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Graph Attention Networks v2 (GATv2)

This is a PyTorch implementation of the GATv2 operator from the paper How Attentive are Graph Attention Networks?.

GATv2s work on graph data similar to GAT. A graph consists of nodes and edges connecting nodes. For example, in Cora dataset the nodes are research papers and the edges are citations that connect the papers.

The GATv2 operator fixes the static attention problem of the standard GAT. Static attention is when the attention to the key nodes has the same rank (order) for any query node. GAT computes attention from query node $i$ to key node $j$ as,

e_{i j} = Le a k yReLU (a^{⊤} [W h_{i} ∥ W h_{j}]) = Le a k yReLU (a_{1}^{⊤} W h_{i} + a_{2}^{⊤} W h_{j})

Note that for any query node $i$ , the attention rank ( $a r g sor t$ ) of keys depends only on $a_{2}^{⊤} W h_{j}$ . Therefore the attention rank of keys remains the same (static) for all queries.

GATv2 allows dynamic attention by changing the attention mechanism,

e_{i j} = a^{⊤} Le a k yReLU (W [h_{i} ∥ h_{j}]) = a^{⊤} Le a k yReLU (W_{l} h_{i} + W_{r} h_{j})

The paper shows that GATs static attention mechanism fails on some graph problems with a synthetic dictionary lookup dataset. It's a fully connected bipartite graph where one set of nodes (query nodes) have a key associated with it and the other set of nodes have both a key and a value associated with it. The goal is to predict the values of query nodes. GAT fails on this task because of its limited static attention.

Here is the training code for training a two-layer GATv2 on Cora dataset.

57import torch
58from torch import nn
59
60from labml_helpers.module import Module

#

Graph attention v2 layer

This is a single graph attention v2 layer. A GATv2 is made up of multiple such layers. It takes $h = {h_{1}, h_{2}, \dots, h_{N}}$ , where $h_{i} \in R^{F}$ as input and outputs $h^{'} = {h_{1}^{'}, h_{2}^{'}, \dots, h_{N}^{'}}$ , where $h_{i}^{'} \in R^{F^{'}}$ .

63class GraphAttentionV2Layer(Module):

#

in_features , $F$ , is the number of input features per node
out_features , $F^{'}$ , is the number of output features per node
n_heads , $K$ , is the number of attention heads
is_concat whether the multi-head results should be concatenated or averaged
dropout is the dropout probability
leaky_relu_negative_slope is the negative slope for leaky relu activation
share_weights if set to True , the same matrix will be applied to the source and the target node of every edge

76    def __init__(self, in_features: int, out_features: int, n_heads: int,
77                 is_concat: bool = True,
78                 dropout: float = 0.6,
79                 leaky_relu_negative_slope: float = 0.2,
80                 share_weights: bool = False):

#

90        super().__init__()
91
92        self.is_concat = is_concat
93        self.n_heads = n_heads
94        self.share_weights = share_weights

#

Calculate the number of dimensions per head

97        if is_concat:
98            assert out_features % n_heads == 0

#

If we are concatenating the multiple heads

100            self.n_hidden = out_features // n_heads
101        else:

#

If we are averaging the multiple heads

103            self.n_hidden = out_features

#

Linear layer for initial source transformation; i.e. to transform the source node embeddings before self-attention

107        self.linear_l = nn.Linear(in_features, self.n_hidden * n_heads, bias=False)

#

If share_weights is True the same linear layer is used for the target nodes

109        if share_weights:
110            self.linear_r = self.linear_l
111        else:
112            self.linear_r = nn.Linear(in_features, self.n_hidden * n_heads, bias=False)

#

Linear layer to compute attention score $e_{i j}$

114        self.attn = nn.Linear(self.n_hidden, 1, bias=False)

#

The activation for attention score $e_{i j}$

116        self.activation = nn.LeakyReLU(negative_slope=leaky_relu_negative_slope)

#

Softmax to compute attention $α_{i j}$

118        self.softmax = nn.Softmax(dim=1)

#

Dropout layer to be applied for attention

120        self.dropout = nn.Dropout(dropout)

#

h , $h$ is the input node embeddings of shape [n_nodes, in_features] .
adj_mat is the adjacency matrix of shape [n_nodes, n_nodes, n_heads] . We use shape [n_nodes, n_nodes, 1] since the adjacency is the same for each head. Adjacency matrix represent the edges (or connections) among nodes. adj_mat[i][j] is True if there is an edge from node i to node j .

122    def forward(self, h: torch.Tensor, adj_mat: torch.Tensor):

#

Number of nodes

132        n_nodes = h.shape[0]

#

The initial transformations, $g_{l}_{i}^{k} = W_{l}^{k} h_{i}$ $g_{r}_{i}^{k} = W_{r}^{k} h_{i}$ for each head. We do two linear transformations and then split it up for each head.

138        g_l = self.linear_l(h).view(n_nodes, self.n_heads, self.n_hidden)
139        g_r = self.linear_r(h).view(n_nodes, self.n_heads, self.n_hidden)

#

Calculate attention score

We calculate these for each head $k$ . We have omitted $\cdot^{k}$ for simplicity.

$e_{i j} = a (W_{l} h_{i}, W_{r} h_{j}) = a (g_{l}_{i}, g_{r}_{j})$

$e_{i j}$ is the attention score (importance) from node $j$ to node $i$ . We calculate this for each head.

$a$ is the attention mechanism, that calculates the attention score. The paper sums $g_{l}_{i}$ , $g_{r}_{j}$ followed by a $Le a k yReLU$ and does a linear transformation with a weight vector $a \in R^{F^{'}}$

$e_{i j} = a^{⊤} Le a k yReLU ([g_{l}_{i} + g_{r}_{j}])$ Note: The paper desrcibes $e_{i j}$ as $e_{i j} = a^{⊤} Le a k yReLU (W [h_{i} ∥ h_{j}])$ which is equivalent to the definition we use here.

#

First we calculate $[g_{l}_{i} + g_{r}_{j}]$ for all pairs of $i, j$ .

g_l_repeat gets ${g_{l}_{1}, g_{l}_{2}, \dots, g_{l}_{N}, g_{l}_{1}, g_{l}_{2}, \dots, g_{l}_{N}, ...}$ where each node embedding is repeated n_nodes times.

177        g_l_repeat = g_l.repeat(n_nodes, 1, 1)

#

g_r_repeat_interleave gets ${g_{r}_{1}, g_{r}_{1}, \dots, g_{r}_{1}, g_{r}_{2}, g_{r}_{2}, \dots, g_{r}_{2}, ...}$ where each node embedding is repeated n_nodes times.

182        g_r_repeat_interleave = g_r.repeat_interleave(n_nodes, dim=0)

#

Now we add the two tensors to get ${g_{l}_{1} + g_{r}_{1}, g_{l}_{1} + g_{r}_{2}, \dots, g_{l}_{1} + g_{r}_{N}, g_{l}_{2} + g_{r}_{1}, g_{l}_{2} + g_{r}_{2}, \dots, g_{l}_{2} + g_{r}_{N}, ...}$

190        g_sum = g_l_repeat + g_r_repeat_interleave

#

Reshape so that g_sum[i, j] is $g_{l}_{i} + g_{r}_{j}$

192        g_sum = g_sum.view(n_nodes, n_nodes, self.n_heads, self.n_hidden)

#

Calculate $e_{i j} = a^{⊤} Le a k yReLU ([g_{l}_{i} + g_{r}_{j}])$ e is of shape [n_nodes, n_nodes, n_heads, 1]

200        e = self.attn(self.activation(g_sum))

#

Remove the last dimension of size 1

202        e = e.squeeze(-1)

#

The adjacency matrix should have shape [n_nodes, n_nodes, n_heads] or[n_nodes, n_nodes, 1]

206        assert adj_mat.shape[0] == 1 or adj_mat.shape[0] == n_nodes
207        assert adj_mat.shape[1] == 1 or adj_mat.shape[1] == n_nodes
208        assert adj_mat.shape[2] == 1 or adj_mat.shape[2] == self.n_heads

#

Mask $e_{i j}$ based on adjacency matrix. $e_{i j}$ is set to $- \infty$ if there is no edge from $i$ to $j$ .

211        e = e.masked_fill(adj_mat == 0, float('-inf'))

#

We then normalize attention scores (or coefficients) $α_{i j} = softm a x_{j} (e_{i j}) = \frac{exp ( e _{i j} )}{\sum _{j^{'} \in N_{i}} exp ( e _{i j^{'}} )}$

where $N_{i}$ is the set of nodes connected to $i$ .

We do this by setting unconnected $e_{i j}$ to $- \infty$ which makes $exp (e_{i j}) \sim 0$ for unconnected pairs.

221        a = self.softmax(e)

#

Apply dropout regularization

224        a = self.dropout(a)

#

Calculate final output for each head $h_{i}^{' k} = j \in N_{i} \sum α_{i j}^{k} g_{r}_{j, k}$

228        attn_res = torch.einsum('ijh,jhf->ihf', a, g_r)

#

Concatenate the heads

231        if self.is_concat:

#

$h_{i}^{'} = ∥ ∥_{k = 1}^{K} h_{i}^{' k}$

233            return attn_res.reshape(n_nodes, self.n_heads * self.n_hidden)

#

Take the mean of the heads

235        else:

#

$h_{i}^{'} = \frac{1}{K} k = 1 \sum K h_{i}^{' k}$

237            return attn_res.mean(dim=1)