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8_sse_mx.py
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"""
.. _model-sse:
Stochastic steady-state embedding (SSE)
=======================================
**Author**: Gai Yu, Da Zheng, Quan Gan, Jinjing Zhou, Zheng Zhang
"""
################################################################################################
#
# .. math::
#
# \newcommand{\bfy}{\textbf{y}}
# \newcommand{\cale}{{\mathcal{E}}}
# \newcommand{\calg}{{\mathcal{G}}}
# \newcommand{\call}{{\mathcal{L}}}
# \newcommand{\caln}{{\mathcal{N}}}
# \newcommand{\calo}{{\mathcal{O}}}
# \newcommand{\calt}{{\mathcal{T}}}
# \newcommand{\calv}{{\mathcal{V}}}
# \newcommand{\until}{\text{until}\ }
#
# In this tutorial, you learn how to use the Deep Graph Library (DGL) with MXNet to implement the following:
#
# - Simple, steady-state algorithms with `stochastic steady-state
# embedding <https://www.cc.gatech.edu/~hdai8/pdf/equilibrium_embedding.pdf>`__
# (SSE)
# - Training with subgraph sampling
#
# Subgraph sampling is a technique to scale-up learning to
# gigantic graphs (for example, billions of nodes and edges). Subgraph sampling can apply to
# other algorithms, such as :doc:`Graph convolution
# network <1_gcn>`
# and :doc:`Relational graph convolution
# network <4_rgcn>`.
#
# Steady-state algorithms
# -----------------------
#
# Many algorithms for graph analytics are iterative procedures that
# end when a steady state is reached. Examples include
# PageRank or mean-field inference on Markov random fields.
#
# Flood-fill algorithm
# ~~~~~~~~~~~~~~~~~~~~
#
# A *Flood-fill algorithm* (or *infection* algorithm) can
# also be seen as a procedure. Specifically, the problem is that
# given a graph :math:`\calg = (\calv, \cale)` and a source node
# :math:`s \in \calv`, you need to mark all nodes that can be reached from
# :math:`s`. Let :math:`\calv = \{1, ..., n\}` and let :math:`y_v`
# indicate whether a node :math:`v` is marked. The flood-fill algorithm
# proceeds as follows.
#
# .. math::
#
#
# \begin{alignat}{2}
# & y_s^{(0)} \leftarrow 1 \tag{0} \\
# & y_v^{(0)} \leftarrow 0 \qquad && v \ne s \tag{1} \\
# & y_v^{(t + 1)} \leftarrow \max_{u \in \caln (v)} y_u^{(t)} \qquad && \until \forall v \in \calv, y_v^{(t + 1)} = y_v^{(t)} \tag{2}
# \end{alignat}
#
#
# where :math:`\caln (v)` denotes the neighborhood of :math:`v`, including
# :math:`v` itself.
#
# The flood-fill algorithm first marks the source node :math:`s`, and then
# repeatedly marks nodes with one or more marked neighbors until no node
# needs to be marked, that is, the steady state is reached.
#
# Flood-fill algorithm and steady-state operator
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#
# More abstractly, :math:`\begin{align}
# & y_v^{(0)} \leftarrow \text{constant} \\
# & \bfy^{(t + 1)} \leftarrow \calt (\bfy^{(t)}) \qquad \until \bfy^{(t + 1)} = \bfy^{(t)} \tag{3}
# \end{align}` where :math:`\bfy^{(t)} = (y_1^{(t)}, ..., y_n^{(t)})` and
# :math:`[\calt (\bfy^{(t)})]_v = \hat\calt (\{\bfy_u^{(t)} : u \in \caln (v)\})`.
# In the case of the flood-fill algorithm, :math:`\hat\calt = \max`. The
# condition “:math:`\until \bfy^{(t + 1)} = \bfy^{(t)}`” in :math:`(3)`
# implies that :math:`\bfy^*` is the solution to the problem if and only
# if :math:`\bfy^* = \calt (\bfy^*)`, that is \ :math:`\bfy^*` is steady
# under :math:`\calt`. Thus we call :math:`\calt` the *steady-state
# operator*.
#
# Implementing a flood-fill algorithm
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#
# You can implement flood-fill in DGL with the following code.
import mxnet as mx
import os
import dgl
def T(g):
def message_func(edges):
return {'m': edges.src['y']}
def reduce_func(nodes):
# First compute the maximum of all neighbors...
m = mx.nd.max(nodes.mailbox['m'], axis=1)
# Then compare the maximum with the node itself.
# One can also add a self-loop to each node to avoid this
# additional max computation.
m = mx.nd.maximum(m, nodes.data['y'])
return {'y': m.reshape(m.shape[0], 1)}
g.update_all(message_func, reduce_func)
return g.ndata['y']
##############################################################################
# To run the algorithm, create a ``DGLGraph`` as in the example code here, consisting of two
# disjointed chains, each with ten nodes, and initialize it as specified in
# Eq. :math:`(0)` and Eq. :math:`(1)`.
#
import networkx as nx
def disjoint_chains(n_chains, length):
path_graph = nx.path_graph(n_chains * length).to_directed()
for i in range(n_chains - 1): # break the path graph into N chains
path_graph.remove_edge((i + 1) * length - 1, (i + 1) * length)
path_graph.remove_edge((i + 1) * length, (i + 1) * length - 1)
for n in path_graph.nodes:
path_graph.add_edge(n, n) # add self connections
return path_graph
N = 2 # the number of chains
L = 500 # the length of a chain
s = 0 # the source node
# The sampler (see the subgraph sampling section) only supports
# readonly graphs.
g = dgl.DGLGraph(disjoint_chains(N, L), readonly=True)
y = mx.nd.zeros([g.number_of_nodes(), 1])
y[s] = 1
g.ndata['y'] = y
##############################################################################
# Now apply ``T`` to ``g`` until convergence. You can see that nodes
# reachable from ``s`` are gradually infected (marked).
#
while True:
prev_y = g.ndata['y']
next_y = T(g)
if all(prev_y == next_y):
break
##############################################################################
# The update procedure is visualized as follows:
#
# |image0|
#
# Steady-state embedding
# ----------------------
#
# Neural flood-fill algorithm
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~
#
# Next, you can design a neural network that simulates the
# flood-fill algorithm.
#
# Instead of using :math:`\calt` to update the states of nodes, use
# :math:`\calt_\Theta`, a graph neural network (and
# :math:`\hat\calt_\Theta` instead of :math:`\hat\calt`).
# The state of a node :math:`v` is no longer a Boolean value
# (:math:`y_v`), but, an embedding :math:`h_v` (a vector of some
# reasonable dimension, say, :math:`H`).
# You can also associate a feature vector :math:`x_v` with :math:`v`. For
# the flood-fill algorithm, simply use the one-hot encoding of a
# node’s ID as its feature vector, so that our algorithm can
# distinguish different nodes.
# Only iterate :math:`T` times instead of iterating until the
# steady-state condition is satisfied.
# After iteration, mark the nodes by passing the node embedding
# :math:`h_v` into another neural network to produce a probability
# :math:`p_v` of whether the node is reachable.
#
# Mathematically, :math:`\begin{align}
# & h_v^{(0)} \leftarrow \text{random embedding} \\
# & h_v^{(t + 1)} \leftarrow \calt_\Theta (h_1^{(t)}, ..., h_n^{(t)}) \qquad 1 \leq t \leq T \tag{4}
# \end{align}` where
# :math:`[\calt_\Theta (h_1^{(t)}, ..., h_n^{(t)})]_v = \hat\calt_\Theta (x_u, h_u^{(t)} : u \in \caln (v)\})`.
# :math:`\hat\calt_\Theta` is a two layer neural network, as follows:
#
# .. math::
#
#
# \hat\calt_\Theta (\{x_u, h_u^{(t)} : u \in \caln (v)\})
# = W_1 \sigma \left(W_2 \left[x_v, \sum_{u \in \caln (v)} \left[h_v, x_v\right]\right]\right)
#
# where :math:`[\cdot, \cdot]` denotes the concatenation of vectors, and
# :math:`\sigma` is a nonlinearity, e.g. ReLU. Essentially, for every
# node, :math:`\calt_\Theta` repeatedly gathers its neighbors’ feature
# vectors and embeddings, sums them up, and feeds the result along with
# the node’s own feature vector to a two layer neural network.
#
# Implementation of neural flood-fill
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#
# Like the naive algorithm, the neural flood-fill algorithm can be
# partitioned into a ``message_func`` (neighborhood information gathering)
# and a ``reduce_func`` (:math:`\hat\calt_\Theta`). We define
# :math:`\hat\calt_\Theta` as a callable ``gluon.Block`` as in this example code.
#
import mxnet.gluon as gluon
class FullGraphSteadyStateOperator(gluon.Block):
def __init__(self, n_hidden, activation, **kwargs):
super(FullGraphSteadyStateOperator, self).__init__(**kwargs)
with self.name_scope():
self.dense1 = gluon.nn.Dense(n_hidden, activation=activation)
self.dense2 = gluon.nn.Dense(n_hidden)
def forward(self, g):
def message_func(edges):
x = edges.src['x']
h = edges.src['h']
return {'m' : mx.nd.concat(x, h, dim=1)}
def reduce_func(nodes):
m = mx.nd.sum(nodes.mailbox['m'], axis=1)
z = mx.nd.concat(nodes.data['x'], m, dim=1)
return {'h' : self.dense2(self.dense1(z))}
g.update_all(message_func, reduce_func)
return g.ndata['h']
##############################################################################
# In practice, Eq. :math:`(4)` may cause numerical instability. One
# solution is to update :math:`h_v` with a moving average, as follows:
#
# .. math::
#
#
# h_v^{(t + 1)} \leftarrow (1 - \alpha) h_v^{(t)} + \alpha \left[\calt_\Theta (h_0^{(t)}, ..., h_n^{(t)})\right]_v \qquad 0 < \alpha < 1
#
# Putting these together you have:
#
def update_embeddings(g, steady_state_operator):
prev_h = g.ndata['h']
next_h = steady_state_operator(g)
g.ndata['h'] = (1 - alpha) * prev_h + alpha * next_h
##############################################################################
# The last step involves implementing the predictor.
#
class Predictor(gluon.Block):
def __init__(self, n_hidden, activation, **kwargs):
super(Predictor, self).__init__(**kwargs)
with self.name_scope():
self.dense1 = gluon.nn.Dense(n_hidden, activation=activation)
self.dense2 = gluon.nn.Dense(2) ## binary classifier
def forward(self, h):
return self.dense2(self.dense1(h))
##############################################################################
# The predictor’s decision rule is just a decision rule for binary
# classification.
#
# .. math::
#
#
# \hat{y}_v = \text{argmax}_{i \in \{0, 1\}} \left[g_\Phi (h_v^{(T)})\right]_i \tag{5}
#
# where the predictor is denoted by :math:`g_\Phi` and :math:`\hat{y}_v`
# indicates whether the node :math:`v` is marked or not.
#
# Our implementation can be further accelerated using DGL's :mod:`built-in
# functions <dgl.function>`, which maps
# the computation to more efficient sparse operators in the backend
# framework (e.g., MXNet/Gluon, PyTorch). Please see
# the :doc:`Graph convolution network <1_gcn>` tutorial
# for more details.
#
# Efficient semi-supervised learning on graph
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#
# In this setting, you can observe the entire structure of one fixed graph as well
# as the feature vector of each node. However, you might only have access to the
# labels of some (very few) of the nodes. Train the neural
# flood-fill algorithm in this setting as well.
#
# Initialize feature vectors ``'x'`` and node embeddings ``'h'``
# first.
#
import numpy as np
import numpy.random as npr
n = g.number_of_nodes()
n_hidden = 16
g.ndata['x'] = mx.nd.eye(n, n)
g.ndata['y'] = mx.nd.concat(*[i * mx.nd.ones([L, 1], dtype='float32')
for i in range(N)], dim=0)
g.ndata['h'] = mx.nd.zeros([n, n_hidden])
r_train = 0.2 # the ratio of test nodes
n_train = int(r_train * n)
nodes_train = npr.choice(range(n), n_train, replace=True)
test_bitmap = np.ones(shape=(n))
test_bitmap[nodes_train] = 0
nodes_test = np.where(test_bitmap)[0]
##############################################################################
# Unrolling the iterations in Eq. :math:`(4)`, we have the following
# expression for updated node embeddings:
#
# .. math::
#
#
# h_v^{(T)} = \calt_\Theta^T (h_1^{(0)}, ..., h_n^{(0)}) \qquad v \in \calv \tag{6}
#
# where :math:`\calt_\Theta^T` means applying :math:`\calt_\Theta` for
# :math:`T` times. These updated node embeddings are fed to :math:`g_\Phi`
# as in Eq. :math:`(5)`. These steps are fully differentiable and the
# neural flood-fill algorithm can thus be trained in an end-to-end
# fashion. Denoting the binary cross-entropy loss by :math:`l`, you have a
# loss function in the following form:
#
# .. math::
#
#
# \call (\Theta, \Phi) = \frac1{\left|\calv_y\right|} \sum_{v \in \calv_y} l \left(g_\Phi \left(\left[\calt_\Theta^T (h_1^{(0)}, ..., h_n^{(0)})\right]_v \right), y_v\right) \tag{7}
#
# After computing :math:`\call (\Theta, \Phi)`, you can update
# :math:`\Theta` and :math:`\Phi` using the gradients
# :math:`\nabla_\Theta \call (\Theta, \Phi)` and
# :math:`\nabla_\Phi \call (\Theta, \Phi)`. One problem with Eq.
# :math:`(7)` is that computing :math:`\nabla_\Theta \call (\Theta, \Phi)`
# and :math:`\nabla_\Phi \call (\Theta, \Phi)` requires back-propagating
# :math:`T` times through :math:`\calt_\Theta`, which may be slow in
# practice. So, adopt the following steady-state loss function, which
# only incorporates the last node embedding update in back-propagation:
#
# .. math::
#
#
# \call_\text{SteadyState} (\Theta, \Phi) = \frac1{\left|\calv_y\right|} \sum_{v \in \calv_y} l \left(g_\Phi \left(\left[\calt_\Theta (h_1^{(T - 1)}, ..., h_n^{(T - 1)})\right]_v, y_v\right)\right) \tag{8}
#
# The following implements one step of training with
# :math:`\call_\text{SteadyState}`. Note that ``g`` in the following is
# :math:`\calg_y` instead of :math:`\calg`.
#
def fullgraph_update_parameters(g, label_nodes, steady_state_operator, predictor, trainer):
n = g.number_of_nodes()
with mx.autograd.record():
steady_state_operator(g)
z = predictor(g.ndata['h'][label_nodes])
y = g.ndata['y'].reshape(n)[label_nodes] # label
loss = mx.nd.softmax_cross_entropy(z, y)
loss.backward()
trainer.step(n) # divide gradients by the number of labelled nodes
return loss.asnumpy()[0]
##############################################################################
# You are now ready to implement the training procedure, which is in two
# phases.
#
# - The first phase updates node embeddings several times using
# :math:`\calt_\Theta` to attain an approximately steady state
# - The second phase trains :math:`\calt_\Theta` and :math:`g_\Phi` using
# this steady state.
#
# You update the node embeddings of :math:`\calg` instead of
# :math:`\calg_y` only. The reason lies in the semi-supervised learning
# setting. To do inference on :math:`\calg`, you need node embeddings on
# :math:`\calg` instead of on :math:`\calg_y` only.
#
def train(g, label_nodes, steady_state_operator, predictor, trainer):
# first phase
for i in range(n_embedding_updates):
update_embeddings(g, steady_state_operator)
# second phase
for i in range(n_parameter_updates):
loss = fullgraph_update_parameters(g, label_nodes, steady_state_operator,
predictor, trainer)
return loss
##############################################################################
# Scaling up with stochastic subgraph training
# --------------------------------------------
#
# The computation time per update is linear to the number of edges in a
# graph. If we have a gigantic graph with billions of nodes and edges, the
# update function would be inefficient.
#
# A possible improvement draws an analogy from mini-batch training on large
# datasets. Instead of computing gradients on the entire graph, only
# consider some subgraphs randomly sampled from the labelled nodes.
# Mathematically, you have the following loss function:
#
# .. math::
#
#
# \call_\text{StochasticSteadyState} (\Theta, \Phi) = \frac1{\left|\calv_y^{(k)}\right|} \sum_{v \in \calv_y^{(k)}} l \left(g_\Phi \left(\left[\calt_\Theta (h_1, ..., h_n)\right]_v\right), y_v\right)
#
# where :math:`\calv_y^{(k)}` is the subset sampled for iteration
# :math:`k`.
#
# In this training procedure, you also update node embeddings only on
# sampled subgraphs, which is perhaps not surprising if you know
# stochastic fixed-point iteration.
#
# Neighbor sampling
# ~~~~~~~~~~~~~~~~~
#
# You can use *neighbor sampling* as a subgraph sampling strategy. Neighbor
# sampling traverses small neighborhoods from seed nodes with breadth first search. For
# each newly sampled node, a small subset of neighboring nodes are sampled
# and added to the subgraph along with the connecting edges, unless the
# node reaches the maximum of :math:`k` hops from the seeding node.
#
# The following shows neighbor sampling with two seed nodes at a time, a
# maximum of two hops, and a maximum of three neighboring nodes.
#
# |image1|
#
# DGL supports very efficient subgraph sampling natively. This helps users
# scale algorithms to large graphs. Currently, DGL provides the
# :func:`~dgl.contrib.sampling.sampler.NeighborSampler`
# API, which returns a subgraph iterator that samples multiple subgraphs
# at a time with neighbor sampling.
#
# The following code demonstrates how to use the ``NeighborSampler`` to
# sample subgraphs, and stores the seed nodes of the subgraph in each iteration:
#
nx_G = nx.erdos_renyi_graph(36, 0.06)
G = dgl.DGLGraph(nx_G.to_directed(), readonly=True)
sampler = dgl.contrib.sampling.NeighborSampler(
G, 2, 3, num_hops=2, shuffle=True)
seeds = []
for subg in sampler:
seeds.append(subg.layer_parent_nid(-1))
##############################################################################
# Sample training with DGL
# ~~~~~~~~~~~~~~~~
#
# The code illustrates the training process in mini-batches.
#
class SubgraphSteadyStateOperator(gluon.Block):
def __init__(self, n_hidden, activation, **kwargs):
super(SubgraphSteadyStateOperator, self).__init__(**kwargs)
with self.name_scope():
self.dense1 = gluon.nn.Dense(n_hidden, activation=activation)
self.dense2 = gluon.nn.Dense(n_hidden)
def forward(self, subg):
def message_func(edges):
x = edges.src['x']
h = edges.src['h']
return {'m' : mx.nd.concat(x, h, dim=1)}
def reduce_func(nodes):
m = mx.nd.sum(nodes.mailbox['m'], axis=1)
z = mx.nd.concat(nodes.data['x'], m, dim=1)
return {'h' : self.dense2(self.dense1(z))}
subg.block_compute(0, message_func, reduce_func)
return subg.layers[-1].data['h']
def update_parameters_subgraph(subg, steady_state_operator, predictor, trainer):
n = subg.layer_size(-1)
with mx.autograd.record():
steady_state_operator(subg)
z = predictor(subg.layers[-1].data['h'])
y = subg.layers[-1].data['y'].reshape(n) # label
loss = mx.nd.softmax_cross_entropy(z, y)
loss.backward()
trainer.step(n) # divide gradients by the number of labelled nodes
return loss.asnumpy()[0]
def update_embeddings_subgraph(g, steady_state_operator):
# Note that we are only updating the embeddings of seed nodes here.
# The reason is that only the seed nodes have ample information
# from neighbors, especially if the subgraph is small (e.g. 1-hops)
prev_h = g.layers[-1].data['h']
next_h = steady_state_operator(g)
g.layers[-1].data['h'] = (1 - alpha) * prev_h + alpha * next_h
def train_on_subgraphs(g, label_nodes, batch_size,
steady_state_operator, predictor, trainer):
# To train SSE, we create two subgraph samplers with the
# `NeighborSampler` API for each phase.
# The first phase samples from all vertices in the graph.
sampler = dgl.contrib.sampling.NeighborSampler(
g, batch_size, g.number_of_nodes(), num_hops=1)
sampler_iter = iter(sampler)
# The second phase only samples from labeled vertices.
sampler_train = dgl.contrib.sampling.NeighborSampler(
g, batch_size, g.number_of_nodes(), seed_nodes=label_nodes, num_hops=1)
sampler_train_iter = iter(sampler_train)
for i in range(n_embedding_updates):
subg = next(sampler_iter)
# Currently, subgraphing does not copy or share features
# automatically. Therefore, we need to copy the node
# embeddings of the subgraph from the parent graph with
# `copy_from_parent()` before computing...
subg.copy_from_parent()
update_embeddings_subgraph(subg, steady_state_operator)
# ... and copy them back to the parent graph.
g.ndata['h'][subg.layer_parent_nid(-1)] = subg.layers[-1].data['h']
for i in range(n_parameter_updates):
try:
subg = next(sampler_train_iter)
except:
break
# Again we need to copy features from parent graph
subg.copy_from_parent()
loss = update_parameters_subgraph(subg, steady_state_operator, predictor, trainer)
# We don't need to copy the features back to parent graph.
return loss
##############################################################################
# You can also define a helper function that reports prediction accuracy.
def test(g, test_nodes, predictor):
z = predictor(g.ndata['h'][test_nodes])
y_bar = mx.nd.argmax(z, axis=1)
y = g.ndata['y'].reshape(n)[test_nodes]
accuracy = mx.nd.sum(y_bar == y) / len(test_nodes)
return accuracy.asnumpy()[0], z
##############################################################################
# Some routine preparations for training.
#
lr = 1e-3
activation = 'relu'
subgraph_steady_state_operator = SubgraphSteadyStateOperator(n_hidden, activation)
predictor = Predictor(n_hidden, activation)
subgraph_steady_state_operator.initialize()
predictor.initialize()
params = subgraph_steady_state_operator.collect_params()
params.update(predictor.collect_params())
trainer = gluon.Trainer(params, 'adam', {'learning_rate' : lr})
##############################################################################
# Now train it. As before, nodes reachable from :math:`s` are
# gradually infected, except that in the background is a neural network.
#
n_epochs = 35
n_embedding_updates = 8
n_parameter_updates = 5
alpha = 0.1
batch_size = 64
y_bars = []
for i in range(n_epochs):
loss = train_on_subgraphs(g, nodes_train, batch_size, subgraph_steady_state_operator,
predictor, trainer)
accuracy_train, _ = test(g, nodes_train, predictor)
accuracy_test, z = test(g, nodes_test, predictor)
print("Iter {:05d} | Train acc {:.4} | Test acc {:.4f}".format(i, accuracy_train, accuracy_test))
y_bar = mx.nd.argmax(z, axis=1)
y_bars.append(y_bar)
##############################################################################
# |image2|
#
# In this tutorial, you used a very small example graph to demonstrate the
# subgraph training for easy visualization. Subgraph training actually
# helps you scale to gigantic graphs. For instance,
# scaling SSE to a graph with 50 million nodes and 150 million edges in a
# single P3.8x large instance, and one epoch, only takes about 160 seconds.
#
# For full examples, see `Benchmark SSE on multi-GPUs <https://github.com/dmlc/dgl/tree/master/examples/mxnet/sse>`_ on Github.
#
# .. |image0| image:: https://s3.us-east-2.amazonaws.com/dgl.ai/tutorial/img/floodfill-paths.gif
# .. |image1| image:: https://s3.us-east-2.amazonaws.com/dgl.ai/tutorial/img/neighbor-sampling.gif
# .. |image2| image:: https://s3.us-east-2.amazonaws.com/dgl.ai/tutorial/img/sse.gif