## 3 Materials and Methods  
In this part, we will explain the basic structure and algorithm of a series of models related to our model in Part A; in Part B, we will formally introduce the ARFM model proposed in this paper and give detailed algorithm explanation; in Part C, we will make a summary of the model. 1. Related models and algorithms (1) Deep Crossing. The inputs to the model were a set of individual characteristics. The model has four types of layers: embedding, stacking, residual units, and scoring. The target function is the log loss function; however, the Softmax function or other functions can also be used: \[logloss=-\frac{1}{N}\sum_{i=1}^{N}(y_{i}\log(p_{i})+(1-y_{i})\log(1-p_{i})),\] (2) where \(N\) is the number of samples, \(y_{i}\) is the label for each sample, and \(p_{i}\) is the output of the single-node scoring layer. Regarding the role of each layer in the model, the embedding layer converts sparse class features into dense embedding vectors. Stacking is the stitching of different embedding and numerical features to form a new feature vector that contains all features. The residual unit layer is key to the model. The main structure of the residual unit layer is a multilayer perceptron, which is realized by a multilayer residual network. Its structure is illustrated in Figure 1.  
In Figure 1, \(X\) represents the input vector and \(F(X)\) represents the function that processes the input, namely, the RELU function in this article. The model can capture more information on nonlinear and combination features by cross-comb  
Figure 1: Residual element structure principle. the feature vector through a multilayer residual network, thereby enhancing the expression ability of the model. (2) Wide and Deep  
According to Google, a model can be divided into four stages: sparse features, dense embedding, hidden layers, and output units. Sparse features send wide and deep input data into two parts, followed by three stages. For a wide part, the features are combined through cross-product transformation, and the formula is as follows:  
\[\varnothing_{k}(X)=\prod\
olimits_{i=1}^{d}x_{i}^{c_{ki}}\quad c_{ki}\in\{0,1\}, \tag{3}\]  
where \(X\) is the set of input feature vectors, \(d\) is the total number of features, and \(c_{ki}\) is a Boolean variable. When the ith feature belongs to the \(k\)th combination feature, the value of \(c_{ki}\) is \(1\); otherwise, it is zero. \(x_{i}\) denotes the value of the ith feature. The deep part is the feedforward neural network. For category features, the original input is a feature string and the sparse high-dimensional category features are first converted into low-dimensional dense real vectors, which are typically called embedding vectors. These low-dimensional dense vectors are fed into the hidden layer of the neural network during the forward propagation. Specifically, each hidden layer performs the following calculations:  
\[a^{(l+1)}=f\Big{(}W^{(l)}a^{(l)}+b^{(l)}\Big{)}, \tag{4}\]  
where \(l\) is the number of layers; \(f\) is the activation function, and the RELU function is usually used. \(a^{(l)}\), \(b^{(l)}\) and \(W^{(l)}\) are the activation, bias, and model weights vectors of the \(l\) layer, respectively. Finally, the wide and deep parts are joined together through the full connection layer, and output is through the logical unit. The prediction of the logistic regression model is as follows:  
\[P(Y=1\Big{|}x)=\sigma\Big{(}w_{wide}^{T}[x,\varnothing(x)]+w_{deep}^{T}a^{(l_{ f})}+b\Big{)}, \tag{5}\]  
where \(Y\) is the binary classification label, \(\sigma()\) is the sigmoid function, \(\varnothing(x)\) is the transformation of the original feature vector \(x\), \(b\) is the bias item, \(l_{f}\) means the \(f\)th layer, and \(w_{wide}\) and \(w_{deep}\) are the weights vectors of wide and deep parts, respectively. (3) Deep FM  
As mentioned above, deep FM's improvement on wide and deep lies in replacing the wide part with FM.