## 4 Model Description
### Position-Aware Rating Aggregation  
When predicting the scores for individual items in Section 4.1, there are no supervised signals on individual items for learning. Instead, we use set ratings as supervision in SPL. Therefore, we need to predict the set-level ratings in order to learn the model. Thus, we present a subset selection strategy in Section 4.2 to reproduce the process in which users rate sets according to their item-level preferences. After predicting item-level scores and deciding on the subset, we discuss the way to aggregate item-level scores in the position-aware rating aggregation module. To do so, the aggregation of items in the set is required. Item ratings are generated explicitly, since the aggregation is in the granularity of ratings. Given item ratings, the rating aggregation plays the role of estimating set ratings. In theory, various forms can be defined for the estimation, e.g., regression and neural network. However, these high-level aggregation can introduce certain bias between item ratings and set ratings. Since the ultimate goal of SPL is to estimate item-level preferences, the _unbiased estimation_ is required. In this paper, we propose to select subset of items to estimate the set rating. To ensure the unbiased estimation as well, we define the \(\oplus\) operation as follows:  
\[\oplus_{i=1}^{|S|}\hat{\tau}_{ui}=\frac{1}{\sum_{i=1}^{|S|}x_{ui}^{s}}\sum_{i= 1}^{|S|}x_{ui}^{s}\cdot\hat{r}_{ui}. \tag{6}\]  
where \(\oplus\) is an aggregation operation to combine items in the set. It is straightforward that Equation (6) is unbiased. \(x_{ui}^{s}\) is the variable to indicate whether item \(i\) is selected in the subset or not. While we define \(x_{ui}^{s}\) to indicate whether item \(i\) is selected in the subset or not, the hard selection brings instability for training. To ensure the smoothness during training, one possible way is to learn soft selection, i.e., learning a weight \(w_{ui}\) for each item \(i\) in the set. However, soft selection deviates from the idea of subset learning. Differently from learning soft assignments, we propose to learn a folded selection variable. We define \(x_{ui}^{s}\) in the following way:  
\[x_{ui}^{s}= \begin{cases}w_{ui},&\text{if the item is selected in the subset},\\ 0,&\text{otherwise}.\end{cases}  \tag{7}\]  
Equation (7) means that \(x_{ui}^{s}\) is a learnable weight parameter as long as item \(i\) is selected. \(x_{ui}^{s}\) will be folded if \(i\) is not selected. Learning for each user-item pair \((u,i)\) a separate weight \(w_{ui}\) is infeasible due to the large parameter space. Thus, we propose to learn user-specific weights. We further introduce **personalized positional weights** to aggregate ratings of items in the subset--that is, for each position \(p\) in the set, we will learn a weight \(w_{up}\). The positions of items are determined by sorting items according to their predicted ratings. Since the weights are not specific to items, the parameter space can be reduced greatly. Aside from smoothness, learning positional weights also helps to better fit the set ratings, given item ratings. For example, the positional weights corresponding to larger item ratings can be lowered if a user tends to overrate sets. The positional-aware weights require \(O(n_{u}\cdot n_{s})\) parameter space rather than \(O(n_{u}\cdot n_{i})\) (\(n_{s}\ll n_{i}\)). We illustrate our proposed positional-aware rating aggregation in Figure 4. Given the ratings of items in the selected subset, e.g., \(\{\hat{r}_{u2},\hat{r}_{u4},\hat{r}_{u5}\}\), we first sort the ratings. Through ranking, we can generate the ordered ratings \(\hat{r}_{u4},\hat{r}_{u5},\hat{r}_{u2}\) if \(\hat{r}_{u4}>\hat{r}_{u5}>\hat{r}_{u2}\). Accordingly, we also retrieve three positional weights from \(\mathbf{w}_{u}\); i.e., \(w_{u1},w_{u2},w_{u3}\).