Memory-Efficient Aggregations
=============================

The :class:`~torch_geometric.nn.conv.message_passing.MessagePassing` interface of :pyg:`PyG` relies on a gather-scatter scheme to aggregate messages from neighboring nodes.
For example, consider the message passing layer

.. math::

    \mathbf{x}^{\prime}_i = \sum_{j \in \mathcal{N}(i)} \textrm{MLP}(\mathbf{x}_j - \mathbf{x}_i),

that can be implemented as:

.. code-block:: python

    from torch_geometric.nn import MessagePassing

    x = ...           # Node features of shape [num_nodes, num_features]
    edge_index = ...  # Edge indices of shape [2, num_edges]

    class MyConv(MessagePassing):
        def __init__(self):
            super().__init__(aggr="add")

        def forward(self, x, edge_index):
            return self.propagate(edge_index, x=x)

        def message(self, x_i, x_j):
            return MLP(x_j - x_i)

Under the hood, the :class:`~torch_geometric.nn.conv.message_passing.MessagePassing` implementation produces a code that looks as follows:

.. code-block:: python

    from torch_geometric.utils import scatter

    x = ...           # Node features of shape [num_nodes, num_features]
    edge_index = ...  # Edge indices of shape [2, num_edges]

    x_j = x[edge_index[0]]  # Source node features [num_edges, num_features]
    x_i = x[edge_index[1]]  # Target node features [num_edges, num_features]

    msg = MLP(x_j - x_i)  # Compute message for each edge

    # Aggregate messages based on target node indices
    out = scatter(msg, edge_index[1], dim=0, dim_size=x.size(0), reduce='sum')

While the gather-scatter formulation generalizes to a lot of useful GNN implementations, it has the disadvantage of explicitly materalizing :obj:`x_j` and :obj:`x_i`, resulting in a high memory footprint on large and dense graphs.

Luckily, not all GNNs need to be implemented by explicitly materalizing :obj:`x_j` and/or :obj:`x_i`.
In some cases, GNNs can also be implemented as a simple-sparse matrix multiplication.
As a general rule of thumb, this holds true for GNNs that do not make use of the central node features :obj:`x_i` or multi-dimensional edge features when computing messages.
For example, the :class:`~torch_geometric.nn.conv.GINConv` layer

.. math::

    \mathbf{x}^{\prime}_i = \textrm{MLP} \left( (1 + \epsilon) \cdot \mathbf{x}_i + \sum_{j \in \mathcal{N}(i)} \mathbf{x}_j \right),

is equivalent to computing

.. math::

    \mathbf{X}^{\prime} = \textrm{MLP} \left( (1 + \epsilon) \cdot \mathbf{X} + \mathbf{A}\mathbf{X} \right),

where :math:`\mathbf{A}` denotes a sparse adjacency matrix of shape :obj:`[num_nodes, num_nodes]`.
This formulation allows to leverage dedicated and fast sparse-matrix multiplication implementations.

In :pyg:`null` **PyG >= 1.6.0**, we officially introduce better support for sparse-matrix multiplication GNNs, resulting in a **lower memory footprint** and a **faster execution time**.
As a result, we introduce the :class:`SparseTensor` class (from the :obj:`torch_sparse` package), which implements fast forward and backward passes for sparse-matrix multiplication based on the `"Design Principles for Sparse Matrix Multiplication on the GPU" <https://arxiv.org/abs/1803.08601>`_ paper.

Using the :class:`SparseTensor` class is straightforward and similar to the way :obj:`scipy` treats sparse matrices:

.. code-block:: python

    from torch_sparse import SparseTensor

    adj = SparseTensor(row=edge_index[0], col=edge_index[1], value=...,
                       sparse_sizes=(num_nodes, num_nodes))
    # value is optional and can be None

    # Obtain different representations (COO, CSR, CSC):
    row,    col, value = adj.coo()
    rowptr, col, value = adj.csr()
    colptr, row, value = adj.csc()

    adj = adj[:100, :100]  # Slicing, indexing and masking support
    adj = adj.set_diag()   # Add diagonal entries
    adj_t = adj.t()        # Transpose
    out = adj.matmul(x)    # Sparse-dense matrix multiplication
    adj = adj.matmul(adj)  # Sparse-sparse matrix multiplication

    # Creating SparseTensor instances:
    adj = SparseTensor.from_dense(mat)
    adj = SparseTensor.eye(100, 100)
    adj = SparseTensor.from_scipy(mat)

Our :class:`~torch_geometric.nn.conv.message_passing.MessagePassing` interface can handle both :obj:`torch.Tensor` and :class:`SparseTensor` as input for propagating messages.
However, when holding a directed graph in :class:`SparseTensor`, you need to make sure to input the **transposed sparse matrix** to :func:`~torch_geometric.nn.conv.message_passing.MessagePassing.propagate`:

.. code-block:: python

    conv = GCNConv(16, 32)
    out1 = conv(x, edge_index)
    out2 = conv(x, adj.t())
    assert torch.allclose(out1, out2)

    conv = GINConv(nn=Sequential(Linear(16, 32), ReLU(), Linear(32, 32)))
    out1 = conv(x, edge_index)
    out2 = conv(x, adj.t())
    assert torch.allclose(out1, out2)

To leverage sparse-matrix multiplications, the :class:`~torch_geometric.nn.conv.message_passing.MessagePassing` interface introduces the :func:`~torch_geometric.nn.conv.message_passing.message_and_aggregate` function (which fuses the :func:`~torch_geometric.nn.conv.message_passing.message` and :func:`~torch_geometric.nn.conv.message_passing.aggregate` functions into a single computation step), which gets called whenever it is implemented and receives a :class:`SparseTensor` as input for :obj:`edge_index`.
With it, the :class:`~torch_geometric.nn.conv.GINConv` layer can now be implemented as follows:

.. code-block:: python

    import torch_sparse

    class GINConv(MessagePassing):
        def __init__(self):
            super().__init__(aggr="add")

        def forward(self, x, edge_index):
            out = self.propagate(edge_index, x=x)
            return MLP((1 + eps) x + out)

        def message(self, x_j):
            return x_j

        def message_and_aggregate(self, adj_t, x):
            return torch_sparse.matmul(adj_t, x, reduce=self.aggr)

Playing around with the new :class:`SparseTensor` format is straightforward since all of our GNNs work with it out-of-the-box.
To convert the :obj:`edge_index` format to the newly introduced :class:`SparseTensor` format, you can make use of the :class:`torch_geometric.transforms.ToSparseTensor` transform:

.. code-block:: python

    import torch
    import torch.nn.functional as F

    from torch_geometric.nn import GCNConv
    import torch_geometric.transforms as T
    from torch_geometric.datasets import Planetoid

    dataset = Planetoid("Planetoid", name="Cora", transform=T.ToSparseTensor())
    data = dataset[0]
    >>> Data(adj_t=[2708, 2708, nnz=10556], x=[2708, 1433], y=[2708], ...)


    class GNN(torch.nn.Module):
        def __init__(self):
            super().__init__()
            self.conv1 = GCNConv(dataset.num_features, 16, cached=True)
            self.conv2 = GCNConv(16, dataset.num_classes, cached=True)

        def forward(self, x, adj_t):
            x = self.conv1(x, adj_t)
            x = F.relu(x)
            x = self.conv2(x, adj_t)
            return F.log_softmax(x, dim=1)

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

    def train(data):
        model.train()
        optimizer.zero_grad()
        out = model(data.x, data.adj_t)
        loss = F.nll_loss(out, data.y)
        loss.backward()
        optimizer.step()
        return float(loss)

    for epoch in range(1, 201):
        loss = train(data)

All code remains the same as before, except for the :obj:`data` transform via :obj:`T.ToSparseTensor()`.
As an additional advantage, :class:`~torch_geometric.nn.conv.message_passing.MessagePassing` implementations that utilize the :class:`SparseTensor` class are deterministic on the GPU since aggregations no longer rely on atomic operations.

Notably, the GNN layer execution slightly changes in case GNNs incorporate single or multi-dimensional edge information :obj:`edge_weight` or :obj:`edge_attr` into their message passing formulation, respectively.
In particular, it is now expected that these attributes are directly added as values to the :class:`SparseTensor` object.
Instead of calling the GNN as

.. code-block:: python

    conv = GMMConv(16, 32, dim=3)
    out = conv(x, edge_index, edge_attr)

we now execute our GNN operator as

.. code-block:: python

    conv = GMMConv(16, 32, dim=3)
    adj = SparseTensor(row=edge_index[0], col=edge_index[1], value=edge_attr)
    out = conv(x, adj.t())

.. note::

    Since this feature is still experimental, some operations, *e.g.*, graph pooling methods, may still require you to input the :obj:`edge_index` format.
    You can convert :obj:`adj_t` back to :obj:`(edge_index, edge_attr)` via:

    .. code-block:: python

        row, col, edge_attr = adj_t.t().coo()
        edge_index = torch.stack([row, col], dim=0)

Please let us know what you think of :class:`SparseTensor`, how we can improve it, and whenever you encounter any unexpected behavior.
