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3D Tensor Parallelism

Author: Zhengda Bian, Yongbin Li


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The 3D tensor parallelism 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=XAY = XA as an example. Given P=q×q×qP=q \times q \times q processors (necessary condition), e.g. q=2q=2, we split the input XX and weight AA into

[X000X001X010X011X100X101X110X111] and [A000A001A010A011A100A101A110A111] respectively,\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 XijlX_{ijl} and AljiA_{lji} are stored at processor (i,j,l)(i,j,l), as shown in the figure below.

Then we all-gather XijlX_{ijl} across (i,0...q,l)(i, 0...q,l), as well as AljiA_{lji} across (0...q,j,l)(0...q, j, l). So, we have XilX_{il} and AljA_{lj} on each processor (i,j,l)(i,j,l) to get XilAljX_{il}A_{lj}. Finally, we reduce-scatter the results across (i,j,0...q)(i, j, 0...q) to get YijlY_{ijl}, which forms

Y=[Y000Y001Y010Y011Y100Y101Y110Y111].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 Yijl˙\dot{Y_{ijl}}, and then reduce-scatter the gradient Xil˙=Yij˙AljT\dot{X_{il}}=\dot{Y_{ij}}A_{lj}^T and Alj˙=XilTYij˙\dot{A_{lj}}=X_{il}^T\dot{Y_{ij}}.


Given P=q×q×qP=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.

ComputationMemory (parameters)Memory (activations)Communication (bandwidth)Communication (latency)


Currently the newest version of ColossalAI doesn't support 3D tensor parallelism, but this feature will be integrated into Shardformer in future releases. For more details about ideas and usages of Shardformer, please refer to Shardformer Doc.

For users of older version of ColossalAI, please refer to ColossalAI-Examples - 3D Tensor Parallelism.