Quantitative Analysis of Transformer Architecture and Llama Model Performance

This document presents a comprehensive engineering‑oriented analysis of Transformer‑based large language models. It is organized into four main parts: basic concepts, quantitative analysis of the Transformer architecture, performance metrics, and model variants, followed by a set of practice questions.

1. Basic Knowledge Review – The section introduces tensors as multi‑dimensional arrays, explains how weights, activations, and multi‑head attention tensors are represented (e.g., [H,H], [B,S,H], [B,S,h,d]), and describes matrix multiplication fundamentals, including the computation and memory cost of a matrix product A@[B] = 2·M·K·N FLOPs and (M·K + K·N + M·N)·sizeof(dtype) bytes of memory traffic.

2. Transformer Architecture Quantitative Analysis – The original encoder‑decoder Transformer is described, then the focus shifts to decoder‑only models (e.g., Llama, GPT‑2). Detailed derivations are given for:

Pre‑processing: tokenization and embedding lookup (pure memory access, negligible FLOPs).

Attention: the three‑step computation (QKᵀ, softmax, OV) for both single‑head and multi‑head cases, with explicit FLOP formulas such as 2·B·S·d·S' for single‑head and 2·B·S·S'·h·d for multi‑head.

Feed‑Forward Network (FFN): two matrix multiplications ([B,S,H]@[H,4H] and [B,S,4H]@[4H,H]) plus a ReLU, totaling 8·H·H parameters and 16·B·S·H·H FLOPs per layer.

The total parameter count for a single Transformer layer is expressed as 12·H·H (attention + FFN) plus the word‑embedding matrix V·H. For a model with L layers, N = L·12·H·H + V·H.

3. Performance Evaluation – The document defines the MFU (Model FLOPs Utilization) metric as the ratio of observed throughput to the theoretical maximum (GPU TFLOPs / per‑token FLOPs). It shows how to compute the theoretical FLOPs for training (≈6·N per token) and inference (≈2·N per token) and discusses practical limits (e.g., A800 can only achieve ~80 % of its 312 TFLOPs peak).

Memory consumption is broken down into three components: model parameters (N·sizeof(dtype)), KV‑cache (L·S'·h·d·2·sizeof(dtype)), and intermediate activations (B·S·H·c). The analysis explains why KV‑cache dominates memory usage for long sequences.

4. Parallelism Strategies – From an engineering perspective, data parallelism splits the batch B, sequence parallelism splits the token length S, pipeline parallelism splits the layer depth L, and tensor parallelism splits large weight matrices (e.g., the 4H dimension).

5. Transformer Variants – The text reviews attention optimizations such as Multi‑Query Attention (MQA) and Grouped‑Query Attention (GQA), which reduce KV‑cache size, and Sliding‑Window Attention for long sequences. FFN optimizations like SwiGLU and Mixture‑of‑Experts (MoE) are also discussed, highlighting changes in parameter count and compute.

6. Practice Questions – Four example problems are provided, each with a brief answer that applies the earlier formulas to real‑world scenarios (e.g., estimating MFU for Llama 3 70B training, calculating FLOPs for Llama 2 7B, evaluating latency feasibility for Llama 2 70B inference on 8 × A800 GPUs, and estimating maximum concurrent requests for a 2‑GPU A800 setup).