- [Maximizing Parallelism in Distributed Training for Huge Neural Networks](https://arxiv.org/pdf/2105.14450.pdf)
## Introduction
The [3D tensor parallelism](https://arxiv.org/pdf/2105.14450.pdf) is an approach to parallelize the computation of neural models, hoping to obtain the optimal communication cost.
Let's still take a linear layer $Y = XA$ as an example.
Given $P=q \times q \times q$ processors (necessary condition), e.g. $q=2$, we split the input $X$ and weight $A$ into
Then we all-gather $X_{ijl}$ across $(i, 0...q,l)$, as well as $A_{lji}$ across $(0...q, j, l)$.
So, we have $X_{il}$ and $A_{lj}$ on each processor $(i,j,l)$ to get $X_{il}A_{lj}$.
Finally, we reduce-scatter the results across $(i, j, 0...q)$ to get $Y_{ijl}$, which forms
$$
Y=
\left[\begin{matrix}
Y_{000} & Y_{001} \\
Y_{010} & Y_{011} \\
Y_{100} & Y_{101} \\
Y_{110} & Y_{111} \end{matrix}
\right].
$$
We also need to note that in the backward pass, we need to all-gather the gradient $\dot{Y_{ijl}}$, and then reduce-scatter the gradient $\dot{X_{il}}=\dot{Y_{ij}}A_{lj}^T$ and $\dot{A_{lj}}=X_{il}^T\dot{Y_{ij}}$.
## Efficiency
Given $P=q \times q \times q$ processors, we present the theoretical computation and memory cost, as well as the communication cost based on the ring algorithm in both the forward and backward pass of 3D tensor parallelism.
| Computation | Memory (parameters) | Memory (activations) | Communication (bandwidth) | Communication (latency) |
print_rank_0(f'Weight of the second linear layer: {self.dense_2.weight.shape}')
self.dropout = col_nn.Dropout(0.1)
def forward(self, x):
x = self.dense_1(x)
print_rank_0(f'Output of the first linear layer: {x.shape}')
x = self.activation(x)
x = self.dense_2(x)
print_rank_0(f'Output of the second linear layer: {x.shape}')
x = self.dropout(x)
return x
```
Launch Colossal-AI on 8 GPUs and build the model
```python
parser = colossalai.get_default_parser()
colossalai.launch(config=CONFIG,
rank=args.rank,
world_size=args.world_size,
local_rank=args.local_rank,
host=args.host,
port=args.port)
m = MLP()
```
We will see the shapes of partitioned parameters(e.g. weights) in the MLP model.
```shell
Weight of the first linear layer: torch.Size([128, 256])
Weight of the second linear layer: torch.Size([512, 64])
```
The complete weight of the first linear layer is supposed to have the shape `[256, 1024]`. After the partitioning of 3D parallelism, it becomes `[128, 256]` on each GPU.
Similarly, the second layer partitions the weight `[1024, 256]` into `[512, 64]`.
We can run the model with some random inputs.
```python
from colossalai.context import ParallelMode
from colossalai.core import global_context as gpc
from colossalai.utils import get_current_device
x = torch.randn((16, 256), device=get_current_device())
# partition input
torch.distributed.broadcast(x, src=0)
x = torch.chunk(x, 2, dim=0)[gpc.get_local_rank(ParallelMode.PARALLEL_3D_WEIGHT)]
x = torch.chunk(x, 2, dim=0)[gpc.get_local_rank(ParallelMode.PARALLEL_3D_INPUT)]
x = torch.chunk(x, 2, dim=-1)[gpc.get_local_rank(ParallelMode.PARALLEL_3D_OUTPUT)]
print_rank_0(f'Input: {x.shape}')
x = m(x)
```
Then we can see the shapes of activation results.
```shell
Input: torch.Size([4, 128])
Output of the first linear layer: torch.Size([4, 512])
Output of the second linear layer: torch.Size([4, 128])
```
The activation tensors in 3D parallelism are all split by $q^2$ in the row and $q$ in the column.
E.g. the output of the first linear layer has the shape `[4, 512]`), while the second layer has the output of `[4, 128]`.
Note, although the results of 3D parallelism have the same shape as that of 2.5D parallelism for weights here, the content of each partition is different.
