# An Adversarial Deep Hybrid Model for Text-Aware Recommendation with Convolutional Neural Networks
## 3 Adversarial Deep Collaborative Filtering  
### General Framework  
We deal with binary implicit feedback such as clicks and purchases, where only positive interactions are observed. In particular, we aim to approximate the underlying distribution over user behavioral data by a parameterized model \(p_{\theta}\). Typically, the data distribution can be expressed in three forms: point-wise [2; 3], pair-wise [40; 41; 42] and list-wise [43; 44]. While the point-wise distribution is easy to implement, it ignores the ranking nature of recommendation. On the other hand, the list-wise approaches fail to model inter-list loss and are inefficient on large scale datasets. We therefore focus on learning the pair-wise distribution, which has previously shown to be effective in both non-DL and DL based collaborative filtering models [40; 45; 46].  
Following [40; 41], we construct a pair-wise training set as follows:  
\[S=\{\mathbf{s}=(u,j,j^{\prime})|u\in U\wedge j\in I_{u}\wedge j^{\prime}\in I /I_{u}\}, \tag{1}\]  
where \(U\) and \(I\) represent the whole user and item set, respectively, and \(I_{u}\) represents the set of items that user \(u\) has interacted with. A sample \(\mathbf{s}=(u,j,j^{\prime})\) is used to indicate that user \(u\) prefers item \(j\) over item \(j^{\prime}\)[40; 41]. Accordingly, we use \(p_{\theta}(\mathbf{s}=(u,j,j^{\prime}))\) to model the probability that user \(u\) likes item \(j\) better than item \(j^{\prime}\). Note that \(p_{\theta}\) is not a single distribution but a series of correlated distributions over the training data \(S\), subject to the totality order constraint that \(p_{\theta}(\mathbf{s}=(u,j,j^{\prime}))=1-p_{\theta}(\mathbf{s}=(u,j^{\prime},j))\) when \(j\
eq j^{\prime}\)[40].  
Given the training set \(S\), the learning task is to estimate the parameters of the generator \(p_{\theta}\). Instead of maximizing the likelihood of observing \(S\), we follow the idea of generative adversarial network (GAN) [19] to learn the parameters \(\theta\) by simultaneously training two models: a generative model \(p_{\theta}\) that generates data samples, and a discriminative model \(D_{w}\) parameterized by \(w\) that estimates the probability that a sample comes from the training data rather than the generator. The training procedure is to let the two components play a mini-max game: \(p_{\theta}\) varies its parameters to generate data that maximize the probability of \(D_{w}\) making a mistake, whereas \(D_{w}\) is trained to draw a clear distinction between the ground-truth samples and the generated ones. When the two components reach a balance, we obtain the desired model \(p_{\theta}\) that is able to produce samples indistinguishable from real data by \(D_{w}\).  
Formally, we perform the following min-max optimization over the generator parameters \(\theta\) and discriminator parameters \(w\):  
\[\min_{\theta}\max_{w}\{\mathrm{E}_{\mathbf{s}\sim p_{r}}[\text{log}D_{w}( \mathbf{s})]+\mathrm{E}_{\mathbf{s}\sim p_{\theta}}[\text{log}(1-D_{w}( \mathbf{s}))]\}, \tag{2}\]  
where \(p_{r}\) denotes underlying real data distribution. We discuss the specific parameterization of \(p_{\theta}\) and \(D_{w}\) in the following subsections. We dub the proposed method as Adversarial Deep Hybrid Recommendation (ADHR) model and illustrate its learning framework in Figure 1.