# 3D Tensor Parallelism Author: Zhengda Bian, Yongbin Li **Prerequisite** - [Define Your Configuration](../basics/define_your_config.md) - [Configure Parallelization](../basics/configure_parallelization.md) - [1D Tensor Parallelism](./1D_tensor_parallel.md) - [2D Tensor Parallelism](./2D_tensor_parallel.md) **Example Code** - [ColossalAI-Examples - 3D Tensor Parallelism](https://github.com/hpcaitech/ColossalAI-Examples/blob/main/features/tensor_parallel/README.md) **Related Paper** - [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 $$ \left[\begin{matrix} X_{000} & X_{001} \\ X_{010} & X_{011} \\ X_{100} & X_{101} \\ X_{110} & X_{111} \end{matrix} \right] \text{~and~} \left[\begin{matrix} A_{000} & A_{001} & A_{010} & A_{011} \\ A_{100} & A_{101} & A_{110} & A_{111} \end{matrix} \right] \text{~respectively,}$$ where each $X_{ijl}$ and $A_{lji}$ are stored at processor $(i,j,l)$, as shown in the figure below.
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) | | :-: | :-: | :-: | :-: | :-: | | $O(1/q^3)$ | $O(1/q^3)$ | $O(1/q^3)$ | $O(6(q-1)/q^3)$ | $O(6(q-1))$ | ## Usage To enable 3D tensor parallelism for our model, e.g. on 8 GPUs, we need to configure the parallelism setting as below. ```python CONFIG = dict(parallel=dict( data=1, pipeline=1, tensor=dict(size=8, mode='3d'), )) ``` Then Colossal-AI will automatically apply 3D parallelism to all the layers from `colossalai.nn`. Let's define a model that consists of a two-layer multi-layer perceptron (MLP) as below. ```python import colossalai import colossalai.nn as col_nn import torch from colossalai.utils import print_rank_0 class MLP(torch.nn.Module): def __init__(self, dim: int = 256): super().__init__() intermediate_dim = dim * 4 self.dense_1 = col_nn.Linear(dim, intermediate_dim) print_rank_0(f'Weight of the first linear layer: {self.dense_1.weight.shape}') self.activation = torch.nn.GELU() self.dense_2 = col_nn.Linear(intermediate_dim, dim) 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.