# Technical Document Extraction: Roofline Model (Llama 13B, A6000) ## 1. Document Header * **Title:** Roofline Model (Llama 13B, A6000) * **Language:** English ## 2. Chart Specifications This is a **Roofline Model** chart, a standard visualization used to represent the performance limits of a computing system (NVIDIA A6000 GPU) running a specific workload (Llama 13B model). ### Axis Definitions * **Y-Axis (Vertical):** Performance (FLOP/s) * **Scale:** Logarithmic (Base 10) * **Range:** 10G to ~300T * **Major Markers:** 10G, 100G, 1T, 10T, 100T * **X-Axis (Horizontal):** Operational Intensity (FLOP/Byte) * **Scale:** Logarithmic (Base 10) * **Range:** ~0.6 to 10k * **Major Markers:** 1, 10, 1k, 10k ### Legend and Thresholds The legend is located in the bottom-right quadrant of the plot area. | Legend Item | Color/Style | Description | Value/Threshold | | :--- | :--- | :--- | :--- | | **768GB/s** | Blue Dashed Line | Memory Bandwidth Limit (Slope) | 768 GB/s | | **181 TFLOP/s** | Red Dashed Line | Peak Compute Performance (Ceiling) | 181 TFLOP/s | | **qkv mlp init** | Blue 'x' | Data points for QKV/MLP initialization | High Intensity/High Perf | | **qkv mlp ar** | Orange 'x' | Data points for QKV/MLP auto-regressive | Mid Intensity/Mid Perf | | **up/gate/down init** | Green 'x' | Data points for Up/Gate/Down initialization | High Intensity/High Perf | | **up/gate/down ar** | Red 'x' | Data points for Up/Gate/Down auto-regressive | Low Intensity/Low Perf | | **qk/pv init** | Purple 'x' | Data points for QK/PV initialization | Mid Intensity/Mid Perf | | **qk/pv ar** | Brown 'x' | Data points for QK/PV auto-regressive | Low Intensity/Low Perf | --- ## 3. Component Analysis ### The "Roofline" Structure 1. **Memory-Bound Region (The Slope):** Represented by the blue dashed line starting from the bottom left. It follows the formula $Performance = Bandwidth \times Intensity$. Any data point sitting on or near this line is limited by how fast data can be moved from memory (768 GB/s). 2. **Compute-Bound Region (The Ceiling):** Represented by the horizontal red dashed line at the top. It represents the hardware's maximum theoretical throughput (181 TFLOP/s). 3. **Ridge Point:** The intersection of the two lines occurs at an Operational Intensity of approximately **235 FLOP/Byte** (indicated by a vertical green dashed line). ### Data Series Trends and Distribution * **Initialization (init) Phases:** * **Trend:** These points (Blue, Green, Purple 'x') cluster toward the right side of the graph (High Operational Intensity). * **Observation:** Most "init" points for `qkv mlp` and `up/gate/down` are located on the horizontal "ceiling," meaning they are compute-bound and utilizing the GPU's maximum TFLOP/s. * **Auto-regressive (ar) Phases:** * **Trend:** These points (Orange, Red, Brown 'x') cluster toward the left side of the graph (Low Operational Intensity). * **Observation:** These points follow the diagonal blue dashed line. This indicates that the auto-regressive decoding phase of the Llama 13B model is strictly memory-bandwidth bound, operating significantly below the peak TFLOP/s of the A6000. --- ## 4. Data Point Extraction (Approximate Values) | Category | Operational Intensity (FLOP/Byte) | Performance (FLOP/s) | Bottleneck | | :--- | :--- | :--- | :--- | | **up/gate/down ar** | ~1 to 10 | 100G to 5T | Memory (768GB/s) | | **qk/pv ar** | ~1 (Vertical stack) | 40G to 800G | Memory (768GB/s) | | **qkv mlp ar** | ~2 to 40 | 1T to 20T | Memory (768GB/s) | | **qk/pv init** | ~40 to 150 | 10T to 80T | Transition/Memory | | **qkv mlp init** | ~800 to 3k | ~150T to 181T | Compute (181 TFLOP/s) | | **up/gate/down init** | ~200 to 4k | ~100T to 181T | Compute (181 TFLOP/s) | ## 5. Summary of Findings The chart demonstrates that for a Llama 13B model on an A6000 GPU: 1. **Initialization** is highly efficient and hits the hardware's compute ceiling (181 TFLOP/s). 2. **Auto-regressive decoding** is inefficient in terms of raw compute utilization because it is bottlenecked by the 768 GB/s memory bandwidth. 3. The **qk/pv ar** (Brown 'x') operations are the least efficient, clustered at the lowest operational intensity (~1 FLOP/Byte).

# Roofline Model Analysis (Llama13B, A6000) ## Title **Roofline Model (Llama 13B, A6000)** ## Axes - **X-axis**: Operational Intensity (FLOP/Byte) - Range: 1 to 10,000 (logarithmic scale) - Key marker: Green dashed vertical line at **100 FLOP/Byte** - **Y-axis**: Performance (FLOP/s) - Range: 10G to 100T (logarithmic scale) - Key markers: - Red dashed horizontal line at **181 TFLOP/s** - Blue dashed horizontal line at **768GB/s** ## Legend | Symbol | Label | Color | Marker | |--------|------------------------|--------|--------| | Blue | 768GB/s | Dashed | N/A | | Red | 181 TFLOP/s | Dashed | N/A | | Blue | qkv mlp init | Cross | X | | Orange | qkv mlp ar | Cross | X | | Green | up/gate/down init | Cross | X | | Red | up/gate/down ar | Cross | X | | Purple | qk/pv init | Cross | X | | Brown | qk/pv ar | Cross | X | ## Data Points - **qkv mlp init** (Blue crosses): - Clustered below 181 TFLOP/s line, increasing with operational intensity. - **qkv mlp ar** (Orange crosses): - Similar trend to qkv mlp init, slightly higher performance at mid-range intensities. - **up/gate/down init** (Green crosses): - Highest performance among non-ar operations, approaching 181 TFLOP/s at 100 FLOP/Byte. - **up/gate/down ar** (Red crosses): - Dominates performance above 100 FLOP/Byte, consistently near 181 TFLOP/s. - **qk/pv init** (Purple crosses): - Lower performance than qkv operations, plateauing below 10T FLOP/s. - **qk/pv ar** (Brown crosses): - Similar to qk/pv init, with minimal improvement at higher intensities. ## Key Trends 1. **Performance Ceiling**: - The red dashed line (181 TFLOP/s) acts as a theoretical maximum for most operations. - Only **up/gate/down ar** (red crosses) approaches this limit at high operational intensities (>100 FLOP/Byte). 2. **Memory Bandwidth Constraint**: - The blue dashed line (768GB/s) represents memory bandwidth. Most data points fall below this line, indicating compute-bound operations. 3. **Operational Intensity Threshold**: - The green dashed line at 100 FLOP/Byte marks a critical transition point. - Above this threshold, **up/gate/down ar** achieves near-peak performance, while other operations plateau. 4. **Operation Efficiency**: - **qkv mlp ar** and **up/gate/down ar** show the highest compute-to-memory efficiency. - **qk/pv** operations (init/ar) are less efficient, remaining in the lower-left quadrant. ## Observations - **Scalability**: - Performance scales logarithmically with operational intensity for most operations. - **Bottlenecks**: - Memory-bound operations (e.g., qk/pv) are limited by bandwidth (768GB/s). - Compute-bound operations (e.g., up/gate/down ar) hit the 181 TFLOP/s ceiling.