1. Introduction
In recent years, mobile payment services (e.g., Alipay Pay ^{1}^{1}1https://en.wikipedia.org/wiki/Alipay operated by Ant Financial, or Apple Pay ^{2}^{2}2https://en.wikipedia.org/wiki/Apple_Pay operated by Apple, etc.) have been playing much more important roles in users’ daily lives. One major commercial goal of these mobile payment operators is to attract more merchants and customers to engage in their mobile payment services, instead of using traditional payment services.
To encourage certain mobile payment activities, the operators of the payment services would need to launch effective marketing campaigns. One way of promoting the payment activities is to offer the merchants with some kinds of incentives, e.g. commissions, coupons, valueadded services and so on, when the merchants can attract customers to pay through the specific payment services.
Taking the above mobile payment marketing as a concrete example at Alipay, each merchant of Alipay is assigned with a unique incentive QR Code^{3}^{3}3Check https://intl.alipay.com/doc/redpacket/ctmz6b for more details. The incentive in this campaign is in the form of operation commissions. The merchant can earn an operation commission from each customer’s redemption of the Alipay Red Packet QR Code (a specific form of incentive QR Code). and is encouraged to ask their customers to scan, so as to share the incentives to the customers. The customers can redeem the shared incentives, similar to any other credit card points, only by making a proper payment through Alipay. At the same time, each successful redemption made by the customer can award an equivalent amount of incentives as commissions to the merchant who shares the incentive. With the incentives propagated and redeemed under such a mechanism (i.e. incentives first be shared from merchants to customers, then be redeemed through a proper payment), the more proper payments customers can accomplish, the more commissions merchants can gain. We illustrate the marketing campaign in Figure 1.
The amount of commissions assigned to merchants are simply rulebased in traditional marketing strategies (Clow, 2004). However, in practice different merchants could response differently in a marketing campaign. For instance, for merchants with limited enthusiasms or abilities, the number of payments they can attract may remain unchanged no matter what amount of commissions they get, while others may show strong positive correlations between the commissions they acquired and the number of payments they can attract. That is, the merchants’ sensitivities to the incentives could vary a lot. On the other hand, the budget of a marketing campaign is always limited. Therefore, in order to maximize the commercial objective of a payment marketing campaign under a limited budget, incentive optimization is the key to success. To summarize, the operators should be able to target the merchants who show strong positive correlations between incentives (e.g. commissions) and objectives (e.g. the number of payments), and spend more budgets on those sensitive merchants.
However, modeling the correlations between incentives and objectives for each merchant is nontrivial. The effect of the treatment^{4}^{4}4We define one treatment in our case as a choice of a certain amount of incentive selected from a set of candidates. For example, award “1 Chinese Yuan for every proper payment the merchant has attracted” is a treatment in case we have a set of candidates, say in Chinese Yuan. on each merchant should be observed in a long term because in practice each merchant could only be aware of the treatment based on an accumulation of incentives in a long enough period. Due to the limited budgets, what we can do is to conduct “treatmentlimited” online experiments by assigning a fixed treatment (say 1 Chinese Yuan) randomly sampled from a set of limited treatments ) to each merchant, and to observe the objectives (say the number of payments) the merchant can attract after then. That is, it is infeasible to collect the whole objectiveincentive curve empirically for any specific merchant (i.e. all the objectives one merchant can achieve under all the treatments
). As such, we summarize the following two challenges. First, we require a representation learning method that can help embed merchants with similar “sensitivities to the incentives” to the similar representations in the vector space, thus can help collect statistical significant data for estimating the objectiveincentive curve for a group of similar merchants. Second, the objectiveincentive curve of each merchant we need to estimate could be arbitrarily complex in welldefined function spaces
(Hofmann et al., 2008; Tibshirani et al., 2014). The variance of the estimation could be large if we do not place any reasonable priors on our model.
In this paper, we present a graph representation learning approach to incentive optimization of merchants in the payment marketing scenario. First, based on the intuition that the customers who interacted with are strong signals to indicate whether a merchant has high potentials to promote the desired payments or not, we analyze the sensitivities of merchants located in different regions in section 3.2.1
. This motivates us to characterize each merchant by considering the distribution of customers who made payments with the merchant, thus resort to a variant of classic graph representation learning approach. Second, we analyze the objectiveincentive curves using random samples from real data, and show that the curve should lie in linear, monotonic function spaces. That is, if we let the number of transactions be the commercial objective, the number of transactions should increase linearly with the incentives. Extensive experiments on real data show that our approach is effective. Finally, we formalize the final decision making process under certain budget (assign the optimal amount of incentive to each merchant) as a linear programming problem, and show our online experimental results at Alipay.
2. Background
In this section, we briefly discuss related literatures to our problems. Basically, we would first talk about related problems such as price optimization, and then review the literatures related to graph representation learning approaches that forms the basis of our approach.
2.1. Price Optimization
Basically, price optimization uses data analysis techniques to address two problems. (1) Understanding how customers will react to different pricing strategies for products and services. (2) Finding the best prices for a given company, considering its commercial goals. Price optimization techniques can help retailers evaluate the potential impact of sales promotions or estimate the right price for each product if they want to sell it in a certain period of time.
Basically, the approaches to price optimization consist of two stages. (1) Utilizing machine learning techniques to estimate sales quantity of products
(Ito and Fujimaki, 2017), sale probabilities from partially observable market data
(Schlosser and Boissier, 2018), or historical lost sales and predict future demand of new products (Ferreira et al., 2015). (2) With those estimates, one can formalize optimization problems for the commercial goals to get the optimal strategies.Similar to price optimization, mobile payment services play the similar roles as retailers, and aim to promote their products, i.e. specific payments in our cases. Most of existing works exploit linear models to estimate the sales (Gallego and Wang, 2014; Ito and Fujimaki, 2017). In our setting we are required to estimate the sensitivities to different treatments for a huge number of merchants with limited samples, which have few related works to our best of knowledge. We contribute a brand new model for estimating the sensitivities of each merchant (product).
2.2. Graph Representation Learning
In this section, we review the literatures related to graph representation learning (Hamilton et al., 2017b), especially graph neural networks approaches that aim to encode the patterns of nodes’ subgraphs as latent features.
Assuming an undirected graph with nodes , edges , the sparse adjacency matrix , a matrix of node features , and graph Laplacian operator (Chung, 1997). The approaches aim to learn to aggregate a subgraph of neighbors associated with each target node. For instance, Hamilton et al. (Hamilton et al., 2017a) proposed GraphSAGE, that defines a set of aggregator functions in the following framework:
(1) 
where denotes the th hidden layer with , is the layerspecific parameters,
denotes the activation function, and
is a predefined aggregator function over the graph, that could be mean, max pooling operators. By stacking the graph aggregation layers
times, each node can aggregate signals from its neighbors to hops.To alleviate the limited representation capacity by doing aggregation only on “predefined” subgraphs (Liu et al., 2018), Veličković et al. (Veličković et al., 2017) proposed attention based mechanisms to parameterize the aggregator functions. Liu et al. (Liu et al., 2018) propose an adaptive path layer to further adaptively filter the receptive field of each node while doing aggragation on the subgraphs. Such methods have proven to achieve stateoftheart results on several public datasets. For users who are interested in the detailed literatures please follow the comprehensive surveys (Wu et al., 2019).
In our scenario, we exploit graph neural networks to embed each merchant by their customers who made payments in the transaction networks. Such encoding can help describe the characteristics of customers each merchant makes transactions with, so as to evaluate the ability a merchant may possess, and to describe the similarity of merchants’ responses to different incentives. Intuitively, merchants that have diverse customers could possibly have much stronger abilities to attract more customers compared with merchants that have limited customers. Two merchants physically located in the same region should share similar customers, thus they may have similar enthusiasms to share the incentives. Such patterns can be well modeled based on the graph learning approaches atop of the transaction network discussed later.
3. Our Methodology
In this section, we first introduce our online experiments to collect samples. After then, we present a variant of graph neural networks for incentive optimization. Finally, we formalize the optimal strategy of treatments as a linear programming problem, based on the estimates of merchants’ personalized objectiveincentive curve (or say sensitivities to the marketing campaign), in a limited budget online setting.
3.1. Online Experiments and Analyses
We aim to target those merchants who are able to boost the future commercial objectives (e.g. the number of transactions using Alipay in our case) as the operators stimulate greater incentives. That is, we need to discern merchants with different sensitivities to the incentives. To obtain the sensitivities of each merchant to the marketing campaign, we need to estimate the so called objectiveincentive curve of each merchant. The curve describes the mapping between incentives under different treatments and the commercial objectives that a merchant can achieve after receiving the treatment. Accurate estimations of the curves can help operators to make decisions by choosing the best strategies.
We choose to launch online experiments by randomly choosing a certain treatment for each merchant in the online environment at Alipay, and observe the commercial objectives each merchant can achieve. The online experiment involves millions of merchants and lasts for several days. The possible treatment is sampled from a set of incentives in Chinese Yuan. Hence, we can assume that the merchants who show similar sensitivities should be uniformly sampled over all the treatments.
However, we can only observe one merchant’s future objective under one treatment, while we are aiming to estimate the whole objectiveincentive curve of each merchant. That means we need to represent merchants with similar capacities and enthusiasms similarly, and have to place appropriate priors to constrain the functions estimating the curve.
First, the abilities of propagating incentives and attracting proper payments can be naturally described through the distribution of customers with which the merchants are interacting. For example, merchants that have transactions with diverse customers are more competitive compared with merchants who have limited customers. Merchants make transactions with the same group of customers could possibly be located together, hence their enthusiasms of marketing the payment services may be similar. To illustrate the relations between locations and sensitivities of merchants, we group the merchants according to regions, and calculate the “sensitivities of regions” denoted by , where means the number of payments averaged over all merchants from region with commission . We plot the “sensitivities of regions” in figure 2. The figure shows that adjacent regions tend to share similar sensitivities. This led us to a representation learning approach based on graph neural networks atop of transaction networks.
Second, we show the expected objectiveincentive curve using observations from our online experiments in Figure 4
. The figure shows that the expected mapping lies in a linear, monotonic function space. In the next section, we will model such mapping using a linear transformation given the representation of each merchant, where the derived “gradient” will be naturally used to characterize the sensitivity of each merchant.
3.2. Models
In this section, we present our models for incentive optimization based on the samples observed from our online experiments. Estimating the objectiveincentive curve for each merchant is nontrivial because this requires a huge number of samples. We propose two methods to alleviate the potential high variance in estimation. First, we resort to graph neural network based methods, and hopefully can embed merchants with similar characteristics to similar embeddings in the vector spaces, thus results into statistical significant amount of observations with various treatment for a group of similar merchants. Second, we propose a monotonic linear transformation to map the graph embeddings learned for each merchant to the final objectiveincentive curve. Our endtoend model jointly optimizes the graph neural networks and the monotonic linear transformation.
3.2.1. Merchant Embedding based on Graph Neural Networks
In this part, we introduce how to embed merchants based on transaction graphs.
As we discussed above in section 3.1, the merchants’ distribution of customers could be represented from different dimensions, e.g. the location, the diversity of customers, and so on. Therefore, we build a graph based on payments to connect merchants and customers.
Dataset  # Node feature dim  # Edge feature dim  # Labeled merchants  

Dataset 1  
Dataset 2 
Assuming an undirected graph with nodes , edges . Each node could be a merchant, a customer, or plays both roles. Node has a neighbor (i.e. there exists an edge ) if there exists a proper payment where j acts as customer and i as merchant in the past several days. We assume the equivalent adjacent matrix of graph , where means there is an edge existed in graph . Assuming as the nodes’ features in dimensions. Assuming denotes the dimensional features, describing the payment related informations through the edges. We have the following graph neural network layers to iterate times (thus propagate neighbors’ signals in hops), and have the output encoding for node as :
(2) 
where denotes node ’s itermediate embedding at the th hidden layer, denotes the activation functions, and is the aggregator function parameterized by as defined in (Liu et al., 2018); , , , and are the parameters. Note that for each merchant our graph neural networks not only capture the distribution over neighbors (i.e. distribution of merchants’ customers), but also the distribution over edges (i.e. the distribution of payments between the merchant and its customers).
3.2.2. Monotonic Linear Mapping
Given the embedding of each merchant based on graph neural networks convolved on the transaction networks, we are able to model the objectiveincentive curves. Based on the analyses in section 3.1, we have the following mapping:
(3) 
where denotes the estimation of the future objective of merchant under treatment , and and are parameters to transform the merchant embeddings to gradient (slope) and intercept. The activation function is used to guarantee the monotonic linear mapping between treatment and objective .
Note that the inferred gradient of each merchant in our model can be used to measure the sensitivity to incentive of merchant . A small value of indicates that is less sensitive to the marketing, while a greater value of indicates the opposite.
To summarize, we have the following objective to optimize:
(4) 
where denotes the mean absolute error, and denotes the observed commercial objective of merchant under treatment . We aim to optimize parameters in an endtoend model. We illustrate the architecture of our model in Figure 3. In practice we use the ADAM (Kingma and Ba, 2014)
to optimize the above objective in a stochastic minibatch manner, and the model is implemented with tensorflow
(Abadi et al., 2016).3.3. Optimal Incentive as Linear Programming
As other price optimization problems, in this section, we show how we formalize incentive optimization as a linear programming problem given the estimates of .
In the online environment, we need to choose the best treatment for every merchant. Since the estimates of each merchant’s “gradient” is positive due to the activation function, the best strategy is always choosing the maximum treatment for all the merchants, so that we can maximize the commercial objective. However, under a limited budget, the strategy should consider the value of inferred gradient. Let us denote the best strategy for merchant is treatment , then we have and for the rest strategies. We formalize the following optimization problem:
(5)  
(6) 
then we have the optimal solution for each , where denotes the dual optimal.
We test our strategies in the online environment at Alipay, and show the online results in section 4.4.
4. Experimental Results
In this section, we conduct extensive empirical experiments to study the performance of our models. We first analyze the offline results based on the dataset collected in our online experiments. Next, we show the online results after deploying our approach at Alipay compared with a multilayer perceptron neural network model, using the same optimal strategy as described in section
3.3.Note that due to the policy of information sensitivity at Alipay, we will not reveal the detailed numbers that could be sensitive.
4.1. Experimental Settings
In this section, we will introduce our dataset for learning the model, and related experimental settings.
4.1.1. Dataset
Our two experimental datasets are collected from our online experiments (mentioned in section 3.1) launched at Alipay in two seperated time periods (15 consecutive days of each), respectively. To collect each of the experimental datasets, of random selected merchants were assigned to an experimental bucket that places a fixed treatment. We set up a total of experimental buckets for observing merchants with different treatments. That is, each bucket corresponds to a distinct treatment in our online experiments, and is responsible for providing the merchants under control a fixed amount of commissions. Thus, we can assume that the merchants with different sensitivities to the marketing campaign are randomly sampled in different buckets with different treatments.
There are more than 2 millions of labeled merchants with observed commercial objects (labels or measures include: the number of payments the merchant can accomplish in the next 3 days, the number of days the merchant have payments in next 3 days) under given treatments. We aim to regress over the commercial objectives of each merchant. We build our model on top of the transaction network. The edges of the transaction network consist of the transactions associated with those labeled merchants in 2hops. That is, the transactions made between the labeled merchants and customers directly, and all the transactions made by those customers. This leads to more than 90 millions of nodes (including merchants and customers) and hundred millions of edges (transactions). We summarize two datasets used for offline experiments in table 1. We collect two different periods of data from online experiments for evaluating the robustness of our results.
4.1.2. Comparison Methods
To verify the effectiveness of our proposed method, we compare our approach with classic regression approaches. It is well known that tree based models (Chen and Guestrin, 2016), linear models, and neural networks (Schmidhuber, 2015)
are widely used for regression problems. Because the majority of features we built are sparse features, and tree based models cannot exploit sparse features well, we choose linear regression and DNN model (utilizing a multilayer perceptron architecture) with the monotonic linear mapping as the output layer introduced in section
3.2.2. We name our proposed model as GE model in the following experiments.For DNN, we set the depth of the neural architecture as 2 with latent embedding size as 256. We set the depth of our graph neural network architecture as 2 ( so that each labeled merchant can aggregate signals from 1hop neighbors, i.e. their directed customers, and 2hop neighbors, i.e. those merchants who shared the same group of customers) with latent embedding size as 256 as well. The rest hyperparameters (such as learning rate and regularizers) of the comparison methods are tunned by grid search.
We randomly sample 80% of the merchants for training and the remaining for testing respectively from each dataset.
4.2. Regression Results
We conduct experiments on the realworld dataset collected from the online experiments run by Alipay as described above. To evaluate the performance of the models, we choose MAE (Mean Absolute Error) and MSE (Mean Square Error) as our metrics, which are commonly used for evaluating regression methods. Even though such metrics cannot directly measure how well our approach can fit the sensitivities of each merchant, they still reflect how well the models fit our observed measures of each merchant under specific treatments.
Dataset 1  MAE  MSE 

LR  0.1432  0.0415 
DNN  0.1404  0.0433 
GE  0.1357  0.0400 
Dataset 2  MAE  MSE 
LR  0.1441  0.0417 
DNN  0.1409  0.0441 
GE  0.1361  0.0408 
As shown in Table 2, DNN model outperforms Linear Regression in terms of MAE, however is doing worse on MSE. This shows that DNN model is more robust due to the final monotonic linear mapping layer (i.e. successfully reduce the risk of high variance), but may mislead the characteristics of different merchants due to the limited representation learning capacity (i.e. cannot well captured merchants with subtle differences). Our GE model produces much better estimates compared with all the other comparison approaches in terms of both MAE and MSE, which implies that graph representation indeed help the representations of merchants and simultaneously depict the merchant’s reactions to treatments.
4.3. ObjectiveIncentive Curves Analyses
In this section, we analyze the estimates of so called “gradient” defined in Eq. (3). “Gradient” can be used to depict how the merchant is sensitive to the marketing campaign, and describe the objectiveincentive curve. An accurate estimation of “gradient” of each merchant is exactly our goal.
Different from the quantitative results reported in section 4.2, the estimated “gradient” of each merchant is hard to evalute directly. However, a reasonable assumption is that if a model is well estimated, the group of merchants with larger estimated “gradient” should be able to achieve much better results on expectation in terms of commercial objectives, compared with those merchants with smaller estimated “gradient”, while all the merchants are under the same treatment of strong incentives.
We denote as merchants’ commercial objective under a treatment with a larger amount of incentives, and as that under a treatment with a lower amount of incentives. We denote the uplift gain as . The uplift gain for the merchants who are sensitive to incentives (i.e. with greater “gradient”) should be comparatively greater than that for less sensitive merchants (i.e. with smaller “gradient”).
To conduct the experiments, we first infer the “gradient” for each merchant in the test data. By sorting the merchants by “gradient” in decending order, we can separate the merchants into two groups, the incentive sensitive group and incentive insensitive group, denoted as and respectively. To evaluate an incentive optimization model, we need to inspect if the and are separated properly.
For the incentive sensitive group of merchants , we calculate the corresponding uplift gain as , where and denote the commercial goals of merchants under treatment of high and low incentives respectively. Similarly we have as the uplift gain obtained from insensitve group of merchants. The greater the difference of the uplift gains is, the better the model should be. We illustrate the values of and inferred from both DNN and GE models in Fig. 5. We show the value of using brown bars, and the value of with blue bars. The gap between two bars of GE model is significantly greater than that of dnn model, which implies that the GE model has learned better merchants’ sensitivity to incentives.
Note that the linear regression model cannot produce personalized “gradient”, and we will not report the results accordingly.
Level of Incentive Sensitivity (desc)  DNN  GE 

20%  5.851  5.935 
20% 40%  2.657  4.749 
40% 60%  3.619  3.912 
60% 80%  3.954  3.782 
80%  2.694  2.157 
For DNN and GE model, we sort the merchants in test data by the inferred “gradient” in decending order, and separate the merchants into five groups equally. We show the uplift gain of each group in Table 3. For the most sensitive merchants, GE model produces a better uplift gain compared with the DNN model. For the least sensitive 20% merchants, uplift gain produced by GE model is relatively smaller compared with the DNN model, which implies that GE model inferred a relative flat objectiveincentive curve for those insensitive merchants.
Models  Cost (%)  Objective1(%)  Objective2(%)  

Baseline        
GE 



4.4. Online Results
We deployed our model in the mobile payment marketing scenario at Alipay with a standard A/B testing configuration. We conduct our A/B testing experiments beginning with a relatively small traffic, i.e. influencing 1% of merchants, and observe the measures in consecutive 5 days. After that, we gradually increase the traffic from 1%, to 2.5%, 5%, 15%, and observe the measure in 5 days respectively. Finally, we conduct the A/B experiments with traffic 30% from January 10th to January 14th, and make the final decision on deploying the model at Alipay. The online results lasting nearly one month consistently show that our GE model outperforms the comparison model while we were increasing the traffic.
There exists millions of users who visit our App during the period while we were conducting A/B experiments with 30% traffic. In addition to the cost of the marketing campaign, two commercial objectives (Objective 1: the number of payments averaged over merchants, Objective 2: the number of days having at least one payments averaged over merchants) are shown as metrics in our experiments. Table 4
shows the relative improvements of GE model compared to the DNN model using 30% of traffic. Along with the relative improvements, we also show the confidence intervals with 95% confidence level. Compared with the DNN model, our proposed approach significantly saved 2.71% cost while improving the two commercial objectives statistical significantly (with pvalue far less than 0.05).
Figure 6 shows the trends of relative improvements on the two commercial goals by applying the strategies optimized using our approach. The figure tells that our approach has been gradually improving the overall commercial goals given our optimal strategies. This is because each merchant is aware of the treatment after they have observed an accumulation of operation commissions.
5. Conclusions
In this paper, we show our best experiences in the mobile payment marketing problem at Alipay. To our best knowledge, this is the first graph neural network based marketing experience shared by industry and has been deployed on the realworld large scale marketing system. To promote the mobile payment activities, we propose to identify merchants that are sensitive to the marketing campaigns based on a graph representation learning approach over transaction networks. We further place a monotonic linear mapping function to reduce the potential high variance due to limited samples of treatments. We develop uplift gains as a novel metric to measure the goodness of our model, that saves the cost of deploying and observing noneffective models online. Our online results in a standard A/B testing configuration lasting nearly a month shows that our approach is effective.
References
 (1)
 Abadi et al. (2016) Martín Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, and Michael Isard. 2016. TensorFlow: a system for largescale machine learning. (2016).
 Chen and Guestrin (2016) Tianqi Chen and Carlos Guestrin. 2016. Xgboost: A scalable tree boosting system. In Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining. ACM, 785–794.
 Chung (1997) Fan RK Chung. 1997. Spectral graph theory. Number 92. American Mathematical Soc.
 Clow (2004) Kenneth E Clow. 2004. Integrated advertising, promotion, and marketing communications. Pearson Education India.
 Ferreira et al. (2015) Kris Johnson Ferreira, Bin Hong Alex Lee, and David SimchiLevi. 2015. Analytics for an online retailer: Demand forecasting and price optimization. Manufacturing & Service Operations Management 18, 1 (2015), 69–88.

Gallego and Wang (2014)
Guillermo Gallego and
Ruxian Wang. 2014.
Multiproduct price optimization and competition under the nested logit model with productdifferentiated price sensitivities.
Operations Research 62, 2 (2014), 450–461.  Hamilton et al. (2017a) William L Hamilton, Rex Ying, and Jure Leskovec. 2017a. Inductive Representation Learning on Large Graphs. arXiv preprint arXiv:1706.02216 (2017).
 Hamilton et al. (2017b) William L Hamilton, Rex Ying, and Jure Leskovec. 2017b. Representation Learning on Graphs: Methods and Applications. arXiv preprint arXiv:1709.05584 (2017).
 Hofmann et al. (2008) Thomas Hofmann, Bernhard Schölkopf, and Alexander J Smola. 2008. Kernel methods in machine learning. The annals of statistics (2008), 1171–1220.
 Ito and Fujimaki (2017) Shinji Ito and Ryohei Fujimaki. 2017. Optimization beyond prediction: Prescriptive price optimization. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, 1833–1841.
 Kingma and Ba (2014) Diederik Kingma and Jimmy Ba. 2014. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014).
 Liu et al. (2018) Ziqi Liu, Chaochao Chen, Longfei Li, Jun Zhou, Xiaolong Li, Le Song, and Yuan Qi. 2018. Geniepath: Graph neural networks with adaptive receptive paths. arXiv preprint arXiv:1802.00910 (2018).
 Schlosser and Boissier (2018) Rainer Schlosser and Martin Boissier. 2018. Dynamic pricing under competition on online marketplaces: A datadriven approach. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. ACM, 705–714.
 Schmidhuber (2015) Jürgen Schmidhuber. 2015. Deep learning in neural networks: An overview. Neural networks 61 (2015), 85–117.
 Tibshirani et al. (2014) Ryan J Tibshirani et al. 2014. Adaptive piecewise polynomial estimation via trend filtering. The Annals of Statistics 42, 1 (2014), 285–323.
 Veličković et al. (2017) Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. 2017. Graph Attention Networks. arXiv preprint arXiv:1710.10903 (2017).
 Wu et al. (2019) Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S Yu. 2019. A comprehensive survey on graph neural networks. arXiv preprint arXiv:1901.00596 (2019).
Comments
There are no comments yet.