NodeFlow and Sampling

Author: Ziyue Huang, Da Zheng, Quan Gan, Jinjing Zhou, Zheng Zhang

Graph convolutional network ~~~

In an \(L\)-layer graph convolution network (GCN), given a graph \(G=(V, E)\), represented as an adjacency matrix \(A\), with node features \(H^{(0)} = X \in \mathbb{R}^{|V| \times d}\), the hidden feature of a node \(v\) in \((l+1)\)-th layer \(h_v^{(l+1)}\) depends on the features of all its neighbors in the previous layer \(h_u^{(l)}\):

\[z_v^{(l+1)} = \sum_{u \in \mathcal{N}(v)} \tilde{A}_{uv} h_u^{(l)} \qquad h_v^{(l+1)} = \sigma ( z_v^{(l+1)} W^{(l)})\]

where \(\mathcal{N}(v)\) is the neighborhood of \(v\), \(\tilde{A}\) could be any normalized version of \(A\) such as \(D^{-1} A\) in Kipf et al., \(\sigma(\cdot)\) is an activation function, and \(W^{(l)}\) is a trainable parameter of the \(l\)-th layer.

In the node classification task you minimize the following loss:

\[\frac{1}{\vert \mathcal{V}_\mathcal{L} \vert} \sum_{v \in \mathcal{V}_\mathcal{L}} f(y_v, z_v^{(L)})\]

where \(y_v\) is the label of \(v\), and \(f(\cdot, \cdot)\) is a loss function, e.g., cross entropy loss.

While training GCN on the full graph, each node aggregates the hidden features of its neighbors to compute its hidden feature in the next layer.

In this tutorial, you run GCN on the Reddit dataset constructed by Hamilton et al., wherein the nodes are posts and edges are established if two nodes are commented by a same user. The task is to predict the category that a post belongs to. This graph has 233,000 nodes, 114.6 million edges and 41 categories. First load the Reddit graph.

import numpy as np
import dgl
import dgl.function as fn
from dgl import DGLGraph
from dgl.data import RedditDataset
import mxnet as mx
from mxnet import gluon

# Load MXNet as backend
dgl.load_backend('mxnet')

# load dataset
data = RedditDataset(self_loop=True)
train_nid = mx.nd.array(np.nonzero(data.train_mask)[0]).astype(np.int64)
features = mx.nd.array(data.features)
in_feats = features.shape[1]
labels = mx.nd.array(data.labels)
n_classes = data.num_labels

# construct DGLGraph and prepare related data
g = DGLGraph(data.graph, readonly=True)
g.ndata['features'] = features

Out:

Finished data loading.
  NumNodes: 232965
  NumEdges: 114848857
  NumFeats: 602
  NumClasses: 41
  NumTrainingSamples: 153431
  NumValidationSamples: 23831
  NumTestSamples: 55703

Here you define the node UDF, which has a fully-connected layer:

class NodeUpdate(gluon.Block):
    def __init__(self, in_feats, out_feats, activation=None):
        super(NodeUpdate, self).__init__()
        self.dense = gluon.nn.Dense(out_feats, in_units=in_feats)
        self.activation = activation

    def forward(self, node):
        h = node.data['h']
        h = self.dense(h)
        if self.activation:
            h = self.activation(h)
        return {'activation': h}

In DGL, you implement GCN on the full graph with update_all in DGLGraph. The following code performs two-layer GCN on the Reddit graph.

# number of GCN layers
L = 2
# number of hidden units of a fully connected layer
n_hidden = 64

layers = [NodeUpdate(g.ndata['features'].shape[1], n_hidden, mx.nd.relu),
          NodeUpdate(n_hidden, n_hidden, mx.nd.relu)]
for layer in layers:
    layer.initialize()

h = g.ndata['features']
for i in range(L):
    g.ndata['h'] = h
    g.update_all(message_func=fn.copy_src(src='h', out='m'),
                 reduce_func=fn.sum(msg='m', out='h'),
                 apply_node_func=lambda node: {'h': layers[i](node)['activation']})
    h = g.ndata.pop('h')

NodeFlow

As the graph scales up to billions of nodes or edges, training on the full graph would no longer be efficient or even feasible.

Mini-batch training allows you to control the computation and memory usage within some budget. The training loss for each iteration is

\[\frac{1}{\vert \tilde{\mathcal{V}}_\mathcal{L} \vert} \sum_{v \in \tilde{\mathcal{V}}_\mathcal{L}} f(y_v, z_v^{(L)})\]

where \(\tilde{\mathcal{V}}_\mathcal{L}\) is a subset sampled from the total labeled nodes \(\mathcal{V}_\mathcal{L}\) uniformly at random.

Stemming from the labeled nodes \(\tilde{\mathcal{V}}_\mathcal{L}\) in a mini-batch and tracing back to the input forms a computational dependency graph (a directed acyclic graph [DAG]), which captures the computation flow of \(Z^{(L)}\).

In the example below, a mini-batch to compute the hidden features of node D in layer 2 requires hidden features of A, B, E, G in layer 1, which in turn requires hidden features of C, D, F in layer 0.

For that purpose, you define NodeFlow to represent this computation flow.

NodeFlow is a type of layered graph, where nodes are organized in \(L + 1\) sequential layers, and edges only exist between adjacent layers, forming blocks. You construct NodeFlow backwards, starting from the last layer with all the nodes whose hidden features are requested. The set of nodes the next layer depends on forms the previous layer. An edge connects a node in the previous layer to another in the next layer if the latter depends on the former. Repeat such process until all \(L + 1\) layers are constructed. The feature of nodes in each layer, and that of edges in each block, are stored as separate tensors.

NodeFlow provides block_compute for per-block computation, which triggers computation and data propogation from the lower layer to the next upper layer.

Neighbor sampling

Real-world graphs often have nodes with large degree, meaning that a moderately deep (e.g., three layers) GCN would often depend on input features of the entire graph, even if the computation only depends on outputs of a few nodes, hence its cost-ineffectiveness.

Sampling methods mitigate this computational problem by reducing the receptive field effectively. Fig-c above shows one such example.

Instead of using all the \(L\)-hop neighbors of a node \(v\), Hamilton et al. propose neighbor sampling, which randomly samples a few neighbors \(\hat{\mathcal{N}}^{(l)}(v)\) to estimate the aggregation \(z_v^{(l+1)}\) of its total neighbors \(\mathcal{N}(v)\) in \(l\)-th GCN layer, by an unbiased estimator \(\hat{z}_v^{(l+1)}\)

\[\hat{z}_v^{(l+1)} = \frac{\vert \mathcal{N}(v) \vert }{\vert \hat{\mathcal{N}}^{(l)}(v) \vert} \sum_{u \in \hat{\mathcal{N}}^{(l)}(v)} \tilde{A}_{uv} \hat{h}_u^{(l)} \qquad \hat{h}_v^{(l+1)} = \sigma ( \hat{z}_v^{(l+1)} W^{(l)} )\]

Let \(D^{(l)}\) be the number of neighbors to be sampled for each node at the \(l\)-th layer, then the receptive field size of each node can be controlled under \(\prod_{i=0}^{L-1} D^{(l)}\) by neighbor sampling.

You then implement neighbor sampling by NodeFlow:

class GCNSampling(gluon.Block):
    def __init__(self,
                 in_feats,
                 n_hidden,
                 n_classes,
                 n_layers,
                 activation,
                 dropout,
                 **kwargs):
        super(GCNSampling, self).__init__(**kwargs)
        self.dropout = dropout
        self.n_layers = n_layers
        with self.name_scope():
            self.layers = gluon.nn.Sequential()
            # input layer
            self.layers.add(NodeUpdate(in_feats, n_hidden, activation))
            # hidden layers
            for i in range(1, n_layers-1):
                self.layers.add(NodeUpdate(n_hidden, n_hidden, activation))
            # output layer
            self.layers.add(NodeUpdate(n_hidden, n_classes))

    def forward(self, nf):
        nf.layers[0].data['activation'] = nf.layers[0].data['features']
        for i, layer in enumerate(self.layers):
            h = nf.layers[i].data.pop('activation')
            if self.dropout:
                h = mx.nd.Dropout(h, p=self.dropout)
            nf.layers[i].data['h'] = h
            # block_compute() computes the feature of layer i given layer
            # i-1, with the given message, reduce, and apply functions.
            # Here, you essentially aggregate the neighbor node features in
            # the previous layer, and update it with the `layer` function.
            nf.block_compute(i,
                             fn.copy_src(src='h', out='m'),
                             lambda node : {'h': node.mailbox['m'].mean(axis=1)},
                             layer)
        h = nf.layers[-1].data.pop('activation')
        return h

DGL provides NeighborSampler to construct the NodeFlow for a mini-batch according to the computation logic of neighbor sampling. NeighborSampler returns an iterator that generates a NodeFlow each time. This function has many options to give users opportunities to customize the behavior of the neighbor sampler, including the number of neighbors to sample or the number of hops to sample, for example. Please see its API document for more details.

# dropout probability
dropout = 0.2
# batch size
batch_size = 1000
# number of neighbors to sample
num_neighbors = 4
# number of epochs
num_epochs = 1

# initialize the model and cross entropy loss
model = GCNSampling(in_feats, n_hidden, n_classes, L,
                    mx.nd.relu, dropout, prefix='GCN')
model.initialize()
loss_fcn = gluon.loss.SoftmaxCELoss()

# use adam optimizer
trainer = gluon.Trainer(model.collect_params(), 'adam',
                        {'learning_rate': 0.03, 'wd': 0})

for epoch in range(num_epochs):
    i = 0
    for nf in dgl.contrib.sampling.NeighborSampler(g, batch_size,
                                                   num_neighbors,
                                                   neighbor_type='in',
                                                   shuffle=True,
                                                   num_hops=L,
                                                   seed_nodes=train_nid):
        # When `NodeFlow` is generated from `NeighborSampler`, it only contains
        # the topology structure, on which there is no data attached.
        # Users need to call `copy_from_parent` to copy specific data,
        # such as input node features, from the original graph.
        nf.copy_from_parent()
        with mx.autograd.record():
            # forward
            pred = model(nf)
            batch_nids = nf.layer_parent_nid(-1).astype('int64')
            batch_labels = labels[batch_nids]
            # cross entropy loss
            loss = loss_fcn(pred, batch_labels)
            loss = loss.sum() / len(batch_nids)
        # backward
        loss.backward()
        # optimization
        trainer.step(batch_size=1)
        print("Epoch[{}]: loss {}".format(epoch, loss.asscalar()))
        i += 1
        # You only train the model with 32 mini-batches just for demonstration.
        if i >= 32:
            break

Out:

Epoch[0]: loss 3.7310597896575928
Epoch[0]: loss 3.0101420879364014
Epoch[0]: loss 2.585841655731201
Epoch[0]: loss 2.159868001937866
Epoch[0]: loss 1.9110859632492065
Epoch[0]: loss 1.7267630100250244
Epoch[0]: loss 1.5326182842254639
Epoch[0]: loss 1.606508731842041
Epoch[0]: loss 1.4062317609786987
Epoch[0]: loss 1.2362780570983887
Epoch[0]: loss 1.3502916097640991
Epoch[0]: loss 1.2101963758468628
Epoch[0]: loss 1.3546274900436401
Epoch[0]: loss 1.0514274835586548
Epoch[0]: loss 1.099158525466919
Epoch[0]: loss 1.0870254039764404
Epoch[0]: loss 1.082731008529663
Epoch[0]: loss 1.0594552755355835
Epoch[0]: loss 1.0993865728378296
Epoch[0]: loss 1.0266661643981934
Epoch[0]: loss 1.0138708353042603
Epoch[0]: loss 0.9079034924507141
Epoch[0]: loss 0.9981871247291565
Epoch[0]: loss 1.066437005996704
Epoch[0]: loss 1.012518048286438
Epoch[0]: loss 1.0037081241607666
Epoch[0]: loss 0.9039970636367798
Epoch[0]: loss 0.9596415162086487
Epoch[0]: loss 0.8718827962875366
Epoch[0]: loss 0.9065074324607849
Epoch[0]: loss 0.8950481414794922
Epoch[0]: loss 0.8208798766136169

Control variate

The unbiased estimator \(\hat{Z}^{(\cdot)}\) used in neighbor sampling might suffer from high variance, so it still requires a relatively large number of neighbors, e.g. \(D^{(0)}=25\) and \(D^{(1)}=10\) in Hamilton et al.. With control variate, a standard variance reduction technique widely used in Monte Carlo methods, 2 neighbors for a node seems sufficient.

Control variate method works as follows: Given a random variable \(X\) and you wish to estimate its expectation \(\mathbb{E} [X] = \theta\), it finds another random variable \(Y\) which is highly correlated with \(X\) and whose expectation \(\mathbb{E} [Y]\) can be easily computed. The control variate estimator \(\tilde{X}\) is

\[\tilde{X} = X - Y + \mathbb{E} [Y] \qquad \mathbb{VAR} [\tilde{X}] = \mathbb{VAR} [X] + \mathbb{VAR} [Y] - 2 \cdot \mathbb{COV} [X, Y]\]

If \(\mathbb{VAR} [Y] - 2\mathbb{COV} [X, Y] < 0\), then \(\mathbb{VAR} [\tilde{X}] < \mathbb{VAR} [X]\).

Chen et al. proposed a control variate based estimator used in GCN training, by using history \(\bar{H}^{(l)}\) of the nodes which are not sampled, the modified estimator \(\hat{z}_v^{(l+1)}\) is

\[\begin{split}\hat{z}_v^{(l+1)} = \frac{\vert \mathcal{N}(v) \vert }{\vert \hat{\mathcal{N}}^{(l)}(v) \vert} \sum_{u \in \hat{\mathcal{N}}^{(l)}(v)} \tilde{A}_{uv} ( \hat{h}_u^{(l)} - \bar{h}_u^{(l)} ) + \sum_{u \in \mathcal{N}(v)} \tilde{A}_{uv} \bar{h}_u^{(l)} \\ \hat{h}_v^{(l+1)} = \sigma ( \hat{z}_v^{(l+1)} W^{(l)} )\end{split}\]

This method can also be conceptually implemented in DGL as shown here.

have_large_memory = False
# The control-variate sampling code below needs to run on a large-memory
# machine for the Reddit graph.
if have_large_memory:
    g.ndata['h_0'] = features
    for i in range(L):
        g.ndata['h_{}'.format(i+1)] = mx.nd.zeros((features.shape[0], n_hidden))
    # With control-variate sampling, you only need to sample two neighbors to train GCN.
    for nf in dgl.contrib.sampling.NeighborSampler(g, batch_size, expand_factor=2,
                                                   neighbor_type='in', num_hops=L,
                                                   seed_nodes=train_nid):
        for i in range(nf.num_blocks):
            # aggregate history on the original graph
            g.pull(nf.layer_parent_nid(i+1),
                   fn.copy_src(src='h_{}'.format(i), out='m'),
                   lambda node: {'agg_h_{}'.format(i): node.mailbox['m'].mean(axis=1)})
        nf.copy_from_parent()
        h = nf.layers[0].data['features']
        for i in range(nf.num_blocks):
            prev_h = nf.layers[i].data['h_{}'.format(i)]
            # compute delta_h, the difference of the current activation and the history
            nf.layers[i].data['delta_h'] = h - prev_h
            # refresh the old history
            nf.layers[i].data['h_{}'.format(i)] = h.detach()
            # aggregate the delta_h
            nf.block_compute(i,
                             fn.copy_src(src='delta_h', out='m'),
                             lambda node: {'delta_h': node.data['m'].mean(axis=1)})
            delta_h = nf.layers[i + 1].data['delta_h']
            agg_h = nf.layers[i + 1].data['agg_h_{}'.format(i)]
            # control variate estimator
            nf.layers[i + 1].data['h'] = delta_h + agg_h
            nf.apply_layer(i + 1, lambda node : {'h' : layer(node.data['h'])})
            h = nf.layers[i + 1].data['h']
        # update history
        nf.copy_to_parent()

You can see full example here, MXNet code and PyTorch code.

Below shows the performance of graph convolution network and GraphSage with neighbor sampling and control variate sampling on the Reddit dataset. Our GraphSage with control variate sampling, when sampling one neighbor, can achieve over 96 percent test accuracy.

More APIs

In fact, block_compute is one of the APIs that comes with NodeFlow, which provides flexibility to research new ideas. The computation flow underlying a DAG can be executed in one sweep, by calling prop_flows.

prop_flows accepts a list of UDFs. The code below defines node update UDFs for each layer and computes a simplified version of GCN with neighbor sampling.

apply_node_funcs = [
    lambda node : {'h' : layers[0](node)['activation']},
    lambda node : {'h' : layers[1](node)['activation']},
]
for nf in dgl.contrib.sampling.NeighborSampler(g, batch_size, num_neighbors,
                                               neighbor_type='in', num_hops=L,
                                               seed_nodes=train_nid):
    nf.copy_from_parent()
    nf.layers[0].data['h'] = nf.layers[0].data['features']
    nf.prop_flow(fn.copy_src(src='h', out='m'),
                 fn.sum(msg='m', out='h'), apply_node_funcs)

Internally, prop_flow triggers the computation by fusing together all the block computations, from the input to the top. The main advantages of this API are 1) simplicity, 2) allowing more system-level optimization in the future.