# Dual Adversarial Variational Embedding for Robust Recommendation

Qiaomin Yi, Ning Yang, Philip S. Yu  Qiaomin Yi is with the School of Computer
Science, Sichuan University, China. E-mail<EMAIL_ADDRESS>Ning Yang
is the corresponding author and with the School of Computer Science, Sichuan
University, China. E-mail<EMAIL_ADDRESS>Philip S. Yu is with the
Department of Computer Science, University of Illinois at Chicago, USA.
E-mail<EMAIL_ADDRESS>
###### Abstract

Robust recommendation aims at capturing true preference of users from noisy
data, for which there are two lines of methods have been proposed. One is
based on noise injection, and the other is to adopt the generative model
Variational Auto-encoder (VAE). However, the existing works still face two
challenges. First, the noise injection based methods often draw the noise from
a fixed noise distribution given in advance, while in real world, the noise
distributions of different users and items may differ from each other due to
personal behaviors and item usage patterns. Second, the VAE based models are
not expressive enough to capture the true preference since VAE often yields an
embedding space of a single modal, while in real world, user-item interactions
usually exhibit multi-modality on user preference distribution. In this paper,
we propose a novel model called Dual Adversarial Variational Embedding (DAVE)
for robust recommendation, which can provide personalized noise reduction for
different users and items, and capture the multi-modality of the embedding
space, by combining the advantages of VAE and adversarial training between the
introduced auxiliary discriminators and the variational inference networks.
The extensive experiments conducted on real datasets verify the effectiveness
of DAVE on robust recommendation.

###### Index Terms:

Robust Recommendation, Adversarial Variational Embedding, Adversarial Training

##  1 Introduction

Recommender systems have been attracting growing interest of researchers due
to their vital role in various online applications, such as e-commerce and
social media. In recommender systems, recommendation models are learned from
historical interaction (feedback) data which are often seen as noise-free by
most of the existing works. In big data era, however, data are usually full of
noise. For example, one click of a user might be a random operation which
cannot represent the true preference of the user. Noisy data will cause the
recommendation models without robustness to weak generalizability and
inability to capture true preference from data with even small perturbations
[32].

Recently, a few methods have been proposed for robust recommendation, which
roughly follow two lines. One line of the existing works improves the
robustness of recommendation models by injecting extra noise to training data
or model parameters during training process [30, 9, 32, 14], while the other
line adopts a generative process to learn powerful recommendation models with
noise tolerability [31, 16, 25]. However, robust recommendation is still far
from being well solved partly due to the following challenges.

  * •

Personalized Noise Reduction In the line of the existing works that obtain
robustness by injecting extra noise to training data or model parameters, the
added noise is often drawn from a fixed probability distribution shared by
different users [30, 9, 32, 14], where the underlying assumption is that data
of different users have the same noise level. In real world, however, the
noise in data of different users and items has different distribution, due to
different behaviors and item usage patterns. Therefore, as part of
personalization, robust recommendation is expected to provide personalized
noise reduction with adaptability to different noise distributions.

  * •

Multimodal Distribution of Preference Inspired by the success of Variational
Auto-encoder (VAE) in computer vision, one line of the existing works on
robust recommendation captures user preference by latent embeddings generated
from VAE based models [15, 16, 1, 25]. However, recent studies show that VAE
tends to yield a latent space with a single modal that is not expressive
enough to capture the true posterior distribution of embeddings [19]. In the
context of recommender systems, user-item interactions often exhibit multi-
modality on user preference distribution, i.e., different users have different
preference distributions around different modes. For example, in music
recommender systems, users’ preferences to music styles can be separated into
multiple clusters each of which can be viewed as a unique distribution, say, a
Gaussian distribution with its unique mean (a specific music style) and
variance. Such multi-modality means that users’ preferences should be
approximated with multiple distributions rather than with only one as the
existing works did. Therefore, to improve the robustness of the embedding
learning for users and items, we need a model expressive enough to capture the
multi-modality of preference.

In this paper, to address the above challenges, we propose a novel model
called Dual Adversarial Variational Embedding (DAVE) for robust
recommendation. The main idea of DAVE is to adaptively capture the different
noise distributions of different users or items and the multi-modality of
preference with an inferred distribution unique to a user or an item, using
variational inference combined with adversarial training. At first, to provide
personalized noise reduction for different users and items, DAVE introduces
two VAEs to infer a unique latent distribution for each user and item in a
dual form, respectively, from which user and item embeddings against noise can
be drawn and then fed into a neural collaborative filtering network [10] for
subsequent preference prediction. Here the advantages are two-fold. The first
advantage is that by the power of variational inference, DAVE can adaptively
generate a unique embedding distribution for each user and item for their
personalized noise reduction, instead of manually setting a fixed noise level.
The second advantage is that to enhance the robustness of embeddings, the
noise is modeled with the variance of the inferred distributions, which can be
viewed as corrupting the latent space by learning the stochastic noise of
user-item interactions, unlike the traditional methods where embeddings of
users and items are essentially a point estimation. In DAVE, the VAEs are
trained jointly with the subsequent neural collaborative filtering network,
where the objectives are minimizing the decoding error of VAE and the
prediction error of neural collaborative filtering simultaneously.

To improve the expressiveness of DAVE for capturing the preference multi-
modality, inspired by the idea of Adversarial Variational Bayes [19], we
further introduce two auxiliary discriminators together with the inference
networks of the VAEs to form two Generative Adversarial Networks (GAN), which
can regularize the learning of the inference networks of the VAEs for users
and items, respectively. Under the framework of GAN, the discriminator, which
estimates the probabilities of sampling from the true distribution, and the
generator, which captures the underlying data distribution, are jointly
trained in an adversarial fashion. In DAVE, the inference network of VAE plays
the role of generator. Unlike traditional VAE training where an explicit
representation of the posterior distribution is required for the computation
of the KL-divergence, the adversarial training between the inference network
and the auxiliary discriminator can approximate the KL-divergence
regularization for any complex posterior distributions, without the need for
explicitly representing the posterior distribution with parametric expression.
Such flexibility enables DAVE to capture the multi-modality of user
preferences by inferring complex posterior distributions that are multimodal
and cannot be explicitly formulated.

The main contributions of this paper are summarized as follows:

  1. (1)

We propose a novel model called Dual Adversarial Variational Embedding (DAVE)
for robust recommendation, which can provide personalized noise reduction for
different users and items, and capture the multi-modality of preference.

  2. (2)

For the personalized noise reduction, we introduce two VAEs, which are jointly
learned with a neural collaborative filtering network, to infer a unique
embedding distribution for each user and item, respectively. Due to the
variational inference power of VAE, the noise levels of different users or
items can be adaptively captured by their own distributions from which robust
embeddings can be drawn.

  3. (3)

To capture the multi-modality of preference, we introduce two auxiliary
discriminators for user and item, respectively, to regularize the learning of
the inference networks via an adversarial training, which endows DAVE with the
flexibility to infer the preference distributions with multi-modality.

  4. (4)

We conduct extensive experiments on real world datasets and the experimental
results verify the effectiveness of the proposed model.

The rest of this paper is organized as follows. In Section 2, we introduce the
preliminaries and formally define the target problem. We present the details
of DAVE in Section 3. In Section 4, we empirically evaluate the performance of
DAVE over real world datasets, verify the robustness and expressiveness of
DAVE, and analyze the influence of hyper-parameters. At last, we briefly
review the related works in Section 5 and conclude in Section 6.

##  2 Preliminaries and Problem Definition

###  2.1 Notation Definition

Let $\bm{U}$ be the set of $N$ users, and $\bm{V}$ be the set of $M$ items. We
define the user-item interaction matrix as $\bm{R}\in\mathbb{R}^{N\times M}$
based on implicit feedbacks (e.g., clicking, buying, or commenting an item),
where $R_{uv}$ = 1 if the interaction between user $u\in\bm{U}$ and item
$v\in\bm{V}$ is observed, otherwise $R_{uv}$ = 0. We associate each user
$u\in\bm{U}$ with two vectors. One is the user interaction vector
$\bm{u}\in\\{0,1\\}^{M}$, which is the transpose of $u$-th row of $\bm{R}$,
and the other is the user embedding $\bm{x}_{u}\in\mathbb{R}^{d}$, where $d$
is the dimensionality of user latent representation. Similarly, each item
$\mathnormal{v}$$\in$$\bm{V}$ are also associated with two vectors. One is the
item interaction vector $\bm{v}\in\\{0,1\\}^{N}$, which corresponds to the
$v$-th column of $\bm{R}$, and the other is the item embedding
$\bm{y}_{v}\in\mathbb{R}^{d}$. The set of items that a user $u$ has interacted
with is denoted by $\bm{V}_{\text{u}}$, i.e.,
$\bm{V}_{\text{u}}=\\{v|R_{uv}=1\\}$. Its complementary set is denoted by
$\bm{\overline{V}}_{\text{u}}$, i.e.,
$\bm{\overline{V}}_{\text{u}}=\bm{V}\setminus\bm{V}_{\text{u}}=\\{v|R_{uv}=0\\}$,
which is the set of items that $u$ has not interacted with.

Figure 1: The architecture of DAVE.

###  2.2 Problem Definition

Given a user set $\bm{U}$, an item set $\bm{V}$, and the observed interaction
matrix $\bm{R}$, our goal is to recommend to a specific user $u\in\bm{U}$ an
item $v\in\bm{\overline{V}}_{\text{u}}$ with maximal predicted
$\widehat{R}_{uv}$. The predicted score $\widehat{R}_{uv}$ is constrained to
the range [0, 1], which represents the probability of $u$ will interact with
$v$.

TABLE I: Notations Symbol | Description
---|---
$\bm{u}$ | user interaction vector
$\bm{v}$ | item interaction vector
$\bm{x}_{u}$ | embedding of user $u$
$\bm{y}_{v}$ | embedding of item $v$
$\widehat{\bm{u}}$ | reconstructed user interaction vector
$\widehat{\bm{v}}$ | reconstructed item interaction vector
$p_{\text{u}}(\cdot)$ | user prior
$p_{\text{v}}(\cdot)$ | item prior
$q_{\text{u}}(\cdot|\bm{u})$ | posterior of user embeddings
$q_{\text{v}}(\cdot|\bm{v})$ | posterior of item embeddings
$g_{\text{u}}(\cdot|\bm{x}_{u})$ | distribution of reconstructed user interaction vectors
$g_{\text{v}}(\cdot|\bm{y}_{v})$ | distribution of reconstructed item interaction vectors
$Q_{\text{u}}(\cdot;\Phi_{\text{u}})$ | user inference network with parameters $\Phi_{\text{u}}$
$Q_{\text{v}}(\cdot;\Phi_{\text{v}})$ | item inference network with parameters $\Phi_{\text{v}}$
$D_{\text{u}}(\cdot;\Psi_{\text{u}})$ | user discriminator with parameters $\Psi_{\text{u}}$
$D_{\text{v}}(\cdot;\Psi_{\text{v}})$ | item discriminator with parameters $\Psi_{\text{v}}$
$G_{\text{u}}(\cdot;\Theta_{\text{u}})$ | user probabilistic decoder with parameters $\Theta_{\text{u}}$
$G_{\text{v}}(\cdot;\Theta_{\text{v}})$ | item probabilistic decoder with parameters $\Theta_{\text{v}}$
$\epsilon_{u}$ | auxiliary noise for user embedding
$\epsilon_{v}$ | auxiliary noise for item embedding

##  3 The Proposed Method

In this section, we first present the architecture of the proposed Dural
Adversarial Variational Embedding (DAVE) model, and then describe its
optimization objective and learning in detail.

###  3.1 Architecture of DAVE

Figure 1 shows the architecture of DAVE, where the main notations are
summarized in Table I. From the Vertical view, we can see that DAVE comprises
two dual parts including the User Adversarial Embedding (UserAVE) and the Item
Adversarial Embedding (ItemAVE), which respectively take a user interaction
vector $\bm{u}$ and an item interaction vector $\bm{v}$ as inputs, and
generate the user embedding $\bm{x}_{u}$ and the item embedding $\bm{y}_{v}$.
Once the user and item embeddings are prepared, DAVE will make the rating
prediction $\widehat{R}_{uv}$ by feeding the embeddings into a neural
collaborative filtering function $F(\bm{x}_{u},\bm{y}_{v})$ which is realized
by an MLP network [10].

In Figure 1, the UserAVE consists of three parts, (1) user inference network
($Q_{\text{u}}$), which takes the user interaction vector ($\bm{u}$) to
generate the user embedding ($\bm{x}_{u}$), (2) user probabilistic decoder
($G_{\text{u}}$), which reconstructs the the user interaction vector
($\widehat{\bm{u}}$) from the user embedding ($\bm{x}_{u}$), and (3) the user
discriminator ($D_{\text{u}}$) to help the user inference network in (1) to
generate more robust user embedding.

In particular, UserAVE uses a variational inference network
$Q_{\text{u}}(\bm{u};\Phi_{\text{u}})$ with parameters $\Phi_{\text{u}}$ to
infer a unique posterior distribution $q_{\text{u}}(\bm{x}_{u}|\bm{u})$ of the
latent representation for each user $u$, from which the embedding $\bm{x}_{u}$
of user $u$ can be drawn out. The inference network
$Q_{\text{u}}(\bm{u};\Phi_{\text{u}})$ is trained jointly with the
probabilistic decoder $G_{\text{u}}(\bm{x}_{u};\Theta_{\text{u}})$ with
parameters $\Theta_{\text{u}}$, which reconstructs the user interaction vector
$\bm{u}$ as $\widehat{\bm{u}}$ with the probability
$g_{\text{u}}(\bm{u}|\bm{x}_{u})$ that minimizes the reconstruction error. Due
to the merit of variational inference, UserVAE can capture the different noise
distributions of the interaction data for different users by the variance of
their unique posterior distribution. Note that the dashed arrow represents
sampling auxiliary noise $\epsilon_{u}$ from the standard normal distribution,
which will be used for the reparameterization trick [12, 23] to optimize the
inference network.

As we have mentioned before, due to the nature of VAE, the inferred posterior
distributions $q_{\text{u}}(\bm{x}_{u}|\bm{u})$ for different users tend to
lie around a single mode [12], which might make the embedding spaces of
different users indistinguishable from each other and consequently fail to
accurately capture the user personalized preference. To improve the
expressiveness of the user embeddings, we further introduce an auxiliary
discriminator $D_{\text{u}}(\bm{x}_{u};\Psi_{\text{u}})$ with parameters
$\Psi_{\text{u}}$, which together with the generator
$Q_{\text{u}}(\bm{u};\Phi_{\text{u}})$ forms a Generative Adversarial Network
(GAN). The optimization objective of
$D_{\text{u}}(\bm{x}_{u};\Psi_{\text{u}})$ is to distinguish the user
embeddings drawn from two distributions: the inferred posterior distribution
$q_{\text{u}}(\bm{x}_{u}|\bm{u})$ unique to each user, and a given prior
distribution $p_{\text{u}}(\bm{x}_{u})\sim\mathcal{N}(\mathbf{0},\mathbf{I})$
shared among different users. As we will see in the experiments, the
adversarial training of the discriminator
$D_{\text{u}}(\bm{x}_{u};\Psi_{\text{u}})$ against the generator
$Q_{\text{u}}(\bm{u};\Phi_{\text{u}})$ can approximate the KL-divergence
regularization between any complex posterior distributions and the prior
distribution, which offers DAVE the flexibility without the need to make an
explicit assumption about the posterior distribution, and enables DAVE to
infer complex posterior distributions that are multimodal and cannot be
formulated explicitly.

As the dual part, the structure of ItemAVE is similar to that of UserAVE.
Particularly, ItemAVE also generates the robust item embeddings $\bm{y}_{v}$
via a VAE with item interaction vector $\bm{v}$ as input, where the
variational inference network $Q_{\text{v}}(\bm{v};\Phi_{\text{v}})$ with
parameters $\Phi_{\text{v}}$, the probabilistic decoder
$G_{\text{v}}(\bm{y}_{v};\Theta_{V})$ with parameters $\Theta_{V}$, and the
auxiliary noise variable $\epsilon_{v}$ are the counterparts of
$Q_{\text{u}}(\bm{u};\Phi_{\text{u}})$,
$G_{\text{u}}(\bm{x}_{u};\Theta_{\text{u}})$, and $\epsilon_{u}$ in UserAVE.
Again due to the variational inference power, the item embeddings will also
benefit from the item posterior distribution $q_{\text{v}}(\bm{y}_{v}|\bm{v})$
unique to each item $v$, which makes them adaptable to the different noise
distributions in interaction data of different items. Similar to UserAVE,
ItemAVE introduces an auxiliary discriminator
$D_{\text{v}}(\bm{y}_{v};\Psi_{\text{v}})$ with parameters $\Psi_{\text{v}}$,
of which the role is also to distinguish the item embeddings drawn from each
inferred unique posterior distributions $q_{\text{v}}(\bm{y}_{v}|\bm{v})$ from
those drawn from a given prior distribution
$p_{\text{v}}(\bm{y}_{v})\sim\mathcal{N}(\mathbf{0},\mathbf{I})$. Similar to
UserAVE, the multi-modality of the inferred posterior embedding distributions
of different items can also be captured due to the flexibility offered by the
adversarial training of the $D_{\text{v}}(\bm{y}_{v};\Psi_{\text{v}})$ and
$Q_{\text{v}}(\bm{v};\Phi_{\text{v}})$.

###  3.2 Objective Function

As UserAVE, ItemAVE, and the neural collaborative filtering network
$F(\bm{x}_{u},\bm{y}_{v})$ will be jointly trained in an end-to-end fashion,
the overall objective function of DAVE is defined as

$\begin{split}\mathcal{L}=\mathcal{L}_{\text{u}}+\mathcal{L}_{\text{v}}+\mathcal{L}_{\text{f}},\end{split}$
(1)

where $\mathcal{L}_{\text{u}}$, $\mathcal{L}_{\text{v}}$, and
$\mathcal{L}_{\text{f}}$ are the objective functions of UserAVE, ItemAVE, and
$F(\bm{x}_{u},\bm{y}_{v})$, respectively, which will be detailed in the
following subsections.

####  3.2.1 Objective Function of UserAVE

As we have mentioned early, UserAVE consists of a VAE and an auxiliary GAN,
and therefore its optimization objective is

$\mathcal{L}_{\text{u}}=\mathcal{L}_{\text{u}}^{\text{VAE}}+\mathcal{L}_{\text{u}}^{\text{D}},$
(2)

where $\mathcal{L}_{\text{u}}^{\text{VAE}}$ and
$\mathcal{L}_{\text{u}}^{\text{D}}$ are the objective functions of VAE and the
auxiliary GAN in UserAVE, respectively.

Traditional objective function of VAE usually regularizes the variational
inference network $Q_{\text{u}}$ using KL-divergence, which cannot serve our
purpose to learn multimodal embedding space as it will cause the inferred
posterior distributions $q_{\text{u}}(\bm{x}_{u}|\bm{u})$ indistinguishable
for different $\bm{u}$. Our overall idea to overcome this issue is to instead
regularize $Q_{\text{u}}$ via an adversarial training between it and the
auxiliary discriminator $D_{\text{u}}$. Due to such adversarial
regularization, the inferred posterior distributions
$q_{\text{u}}(\bm{x}_{u}|\bm{u})$ of different users can stay away from each
other, which benefits the learning of multimodal embedding space.

As the observed data are just the user interaction vectors $\bm{u}$, we will
learn the parameters of VAE, $\Phi_{\text{u}}$ and $\Theta_{\text{u}}$, by
maximizing the log-likelihood $\log p(\bm{u})$, where $p(\bm{u})$ is the
distribution of user interaction data. By applying Variational Bayes and
Jensen’s inequality [12], for a specific user $u$ we have

$\begin{split}\log p(\bm{u})\geq\mathbb{E}_{\bm{x}_{u}\sim
q_{\text{u}}(\bm{x}_{u}|\bm{u})}&[\log g_{\text{u}}(\bm{u}|\bm{x}_{u})]\\\
&-\text{KL}(q_{\text{u}}(\bm{x}_{u}|\bm{u})\parallel
p_{\text{u}}(\bm{x_{\text{u}}})),\end{split}$ (3)

where the right side is the evidence lower bound (ELBO) [3] also known as
variational lower bound, and $p_{\text{u}}(\bm{x_{\text{u}}})$ is the Gaussian
prior of $\bm{x}_{u}$. In the ELBO, the variational distribution
$q_{\text{u}}(\bm{x}_{u}|\bm{u})$ is also a Gaussian distribution with mean
$\bm{\mu}_{u}$ and variance $\bm{\sigma}_{\text{u}}^{2}$, which are outputs of
the inference network $q_{\text{u}}(\bm{u};\Phi_{\text{u}})$, and
$g_{\text{u}}(\bm{u}|\bm{x}_{u})$ is the reconstruction probability dependent
on the decoder $G_{\text{u}}(\Theta_{\text{u}})$. In variational inference,
maximizing the log-likelihood is reduced to the maximization of ELBO over user
set $\bm{U}$, i.e.,

$\begin{split}\mathcal{L}_{\text{u}}^{\text{VAE}}(\Theta_{\text{u}},\Phi_{\text{u}})=\mathbb{E}_{\bm{u}\sim
p(\bm{u})}&\Big{[}\mathbb{E}_{\bm{x}_{u}\sim
q_{\text{u}}(\bm{x}_{u}\mid\bm{u})}[\log
g_{\text{u}}(\bm{u}\mid\bm{x}_{u})]\\\
&-\text{KL}(q_{\text{u}}(\bm{x}_{u}\mid\bm{u})\parallel
p_{\text{u}}(\bm{x}_{u}))\Big{]}.\end{split}$ (4)

Note that the reconstruction probability $g_{\text{u}}(\bm{u}\mid\bm{x}_{u})$
implicitly represents the negative reconstruction error [12], and therefore,
maximizing its expectation is equivalent to minimizing the expected
reconstruction error.

As the KL-divergence is

$\mathbb{E}_{\bm{x}_{u}\sim q_{\text{u}}(\bm{x}_{u}\mid\bm{u})}[\log
q_{\text{u}}(\bm{x}_{u}|\bm{u})-\log p_{\text{u}}(\bm{x}_{u})],$ (5)

we can rewrite the objective function (4) as

$\begin{split}\mathcal{L}_{\text{u}}^{\text{VAE}}(\Theta_{\text{u}},\Phi_{\text{u}})=\mathbb{E}_{\bm{u}\sim
p(\bm{u})}&\mathbb{E}_{\bm{x}_{u}\sim
q_{\text{u}}(\bm{x}_{u}\mid\bm{u})}\big{[}\log
g_{\text{u}}(\bm{u}\mid\bm{x}_{u})\\\ &+\log p_{\text{u}}(\bm{x}_{u})-\log
q_{\text{u}}(\bm{x}_{u}|\bm{u})\big{]}.\end{split}$ (6)

When the variational posterior distribution $q_{\text{u}}(\bm{x}_{u}|\bm{u})$
has explicit representation that is tractable like Gaussian distribution, we
can directly compute the objective function for maximization. However, recent
studies [27, 19, 11] show that if the VAE is optimized only according to
Equation (6), the KL-divergence will make the variational posterior
distributions $q_{\text{u}}(\bm{x}_{u}|\bm{u})$ of different users all close
to the same prior $p_{\text{u}}(\bm{x}_{u})$, which results in inferior user
embeddings that are not expressive enough to capture the multi-modality of the
preference distributions of users. Therefore, to improve the expressiveness of
the variational inference network of UserAVE, we assume
$q_{\text{u}}(\bm{x}_{u}|\bm{u})$ is implicit and introduce an auxiliary
discriminator $D_{\text{u}}(\bm{x}_{u};\Psi_{\text{u}})$ together with the
inference network $Q_{\text{u}}(\bm{u};\Phi_{\text{u}})$ as generator to form
a GAN to help the inference of the implicit posterior distribution
$q_{\text{u}}(\bm{x}_{u}|\bm{u})$. It is easy to define the objective function
of the discriminator as

$\begin{split}\mathcal{L}_{\text{u}}^{\text{D}}(\Psi_{\text{u}})&=\mathbb{E}_{\bm{u}\sim
p_{(}\bm{u})}\mathbb{E}_{\bm{x}_{u}\sim
p_{\text{u}}(\bm{x}_{u})}\log\sigma(D_{\text{u}}(\bm{x}_{u};\Psi_{\text{u}}))\\\
&+\mathbb{E}_{\bm{u}\sim p(\bm{u})}\mathbb{E}_{\bm{x}_{u}\sim
q_{\text{u}}(\bm{x}_{u}\mid\bm{u})}\log(1-\sigma(D_{\text{u}}(\bm{x}_{u};\Psi_{\text{u}})),\end{split}$
(7)

where $\sigma(\cdot)$ is the sigmoid function. As we treat the samples from
the prior as true while ones from the variational posterior as fake, it is
easy to show that when the generator $Q_{\text{u}}(\bm{u};\Phi_{\text{u}})$ is
fixed (and equivalently, $q_{\text{u}}(\bm{x}_{u}\mid\bm{u})$ is fixed),
$\mathcal{L}_{\text{u}}^{\text{D}}$ achieves its maximum at
$\Psi^{*}_{\text{u}}$ such that

$D_{\text{u}}(\bm{x}_{u};\Psi^{*}_{\text{u}})=\log
p_{\text{u}}(\bm{x}_{u})-\log q_{\text{u}}(\bm{x}_{u}\mid\bm{u}),$ (8)

where $\Psi^{*}_{\text{u}}$ represents the optimal parameters of the
discriminator [7]. Therefore, Equation (6) can be further rewritten as

$\begin{split}\mathcal{L}_{\text{u}}^{\text{VAE}}(\Theta_{\text{u}},\Phi_{\text{u}})=\mathbb{E}_{\bm{u}\sim
p(\bm{u})}&\mathbb{E}_{\bm{x}_{u}\sim
q_{\text{u}}(\bm{x}_{u}\mid\bm{u})}\big{[}\log
g_{\text{u}}(\bm{u}\mid\bm{x}_{u})\\\
&+D_{\text{u}}(\bm{x}_{u};\Psi^{*}_{\text{u}})\big{]}.\end{split}$ (9)

As we have mentioned before, the insight here is that although the adversarial
training between $D_{\text{u}}$ and $Q_{\text{u}}$ will also cause the
posterior distributions inferred by $Q_{\text{u}}$ to be close to the prior,
which exactly approximates the regularization that minimizes the KL-divergence
of the posterior distribution to the prior, it obviates the requirement to
explicitly represent the posterior distribution. This is in contrast with the
existing VAE based models. Such flexibility enables DAVE to infer any complex
posterior distributions that are multimodal and cannot be formulated with a
simple parametric expression, with the result that the drawn user embeddings
have chance to stay far away from each other, which benefits the capturing of
diverse user preference distributions.

The user embeddings $\bm{x}_{u}$ are generated by sampling from
$q_{\text{u}}(\bm{x}_{u}|\bm{u})$, which, however, makes the objective
functions not differentiable. Using the reparameterization trick [12], we draw
an auxiliary noise $\bm{\epsilon}_{u}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$
and instead generate the user embeddings by a composite function
$\phi_{\text{u}}(\bm{u},\bm{\epsilon}_{u})=\phi_{\text{u}}(Q_{\text{u}}(\bm{u};\Phi_{\text{u}}),\bm{\epsilon}_{u})$.
Then Equations (7) and (9) can be finally rewritten as follows:

$\begin{split}\mathcal{L}_{\text{u}}^{\text{D}}&(\Psi_{\text{u}})=\mathbb{E}_{\bm{u}\sim
p_{(}\bm{u})}\mathbb{E}_{\bm{x}_{u}\sim
p_{\text{u}}(\bm{x}_{u})}\log\sigma(D_{\text{u}}(\bm{x}_{u};\Psi_{\text{u}}))\\\
&+\mathbb{E}_{\bm{u}\sim
p(\bm{u})}\mathbb{E}_{\bm{\epsilon}_{u}\sim\mathcal{N}(\mathbf{0},\mathbf{I})}\log(1-\sigma(D_{\text{u}}(\phi_{\text{u}}(\bm{u},\bm{\epsilon}_{u});\Psi_{\text{u}})),\end{split}$
(10)

and

$\begin{split}\mathcal{L}_{\text{u}}^{\text{VAE}}(\Theta_{\text{u}},\Phi_{\text{u}})=\mathbb{E}_{\bm{u}\sim
p(\bm{u})}&\mathbb{E}_{\bm{\epsilon}_{u}\sim\mathcal{N}(\mathbf{0},\mathbf{I})}\big{[}\log
g_{\text{u}}(\bm{u}\mid\phi_{\text{u}}(\bm{u},\bm{\epsilon}_{u}))\\\
&+D_{\text{u}}(\phi_{\text{u}}(\bm{u},\bm{\epsilon}_{u});\Psi^{*}_{\text{u}})\big{]}.\end{split}$
(11)

In the experiments of this paper, we define the reparameterization function as
$\phi_{\text{u}}(\bm{u},\bm{\epsilon}_{u})=\bm{\mu}_{u}+\bm{\sigma}_{u}\odot\bm{\epsilon}_{u}$,
where $\odot$ is element-wise product, $\bm{\mu}_{u}$ and
$\bm{\sigma}^{2}_{u}$ are the mean and variance of
$q_{\text{u}}(\bm{x}_{u}|\bm{u})$, respectively. Note that $\bm{\mu}_{u}$ and
$\bm{\sigma}^{2}_{u}$ are outputs of the inference network
$Q_{\text{u}}(\bm{u};\Phi_{\text{u}})$.

####  3.2.2 Objective Function of ItemAVE

As ItemAVE is the dual part of UserAVE, its objective function can be
similarly defined as

$\mathcal{L}_{\text{v}}=\mathcal{L}_{\text{v}}^{\text{VAE}}+\mathcal{L}_{\text{v}}^{\text{D}},$
(12)

where $\mathcal{L}_{\text{v}}^{\text{VAE}}$ and
$\mathcal{L}_{\text{v}}^{\text{D}}$ are the objective functions of VAE and the
auxiliary GAN in ItemAVE, respectively. Through a similar derivation,
$\mathcal{L}_{\text{v}}^{\text{VAE}}$ and $\mathcal{L}_{\text{v}}^{\text{D}}$
can be respectively defined as follows:

$\begin{split}\mathcal{L}_{\text{v}}^{\text{D}}&(\Psi_{\text{v}})=\mathbb{E}_{\bm{v}\sim
p_{(}\bm{v})}\mathbb{E}_{\bm{y}_{v}\sim
p_{\text{v}}(\bm{y}_{v})}\log\sigma(D_{\text{v}}(\bm{y}_{v};\Psi_{\text{v}}))\\\
&+\mathbb{E}_{\bm{v}\sim p(\bm{v})}\mathbb{E}_{\bm{y}_{v}\sim
q_{\text{v}}(\bm{y}_{v}\mid\bm{v})}\log(1-\sigma(D_{\text{v}}(\phi_{\text{v}}(\bm{v},\bm{\epsilon}_{v});\Psi_{\text{v}}))\end{split}$
(13)

and

$\begin{split}\mathcal{L}_{\text{v}}^{\text{VAE}}(\Theta_{\text{v}},\Phi_{\text{v}})=\mathbb{E}_{\bm{v}\sim
p(\bm{v})}&\mathbb{E}_{\bm{\epsilon}_{v}\sim\mathcal{N}(\mathbf{0},\mathbf{I})}\big{[}\log
g_{\text{v}}(\bm{v}\mid\phi_{\text{v}}(\bm{v},\bm{\epsilon}_{v}))\\\
&+D_{\text{v}}(\phi_{\text{v}}(\bm{v},\bm{\epsilon}_{v});\Psi^{*}_{\text{v}})\big{]},\end{split}$
(14)

where auxiliary noise
$\bm{\epsilon}_{v}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ and
$\Psi^{*}_{\text{v}}$ is the optimal parameters of the discriminator
$D_{\text{v}}(\bm{y}_{v};\Psi_{\text{v}})$. The reparameterization function is
defined as
$\phi_{\text{v}}(\bm{v},\bm{\epsilon}_{v})=\bm{\mu}_{v}+\bm{\sigma}_{v}\odot\bm{\epsilon}_{v}$,
where $\bm{\mu}_{v}$ and $\bm{\sigma}^{2}_{v}$ are the mean and variance of
$q_{\text{v}}(\bm{y}_{v}|\bm{v})$, respectively, which are outputs of the
inference network $Q_{\text{v}}(\bm{v};\Phi_{\text{v}})$ of ItemAVE.

####  3.2.3 Objective Function of Prediction

For a pair of user $u$ and $v$, once their embeddings $\bm{x}_{u}$ and
$\bm{y}_{v}$ are generated, DAVE will predict a score $\widehat{R}_{uv}$ of
$v$ given by $u$, by feeding the embeddings into the neural collaborative
filtering function $F(\bm{x}_{u},\bm{y}_{v})$ which is implemented as the
following MLP network [10]:

$\begin{split}&\bm{h}_{uv}=a_{L-1}(\bm{W}_{L-1}(...a_{1}(\bm{W}_{1}(\bm{x}_{u}\odot\bm{y}_{v})+\bm{b}_{1})...)+\bm{b}_{L-1})\\\
&\widehat{R}_{uv}=a_{L}(\bm{W}_{L}\bm{h}_{uv}+\bm{b}_{L}),\end{split}$ (15)

where $L$ is the number of layers, $\bm{W}_{i}$, $\bm{b}_{i}$, $a_{i}$ ($1\leq
i\leq L$) denote the weight matrix, bias vector, and activation function of
the $i$-th layer, respectively. In this paper, we choose ReLU for the
activation functions $a_{i}$ ($1\leq i\leq L-1$) of the hidden layers, and
Sigmoid function for the activation function $a_{L}$ of the output layer.

We reduce the score prediction to a binary classification over implicit
feedback matrix $\bm{R}$, of which the optimization objective can be defined
as to maximize the following log-likelihood function:

$\begin{split}\mathcal{L}_{\text{f}}(\Omega)=\sum_{u\in\bm{U},v\in\bm{V}}R_{uv}\log\widehat{R}_{uv}+(1-R_{uv})\log(1-\widehat{R}_{uv}),\end{split}$
(16)

where $\Omega=\\{\bm{W}_{i},\bm{b}_{i},1\leq i\leq L\\}$ is the parameters
that need to be learned. It is easy to show that maximizing the likelihood
$\mathcal{L}_{\text{f}}(\Omega)$ equivalently minimizes the classification
error.

Algorithm 1 Learning DAVE.

0: User-item interaction matrix $\bm{R}$, batchsize $B$, dimensionality of
embedding $d$.

0: DAVE parameters
$\Phi_{\text{u}},\Phi_{\text{v}},\Theta_{\text{u}},\Theta_{\text{v}},\Psi_{\text{u}},\Psi_{\text{v}},\Omega$.

1: Initialize the parameters.

2: repeat

3: Sample a mini-batch $\\{(\bm{u},\bm{v})\\}$ of size $B$.

4: Fixing $Q_{\text{u}}$, $G_{\text{u}}$, $Q_{\text{v}}$, $G_{\text{v}}$, and
$F$, generate the uer embedding $\bm{x}_{u}$ and the item embedding
$\bm{y}_{v}$ for each pair $(\bm{u},\bm{v})$ in the mini-batch by inference
networks $Q_{\text{u}}$ and $Q_{\text{v}}$, respectively.

5: For each $\bm{u}$, sample a $d$-dimensional vectors $\bm{x}^{\prime}_{u}$
from user prior $p_{\text{u}}(\bm{x}_{u})$.

6: For each $\bm{v}$, sample a $d$-dimensional vectors $\bm{y}^{\prime}_{v}$
from item prior $p_{\text{v}}(\bm{y}_{v})$.

7: Update the parameters $\Psi_{\text{u}}$ of $D_{\text{u}}$ with the gradient
of $\mathcal{L}_{\text{u}}^{\text{D}}(\Psi_{\text{u}})$ (Equation (10)), using
$\\{\bm{x}_{u}\\}$ as fake examples and $\\{\bm{x}^{\prime}_{u}\\}$ as real
examples.

8: Update the parameters $\Psi_{\text{v}}$ of $D_{\text{v}}$, using the
gradient of $\mathcal{L}_{\text{v}}^{\text{D}}(\Psi_{\text{u}})$ (Equation
(13)), using $\\{\bm{y}_{v}\\}$ as fake examples and
$\\{\bm{y}^{\prime}_{v}\\}$ as real examples.

9: Fixing $D_{\text{u}}$ and $D_{\text{v}}$, jointly update
$\Phi_{\text{u}},\Phi_{\text{v}},\Theta_{\text{u}},\Theta_{\text{v}}$, and
$\Omega$, with the gradient of the sum of
$\mathcal{L}_{\text{u}}^{\text{VAE}}$ (Equation (11)),
$\mathcal{L}_{\text{v}}^{\text{VAE}}$ (Equation (14)), and
$\mathcal{L}_{\text{f}}$ (Equation (16).

10: until convergence.

###  3.3 Model Learning

DAVE will be trained with the following objective

$\max_{\Phi_{\text{u}},\Phi_{\text{v}},\Theta_{\text{u}},\Theta_{\text{v}},\Psi_{\text{u}},\Psi_{\text{v}},\Omega}\mathcal{L},$
(17)

where
$\mathcal{L}=\mathcal{L}_{\text{u}}^{\text{VAE}}+\mathcal{L}_{\text{u}}^{\text{D}}+\mathcal{L}_{\text{v}}^{\text{VAE}}+\mathcal{L}_{\text{v}}^{\text{D}}+\mathcal{L}_{\text{f}}$.

To fulfill the adversarial training, the overall training process consists of
the following two alternate steps:

  * •

Step 1: Fixing $Q_{\text{u}}$, $G_{\text{u}}$, $Q_{\text{v}}$, $G_{\text{v}}$,
and the neural collaborative filtering network $F$, optimizing $D_{\text{u}}$
and $D_{\text{v}}$ with respect to $\mathcal{L}^{\text{D}}_{\text{u}}$
(Equation (10)) and $\mathcal{L}^{\text{D}}_{\text{v}}$ (Equation (13)),
respectively;

  * •

Step 2: Fixing $D_{\text{u}}$ and $D_{\text{v}}$, jointly training
$(Q_{\text{u}},G_{\text{u}})$, $(Q_{\text{v}},G_{\text{v}})$, and $F$, with
respect to $\mathcal{L}_{\text{u}}^{\text{VAE}}$ (Equation (11)),
$\mathcal{L}_{\text{v}}^{\text{VAE}}$ (Equation (14)), and
$\mathcal{L}_{\text{f}}$ (Equation (16)), respectively.

Note that at each iteration, the discriminators $D_{\text{u}}$ and
$D_{\text{v}}$ should be updated before the variational inference networks
$Q_{\text{u}}$, $Q_{\text{v}}$, because $\mathcal{L}_{\text{u}}^{\text{VAE}}$
and $\mathcal{L}_{\text{v}}^{\text{VAE}}$ depend on the optimal $D_{\text{u}}$
and $D_{\text{v}}$ so far. It is also worth noting that in Step 2, the two
VAEs, $(Q_{\text{u}},G_{\text{u}})$ and $(Q_{\text{v}},G_{\text{v}})$, are
trained jointly with the neural collaborative filtering network, by which the
reconstruction error and prediction error can be minimized simultaneously. The
joint training can take advantage of multi-task learning which makes the
supervision signal of $R_{uv}$ able to be propagated back to the inference
networks $Q_{\text{u}}$ and $Q_{\text{v}}$. The training procedure is
summarized in Algorithm 1 which iteratively updates the parameters of DAVE
using mini-batch stochastic gradient ascent.

##  4 Experiments

The experiments mainly aim to answer the following research questions:

  1. RQ1

How does DAVE perform as compared with state-of-the-art recommendation
methods?

  2. RQ2

How is the robustness of DAVE?

  3. RQ3

How is the expressiveness of DAVE?

  4. RQ4

How do the hyper-parameters, embedding dimensionality and negative sampling
ratio, affect the performance of DAVE?

Since we use implicit feedback data, as most of the existing work did [10, 30,
32, 9], we will evaluate DAVE over top-$k$ recommendation task.

###  4.1 Experimental Setting

####  4.1.1 Datasets

We conduct experiments on five publicly available datasets: Yelp
111https://github.com/hexiangnan/sigir16-eals, Digital Music
222https://nijianmo.github.io/amazon/index.html, MovieLens 1M
333https://github.com/hexiangnan/neural_collaborative_filtering (ML-1M),
MovieLens 100K 444https://grouplens.org/datasets/MovieLens/100k/ (ML-100k) and
Pinterest 555https://github.com/hexiangnan/neural_collaborative_filtering,
which are summarized in Table II. The first four datasets provide users’
explicit ratings on items, so we transform them into implicit data, where each
entry is marked as 1 if the rating is observed, otherwise 0. Specially, there
are at least 20 ratings for each user in the two MovieLens datasets, while in
Yelp, we only retain the users who have at least 10 interactions, due to the
higher sparsity of Yelp. In Yelp, a user may rate an item many times. These
repetitive ratings count only once in the building of the interaction matrix,
which can prevent an interaction from appearing in both the training set and
the testing set. Digital Music is a public dataset collected from Amazon.
Since it is highly sparse, we only retain the users and items with at lest 5
ratings, which results in a subset that contains 9,906 users and 12,381 items.
Pinterest is a dataset consisting of implicit feedbacks, which has been used
to evaluate collaborative recommendations on images [10, 9]. In Pinterest, an
interaction represents a user has pinned an image to his/her board.

####  4.1.2 Evaluation Protocol

To evaluate the performance of DAVE, we adopt the leave-one-out method, which
is widely used in top-$k$ recommendation evaluation [32, 10, 9]. Specifically,
in Yelp, MovieLens 1M and MovieLens 100K, for each user in a dataset, we leave
out the latest user-item interaction to form the testing set and use the
remaining interactions to form the training set. In Digital Music and
Pinterest, since each rating or pin has no timestamp, we randomly leave out
one user-item interaction for each user to form the testing set. Note that we
also randomly set aside one interaction for each user to form the validation
set for the tuning of hyper-parameters. Since it is too time-consuming to rank
all items for every user during testing, we follow the common strategy [32,
10] that we will check whether the testing item, which the user rated, is
ranked ahead of 99 unrated items which are randomly selected from the datasets
in advance. The performance of the ranked list is judged by Hit Ratio (HR) and
Normalized Discounted Cumulative Gain (NDCG). The HR@$k$ is the ratio of the
ranking list that the testing item is ranked in the first $k$ positions, while
the NDCG accounts for the position of the hit, which assigns higher weight to
hits at higher positions. For both metrics, larger values indicate better
performance. We will report the results of $k=5,10,20$ on five datasets.

TABLE II: Statistics of Datasets Dataset | #Interactions | #Items | #Users | Sparsity
---|---|---|---|---
Yelp | 730,790 | 25,815 | 25,677 | 99.89%
Digital Music | 123,518 | 12,381 | 9,906 | 99.90%
MovieLens 1M | 1,000,209 | 3,706 | 6,040 | 95.53%
MovieLens 100K | 100,000 | 1,682 | 943 | 93.69%
Pinterest | 1,500,809 | 9,916 | 55,187 | 99.73%
TABLE III: Comparison of Baselines Baselines Characteristics |  User Embedding |  Item Embedding |  Noise-tolerant |  Personalized noise reduction |  Embedding Distribution |  Multi-Modality
---|---|---|---|---|---|---
CDAE |  ✓ |  |  ✓ |  |  |
APR |  ✓ |  ✓ |  ✓ |  |  |
ACAE |  ✓ |  |  ✓ |  |  |
NeuMF |  ✓ |  ✓ |  |  |  |
CFGAN |  ✓ |  |  |  |  |
AVB |  ✓ |  |  |  |  |  ✓
VAEGAN |  ✓ |  |  |  |  |  ✓
CVAE-GAN |  ✓ |  |  ✓ |  ✓ |  ✓ |
RecVAE |  ✓ |  |  ✓ |  ✓ |  ✓ |
DAVE-adv |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |
DAVE+aae |  ✓ |  ✓ |  ✓ |  |  |
DAVE |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  ✓

####  4.1.3 Baselines

We will compare DAVE with the following advanced methods, whose
characteristics are shown in Table 3.

  * •

NeuMF [10]: NeuMF is a general framework NCF for collaborative filtering based
on neural networks. It employs a Multi-Layer Perceptron (MLP) to model non-
linear user-item interactions between latent features of users and items.

  * •

CDAE [30]: CADE is a Denoising Auto-encoder based collaborative filtering
framework for top-$k$ recommendation. By utilizing denoising technique, CDAE
can learn robust latent representations of corrupted user-item interactions
for recommendation.

  * •

CFGAN [5]: CFGAN is a GAN-based collaborative filtering framework, where a
real value vector-wise adversarial training is introduced to improve the
representation learning of the users or items.

  * •

APR [9]: APR is an Adversarial Personalized Ranking framework, which enhances
the pairwise ranking method BPR [21] by adversarial training. Particularly,
APR offers the robustness at the level of model parameters rather than model
input, by injecting adversarial noise to parameters of BPR model [21] during
the adversarial training.

  * •

ACAE [32]: ACAE is a general adversarial training framework for neural
network-based recommendation models, which also applies adversarial training
for improving the robustness of recommendations.

  * •

AVB [19]: AVB is a technique for training variational auto-encoders with
arbitrarily expressive inference models based on adversarial training, which
introduces an auxiliary discriminative network that allows to rephrase the
maximum-likelihood problem as a two-player game.

  * •

VAEGAN [31]: VAEGAN is a Collaborative Filtering Framework based on
Adversarial Variational auto-encoders, which utilizes a flexible black-box
inference model as well as adversarial training to train VAEs for implicit
variational inference.

  * •

CVAE-GAN [2]: CVAE-GAN is variational generative adversarial networks, which
is a general learning framework that combines a variational auto-encoder with
a generative adversarial network.

  * •

RecVAE [25]: RecVAE is a Recommender VAE model with a new architecture for the
encoder network that can be trained with corrupted implicit user-item
interaction vectors.

####  4.1.4 Parameter Setting

The hyper-parameters are tuned on validation sets. We set the batch size to
256 for MovieLens 100K, MovieLens 1M and Pinterest, and 128 for Yelp and
Digital Music. The negative sampling ratio is set to 4 for Yelp and MovieLens
100K, 3 for MovieLens 1M, and 2 for Digital Music and Pinterest. The embedding
dimensionality is set to 32 for Yelp, and 64 for MovieLens 1M, MovieLens 100K,
Digsital Music and Pinterest. We use Adam to learn the VAEs
$(Q_{\text{u}},G_{\text{u}})$ and $(Q_{\text{v}},G_{\text{v}})$ and the neural
collaborative filtering network $F$, and use RMSprop to learn the
discriminators $D_{\text{u}}$ and $D_{\text{v}}$, where learning rate is set
to 0.0001. For $D_{\text{u}}$ and $D_{\text{v}}$, we set the number of hidden
layers to 2 and the numbers of hidden nodes are respectively 50 and 100. For
$F$, we set three hidden layers each of which consists of 32 hidden nodes. We
use standard Gaussian distribution as the prior of the embeddings.

###  4.2 Experimental Analysis

####  4.2.1 Recommendation Performance (RQ1)

Tables IV, V, VI, VII and VIII show the results of top-$k$ recommendation on
the five datasets, respectively, where $k=\\{5,10,20\\}$.

TABLE IV: Recommendation Performance on MovieLens 100K. The best runs per metric are marked in boldface. The best runs per metric among robust recommendation methods CDAE, APR, and ACAE are underlined. | MovieLens 100K
---|---
| HR | HR | HR | NDCG | NDCG | NDCG
| @5 | @10 | @20 | @5 | @10 | @20
CDAE | 0.4284 | 0.6331 | 0.7996 | 0.2855 | 0.3511 | 0.3934
APR | 0.4772 | 0.6755 | 0.8261 | 0.3253 | 0.3896 | 0.4276
ACAE | 0.4602 | 0.6437 | 0.8049 | 0.3107 | 0.3697 | 0.4106
CFGAN | 0.2810 | 0.4422 | 0.632 | 0.1921 | 0.2438 | 0.2913
NeuMF | 0.4645 | 0.6257 | 0.7943 | 0.3183 | 0.3704 | 0.4128
AVB | 0.3648 | 0.5514 | 0.7328 | 0.2402 | 0.3000 | 0.3620
VAEGAN | 0.3107 | 0.4634 | 0.6459 | 0.2044 | 0.2531 | 0.2992
CVAE-GAN | 0.2609 | 0.4008 | 0.5832 | 0.1673 | 0.2117 | 0.2571
RecVAE | 0.4793 | 0.6448 | 0.8028 | 0.3216 | 0.3753 | 0.4157
DAVE-adv | 0.4634 | 0.6288 | 0.7794 | 0.3193 | 0.3727 | 0.4108
DAVE+aae | 0.4942 | 0.6776 | 0.8282 | 0.3317 | 0.3907 | 0.4291
DAVE | 0.4995 | 0.6723 | 0.8293 | 0.3415 | 0.3971 | 0.4369
TABLE V: Recommendation Performance on MovieLens 1M. The best runs per metric are marked in boldface. The best runs per metric among robust recommendation methods CDAE, APR, and ACAE are underlined. | MovieLens 1M
---|---
| HR | HR | HR | NDCG | NDCG | NDCG
| @5 | @10 | @20 | @5 | @10 | @20
CDAE | 0.4343 | 0.6134 | 0.7882 | 0.2948 | 0.3527 | 0.3970
APR | 0.4603 | 0.6396 | 0.8167 | 0.3148 | 0.3728 | 0.4176
ACAE | 0.5002 | 0.6649 | 0.8164 | 0.3473 | 0.4004 | 0.4388
CFGAN | 0.3070 | 0.4594 | 0.6339 | 0.2077 | 0.2568 | 0.3007
NeuMF | 0.5089 | 0.6833 | 0.8321 | 0.3562 | 0.4124 | 0.4503
AVB | 0.3891 | 0.5705 | 0.7500 | 0.2582 | 0.3167 | 0.4576
VAEGAN | 0.3166 | 0.4652 | 0.6512 | 0.2118 | 0.2596 | 0.3064
CVAE-GAN | 0.2877 | 0.4348 | 0.6308 | 0.1879 | 0.2365 | 0.2847
RecVAE | 0.5371 | 0.6993 | 0.8467 | 0.3729 | 0.4257 | 0.4631
DAVE-adv | 0.4752 | 0.6575 | 0.8270 | 0.3242 | 0.3832 | 0.4261
DAVE+aae | 0.5248 | 0.6909 | 0.8397 | 0.3625 | 0.4162 | 0.4541
DAVE | 0.5417 | 0.7185 | 0.8518 | 0.3761 | 0.4334 | 0.4671
TABLE VI: Recommendation Performance on Yelp. The best runs per metric are marked in boldface. The best runs per metric among robust recommendation methods CDAE, APR, and ACAE are underlined. | Yelp
---|---
| HR | HR | HR | NDCG | NDCG | NDCG
| @5 | @10 | @20 | @5 | @10 | @20
CDAE | 0.3231 | 0.4444 | 0.5963 | 0.2289 | 0.2680 | 0.3064
APR | 0.6494 | 0.7920 | 0.9048 | 0.4810 | 0.5274 | 0.5560
ACAE | 0.6125 | 0.7569 | 0.8746 | 0.4527 | 0.4996 | 0.5294
CFGAN | 0.3027 | 0.4252 | 0.5626 | 0.2110 | 0.2504 | 0.2850
NeuMF | 0.6529 | 0.7838 | 0.8793 | 0.4836 | 0.5262 | 0.5505
AVB | 0.3244 | 0.4476 | 0.5944 | 0.2297 | 0.2694 | 0.3064
VAEGAN | 0.3273 | 0.4506 | 0.6044 | 0.2320 | 0.2717 | 0.3105
CVAE-GAN | 0.3227 | 0.4495 | 0.5976 | 0.2293 | 0.2701 | 0.3074
RecVAE | 0.6464 | 0.7843 | 0.8936 | 0.4866 | 0.5313 | 0.5590
DAVE-adv | 0.6192 | 0.7643 | 0.8744 | 0.4504 | 0.4976 | 0.5256
DAVE+aae | 0.6687 | 0.8022 | 0.9025 | 0.4980 | 0.5415 | 0.5670
DAVE | 0.6688 | 0.8032 | 0.9015 | 0.5018 | 0.5456 | 0.5706
TABLE VII: Recommendation Performance on Digital Music. The best runs per metric are marked in boldface. The best runs per metric among robust recommendation methods CDAE, APR, and ACAE are underlined. | Digital Music
---|---
| HR | HR | HR | NDCG | NDCG | NDCG
| @5 | @10 | @20 | @5 | @10 | @20
CDAE | 0.2018 | 0.3001 | 0.4215 | 0.1375 | 0.1692 | 0.1997
APR | 0.4861 | 0.6145 | 0.7435 | 0.3500 | 0.3908 | 0.4235
ACAE | 0.4562 | 0.5930 | 0.7319 | 0.3355 | 0.3796 | 0.4147
CFGAN | 0.1978 | 0.2896 | 0.4066 | 0.1340 | 0.1636 | 0.1931
NeuMF | 0.3534 | 0.4707 | 0.6049 | 0.2597 | 0.2974 | 0.3312
AVB | 0.2000 | 0.2956 | 0.4164 | 0.1367 | 0.1675 | 0.1980
VAEGAN | 0.2081 | 0.3059 | 0.4348 | 0.1416 | 0.1730 | 0.2054
CVAE-GAN | 0.2058 | 0.3048 | 0.4284 | 0.1402 | 0.1721 | 0.2032
RecVAE | 0.4128 | 0.5226 | 0.6504 | 0.3190 | 0.3544 | 0.3866
DAVE-adv | 0.4328 | 0.5680 | 0.7156 | 0.3131 | 0.3566 | 0.3939
DAVE+aae | 0.4760 | 0.6192 | 0.7594 | 0.3420 | 0.3883 | 0.4239
DAVE | 0.4872 | 0.6269 | 0.7651 | 0.3555 | 0.4007 | 0.4357

At first, from Tables IV, V, VI, VII and VIII, we can see that on MovieLens
100K, MovieLens 1M, Yelp and Digital Music, DAVE shows better performance than
the robust recommendation methods CDAE, APR and ACAE with respect to HR@$k$,
except for HR@10 on MovieLens 100K and HR@20 on Yelp.

We can also note that DAVE consistently outperforms CDAE, APR and ACAE with
respect to NDCG@$k$ on five datasets. In particular, on MovieLens 100K,
compared with the most competitive method APR, DAVE increases the NDCG@5 by
5%, NDCG@10 by 1.9%, and NDCG@20 by 2.2% (see Table IV); on MovieLens 1M,
compared to the most competitive method ACAE, DAVE increases the NDCG@5 by
8.3%, NDCG@10 by 8.2%, and NDCG@20 by 6.4% (see Table V); on Yelp, compared to
the best competitor APR, DAVE increases the NDCG@5 by 4.3%, NDCG@10 by 3.5%,
and NDCG@20 by 2.6% (see Table VI); on Digital Music, compared to the best
competitor APR, DAVE increases the NDCG@5 by 1.5%, NDCG@10 by 1.5%, and
NDCG@20 by 2.8% (see Table VII); on Pinterest, compared to the best competitor
APR, DAVE increases the NDCG@5 by 1.3%, NDCG@10 by 0.9%, and NDCG@20 by 1%
(see Table VIII). We argue that these improvements are mainly due to the
better expressiveness of DAVE. Unlike the existing robust recommendation
methods that assume user preference distribution is a single modal, DAVE is
able to handle the multi-modality of user-item interaction data so that true
user preference distributed around different modes can be captured. At last,
we also note the exception that DAVE is slightly inferior to APR with respect
to HR@$k$ on Pinterest. This is partly because in Pinterest the number of
users is far more than the number of items. Such unbalance reduces the
preference diversity revealed by the data and consequently hinders DAVE from
best capturing the multi-modality of the preference distributions.

We can also see that DAVE outperforms NeuMF and CFGAN on all datasets. For
NeuMF, this is because that DAVE can model different noise distributions via
VAE during the representation learning for different users and items, which
leads to more robust embeddings than naive neural collaborative filtering.
Note that CFGAN also combines GAN as well as adversarial training with
collaborative filtering, but it directly generates user-item interaction
vectors for collaborative filtering rather than learns latent representations
for users and items, which is in contrast with DAVE. Although the output of a
certain hidden layer of the generative model of CFGAN can serve as user latent
representation, learning only the user latent representation is not enough to
model the non-linear interactions between users and items, so it is difficult
for CFGAN to effectively capture the preference of user to item, which leads
CFGAN to almost the worst performance.

Finally, we can observe that DAVE outperforms AVB, VAEGAN, CVAE-GAN and RecVAE
on all datasets. Although AVB and VAEGAN also focus on tackling the single-
modality problem of VAE by utilizing GAN as well as adversarial training, they
cannot provide personalized noise reduction for different users. RecVAE
improves VAE with a new architecture for the encoder, but it cannot capture
multimodal preference distributions. In addition, AVB, VAEGAN, CVAE-GAN and
RecVAE learn latent representations only for users, which are not enough to
model the non-linear interactions between users and items. DAVE infers
distributions over embeddings for both users and items by two VAEs, which
combines the advantages of the user-based methods and item-based methods for
collaborative filtering.

TABLE VIII: Recommendation Performance on Pinterest. The best runs per metric are marked in boldface. The best runs per metric among robust recommendation methods CDAE, APR, and ACAE are underlined. | Pinterest
---|---
| HR | HR | HR | NDCG | NDCG | NDCG
| @5 | @10 | @20 | @5 | @10 | @20
CDAE | 0.3303 | 0.4814 | 0.6395 | 0.2174 | 0.2662 | 0.3063
APR | 0.7246 | 0.8884 | 0.9704 | 0.5157 | 0.5691 | 0.5902
ACAE | 0.7086 | 0.8756 | 0.9663 | 0.5024 | 0.5569 | 0.5802
CFGAN | 0.1596 | 0.2628 | 0.4118 | 0.1016 | 0.1348 | 0.1722
NeuMF | 0.6890 | 0.8664 | 0.9619 | 0.4831 | 0.5410 | 0.5656
AVB | 0.2731 | 0.4611 | 0.6452 | 0.1407 | 0.2013 | 0.2480
VAEGAN | 0.1778 | 0.2901 | 0.4576 | 0.1134 | 0.1494 | 0.1915
CVAE-GAN | 0.1739 | 0.2863 | 0.4530 | 0.1108 | 0.1469 | 0.1887
RecVAE | 0.6851 | 0.8371 | 0.9341 | 0.5030 | 0.5525 | 0.5773
DAVE-adv | 0.7020 | 0.8698 | 0.9617 | 0.5016 | 0.5563 | 0.5800
DAVE+aae | 0.6864 | 0.8576 | 0.9562 | 0.4892 | 0.5450 | 0.5704
DAVE | 0.7219 | 0.8798 | 0.9663 | 0.5226 | 0.5741 | 0.5963

(a) HR@5 (b) NDCG@5

(c) HR@10 (d) NDCG@10

Figure 2: Noise tolerability on MovieLens 100K.

####  4.2.2 Noise Tolerability (RQ2)

Now we investigate the robustness of DAVE by comparing it with its variant
DAVE+aae over MovieLens 100K. We also compare DAVE with RecVAE, ACAE, CDAE,
and CAVE-GAN since they can address noise data too. Here we want to verify
DAVE has better recommendation performance as well as better noise
tolerability in the face of noisy interaction. For each user $u$ and item $v$
in the testing set, we intentionally inject some noise to the data through two
steps: first randomly choose a noise level (i.e., the ratio of the noisy
interactions) and then randomly flipping over some entries of their
interaction vectors $\bm{u}$ and $\bm{v}$ with respect to the chosen noise
level, by which we simulate the scenario that different users or items have
different noise levels. Figure 2 reports the results in terms of HR@$k$ and
NDCG@$k$ with $k=5,10$ at the noise levels 0.1, 0.3, 0.5, 0.7, and 0.9.

From Figure 2 we can observe that the performance of all methods degrades as
noise level increases. At different noise levels, however, DAVE consistently
exhibits better performance than all the alternative methods, and the greater
the noise level (i.e., the more the noise added), the bigger the gap between
DAVE and the alternative methods. DAVE+aae generates the user or item
embeddings with a point estimate produced by Adversarial Auto-encoder (AAE)
[17]. In contrast, DAVE generates the embedding for a user or an item by
sampling from an inferred embedding distribution unique to that user or item.
The result shows that inferring unique embedding distribution for different
users and items brings DAVE the better noise tolerability. Similarly, DAVE
shows much better noise tolerability than RecVAE, ACAE, CDAE, and CAVE-GAN as
it can generate more expressive preference embeddings due to its adaptability
to the different noise distributions and the ability to capture the multi-
modality of the preference distributions.

####  4.2.3 Model Expressiveness (RQ3)

In the experiments, the posterior distributions of embeddings unique to
different users and items are Gaussian. For a user $u$, the variational
inference network $Q_{\text{u}}$ in UserAVE will generate a pair of mean and
standard deviation, $(\bm{\mu}_{u},\bm{\sigma}_{u})$, which defines the
posterior distributions of embeddings of that user. Similarly, for an item
$v$, the variational inference network $Q_{\text{v}}$ in ItemAVE will generate
the pair $(\bm{\mu}_{v},\bm{\sigma}_{v})$ to define the posterior
distributions of embeddings of that item. To evaluate the expressiveness of
DAVE, we will check the distributions of $(\bm{\mu}_{u},\bm{\sigma}_{u})$ and
$(\bm{\mu}_{v},\bm{\sigma}_{v})$ inferred by DAVE and DAVE-adv on MovieLens
100K. Particularly, for each pair $(\bm{\mu}_{u},\bm{\sigma}_{u})$, we
concatenate $\bm{\mu}_{u}$ and $\bm{\sigma}_{u}$ to form a new vector to
represent the posterior distribution defined by
$(\bm{\mu}_{u},\bm{\sigma}_{u})$, and then visualize the distribution of these
new vectors in a 2-dimensional space using t-SNE algorithm [28]. The same
process is also applied to each $(\bm{\mu}_{v},\bm{\sigma}_{v})$.

Figures 3 and 4 show the visualization of the posterior embedding
distributions of 943 users and 1682 items in MovieLens 100K, respectively,
where a point represents the 2-dimensional projection of a concatenating
vector and the points belonging to the same cluster are of the same color.

(a) DAVE (b) DAVE-adv

Figure 3: Visualization of the distribution of the posterior distributions of
user embeddings learned by (a) DAVE and (b) DAVE-adv in MovieLens 100K.

(a) DAVE (b) DAVE-adv

Figure 4: Visualization of the distribution of the posterior distributions of
item embeddings learned by (a) DAVE and (b) DAVE-adv in MovieLens 100K.

We can see that the points representing the posterior embedding distributions
inferred by DAVE are obviously separated into multiple clusters
(distributions) shown in Figures 3(a) and 4(a), and the gaps between the
clusters inferred by DAVE are far more significant than the gaps between the
clusters inferred by DAVE-adv shown in Figures 3(b) and 4(b). We argue that
such difference is caused by the different effects offered by KL-divergence
and adversarial training. In DAVE-adv, in order to compute the KL-divergence,
the posterior distribution is explicitly represented with a Gaussian, and
minimizing the KL-divergence encourages the posterior distributions to be
close to a common prior distribution, which consequently makes the posterior
distributions inferred by DAVE-adv tend to be single modal and unexpressive.
On the contrary, although the adversarial training in DAVE approximates the
minimizing of KL-divergence, it offers the flexibility without the requirement
to explicitly represent the posterior distribution. Such flexibility makes it
possible for DAVE to infer complex posterior distributions that are multimodal
and cannot to be explicitly formulated, which improves the expressiveness of
the user embeddings drawn from the inferred posterior distributions and
benefits the capturing of the diversity of user preference.

(a) HR@10 (b) NDCG@10

Figure 5: Tuning of embedding dimensionality.

####  4.2.4 Tuning of Hyper-parameters (RQ4)

Now we tune two hyper-parameters of DAVE, embedding dimensionality and
negative sampling ratio, in terms of HR@10 and NDCG@10 over five validation
sets. The results are shown in Figures 5 and 6, respectively.

From Figure 5, we can see that on MovieLens 100K, MovieLens 1M, Digital Music
and Pinterest, the optimal embedding dimensionality is 64, while on Yelp it is
32. Basically the performance of DAVE improves first with the increase of
embedding dimensionality, then degrades due to overfitting incurred by
excessive embedding dimensionality. We also note that on Yelp, after the
optimal embedding dimensionality 32, the performance of DAVE is on downward
trend with some fluctuates that might be incurred by random initialization of
parameters.

Figure 6 shows the effect of negative sampling ratio on the performance of
DAVE. We can see that with the increase of negative sampling ratio from 1 to
5, the HR@10 and NDCG@10 grow first, then drop. On Yelp and MovieLens 100K,
DAVE achieves the best performance at the negative sampling ratio of 4, and on
Digital Music and Pinterest, DAVE achieves the best performance at the
negative sampling ratio of 2 while on MovieLens 1M, DAVE achieves the best
performance at the negative sampling ratio of 3. The observation also implies
that excessively high negative sampling ratio may mistakenly lead to more
false-negative samples, which results in reduced robustness and weaker
generalization performance of DAVE.

(a) HR@10 (b) NDCG@10

Figure 6: Tuning of negative sampling ratio.

##  5 RELATED WORK

In this section, we briefly review related work on the traditional recommender
systems and the robust recommendation.

###  5.1 Traditional Recommender Systems

In traditional recommender systems, collaborative filtering is the most widely
used technique for personalized recommendation, which aims to predict user
preference from historical user-item interactions, with a learnable
interaction function of informative representations of users and items that
capture the collaborative signals, i.e., similar users behave similarly [13,
22]. Early matrix factorization (MF) based techniques linearly model the user-
item interaction with inner product of user and item embeddings that are
extracted from factor matrices [13, 20]. To overcome the drawbacks of the MF
based models that oversimplify the nonlinearity of user-item interactions,
recently, various kinds of deep learning based models have been proposed to
learn comprehensive representations for users and items, and capture the
nonlinear user/item relationships [33]. For example, Cheng et al. [6] and He
et al. [10] propose the Wide&Deep model and the Neural Collaborative Filtering
model, respectively, which can model the nonlinearity of user-item
interactions with a multilayer perceptron. However, the traditional
recommender systems often assume the user-item interaction data are noise-
free, and lack the consideration on robustness of the models, which makes them
likely fail to capture users’ true preference from the data with perturbations
[32].

###  5.2 Robust Recommendation

The existing works on robust recommendation roughly follow two lines, where
one line is to improve the recommendation robustness by injecting noise to
input or model parameters during model training, and the other line is to
adopt a generative model like VAE to infer a latent representation space from
which robust embeddings of users and items can be generated.

###  5.3 Noise Injection Based methods for Robust Recommendation

The existing methods for robust recommendation [30, 9, 32, 14] often inject
extra noise to training data or model parameters to deal with noisy user-item
interactions, which roughly fall into two classes. One class of methods, such
as CDAE [30], use Denoising Auto-encoder (DAE) [29] for generating robust
embeddings of users and items, which adds random drop-out noise in user-item
interaction vectors and trains an auto-encoder based on intentionally
corrupted input with the objective of minimizing reconstruction errors.

The other class of the noise based methods introduces adversarial noise as
well as adversarial training to improve the model robustness [7, 9, 26, 32].
He et al. propose an Adversarial Personalized Ranking (APR) model which can
enhance the pairwise ranking method BPR [21] by performing adversarial
training [9]. Tang et al. propose an Adversarial Multimedia Recommendation
(AMR) model for robust recommendation of images, which is trained to defend an
adversary of perturbations to the target image [26]. Yuan et al. propose a
general adversarial training framework, which can improve both the robustness
and the overall performance of NN-based recommendation models [32].

There are two main defects in the above two classes of noise injection based
methods. First, the model robustness depends on a fixed noise injection level
set beforehand, which ignores the personalization of the noise reduction for
different users. Second, for the adversarial noise based methods, it is hard
to choose a proper adversarial noise level for the tradeoff between the
overall performance and the robustness of the models, and an over strong
adversarial noise level may impair the recommendation performance of the
models.

###  5.4 Variational Auto-encoder Based methods for Robust Recommendation

Recently, due to the impressive power of VAE in representation learning in the
fields of computer vision and network embedding [18, 4, 24], a line of VAE
based methods have been proposed for robust collaborative filtering [1, 8, 15,
25, 16]. For example, He et al. propose an additional variational auto-encoder
which can generate robust embeddings encoding side information of items,
including content information and tag information [8]. Li et al. propose a
collaborative variational auto-encoder (CVAE) for robust recommendation of
multimedia, where VAE is used to generate latent representations for
multimedia content [15]. Shenbin et al. propose a Recommender VAE (RecVAE)
model with a new architecture for the encoder network that can be trained with
corrupted implicit user-item interaction vectors [25]. However, as we have
mentioned before, VAE based methods likely leads to less expressive models
that are unable to handle the multi-modality of the distributions of user
preference. At the same time, Makhzani et al. propose the Adversarial Auto-
encoder (AAE) model [17] which can be used for variational inference. Similar
to our model, AAE also uses an adversarial training to regularize the
variational inference. However, different from our model where the posterior
of each user is separately regularized (Equation (5)), AAE regularizes the
aggregated (averaged) posterior $q(\bm{x})$ to be close to the prior, i.e.,
minimizes

$\text{KL}\big{(}q(\bm{x})=\int_{\bm{u}}q(\bm{x}|\bm{u})p(\bm{u}),p(\bm{x})\big{)},$
(18)

which deviates from the VAE’s optimization objective of improving ELBO and
limits its ability to capture multi-modality.

##  6 CONCLUSION

To overcome the defects of the existing methods for robust recommendation, we
propose a novel Dual Adversarial Variational Embedding (DAVE) model which is
able to provide the personalized noise reduction and capture the multi-
modality of the preference distributions, by combining the advantages of VAE
and adversarial training. Particularly, to provide the personalized noise
reduction for different users and items, we introduce two VAEs, to infer a
unique embedding distribution for each user and item, respectively. Due to the
variational inference power of VAEs, the different noise levels of users and
items can be adaptively captured by their own embedding distributions from
which robust embeddings can be drawn. To improve the model expressiveness, we
further introduce two GANs to DAVE. Due to the regularization offered by the
adversarial training between the discriminators and the variational inference
networks, DAVE is expressive enough to approximate the preference
distributions with multi-modality. At last, the extensive experiments
conducted on real datasets verify the effectiveness of DAVE on robust
recommendations.

## Acknowledgment

This work is supported by National Natural Science Foundation of China under
grant 61972270. This work is also supported in part by NSF under grants
III-1763325, III-1909323, and SaTC-1930941.

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|  Qiaomin Yi obtained her bachelor’s degree from the School of Information
Engineering, Northwest A&F University, China, in 2018. She is now pursuing the
master’s degree in the School of Computer Science, Sichuan University, China.
Her research interests include data mining and recommender systems.
---|---
|  Ning Yang is an associate professor at Sichuan University, China. He
obtained his Ph.D degree in Computer Science from Sichuan University in 2010.
His research interests include recommender systems and social media mining.
---|---
|  Philip S. Yu received the PhD degree in electrical engineering from
Stanford University. He is a distinguished professor in computer science at
the University of Illinois at Chicago and is also the Wexler chair in
information technology. His research interests include big data, data mining,
and social computing. He is a fellow of the ACM and the IEEE.
---|---