Causal Disentangled Variational Auto-Encoder for Preference Understanding in Recommendation

Let $u\in\{1,....,U\}$ and $i\in\{1,....,I\}$ index users and items, respectively. For the recommender system, the dataset $\mathcal{D}$ of user behavior consists of $U$ users and $I$ items interactions. For a user $u$, the historical interactions $D_{u}=\{x_{u,i}:x_{u,i}\in\{0,1\}\}$ is a multi-hot vector, where $x_{u,i}=0$ represents that there is no recorded interaction between user $u$ and item $i$, and $x_{u,i}=1$ represents an interaction between the user $u$ and item $i$, such as click. For notation brevity, we use $\mathbf{x}_{u}$ denotes all the interactions of the user $u$, that is $\mathbf{x}_{u}=\{x_{u,i}:x_{u,i}=1\}$. Users may have very diverse interests, and interact with items that belong to many high-level concepts, such as preferred film directors, actors, genres, and year of production. We aim to learn disentangled representations from user behavior that reflect the user preference related to different high-level concepts and reflect the causal relationship between these concepts.

Consider $k$ high-level concepts in observations to formalize causal representation. These concepts are thought to have causal relationships with one another in a manner that can be described by a Directed Acyclic Graph (DAG), which can be represented as an adjacency matrix, denoted by $A$. To construct this causal structure, we introduce a causal layer in our framework. This layer is specifically designed to implement the nonlinear Structural Causal Model (SCM) as proposed by (Yu et al., 2019):

where $A$ is the weighted adjacency matrix among the k elements of z, $\epsilon$ is the exogenous variables that $\epsilon\sim\mathcal{N}(0,I)$, $g$ is nonlinear element-wise transformations. The set of parameters of $A$ and $g$ are denoted by $\alpha=(A,g)$. Additionally, $A_{ij}$ is non-zero if and only if $[z]_{i}$ is a parent of $[z]_{j}$, and the corresponding binary adjacency matrix is denoted by $I_{A}=I(A\neq 0)$, where $I(\cdot)$ is an element-wise indicator function. When $g$ is invertible, Equation 2 can be rephrased as follows:

which implies that after undergoing a nonlinear transformation of $g$, the factors $\mathbf{z}$ satisfy a linear SCM. To ensure disentanglement, we use the labels of the concepts to be additional information, denoted as $c$. The additional information $c$ is used to learn the weights of the non-zero elements in the prior adjacency matrix, which represent the sign and scale of causal effects.

Our model is built upon the generative framework of the Variational Autoencoder (VAE) and introduces a causal layer to describe the SCMs. Let $E$ and $D$ denote the encoder and decoder, respectively. We use $\theta$ to denote the set $(E,D,\alpha,\lambda)$ that contains all the trainable parameters of our model. For a user $u$, our generative model parameterized by $\theta$ assumes that the observed data are generated from the following distribution:

where

Assuming the encoding process $\mathbf{\epsilon}=E(\mathbf{x}_{u},\mathbf{c})+\zeta$, where $\zeta$ is the vectors of independent noise with probability density $p_{\zeta}$. The inference models that take into account the causal structure can be defined as:

where $q_{\phi}(\mathbf{\epsilon},\mathbf{z}_{u}|\mathbf{x}_{u},\mathbf{c})$ is the approximate posterior distribution, parameterized by $\phi$, that models the distribution of the latent representation given the user’s behavior data $\mathbf{x}_{u}$ and additional information $\mathbf{c}$. Since $\epsilon$ and $z$ have a one-to-one correspondence, we can simplify the variational posterior distribution as follows:

where $\delta(\cdot)$ is the Dirac delta function. And we define the joint prior distribution for latent variables $\epsilon$ and $z$ as:

where $p_{\epsilon}(\epsilon)=\mathcal{N}(0,I)$ and $p_{\theta}(\mathbf{z}_{u}|\mathbf{c})$ is a factorized Gaussian distribution such that:

where $p_{\theta}(z_{u}^{(i)}|c_{i})=\mathcal{N}(\lambda_{1}(c_{i}),\lambda_{2}^{2}(c_{i}))$. $\lambda_{1}$ and $\lambda_{2}$ are arbitrary functions.

Given the representation of a user, denoted by $\mathbf{z_{u}}$, the decoder’s goal is to predict the item, out of a total of $I$ items, that the user is most likely to click. Let $D$ denotes the decoder. We assume the decoding processes $\quad\mathbf{x}=D(\mathbf{z})+\xi$, where $\xi$ is the vectors of independent noise with probability density $q_{\xi}$. Then we define the generative model parameterized by parameters $\theta$ as follows:

Our objective is to learn the parameters $\phi$ and $\theta$ that maximize the evidence lower bound (ELBO) of $\Sigma_{u}\ln{p_{\theta}(\mathbf{x}_{u})}$. The ELBO is defined as the expectation of the log-likelihood with respect to the approximate posterior $q_{\phi}(\mathbf{\epsilon},\mathbf{z}_{u}|\mathbf{x}_{u},\mathbf{c})$, where $\mathbf{z}_{u}$ and $\epsilon$ are the latent variables:

Based on the definitions of the approximate posterior in Equation 6 and the prior distribution in Equation 7, ELBO defined in Equation 11 can be expressed in a neat form as follows:

Aside from disentangling high-level concepts, we are also interested in capturing the user’s specific preference for fine-grained level factors within different concepts, such as action or funny films in the film genre. Specifically, we aim to enforce statistical independence between the dimensions of the latent representation, so that each dimension describes a single factor, which can be formulated as forcing the following:

Follow the idea in (Ma et al., 2019) that $\beta$-VAE can be used to encourage independence between the dimensions. By varying the value of $\beta$, the model can be encouraged to learn more disentangled representations, where each latent dimension captures a single, independent underlying factor (Higgins et al., 2017). As a result, we amplify the regularization term by a factor of $\beta\gg 1$, resulting in the following ELBO:

To ensure the learning of causal structure and causal representations, we include a form of supervision during the training process of the model. The first component of this supervision involves utilizing the extra information $c$ to establish a constraint on the weighted adjacency matrix $A$. This constraint ensures that the matrix accurately reflects the causal relationships between the labels:

where $\sigma(\cdot)$ is the sigmoid function and $\kappa_{1}$ is a small positive constant value. The second component of supervision constructs a constraint on learning the latent causal representation $z$:

where $\kappa_{2}$ is a small positive constant value. Therefore, we have the following training objective:

where $\gamma_{1}$ and $\gamma_{2}$ are regularization hyperparemeters.

Our experiments were conducted on four datasets, which included a combination of real-world datasets. The largescale Netflix Prize dataset and three MovieLens datasets of different scales (i.e., ML-100k, ML-1M, and ML-20M) were used following the same methodology as MacridVAE. To binarize these four datasets, we only kept ratings of four or higher and users who had watched at least five movies. We choose four causally related concepts: (DIRECTOR $\rightarrow$ FILM GENRE), (FILM GENRE $\rightarrow$ ACTOR), (PRODUCTION YEAR). We compare the proposed approach with four existing baselines:

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  MacridVAE (Ma et al., 2019) is a disentangled representation learning method for the recommendation.

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  $\beta$-MultVAE (Liang et al., 2018) and MultiDAE (Liang et al., 2018) are VAE based representation learning method for reommendation.

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  DGCF (Wang et al., 2020) is a disentangled graph-based method for collaborative filtering.

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  SEM-MacridVAE (Wang et al., 2023) is the extension of MacridVAE by introducing semantic information.

The evaluation metric used is nDCG and recall, which is the same as (Wang et al., 2023).

Overall Comparison. The overall comparison can be found on Table 1. We can see that the proposed method generally outperformed all of the existing works. It demonstrates that the proposed causal disentanglement representation works better than traditional disentanglement representation.