# Technical Document Extraction: Roofline Model (Llama 13B, A40) ## 1. Document Header * **Title:** Roofline Model (Llama 13B, A40) * **Subject:** Performance analysis of the Llama 13B model running on an NVIDIA A40 GPU. ## 2. Chart Specifications The image is a **Roofline Chart**, a standard visualization used to identify performance bottlenecks in computing workloads. ### Axis Definitions * **Y-Axis (Vertical):** Performance (FLOP/s) * **Scale:** Logarithmic (Base 10). * **Markers:** 10G, 100G, 1T, 10T, 100T. * **X-Axis (Horizontal):** Operational Intensity (FLOP/Byte) * **Scale:** Logarithmic (Base 10). * **Markers:** 1, 10, 100, 1k (1,000), 10k (10,000). ### Legend and Thresholds The legend is located in the bottom-right quadrant of the chart area. | Legend Item | Color/Style | Description | Value/Threshold | | :--- | :--- | :--- | :--- | | **696GB/s** | Blue Dashed Line | Memory Bandwidth Limit | Slopes upward at 45° (log-log) | | **149.7 TFLOP/s** | Red Dashed Line | Peak Compute Performance | Horizontal line at ~1.5e14 | | **Ridge Point** | Green Vertical Dashed | Intersection of Bandwidth and Compute | ~215 FLOP/Byte | | **qkv mlp init** | Blue 'x' | Initialization phase for QKV/MLP layers | Compute-bound region | | **qkv mlp ar** | Orange 'x' | Autoregressive phase for QKV/MLP layers | Memory-bound region | | **up/gate/down init** | Green 'x' | Initialization phase for Up/Gate/Down layers | Compute-bound region | | **up/gate/down ar** | Red 'x' | Autoregressive phase for Up/Gate/Down layers | Memory-bound region | | **qk/pv init** | Purple 'x' | Initialization phase for QK/PV layers | Transition region | | **qk/pv ar** | Brown 'x' | Autoregressive phase for QK/PV layers | Memory-bound region | --- ## 3. Component Analysis and Data Trends ### Region 1: Memory-Bound (Left of the Ridge Point) * **Visual Trend:** Data points follow the upward slope of the blue dashed line (696GB/s). * **Observations:** * **Autoregressive (ar) phases** (Orange, Red, Brown 'x' marks) are clustered here. * **qk/pv ar (Brown):** Lowest operational intensity (~1 FLOP/Byte), resulting in the lowest performance (between 50G and 1T FLOP/s). * **up/gate/down ar (Red) & qkv mlp ar (Orange):** Higher intensity than qk/pv ar, ranging from ~2 to ~40 FLOP/Byte. Performance scales linearly with intensity along the bandwidth limit. ### Region 2: Compute-Bound (Right of the Ridge Point) * **Visual Trend:** Data points flatten out and cluster along the horizontal red dashed line (149.7 TFLOP/s). * **Observations:** * **Initialization (init) phases** (Blue, Green 'x' marks) are clustered here. * **up/gate/down init (Green):** High operational intensity (approx. 400 to 4,000 FLOP/Byte). These points are pinned against the 149.7 TFLOP/s ceiling, indicating maximum hardware utilization. * **qkv mlp init (Blue):** Similar to the green series, these points sit at the peak performance ceiling with intensities between 1,000 and 3,000 FLOP/Byte. ### Region 3: Transition/Ridge Area * **Visual Trend:** Points begin to deviate from the bandwidth slope and curve toward the compute ceiling. * **Observations:** * **qk/pv init (Purple):** These points bridge the gap, located between 40 and 150 FLOP/Byte. They show performance increasing from 10T to nearly 100T FLOP/s but remain below the absolute peak. --- ## 4. Summary of Findings 1. **Hardware Limits:** The A40 GPU used for Llama 13B has a peak throughput of **149.7 TFLOP/s** and a memory bandwidth of **696 GB/s**. 2. **Bottleneck Identification:** * **Initialization phases** are **Compute-Bound**. They efficiently utilize the GPU's processing power because they have high operational intensity. * **Autoregressive phases** (the generation step) are **Memory-Bound**. Their performance is limited by how fast data can be moved from memory, not by the GPU's calculation speed. 3. **Critical Threshold:** The "Ridge Point" occurs at approximately **215 FLOP/Byte**. Any operation with an intensity lower than this will be limited by memory bandwidth.

# Technical Document Extraction: Roofline Model (Llama 13B, A40) ## Chart Overview - **Title**: Roofline Model (Llama 13B, A40) - **Type**: Scatter plot with trend lines - **Purpose**: Visualizes performance (FLOP/s) vs. operational intensity (FLOP/Byte) for different computational operations ## Axes - **X-axis (Operational Intensity)**: - Label: "Operational Intensity (FLOP/Byte)" - Range: 1 to 10,000 (logarithmic scale) - Tick markers: 1, 10, 100, 1k - **Y-axis (Performance)**: - Label: "Performance (FLOP/s)" - Range: 10G to 100T (logarithmic scale) - Tick markers: 10G, 100G, 1T, 10T, 100T ## Legend - **Location**: Bottom-right corner - **Color/Marker Legend**: - **Blue dashed line**: 696GB/s (theoretical memory bandwidth limit) - **Red dashed line**: 149.7 TFLOP/s (theoretical compute limit) - **Blue x**: qkv mlp init - **Orange x**: qkv mlp ar - **Green x**: up/gate/down init - **Red x**: up/gate/down ar - **Purple x**: qk/pv init - **Brown x**: qk/pv ar ## Key Trends 1. **Theoretical Limits**: - Horizontal red line at 149.7 TFLOP/s (compute-bound ceiling) - Vertical green line at 696GB/s (memory-bound ceiling) 2. **Data Series Trends**: - **qkv mlp init** (blue x): - Starts at ~1 FLOP/Byte, 100G FLOP/s - Ends at ~1000 FLOP/Byte, 100T FLOP/s - Linear upward trajectory - **qkv mlp ar** (orange x): - Starts at ~10 FLOP/Byte, 100G FLOP/s - Ends at ~1000 FLOP/Byte, 100T FLOP/s - Linear upward trajectory - **up/gate/down init** (green x): - Starts at ~100 FLOP/Byte, 10T FLOP/s - Ends at ~1000 FLOP/Byte, 100T FLOP/s - Linear upward trajectory - **up/gate/down ar** (red x): - Starts at ~10 FLOP/Byte, 10T FLOP/s - Ends at ~1000 FLOP/Byte, 100T FLOP/s - Linear upward trajectory - **qk/pv init** (purple x): - Starts at ~10 FLOP/Byte, 10T FLOP/s - Ends at ~1000 FLOP/Byte, 100T FLOP/s - Linear upward trajectory - **qk/pv ar** (brown x): - Starts at ~1 FLOP/Byte, 100G FLOP/s - Ends at ~1000 FLOP/Byte, 100T FLOP/s - Linear upward trajectory ## Spatial Grounding - **Legend Position**: Bottom-right quadrant - **Data Point Verification**: - All blue x markers align with "qkv mlp init" legend - All orange x markers align with "qkv mlp ar" legend - All green x markers align with "up/gate/down init" legend - All red x markers align with "up/gate/down ar" legend - All purple x markers align with "qk/pv init" legend - All brown x markers align with "qk/pv ar" legend ## Critical Observations 1. All operations approach the theoretical compute limit (149.7 TFLOP/s) as operational intensity increases 2. Memory-bound operations (qkv mlp init, qk/pv ar) start at lower operational intensities 3. Compute-bound operations (up/gate/down init, qk/pv init) require higher operational intensities to reach peak performance 4. The roofline model demonstrates clear separation between memory-bound and compute-bound operations ## Data Table Reconstruction | Operation Type | Initial Intensity (FLOP/Byte) | Initial Performance (FLOP/s) | Final Intensity (FLOP/Byte) | Final Performance (FLOP/s) | |----------------------|-------------------------------|------------------------------|-----------------------------|----------------------------| | qkv mlp init | 1 | 100G | 1000 | 100T | | qkv mlp ar | 10 | 100G | 1000 | 100T | | up/gate/down init | 100 | 10T | 1000 | 100T | | up/gate/down ar | 10 | 10T | 1000 | 100T | | qk/pv init | 10 | 10T | 1000 | 100T | | qk/pv ar | 1 | 100G | 1000 | 100T | ## Trend Verification - All data series show linear scaling with operational intensity - Memory-bound operations (blue x, brown x) cluster near the memory bandwidth limit (696GB/s) - Compute-bound operations (green x, purple x) cluster near the compute limit (149.7 TFLOP/s) - All operations converge at the theoretical maximum performance point (1000 FLOP/Byte, 100T FLOP/s)