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1D Tensor Parallelism
Author: Zhengda Bian, Yongbin Li
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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.
| Computation | Memory (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
1D tensor parallelism is implemented by Shardformer feature in the newest version of ColossalAI.
For more details about ideas and usages of Shardformer, please refer to Shardformer Doc.