]}

# Accurate Cold-start Bundle Recommendation via
Popularity-based Coalescence and Curriculum Heating

Hyunsik Jeon  UC San DiegoSan Diego, CAUSA<EMAIL_ADDRESS>,  Jong-eun Lee
Seoul National UniversitySeoulSouth Korea<EMAIL_ADDRESS>,  Jeongin Yun
Seoul National UniversitySeoulSouth Korea<EMAIL_ADDRESS>and  U Kang
Seoul National UniversitySeoulSouth Korea<EMAIL_ADDRESS>
(2024)

###### Abstract.

How can we accurately recommend cold-start bundles to users? The cold-start
problem in bundle recommendation is critical in practical scenarios since new
bundles are continuously created for various marketing purposes. Despite its
importance, no previous studies have addressed cold-start bundle
recommendation. Moreover, existing methods for cold-start item recommendation
overly rely on historical information, even for unpopular bundles, failing to
tackle the primary challenge of the highly skewed distribution of bundle
interactions. In this work, we propose CoHeat (Popularity-based Coalescence
and Curriculum Heating), an accurate approach for the cold-start bundle
recommendation. CoHeat tackles the highly skewed distribution of bundle
interactions by incorporating both historical and affiliation information
based on the bundle’s popularity when estimating the user-bundle relationship.
Furthermore, CoHeat effectively learns latent representations by exploiting
curriculum learning and contrastive learning. CoHeat demonstrates superior
performance in cold-start bundle recommendation, achieving up to $193\%$
higher nDCG$@20$ compared to the best competitor.

cold-start bundle recommendation; curriculum learning; contrastive learning

††copyright: acmcopyright††journalyear: 2024††doi:
XXXXXXX.XXXXXXX††conference: Proceedings of The ACM Web Conference 2024; May
13–17, 2024; Singapore††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs:
Information systems Recommender systems

##  1\. Introduction

_How can we accurately recommend cold-start bundles to users?_ Bundle
recommendation has garnered significant attention in both academia and
industry since it enables providers to offer items to users with one-stop
convenience (Ma et al., 2022). In particular, recommending new bundles to
users (i.e. cold-start bundle recommendation) is important in practical
scenarios because the new bundles are constantly created for various marketing
purposes (Chen et al., 2019b).

In recent years, bundle recommendation has seen advancements through matrix
factorization-based approaches (Pathak et al., 2017; Cao et al., 2017; Chen et
al., 2019b) and graph learning-based approaches (Deng et al., 2020; Chang et
al., 2020; Ma et al., 2022). However, they have been developed for a warm-
start setting, where all bundles possess historical interactions with users.
Consequently, they fail to effectively perform in a cold-start setting, where
certain bundles are devoid of historical interactions. This is because warm-
start methods rely highly on historical information to learn bundle
representations. On the other hand, the cold-start problem in item
recommendation has been extensively studied, with a focus on aligning behavior
representations with content representations. For instance, generative methods
have aimed to model the generation of item behavior representations using mean
squared error (van den Oord et al., 2013) and adversarial loss (Chen et al.,
2022). Dropout-based methods (Volkovs et al., 2017; Zhu et al., 2020) have
aimed to bolster robustness to behavior information by randomly dropping the
behavior embedding in the training phase. More recently, contrastive learning-
based methods (Wei et al., 2021; Zhou et al., 2023) have shown superior
performance by reducing the discrepancy between the distributions of behavior
and content information of items. However, none of the existing works have
explicitly considered the skewed distribution of interactions which is a
pivotal aspect in bundle recommendation as shown in Figure 1(a). For unpopular
bundles, aligning behavior representations from insufficient historical
information with content representations amplifies inherent biases and makes
it difficult to learn meaningful representations; this results in sacrificing
the performance on a warm-start setting to improve the performance on a cold-
start setting (see Figure 2).

(a) Extremely skewed distribution of bundle interactions.

(b) Insufficient information from history-view.

(c) Sufficient information from affiliation-view.

Figure 1.  (a) Extremely skewed distribution of bundle interactions in real-
world datasets (data statistics are summarized in Table 1). (b-c) For an
unpopular bundle, history-view provides insufficient information while
affiliation-view provides sufficient information.

(a) Youshu

(b) NetEase

(c) iFashion

Figure 2. Performance comparison between CoHeat and competitors on three real-
world datasets: Youshu, NetEase, and iFashion. The performance is evaluated
through Recall@20 for all experiments. We mark cold-start methods as orange,
and warm-start methods as red. CoHeat demonstrates superior performance over
existing methods in both cold and warm settings, with a notable advantage in
outperforming competitors.

In this paper, we propose CoHeat (Popularity-based Coalescence and Curriculum
Heating), an accurate method for cold-start bundle recommendation. CoHeat
constructs representations of users and bundles using two distinct graph-based
views: history-view and affiliation-view. The history-view graph is grounded
in historical interactions between users and bundles, whereas the affiliation-
view graph captures information rooted in bundle affiliations. To handle the
extremely skewed distribution, CoHeat strategically leverages both views in
its predictions, emphasizing affiliation-view for less popular bundles since
they provide richer information than the sparse history-view, as shown in
Figures 1(b) and 1(c). In addition, to effectively learn the affiliation-view
representations which are fully used for cold-start bundles, CoHeat exploits a
curriculum learning approach that gradually shifts the training focus from the
history-view to the affiliation-view. CoHeat further exploits a contrastive
learning approach to align the representations of the two views effectively.

Our contributions are summarized as follows:

  * •

Problem. To our knowledge, this is the first work that tackles the cold-start
problem in bundle recommendation, a challenging problem of significant impact
in real-world scenarios.

  * •

Method. We propose CoHeat, an accurate method for cold-start bundle
recommendation. CoHeat effectively treats the extremely skewed distribution of
interactions in order to accurately recommend cold-start bundles based on
their affiliations.

  * •

Experiments. We experimentally show that CoHeat provides the state-of-the-art
performance achieving up to 193% higher nDCG$@20$ compared to the best
competitor in cold-start bundle recommendation (see Figure 2).

##  2\. Preliminaries

###  2.1. Problem Definition

The problem of cold-start bundle recommendation is defined as follows. Let
$\mathcal{U}$, $\mathcal{B}$, and $\mathcal{I}$ be the sets of users, bundles,
and items, respectively. Among the bundles,
$\mathcal{B}_{w}\subset\mathcal{B}$ refers to the warm-start bundles that have
at least one historical interaction with users, while
$\mathcal{B}_{c}=\mathcal{B}\setminus\mathcal{B}_{w}$ represents the cold-
start bundles that lack any historical interaction with users. The observed
user-bundle interactions, user-item interactions, and bundle-item affiliations
are respectively defined as
$\mathcal{X}=\\{(u,b)|u\in\mathcal{U},b\in\mathcal{B}_{w}\\}$,
$\mathcal{Y}=\\{(u,i)|u\in\mathcal{U},i\in\mathcal{I}\\}$, and
$\mathcal{Z}=\\{(b,i)|b\in\mathcal{B},i\in\mathcal{I}\\}$. Given
$\\{\mathcal{X},\mathcal{Y},\mathcal{Z}\\}$, our goal is to recommend $k$
bundles from $\mathcal{B}$ to each user $u\in\mathcal{U}$. Note that the given
interactions are observed only for warm bundles but the objective includes
recommending also cold bundles to users.

The central challenge in cold-start bundle recommendation, compared to
traditional bundle recommendation, lies in accurately predicting the
relationship between a user $u\in\mathcal{U}$ and a cold-start bundle
$b\in\mathcal{B}_{c}$ in the absence of any historical interactions of $b$.
Hence, the crux of addressing the problem is to effectively estimate the
representations of cold-start bundles using their affiliation information.

Figure 3.  Overview of CoHeat (see Section 3 for details).

###  2.2. Curriculum Learning

Curriculum learning, inspired by human learning, structures training from
simpler to more complex tasks, unlike standard approaches that randomize task
order (Bengio et al., 2009; Wang et al., 2022a). Its effectiveness has been
proven in various domains, including computer vision (Yu et al., 2020; Zhang
et al., 2021), natural language processing (Zhan et al., 2021; Gong et al.,
2021), robotics (He et al., 2020b; Manela and Biess, 2022), and recommender
systems (Chen et al., 2021a, b).

In this work, we harness curriculum learning to enhance the learning process
of user-bundle relationships. We initiate with a focus on the more
straightforward history-view embeddings and then progressively shift attention
to the intricate affiliation-view embeddings. This strategy stems from the
ease of learning history-view embeddings, which directly capture collaborative
signals from historical interactions. In contrast, affiliation-view embeddings
are more complicated due to their dependence on the representations of
affiliated items.

###  2.3. Contrastive Learning

Contrastive learning aims to learn meaningful embeddings by distinguishing
between similar and dissimilar data samples. It has consistently demonstrated
superior performance across a range of research fields, including computer
vision (Wang and Isola, 2020; Robinson et al., 2021), natural language
processing (Yan et al., 2021; Gao et al., 2021), and recommender systems (Wang
et al., 2022b; Cai et al., 2023). Specifically, CrossCBR (Ma et al., 2022)
recently achieved a good performance in bundle recommendation by regularizing
embeddings of users and bundles using InfoNCE (van den Oord et al., 2018)
between history-view and affiliation-view.

However, CrossCBR aligns the two views while equally treating them in
prediction. In contrast, our work adaptively modulates the weights of these
views based on bundle popularity, thereby facilitating information transfer
from the more informative view to its sparser counterpart. Additionally, in
our contrastive learning approach, we utilize the alignment and uniformity
loss (Wang and Isola, 2020). This has been shown to surpass InfoNCE in various
applications (Wang and Isola, 2020; Wang et al., 2022b), as it directly
optimizes the core perspectives of contrastive learning.

##  3\. Proposed Method

###  3.1. Overview

We address the following challenges to achieve high performance on cold-start
bundle recommendation.

Handling highly skewed interactions. Previous works overly depend on history-
view representations, which are unreliable if bundles have sparse
interactions. How can we effectively learn the representations from highly
skewed interactions?

Effectively learning affiliation-view representations. Despite the ample
information provided by the affiliation-view, multiple items in a bundle
complicate learning of these representations. How can we effectively learn the
affiliation-view representations?

Bridging the gap between two view representations. Aligning history-view and
affiliation-view is crucial, as we estimate future interactions of cold
bundles only using their affiliations. How can we effectively reconcile these
two view representations? To address these challenges, we propose CoHeat
(Popularity-based Coalescence and Curriculum Heating) with the following main
ideas.

Popularity-based coalescence. For the score between users and bundles, we
propose the coalescence of two view scores, with less popular bundles relying
more on affiliation-view scores and less on history-view scores.

Curriculum heating. We propose a curriculum learning approach that focuses
initially on training representations using the history-view, gradually
shifting the focus to the affiliation-view.

Representation alignment and uniformity. We exploit a representation alignment
and uniformity approach to effectively reconcile the history-view and
affiliation-view representations.

Figure 3 depicts the schematic illustration of CoHeat. Given user-bundle
interactions, user-item interactions, and bundle-item affiliations, CoHeat
forms two graph-based views. Then, it predicts user-bundle scores by
coalescing scores from both views based on bundle popularity. During training,
CoHeat prioritizes history-view initially, transitioning progressively to
affiliation-view via curriculum heating. CoHeat also exploits alignment and
uniformity loss to regularize both views.

###  3.2. Two Graph-based Views

The objective of bundle recommendation is to estimate the relationship between
users and bundles by learning their latent representations. We utilize graph-
based representations of users and bundles to fully exploit the given user-
bundle interactions, user-item interactions, and bundle-item affiliations. We
construct history-view and affiliation-view graphs and use LightGCN (He et
al., 2020a) to obtain embeddings of users and bundles (Ma et al., 2022).

History-view representation and score. In history-view, we aim to capture the
behavior signal between users and bundles. Specifically, we construct a
bipartite graph using user-bundle interactions, and propagate the historical
information using a LightGCN. The $k$’th layer of the LightGCN is computed as
follows:

(1)
$\small\begin{split}\mathbf{h}_{u}^{(k)}=\sum_{b\in\mathcal{N}_{u}}\frac{1}{\sqrt{|\mathcal{N}_{u}|}\sqrt{|\mathcal{N}_{b}}|}\mathbf{h}_{b}^{(k-1)},\\\
\mathbf{h}_{b}^{(k)}=\sum_{u\in\mathcal{N}_{b}}\frac{1}{\sqrt{|\mathcal{N}_{b}|}\sqrt{|\mathcal{N}_{u}}|}\mathbf{h}_{u}^{(k-1)},\end{split}$

where $\mathbf{h}_{u}^{(k)},\mathbf{h}_{b}^{(k)}\in\mathbb{R}^{d}$ are the
embeddings of user $u$ and bundle $b$ at $k$’th layer, respectively;
$\mathcal{N}_{u}$ and $\mathcal{N}_{b}$ are the sets of user $u$’s neighbors
and bundle $b$’s neighbors in the user-bundle graph, respectively.
$\mathbf{h}_{u}^{(0)},\mathbf{h}_{b}^{(0)}\in\mathbb{R}^{d}$ are randomly
initialized before the training of the model. We obtain the history-view
representations of user $u$ and bundle $b$ by aggregating the embeddings from
all layers with a weighting approach that places greater emphasis on the lower
layers as follows:

(2)
$\small\mathbf{h}_{u}=\sum_{k=0}^{K}\frac{1}{k+1}\mathbf{h}_{u}^{(k)},\mathbf{h}_{b}=\sum_{k=0}^{K}\frac{1}{k+1}\mathbf{h}_{b}^{(k)},$

where $\mathbf{h}_{u},\mathbf{h}_{b}\in\mathbb{R}^{d}$ are the history-view
embeddings of user $u$ and bundle $b$, respectively; $K$ denotes the last
layer. Finally, the history-view score between user $u$ and bundle $b$ is
defined as $h_{ub}=\mathbf{h}_{u}^{\top}\mathbf{h}_{b}$.

Affiliation-view representation and score. In affiliation-view, we aim to
learn the relationship between users and bundles from the perspective of item
affiliations. Specifically, we construct a bipartite graph using user-item
interactions, and propagate the historical information using another LightGCN.
Then, we obtain bundle representations by aggregating the affiliated items’
representations. The $k$’th layer of the LightGCN is computed as follows:

(3)
$\small\begin{split}\mathbf{a}_{u}^{(k)}=\sum_{i\in\mathcal{N}^{\prime}_{u}}\frac{1}{\sqrt{|\mathcal{N}^{\prime}_{u}|}\sqrt{|\mathcal{N}_{i}}|}\mathbf{a}_{i}^{(k-1)},\\\
\mathbf{a}_{i}^{(k)}=\sum_{u\in\mathcal{N}_{i}}\frac{1}{\sqrt{|\mathcal{N}_{i}|}\sqrt{|\mathcal{N}^{\prime}_{u}}|}\mathbf{a}_{u}^{(k-1)},\end{split}$

where $\mathbf{a}_{u}^{(k)},\mathbf{a}_{i}^{(k)}\in\mathbb{R}^{d}$ are the
embeddings of user $u$ and item $i$ at $k$’th layer, respectively;
$\mathcal{N}^{\prime}_{u}$ and $\mathcal{N}_{i}$ are the sets of user $u$’s
neighbors and item $i$’s neighbors in the user-item graph, respectively.
$\mathbf{a}_{u}^{(0)},\mathbf{a}_{i}^{(0)}\in\mathbb{R}^{d}$ are randomly
initialized before the training. We obtain the affiliation-view
representations of user $u$ and item $i$ by aggregating the embeddings from
all layers with a weighting approach as follows:

(4)
$\small\mathbf{a}_{u}=\sum_{k=0}^{K}\frac{1}{k+1}\mathbf{a}_{u}^{(k)},\mathbf{a}_{i}=\sum_{k=0}^{K}\frac{1}{k+1}\mathbf{a}_{i}^{(k)},$

where $\mathbf{a}_{u},\mathbf{a}_{i}\in\mathbb{R}^{d}$ are the affiliation-
view embeddings of user $u$ and item $i$, respectively; $K$ indicates the last
layer. We then obtain the affiliation-view representations of bundle $b$ by an
average pooling as
$\mathbf{a}_{b}=\frac{1}{|\mathcal{N}^{\prime}_{b}|}\sum_{i\in\mathcal{N}^{\prime}_{b}}\mathbf{a}_{i},$
where $\mathcal{N}^{\prime}_{b}$ is the set of bundle $b$’s affiliated items.
Finally, the affiliation-view score between user $u$ and bundle $b$ is defined
as $a_{ub}=\mathbf{a}_{u}^{\top}\mathbf{a}_{b}$.

###  3.3. Popularity-based Coalescence

For recommending bundles to users, our objective is to estimate the final
score $\hat{y}_{ub}\in\mathbb{R}$ between user $u$ and bundle $b$ using scores
$h_{ub}$ and $a_{ub}$, derived from the two distinct views. However, real-
world datasets present an inherent challenge of handling the extremely skewed
distribution of interactions between users and bundles, as illustrated in
Figure 1(a). While both views are informative, many unpopular bundles are
underrepresented in the history-view due to the insufficient interactions as
illustrated in Figure 1(b). In contrast, they are often sufficiently
represented in the affiliation-view, as depicted in Figure 1(c). A uniform
weighting strategy for both views, as in CrossCBR, risks amplifying biases
inherent to the history-view, especially for the unpopular bundles. This
predicament is further exacerbated for cold-start bundles devoid of history-
view data.

To deal with this challenge, we propose two desired properties for the user-
bundle relationship score $\hat{y}_{ub}$.

######  Property 1 (History-view influence mitigation):

The influence of history-view score should be mitigated as a bundle’s
interaction number decreases, i.e. $\frac{\partial\hat{y}_{ub}}{\partial
h_{ub}}<\frac{\partial\hat{y}_{ub^{\prime}}}{\partial h_{ub^{\prime}}}$ if
$n_{b}<n_{b^{\prime}}$ where $n_{b}$ is the number of interactions of bundle
$b$.

######  Property 2 (Affiliation-view influence amplification):

The influence of affiliation-view score should be amplified as a bundle’s
interaction number decreases, i.e. $\frac{\partial\hat{y}_{ub}}{\partial
a_{ub}}>\frac{\partial\hat{y}_{ub^{\prime}}}{\partial a_{ub^{\prime}}}$ if
$n_{b}<n_{b^{\prime}}$ where $n_{b}$ is the number of interactions of bundle
$b$.

Properties 1 and 2 are crucial in achieving a balanced interplay between the
history-view and affiliation-view scores based on bundle popularities.
Specifically, they ensure a heightened emphasis on the affiliation-view over
the history-view for less popular bundles.

We propose the user-bundle relationship score $\hat{y}_{ub}$ that satisfies
the two desired properties by weighting the two scores $h_{ub}$ and $a_{ub}$
based on bundle popularities as follows:

(5) $\small\hat{y}_{ub}=\gamma_{b}h_{ub}+(1-\gamma_{b})a_{ub},$

where $\gamma_{b}\in[0,1]$, which is defined in the next subsection, denotes a
weighting coefficient such that $\gamma_{b}>\gamma_{b^{\prime}}$ if
$n_{b}>n_{b^{\prime}}$. A smaller value of $\gamma_{b}$ (i.e. a smaller value
of $n_{b}$) ensures that the score $\hat{y}_{ub}$ is predominantly influenced
by the affiliation-view score $a_{ub}$. We show in Lemmas 3.1 and 3.2 that
Equation (5) satisfies all the desired properties.

###### Lemma 3.1.

Equation (5) satisfies Property 1.

###### Proof.

$\frac{\partial\hat{y}_{ub}}{\partial h_{ub}}=\gamma_{b}$. Thus,
$\frac{\partial\hat{y}_{ub}}{\partial
h_{ub}}<\frac{\partial\hat{y}_{ub^{\prime}}}{\partial h_{ub^{\prime}}}$ if
$n_{b}<n_{b^{\prime}}$ because $\gamma_{b}<\gamma_{b^{\prime}}$. ∎

###### Lemma 3.2.

Equation (5) satisfies Property 2.

###### Proof.

$\frac{\partial\hat{y}_{ub}}{\partial a_{ub}}=1-\gamma_{b}$. Thus,
$\frac{\partial\hat{y}_{ub}}{\partial
a_{ub}}>\frac{\partial\hat{y}_{ub^{\prime}}}{\partial a_{ub^{\prime}}}$ if
$n_{b}<n_{b^{\prime}}$ because $1-\gamma_{b}>1-\gamma_{b^{\prime}}$. ∎

###  3.4. Curriculum Heating

Despite the ample information provided by the affiliation-view, multiple items
in a bundle complicate the learning of affiliation-view representations. This
difficulty arises because accurate representation of a bundle necessitates
well-represented embeddings of its all affiliated items. On the other side,
the history-view representation is relatively straightforward to learn. This
simplicity arises because we encapsulate each bundle’s historical
characteristics into a single embedding rather than understanding the
intricate composition of the bundle.

Hence, we modify Equation (5) by exploiting a curriculum learning approach
that focuses initially on training history-view representations, and gradually
shifts the focus to the affiliation-view representations as follows:

(6)
$\small\hat{y}_{ub}^{(t)}=\gamma_{b}^{(t)}h_{ub}+(1-\gamma_{b}^{(t)})a_{ub},$

where $\hat{y}_{ub}^{(t)}\in\mathbb{R}$ is the estimated relationship score
between user $u$ and bundle $b$ at epoch $t$. $\gamma_{b}^{(t)}\in\mathbb{R}$
is defined as $\gamma_{b}^{(t)}=\tanh(\frac{n_{b}}{\psi^{(t)}})$, where
$n_{b}$ is the number of interactions of bundle $b$, and $\psi^{(t)}>0$ is the
temperature at epoch $t$. Note that $\gamma_{b}^{(t)}$ lies within the
interval $[0,1]$ because $\frac{n_{b}}{\psi^{(t)}}\geq 0$. Then, we
incrementally raise the temperature $\psi^{(t)}$ up to the maximum temperature
as follows:

(7) $\small\psi^{(t)}=\epsilon^{t/T},t:0\rightarrow T,$

where $t,T\in\mathbb{R}$ are the current and the maximum epochs of the
training process, and $\epsilon>1$ is the hyperparameter of the maximum
temperature. In the initial epochs of training, $\gamma_{b}^{(t)}$ is large
since $t$ is small. As a result, the score $\hat{y}_{ub}^{(t)}$ relies more
heavily on $h_{ub}$ than $a_{ub}$. However, as the training progresses and $t$
increases, $\gamma_{b}^{(t)}$ diminishes, shifting the emphasis from $h_{ub}$
to $a_{ub}$. This heating mechanism is applied to all bundles regardless of
their popularity. Furthermore, we show in Lemmas 3.3 and 3.4 that Equation (6)
still satisfies the two desired properties.

###### Lemma 3.3.

Equation (6) satisfies Property 1.

###### Proof.

$\frac{\partial\hat{y}^{(t)}_{ub}}{\partial
h_{ub}}=\tanh(\frac{n_{b}}{\psi^{(t)}})$. Thus,
$\frac{\partial\hat{y}^{(t)}_{ub}}{\partial
h_{ub}}<\frac{\partial\hat{y}^{(t)}_{ub^{\prime}}}{\partial h_{ub^{\prime}}}$
if $n_{b}<n_{b^{\prime}}$ because $\psi^{(t)}$ is the same for all bundles at
epoch $t$ and $tanh(\cdot)$ is an increasing function. ∎

###### Lemma 3.4.

Equation (6) satisfies Property 2.

###### Proof.

$\frac{\partial\hat{y}^{(t)}_{ub}}{\partial
a_{ub}}=1-\tanh(\frac{n_{b}}{\psi^{(t)}})$. Thus,
$\frac{\partial\hat{y}^{(t)}_{ub}}{\partial
a_{ub}}>\frac{\partial\hat{y}^{(t)}_{ub^{\prime}}}{\partial a_{ub^{\prime}}}$
if $n_{b}<n_{b^{\prime}}$ because $\psi^{(t)}$ is the same for all bundles at
epoch $t$ and $1-tanh(\cdot)$ is a decreasing function. ∎

###  3.5. Representation Alignment and Uniformity

While the history-view and affiliation-view are crafted to capture distinct
representations, aligning the two views is essential, especially when
predicting future interactions of cold bundles solely based on affiliation-
view representations. To achieve this, we exploit a contrastive learning-based
approach that reconciles the two views. Specifically, we use the alignment and
uniformity loss (Wang and Isola, 2020) as a regularization for the
representations of the two views. We firstly $l_{2}$-normalize the embeddings
of the two views as follows:

(8)
$\small\tilde{\mathbf{h}}_{u}=\frac{\mathbf{h}_{u}}{\|\mathbf{h}_{u}\|_{2}},\tilde{\mathbf{a}}_{u}=\frac{\mathbf{a}_{u}}{\|\mathbf{a}_{u}\|_{2}},\tilde{\mathbf{h}}_{b}=\frac{\mathbf{h}_{b}}{\|\mathbf{h}_{b}\|_{2}},\tilde{\mathbf{a}}_{b}=\frac{\mathbf{a}_{b}}{\|\mathbf{a}_{b}\|_{2}},$

where $\mathbf{h}_{u},\mathbf{h}_{b}\in\mathbb{R}^{d}$ are history-view
representations of user $u$ and bundle $b$, respectively;
$\mathbf{a}_{u},\mathbf{a}_{b}\in\mathbb{R}^{d}$ are affiliation-view
representations of user $u$ and bundle $b$, respectively. Then, we define an
alignment loss as follows:

(9) $\small l_{align}=\mathop{\mathbb{E}}_{u\sim
p_{user}}\|\tilde{\mathbf{h}}_{u}-\tilde{\mathbf{a}}_{u}\|_{2}^{2}+\mathop{\mathbb{E}}_{b\sim
p_{bundle}}\|\tilde{\mathbf{h}}_{b}-\tilde{\mathbf{a}}_{b}\|_{2}^{2},$

where $p_{user}$ and $p_{bundle}$ are the distributions of users and bundles,
respectively. The alignment loss makes the embeddings of the two views close
to each other for each user and bundle. We also define a uniformity loss as
follows:

(10)
$\small\begin{split}l_{uniform}&=\log\mathop{\mathbb{E}}_{u,u^{\prime}\sim
p_{user}}e^{-2\|\tilde{\mathbf{h}}_{u}-\tilde{\mathbf{h}}_{u^{\prime}}\|_{2}^{2}}\\\
&+\log\mathop{\mathbb{E}}_{u,u^{\prime}\sim
p_{user}}e^{-2\|\tilde{\mathbf{a}}_{u}-\tilde{\mathbf{a}}_{u^{\prime}}\|_{2}^{2}}\\\
&+\log\mathop{\mathbb{E}}_{b,b^{\prime}\sim
p_{bundle}}e^{-2\|\tilde{\mathbf{h}}_{b}-\tilde{\mathbf{h}}_{b^{\prime}}\|_{2}^{2}}\\\
&+\log\mathop{\mathbb{E}}_{b,b^{\prime}\sim
p_{bundle}}e^{-2\|\tilde{\mathbf{a}}_{b}-\tilde{\mathbf{a}}_{b^{\prime}}\|_{2}^{2}},\end{split}$

where $u^{\prime}$ and $b^{\prime}$ denote a user and a bundle distinct from
$u$ and $b$, respectively. The uniformity loss ensures distinct
representations for different users (or bundles) by scattering them across the
space. Finally, we define the contrastive loss for the two views as follows:

(11) $\mathcal{L}_{AU}=l_{align}+l_{uniform}.$

###  3.6. Objective Function and Training

To effectively learn the user-bundle relationship, we utilize Bayesian
Personalize Ranking (BPR) loss (Rendle et al., 2009), which is the most widely
used loss owing to its powerfulness, as follows:

(12) $\small\mathcal{L}_{BPR}^{(t)}=\mathop{\mathbb{E}}_{(u,b^{+},b^{-})\sim
p_{data}}-\ln\sigma(\hat{y}_{ub^{+}}^{(t)}-\hat{y}_{ub^{-}}^{(t)}),$

where $p_{data}$ is the data distribution of user-bundle interactions, with
$u$ denoting a user, $b^{+}$ indicating a positive bundle, and $b^{-}$
representing a negative bundle. We define the final objective function as
follows:

(13)
$\small\mathcal{L}^{(t)}=\mathcal{L}_{BPR}^{(t)}+\lambda_{1}\mathcal{L}_{AU}+\lambda_{2}\|\Theta\|_{2},$

where $\lambda_{1},\lambda_{2}\in\mathbb{R}$ are balancing hyperparameters for
the terms, and $\Theta$ denotes trainable parameters of CoHeat. For the
distributions $p_{user}$ and $p_{bundle}$, we use in-batch sampling which
selects samples from the training batch of $p_{data}$ rather than the entire
dataset. This approach has empirically demonstrated to mitigate the training
bias in prior studies (Zhou et al., 2021; Wang et al., 2022b). All the
parameters are optimized in an end-to-end manner through the optimization. We
also adopt an edge dropout (Wu et al., 2021; Ma et al., 2022) while training
to enhance the performance robustness.

##  4\. Experiments

Table 1. Summary of three real-world datasets where “dens.” denotes the density of a matrix. Dataset | Users | Bundles | Items | User-bundle (dens.) | User-item (dens.) | Bundle-item (dens.) | Avg. size of bundle
---|---|---|---|---|---|---|---
Youshu111https://github.com/mysbupt/CrossCBR | 8,039 | 4,771 | 32,770 | 51,377 (0.13%) | 138,515 (0.05%) | 176,667 (0.11%) | 37.03
NetEase111https://github.com/mysbupt/CrossCBR | 18,528 | 22,864 | 123,628 | 302,303 (0.07%) | 1,128,065 (0.05%) | 1,778,838 (0.06%) | 77.80
iFashion111https://github.com/mysbupt/CrossCBR | 53,897 | 27,694 | 42,563 | 1,679,708 (0.11%) | 2,290,645 (0.10%) | 106,916 (0.01%) | 3.86
Table 2. Performance comparison of CoHeat and baseline cold-start methods on three real-world datasets.  Model | Youshu | NetEase | iFashion
---|---|---|---
Recall$@20$ | nDCG$@20$ | Recall$@20$ | nDCG$@20$ | Recall$@20$ | nDCG$@20$
_Cold_ | _Warm_ | _All_ | _Cold_ | _Warm_ | _All_ | _Cold_ | _Warm_ | _All_ | _Cold_ | _Warm_ | _All_ | _Cold_ | _Warm_ | _All_ | _Cold_ | _Warm_ | _All_
DropoutNet (Volkovs et al., 2017) | .0022 | .0336 | .0148 | .0007 | .0153 | .0055 | .0028 | .0154 | .0046 | .0015 | .0078 | .0024 | .0009 | .0060 | .0039 | .0008 | .0045 | .0027
CB2CF (Barkan et al., 2019) | .0012 | .0258 | .0028 | .0007 | .0208 | .0021 | .0016 | .0049 | .0027 | .0006 | .0027 | .0014 | .0009 | .0057 | .0066 | .0006 | .0043 | .0048
Heater (Zhu et al., 2020) | .0016 | .1753 | .0541 | .0007 | .0826 | .0286 | .0021 | .0125 | .0102 | .0010 | .0064 | .0054 | .0015 | .0217 | .0123 | .0010 | .0151 | .0083
GAR-CF (Chen et al., 2022) | .0015 | .1688 | .0529 | .0011 | .0726 | .0317 | .0010 | .0063 | .0014 | .0005 | .0035 | .0008 | .0013 | .0203 | .0090 | .0013 | .0143 | .0055
GAR-GNN (Chen et al., 2022) | .0013 | .0835 | .0358 | .0006 | .0569 | .0178 | .0009 | .0056 | .0027 | .0003 | .0030 | .0012 | .0065 | .0172 | .0126 | .0030 | .0107 | .0087
CVAR (Zhao et al., 2022) | .0008 | .1958 | .0829 | .0002 | .1112 | .0533 | .0002 | .0308 | .0156 | .0001 | .0154 | .0084 | .0007 | .0220 | .0125 | .0004 | .0152 | .0084
CLCRec (Wei et al., 2021) | .0137 | .0626 | .0367 | .0087 | .0317 | .0194 | .0136 | .0407 | .0259 | .0075 | .0215 | .0138 | .0053 | .0203 | .0126 | .0043 | .0135 | .0085
CCFCRec (Zhou et al., 2023) | .0044 | .1554 | .0702 | .0022 | .0798 | .0425 | .0007 | .0265 | .0130 | .0004 | .0128 | .0068 | .0005 | .0439 | .0252 | .0003 | .0304 | .0172
CoHeat (ours) | .0183 | .2804 | .1247 | .0105 | .1646 | .0833 | .0191 | .0847 | .0453 | .0093 | .0455 | .0264 | .0170 | .1156 | .0658 | .0096 | .0876 | .0504

In this section, we perform experiments to answer the following questions.

  * Q1.

Comparison with cold-start methods. Does CoHeat show superior performance in
comparison to other cold-start methods in bundle recommendation?

  * Q2.

Comparison with warm-start methods. Does CoHeat show similar performance in
warm-start bundle recommendation compared with baselines, although CoHeat is a
cold-start bundle recommendation method?

  * Q3.

Ablation study. How do the main ideas of CoHeat affect the performance?

  * Q4.

Effect of the maximum temperature. How does the maximum temperature
$\epsilon$, the critical hyperparameter, affect the performance of CoHeat?

###  4.1. Experimental Setup

Datasets. We use three real-world bundle recommendation datasets as summarized
in Table 1. Youshu (Chen et al., 2019b) comprises bundles of books sourced
from a book review site; NetEase (Cao et al., 2017) features bundles of music
tracks from a cloud music service; iFashion (Chen et al., 2019a) consists of
bundles of fashion items from an outfit sales platform.

Baseline cold-start methods. We compare CoHeat with existing cold-start item
recommendation methods because they can be easily adapted for bundle
recommendation by considering bundle-item affiliations as content information.
DropoutNet (Volkovs et al., 2017) is a robustness-based method with a dropout
operation. CB2CF (Barkan et al., 2019) and Heater (Zhu et al., 2020) are
constraint-based methods that regularize the alignment. GAR (Chen et al.,
2022) is a generative method with two variants GAR-CF and GAR-GNN. CVAR (Zhao
et al., 2022) is another generative method with a conditional decoder. CLCRec
(Wei et al., 2021) and CCFCRec (Zhou et al., 2023) are contrastive learning-
based methods. We use bundle-item multi-hot vectors as their content
information.

Baseline warm-start methods. We also compare CoHeat with previous warm-start
recommendation methods. MFBPR (Rendle et al., 2009) and LightGCN (He et al.,
2020a) are item recommendation methods with the modelings of matrix
factorization and graph learning, respectively. SGL (Wu et al., 2021), SimGCL
(Yu et al., 2022), and LightGCL (Cai et al., 2023) are the improved methods of
item recommendation with contrastive learning approaches. DAM (Chen et al.,
2019b) is a bundle recommendation method with the modeling of matrix
factorization. BundleNet (Deng et al., 2020), BGCN (Chang et al., 2020, 2023),
and CrossCBR (Ma et al., 2022) are other bundle recommendation methods with
the modeling of graph learning.

Evaluation metrics. We use Recall$@k$ and nDCG$@k$ metrics as in previous
works (Wei et al., 2021; Ma et al., 2022). Recall$@k$ measures the proportion
of relevant items in the top-$k$ list, while nDCG$@k$ weighs items by their
rank. We set $k$ to $20$. In tables, bold and underlined values indicate the
best and second-best results, respectively.

Experimental process. We conduct experiments in warm-start, cold-start, and
all-bundle scenarios as in previous works (Wei et al., 2021). For the warm-
start scenario, interactions are split into 7:1:2 subsets for training,
validation, and testing. In the cold-start scenario, bundles are split in
7:1:2 ratio. In the all-bundle scenario, interactions are split in 7:1:2 ratio
with a half for warm-start and the other half for cold-start bundles. We
report the best Recall$@20$ and nDCG$@20$ within 100 epochs, averaged over
three runs.

Hyperparameters. We utilize the baselines with their official implementations
and use their reported best hyperparameters. We implement CoHeat with PyTorch.
We set the dimensionality $d$ of node embeddings as $64$. The other
hyperparameters are grid-searched: the learning rate in {$0.001$, $0.0001$,
$0.00001$}, $\lambda_{1}$ in {$0.1$, $0.2$, $0.5$, $1.0$}, $\lambda_{2}$ in
{$0.00004$, $0.0001$, $0.0004$, $0.001$}, $K$ in {$1$, $2$}, and the maximum
temperature in {$10^{1}$, $10^{2}$, $10^{3}$, $10^{4}$, $10^{5}$, $10^{6}$}.

###  4.2. Comparison with Cold-start Methods (Q1)

In Table 2, we compare CoHeat with baseline cold-start methods. The results
show that CoHeat consistently surpasses the baselines across all datasets and
settings, verifying its superiority. Notably, CoHeat achieves 193% higher
nDCG@$20$ compared to CCFCRec, the best competitor, on the iFashion dataset in
the all-bundle scenario.

###  4.3. Comparison with Warm-start Methods (Q2)

Table 3 compares CoHeat with baseline warm-start methods in the warm-start
scenario. Even though CoHeat is primarily designed for cold-start bundle
recommendation, it surpasses all the baselines in the warm-start scenario.
This indicates CoHeat effectively learns representations from both history-
view and affiliation-view by treating the extremely skewed distribution of
user-bundle interactions. For the baselines, the performance improves when
contrastive learning is used as exemplified in SGL, SimGCL, LightGCL, and
CrossCBR. Additionally, graph-based models such as LightGCN, SGL, SimGCL,
LightGCL, BundleNet, BGCN, and CrossCBR excel over other non-graph-based
models. In light of these observations, CoHeat strategically exploits a graph-
based modeling approach and harnesses the power of contrastive learning. This
makes CoHeat robustly achieve the highest performance across diverse
scenarios.

Table 3. Performance comparison of CoHeat and baseline warm-start methods on three real-world datasets.  Model | Youshu | NetEase | iFashion
---|---|---|---
Recall | nDCG | Recall | nDCG | Recall | nDCG
$@20$ | $@20$ | $@20$ | $@20$ | $@20$ | $@20$
MFBPR (Rendle et al., 2009) | .1959 | .1117 | .0355 | .0181 | .0752 | .0542
LightGCN (He et al., 2020a) | .2286 | .1344 | .0496 | .0254 | .0837 | .0612
SGL (Wu et al., 2021) | .2568 | .1527 | .0687 | .0368 | .0933 | .0690
SimGCL (Yu et al., 2022) | .2691 | .1593 | .0710 | .0377 | .0919 | .0677
LightGCL (Cai et al., 2023) | .2712 | .1607 | .0722 | .0388 | .0943 | .0686
DAM (Chen et al., 2019b) | .2082 | .1198 | .0411 | .0210 | .0629 | .0450
BundleNet (Deng et al., 2020) | .1895 | .1125 | .0391 | .0201 | .0626 | .0447
BGCN (Chang et al., 2020, 2023) | .2347 | .1345 | .0491 | .0258 | .0733 | .0531
CrossCBR (Ma et al., 2022) | .2776 | .1641 | .0791 | .0433 | .1133 | .0875
CoHeat (ours) | .2804 | .1646 | .0847 | .0455 | .1156 | .0876

###  4.4. Ablation Study (Q3)

Table 4 provides an ablation study that compares CoHeat with its three
variants CoHeat-_PC_ , CoHeat-_CH_ , and CoHeat-_AU_. This study is conducted
in the cold-start scenario, which is the primary focus of our work. In CoHeat-
_PC_ , we remove the influence of popularity-based coalescence by setting the
value of $\gamma_{b}^{(t)}$ in Equation (5) to a constant $0.5$. For CoHeat-
_CH_ , we exploit an anti-curriculum learning strategy. The temperature in
Equation (7) is defined as $t:T\rightarrow 0$, initiating the learning process
with the affiliation-view and gradually shifting the focus to the history-
view. For CoHeat-_AU_ , we omit $\mathcal{L}_{AU}$ from Equation (13), thereby
excluding the contrastive learning between the two views. As shown in the
table, CoHeat consistently outperforms all the variants, which verifies all
the main ideas help improve the performance. In particular, CoHeat-_PC_ shows
a severe performance drop, justifying the importance of satisfying Properties
1 and 2 when addressing the extreme skewness inherent in cold-start bundle
recommendation.

###  4.5. Effect of the Maximum Temperature (Q4)

The maximum temperature $\epsilon$ in Equation (7) is the most influential
hyperparameter of CoHeat since it directly affects both popularity-based
coalescence and curriculum heating. Accordingly, we analyze the influence of
$\epsilon$ in cold-start scenario on real-world datasets, as depicted in
Figure 4. As shown in the figure, CoHeat shows low performance for the extreme
low temperature because the representations of affiliation-view are not
sufficiently learned. For the extreme high temperature, the performance
degrades because the speed of the curriculum is too fast to fully learn the
representation of the two views. As a result, we set $\epsilon$ to $10^{4}$
for all datasets since it shows the best performance.

##  5\. Related Works

Bundle recommendation. Our work focuses on the cold-start problem in bundle
recommendation. Previous works can be categorized based on their modeling
structures: matrix factorization-based models (Pathak et al., 2017; Cao et
al., 2017; Chen et al., 2019b) and graph learning-based models (Deng et al.,
2020; Chang et al., 2020, 2023; Ma et al., 2022). Such methods operate under
the assumption that all bundles have historical interactions, which makes them
ill-suited for tackling the cold-start problem. However, in real-world
scenarios, new bundles are introduced daily, leading to an inherent cold-start
challenge. Our work addresses this significant yet overlooked issue,
recognizing its potential impact on the field.

Cold-start recommendation. The cold-start problem, a longstanding challenge in
recommender systems, focuses on recommending cold-start items that have yet to
be interacted with users. Existing works are mainly divided into generative
methods (van den Oord et al., 2013; Chae et al., 2019; Sun et al., 2020; Chen
et al., 2022; Zhao et al., 2022), dropout-based methods (Volkovs et al., 2017;
Du et al., 2020; Shi et al., 2019), meta-learning methods (Pan et al., 2019),
and constraint-based methods (Barkan et al., 2019; Zhu et al., 2020; Wei et
al., 2021; Zhou et al., 2023). However, such prior works have not explicitly
addressed the highly skewed distribution of interactions, a critical aspect in
bundle recommendation. Thus, our work excels over these methods in cold-start
bundle recommendation by effectively considering the skewed distribution
during training.

##  6\. Conclusion

Table 4.  Ablation study of CoHeat in cold-start scenario which is our main target.  Model | Youshu | NetEase | iFashion
---|---|---|---
Recall | nDCG | Recall | nDCG | Recall | nDCG
$@20$ | $@20$ | $@20$ | $@20$ | $@20$ | $@20$
CoHeat-_PC_ | .0000 | .0000 | .0000 | .0000 | .0000 | .0000
CoHeat-_CH_ | .0177 | .0087 | .0176 | .0087 | .0164 | .0093
CoHeat-_AU_ | .0069 | .0031 | .0029 | .0013 | .0013 | .0005
CoHeat (ours) | .0183 | .0105 | .0191 | .0093 | .0170 | .0096

(a) $\epsilon$ vs. Recall$@20$

(b) $\epsilon$ vs. nDCG$@20$

Figure 4. Effect of the maximum temperature $\epsilon$.

We propose CoHeat, an accurate method for cold-start bundle recommendation.
CoHeat strategically leverages history and affiliation views to handle the
extremely skewed distribution of bundle interactions. By emphasizing the
affiliation-view for less popular bundles, CoHeat effectively captures richer
information than the often sparse history-view. The incorporation of
curriculum learning further enhances the learning process, starting with the
simpler history-view embeddings and gradually transitioning to the more
intricate affiliation-view embeddings. In addition, the contrastive learning
of CoHeat bolster the learning of representations of the two views. Extensive
experiments show that CoHeat provides the state-of-the-art performance in
cold-start bundle recommendation, achieving up to $193\%$ higher nDCG$@20$
compared to the best competitor.

###### Acknowledgements.

This work was supported by Jung-Hun Foundation. This work was also supported
by Institute of Information & communications Technology Planning & Evaluation
(IITP) grant funded by the Korea government(MSIT) [NO.2021-0-01343, Artificial
Intelligence Graduate School Program (Seoul National University)] and
[NO.2021-0-02068, Artificial Intelligence Innovation Hub (Artificial
Intelligence Institute, Seoul National University)]. The Institute of
Engineering Research and ICT at Seoul National University provided research
facilities for this work. U Kang is the corresponding author.

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