Learning Graphs Representations Using Recurrent Graph Convolution Networks For Forecasting Ethereum Prices
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The document discusses the extension of graph convolution networks into recurrent architectures for forecasting Ethereum prices. It details various experiments, methodologies, and performance metrics, highlighting the superiority of recurrent graph neural networks and deep learning models in the context of price prediction. Additionally, various techniques and algorithms related to graph convolution, recurrent neural networks, and deep reinforcement learning are explored in the context of financial trading and model optimization.
![Reinforcement Learning
Observation
Action
Value – Maps state, action pair to expected future reward
𝑸 𝒔, 𝒂 ≈ 𝔼 𝑹 𝒕+𝟏 + 𝑹 𝒕+𝟐 + 𝑹 𝒕+𝟑 + … 𝑺𝒕 = 𝒔, 𝑨 𝒕 = 𝒂]
Optimal Value – Bellman Equation 1957
𝑸∗
𝒔, 𝒂 ≈ 𝔼 𝑹 𝒕+𝟏 𝑸∗
(𝑺𝒕+𝟏, 𝒃) 𝑺 𝒕 = 𝒔, 𝑨 𝒕 = 𝒂]
TD Algorithm – Watkins 1989
𝑸 𝒕+𝟏 𝑺𝒕, 𝑨 𝒕 = 𝑸 𝒕 𝑺 𝒕, 𝑨 𝒕 + 𝜶(𝑹 𝒕+𝟏 + γmax
𝑎
𝑸 𝒕 𝑺𝒕+𝟏, 𝑨 𝒕 − 𝑸 𝒕 𝑺𝒕, 𝑨 𝒕 ]
RECURRENT GRAPH NEURAL NETWORKS](https://image.slidesharecdn.com/rgcn-180727132411/85/Learning-Graphs-Representations-Using-Recurrent-Graph-Convolution-Networks-For-Forecasting-Ethereum-Prices-22-320.jpg)
![def relational_graph_convolution(self, inputs):
features = inputs[0]
A = inputs[1:] # list of basis functions
# convolve
supports = list()
for i in range(support):
supports.append(K.dot(A[i], features))
supports = K.concatenate(supports, axis=1)
output = K.dot(supports, self.W)
𝒉 𝟎
(𝒍+𝟏)
= 𝝈(𝒉 𝟎
(𝒍)
𝒘 𝟎
(𝒍)
𝒊∈𝝒
𝟏
𝒄𝒊,𝒍
𝒘𝒋
(𝒍)
𝒉𝒋
(𝒍)
)
RECURRENT GRAPH NEURAL NETWORKS](https://image.slidesharecdn.com/rgcn-180727132411/85/Learning-Graphs-Representations-Using-Recurrent-Graph-Convolution-Networks-For-Forecasting-Ethereum-Prices-44-320.jpg)
Learning Graphs Representations Using Recurrent Graph Convolution Networks For Forecasting Ethereum Prices
- 1. Recurrent Graph Convolution Networks for Forecasting Ethereum prices ICCS 2018 In collaboration with
- 2. tl;dr: We extended Graph Convolution Networks to be Recurrent over time.
- 3. What is Ethereum - A 100% open source platform to build and distribute decentralized applications - No middle men - Social sites, Financial systems, Voting mechanisms, Games, Reputation Systems - 100% peer to peer, censorship proof - Also a Tradable Asset. RECURRENT GRAPH NEURAL NETWORKS
- 5. EXPERIMENT SETTING Time 0 900 1800 2700 3600 4500 5400 6300 7200 8100 9000 9900 10800 11700 12600 13500 14400 15300 16200 17100 18000 18900 19800 … … … … … … … … Batch - 1 Batch - 2 Batch - 3 Batch - 4 Batch - 5 Batch - 6 Batch - 7 Batch - 8 Batch - 9 … … … Optimization Window Unseen Vector – 60 Min lagged prices Ground Truth – ETH Future 5 Min Prices Batch – Training: 240 Vectors Test: 90 Vectors In Sample Training set: 28.02.18 - 13.05.18 Test set: 13.05.18 – 29.05.18 Out of sample 5 (min)60 (min) Vector Structure Optimization Window Unseen Optimization Window Unseen Optimization Window Unseen Optimization Window Unseen Optimization Window Unseen Optimization Window Unseen Optimization Window Unseen Optimization Window Unseen RECURRENT GRAPH NEURAL NETWORKS
- 6. DEEP LEARNING SUPERIORITY 96.92% Deep Learning 94.9% Human ref: http://www.image-net.org/challenges/LSVRC/ RECURRENT GRAPH NEURAL NETWORKS
- 7. GRADIENT DESCENT 𝐸 = Error of the network 𝑤𝑡 = 𝑤𝑡−1 − 𝛾 𝜕𝐸 𝜕𝑤 𝑊 = Weight matrix representing the filters RECURRENT GRAPH NEURAL NETWORKS
- 8. BackPropagation Legend 𝑥0 𝑓0(𝑥0, 𝑤0) 𝑓1(𝑥1, 𝑤1) 𝑓2(𝑥2, 𝑤2) 𝑓𝑛 𝑥 𝑛, 𝑤 𝑛 = ො𝑦 𝑓𝑛−1(𝑥 𝑛−1, 𝑤 𝑛−1) 𝑓𝑛−2(𝑥 𝑛−2, 𝑤 𝑛−2) 𝑤0 𝑤1 𝑤 𝑛 𝑤 𝑛−1 𝐸 = 𝑙 ො𝑦, 𝑦𝑦 𝑙 ො𝑦, 𝑦 - Loss Function 𝑥0 - Features Vector 𝑥𝑖 - Output of 𝑖 layer 𝑤𝑖 - Weights of 𝑖 layer 𝑦 – Ground Truth ො𝑦 – Model Output 𝐸 – Loss Surface 𝜕𝐸 𝜕𝑥 𝑛 = 𝜕𝑙 ො𝑦, 𝑦 𝜕𝑥 𝑛 𝜕𝐸 𝜕𝑤 𝑛 = 𝜕𝐸 𝜕𝑥 𝑛 𝜕𝑓𝑛 𝑥 𝑛−1, 𝑤 𝑛 𝜕𝑤 𝑛 𝜕𝐸 𝜕𝑥 𝑛−1 = 𝜕𝐸 𝜕𝑥 𝑛 𝜕𝑓𝑛 𝑥 𝑛−1, 𝑤 𝑛 𝑥 𝑛−1 𝑓– Activation Function 𝜕𝐸 𝜕𝑥 𝑛−2 = 𝜕𝐸 𝜕𝑥 𝑛−1 𝜕𝑓𝑛−1 𝑥 𝑛−2, 𝑤 𝑛−1 𝑥 𝑛−2 𝜕𝐸 𝜕𝑤 𝑛−1 = 𝜕𝐸 𝜕𝑥 𝑛−1 𝜕𝑓𝑛 𝑥 𝑛−2, 𝑤 𝑛−1 𝜕𝑤 𝑛−1 … … 𝐹𝑜𝑟𝑤𝑎𝑟𝑑𝑃𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑖𝑜𝑛 𝐵𝑎𝑐𝑘𝑃𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑖𝑜𝑛 1: Forward Propagation 2: Loss Calculation 3: Optimization RECURRENT GRAPH NEURAL NETWORKS
- 9. CONVOLUTION ඵ −∞−∞ ∞∞ 𝑓 𝜏1, 𝜏2 ∙ 𝑔 𝑥 − 𝜏1, 𝑦 − 𝜏2 𝑑𝜏1 𝑑𝜏2 𝑓 𝑥, 𝑦 𝑔 𝑥, 𝑦 𝑓 ∗ 𝑔 RECURRENT GRAPH NEURAL NETWORKS
- 10. ConvNet 𝑥Input ො𝑦 Class Convolution & Maxpooling Convolution & Maxpooling Convolution & Maxpooling Fully Connected RECURRENT GRAPH NEURAL NETWORKS
- 11. F1RMSE PnL(%) Results: 1D-ConvNet 0.9 0.58 -17.317 Results for out of sample simulated trading Simple root mean square error F1-beta score (Harmonic mean of precision and recall) taken as classification decision where the predicted price is greater then the current price +15% transaction fee. Profits and losses (percentage) for out of sample trading. Assuming 15% transaction fee. RECURRENT GRAPH NEURAL NETWORKS
- 12. Recurrent Neural Network -Memory Achieved through feedback -Due to self multiplications, Feedback Weight matrix tend to explode or vanish. -Solution: logistic gating mechanism Keep Gate 1.73 Write Gate Read Gate Input from rest of RNN Output to rest of RNN Input Command Output Cell Gate RECURRENT GRAPH NEURAL NETWORKS
- 13. Recurrent Neural Network 𝒔 𝒕+𝟏𝒔 𝒕 𝒔 𝒕+𝟐…….. ……..Backpropagation Through Time Long Short Term Memory ቁ𝑓𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔 𝑓 𝑦𝑗 𝑙 𝜍 𝑡−1 + 𝜓 𝑓ℎ 𝑡−1 + 𝑏𝑓 ቁ𝜄𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔𝜄 𝑦𝑗 𝑙 𝜍 𝑡−1 + 𝜓𝜄ℎ 𝑡−1 + 𝑏𝜄 ቁ𝑜𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔 𝑜 𝑦𝑗 𝑙 𝜍 𝑡−1 + 𝜓 𝑜ℎ 𝑡−1 + 𝑏 𝑜 ቁ𝜍𝑖 𝑙+1 𝑡 = 𝑡𝑎𝑛ℎ( 𝜔𝜍 𝑦𝑗 𝑙 + 𝜓𝜍ℎ 𝑡−1 + 𝑏𝜍 Forget Gate Output Gate Input Gate Cell RECURRENT GRAPH NEURAL NETWORKS
- 14. F1RMSE PnL(%) Results: LSTM 0.1 0.42 -7.115 Results for out of sample simulated trading Simple root mean square error F1-beta score (Harmonic mean of precision and recall) taken as classification decision where the predicted price is greater then the current price +15% transaction fee. Profits and losses (percentage) for out of sample trading. Assuming 15% transaction fee. RECURRENT GRAPH NEURAL NETWORKS
- 15. INPUT BIDIRECTIONAL GRU RESIDUAL DIALATED CONV1D 𝒈 𝒕+𝟏𝒈 𝒕 𝒈 𝒕+𝟐 𝒈 𝒕+𝟏𝒈 𝒕+𝟐 𝒈 𝒕 TRANSPOSE AXIS=1 tanh softmax tanh softmax tanh softmax HARD ATTENTION
- 16. F1RMSE PnL(%) Results: CNN-LSTM 0.05 0.53 -7.461 Results for out of sample simulated trading Simple root mean square error F1-beta score (Harmonic mean of precision and recall) taken as classification decision where the predicted price is greater then the current price +15% transaction fee. Profits and losses (percentage) for out of sample trading. Assuming 15% transaction fee. RECURRENT GRAPH NEURAL NETWORKS
- 18. DEEP LEARNING COMMON STRUCTURES SUPERVISED UNSUPERVISED Perceptron It is a type of linear classifier, a classification algorithm that makes its predictions based on a linear predictor function combining a set of weights with the feature vector. The algorithm allows for online learning, in that it processes elements in the training set one at a time. RECURRENTFEED FORWARD Feed Forward Network sometimes Referred to as MLP, is a fully connected dense model used as a simple classifier. Convolutional Network assume that highly correlated features located close to each other in the input matrix and can be pooled and treated as one in the next layer. Known for superior Image classification capabilities. Simple Recurrent Neural Network is a class of artificial neural network where connections between units form a directed cycle. Hopfield Recurrent Neural Network It is a RNN in which all connections are symmetric. it requires stationary inputs. Long Short Term Memory Network contains gates that determine if the input is significant enough to remember, when it should continue to remember or forget the value, and when it should output Auto Encoder aims to learn a representation (encoding) for a set of data, typically for the purpose of dimensionality reduction. Restricted Boltzmann Machine can learn a probability distribution over its set of inputs.. Deep Belief Net is a composition of simple, unsupervised networks such as restricted Boltzmann machines ,where each sub-network's hidden layer serves as the visible layer for the next. RECURRENT GRAPH NEURAL NETWORKS
- 19. Search Problems = = = RECURRENT GRAPH NEURAL NETWORKS
- 20. Markov Decision Process Action: 𝒂 𝒕 Action: 𝒂 𝒕+𝟏 Reward : 𝒓 𝒕 Reward : 𝒓 𝒕+𝟏 Reward : 𝒓 𝒕+𝟐 𝒔 𝒕+𝟏𝒔 𝒕 𝒔 𝒕+𝟐…….. …….. 𝑆 ≔ {𝑠1, 𝑠2, 𝑠3, … 𝑠 𝑛} 𝐴 ≔ {𝑎1, 𝑎2, 𝑎3, … 𝑎 𝑛} 𝑇(𝑠, 𝑎, 𝑠𝑡+1) 𝑅(𝑠, 𝑎) Set of states Set of Actions Reward Function Transition Function RECURRENT GRAPH NEURAL NETWORKS
- 21. Policy Search 𝒔 𝒕 𝝅 𝑸(𝒔, 𝒂) 𝑸(𝒔, 𝒂) 𝑸(𝒔, 𝒂) 𝑸(𝒔, 𝒂) Policy Expected Reward 𝝅: 𝒔 → 𝒂 The goal will be to Maximize the reward RECURRENT GRAPH NEURAL NETWORKS
- 22. Reinforcement Learning Observation Action Value – Maps state, action pair to expected future reward 𝑸 𝒔, 𝒂 ≈ 𝔼 𝑹 𝒕+𝟏 + 𝑹 𝒕+𝟐 + 𝑹 𝒕+𝟑 + … 𝑺𝒕 = 𝒔, 𝑨 𝒕 = 𝒂] Optimal Value – Bellman Equation 1957 𝑸∗ 𝒔, 𝒂 ≈ 𝔼 𝑹 𝒕+𝟏 𝑸∗ (𝑺𝒕+𝟏, 𝒃) 𝑺 𝒕 = 𝒔, 𝑨 𝒕 = 𝒂] TD Algorithm – Watkins 1989 𝑸 𝒕+𝟏 𝑺𝒕, 𝑨 𝒕 = 𝑸 𝒕 𝑺 𝒕, 𝑨 𝒕 + 𝜶(𝑹 𝒕+𝟏 + γmax 𝑎 𝑸 𝒕 𝑺𝒕+𝟏, 𝑨 𝒕 − 𝑸 𝒕 𝑺𝒕, 𝑨 𝒕 ] RECURRENT GRAPH NEURAL NETWORKS
- 23. Gets “Rewards” and Penalties based on it’s success of producing a better generation of models. Father Model Being built, compiled, evaluated and stored for future reconstruction and retraining by a Human. Child Model Deep Reinforcement Learning Deep Meta Learning RECURRENT GRAPH NEURAL NETWORKS
- 25. F1RMSE PnL(%) Results: Deep Meta Learning 0.027 0.68 -3.2 Results for out of sample simulated trading Simple root mean square error F1-beta score (Harmonic mean of precision and recall) taken as classification decision where the predicted price is greater then the current price +15% transaction fee. Profits and losses (percentage) for out of sample trading. Assuming 15% transaction fee. RECURRENT GRAPH NEURAL NETWORKS
- 26. Reward Shaping Random Walk LSTM RECURRENT GRAPH NEURAL NETWORKS
- 28. <hash> <hash> <hash> <hash> <hash> <hash> <hash> <hash> <hash> <hash> <hash> <hash> <Amount> <Amount> <Amount> <Amount> <Amount> <Amount> … … Blockchain Representation
- 29. Learning Graph Representations Random Walks On Graphs Perozzi et al., 2014 Spectral networks Bruna et al., 2013 Marginalized kernels between labeled graphs Kashima, 2013 Graph Neural Networks Gori 2015 Convolutional Networks on Graphs for Learning Molecular Fingerprints Duvenaud, 2015 RECURRENT GRAPH NEURAL NETWORKS
- 30. Spectral Networks Convolution are diagonalized in Fourier Domain: 𝒙 ∗ 𝒉 = 𝓕−𝟏 𝒅𝒊𝒂𝒈 𝓕𝒉 𝓕𝒙 Where 𝓕 𝒌,𝒍 = 𝒆 ( −𝟐𝝅𝒊(𝒌∙𝒍) 𝑵 𝒅 ) Fourier basis can be defined as the eigenbasis of Laplacian operator: ∆𝒙 𝒖 = 𝒋≤𝒅 𝝏 𝟐 𝒙 𝝏𝒖𝒋 𝟐 (𝒖) RECURRENT GRAPH NEURAL NETWORKS
- 31. Laplacian 𝑓 𝑓′ 𝑓′′ 𝛻𝑓 𝛻 ∙ 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝐷𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 RECURRENT GRAPH NEURAL NETWORKS
- 32. Graph Laplacian RECURRENT GRAPH NEURAL NETWORKS
- 33. Graph Convolution Spectral graph convolution multiplication of a signal with a filter in the Fourier space of a graph. Graph Fourier transform multiplication of a graph signal 𝑋(i.e. feature vectors for every node) with the eigenvector matrix 𝑈of the graph Laplacian 𝐿. Graph Laplacian can be easily computed from the symmetrically normalized graph adjacency matrix ҧ𝐴: 𝐿 = 𝐼 − ҧ𝐴 Fourier basis of 𝑿 are Eigenvectors 𝑽 of 𝑳 RECURRENT GRAPH NEURAL NETWORKS
- 34. Spectral Networks Convolution of Graph: 𝒙 ∗ 𝒉 𝒌 = 𝑽𝒅𝒊𝒂𝒈(𝒉)𝑽 𝑻 𝒙 RECURRENT GRAPH NEURAL NETWORKS
- 36. Graphs Isomorphism RECURRENT GRAPH NEURAL NETWORKS
- 37. ConvNet (LeNet5) 𝑥Input ො𝑦 Class Convolution & Maxpooling Convolution & Maxpooling Convolution & Maxpooling Fully Connected RECURRENT GRAPH NEURAL NETWORKS
- 38. ConvNet (LeNet) 𝑥Input ො𝑦 Class Convolution & Maxpooling Convolution & Maxpooling Convolution & Maxpooling Fully Connected Classifier Representation Learning RECURRENT GRAPH NEURAL NETWORKS
- 39. Representation Bank Give me the best representation for “cat” the best representation for “cat” Cat RECURRENT GRAPH NEURAL NETWORKS
- 40. Single CNN layer with 3X3 filter Convolutional Neural Network 2D 1D RECURRENT GRAPH NEURAL NETWORKS
- 41. Single CNN layer with 3X3 filter Euclidian Space Convolution Update for a single pixel -Transform neighbors individually 𝑤𝑖 (𝑙) ℎ𝑖 (𝑙) -Add everything up σ𝑖 𝑤𝑖 (𝑙) ℎ𝑖 (𝑙) -Add everything up ℎ0 (𝑙+1) = 𝜎(σ𝑖 𝑤𝑖 (𝑙) ℎ𝑖 (𝑙) ) 𝒉 𝟎 𝒉 𝟐𝒉 𝟏 𝒉 𝟑 𝒉 𝟒 𝒉 𝟓𝒉 𝟔𝒉 𝟕 𝒉 𝟖 𝑤7 𝑤8 𝑤1 𝑤2 𝑤3 𝑤4 𝑤5𝑤6 RECURRENT GRAPH NEURAL NETWORKS
- 42. Euclidian Space Convolution 𝒉 𝟎 𝒉 𝟐𝒉 𝟏 𝒉 𝟑 𝒉 𝟒 𝒉 𝟓𝒉 𝟔𝒉 𝟕 𝒉 𝟖 𝑤7 𝑤8 𝑤1 𝑤2 𝑤3 𝑤4 𝑤5𝑤6 ℎ0 (𝑙+1) = 𝜎( 𝑖 𝑤𝑖 (𝑙) ℎ𝑖 (𝑙) ) RECURRENT GRAPH NEURAL NETWORKS
- 43. Graph Convolution as Message Passing 𝒉 𝟎 (𝒍+𝟏) = 𝝈(𝒉 𝟎 (𝒍) 𝒘 𝟎 (𝒍) 𝒊∈𝝒 𝟏 𝒄𝒊,𝒍 𝒘𝒋 (𝒍) 𝒉𝒋 (𝒍) ) Propagation rule 𝒘 𝟎 𝒘 𝟏 𝒉𝒊 RECURRENT GRAPH NEURAL NETWORKS
- 44. def relational_graph_convolution(self, inputs): features = inputs[0] A = inputs[1:] # list of basis functions # convolve supports = list() for i in range(support): supports.append(K.dot(A[i], features)) supports = K.concatenate(supports, axis=1) output = K.dot(supports, self.W) 𝒉 𝟎 (𝒍+𝟏) = 𝝈(𝒉 𝟎 (𝒍) 𝒘 𝟎 (𝒍) 𝒊∈𝝒 𝟏 𝒄𝒊,𝒍 𝒘𝒋 (𝒍) 𝒉𝒋 (𝒍) ) RECURRENT GRAPH NEURAL NETWORKS
- 45. GRAPH CONVOLUTIONAL NETWORKS ReLUReLU Input Features for nodes 𝑋 ∈ ℝ 𝑁∗𝐸 Adjacency matrix containing all links መ𝐴 Embeddings Representations that combine features of neighborhood Neighborhood size depends on number of layers RECURRENT GRAPH NEURAL NETWORKS
- 46. Problem Embeddings are not optimized For classification task!
- 47. GRAPH CONVOLUTIONAL NETWORKS ReLUReLU Input Features for nodes 𝑋 ∈ ℝ 𝑁∗𝐸 Adjacency matrix containing all links መ𝐴 Evaluate loss on labeled nodes only ℒ = − 𝐼∈𝑦 𝑖 𝑓=𝐼 𝐹 𝑌𝐼𝑓ln( 𝑖 𝑒 𝑥 𝑖) RECURRENT GRAPH NEURAL NETWORKS
- 48. EXAMPLE OF FORWARD PASS 𝑓( ) = RECURRENT GRAPH NEURAL NETWORKS
- 49. Inits SEMI-SUPERVISED CLASSIFICATION WITH GRAPH CONVOLUTIONAL NETWORKS Move Nodes https://github.com/tkipf/gcn RECURRENT GRAPH NEURAL NETWORKS
- 50. Inits SEMI-SUPERVISED CLASSIFICATION WITH GRAPH CONVOLUTIONAL NETWORKS Move Nodes https://github.com/tkipf/gcn 𝒉 𝟎 (𝒍+𝟏) = 𝝈( 𝒊∈𝝒 𝟏 𝒄𝒊,𝒍 𝒘 𝟏 (𝒍) 𝒉𝒋 (𝒍) ) RECURRENT GRAPH NEURAL NETWORKS
- 51. F1RMSE PnL(%) Results: Graph Convolution 0.037 0.71 0.3 Results for out of sample simulated trading Simple root mean square error F1-beta score (Harmonic mean of precision and recall) taken as classification decision where the predicted price is greater then the current price +15% transaction fee. Profits and losses (percentage) for out of sample trading. Assuming 15% transaction fee. RECURRENT GRAPH NEURAL NETWORKS
- 52. Temporal?
- 53. Recurrent Neural Networks Graph Convolution Networks 𝒘 𝟎 𝒘 𝟏 𝒉𝒊 𝒉 𝟎 (𝒍+𝟏) = 𝝈(𝒉 𝟎 (𝒍) 𝒘 𝟎 (𝒍) 𝒊∈𝝒 𝟏 𝒄𝒊,𝒍 𝒘𝒋 (𝒍) 𝒉𝒋 (𝒍) ) ቁ𝑓𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔 𝑓 𝑦𝑗 𝑙 𝜍 𝑡−1 + 𝜓 𝑓ℎ 𝑡−1 + 𝑏𝑓 ቁ𝜄𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔𝜄 𝑦𝑗 𝑙 𝜍 𝑡−1 + 𝜓𝜄ℎ 𝑡−1 + 𝑏𝜄 ቁ𝑜𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔 𝑜 𝑦𝑗 𝑙 𝜍 𝑡−1 + 𝜓 𝑜ℎ 𝑡−1 + 𝑏 𝑜 ቁ𝜍𝑖 𝑙+1 𝑡 = 𝑡𝑎𝑛ℎ( 𝜔𝜍 𝑦𝑗 𝑙 + 𝜓𝜍ℎ 𝑡−1 + 𝑏𝜍
- 54. Recurrent Neural Networks Graph Convolution Networks 𝒉 𝟎 (𝒍+𝟏) = 𝝈(𝒉 𝟎 (𝒍) 𝒘 𝟎 (𝒍) 𝒊∈𝝒 𝟏 𝒄𝒊,𝒍 𝒘𝒋 (𝒍) 𝒉𝒋 (𝒍) ) ቁ𝑓𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔 𝑓 𝑦𝑗 𝑙 𝜍 𝑡−1 + 𝜓 𝑓ℎ 𝑡−1 + 𝑏𝑓 ቁ𝜄𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔𝜄 𝑦𝑗 𝑙 𝜍 𝑡−1 + 𝜓𝜄ℎ 𝑡−1 + 𝑏𝜄 ቁ𝑜𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔 𝑜 𝑦𝑗 𝑙 𝜍 𝑡−1 + 𝜓 𝑜ℎ 𝑡−1 + 𝑏 𝑜 ቁ𝜍𝑖 𝑙+1 𝑡 = 𝑡𝑎𝑛ℎ( 𝜔𝜍 𝑦𝑗 𝑙 + 𝜓𝜍ℎ 𝑡−1 + 𝑏𝜍 ൱𝑓𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔 𝑓 𝑟∈ℛ ൱ 𝑗∈𝒩𝑖 𝑟 1 𝑐𝑖,𝑟 𝜃𝑟 𝑙 𝑦𝑗 𝑙 + 𝜃0 𝑙 𝑦𝑖 𝑙 𝜍 𝑡−1 + 𝜓 𝑓ℎ 𝑡−1 + 𝑏𝑓 ൱𝜄𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔𝜄 𝑟∈ℛ ൱ 𝑗∈𝒩𝑖 𝑟 1 𝑐𝑖,𝑟 𝜃𝑟 𝑙 𝑦𝑗 𝑙 + 𝜃0 𝑙 𝑦𝑖 𝑙 𝜍 𝑡−1 + 𝜓𝜄ℎ 𝑡−1 + 𝑏𝜄 ൱𝑜𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔 𝑜 𝑟∈ℛ ൱ 𝑗∈𝒩𝑖 𝑟 1 𝑐𝑖,𝑟 𝜃𝑟 𝑙 𝑦𝑗 𝑙 + 𝜃0 𝑙 𝑦𝑖 𝑙 𝜍 𝑡−1 + 𝜓 𝑜ℎ 𝑡−1 + 𝑏 𝑜 ൱𝜍𝑖 𝑙+1 𝑡 = 𝑡𝑎𝑛ℎ( 𝜔𝜍 𝑟∈ℛ ൱ 𝑗∈𝒩𝑖 𝑟 1 𝑐𝑖,𝑟 𝜃𝑟 𝑙 𝑦𝑗 𝑙 + 𝜃0 𝑙 𝑦𝑖 𝑙 + 𝜓𝜍ℎ 𝑡−1 + 𝑏𝜍 Forget Gate Output Gate Input Gate Cell
- 55. RECURRENT GRAPH CONVOLUTIONAL NETWORKS RECURRENT GRAPH NEURAL NETWORKS ൱𝑓𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔 𝑓 𝑟∈ℛ ൱ 𝑗∈𝒩𝑖 𝑟 1 𝑐𝑖,𝑟 𝜃𝑟 𝑙 𝑦𝑗 𝑙 + 𝜃0 𝑙 𝑦𝑖 𝑙 𝜍 𝑡−1 + 𝜓 𝑓ℎ 𝑡−1 + 𝑏𝑓 ൱𝜄𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔𝜄 𝑟∈ℛ ൱ 𝑗∈𝒩𝑖 𝑟 1 𝑐𝑖,𝑟 𝜃𝑟 𝑙 𝑦𝑗 𝑙 + 𝜃0 𝑙 𝑦𝑖 𝑙 𝜍 𝑡−1 + 𝜓𝜄ℎ 𝑡−1 + 𝑏𝜄 ൱𝑜𝑖 𝑙+1 𝑡 = 𝜎𝑔( 𝜔 𝑜 𝑟∈ℛ ൱ 𝑗∈𝒩𝑖 𝑟 1 𝑐𝑖,𝑟 𝜃𝑟 𝑙 𝑦𝑗 𝑙 + 𝜃0 𝑙 𝑦𝑖 𝑙 𝜍 𝑡−1 + 𝜓 𝑜ℎ 𝑡−1 + 𝑏 𝑜 ൱𝜍𝑖 𝑙+1 𝑡 = 𝑡𝑎𝑛ℎ( 𝜔𝜍 𝑟∈ℛ ൱ 𝑗∈𝒩𝑖 𝑟 1 𝑐𝑖,𝑟 𝜃𝑟 𝑙 𝑦𝑗 𝑙 + 𝜃0 𝑙 𝑦𝑖 𝑙 + 𝜓𝜍ℎ 𝑡−1 + 𝑏𝜍 Forget Gate Output Gate Input Gate Cell ℎ 𝑡 ℎ 𝑡+1 ℎ 𝑡+2 ℎ 𝑡+3
- 56. F1RMSE PnL(%) Results: Recurrent Graph Convolution 0.028 0.77 2.4 Results for out of sample simulated trading Simple root mean square error F1-beta score (Harmonic mean of precision and recall) taken as classification decision where the predicted price is greater then the current price +15% transaction fee. Profits and losses (percentage) for out of sample trading. Assuming 15% transaction fee. RECURRENT GRAPH NEURAL NETWORKS
- 57. STRATEGY GRADIENT? 𝜕( 𝜃) 𝜕𝜃 − 𝜕( 𝜑) 𝜕𝜑 Returns Risk RECURRENT GRAPH NEURAL NETWORKS
- 58. Graph Auto Encoders 𝑨 – input graph 𝒙 – input node 𝑨 – output graph Useful for predicting connectivity links RECURRENT GRAPH NEURAL NETWORKS
- 59. Recommender Systems Users Items Graph Representation Graph Prediction Graph AutoEncoder RECURRENT GRAPH NEURAL NETWORKS
- 60. 𝜎 μ Simulation TRADING STRATEGY GRADIENTS 𝑖=1 𝑛 𝜎𝑖 2 + 𝜇𝑖 2 − log 𝜎𝑖 − 1 | ො𝑦 − 𝑦| 2 2 Action: 𝒂 𝒕+𝟏 RECURRENT GRAPH NEURAL NETWORKS
- 61. F1RMSE PnL(%) Results: Recurrent Graph Auto Encoder 0.024 0.86 5.6 Results for out of sample simulated trading Simple root mean square error F1-beta score (Harmonic mean of precision and recall) taken as classification decision where the predicted price is greater then the current price +15% transaction fee. Profits and losses (percentage) for out of sample trading. Assuming 15% transaction fee. RECURRENT GRAPH NEURAL NETWORKS
- 62. Conclusions -Deep Learning works well on Euclidean data. -Attempts to utilize DL for Non-Euclidean are starting to become viable. -Reward shaping and drifted metrics are extremely misleading. -After trying heavily we conclude that Aggregated data (prices) of Ethereum is insufficient when trying to forecast behavior. -We introduce a novel layer: Recurrent Graph Convolution and demonstrate How this approach yield “tradable” results. RECURRENT GRAPH NEURAL NETWORKS
- 63. FIN