Billion-user Customer Lifetime Value Prediction: An Industrial-scale Solution from Kuaishou

The existing LTV prediction methods are mainly divided into two lines. The first is based on historical data or classical probability and statistics model. The RFM framework (Fader et al., 2005), groups users based on recency, frequency, and monetary value of historical consumption, to roughly calculate how long or how often the users buy or consume something. The basic assumption it follows is that the users who tend to consume more frequently, will be more likely to consume again in the near future if they have relatively higher consumption recently. The BTYD (Wadsworth, 2012) family is a very classic probability model for user repeat purchase/churn, which assumes that both users churn and purchase behavior follow some sort of stochastic process, one of the most well-known solutions is the Pareto/NBD (Fader et al., 2005) model, which is commonly used in non-contractual, continuous consumption scenarios (that is to say, customers may purchase at any time). This method will model two-parameter distributions, Pareto distribution is used for binary classification, predicting whether users are still active, and negative binomial distribution is used to estimate the frequency of purchases. Most of these methods are coarse-grained modeling of the consumption habits of users, such as whether to purchase it, the frequency of purchases, etc, they do not provide fine-grained modeling of the amount spent or the specific value that users will contribute to the platform over a long period of time. The second is to model LTV directly through machine learning or deep learning. Early methods rely more on hand-crafted features and tree-structured models. Recently, deep learning technology has also been applied to LTV prediction. (Drachen et al., 2018) is a two-stage XGBoost model, which first identifies high-quality users and then predicts their LTV. (Vanderveld et al., 2016) firstly splits users into several groups and performs random forest prediction within each group. ZILN (Wang et al., 2019) assumes that it obeys the log-normal distribution, and exploits the advantages of DNN in feature intersection and information capture to fit the mean and standard deviation of the distribution, and finally the expected value of the log-normal distribution is used as an estimate of LTV, but the distribution assumption is so restrictive, which leads to limitations in its application. (Chen et al., 2018) shows that CNN is more efficient than MLP to model time series data for LTV prediction. (Xing et al., 2021) focuses on how to use the wavelet transform and GAT to learn more reliable user representation from the sparse and changeable historical behavior sequence of users. However, they are still based on simple MSE loss to fit LTV distribution.

In the real world, most users are low-value contributors, so the distribution of LTV data is extremely imbalanced, of which long-tailed distributions are the most common. As we all know, modeling long-tailed distributions is notoriously difficult to solve. Research in this area is roughly divided into three categories: Class Re-Balancing, Information Augmentation and Module Improvement (Zhang et al., 2021).

In Class Re-Balancing, one of the most widely used methods to solve the problem of sample imbalance is Re-Sampling, including Random Over-Sampling (ROS) and Random Under-Sampling (RUS). ROS randomly repeats the samples of the tail class, while RUS randomly discards the samples of the head class. When the class is extremely imbalanced, ROS tends to overfit the tail class, while RUS will reduce the model performance of the head class. Another conventional approach to this problem is Cost-Sensitive Learning, also known as Re-Weight. This method assigns larger weights to minor samples and smaller weights to major samples. However, the weights often depend on the prior distribution of the sample label, and when the distribution changes, the sample weights need to be re-determined.

Information Augmentation mainly alleviates the long-tailed problem through transfer learning and data augmentation. For transfer learning, it includes methods such as head-to-tail knowledge transfer (Yin et al., 2019), knowledge distillation (He et al., 2021) and model pre-training (Erhan et al., 2010), while data augmentation is to alleviate the problem of data imbalance by enhancing quantity and quality of sample sets, but the quality of expanded data is difficult to guarantee.

Module Improvement is to improve the modeling ability of long-tailed distribution from the perspective of network structure optimization. It can be roughly divided into the following categories: (1). Improving feature extraction through representation learning. (2). Designing classifiers to enhance the classification ability of the model. (3). Decoupling feature extraction and classifier classification. (4). Ensemble learning to improve the entire architecture.

Our work continues the line of modeling LTV through machine learning/deep learning. We effectively model LTV from the perspective of imbalanced distribution processing, and model the ordered dependencies of multi-time-span LTVs for the first time, making important technical contributions to the field of LTV modeling.

Given the feature vector x under a fixed time window of the user (e.g. 7 days), predicting the value it will bring to the platform in the next $N$ days ($LTV_{N}$). Obviously, $LTV_{N-\Delta}$ $\leq$ $\cdots$ $\leq$ $LTV_{N}$ $\leq$ $\cdots$ $\leq$ $LTV_{N+\Delta}$. The multi-task framework needs to predict LTV of $T$ different time spans based on the input feature x at the same time:

The features of model input mainly include user portrait characteristics, historical behavior information during a period of time window (e.g. 7 days), etc. For the historical behavior information of users, we organize them into day-level sequence format, such as ”active duration”: [13, 7, 70, 23, 44, 12, 9], which represents the active minutes one user interacts with the Kuaishou APP per day in the past 7 days. From the practical effect, the behavior features of day-level sequence format can bring more information gain. At the same time, in order to improve the generalization ability of the model, we do not consider highly personalized features such as $user\_id$, $item\_id$ in feature selection. In offline data analysis, we found that the LTV of some business scenarios has typical cyclical fluctuation due to user’s periodical behavior. For example, the consumption tendency of users on weekends increases, while decreases on weekdays. In addition, LTV is also more sensitive to seasonal or abnormal events. For instance, during holidays or e-commerce shopping festivals, the user’s consumption tendency increases significantly. The model needs to capture these signals, so we collect these periodic, seasonal or abnormal event information as parts of the features input to the model. Whether in marketing, advertising or other businesses, we often pay attention to the LTV prediction performance of users from different channels, therefore, some channel-related information can be added to enhance the model’s ability to predict LTV of channel dimension. Drawing on the idea of the RFM method, we also introduce some recency, frequency, and monetary value related to the user’s historical consumption as features.

As shown in Figure 2, given the input features x, we embed each entry $x_{i}$ ( $x_{i}$ $\in$ x, 1 $\leq$ $|\textbf{x}|$) into a low dimension dense vector representation $v_{i}$ via a Shared Embedding Layer. Among them, the features of the dense type will be discretized firstly, for some dense features of long-tailed distribution, we tend to process at equal frequencies. The embedding vector of each feature entry will be concatenated together as the final user representation $v$ = [$v_{1}$;$v_{1}$;$\cdots$;$v_{|\textbf{x}|}$], where [$\cdot$;$\cdot$] denotes the concatenation of two vectors. Better user representation can be obtained through more complex feature intersections (e.g. DeepFM (Guo et al., 2017), DCN (Wang et al., 2017)) or user behavior sequence modeling (e.g. Transformer (Vaswani et al., 2017), Bert (Devlin et al., 2018)), but it is not the focus of this paper.

For each specific LTV prediction task, we design a novel module MDME (Multi Distribution Multi Experts), of which the core idea is Divide-and-Conquer. By decomposing the complex problem into several simpler sub-problems, each of them can be broken down easily. It is true that modeling LTV directly will bring a lot of troubles, among which the most important one is that, due to the imbalance distribution of samples, it is brutal to learn the tailed samples well. The sample distribution mentioned here refers to the distribution of LTV value. In theory, it is less challenging for the model to learn from a more balanced sample distribution than from an imbalanced one (Xiang et al., 2020). Inspired by this, we try to ”cut” the entire sample set into segments according to the LTV distribution, so that the imbalanced degree of LTV distribution in each segment is greatly alleviated.

Figure 3 shows the structure of MDME. The whole end-to-end learning process consists of three stages. Firstly, the Distribution Segmentation Module (DSM) realizes the segmentation of LTV distribution, that is, sub-distribution multi-classification. Specifically, the sample set is divided into several sub-distributions according to the LTV value. For some LTV values that account for a very high proportion in the sample set, they can be regarded as an independent sub-distribution, just like 0 in the zero-inflated long-tailed distribution. After distribution segmentation, each sample will only belong to one of the sub-distributions. We need to learn the mapping relationship between the samples and the sub-distributions, which is a typical multi-classification problem. At this stage, the sample label is defined as the sub-distribution number. Secondly, we continue to adhere to the basic principle of Divide-and-Conquer, and use the Sub-Distribution modeling Module (SDM) to further break each sub-distribution into several buckets according to the actual LTV value of the samples within the sub-distribution, then we transform the sub-distribution modeling into the problem of multi-bucket classification. In this way, we can adjust the number of samples falling into each bucket by tuning the bucket width, thus, the number of samples in each bucket is approximately equal. The sample label at this stage is the bucket number. Similar to distribution segmentation, the LTV value with a high proportion of samples in the sub-distribution can be used as an independent bucket number, that is, the corresponding bucket width is 1. After two stages of label redefinition, the modeling difficulty of the entire LTV distribution is greatly mitigated and the modeling granularity has been reduced to each bucket. At this time, the LTV distribution of samples in each bucket has been relatively balanced. The last stage is to learn a bias in the bucket to achieve fine-grained LTV modeling in the bucket, which is also processed by SDM. We perform min-max normalization on the LTV value of the samples in the bucket, so that the range of LTV value is compressed to between 0 and 1, which we define as the bias coefficient. The bias coefficient is then regressed based on MSE. Since MSE is very sensitive to extreme values, normalization can restrict the value range of label and the magnitude of loss, so as to reduce the interference to the learning of other tasks. In summary, for each sample, the target sub-distribution is first determined by DSM, then SDM corresponding to the target sub-distribution estimates the target bucket and the bias coefficient in the bucket to obtain the final LTV. The whole process realizes the prediction of LTV from coarse-grained to fine-grained.

DSM includes the Distribution Classification Tower (DCT) and the Distribution Ordinal Tower (DOT), SDM consists of Bucket Classification Tower (BCT), Bucket Ordinal Tower (BOT) and Bucket Bias Tower (BBT), of which DCT and BCT have similar structures, they respectively implement multi-classification of sub-distribution and bucket multi-classification within sub-distribution. The activation function of the output layer in both DCT and BCT is softmax, which generates sub-distribution multinomial distribution and bucket multinomial distribution. We use the estimated probability of each sub-distribution in DCT as the weight of the bucket multinomial distribution within each sub-distribution to obtain a normalized bucket multinomial distribution of the entire LTV, denoted as $o_{t}$, where $t$ represents the $t_{th}$ LTV prediction task. The ODMN framework performs linear transformation on the normalized bucket multinomial distribution of the upstream output through Mono Unit, and directly adds the transformation result to the output logits of downstream DCT and BCT, which can affect the output distribution of the downstream task, so that once the normalized bucket multinomial distribution of the upstream output shifts, the downstream task can capture this signal in time, and the output distribution will also shift in the same direction accordingly. The normalized bucket multinomial distribution and Mono Unit will be introduced in next subsection. In addition, there is obviously a relative order relationship between sub-distributions and between buckets. We found that introducing a module to explicitly learn this relationship can further improve the accuracy of classification and ranking. Therefore, the DOT and the BOT are designed based on Ordinal Regression (Fu et al., 2018) to model this kind of order relationship, of which, the output layer activation function is set to sigmoid, and the estimated values of DOT and BOT will be used as soft label to guide the learning of DCT and BCT through distillation respectively. After DCT and BCT determining the target sub-distribution and the corresponding target bucket, it’s time for BBT to implement the fine-grained regression by fitting the normalized bias of LTV in each bucket based on MSE, the number of logits in the output layer of BBT is set to the number of buckets whose width is greater than 1, and sigmoid is selected as the activation function of the output layer to limit the estimated value between zero and one. Assuming that the $t_{th}$ task divides LTV into $S_{t}$ sub-distributions, for each one $s_{t}$(1 $\leq$ $s_{t}$ $\leq$ $S_{t}$), the output function of each tower module is defined as:

The structure of MDME realizes LTV modeling from coarse-grained to fine-grained. Given the user representation vector, the sub-distribution with the highest probability predicted by DCT module is set as the target sub-distribution, and then the BCT module of the target sub-distribution outputs the target bucket with the highest probability. At the same time, we obtain the left boundary and bucket width of the target bucket. Finally, the BBT module outputs the bias coefficient in the target bucket, and we can calculate the estimated LTV value:

$u_{t}$ is the bucket number which has the maximum probability in the distribution $q_{argmax(p_{t}^{c})}^{c}$. It is worth to mention that the MDME module can be disassembled as an independent model to estimate LTV, and still achieves good performance.

The LTVs of different time spans satisfy the ordered relationship, such as $ltv_{30}$ $\leq$ $ltv_{90}$ $\leq$ $ltv_{180}$ $\leq$ $ltv_{365}$, which is determined by business. The conventional modeling strategies are to utilize an independent model to estimate one target, or simply to learn LTVs of multiple time spans at the same time based on multi-task learning. However, they do not make full use of the ordered dependencies between LTVs of different time spans. We believe that modeling this ordered dependencies relationship can effectively improve model performance. Here, through several multi-layer perceptrons with non-negative hidden layer parameters, which we call Mono Unit, the output layers of the upstream and downstream LTV prediction tasks are connected in series, so that the distribution information output by the previous task can monotonically and directly affect the output distribution of the latter task. The reason why Mono Unit connects with the output layer of each task network is that, the closer to the output layer, the richer theoretically the task-related information the hidden layer could get. (Xi et al., 2021).

Specifically, the task needs to output a normalized bucket multinomial distribution. The MDME module performs ”coarse-to-fine-grained” processing on the imbalanced LTV distribution. Through distribution segmentation, and sub-distribution bucketing, it greatly reduces the LTV modeling difficulty. These two stages transform LTV modeling into a bucket multi-classification problem, we can calculate the normalized bucket multinomial distribution $o_{t}$ based on the multinomial distributions output by these two stages.

The normalized bucket multinomial distribution of the output of the $t-1_{th}$ task directly performs the ”Add” operation with the output logits of the DCT module and the output logits of the $S_{t}$ BCT modules respectively after the processing of ($S_{t}$+1) Mono Units. Mono Unit can capture the shifting trend of the output multinomial distribution of upstream tasks, and then affect the output distribution of downstream tasks. In theory, the modeling difficulty of short-term LTV is less than that of long-term LTV. Through the multinomial distribution mapping and capturing mechanism, short-term LTV can be used to assist the modeling of long-term LTV, and the difficulty of modeling subsequent tasks will be reduced. Meanwhile, it is necessary to do gradient truncation between upstream and downstream tasks, in order to diminish the impact of downstream task on upstream one.

ODMN framework further improves the performance and prediction accuracy by modeling the ordered dependencies of LTVs in different time spans, and achieves soft monotonic constraints. From the perspective of modeling difficulty, the design of Mono Unit enables simple tasks to assist the learning of complex tasks, thereby easing the modeling difficulty of long-term LTV.

For each MDME module, the losses associated with DSM module include the sub-distribution multi-classification cross-entropy loss $\mathcal{L}_{t}^{c}$, the sub-distribution Ordinal Regression loss $\mathcal{L}_{t}^{o}$, and the sub-distribution distillation classification cross-entropy loss $\mathcal{L}_{t}^{dis}$. The losses of each SDM module consists of the bucket multi-classification cross-entropy loss $\mathcal{L}_{s_{t}}^{c}$, the bucket Ordinal Regression loss $\mathcal{L}_{s_{t}}^{o}$, the bucket distillation classification cross-entropy loss $\mathcal{L}_{s_{t}}^{dis}$, and the bucket bias regression loss $\mathcal{L}_{s_{t}}^{b}$. The definition of the Ordinal Regression loss is as follows:

$B$ is the number of samples in a batch, $U$ is the size of the output logits, $y_{t}$ is the real label corresponding to a sample, and $\hat{l}$ is the estimated value of Ordinal Regression. $\eta$(·) is an indicator function such that $\eta$(true) = 1 and $\eta$(false) = 0, u is bucket number. $\mathcal{P}^{u}$ is the probability that the predicted ordinal regression value is greater than actual bucket number. It should be noted that the learning of DSM module needs to use all samples, but for the SDM module, only the samples belonging to the sub-distribution or the buckets are utilized to isolate the training samples. Mono Unit captures the shifting trend of the bucket multinomial distribution output by the upstream task, then affects the output distribution of the downstream task, thus realizing soft monotonic constraints in a coarse-grained form. In order to further model the monotonic constraint relationship between upstream and downstream tasks, we perform fine-grained calibration on the estimated LTV values of each task. Specifically, if the short-term LTV estimated by the upstream task is larger than the long-term LTV estimated by the adjacent downstream tasks , a penalty loss is introduced:

In summary, the loss function of ODMN is defined as follows, $\alpha$ controls the strength of the fine-grained ordered dependency calibration loss, $\beta$ and $\gamma_{s_{t}}$ determines the contribution of each distillation loss:

We adopt Normalized Rooted Mean Square Error (NRMSE) and Normalized Mean Average Error (NMAE) defined in (Xing et al., 2021) as parts of the evaluation metrics. The absolute value of Mean Bias Error (AMBE) is another important metric, which can capture the average bias in the prediction. Note that, the lower these three metrics are, the performance is better. However, the above metrics can not measure the model’s ability to differentiate high-value users from low-value users, nor can they reflect how well the model fits the real imbalanced distribution.
ZILN (Wang et al., 2019) introduce the normalized model Gini based on the Lorenz Curve and the Gini Coefficient. Among them, the Lorenz Curve (Figure 4) is an intuitive tool to describe the distribution of high and low value users. The horizontal axis of the Lorenz Curve is the cumulative percentage of the number of users, and the vertical axis is the cumulative percentage of LTV contributed by users. The fit degree of the model estimated value curve and the true value curve reflects the fitting ability of the model to the imbalanced distribution. Gini Coefficient is a discriminative measure. The larger the estimated Gini Coefficient, the better the model’s ability of discrimination. For an imbalanced dataset, assuming that the total sample size is $N$, $N_{i}$ is the sample size of the class $i$, and $C$ is the total class size, then Gini Coefficient can be defined as:

The ratio between the model Gini coefficient and the true Gini coefficient yields the normalized model Gini coefficient which measure fit degree of the estimated value curve and the true value curve. However, the normalized model Gini coefficient ignores that curves might crossover. As shown in Figure 4, when the ratio is 1, the curves might not be overlapped.

In this paper, we propose a new evaluation metric called Mutual Gini, which can quantitatively measure the difference between the curves based on the Lorenz Curve. As shown in Figure 4, the green curve is the Lorenz Curve of the real label, and the red one belongs to the estimated value, the Mutual Gini is defined as the area $A$ between the green curve and the red one. The smaller the Mutual Gini, the better the model fits the real imbalanced distribution. The calculation of Mutual Gini is as follows:

$\Phi(x)$ and $\Psi(x)$ are the Lorenz Curves of the true label and the estimated value respectively. As we can see, the definition of Mutual Gini helps us to measure the overall accuracy of the LTV distribution, rather than the point-wise loss of LTV prediction. This is very critical as we usually rely on this distribution to make real-world operation decisions.

Here, we choose two well-known methods as baselines, namely (1) Two-Stage (Drachen et al., 2018), (2) ZILN (Wang et al., 2019). The reason why TSUR (Xing et al., 2021) and other approaches are not utilized as baselines is that, they focus on user representation learning, which can be compatible with our framework for better model performance.

• •

  Two-Stage (Drachen et al., 2018) uses a two-step process to predict LTV, it first classifies whether a user was premium or not, followed by predicting the monetary value that the user brings.

• •

  ZILN (Wang et al., 2019) assumes the zero-inflated long-tailed LTV obeys the log-normal distribution, and takes advantages of DNN in feature intersection and information capture to fit the mean and standard deviation of the distribution, and finally the expected value of the log-normal distribution is used as the estimated LTV.

Table 1 presents the results of all the baselines and our model on $ltv_{30}$, $ltv_{90}$, $ltv_{180}$, $ltv_{365}$ prediction. Firstly, Two-Stage method is the worst on every evaluation metric. Secondly, ZILN works better, but its performance on the AMBE is poor, and both ZILN and Two-Stage methods performs poorly on Mutual Gini, which means that they can not fit the imbalanced LTV distribution well, as can be seen from the Gini Coefficient, although we do not think the Gini Coefficient to be a good metric of model performance. Finally, compared to the above two methods, ODMN has a significant improvement in all metrics, which shows the best performance.

In this section, we conduct ablation experiments to verify the impact of different modules in our model. The experiments are carried out in two directions, one is the single time granularity LTV modeling for the purpose of MDME module validation, and the other is the validation of ODMN framework to model multi-time-span LTVs.

To further verify the effectiveness of the model, we conduct A/B tests in the advertising for Kuaishou user growth. Specifically, we take 10$\%$ of the traffic as the control group and 10$\%$ of the traffic as the experimental group, use different models to estimate the $ltv_{30}$ of the users, and deliver ads to users with highest ROI. For the convenience of observing the experimental results, the modeling label in $ltv_{30}$ estimation is set as the value that the user will bring to the platform after 30 days. The delivery strategy afterwards is the same (omitted to company privacy) to the two groups. We accumulate experimental data for 7 days, 14 days, and 30 days, and calculate the ROI (Return on Investment, omitted to company privacy) for both groups. In terms of model, we choose ZILN with the state-of-the-art performance as the baseline, and use our method as the model of the experimental group.

For company privacy, only the ROI improvement value of our method relative to the baseline is provided in Table 4. it is obvious that our method performs better, which further demonstrates the effectiveness of the proposed model.

Our model has been fully deployed in the Kuaishou user growth business. We adopts the method of day-level training and prediction of full volume users. Then, we save the user’s estimated LTV in the cache for online real-time acquisition.