Version: v0.1.6

# 1D Tensor Parallelism

Author: Zhengda Bian, Yongbin Li

Prerequisite

Example Code

Related Paper

## Introduction​

Tensor parallelism partitions model weights across multiple devices in order to reduce memory load. An efficient 1D tensor parallelism implementation was introduced by Megatron-LM.

Let's take a linear layer as an example, which consists of a GEMM $Y = XA$. Given 2 processors, we split the columns of $A$ into $[A_1 ~ A_2]$, and calculate $Y_i = XA_i$ on each processor, which then forms $[Y_1 ~ Y_2] = [XA_1 ~ XA_2]$. This is called a column-parallel fashion.

When a second linear layer $Z=YB$ follows the column-parallel one, we split $B$ into $\left[\begin{matrix} B_1 \\ B_2 \end{matrix} \right]$, which is called a row-parallel fashion. To calculate $Z = [Y_1 ~ Y_2] \left[\begin{matrix} B_1 \\ B_2 \end{matrix} \right]$, we first calculate $Y_iB_i$ on each processor, then use an all-reduce to aggregate the results as $Z=Y_1B_1+Y_2B_2$.

We also need to note that in the backward pass, the column-parallel linear layer needs to aggregate the gradients of the input tensor $X$, because on each processor $i$ we only have $\dot{X_i}=\dot{Y_i}A_i^T$. Thus, we apply an all-reduce across the processors to get $\dot{X}=\dot{Y}A^T=\dot{Y_1}A_1^T+\dot{Y_2}A_2^T$.

## Efficiency​

Given $P$ 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 1D tensor parallelism.

ComputationMemory (parameters)Memory (activations)Communication (bandwidth)Communication (latency)
$O(1/P)$$O(1/P)$$O(1)$$O(2(P-1)/P)$$O(2(P-1))$

## Usage​

To enable 1D tensor parallelism for our model, e.g. on 2 GPUs, we need to configure the parallism setting as below.

CONFIG = dict(parallel=dict(    data=1,    pipeline=1,    tensor=dict(size=2, mode='1d'),))

Then Colossal-AI will automatically apply 1D 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.

import colossalaiimport colossalai.nn as col_nnimport torchfrom colossalai.utils import print_rank_0class 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.transpose(0, 1).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.transpose(0, 1).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 2 GPUs and build the model.

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.

Weight of the first linear layer: torch.Size([256, 512])Weight of the second linear layer: torch.Size([512, 256])

The complete weight of the first linear layer is supposed to have the shape [256, 1024]. After the column-parallel partitioning, it becomes [256, 512]. Similarly, the second row-parallel layer partitions the weight [1024, 256] into [512, 256].

We can run the model with some random inputs.

from colossalai.utils import get_current_devicex = torch.randn((16, 256), device=get_current_device())torch.distributed.broadcast(x, src=0)  # synchronize inputx = m(x)

Then we can see the shapes of activation results.

Output of the first linear layer: torch.Size([16, 512])Output of the second linear layer: torch.Size([16, 256])

The output of the first linear layer is split into 2 partitions (each has the shape [16, 512]), while the second layer has identical outputs across the GPUs.