## Heatmap: Recompute Cost vs. Batch Size and Window Size ### Overview The image is a heatmap visualizing the recompute cost (in percentage) for different combinations of batch sizes and window sizes. The heatmap uses a color gradient from blue to orange, where blue represents lower recompute costs and orange represents higher costs. The x-axis represents window size, and the y-axis represents batch size. ### Components/Axes * **X-axis:** Window Size, with values 16, 32, 64, 128, 256, and 512. * **Y-axis:** Batch Size, with values 1, 2, 4, 8, 16, and 32. * **Color Legend (Right):** Recompute Cost (%), ranging from 5% (blue) to 45% (orange). The legend has markers at 5%, 10%, 15%, 20%, 25%, 30%, 35%, 40%, and 45%. ### Detailed Analysis The heatmap presents recompute cost percentages for various batch size and window size combinations. The values are displayed within each cell of the heatmap. Here's a breakdown of the values: * **Batch Size 32:** * Window Size 16: 2.82% (Blue) * **Batch Size 16:** * Window Size 16: 3.09% (Blue) * Window Size 32: 6.40% (Blue) * **Batch Size 8:** * Window Size 16: 3.11% (Blue) * Window Size 32: 6.49% (Blue) * Window Size 64: 13.83% (Blue) * **Batch Size 4:** * Window Size 16: 2.97% (Blue) * Window Size 32: 6.47% (Blue) * Window Size 64: 14.09% (Blue) * Window Size 128: 27.85% (Tan) * **Batch Size 2:** * Window Size 16: 3.31% (Blue) * Window Size 32: 6.03% (Blue) * Window Size 64: 13.86% (Blue) * Window Size 128: 28.96% (Tan) * Window Size 256: 42.28% (Orange) * **Batch Size 1:** * Window Size 16: 3.44% (Blue) * Window Size 32: 6.81% (Blue) * Window Size 64: 12.42% (Blue) * Window Size 128: 24.92% (Blue-Tan) * Window Size 256: 46.41% (Orange) * Window Size 512: 42.33% (Orange) ### Key Observations * The recompute cost generally increases as both the batch size decreases and the window size increases. * Lower batch sizes (1, 2, 4) show a more significant increase in recompute cost as the window size increases compared to higher batch sizes (8, 16, 32). * The lowest recompute costs are observed with larger batch sizes (16, 32) and smaller window sizes (16, 32). * The highest recompute costs are observed with the smallest batch size (1) and larger window sizes (256, 512). ### Interpretation The heatmap illustrates the trade-offs between batch size, window size, and recompute cost. Smaller batch sizes often lead to increased recomputation, especially when combined with larger window sizes. This suggests that the computational overhead of recomputing activations becomes more significant when processing smaller chunks of data over larger contexts. Conversely, larger batch sizes reduce the need for frequent recomputation, resulting in lower costs, particularly with smaller window sizes. The data suggests that optimizing both batch size and window size is crucial for minimizing recompute costs in a given system or model.

\n ## Heatmap: Recompute Cost vs. Batch Size and Window Size ### Overview This image presents a heatmap visualizing the relationship between "Window Size" and "Batch Size" in relation to "Recompute Cost (%)". The heatmap displays a grid of cells, each colored according to the corresponding recompute cost percentage. The color scale ranges from approximately 2.82% (dark purple) to 46.41% (dark orange). ### Components/Axes * **X-axis:** "Window Size" with markers at 16, 32, 64, 128, 256, and 512. * **Y-axis:** "Batch Size" with markers at 1, 2, 4, 8, 16, and 32. * **Color Scale (Legend):** Located on the right side of the heatmap, representing "Recompute Cost (%)". The scale ranges from approximately 5% (light blue) to 45% (dark orange). * **Data Cells:** Each cell represents a specific combination of Window Size and Batch Size, with the recompute cost percentage displayed within the cell. ### Detailed Analysis The heatmap contains the following data points: * **Batch Size = 32:** * Window Size = 16: 2.82% * Window Size = 32: 6.40% * **Batch Size = 16:** * Window Size = 16: 3.09% * Window Size = 32: 6.49% * Window Size = 64: 13.83% * **Batch Size = 8:** * Window Size = 16: 3.11% * Window Size = 32: 6.47% * Window Size = 64: 14.09% * Window Size = 128: 27.85% * **Batch Size = 4:** * Window Size = 16: 2.97% * Window Size = 32: 6.47% * Window Size = 64: 13.86% * Window Size = 128: 28.96% * Window Size = 256: 42.28% * **Batch Size = 2:** * Window Size = 16: 3.31% * Window Size = 32: 6.03% * Window Size = 64: 12.42% * Window Size = 128: 24.92% * Window Size = 256: 46.41% * Window Size = 512: 42.33% * **Batch Size = 1:** * Window Size = 16: 3.44% * Window Size = 32: 6.81% * Window Size = 64: 12.42% * Window Size = 128: 24.92% * Window Size = 256: 46.41% * Window Size = 512: 42.33% **Trends:** * As Window Size increases, Recompute Cost generally increases for a given Batch Size. * For smaller Window Sizes (16, 32), Recompute Cost remains relatively low across all Batch Sizes. * The highest Recompute Costs are observed with Window Size = 256 and Batch Size = 2, and Window Size = 256 and Batch Size = 1, both at approximately 46.41%. * Recompute Cost decreases when Window Size = 512 and Batch Size = 1 or 2. ### Key Observations * The lowest recompute cost is 2.82% when Batch Size is 32 and Window Size is 16. * There is a noticeable increase in recompute cost as the window size increases beyond 128, especially for smaller batch sizes. * The recompute cost appears to plateau or even decrease slightly for very large window sizes (512), potentially indicating diminishing returns or optimization benefits. ### Interpretation The heatmap demonstrates the trade-off between Window Size, Batch Size, and Recompute Cost. Larger Window Sizes generally lead to higher recompute costs, likely due to the increased computational complexity of processing larger data segments. However, the decrease in recompute cost at a Window Size of 512 suggests that there might be a point where larger windows become more efficient, possibly due to better utilization of parallel processing or optimized algorithms. The Batch Size also influences the recompute cost, but its effect is less pronounced than that of the Window Size. The optimal combination of Window Size and Batch Size depends on the specific application and the available computational resources. This data suggests that for minimizing recompute cost, smaller window sizes and moderate batch sizes are preferable. However, if computational resources are abundant, larger window sizes might be acceptable or even beneficial. The data suggests a non-linear relationship between the parameters. A simple linear model would not accurately capture the observed behavior, particularly the plateauing effect at larger window sizes. Further investigation might be needed to understand the underlying reasons for this behavior and to identify the optimal parameter settings for different scenarios.

\n ## Heatmap: Recompute Cost (%) by Batch Size and Window Size ### Overview The image is a heatmap visualizing the "Recompute Cost (%)" as a function of two variables: **Batch Size** (vertical axis) and **Window Size** (horizontal axis). The cost is represented by a color gradient, with a corresponding color bar legend on the right. The data forms a triangular pattern, with values only present where the Window Size is greater than or equal to the Batch Size. ### Components/Axes * **Y-Axis (Vertical):** Labeled **"Batch Size"**. The categories, from bottom to top, are: `1`, `2`, `4`, `8`, `16`, `32`. * **X-Axis (Horizontal):** Labeled **"Window Size"**. The categories, from left to right, are: `16`, `32`, `64`, `128`, `256`, `512`. * **Color Bar Legend:** Positioned vertically on the right side of the chart. It is labeled **"Recompute Cost (%)"**. The scale runs from approximately **5%** (dark blue) at the bottom to **45%** (bright orange) at the top, with intermediate tick marks at 10%, 15%, 20%, 25%, 30%, 35%, and 40%. * **Data Grid:** A 6x6 grid where each cell contains a percentage value. Cells are colored according to the legend. Cells above the main diagonal (where Window Size < Batch Size) are empty (white). ### Detailed Analysis The following table reconstructs the data from the heatmap. Each cell value represents the Recompute Cost (%) for the corresponding Batch Size (row) and Window Size (column). | Batch Size \ Window Size | 16 | 32 | 64 | 128 | 256 | 512 | | :----------------------- | :------ | :------ | :------ | :------ | :------ | :------ | | **32** | 2.82% | | | | | | | **16** | 3.09% | 6.40% | | | | | | **8** | 3.11% | 6.49% | 13.83% | | | | | **4** | 2.97% | 6.47% | 14.09% | 27.85% | | | | **2** | 3.31% | 6.03% | 13.86% | 28.96% | 42.28% | | | **1** | 3.44% | 6.81% | 12.42% | 24.92% | 46.41% | 42.33% | **Trend Verification:** * **Horizontal Trend (Fixed Batch Size):** For any given batch size, the recompute cost **increases significantly** as the window size increases. The color shifts from blue to orange moving right along any row. * **Vertical Trend (Fixed Window Size):** For any given window size, the recompute cost generally **decreases slightly or remains stable** as the batch size increases. The color becomes slightly cooler (less orange) moving up any column. ### Key Observations 1. **Maximum Cost:** The highest recorded recompute cost is **46.41%**, occurring at **Batch Size = 1** and **Window Size = 256**. 2. **Minimum Cost:** The lowest recorded recompute cost is **2.82%**, occurring at **Batch Size = 32** and **Window Size = 16**. 3. **Cost Gradient:** The most dramatic cost increase occurs when moving from a Window Size of 64 to 128. For example, at Batch Size 4, the cost jumps from 14.09% to 27.85%. 4. **Anomaly at Batch Size 1:** The cost pattern for Batch Size = 1 is non-monotonic. It peaks at Window Size 256 (46.41%) and then **decreases** to 42.33% at Window Size 512, which is unique in the dataset. 5. **Data Sparsity:** The upper-right triangle of the matrix is empty, indicating that configurations where the Window Size is smaller than the Batch Size were not measured or are not applicable. ### Interpretation This heatmap demonstrates the computational trade-offs in a system where "recompute cost" is a critical metric, likely related to memory optimization techniques in machine learning (e.g., activation checkpointing). The data suggests: * **Window Size is the Dominant Factor:** Increasing the window size has a much more severe impact on recompute cost than decreasing the batch size. This implies the system's memory or computational overhead scales poorly with sequence length (window size). * **Efficiency at Larger Batches:** For a fixed, large window size (e.g., 128 or 256), using a larger batch size (e.g., 8 or 16) results in a lower *percentage* recompute cost. This could indicate better amortization of fixed overheads or more efficient parallelization. * **Practical Implication:** To minimize recompute cost, one should prioritize using the smallest feasible window size. If a large window is necessary, pairing it with a larger batch size can mitigate the cost percentage, though the absolute cost will still be high. * **The Batch Size=1 Anomaly:** The drop in cost from window size 256 to 512 for batch size 1 is intriguing. It may point to a different code path, a hardware utilization threshold, or a measurement artifact for that specific, often challenging, configuration. **In summary, the chart provides a clear quantitative guide for optimizing system parameters: recompute cost is highly sensitive to window size and is generally lower for larger batch sizes within the measured, applicable configurations.**

## Heatmap: Recompute Cost by Batch Size and Window Size ### Overview The image is a heatmap visualizing recompute costs as percentages, with batch sizes (1, 2, 4, 8, 16, 32) on the y-axis and window sizes (16, 32, 64, 128, 256, 512) on the x-axis. Colors range from blue (low cost) to orange (high cost), with numerical values embedded in each cell. ### Components/Axes - **Y-Axis (Batch Size)**: Labeled "Batch Size" with values: 1, 2, 4, 8, 16, 32. - **X-Axis (Window Size)**: Labeled "Window Size" with values: 16, 32, 64, 128, 256, 512. - **Color Bar**: Right-aligned, labeled "Recompute Cost (%)" with a gradient from blue (5%) to orange (45%). - **Numerical Values**: Percentages in each cell, e.g., "2.82%" (top-left) and "42.33%" (bottom-right). ### Detailed Analysis - **Batch Size 32**: - 16: 2.82% (blue) - 32: 6.40% (light blue) - 64: 13.83% (medium blue) - 128: 27.85% (light orange) - 256: 42.28% (orange) - 512: 42.33% (orange) - **Batch Size 16**: - 16: 3.09% (blue) - 32: 6.40% (light blue) - 64: 13.83% (medium blue) - 128: 27.85% (light orange) - 256: 42.28% (orange) - 512: 42.33% (orange) - **Batch Size 8**: - 16: 3.11% (blue) - 32: 6.49% (light blue) - 64: 13.83% (medium blue) - 128: 27.85% (light orange) - 256: 42.28% (orange) - 512: 42.33% (orange) - **Batch Size 4**: - 16: 2.97% (blue) - 32: 6.47% (light blue) - 64: 14.09% (medium blue) - 128: 27.85% (light orange) - 256: 42.28% (orange) - 512: 42.33% (orange) - **Batch Size 2**: - 16: 3.31% (blue) - 32: 6.03% (light blue) - 64: 13.86% (medium blue) - 128: 28.96% (light orange) - 256: 42.28% (orange) - 512: 42.33% (orange) - **Batch Size 1**: - 16: 3.44% (blue) - 32: 6.81% (light blue) - 64: 12.42% (medium blue) - 128: 24.92% (light orange) - 256: 46.41% (orange) - 512: 42.33% (orange) ### Key Observations 1. **Increasing Trends**: Recompute costs rise with larger batch and window sizes. The highest costs (46.41%) occur at batch size 1 and window size 256. 2. **Color Consistency**: Blue shades dominate smaller batch/window sizes, while orange shades appear for larger values, aligning with the legend. 3. **Anomalies**: Batch size 1 has the highest cost at window size 256 (46.41%), despite smaller batch size, suggesting a non-linear interaction between parameters. ### Interpretation The data demonstrates that recompute costs scale non-linearly with both batch and window sizes. Larger batch sizes (e.g., 32) paired with large window sizes (e.g., 512) result in costs exceeding 40%, while smaller configurations (e.g., batch 32, window 16) remain below 5%. The anomaly at batch size 1 and window 256 suggests that smaller batches may exacerbate recompute costs when combined with very large windows, possibly due to memory constraints or algorithmic inefficiencies. This highlights the importance of balancing batch and window sizes to optimize recompute costs in distributed training or parallel processing workflows.