Scaling GNNs via Neighbor Sampling
==================================

One of the challenges of Graph Neural Networks is to scale them to large graphs, *e.g.*, in industrial and social applications.
Traditional deep neural networks are known to scale well to large amounts of data by decomposing the training loss into individual samples (called a *mini-batch*) and approximating exact gradients stochastically.
In contrast, applying stochastic mini-batch training in GNNs is challenging since the embedding of a given node depends recursively on all its neighbor’s embeddings, leading to high inter-dependency between nodes that grows exponentially with respect to the number of layers.
This phenomenon is often referred to as *neighbor explosion*.
As a simple workaround, GNNs are typically executed in a full-batch fashion (see `here <https://github.com/pyg-team/pytorch_geometric/blob/master/examples/gcn.py>`_ for an example), where the GNN has access to all hidden node representations in all its layers.
However, this is not feasible in large-scale graphs due to memory limitations and slow convergence.

Scalability techniques are indispensable for applying GNNs to large-scale graphs in order to alleviate the neighbor explosion problem induced by mini-batch training, *i.e.* **node-wise**, **layer-wise** or **subgraph-wise** sampling techniques, or to **decouple propagations from predictions**.
In this tutorial, we take a closer look at the most common node-wise sampling approach, originally introduced in the `"Inductive Representation Learning on Large Graphs" <https://arxiv.org/abs/1706.02216>`_ paper.

Neighbor Sampling
-----------------

:pyg:`PyG` implements neighbor sampling via its :class:`torch_geometric.loader.NeighborLoader` class.
Neighbor sampling works by recursively sampling a fixed number of at most :math:`k` neighbors for a node :math:`v \in \mathcal{V}`, *i.e.* :math:`\tilde{\mathcal{N}}(v) \subset \mathcal{N}(v)` with :math:`|\tilde{\mathcal{N}}| \le k`, leading to an overall bounded :math:`L`-hop neighborhood size of :math:`\mathcal{O}(k^L)`.
That is, starting from a set of seed nodes :math:`\mathcal{B} \subset \mathcal{V}`, we sample at most :math:`k` neighbors for every node in :math:`v \in \mathcal{B}`, and then proceed to sample neighbors for every sampled node in the previous hop, and so on.
The resulting graph structure holds a **directed** :math:`L`-hop subgraph around every node in :math:`v \in \mathcal{B}`, for which it is guaranteed that every node has at least one path of at most length :math:`L` to at least one of the seed nodes in :math:`\mathcal{B}`.
As such, a message passing GNN with :math:`L` layers will incorporate the full set of sampled nodes in its computation graph.

.. figure:: ../_static/thumbnails/neighbor_loader.png
  :align: center
  :width: 40%

|

It is important to note that neighbor sampling can only mitigate the neighbor explosion problem to some extend since the overall neighborhood size still increases exponentially with the number of layers.
As a result, sampling for more than two or three iterations is generally not feasible.

Often times, the number of sampled hops and the number of message passing layers is kept in sync.
Specifically, it is very wasteful to sample for more hops than there exist message passing layers since the GNN will never be able to incorporate the features of the nodes sampled in later hops into the final node representation of its seed nodes.
However, it is nonetheless possible to utilize deeper GNNs, but one needs to be careful to convert the sampled subgraph into a bidirectional variant to ensure correct message passing flow.
:pyg:`PyG` provides support for this via an additional argument in :class:`~torch_geometric.loader.NeighborLoader`, while other mini-batch techniques are designed for this use-case out-of-the-box, *e.g.*, :class:`~torch_geometric.loader.ClusterLoader`, :class:`~torch_geometric.loader.GraphSAINTSampler` and :class:`~torch_geometric.loader.ShaDowKHopSampler`.

Basic Usage
-----------

.. note::

    In this section of the tutorial, we will learn how to utilize the :class:`~torch_geometric.nn.models.Node2Vec` class of :pyg:`PyG` to train GNNs on single graphs in a mini-batch fashion.
    A fully working example on large-scale real-world data is available in `examples/reddit.py <https://github.com/pyg-team/pytorch_geometric/blob/master/examples/reddit.py>`_.

The :class:`~torch_geometric.loader.NeighborLoader` is initialized from a :pyg:`PyG` :class:`~torch_geometric.data.Data` or :class:`~torch_geometric.data.HeteroData` object and defines how sampling should be performed:

* :obj:`input_nodes` defines the set of seed nodes from which we want to start sampling from.
* :obj:`num_neighbors` defines the number of neighbors to sample for each node in each hop.
* :obj:`batch_size` defines the size of seed nodes we want to consider at once.
* :obj:`replace` defines whether to sample with or without replacement.
* :obj:`shuffle` defines whether seed nodes should be shuffled at every epoch.

.. code-block:: python

    import torch
    from torch_geometric.data import Data
    from torch_geometric.loader import NeighborLoader

    x = torch.randn(8, 32)  # Node features of shape [num_nodes, num_features]
    y = torch.randint(0, 4, (8, ))  # Node labels of shape [num_nodes]
    edge_index = torch.tensor([
        [2, 3, 3, 4, 5, 6, 7],
        [0, 0, 1, 1, 2, 3, 4]],
    )

    #   0  1
    #  / \/ \
    # 2  3  4
    # |  |  |
    # 5  6  7

    data = Data(x=x, y=y, edge_index=edge_index)

    loader = NeighborLoader(
        data,
        input_nodes=torch.tensor([0, 1]),
        num_neighbors=[2, 1],
        batch_size=1,
        replace=False,
        shuffle=False,
    )

Here, we initialize the :class:`~torch_geometric.loader.NeigborLoader` to sample subgraphs for the first two nodes, where we want to sample 2 neighbors in the first hop, and 1 neighbor in the second hop.
Our :obj:`batch_size` is set to :obj:`1`, such that :obj:`input_nodes` will be split into chunks of size :obj:`1`.

In the execution of :class:`~torch_geometric.loader.NeighborLoader`, we expect that the seed node :obj:`0` samples nodes :obj:`2` and :obj:`3` in the first hop. In the second hop, node :obj:`2` samples node :obj:`5`, and node :obj:`3` samples node :obj:`6`.
Let's confirm by looking at the output of the :obj:`loader`:

.. code-block:: python

    batch = next(iter(loader))

    batch.edge_index
    >>> tensor([[1, 2, 3, 4],
                [0, 0, 1, 2]])

     batch.n_id
     >>> tensor([0, 2, 3, 5, 6])

     batch.batch_size
     >>> 1

The :class:`~torch_geometric.loader.NeighborLoader` will return a :class:`~torch_geometric.data.Data` object, which contains the following attributes:

* :obj:`batch.edge_index` contain the edge indices of the subgraph.
* :obj:`batch.n_id` contains the original node indices of all the sampled nodes.
* :obj:`batch.batch_size` contains the number of seed nodes/the batch size.

In addition, node and edge features will be filtered to only contain the features of sampled nodes/edges, respectively.

Importantly, :obj:`batch.edge_index` contains the sampled subgraph with relabeled node indices, such that its indices range from :obj:`0` to :obj:`batch.num_nodes - 1`.
If you want to reconstruct the original node indices of :obj:`batch.edge_index`, do:

.. code-block:: python

    batch.n_id[batch.edge_index]
    >>> tensor([[2, 3, 5, 6],
                [0, 0, 2, 3]])

Furthermore, while :class:`~torch_geometric.loader.NeighborLoader` starts sampling *from* seed nodes, the resulting subgraph will hold edges that point *to* the seed nodes.
This aligns well with the default :pyg:`PyG` message passing flow from source to destination nodes.

Lastly, nodes in the output of :class:`~torch_geometric.loader.NeighborLoader` are guaranteed to be sorted.
In particular, the first :obj:`batch_size` sampled nodes will exactly match with the seed nodes that were used for sampling:

.. code-block:: python

    batch.n_id[:batch.batch_size]
    >>> tensor([0])

Afterwards, we can use :class:`~torch_geometric.loader.NeighborLoader` as a data loading routine to train GNNs on large-scale graphs in mini-batch fashion.
For this, let's create a simple two-layer :class:`~torch_geometric.nn.models.GraphSAGE` model:

.. code-block:: python

    from torch_geometric.nn import GraphSAGE

    device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')

    model = GraphSAGE(
        in_channels=32,
        hidden_channels=64,
        out_channels=4,
        num_layers=2
    ).to(device)

    optimizer = torch.optim.Adam(model.parameters(), lr=0.01)

We can now combine the :obj:`loader` and :obj:`model` to define our training routine:

.. code-block:: python

    import torch.nn.functional as F

    for batch in loader:
        optimizer.zero_grad()
        batch = batch.to(device)
        out = model(batch.x, batch.edge_index)

        # NOTE Only consider predictions and labels of seed nodes:
        y = batch.y[:batch.batch_size]
        out = out[:batch.batch_size]

        loss = F.cross_entropy(out, y)
        loss.backward()
        optimizer.step()

The training loop follows a similar design to any other :pytorch:`PyTorch` training loop.
The only important difference is that by default the model will output a matrix of shape :obj:`[batch.num_nodes, *]`, while we are only interested in the predictions of the seed nodes.
As such, we can use efficient slicing both on the node predictions and the ground-truth information :obj:`batch.y` to only obtain predictions and ground-truth information of actual seed nodes.
This ensures that we are only making use of the first :obj:`batch_size` many nodes for loss and metric computation.

Hierarchical Extension
----------------------

A drawback of :class:`~torch_geometric.loader.Neighborloader` is that it computes a representations for *all* sampled nodes at *all* depths of the network.
However, nodes sampled in later hops no longer contribute to the node representations of seed nodes in later GNN layers, thus performing useless computation.
:class:`~torch_geometric.loader.NeighborLoader` will be marginally slower since we are computing node embeddings for nodes we no longer need.
This is a trade-off we make to obtain a clean, modular and experimental-friendly GNN design, which does not tie the definition of the model to its utilized data loader routine.
The `Hierarchical Neighborhood Sampling <../advanced/hgam.html>`__ tutorial shows how to eliminate this overhead and speed up training and inference in mini-batch GNNs further.

Advanced Options
----------------

:class:`~torch_geometric.loader.NeighborLoader` provides many more features for advanced usage.
In particular,

* :class:`~torch_geometric.loader.NeighborLoader` supports both sampling on homogeneous and heterogeneous graphs out-of-the-box.
  For sampling on heterogeneous graphs, simply initialize it with a :class:`~torch_geometric.data.HeteroData` object.
  Sampling on heterogeneous graphs via :class:`~torch_geometric.loader.NeighborLoader` allows for fine-granular control of sampling parameters, *e.g.*, it allows to specify the number of neighbors to sample for each edge type individually.
  Take a look at the `Heterogeneous Graph Learning <../advanced/heterogeneous.html>`__ tutorial for additional information.

* By default, :class:`~torch_geometric.loader.NeighborLoader` fuses sampled nodes across different seed nodes into a single subgraph.
  This way, shared neighbors of seed nodes will not be duplicated in the resulting subgraph and hence save memory.
  You can disable this behavior by passing the :obj:`disjoint=True` option to the :class:`~torch_geometric.loader.NeighborLoader`.

* By default, the subgraphs returned from :class:`~torch_geometric.loader.NeighborLoader` will be **directed**, which restricts its use to GNNs with equal depth to the number of sampling hops.
  If you want to utilize deeper GNNs, specify the :obj:`subgraph_type` option.
  If set to :obj:`"bidirectional"`, sampled edges are converted to bidirectional edges.
  If set to :obj:`"induced"`, the returned subgraph will contain the induced subgraph of all sampled nodes.

* :class:`~torch_geometric.loader.NeighborLoader` is designed to perform sampling from individual seed nodes.
  As such, it is not directly applicable in a link prediction scenario.
  For this use-cases, we developed the :class:`~torch_geometric.loader.LinkNeighborLoader`, which expects a set of input edges, and will return subgraphs that were created via neighbor sampling from both source and destination nodes.
