The issue of class imbalance exists in many areas [17,18], and has a great impact on the prediction performance of machine learning algorithms [31–33] and a nontrivial impact on the RS [34,35]. Class and feature imbalance in e-commerce has attracted the attention of many experts [36,37].

Regarding class imbalance, traditional machine learning algorithms often apply preprocessing to increase or reduce the number of positive or negative instances, respectively [22,23]. To address the problem of sample redundancy or outliers, Wei et al. [38] presented an improved stochastic synthetic minority oversampling technique that applies ascending operations to rank the majority class of samples and assigns weights through kernel density estimation. Hoyos-Osorio et al. [39] proposed an undersampling method for imbalanced data classification based on information-theoretic learning, which selects the most relevant examples from the majority class to enhance classification performance in imbalanced data scenarios. Preprocessing methods, such as oversampling and undersampling, increase computational complexity and cause overfitting by replicating minority classes, or lose information by reducing the size of the majority class [40]. Therefore, Lin et al. [41] investigated the effect of hybrid combinations of undersampling and oversampling methods of different order on 44 different class-imbalanced datasets. Cost-sensitive algorithms that assume higher costs for misclassification of minority classes are gaining attention since they do not change the original data distribution [24,25]. Cost-sensitive deep neural networks that learn weights for different classes [15,25,42–44] or employ new loss functions [45] have been proposed. However, current cost-sensitive algorithms do not consider the confidence level, which represents the accuracy that a sample point is correctly classified. Lin et al. [20] proposed focal loss to solve the dense object detection problem, and were the first to mention penalization of easy samples. Lu et al. [21] proposed shrinkage loss, which penalizes easy samples without losing hard samples. However, the complexity of the calculation and stability of the training process should also be considered. We focus on suppression loss to solve the imbalance issue, with faster calculation and stabler training under the premise of better results.

Traditional feature selection and feature fusion are common processing methods to address feature imbalance [26–28]. However, feature selection requires the discarding of some features with a low contribution to the result, which will affect the integrity of features [17,18]. LAFAM can well solve the feature imbalance issue, as discussed in Section 3.

Almost all e-commerce platforms use product recommendation systems [46–48], which can often help customers find items of interest and thereby contribute to boosting sales. Traditional recommendation systems can be categorized as content-based recommendation systems (CBRSs), collaborative filtering recommendation systems (CFRSs), and hybrids [49,50]. A CBRS generates preferences based on a user’s profile and product features [51,52]. A CFRS uses a similarity matrix to generate preferences based on the rating of neighbor users and items [1,53]. The hybrid framework combines CBRS and CFRS to achieve precise performance by reducing the drawbacks of conventional techniques [54,55]. Although classical RS methods have achieved remarkable success, they suffer from issues such as cold starts and data sparsity [1–4,56].

With recent deep learning achievements in applications such as natural language processing (NLP), machine translation, and computer vision (CV), machine learning models have been exploited for RSs, bringing more capabilities by addressing the challenges of traditional RS models. Compared to the traditional recommendation architectures, deep learning-based RS models provide better representation learning of user-item interactions. Multi-layer perceptron (MLP) can model data with simple correlation to enhance nonlinear transformation, but high complexity and slow convergence limit its performance [56]. Convolutional neural network (CNN) is powerful for feature extraction of contextual information, but it requires high parameterization tuning [57]. Recurrent neural network (RNN) is specifically used to model sequential data [58]. However, it suffers from the exploding or vanishing gradient, which makes it difficult to train when incorporating temporal layers to capture sequential information. As deep learning models are increasingly adopted in recommendation systems, explainability has become a critical concern for researchers and practitioners [59,60]. Explainable artificial intelligence (XAI) [61] is being progressively integrated into recommendation models to assist users and developers in better understanding how predictions are made.

Since imbalance has a great impact on the prediction results of machine learning algorithms (as described in Section 2.1), it should be considered when designing an RS. Hence, we propose LAFAM as a recommendation system, which focuses on solving the impact of imbalance on machine learning algorithms and RSs.

We first utilize PS-smart [62], XGBoost [63], and GBDT [64] to determine and rank the feature importance. These algorithms can effectively prioritize features that contribute most to model performance, thereby mitigating the potential negative impact of less informative features that could exacerbate feature imbalance. Based on the rankings provided by these three methods, we then calculate the weighted feature importance scores to produce a final ranked list of features. To optimize the output efficiency of network, we truncate the ranked list to discard features with insignificant contributions. The exact truncation position should be adapted to the specific problem at hand, aiming to maximize efficiency and minimize computational resources and time without compromising output quality. This step is particularly useful for addressing feature imbalance, as it reduces the influence of redundant or noisy features, thus helping the model focus on the most relevant information.

Furthermore, we categorize the features into dense features, sparse features, and sequential features to better align with the network operations. Different types of features often exhibit distinct distributional characteristics, and proper categorization helps optimize the handling of imbalanced features. The detailed classification process and corresponding sub-networks will be explained in Section 4.2. It is important to note that for addressing different problems, the classification approach should be adjusted according to the types of subsequent sub-networks used. This targeted feature preprocessing approach ensures that imbalanced features are treated in a way that maximizes efficiency and minimizes the risk of performance degradation due to feature imbalances.

LAFAM can perform independent sub-network training on important features, which are obtained through feature preprocessing, with learning networks depending on features. A supervision network learns the contribution weight of each sub-network. More important features contribute more weight to the result, which increases their influence and improves accuracy. Fig. 2 shows the structure of the model, in which different features are input to N sub-networks, and the supervision network learns their output weights.

Specifically, we set up several sub-networks to learn the features, and the number and type of sub-networks can be adjusted according to different problems. The sub-networks shown in Fig. 2 are of different types. In our model, the output of each network is a score between 0 and 1, which indicates the sample’s probability of becoming a hot sale product learning within each respective network. A score of 0 signifies the lowest potential to be a hot sale product, and a score of 1 signifies the highest potential.

For each sub-network, the larger the loss value the lower the confidence of the current output. It is difficult to judge a sub-network based on features with low confidence, and these are usually discarded by the traditional methods. However, any feature will have an impact on the prediction result, and we should not discard a feature that is difficult to judge or makes a small contribution. For example, in e-commerce recommendations, commission, and rebate characteristics are often difficult to judge by the predictive network, but they are important. In our network, confidence means that the network should strengthen the learning of the sub-network, so its loss value accounts for a larger proportion of the loss of the entire network. Therefore, we calculate the softmax value of the loss value output by the network by Eq. (1), and determine the proportion of its contribution to the overall loss. At the end of the network, the weighted sum of the loss values obtains the overall loss.

For a given li,i∈(0,1,2,…,N), which is the output loss of sub-networks, we estimate the probability p(ε=j|li) for each category j with softmax, i.e., we estimate the probability of each classification result of ε. Let ε→=[ε0,ε1,ε2,…,εN] be a vector of weights of output values from each network. When we obtain the predicted value through the network, we hope that a result with high confidence will dominate. Therefore, in calculating the predicted value, we use the opposite coefficient εi as the weight of output from sub-networks: (1)εi=eli∑n=0NelN=eliel0+el1+⋯+elN,i=0,1,2,…,N

Then, we design a supervised network whose output is the ratio of each subnetwork output to the total output. The labels used for training are the weight vectors ε′→=[ε1′,ε2′,ε3′,…,εN′] about li obtained after a softmax calculation based on the loss values. We substitute the parameters shown in Fig. 2 in the formula, which simplifies the ratio of each category to: (2)εi′=e−li∑n=0Ne−lN=e−lie−l0+e−l1+⋯+e−lN,i=0,1,2,…,N

In the training process, we need to fit ε′→ and ε→ to make these two probability distributions infinitely close. Therefore, the final loss function lra is: (3)lra=λlfinal+μlsubnet+ηlsuperv=λ∥yfinal−y0∥2+με→∥ysubnet−y0∥2+η∥ysuperv→−ε′→∥2=λ∥yfinal−y0∥2+μ∑j=0NeljT⋅∥yj−y0∥2∑k=0NelkT+η∑j=0N∥ysuperv→−(1−elj∑k=1Nelk)∥2where λ, μ, and η are network parameters that can be set and adjusted using grid search or other tuning methods; yfinal is the final output of the network overall; y0 is the label of the original sample; ysubnet={y1,y2,…,yN} is the output of the sub-network; ysuperv→ is the output of the supervision network; N is the number of feature groups, which in this task is 4; and T is the temperature coefficient used in softmax operation, and it can adjust the difference between the values calculated by softmax.

In general, T≥1. The smaller T is, the steeper the softmax curve is, and the bigger the difference between the output values will be. The larger T is, the smoother the softmax curve is, and the difference between the output values will be small. In this way, we control the balance of weighting the sub-network loss li by the coefficient T.

We propose suppression loss to solve the class imbalance issue, which influences machine learning algorithms to classify inputs in the positive (majority) class every time to achieve high prediction accuracy. If we train the deep learning model with traditional loss functions (e.g., square loss, log loss), the loss value of positive samples will account for a larger proportion of the total loss. However, in practice, we are often more concerned with the accuracy of the negative class (precision or F-measure). For example, in our scenario, hot sale products are a minority, but traditional deep learning-based models tend to neglect the minority class and output general products as the prediction result. Therefore, regarding class imbalance, we want negative classes to contribute more loss value. Hence, we propose suppression loss: (4)Lsp(yfinal,y0)=[α(yfinal+β)2+2α(yfinal+β)+12]×[θ(|yfinal−y0|+ρ)2+2θ||yfinal−y0|+ρ|+12]×|yfinal−y0|γwhere |yfinal−y0| is the absolute difference between the estimated probability yfinal and its true label y0; α, β, θ, ρ, and γ are the parameters of the model, which can be set according to the practical dataset problem through the grid search method.

The suppression loss function consists of three parts. The first part is the suppression of the loss contribution to the large category of samples through a function. The second part is to suppress the easy sample contribution by using the same function and different parameters. The last part is the suppression of the easy sample contribution employing a high-power function that expands the output disparity. The feasibility of suppression loss is explained below, and a more detailed process can be found in the Appendix (Figs. A1 and A2).

Firstly, we need a function fsp(x) to meet the following conditions:

The value range of fsp(x) is (0,1) or (0,1].

We find that the simple function f(x)=x1+|x| has the above properties. This is an S-type function, with similar properties to a sigmoid function. However, f(x) is more concise and less complex, as shown in Fig. 3a. The function fsp(x)=α(x+β)2+2α|x+β|+12 is obtained by linear transformation (translation and compression). Its value range is (0,1), and its slope and the point over 0.5 can be adjusted by changing α and β. Fig. 3a is the original graph of f(x)=x1+|x|. Fig. 3b is a function diagram for controlling both the independent and dependent variables within the range of 0–1 after compression and translation. Fig. 3c is a graph of fsp(x) under different parameters, from which it can be seen that the larger α is, the larger the slope is, i.e., easily classified data can be better penalized. Fig. 3d is a comparative diagram of three loss function adjustment factors. When x is closer to 0, the smaller the suppression loss the stronger the penalty of easily classified data. The larger x is, the closer the function value is to 1. The function hardly punishes sufficiently large x. It can be seen from Fig. 3d that focal loss still inhibits large x values. The two-parameter control of shrinkage loss is not as accurate as the three-parameter control of suppression loss. Suppression loss has a simpler functional form than shrinkage loss (fsp(⋅) is simple, but fshrinkage(⋅) is elementary), which will greatly help its stability. We provide verification results in the following experiment.

When applied to classification problems, it is assumed that the label of the positive class is 0. In this case, the closer the predicted value is to 0, the more its loss is penalized, i.e., this kind of sample contributes less to the overall loss. If the label is 1, then the smaller the predicted value the larger its loss, and the less it is penalized in the function, so this kind of sample contributes greatly to the overall loss.

According to the focal loss [20], we superimpose fsp(⋅) on lγ. In this way, we can penalize the sample points of positive and easy classes. As shown in Table 1, easy positive sample points are penalized by fsp(⋅) and γ at the same time, i.e., they are restrained to the greatest extent. Hard positive sample points are penalized by fsp(⋅). Easy negative sample points are penalized by γ, while hard negative sample points are barely penalized. This achieves the goal of making hard negative sample points contribute the most loss. The accuracy of the classification of hard negative sample points determines the final loss value of the network. The more mistakes in hard negative sample points, the greater the loss. The network pays more attention to hard negative sample points. Therefore, the classification rate of negative classes can be improved.

We use the Tmall (Taobao) non-public dataset and the Kaggle public dataset Corporación Favorita Grocery Sales Forecasting (CFGSF) for experimental verification.

Tmall: The dataset is 10.7 GB in size and contains 51,134,193 rows of data, with 453 data features from August 2019 to August 2020. The dataset contains information such as product name, product category, product price, coupon price, historical sales volume, highest single-day sales, and single-day average sales.

Table 2 presents some examples of sales records from the CFGSF dataset. After merging, extracting, cleaning, and screening to ensure that abnormal data and noise would not affect the classification accuracy of the neural network, the dataset contains 70,205,249 rows of data with a dataset size of 13.93 GB. To put the forecasting problem into practice, we transform the forecasting of commodity sales from a regression problem to a classification problem. After normalizing the sales volume of the training set, we select 0.15 as the threshold to define the two classifications. After classification, the data volumes of categories 0 and 1 are 70,184,766 and 20,483, respectively, and the imbalance ratio is 3426:1.

With a simple MoE structure, the importance weight distribution of the softmax gate learning model on experts of different feature groups is shown in Fig. 4a. The weight of time series features is high, followed by the importance of commission and voucher features. This is consistent with the conclusion from the PS-smart that the future sales volume of most samples is highly correlated with the sales volume of N days in recent history. Fig. 4b shows the learning result based on LAFAM. It has a high learning fit for a large number of simple samples, with high importance of historical time series features and insufficient learning of other features. We observe that the latter’s dependence weight distribution on different features is more even so that other basic feature models can also be fully learned, thereby obtaining better recommendation results.

Therefore, in the process of realizing the model, we divide the features into four parts. The model application structure diagram is shown in Fig. 5. The basic feature depicts the basic attributes of commodities (including historical prices, historical sales, and other statistical features). The second and third parts are the coupling and commission features, which are the key factors to describe whether goods can become hot sale products. The fourth part is the sequence feature, i.e., the time series of daily transactions of commodities in the past three months. The features are categorized and fed into four distinct sub-networks.

All features are processed using the deep & cross network (DCN), a commonly employed approach in recommendation systems, which includes 3 deep layers with ReLU activation functions and 2 cross layers. The cross layers are specifically designed to learn feature interactions in a more efficient manner, capturing bounded-degree interactions between features at different levels. This structure is highly effective for both vertical recommendations and wide-ranging interest exploration, and it offers high computational efficiency, making it an ideal choice for learning from all features. Coupling and commission features are handled using a 3-layer MLP with ReLU activation function, which can extract high-dimensional features from the samples, making it suitable for recommendation predictions. Sequence features are processed using a 2-layer gated recurrent unit (GRU), each layer with 64 hidden units and using the tanh activation function. The GRU is a variant of the long short-term memory (LSTM) network, itself an evolved form of the RNN, known for its excellent predictive performance. Compared to the LSTM, the GRU has a simpler structure and effectively addresses the long-term dependency problem of RNNs. This results in high computational efficiency and excellent accuracy, making the GRU well-suited for learning from sequence features. For the training process, we employ the Adam optimizer with a learning rate of 0.01 and a batch size of 64. The model is trained for 50 epochs, with the suppression loss used to compute the final loss value. The parameter settings for the LAFAM framework are determined based on our own experimental results to ensure optimal performance for the given dataset and task.

Combined with the theoretical derivation and practical application, we improve and optimize the network as follows:

We combine the idea of MoE to build an expert network for different feature groups. We combine the expert results through the softmax gate layer and use the loss aware method for learning. In the training process, the output result yi of each expert is calculated with the true value of loss in the training stage, which is recorded as l0, l1, l2, l3, and the distribution of different losses is obtained through a softmax layer. Assuming that each expert learns well and tries to accurately predict the final value yfinal, the distribution of loss represents the importance of different feature groups to the prediction results. Therefore, the difference in attention results of samples in different feature groups is strengthened in the training stage.

We evaluate the performance of the LAFAM network, suppression loss, and the RS by controlling variables, as shown in Fig. 6. We first assess the performance of LAFAM according to regression problems. Since the output of LAFAM is a score (as shown in Fig. 1), which belongs to [0,1], the higher the score, the more likely a product is to become a hot sale product. Then, we use the parameters obtained through a grid search to verify the performance of suppression loss, and we validate the performance of LAFAM and suppression loss from a classification problem perspective. Finally, we verify the effect of the entire RS, i.e., the fusion of LAFAM and suppression loss.

In the experiments, our proposed LAFAM can calculate the probability of a product being a hot sale product. In regression problems, the performance of LAFAM is assessed with four evaluation metrics, which are Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Weighted Mean Absolute Percentage Error (WMAPE), and Top 50/100/200 Hit Rate (HR@50/100/200). In sorting problems, we use Precision, Accuracy, and F-measure to evaluate the effectiveness of LAFAM and suppression loss.

RMSE presents the gap between the predicted value by the model and the true value: (5)RMSE=1n∑i=1n|y^i−yi|where n is the number of samples, y^i is the predicted output, and yi is the true label.

MAE directly calculates the absolute value of the error between the predicted value and the true value: (6)MAE=1n∑i=1n|y^i−yi|

WMAPE weights the prioritized products so as to bias the prediction error towards those products: (7)WMAPE=∑i=1n|y^i−yi|∑i=1nyi

HR is a recall-based metric, which is defined as: (8)HR@k=NkGTwhere GT denotes the number of test sets, and Nk denotes the sum of test sets in each user’s Top k recommended products.

Accuracy indicates the percentage of the number of correctly categorized samples to the total number of samples: (9)Accuracy=TP+TNTP+TN+FP+FNwhere TP indicates that the final prediction of the positive sample is positive, TN demonstrates that the final prediction of the negative sample is negative, FP means that the negative sample ends up with a positive prediction, FN implies that the positive sample ends up with a negative prediction.

Precision denotes the proportion of correctly categorized positive samples out of all categorized positive samples: (10)Precision=TPTP+FP

Recall represents how many of all positive samples are correctly categorized by the model. F-measure is the harmonic mean of Precision and Recall: (11)Recall=TPTP+FN (12)F−measure=(1+β2)×Recall×Precisionβ2×Recall+Precisionwhere β is a coefficient regulating the weight of Precision and Recall, typically taken as 1.