\beta_{1}=0.9,\beta_{2}=0.98$ and $\epsilon=1\times 10^{-8}$, the learning
rate of the wrapped optimizer is adjusted from epoch to epoch. In the
adversarial optimization procedure, two optimizers are adopted for both the
critic step, which optimize the domain discriminator loss, and the generator
step, which optimize the negative discriminator loss along with the
reconstruction and $l_{2}$ constraint loss on overlapped users. We refer these
two optimizers as opt_dis and opt_gen, respectively. For these two optimizers
we adopt a learning rate of $1\times 10^{-4}$, $\beta_{1}=0.5$,
$\beta_{2}=0.9$ and $\epsilon=1\times 10^{-8}$. Finally, in the fine-tune
stage, another ADAM optimizer with an learning of $1\times 10^{-3}$,
$\beta_{1}=0.9$, $\beta_{2}=0.9$ and $\epsilon=1\times 10^{-8}$ is adopted.

Sampled Soft-max: As defined in the paper, the reconstruction loss of the
autoencoder is given by:

(14) $\displaystyle\mathcal{L}_{rec}(s^{i},\hat{s}^{i})$ $\displaystyle=-\log
p_{\theta}(\hat{s}^{i}|s^{i})$ $\displaystyle=-\sum_{t=1}^{N}\log
p_{\theta}(\hat{v}_{t}^{i}|\hat{v}_{<t}^{i},\bm{h}^{i}).$

Following previous studies, such as BERT, we use cross-entropy to model the
probability distribution $p_{\theta}$ over all possible outputs (the entire
item set) at each position of the reconstructed sequence. However, this may
cause memory error when the number of items is huge, for example, we have
hundreds of thousands of items in our collected datasets. To solve this
problem, we adopted the sampled soft-max to avoid memory issues. Specifically,
we first randomly select neg_softmax negative samples and we perform softmax
and calculate the probability distribution $p_{\theta}$ over the sampled
negative samples and the target sample (the ground truth next-item). Finally,
cross-entropy is calculated over the sampled items and the ground-truth item.
This way, we can reduce the calculation burden and avoid memory error on the
dataset with a large number of items.The number of chosen negative samples is
set to be 20 on all the public datasets.On the collected dataset, it is set to
be 10.

Gradient penalty: We also adopted the gradient penalty to further stabilize
the training of the adversarial networks. The number of critic steps in each
adversarial iteration CRITIC_ITERS is set to be 5.

###  6.2. Implementations of Baselines

Most of the single domain baseline methods have open-sourced their
implementation. For a fair comparison, for any shared hyper-parameters, for
example, the dimensionality of the item and user embedding, the number of
transformer layers for the SAS and Bert4Rec model, we set them to be the same
as what we used in our model (please refer the previous section). For any
other model-specific hyper-parameters, we use the default value provided in
their opened source code. In summary, we have referred to the following open-
sourced projects for all the baselines:

  * •

BPRMF: https://github.com/guoyang9/BPR-pytorch

  * •

SAS: https://github.com/kang205/SASRec

  * •

CMF: Minor revision from the BPRMF code.

  * •

DOML: https://github.com/lpworld/DOML

For those baselines that haven’t open sourced their code, we either
implemented it by ourselves, for example, MFEM is the combination of BPRMF and
a mapping function of user embeddings from the source domain to the target
domain, or request the authors for their code and adapt it to our problem, for
example, we have implemented our version of CnGAN based on the requested code
from the authors of CnGAN. The implementations of baselines are all included
in our code package111https://github.com/Chain123/RecGURU.

Note that, the origin DOML is designed for cross-domain rating prediction,
however, in this paper, we focus on the implicit recommendation. Therefore, we
made a minor modification on the origin code and the MSE with the BPR-loss.

###  6.3. Dataset and Pre-processing

####  6.3.1. The Amazon datasets

We applied the following pre-processing steps on all four Amazon datasets.
First, we selected all the ratings with a score larger than 2, thus, all the
interactions in the selected dataset are supposed to be positive feedback. We
further ignore the review scores for the implicit feedback-based
recommendations. Second, we get the 5-core dataset based on the selected
interactions. Third, we re-code all the items from 1 to the total number of
items in each dataset. Finally, we extracted the user IDs for all the
overlapped users in the two cross-domain scenarios, i.e. “Sport-Cloth” and
“Movie-Book”.

####  6.3.2. The Collected dataset

222https://drive.google.com/file/d/1NbP48emGPr80nL49oeDtPDR3R8YEfn4J/view?usp=sharing

To facilitate the recommendation in video streaming and short videos, we
collect two sequential video recommendation datasets from two real-world
applications, “Wesee” and “Tencent Video”, which provide all kinds of short
videos and dramas to hundreds of billions of users.“Wesee” is a Chinese video
streaming website providing popular movies and all kinds of TV shows as well
as short videos to over one hundred million users. On the other hand, “Tencent
Video” is a short video sharing platform with over ten million daily active
users.

Note that, in the real scenario, there are only a few portions of overlapped
users between these two applications, however, here we chose most of the
overlapped users due to: first, there are still hundreds of thousands of
overlapped users, second, with high overlapping rate, we are able to adjust
the overlapping rate manually for stress tests.

To form a cross-domain sequential recommendation scenario, the two datasets
are collected within the same time period so that user preferences are
consistent during dataset collection, and thus can be transferred between
domains for cross-domain collaboration. For each application, interaction
information such as action, item ID, and time-stamp is collected for all
users.We consider interactions with “click”, “finish” (which means finish
watching the whole video) as positive interactions between users and video
items, all other interactions are ignored. For a given user, its behavior in
three consecutive days is collected and stored as three lists: action list,
item ID list, and time-stamp list. Then we remove entries with bad user IDs
and item IDs, such as empty or garbled codes. Furthermore, to avoid duplicated
redundancies, if an item appears in a raw in the behavior sequence for the
same user, we drop all the redundant repeats and only keep one copy. Finally,
users with behavior sequences of length larger than 5 are used in our paper.
Detailed statistics of these two datasets are given in our submission.

Table 4. Dataset statistics of the four collected cross-domain scenarios with the portion of overlapped users ranging from 10% to 75%.  Scenario | Dataset | #User | #Item | Avg.Seqlen | #Overlap
---|---|---|---|---|---
10% | Wesee | 1203194 | 288155 | 16.16 | 194756
Tencent Video | 1434999 | 1311440 | 26.46
30% | Wesee | 1398645 | 303942 | 16.18 | 585580
Tencent Video | 1630372 | 1356831 | 26.49
50% | Wesee | 1594256 | 316338 | 16.19 | 976556
Tencent Video | 1825737 | 1397521 | 26.51
75% | Wesee | 1838276 | 329679 | 16.19 | 1464731
Tencent Video | 2069892 | 1438803 | 26.53

Datasets with different portions of overlapped users. To test the robustness
of our method under different portions of user overlapping scenarios, we form
four new datasets from the originally collected dataset by randomly sampling
the overlapping users. Table 4 provides an overview of the four manufactured
datasets.?As we did not include users real ID to protect their privacy,For
each overlapped user we generate a random number to decide whether to keep it
as an overlapped user or put it to a specific domain, i.e. “Wesee” or “Tencent
Video”. If an overlapped user is put to a specific domain as a “non-
overlapped” user, its behavior sequence in another domain is removed.
Therefore, in the different manufactured datasets, we would have a slightly
different number of items.


[a]Thomas Spriggs

# Thermal interquark potentials for bottomonium using NRQCD from the HAL QCD
method

Chris Allton  Timothy Burns  Seyong Kim

###### Abstract

We report our preliminary progress in the calculation of the interquark
potential of bottomonium at non-zero temperature using the HAL QCD method. We
use NRQCD correlation functions of non-local mesonic $S$-wave states to obtain
the central potential as a function of temperature. These results have been
obtained using our anisotropic 2+1 flavour "Generation 2" FASTSUM ensembles.

##  1 Introduction

The interquark potential of quarkonia is one of the first quantities studied
in the quest for a deeper understanding of the nature of the strong
interaction. Pioneering studies include [1] where the Cornell potential was
used to calculate the spectrum of charmonium states using a Quantum Mechanical
formalism. In thermal QCD, the temperature dependence of the interquark
potential results in quarkonium states melting at different temperatures [2].
These considerations strongly motivate a study of the thermal behaviour of the
quarkonia interquark potential.

Heavy quarks interacting via QCD can be approximated using the non-
relativistic approach, NRQCD, which allows a significant simplification. NRQCD
calculations of bottomonia are typically accurate at the few percent level. In
this work we use NRQCD to determine the interquark potential in bottomonia
using the HAL QCD approach. Correlation functions of bottomonia operators are
studied where the quark and antiquark are non-local, and this allows a proxy
for the wavefunction to be calculated. Using this wavefunction in the
Schrödinger equation leads to the interquark potential. We find indications of
the weakening of the potential as the temperature increases, as expected. This
work extends previous studies of the interquark potential by the FASTSUM
Collaboration in the charmonium system [3, 4].

##  2 NRQCD correlation functions and lattice setup

NRQCD is an effective theory with a power counting in the heavy quark
velocity, $v$. In this theory, the heavy quark and antiquark fields decouple
and so virtual heavy quark-antiquark loops cannot form. The NRQCD quark
propagator is calculated via an initial value problem, rather than via a
matrix inversion as is the case for relativistic quarks. NRQCD is particularly
amenable for lattice simulations because mesonic correlation functions do not
have “backward movers” which complicate the study of QCD mesons.

Our NRQCD formulation incorporates both ${\cal O}(v^{4})$ and the leading
spin-dependent corrections. The $b$-quark mass is tuned by setting the
“kinetic” mass (i.e. from the dispersion relation) of the spin-averaged $1S$
states to its experimental value. Full details of our NRQCD setup appear in
[5].

All our results were obtained using our FASTSUM $N_{f}=2+1$ flavour
“Generation 2” ensembles which have the parameters listed in Table 1.

$N_{\tau}$ | 16 | 20 | 24 | 28 | 32 | 36 | 40
---|---|---|---|---|---|---|---
T [MeV] | 352 | 281 | 235 | 201 | 176 | 156 | 141
$N_{\text{configurations}}$ | 1050 | 950 | 1000 | 1000 | 1000 | 500 | 500
Table 1: An overview of the FASTSUM Generation 2 correlation functions used in
this work. Lattice volumes are $(24a_{s})^{3}\times(N_{\tau}a_{\tau})$ with
$a_{s}=0.1227(8)$fm and $a_{\tau}=35.1(2)$am. For these ensembles with a pion
mass of $M_{\pi}=384(4)$MeV, the pseudo-critical temperature T${}_{\rm
pc}=181(1)$MeV [6].

##  3 HAL QCD Method

We follow the HAL QCD time-dependent method to extract the interquark
potential [7, 8]. A key quantity for the HAL QCD method is the Nambu Bethe
Salpeter (NBS) wave function, $\psi_{i}(\textbf{r})=\langle
0|J(\textbf{r})|i\rangle$, i.e. the overlap of the non-local mesonic operator
$J(\textbf{r})$ between the vacuum and the bottomonium state $|i\rangle$. The
mesonic operator $J(\textbf{r})$ is defined

$J_{\Gamma}(x;\textbf{r})=\overline{Q}(x)\,U(x,x+\textbf{r})\,\Gamma
Q(x+\textbf{r})$

and thus probes the bottomium state with a displacement of r between its two
constituent quarks. $\Gamma$ is a Dirac matrix chosen to have the desired
quantum numbers appropriate for either the $\Upsilon$ $(\Gamma=\gamma_{i})$ or
$\eta_{b}$ $(\Gamma=\gamma_{5})$ states, and $U(x,x+\textbf{r})$ is the gauge
connection between $x$ and $x+\textbf{r}$.

We calculate the zero-momentum correlation function

$G_{\Gamma}(\textbf{r},\tau)=\sum_{\textbf{x}}\langle
J_{\Gamma}(\textbf{x},\tau;\textbf{r})J_{\Gamma}^{\dagger}(0;\textbf{0})\rangle=\sum_{i}\frac{\psi_{i}(\textbf{r})\psi_{i}^{*}(\textbf{0})}{2E_{i}}e^{-E_{i}\tau}=\sum_{i}\Psi_{i}(\textbf{r})e^{-E_{i}\tau}.$

The sum over states $i$ is the usual spectral representation, and for
convenience we’ve defined

$\Psi_{i}(\textbf{r})=\frac{\psi_{i}(\textbf{r})\psi_{i}^{*}(\textbf{0})}{2E_{i}}.$

Since we are treating the bottom quark nonrelativistically, we can assume that
$\Psi_{i}(\textbf{r})$ obeys the time independent Schrodinger equation in
Euclidean space-time,

$\bigg{(}-\frac{\nabla^{2}_{r}}{2\mu}+V_{\Gamma}(r)\bigg{)}\Psi_{i}(\textbf{r})=E_{i}\Psi_{i}(\textbf{r}),$

where $V_{\Gamma}(r)$ is the interquark potential for the channel $\Gamma$,
$\mu$ is the reduced mass, and we restrict to S-wave states. Since the
correlation function, $G(\tau)$, is a linear combination of
$\Psi_{i}(\textbf{r})$, we find that it satisfies the Schrödinger equation,

$\bigg{(}-\frac{\nabla^{2}_{r}}{2\mu}+V_{\Gamma}(r)\bigg{)}G_{\Gamma}(\textbf{r},\tau)=-\frac{\partial
G_{\Gamma}(\textbf{r},\tau)}{d\tau}.$ (1)

We use eq(1) to extract the potential, $V_{\Gamma}$ from
$G_{\Gamma}(\textbf{r},\tau)$. We note that the NRQCD case considered here has
a particularly simple form because there are no backward movers. This
contrasts with the relativistic case where there are backward movers which
need to be considered [3, 4].

We use finite derivatives to approximate the Laplacian and the temporal
derivative. Because we consider $S$-wave states with rotational symmetry, the
Laplacian in spherical coordinates can be approximated by

$\nabla_{r}^{2}f(r)=\frac{\partial^{2}f}{\partial r}+\frac{2}{r}\frac{\partial
f}{dr}\approx\bigg{(}\frac{f(r+a_{s})-2f(r)+f(r-a_{s})}{a_{s}^{2}}+\frac{f(r+a_{s})-f(r-a_{s})}{ra_{s}}\bigg{)}.$
(2)

The time derivative is similarly approximated by

$\frac{\partial f}{\partial\tau}\approx\bigg{(}\frac{f(\tau+a_{\tau})-f(\tau-
a_{\tau})}{2a_{\tau}}\bigg{)}.$ (3)

Using the leading order terms in the velocity expansion of the interquark
potential for $S$-wave states [9], the central potential can be defined in
terms of the potential from the pseudoscalar (i.e. the $\eta_{b}$) and vector
($\Upsilon$) channels’ potentials,

$V_{c}(r)=\frac{1}{4}V_{\text{PS}}(r)+\frac{3}{4}V_{\text{V}}(r).$ (4)

The spin-dependent potential is also accessible to us from these two channels,
but is not considered here. Higher order terms in the potential such as the
spin-orbit term are also not studied because they require channels with
orbital angular momentum.

##  4 Results

###  4.1 Derivatives

We begin by separately studying the spatial and temporal derivatives in eq(1).
Figure 1 shows the spatial derivative, i.e. the kinetic contribution for the
$\Upsilon$ at two indicative temperatures, $T=141$MeV (left) and $T=352$MeV
(right). For both temperatures the $\tau$ dependence becomes more noticeable
at larger $r$, as does the size of the statistical errors. The increase in
noise at larger $r$ is to be expected for point-split lattice correlation
functions since points close together are correlated and so fluctuations
increase with $r$.

Figure 1: The kinetic (i.e. spatial derivative) contribution to the $\Upsilon$
potential plotted against imaginary time for $T=141$MeV (left) and $T=352$MeV
(right). For both temperatures the noise increases with distance, $r/a_{s}$.
The $T=352$MeV data shows larger variation and the axis scale is adjusted to
reflect this.

Figure 2: The temporal derivative contribution to the $\Upsilon$ potential
plotted against imaginary time for $T=141$MeV (left) and $T=352$MeV (right).

In Fig. 2 we show the temporal derivative term from eq(1) for the same two
temperatures. This again shows the variation with $r$, although this is less
than in the spatial derivative case. We also note that there is a plateau at
large $\tau$ visible for the $T=141$MeV case. This is to be expected, because
at large $\tau$, the time derivative asymptotes to the ground state mass.

Comparing the two derivative from figs.1 & 2, we see that the spatial
derivative is numerically larger.

###  4.2 Potentials for the $\eta_{b}$ and $\Upsilon$ Channels

We combine the spatial and temporal derivatives for the $\eta_{b}$ and
$\Upsilon$ channels in eq(1) to obtain the the potentials for those channels,
$V_{\eta_{b}}$ and $V_{\Upsilon}$. The $\Upsilon$ case is plotted in fig.3.
Note $V_{\eta_{b}}$ and $V_{\Upsilon}$ are explicit functions of $\tau$ in the
time-dependent HAL QCD method, due to the way these potentials are derived.
Ideally, they should be constant functions w.r.t. $\tau$, but as can be seen
from fig.3, this is not the case, except at large $\tau$, or for small $r$. We
will investigate this $\tau$-dependency in future work by considering lattice
derivatives which are more sophisticated than those in eqs(2) & (3). We note
that the results have the smallest systematics for small $r$ values. We obtain
our final estimate of the each channel’s potential by averaging over a time
window as discussed in the next section.

Figure 3: The $\Upsilon$ potential plotted against imaginary time for
$T=141$MeV (left) and $T=352$MeV (right).

###  4.3 Central Potential

We average the potentials for the $\eta_{b}$ and $\Upsilon$ channels over the
time interval $\tau\in[\tau_{1},\tau_{2}]$ (as discussed below) and then
combine them to obtain the central potential, $V_{C}(r)$ using eq(4). Figure 4
plots $V_{C}(r)$ for all temperatures studied (see Table 1). Due to the
periodic boundary conditions in the spatial direction, there are only 13
distinct lattice points in the spatial direction. However, the noise grows too
quickly for points $r>7a_{s}$ to be considered.

Our aim is to determine the interquark potential as a function of temperature.
In order to disentangle possible systematic effects from thermal effects, we
use the same time window $[\tau_{1},\tau_{2}]$ for neighbouring temperatures.
This ensures that the fitting procedure is identical for both of these
temperatures, and so any variation in the potential can be ascribed to a
thermal, rather than systematic effect. The data plotted in fig.4 follows this
procedure. The points are off-set horizontally for clarity. We see that the
potential at large distances $r\gtrsim 0.8$fm have large errors and become
unstable for the hottest temperatures, but those for distances $r\lesssim
0.8$fm have more modest errors and are predictive.

From the plot, we see indications of thermal effects which are best seen in
the insert for the two distances $r\approx 0.4$ & 0.5fm. When considering
neighbouring temperatures which share the same time window
$[\tau_{1},\tau_{2}]$ we see that there is a clear trend towards a flattening
of $V_{C}$ as $T$ increases. This confirms our expectation that the interquark
potential is temperature dependent and becomes weaker with increasing
temperature.

Figure 4: The central interquark potential in the bottomonium system plotted
against quark separation $r$ for a range of temperatures. The points are
offset horizontally for clarity. The potentials for each temperature are
obtained by averaging two time ranges $[\tau_{1},\tau_{2}]$ as indicated in
the legend, chosen so that they are identical for neighbouring temperatures.
This allows thermal effects to be disentangled from fitting systematics as
discussed in the text. The insert shows a closeup of two $r$ values which
indicates a thermal effect.

##  5 Summary

These proceedings present a calculation of the thermal interquark potential in
the bottomonium system using the HAL QCD method with NRQCD quarks. Our fastsum
Collaboration’s anistropic "Generation 2" ensembles were used. We find
indications of thermal effects in the central potential, $V_{C}(r)$, observing
the expected flattening of the potential as the temperature increases. Future
work will use lattice derivatives which have smaller discretisation errors,
and a momentum space approach which will allow us to calculate the potential
at all spatial displacements. We will also study higher order terms in the
potential.

## Acknowledgements

This work is supported by STFC grant ST/T000813/1. SK is supported by the
National Research Foundation of Korea under grant NRF-2021R1A2C1092701 funded
by the Korean government (MEST). This work used the DiRAC Extreme Scaling
service at the University of Edinburgh, operated by the Edinburgh Parallel
Computing Centre on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk).
This equipment was funded by BEIS capital funding via STFC capital grant
ST/K000411/1, STFC capital grant ST/H008845/1, and STFC DiRAC Operations
grants ST/K005804/1 and ST/K005790/1. DiRAC is part of the National
e-Infrastructure. This work was performed using PRACE resources at Cineca via
grants 2011040469 and 2012061129. We acknowledge the support of the
Supercomputing Wales project, which is part-funded by the European Regional
Development Fund (ERDF) via Welsh Government.

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