Proofs for Parallelization Strategies

This appendix provides a detailed mathematical analysis of the parallelization strategies discussed in the main text. We will delve into the theoretical underpinnings of Model Parallelism, Pipeline Parallelism, and Tensor Parallelism, providing proofs for their efficacy under certain conditions.

Model Parallelism

Model parallelism involves executing different parts of a model on separate GPUs. The goal is to balance the computational load and minimize inter-GPU communication.

Proof of Load Balance:

Let $L$ be the total number of layers in an LLM, and $G$ be the number of GPUs available. When using model parallelism, the layers are distributed such that each GPU $g$ gets $\frac{L}{G}$ layers. The load balance can be mathematically expressed as:

$\left| \sum_{i \in GPU_g} C_i - \frac{1}{G} \sum_{i=1}^{L} C_i \right| \leq \epsilon$

Where $C_i$ is the computational complexity of layer $i$, and $\epsilon$ is a small constant representing the allowable imbalance.

Pipeline Parallelism

Pipeline parallelism processes multiple instances of the model simultaneously, with each instance going through different stages of the pipeline.

Proof of Increased Throughput:

Consider $P$ parallel instances of an LLM, each with $S$ stages. The throughput $T$ is given by:

$T = \frac{P \times S}{\text{Total time per instance}}$

Assuming that the stages are perfectly balanced, the total time per instance is the time of the longest stage. If we denote the time taken by the longest stage as $s_{max}$, the throughput can be simplified to:

$T = \frac{P \times S}{s_{max}}$

This shows that the throughput is directly proportional to the number of parallel instances and stages.

Tensor Parallelism

Tensor parallelism involves splitting the input tensors across multiple GPUs, reducing the memory footprint on each GPU.

Proof of Memory Reduction:

Let $T$ be a tensor of size $D \times N$ that needs to be processed by an LLM. When split across $G$ GPUs using tensor parallelism, each GPU processes a sub-tensor $T_g$ of size $\frac{D}{G} \times N$. The total memory required before and after splitting is:

$\text{Memory}_{\text{before}} = D \times N$

$\text{Memory}_{\text{after}} = G \times \left( \frac{D}{G} \times N \right) = D \times N$

Despite the total memory remaining the same, the memory footprint on each individual GPU is reduced, which can be critical when dealing with memory constraints.

Analysis of Communication Overhead

In all parallelization strategies, communication overhead is a critical factor that can affect the overall performance.

Proof of Communication Overhead in Model Parallelism:

Let $C_{comm}$ be the communication overhead per layer when using model parallelism. The total communication overhead $O_{comm}$ for a model with $L$ layers is:

$O_{comm} = L \times C_{comm}$

This overhead must be minimized for efficient parallel execution. Techniques such as gradient aggregation, where gradients from different GPUs are combined before communication, can help reduce this overhead.

Conclusion

The proofs provided in this appendix serve to illustrate the theoretical basis for the parallelization strategies discussed. They highlight the importance of balancing computational load, minimizing communication overhead, and effectively managing memory in the deployment of LLMs on mid-range GPUs. These principles are fundamental in the design of efficient and scalable LLM inference systems.