Theoretical Analysis and Performance

The theoretical analysis of parallelization and memory management strategies is crucial for understanding their impact on LLM inference performance. This section delves into the mathematical modeling and analysis of the strategies discussed earlier, providing insights into their efficiency and potential benefits.

Performance Limits of Parallelized LLM Inference

The performance of parallelized LLM inference is bounded by the slowest component in the pipeline, often referred to as the "critical path." The critical path is influenced by the parallelization strategy employed. For instance, in model parallelism, the critical path is determined by the maximum latency across all parallelized layers.

Let $T_i$ be the time taken to process layer $i$ in parallel, and $T_{max}$ be the maximum of $T_i$ for all layers. The throughput $\Theta$ of the parallelized system is given by:

$\Theta = \frac{1}{T_{max}}$

In an ideal scenario with no communication overhead, the throughput would be inversely proportional to the latency of the slowest layer. However, in practice, communication overhead $O_{comm}$ must be considered, leading to an effective throughput $\Theta_{eff}$ :

$\Theta_{eff} = \Theta - O_{comm}$

Optimal Parallelization Strategies

Optimizing parallelization strategies involves finding a balance between computational load and communication overhead. The optimal strategy minimizes the total execution time $E_{total}$, which includes both computation $C_{comp}$ and communication $C_{comm}$ times:

$E_{total} = C_{comp} + C_{comm}$

The computation time $C_{comp}$ can be estimated as the sum of the processing times for all layers or operations. The communication time $C_{comm}$ is influenced by the size of the data being communicated and the bandwidth of the interconnect between GPUs.

Performance Trade-offs in LLM Deployment

There are trade-offs to consider when deploying LLMs on mid-range GPUs. For instance, increasing the parallelism level can lead to higher throughput but may also increase communication overhead. The efficiency of memory management techniques also has a trade-off curve with the complexity of the inference task.

The trade-off can be quantified by analyzing the speedup $S$ gained from parallelization, which is the ratio of the serial execution time $T_{serial}$ to the parallel execution time $T_{parallel}$ :

$S = \frac{T_{serial}}{T_{parallel}}$

Ideally, for 'G' GPUs, a linear speedup is expected:

$S_{ideal} = G$

However, due to overheads, the actual speedup $S_{actual}$ is often less than the ideal speedup. The efficiency $E$ of the parallelization can be calculated as:

$E = \frac{S_{actual}}{G}$