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Question: Inspired by the success of generative adversarial networks (GANs) in image domains, we introduce a novel hierarchical architecture for learning characteristic topological features from a single arbitrary input graph via GANs. The hierarchical architecture consisting of multiple GANs preserves both local and global topological features, and automatically partitions the input graph into representative stages for feature learning. The stages facilitate reconstruction and can be used as indicators of the importance of the associated topological structures. Experiments show that our method produces subgraphs retaining a wide range of topological features, even in early reconstruction stages. This paper contains original research on combining the use of GANs and graph topological analysis. Graphs have great versatility, able to represent complex systems with diverse relationships between objects and data. With the rise of social networking, and the importance of relational properties to the "big data" phenomenon, it has become increasingly important to develop ways to automatically identify key structures present in graph data. Identification of such structures is crucial in understanding how a social network forms, or in making predictions about future network behavior. To this end, a large number of graph analysis methods have been proposed to analyze network topology at the node BID57, community BID41 BID55, and global levels BID64.Each level of analysis is greatly influenced by network topology, and thus far algorithms cannot be adapted to work effectively for arbitrary network structures. Modularity-based community detection BID65 works well for networks with separate clusters, whereas edge-based methods BID37 are suited to dense networks. Similarly, when performing graph sampling, Random Walk (RW) is suitable for sampling paths BID51, whereas Forrest Fire (FF) is useful for sampling clusters BID52. When it comes to graph generation, Watts-Strogatz (WS) graph models BID62 can generate graphs with small world features, whereas Barabsi-Albert (BA) graph models BID31 simulate super hubs and regular nodes according to the scale-free features of the network. However, real-world networks typically have multiple topological features. Considering real-world networks also introduces another issue that traditional graph analysis methods struggle with; having a mere single instance of a graph (e.g. the transaction graph for a particular bank), making it difficult to identify the key topological properties in the first place. In particular, we are interested in both "local topological features" (such as the presence of subgraph structures like triangles) and "global topological features" such as degree distribution. Instead of directly analyzing the entire topology of a graph, GTI first divides the graph into several hierarchical layers. A hierarchical view of a graph can split the graph by local and global topological features, leading to a better understanding of the graph BID63. As different layers have different topological features, GTI uses separate GANs to learn each layer and the associated features. By leveraging GANs renowned feature identification BID42 on each layer, GTI has the ability to automatically capture arbitrary topological features from a single input graph. Figure 1 demonstrates how GTI can learn to reproduce an input graph where a single GAN cannot. In addition to learning topological features from the input graph, the GTI method defines a reconstruction process for reproducing the original graph via a series of reconstruction stages (the number Figure 1 : How GTI recovers the original graph while naive GAN methods do not: The DCGAN output looks like a complete graph, whereas GTI can capture the super-hub structure of node 3 and node 2.of which is automatically learned during training). As stages are ranked in order of their contribution to the full topology of the original graph, early stages can be used as an indicator of the most important topological features. Our focus in this initial work is on the method itself and demonstrating our ability to learn these important features quickly (via demonstrating the retention of identifiable structures and comparisons to graph sampling methods). In this section, we demonstrate the work flow of GTI (see Figure 2), with a particular focus on the GAN, Sum-up and Stage Identification modules. At a high level, the GTI method takes an input graph, learns its hierarchical layers, trains a separate GAN on each layer, and autonomously combines their output to reconstruct stages of the graph. Here we give a brief overview of each module. Hierarchical Identification Module: This module detects the hierarchical structure of the original graph using the Louvain hierarchical community detection method BID33, denoting the number of layers as L. The number of communities in each layer is used as a criterion for how many subgraphs a layer should pass to the next module. Layer Partition Module: Here we partition a given layer into M non-overlapping subgraphs, where M is the number of communities. We do not use the learned communities from the Louvain method as we cannot constrain the size of any community. We instead balance the communities into fixed size subgraphs using the METIS approach BID46.Layer GAN Module: Rather than directly using one GAN to learn the whole graph, we use different GANs to learn features for each layer separately. If we use a single GAN to learn features for the whole graph, some topological features may be diluted or even ignored. This module regenerates subgraphs, with the Layer Regenerate Module generating the adjacency matrix for the corresponding layer. For more detail see Section 2.1.Layer Regenerate Module: Here, for a given layer, the corresponding GAN has learned all the properties of each subgraph, meaning we can use the generator in this GAN to regenerate the topology of the layer by generating M subgraphs of k nodes. Note that this reconstruction only restores edges within each non-overlapping subgraph, and does not include edges between subgraphs. All Layer Sum-up Module: This module outputs a weighted reconstructed graph by summing up all reconstructed layers along with the edges between subgraphs that were not considered in the Regenerate Module. The "weight" of each edge in this module represents its importance to the reconstruction. Indeed, we rely upon these weights to identify the reconstruction stages for the original graph. For more details, see Section 2.2.Stage Identification Module: By analyzing the weighted adjacency matrix of the Sum-up Module, we extract stages for the graph. These stages can be interpreted as steps for a graph reconstruction process. See Section 2.3 for details. Where the generator is a deconvolutional neural network with the purpose of restoring a k × k adjacency matrix from the standard uniform distribution, the discriminator is instead a CNN whose purpose is to estimate if the input adjacency matrix is from a real dataset or from a generator. Here, BN is short for batch normalization which is used instead of max pooling because max pooling selects the maximum value in the feature map and ignores other values, whereas BN will synthesize all available information. LR stands for the leaky ReLU active function (LR = max(x, 0.2 × x)) which we use since 0 has a specific meaning for adjacency matrices. In addition, k represents the size of a subgraph, and F C the length of a fully connected layer. We set the stride for the convolutional/deconvolutional layers to be 2. We adopt the same loss function and optimization strategy (1000 iterations of ADAM BID47) with a learning rate of 0.0002) used in the DCGAN method of BID60. ) to add the graphs from all layers together. re G is the reconstructed adjacency matrix (with input from all layers), G i, i ∈ L is the reconstructed adjacency matrix for each layer (with G representing the full original graph with N nodes), E refers to all the inter-subgraph (community) edges identified by the Louvain method from each hierarchy, and b represents a bias. While each layer of the reconstruction may lose certain edge information, summing up the hierarchical layers along with E will have the ability to reconstruct the entire graph. DISPLAYFORM0 To obtain w and b for each layer, we use Equation 2 as the loss function (where we add = 10 DISPLAYFORM1 to avoid taking log or division by 0), minimizing over 500 iterations of SGD with learning rate 0.1. We note that Equation 2 is similar to a KL divergence, though of course re G and G are not probability distributions. DISPLAYFORM2 re G can be interpreted as representing how each edge contributes to the entire topology. According to these weights we can then divide the network into several stages, with each stage representing a collection of edges greater than a certain weight. We introduce the concept of a "cut-value" to turn re G into a binary adjacency matrix. We observe that many edges in re G share the same weight, which implies these edges share the same importance. Furthermore, the number of unique weights can define different reconstruction stages, with the most important set of edges sharing the highest weight. Each stage will include edges with weights greater than or equal to the corresponding weight of that stage. Hence, we define an ordering of stages by decreasing weight, giving insight on how to reconstruct the original graph in terms of edge importance. We denote the ith largest unique weight-value as CV i (for "cut value") and thereby define the stages as in Equation 3 (an element-wise product), where I[w ≥ CV i] is an indicator function for each weight being equal or larger than the CV i. DISPLAYFORM3 In Section 4, we use synthetic and real networks to show that each stage preserves identifiable topological features of the original graph during the graph reconstruction process. As each stage contains a subset of the original graphs edges, we can interpret each stage as a sub-sampling of the original graph. This allows us to compare with prominent graph sampling methodologies to emphasize our ability to retain important topological features. The importance of deep learning and the growing maturity of graph topology analysis has led to more focus on the ability to use the former for the latter BID54. A number of supervised and semi-supervised learning methods have been developed for graph analysis. A particular focus is on the use of CNNs BID34 BID44 BID39 BID36. These new methods have shown promising for their respective tasks in comparison to traditional graph analysis methods (such as kernel-based methods, graph-based regularization techniques, etc).Since GANs were first introduced BID42, its theory and application has expanded greatly. Many advances in training methods BID38 BID67 BID59 have been proposed in recent years, and this has facilitated their use in a number of applications. For example, GANs have been used for artwork synthesis ), text classification BID56, image-to-image translation BID66, imitation of driver behavior BID49, identification of cancers BID48, and more. The GTI method expands the use of GANs into the graph topology analysis area. A distinguishing feature of our method is that it is an unsupervised learning tool (facilitated by the use of GANs) that leverages the hierarchical structure of a graph. GTI can automatically capture both local and global topological features of a network. To the best of the authors' knowledge, this is the first such unsupervised method. All experiments in this paper were conducted locally on CPU using a Mac Book Pro with an Intel Core i7 2.5GHz processor and 16GB of 1600MHz RAM. Though this limits the size of our experiments in this preliminary work, the extensive GAN literature (see Section 3) and the ability to parallelize GAN training based on hierarchical layers suggests that our method can be efficiently scaled to much larger systems. We use a combination of synthetic and real datasets. Through the use of synthetic datasets with particular topological properties, we are able to demonstrate the retention of these easily identifiable 62, 3.87, 26.64, 27.98, 31.79, 32.42, 34.22, 34.65, 34.80, 64.06, 76.81, 100 P2P 3334 6627 7 49.04, 53.90, 70.32, 87.54, 88.40, 89.65, 100 properties across the reconstruction stages. Of course, in real-world applications we do not know the important topological structures a priori, and so also demonstrate our method on a number of real-world datasets of varying sizes. We use the ER graph model BID40, the BA graph model BID31, the WS graph model BID63, and the Kronecker graph model BID53 to generate our synthetic graphs. The varying sizes of our synthetic graphs (as well as our real-world datasets) are outlined in TAB0. The ER (p = 0.2), WS (k = 2, p = 0.1) and BA (m = 2) graphs were generated using the NetworkX package BID43. The Kronecker graph was generated using the krongen package of the SNAP project 1.For real datasets, we use data available from the Stanford Network Analysis Project BID50. In particular, we use the Facebook network, the wiki-Vote network, and the P2P-Gnutella network. The Facebook dataset consists of "friends lists", collected from survey participants according to the connections between user-accounts on the online social network. It includes node features, circles, and ego networks; all of which has been anonymized by replacing the Facebookinternal ids. Wiki-vote is a voting network (who votes for whom etc) that is used by Wikipedia to elect page administrators; P2P-Gnutella is a peer-to-peer file-sharing network: Nodes represent hosts in the Gnutella network topology, with edges representing connections between the hosts. RoadNet is a connected component of the road network of Pennsylvania. Intersections and endpoints are represented by nodes, and the roads connecting them are edges. Here we use two examples to demonstrate how GTI retains important local topological structure during each reconstruction stage. We demonstrate the reconstruction process of a BA network in Figure 4, with the top row demonstrating the entire reconstruction process of the full network. We clearly observe that each reconstructed network becomes denser and denser as additional stages are added. The bottom row of Figure 4 shows the subgraphs corresponding to nodes 0 to 19 at each reconstruction stage. We observe that these subgraphs retain the most important feature of the original subgraph (the star structures at node 0), even during the first reconstruction stage. Road Network Stages Analysis: We observe in TAB0 that the retained edge percentages of the RoadNet reconstruction decrease more consistently with each stage than in the BA network. This is reasonable, because geographical distance constraints naturally in fewer super hubs, with each node having less variability in its degree. In Figure 5, we observe the reconstruction of the full network, and the node 0 to node 19 subgraph of RoadNet. We observe in the bottom row of Figure 5 that the dominant cycle structure of the original node 0-19 subgraph clearly emerges. We also observe an interesting property of the stages of the original graph in the top row of Figure 5. As SNAP does not provide the latitude and longitude of nodes, we cannot use physical location. We instead calculate the modularity of each stage, where modularity represents the tightness of the community BID58. We found that the modularity deceases from 0.98 to 0.92 approximately linearly. This indicates that GTI views the dense connections between local neighborhoods as a particularly representative topological property of road networks, as such clusters are formed before links between clusters. In the previous section, we demonstrated GTI's ability to the preserve local topological features. Here we focus on degree distribution and the distribution of cluster coefficients, global topological features. Figure 6 and Figure 7 respectively show the log-log degree distributions and log-log cluster coefficient distributions for each of the datasets given in TAB0, where the horizontal axis represents the ordered degrees (ordered cluster coefficient), and the vertical axis the corresponding density. The red line is used to demonstrate the degree distribution (cluster coefficient distribution) of the original graph, with the distributions of the ordered stages represented by a color gradient from green to blue. We observe that with the exception of the ER network, the degree distributions and the cluster coefficient distributions of early stages are similar to the original graphs, and only become more accurate as we progress through the reconstruction stages. Although the degree distributions and cluster coefficient distributions for the early stages of the ER network reconstruction are shifted, we observe that GTI quickly learns the Poisson like shape in degree distribution, and also learns the "peak-like" shape in the cluster coefficient distribution. This is particularly noteworthy given that the ER model has no true underlying structure (as graphs are chosen uniformly at random). Finally, we note that GTI quickly learns that the cluster coefficient of the WS network is zero. The graphs generated by GTI can be considered as samples of the original graph in the sense that they are representative subgraphs of a large input graph. We compare the performance of GTI with that of other widely used graph sampling algorithms (Random Walk, Forest Fire and Random Jump) with respect to the ability to retain topological structures BID51.We demonstrate this through the subgraph structures of the BA and Facebook datasets, comparing stage 1 of GTI against the graph sampling algorithms (designed to terminate with the same number Log of Degrees Log of Density of nodes as the GTI stage). We take out each of subgraphs from BA and Facebook network (nodes 0-19 and nodes 0-49) to visually compare the ability of the first stage of GTI to retain topological features in comparison to the three graph sampling methods. In Figure 8, we observe that stage 1 of GTI has retained a similar amount of structure in the 20 node BA subgraph as Forest Fire BID52, while demonstrating considerably better retention than either Random Walk or Random Jump. However, for 50 node BA subgraph, only GTI has the ability to retain the two super hubs present in the original graph. In Figure 9, we observe that GTI demonstrates vastly superior performance to the other methods when run on the Facebook dataset, which has a number of highly dense clusters with very sparse inter-cluster connections. DISPLAYFORM0 DISPLAYFORM1 Original Graph GTI Random Walk represents the node-node similarity with mismatch penalty BID45 between node i ∈ G L−1 and node j ∈ G. Here, the average node-node similarity indicates how G L−1 resembles G.For comparison, we use the same model parameters as before to generate 100 examples of BA, ER, Kronecker, and WS networks. These newly generated networks serve as base models to help us evaluate the similarity between the penultimate stage of GTI and the original graph, as well as the effectiveness of the sampling methods. Unlike the sampling methods, the of TAB1 demonstrate a consistent ability to retain topological features across a variety of different graph types. This paper leveraged the success of GANs in (unsupervised) image generation to tackle a fundamental challenge in graph topology analysis: a model-agnostic approach for learning graph topological features. By using a GAN for each hierarchical layer of the graph, our method allowed us to reconstruct diverse input graphs very well, as well as preserving both local and global topological features when generating similar (but smaller) graphs. In addition, our method identifies important features through the definition of the reconstruction stages. A clear direction of future research is in extending the model-agnostic approach to allow the input graph to be directed and weighted, and with edge attributes.

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A GAN based method to learn important topological features of an arbitrary input graph.

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We leverage deterministic autoencoders as generative models by proposing mixing functions which combine hidden states from pairs of images. These mixes are made to look realistic through an adversarial framework.


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Question: In Chinese societies, superstition is of paramount importance, and vehicle license plates with desirable numbers can fetch very high prices in auctions. Unlike other valuable items, license plates are not allocated an estimated price before auction. I propose that the task of predicting plate prices can be viewed as a natural language processing (NLP) task, as the value depends on the meaning of each individual character on the plate and its semantics. I construct a deep recurrent neural network (RNN) to predict the prices of vehicle license plates in Hong Kong, based on the characters on a plate. I demonstrate the importance of having a deep network and of retraining. Evaluated on 13 years of historical auction prices, the deep RNN's predictions can explain over 80 percent of price variations, outperforming previous models by a significant margin. I also demonstrate how the model can be extended to become a search engine for plates and to provide estimates of the expected price distribution. Chinese societies place great importance on numerological superstition. Numbers such as 8 (representing prosperity) and 9 (longevity) are often used solely because of the desirable qualities they represent. For example, the Beijing Olympic opening ceremony occurred on 2008/8/8 at 8 p.m., the Bank of China (Hong Kong) opened on 1988/8/8, and the Hong Kong dollar is linked to the U.S. dollar at a rate of around 7.8.License plates represent a very public display of numbers that people can own, and can therefore unsurprisingly fetch an enormous amount of money. Governments have not overlooked this, and plates of value are often auctioned off to generate public revenue. Unlike the auctioning of other valuable items, however, license plates generally do not come with a price estimate, which has been shown to be a significant factor affecting the sale price BID2 BID23. The large number of character combinations and of plates per auction makes it difficult to provide reasonable estimates. This study proposes that the task of predicting a license plate's price based on its characters can be viewed as a natural language processing (NLP) task. Whereas in the West numbers can be desirable (such as 7) or undesirable (such as 13) in their own right for various reasons, in Chinese societies numbers derive their superstitious value from the characters they rhyme with. As the Chinese language is logosyllabic and analytic, combinations of numbers can stand for sound-alike phrases. Combinations of numbers that rhyme with phrases that have positive connotations are thus desirable. For example, "168," which rhythms with "all the way to prosperity" in Chinese, is the URL of a major Chinese business portal (http://www.168.com). Looking at the historical data analyzed in this study, license plates with the number 168 fetched an average price of US$10,094 and as much as $113,462 in one instance. Combinations of numbers that rhyme with phrases possessing negative connotations are equally undesirable. Plates with the number 888 are generally highly sought after, selling for an average of $4,105 in the data, but adding a 5 (rhymes with "no") in front drastically lowers the average to $342.As these examples demonstrate, the value of a certain combination of characters depends on both the meaning of each individual character and the broader semantics. The task at hand is thus closely related to sentiment analysis and machine translation, both of which have advanced significantly in recent years. Using a deep recurrent neural network (RNN), I demonstrate that a good estimate of a license plate's price can be obtained. The predictions from this study's deep RNN were significantly more accurate than previous attempts to model license plate prices, and are able to explain over 80 percent of price variations. There are two immediate applications of the findings in this paper: first, an accurate prediction model facilitates arbitrage, allowing one to detect underpriced plates that can potentially fetch for a higher price in the active second-hand market. Second, the feature vectors extracted from the last recurrent layer of the model can be used to construct a search engine for historical plate prices. Among other uses, the search engine can provide highly-informative justification for the predicted price of any given plate. In a more general sense, this study demonstrates the value of deep networks and NLP in making accurate price predictions, which is of practical importance in many industries and has led to a huge volume of research. As detailed in the following review, studies to date have mostly relied on small, shallow networks. The use of text data is also rare, despite the large amount of business text data available. By demonstrating how a deep network can be trained to predict prices from sequential data, this study provides an approach that may improve prediction accuracy in many industrial applications. License plates have been sold through government auctions in Hong Kong since 1973, and restrictions are placed on the reselling of plates. Between 1997 and 2009, 3,812 plates were auctioned per year, on average. Traditional plates, which were the only type available before September 2006, consist of either a two-letter prefix or no prefix, followed by up to four digits (e.g., AB 1, LZ 3360, or 168). Traditional plates can be divided into the mutually exclusive categories of special plates and ordinary plates. Special plates are defined by a set of legal rules and include the most desirable plates.1 Ordinary plates are issued by the government when a new vehicle is registered. If the vehicle owner does not want the assigned plate, she can return the plate and bid for another in an auction. The owner can also reserve any unassigned plate for auction. Only ordinary plates can be resold. In addition to traditional plates, personalized plates allow vehicle owners to propose the string of characters used. These plates must then be purchased from auctions. The data used in this study do not include this type of plate. Auctions are open to the public and held on weekends twice a month by the Transport Department. The number of plates to be auctioned ranged from 90 per day in the early years to 280 per day in later years, and the list of plates available is announced to the public well in advance. The English oral ascending auction format is used, with payment settled on the spot, either by debit card or check. Most relevant to the current study is the limited literature on the modeling price of license plates, which uses hedonic regressions with a larger number of handcrafted features BID31 BID32 BID24. These highly ad-hoc models rely on handcrafted features, so they adapt poorly to new data, particularly if they include combinations of characters not previously seen. In contrast, the deep RNN considered in this study learns the value of each combination of characters from its auction price, without the involvement of any handcrafted features. The literature on using neural networks to make price predictions is very extensive and covers areas such as stock prices BID3 BID25 BID13 BID8, commodity prices BID19 BID20 BID10, real estate prices BID9 BID11 BID33, electricity prices BID30 BID10, movie revenues BID28 BID34 BID36 BID12, automobile prices BID17 ) and food prices BID15. Most studies focus on numeric data and use small, shallow networks, typically using a single hidden layer of fewer than 20 neurons. The focus of this study is very different: predicting prices from combinations of alphanumeric characters. Due to the complexity of this task, the networks used are much larger (up to 1,024 hidden units per layer) and deeper (up to 9 layers).The approach is closely related to sentiment analysis. A particularly relevant line of research is the use of Twitter feeds to predict stock price movements BID6 BID4 BID26, although the current study has significant differences. A single model is used in this study to generate predictions from character combinations, rather than treating sentiment analysis and price prediction as two distinct tasks, and the actual price level is predicted rather than just the direction of price movement. This end-to-end approach is feasible because the causal relationship between sentiment and price is much stronger for license plates than for stocks. Finally, BID0 utilizes a Long-Short-Term Memory (LSTM) network to study the collective price movements of 10 Japanese stocks. The neural network in that study was solely used as a time-series model, taking in vectorized textual information from two simplier, non-neural-networkbased models. In contrast, this study utilizies a neural network directly on textual information. Deep RNNs have been shown to perform very well in tasks that involve sequential data, such as machine translation BID7 BID35 BID1 and classification based on text description BID14, and are therefore used in this study. Predicting the price of a license plate is relatively simple: the model only needs to predict a single value based on a string of up to six characters. This simplicity makes training feasible on the relatively small volume of license plate auction data used in the study, compared with datasets more commonly used in training deep RNN. The input from each sample is an array of characters (e.g., ["X," "Y," "1," "2," "8"]), padded to the same length with a special character. Each character s t is converted by a lookup table g to a vector representation h t 0, known as character embedding: DISPLAYFORM0 The dimension of the character embedding, n, is a hyperparameter. The values h t 0,1,..., h t 0,n are initialized with random values and learned through training. The embedding is fed into the neural network sequentially, denoted by the time step t. The neural network consists of multiple bidirectional recurrent layers, followed by one or more fully connected layers BID27. The bidirectionality allows the network to access hidden states from both the previous and next time steps, improving its ability to understand each character in context. The network also uses batch normalization, which has been shown to speed up convergence BID22.Each recurrent layer is implemented as follows: BatchNorm transformation, andx is the within-mini-batch-standardized version of x. 2 W, U, γ and β are weights learnt by the network through training. DISPLAYFORM1 The fully connected layers are implemented as DISPLAYFORM2 except for the last layer, which is implemented as DISPLAYFORM3 b l is a bias vector learnt from training. The outputs from all time steps in the final recurrent layer are added together before being fed into the first fully connected layer. To prevent overfitting, dropout is applied after every layer except the last BID16.The model's hyperparameters include the dimension of character embeddings, number of recurrent layers, number of fully connected layers, number of hidden units in each layer, and dropout rate. These parameters must be selected ahead of training. The data used are the Hong Kong license plate auction from January 1997 to July 2010, obtained from the HKSAR government. The data contain 52,926 auction entries, each consisting of i. the characters on the plate, ii. the sale price (or a specific symbol if the plate was unsold), and iii. the auction date. Figure 2 plots the distribution of prices within the data. The figure shows that the prices are highly skewed: while the median sale price is $641, the mean sale price is $2,073. The most expensive plate in the data is "12," which was sold for $910,256 in February 2005. To compensate for this skewness, log prices were used in training and inference. Ordinary plates start at a reserve price of HK$1,000 ($128.2), with $5,000 ($644.4) for special plates. The reserve prices mean that not every plate is sold, and 5.1 percent of the plates in the data were unsold. As these plates did not possess a price, we followed previous studies in dropping them from the dataset, leaving 50,698 entries available for the experiment. The finalized data were divided into three parts, in two different ways: the first way divided the data randomly, while the second divided the data sequentially into non-overlapping parts. The second way creates a more realistic scenario, as it represents what a model in practical deployment would face. It is also a significantly more difficult scenario: because the government releases plates alphabetically through time, plates that start with later alphabets would not be available in sequentiallysplit data. For example, plates that start with "M" were not available before 2005, and plates that 2 Specifically,xi = is a small positive constant that is added to improve numerical stability, set to 0.0001 for all layers.start with "P" would not until 2010. It is therefore very difficult for a model trained on sequentiallysplit data to learn the values of plates starting with later alphabets. In both cases, training was conducted with 64 percent of the data, validation was conducted with 16 percent, and the remaining 20 percent served as the test set. I conducted a grid search to investigate the properties of different combinations of hyperparameters, varying the dimension of character embeddings, the number of recurrent layers, the number of fully connected layers FIG0, the number of hidden units in each layer and the dropout rate (0, .05, .1). A total of 1080 sets of hyperparameters were investigated. The grid search was conducted in three passes: In the first pass, a network was trained for 40 epochs under each set of hyperparameters, repeated 4 times. In the second pass, training was repeated 10 times for each of the 10 best sets of hyperparameters from the first pass, based on median validation RMSE. In the final pass, training was repeated for 30 times under the best set of hyperparameters from the second pass, again based on median validation RMSE. Training duration in the second and the third passes was 120 epochs. During each training session, a network was trained under mean-squared error with different initializations. An Adam optimizer with a learning rate of 0.001 was used throughout BID18. After training was completed, the best state based on the validation error was reloaded for inference. Training was conducted with four of NVIDIA GTX 1080s. To fully use the GPUs, a large mini-batch size of 2,048 was used.3 During the first pass, the median training time on a single GPU ranged from 8 seconds for a 2-layer, 64-hidden-unit network with an embedding dimension of 12, to 1 minute 57 seconds for an 8-layer, 1,024-hidden-unit network with an embedding dimension of 24, and to 7 minutes 50 seconds for a 12-layer 2,048-hidden-unit network with an embedding dimension of 256.Finally, I also trained recreations of models from previous studies as well as a series of fullyconnected networks and character n-gram models for comparison. Given that the maximum length of a plate is six characters, for the n-gram models I focused on n ≤ 4, and in each case calculated a predicted price based on the median and mean of k closest neighbors from the training data, where k = 1, 3, 5, 10, 20. TAB1 reports the summary statistics for the set of parameters out of the 1080 sets specified in section 5.2, based on the median validation RMSE. The model was able to explain more than 80 percent of the variation in prices when the data was randomly split. As a comparison, BID32 and BID24, which represent recreations of the regression models in BID32 and BID24, respectively, were capable of explaining only 70 percent of the variation at most. 4 The importance of having recurrent layers can be seen from the inferior performance of the fullyconnected network (MLP) with the same embedded dimension, number of layers and neurons as the best RNN model. This model was only capable of explaining less than 66 percent of the variation in prices. In the interest of space, I include only two best-performing n-gram models based on median prices of neighbors. Both models were significantly inferior to RNN and hedonic regressions, being able to explain only 40 percent of the variation in prices. For unigram, the best validation performance was achieved when k = 10. For n > 2, models with unlimited features have very poor performance, as they generate a large number of features that rarely appear in the data. Restricting the number of features based on occurances and allowing a range of n within a single model improve performance, but never surpassing the performance of the simple unigram. The performance of using median price and using mean price are very close, with a difference smaller than 0.05 in all cases. All models took a significant performance hit when the data was split sequentially, with the RNN maintaining its performance lead over other models. The impact was particularly severe for the test set, because it was drawn from a time period furthest away from that of the train set. FIG2 plots the relationship between predicted price and actual price from a representative run of the best model, grouped in bins of HK$1,000 ($128.2). The model performed well for a wide range of prices, with bins tightly clustered along the 45-degree line. It consistently underestimated the price of the most expensive plates, however, suggesting that the buyers of these plates had placed on them exceptional value that the model could not capture. Unlike hedonic regressions, which give the same predictions and achieve the same performance in every run, a neural network is susceptible to fluctuations due to convergence to local maxima. These fluctuations can be smoothed out by combining the predictions of multiple runs of the same model, although the number of runs necessary to achieve good is then a practical concern. 6 PERFORMANCE ENHANCEMENTS 6.1 RETRAINING OVER TIME Over time, a model could conceivably become obsolete if, for example, taste or the economic environment changed. In this section, I investigate the effect of periodically retraining the model with the sequentially-split data. Specifically, retraining was conducted throughout the test data yearly, monthly, or never. The best RNN-only model was used, with the sample size kept constant at 25,990 in each retraining, which is roughly five years of data. The process was repeated 30 times as before. FIG4 plots the median RMSE and R 2, evaluated monthly. For the RNN model with no retraining prediction, accuracy dropped rapidly by both measures. RMSE increases an average of 0.017 per month, while R 2 dropped 0.01 per month. Yearly retraining was significantly better, with a 8.6 percent lower RMSE and a 6.9 percent higher R 2. The additional benefit of monthly retraining was, however, much smaller. Compared with the yearly retraining, there was only a 3.3 percent reduction in the RMSE and a 2.6 percent increase in the explanatory power. The differences were statistically significant. Combining several models is known to improve prediction accuracy. This section considers a combination of the preceding neural network, BID32 DISPLAYFORM0 where y rnn is the prediction of the neural network, y woo the prediction of BID32 )'s regression model with only the license-plate-specific features, and x i a series of additional features, including the year and month of the auction, whether it was an afternoon session, the plate's position within the session's ordering, the existence of a prefix, the number of digits, a log of the local market stock index, and a log of the consumer price index. α, δ and ν were estimated by linear regression on the training data. For this ensemble model, the performance between different retraining frequencies was very close, with a less than 1 percent difference in the RMSE and a less than 2 percent difference in R 2 when going from no retraining to monthly retraining. Nevertheless, the differences remained statistically significant, as retraining every month did improve accuracy. The performance of the ensemble model was also considerably more stable than the RNN alone, with only half of the volatility at every retraining frequency. The primary reason behind this difference was the RNN's inability to account for extreme prices. The ensemble model was able to predict these extreme prices because BID32 handcrafted features specifically for these valuable plates. These suggest that while there is a clear benefit in periodical retraining, this benefit diminishes rapidly beyond a certain threshold. Moreover, while deep RNN generally outperforms handcrafted features, the latter could be used to capture outliers. Compared to models such as regression and n-gram it is relatively hard to understand the rationale behind a RNN model's prediction, given the large number of parameters involved and the complexity of the their interaction. If the RNN model is to be deployed in the field, it would need to be able to explain its prediction in order to convince human users to adopt it in practice. One way to do so is to extract a feature vector for each plate by summing up the output of the last recurrent layer over time. This feature vector is of the same size as the number of neurons in the last layer, which can be fed into a standard k-nearest-neighbor model to provide a "rationale" for the model's prediction. To demonstrate this procedure, I use the best RNN model in TAB1 to generate feature vectors for all training samples. These samples are used to setup a k-NN model. When the user submit a query, a price prediction is made with the RNN model, while a number of examples are provided by the k-NN model as rationale. TAB2 illustrate the outcome of this procedure with three examples. The model was asked to predict the price of three plates, ranging from low to high value. The predicted prices are listed in the Prediction section, while the Historical Examples section lists for each query the top four entries returned by the k-NN model. Notice how the procedure focused on the numeric part for the low-value plate and the alphabetical part for the middle-value plate, reflecting the value of having identical digits and identical alphabets respectively. The procedure was also able to inform the user that a plate has been sold before. Finally, the examples provided for the high-value plate show why it is hard to obtain an accurate prediction for such plates, as the historical prices for similar plates are also highly variable. While the RNN model outputs only a single price estimate, auctions that provide estimates typically give both a high and a low estimate. The k-NN model from the previous section can provide reasonably good estimates for common, low-value plates, but works poorly for rare, high-value plates due to the lack of similar plates in record. To tackle this problem, this section uses a Mixture Density Network (MDN) to estimate the distribution of plate prices BID5. The estimated probability distribution of realized price p for a given predicted pricep is DISPLAYFORM0 where [z 1 (p),..., z 24 (p), µ 1 (p),..., µ 24 (p), σ 1 (p),..., σ 24 (p)] is the output vector from a neural network with a single fully-connected layer of 256 neurons with a single inputp. The network was trained with the Adam optimizer for 5000 epochs, using the log likelihood of the distribution, − log (P (p |p)), as the cost function. FIG8 demonstrates the network's ability to fit the distribution of prices. The estimated density resembles the distribution of common, low-value plates, while producing a density that is noticeably wider than the distribution of actual prices for rare, high-value plates. This study demonstrates that a deep recurrent neural network can provide good estimates of license plate prices, with significantly higher accuracy than other models. The deep RNN is capable of learning the prices from the raw characters on the plates, while other models must rely on handcrafted features. With modern hardware, it takes only a few minutes to train the best-performing model described previously, so it is feasible to implement a system in which the model is constantly retrained for accuracy. A natural next step along this line of research is the construction of a model for personalized plates. Personalized plates contain owner-submitted sequences of characters and so may have vastly more complex meanings. Exactly how the model should be designed-for example, whether there should be separate models for different types of plates, or whether pre-training on another text corpus could help-remains to be studied. I would like to thank Travis Ng for providing the license plate data used in this study.

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Predicting auction price of vehicle license plates in Hong Kong with deep recurrent neural network, based on the characters on the plates.

Contrary to previous beliefs, the training performance of deep networks, when measured appropriately, is predictive of test performance, consistent with classical machine learning theory.