## Heatmap: Comparison of Values Before and After GRPO ### Overview The image presents two heatmaps side-by-side, visualizing numerical values before and after the application of a process labeled "GRPO". The heatmaps share the same axes, representing the "number of x's digits" and "number of y's digits," both ranging from 1 to 10. The color intensity in each cell corresponds to the magnitude of the numerical value within that cell. A white dashed line is drawn on each heatmap, forming an L-shape. ### Components/Axes * **Titles:** "Before GRPO" (left heatmap), "After GRPO" (right heatmap) * **Y-axis Label:** "number of y's digits" (vertical, on both heatmaps) * **X-axis Label:** "number of x's digits" (horizontal, on both heatmaps) * **X-axis Markers:** 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (on both heatmaps) * **Y-axis Markers:** 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (on both heatmaps) * **Data Values:** Numerical values are displayed within each cell of the heatmaps. The color of the cells varies based on the value, ranging from dark red/black (low values) to yellow/white (high values). ### Detailed Analysis **Left Heatmap (Before GRPO):** * **General Trend:** The values are generally lower compared to the right heatmap. The highest values are concentrated in the top-left corner. * **Specific Values:** * (1,1): 38 * (1,2): 33 * (1,3): 29 * (1,4): 19 * (1,5): 15 * (1,6): 12 * (1,7): 9 * (1,8): 6 * (1,9): 4 * (2,1): 19 * (2,2): 15 * (2,3): 4 * (2,4): 5 * (2,5): 5 * (2,6): 4 * (2,7): 5 * (2,8): 3 * (2,9): 1 * (2,10): 1 * (3,1): 6 * (3,2): 7 * (3,3): 4 * (3,4): 9 * (3,5): 6 * (3,6): 4 * (3,7): 6 * (3,8): 4 * (3,9): 2 * (3,10): 5 * (4,1): 5 * (4,2): 5 * (4,3): 6 * (4,4): 5 * (4,5): 4 * (4,6): 5 * (4,7): 5 * (4,8): 4 * (4,9): 5 * (4,10): 6 * (5,1): 6 * (5,2): 8 * (5,3): 5 * (5,4): 4 * (5,5): 3 * (5,6): 3 * (5,7): 5 * (5,8): 4 * (5,9): 6 * (5,10): 4 * (6,1): 6 * (6,2): 7 * (6,3): 4 * (6,4): 4 * (6,5): 4 * (6,6): 3 * (6,7): 4 * (6,8): 4 * (6,9): 5 * (6,10): 7 * (7,1): 6 * (7,2): 4 * (7,3): 4 * (7,4): 3 * (7,5): 4 * (7,6): 5 * (7,7): 5 * (7,8): 6 * (7,9): 7 * (7,10): 6 * (8,1): 5 * (8,2): 7 * (8,3): 5 * (8,4): 6 * (8,5): 5 * (8,6): 7 * (8,7): 10 * (8,8): 9 * (8,9): 8 * (8,10): 6 * (9,1): 6 * (9,2): 7 * (9,3): 8 * (9,4): 8 * (9,5): 10 * (9,6): 14 * (9,7): 10 * (9,8): 8 * (9,9): 5 * (9,10): 7 * (10,1): 5 * (10,2): 6 * (10,3): 9 * (10,4): 10 * (10,5): 10 * (10,6): 9 * (10,7): 9 * (10,8): 7 * (10,9): 8 * (10,10): 6 **Right Heatmap (After GRPO):** * **General Trend:** The values are significantly higher compared to the left heatmap. The heatmap is predominantly yellow and white, indicating high values across the board. * **Specific Values:** * (1,1): 75 * (1,2): 67 * (1,3): 69 * (1,4): 70 * (1,5): 67 * (1,6): 67 * (1,7): 67 * (1,8): 56 * (1,9): 54 * (1,10): 51 * (2,1): 72 * (2,2): 72 * (2,3): 71 * (2,4): 70 * (2,5): 68 * (2,6): 70 * (2,7): 70 * (2,8): 67 * (2,9): 67 * (2,10): 75 * (3,1): 64 * (3,2): 73 * (3,3): 74 * (3,4): 75 * (3,5): 76 * (3,6): 72 * (3,7): 75 * (3,8): 72 * (3,9): 78 * (3,10): 82 * (4,1): 61 * (4,2): 71 * (4,3): 80 * (4,4): 79 * (4,5): 78 * (4,6): 79 * (4,7): 81 * (4,8): 82 * (4,9): 88 * (4,10): 88 * (5,1): 64 * (5,2): 74 * (5,3): 80 * (5,4): 80 * (5,5): 81 * (5,6): 83 * (5,7): 85 * (5,8): 88 * (5,9): 92 * (5,10): 93 * (6,1): 60 * (6,2): 71 * (6,3): 78 * (6,4): 83 * (6,5): 85 * (6,6): 85 * (6,7): 89 * (6,8): 93 * (6,9): 95 * (6,10): 96 * (7,1): 53 * (7,2): 67 * (7,3): 81 * (7,4): 84 * (7,5): 86 * (7,6): 90 * (7,7): 94 * (7,8): 95 * (7,9): 96 * (7,10): 96 * (8,1): 52 * (8,2): 68 * (8,3): 81 * (8,4): 87 * (8,5): 89 * (8,6): 94 * (8,7): 96 * (8,8): 96 * (8,9): 96 * (8,10): 98 * (9,1): 52 * (9,2): 72 * (9,3): 84 * (9,4): 88 * (9,5): 92 * (9,6): 96 * (9,7): 96 * (9,8): 98 * (9,9): 97 * (9,10): 96 * (10,1): 61 * (10,2): 77 * (10,3): 86 * (10,4): 90 * (10,5): 93 * (10,6): 96 * (10,7): 97 * (10,8): 97 * (10,9): 96 * (10,10): 95 ### Key Observations * The "GRPO" process significantly increases the values across all combinations of x and y digits. * The white dashed line seems to separate regions of different performance or behavior, but without further context, its exact meaning is unclear. * The top-left corner shows the most significant improvement after applying "GRPO". ### Interpretation The heatmaps demonstrate the effectiveness of the "GRPO" process in increasing the measured values, whatever they represent. The axes "number of x's digits" and "number of y's digits" likely refer to input parameters or characteristics of a system being evaluated. The "GRPO" process appears to optimize or enhance the system's performance across various combinations of these parameters. The white dashed line might indicate a threshold or boundary related to the application of "GRPO," or it could delineate regions where the process has varying degrees of impact. Further context is needed to fully understand the meaning of the values and the significance of the dashed line.

\n ## Heatmap: Performance Comparison Before and After GRPO ### Overview This image presents two heatmaps, visually comparing a performance metric "Before GRPO" and "After GRPO". Both heatmaps represent the relationship between the number of 'x' digits and the number of 'y' digits, with color intensity indicating the corresponding value. The x-axis represents the number of 'x' digits (ranging from 1 to 10), and the y-axis represents the number of 'y' digits (also ranging from 1 to 10). ### Components/Axes * **Title (Left):** "Before GRPO" * **Title (Right):** "After GRPO" * **X-axis Label (Both):** "number of x's digits" * **Y-axis Label (Both):** "number of y's digits" * **X-axis Markers (Both):** 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 * **Y-axis Markers (Both):** 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 * **Color Scale (Left):** Ranges from dark blue (low values) to red (high values). * **Color Scale (Right):** Ranges from dark blue (low values) to yellow (high values). * **Legend (Implicit):** Color intensity represents the value at each (x, y) coordinate. ### Detailed Analysis or Content Details **Before GRPO (Left Heatmap):** The heatmap shows a generally decreasing trend as both the number of 'x' digits and the number of 'y' digits increase. The highest values are concentrated in the top-left corner. * **(1, 1):** 38 * **(1, 2):** 19 * **(1, 3):** 29 * **(1, 4):** 15 * **(1, 5):** 12 * **(1, 6):** 9 * **(1, 7):** 6 * **(1, 8):** 4 * **(1, 9):** 2 * **(1, 10):** 4 * **(2, 1):** 15 * **(2, 2):** 4 * **(2, 3):** 5 * **(2, 4):** 5 * **(2, 5):** 4 * **(2, 6):** 3 * **(2, 7):** 1 * **(2, 8):** 1 * **(2, 9):** 1 * **(2, 10):** 1 * **(3, 1):** 6 * **(3, 2):** 7 * **(3, 3):** 4 * **(3, 4):** 9 * **(3, 5):** 6 * **(3, 6):** 4 * **(3, 7):** 2 * **(3, 8):** 5 * **(3, 9):** 6 * **(3, 10):** 6 * **(4, 1):** 5 * **(4, 2):** 5 * **(4, 3):** 6 * **(4, 4):** 5 * **(4, 5):** 5 * **(4, 6):** 4 * **(4, 7):** 5 * **(4, 8):** 6 * **(4, 9):** 6 * **(4, 10):** 6 * **(5, 1):** 6 * **(5, 2):** 8 * **(5, 3):** 5 * **(5, 4):** 3 * **(5, 5):** 3 * **(5, 6):** 5 * **(5, 7):** 4 * **(5, 8):** 6 * **(5, 9):** 4 * **(5, 10):** 4 * **(6, 1):** 6 * **(6, 2):** 7 * **(6, 3):** 4 * **(6, 4):** 4 * **(6, 5):** 3 * **(6, 6):** 4 * **(6, 7):** 5 * **(6, 8):** 7 * **(6, 9):** 6 * **(6, 10):** 6 * **(7, 1):** 6 * **(7, 2):** 4 * **(7, 3):** 4 * **(7, 4):** 3 * **(7, 5):** 5 * **(7, 6):** 5 * **(7, 7):** 6 * **(7, 8):** 7 * **(7, 9):** 6 * **(7, 10):** 6 * **(8, 1):** 5 * **(8, 2):** 7 * **(8, 3):** 5 * **(8, 4):** 6 * **(8, 5):** 7 * **(8, 6):** 10 * **(8, 7):** 9 * **(8, 8):** 8 * **(8, 9):** 6 * **(8, 10):** 6 * **(9, 1):** 6 * **(9, 2):** 7 * **(9, 3):** 8 * **(9, 4):** 8 * **(9, 5):** 10 * **(9, 6):** 14 * **(9, 7):** 10 * **(9, 8):** 8 * **(9, 9):** 5 * **(9, 10):** 7 * **(10, 1):** 5 * **(10, 2):** 6 * **(10, 3):** 9 * **(10, 4):** 10 * **(10, 5):** 9 * **(10, 6):** 7 * **(10, 7):** 8 * **(10, 8):** 6 * **(10, 9):** 6 * **(10, 10):** 6 **After GRPO (Right Heatmap):** The heatmap shows a generally increased value across the board, with the highest values concentrated in the bottom-right corner. * **(1, 1):** 75 * **(1, 2):** 69 * **(1, 3):** 70 * **(1, 4):** 67 * **(1, 5):** 67 * **(1, 6):** 56 * **(1, 7):** 54 * **(1, 8):** 51 * **(1, 9):** 53 * **(1, 10):** 51 * **(2, 1):** 72 * **(2, 2):** 71 * **(2, 3):** 70 * **(2, 4):** 68 * **(2, 5):** 70 * **(2, 6):** 67 * **(2, 7):** 67 * **(2, 8):** 75 * **(2, 9):** 75 * **(2, 10):** 75 * **(3, 1):** 73 * **(3, 2):** 74 * **(3, 3):** 75 * **(3, 4):** 76 * **(3, 5):** 72 * **(3, 6):** 75 * **(3, 7):** 78 * **(3, 8):** 82 * **(3, 9):** 82 * **(3, 10):** 82 * **(4, 1):** 71 * **(4, 2):** 80 * **(4, 3):** 79 * **(4, 4):** 78 * **(4, 5):** 79 * **(4, 6):** 81 * **(4, 7):** 82 * **(4, 8):** 88 * **(4, 9):** 88 * **(4, 10):** 88 * **(5, 1):** 74 * **(5, 2):** 80 * **(5, 3):** 81 * **(5, 4):** 83 * **(5, 5):** 85 * **(5, 6):** 88 * **(5, 7):** 92 * **(5, 8):** 93 * **(5, 9):** 93 * **(5, 10):** 93 * **(6, 1):** 71 * **(6, 2):** 78 * **(6, 3):** 85 * **(6, 4):** 85 * **(6, 5):** 89 * **(6, 6):** 93 * **(6, 7):** 95 * **(6, 8):** 96 * **(6, 9):** 96 * **(6, 10):** 96 * **(7, 1):** 53 * **(7, 2):** 67 * **(7, 3):** 84 * **(7, 4):** 86 * **(7, 5):** 90 * **(7, 6):** 94 * **(7, 7):** 95 * **(7, 8):** 96 * **(7, 9):** 96 * **(7, 10):** 96 * **(8, 1):** 52 * **(8, 2):** 68 * **(8, 3):** 81 * **(8, 4):** 87 * **(8, 5):** 89 * **(8, 6):** 94 * **(8, 7):** 96 * **(8, 8):** 96 * **(8, 9):** 96 * **(8, 10):** 96 * **(9, 1):** 52 * **(9, 2):** 72 * **(9, 3):** 84 * **(9, 4):** 88 * **(9, 5):** 92 * **(9, 6):** 96 * **(9, 7):** 98 * **(9, 8):** 97 * **(9, 9):** 96 * **(9, 10):** 96 * **(10, 1):** 77 * **(10, 2):** 86 * **(10, 3):** 90 * **(10, 4):** 93 * **(10, 5):** 97 * **(10, 6):** 97 * **(10, 7):** 96 * **(10, 8):** 95 * **(10, 9):** 95 * **(10, 10):** 95 ### Key Observations * The "After GRPO" heatmap consistently shows higher values than the "Before GRPO" heatmap across all combinations of 'x' and 'y' digits. * Before GRPO, the highest values are concentrated in the upper-left corner, while after GRPO, they shift towards the bottom-right corner. * The color scale on the right heatmap is shifted towards yellow, indicating a general increase in values. ### Interpretation The data suggests that GRPO (presumably a process or intervention) has a positive impact on the performance metric being measured. The shift in the highest values from the top-left to the bottom-right corner indicates that the performance improvement is more pronounced when both the number of 'x' digits and the number of 'y' digits are higher. This could imply that GRPO is particularly effective in scenarios involving more complex inputs or larger datasets. The consistent increase in values across the board suggests a broad and systemic improvement due to GRPO. The heatmap provides a clear visual representation of this improvement, making it easy to identify the areas where GRPO has the greatest effect.

## Heatmap Comparison: Before and After GRPO ### Overview The image displays two side-by-side heatmaps comparing numerical data before and after a process labeled "GRPO." The heatmaps visualize a 10x10 grid of values, where the x-axis represents the "number of x's digits" and the y-axis represents the "number of y's digits," both ranging from 1 to 10. The "Before GRPO" heatmap uses a dark red-to-black color scale, indicating lower values, while the "After GRPO" heatmap uses a yellow-to-green color scale, indicating significantly higher values. ### Components/Axes * **Chart Titles:** * Left Chart: "Before GRPO" (centered above the left heatmap). * Right Chart: "After GRPO" (centered above the right heatmap). * **X-Axis (Both Charts):** Labeled "number of x's digits". Major tick marks and labels are present for integers 1 through 10. * **Y-Axis (Both Charts):** Labeled "number of y's digits". Major tick marks and labels are present for integers 1 through 10. * **Legends/Color Bars:** * **Before GRPO (Right side of left chart):** A vertical color bar ranging from black (bottom, value 0) to bright red (top, value 40). The scale is labeled with values 0, 10, 20, 30, 40. * **After GRPO (Right side of right chart):** A vertical color bar ranging from yellow (bottom, value 40) to dark green (top, value 100). The scale is labeled with values 40, 60, 80, 100. * **Grid Structure:** Both charts are 10x10 grids. Each cell contains a numerical value and is colored according to its value and the corresponding color scale. ### Detailed Analysis **Data Extraction - Before GRPO (Left Heatmap):** The values are generally low, with the highest values concentrated in the top-left corner (low x and y digits). The color scale indicates values from 0 to 40. * **Row y=1:** 38, 33, 29, 19, 15, 12, 9, 6, 4, 2 * **Row y=2:** 19, 15, 4, 5, 4, 5, 4, 3, 1, 1 * **Row y=3:** 6, 7, 4, 9, 6, 4, 4, 2, 4, 5 * **Row y=4:** 5, 5, 6, 5, 4, 5, 4, 4, 5, 6 * **Row y=5:** 6, 8, 5, 4, 3, 3, 5, 4, 4, 6 * **Row y=6:** 6, 7, 4, 4, 3, 4, 4, 4, 5, 7 * **Row y=7:** 6, 4, 4, 3, 4, 5, 5, 6, 6, 7 * **Row y=8:** 5, 7, 5, 6, 5, 6, 7, 9, 10, 8 * **Row y=9:** 6, 7, 8, 8, 10, 14, 10, 8, 5, 7 * **Row y=10:** 5, 6, 9, 10, 9, 9, 7, 8, 6 **Data Extraction - After GRPO (Right Heatmap):** The values are uniformly high, mostly in the 70s, 80s, and 90s. The color scale indicates values from 40 to 100. * **Row y=1:** 75, 82, 88, 80, 80, 76, 79, 82, 44, 51 * **Row y=2:** 72, 72, 71, 70, 70, 79, 79, 82, 75 * **Row y=3:** 73, 74, 75, 76, 72, 75, 79, 78, 82 * **Row y=4:** 73, 81, 80, 79, 78, 79, 81, 82, 88, 87 * **Row y=5:** 74, 80, 80, 81, 83, 85, 88, 92, 93 * **Row y=6:** 77, 82, 83, 85, 85, 87, 89, 93, 95, 96 * **Row y=7:** 78, 81, 84, 86, 90, 94, 95, 96, 96, 98 * **Row y=8:** 79, 82, 84, 88, 92, 96, 96, 98, 97, 96 * **Row y=9:** 77, 82, 84, 88, 92, 96, 96, 98, 97, 96 * **Row y=10:** 77, 86, 88, 90, 93, 96, 97, 96, 98, 95 ### Key Observations 1. **Magnitude Shift:** The most striking observation is the dramatic increase in all values after GRPO. The "Before" values range from 1 to 38, while the "After" values range from 44 to 98. 2. **Pattern Change:** The "Before" heatmap shows a clear diagonal gradient, with values decreasing from the top-left (low x, low y) to the bottom-right (high x, high y). The "After" heatmap shows a much more uniform distribution, with a slight gradient of increasing values from the top-left to the bottom-right. 3. **Outlier:** In the "After GRPO" heatmap, the cell at (x=9, y=1) has a value of 44, which is a significant outlier, being much lower than its neighbors (82 and 51) and the overall trend of high values. 4. **Color Scale Interpretation:** The "Before" scale uses a sequential dark-to-light (black to red) scheme where lighter/brighter indicates higher value. The "After" scale uses a different sequential scheme (yellow to green) where darker green indicates higher value. ### Interpretation This visualization demonstrates the effect of a process called "GRPO" on a performance metric across different problem complexities, defined by the digit lengths of two variables (x and y). * **What the data suggests:** GRPO is highly effective. It transforms a system that performs poorly (scores <40) on most tasks, especially those with higher complexity (more digits), into one that performs excellently (scores >70, often >90) across nearly all task complexities. * **Relationship between elements:** The x and y axes likely represent the difficulty of a computational task (e.g., multiplication, where more digits mean a harder problem). The cell value is a success rate or accuracy score. The "Before" state shows the system struggles significantly as complexity increases. The "After" state shows GRPO has not only raised the baseline performance but also made the system robust to increases in complexity, as evidenced by the high scores in the bottom-right quadrant (high x, high y). * **Notable anomalies:** The outlier at (x=9, y=1) in the "After" chart (value 44) is critical. It indicates a specific, narrow failure mode persists after GRPO: tasks where x has 9 digits and y has 1 digit. This could point to a specific edge case in the algorithm or training data that was not fully addressed. * **Underlying implication:** The shift from a strong negative correlation between complexity and performance to a weak positive correlation suggests GRPO fundamentally changes how the system handles scale, possibly by improving generalization or algorithmic efficiency. The process appears to be a major optimization or training breakthrough.

## Heatmap: Performance Metrics Before and After GRPO ### Overview The image presents two side-by-side heatmaps comparing performance metrics (likely accuracy or success rates) across combinations of "number of x's digits" (1-10) and "number of y's digits" (1-10). The left heatmap shows values "Before GRPO," while the right shows values "After GRPO." Color gradients indicate magnitude, with darker colors representing lower values and lighter colors representing higher values. ### Components/Axes - **X-axis**: "number of x's digits" (1-10) - **Y-axis**: "number of y's digits" (1-10) - **Color Scale**: - **Before GRPO**: Dark red (low) → Black (high) - **After GRPO**: Yellow (low) → Light blue (high) - **Data Values**: Numerical annotations in each cell represent metric values. ### Detailed Analysis #### Before GRPO - **Range**: 1 (minimum) to 38 (maximum) - **Distribution**: - Top-left (1x1): 38 (darkest red) - Bottom-right (10x10): 6 (dark red) - Middle values cluster between 4-15 (e.g., 1x2: 15, 5x5: 5) - **Trend**: Values decrease diagonally from top-left to bottom-right. #### After GRPO - **Range**: 51 (minimum) to 98 (maximum) - **Distribution**: - Top-left (1x1): 75 (light yellow) - Bottom-right (10x10): 98 (light blue) - Middle values cluster between 60-97 (e.g., 5x5: 85, 8x8: 96) - **Trend**: Values increase diagonally from top-left to bottom-right. ### Key Observations 1. **Magnitude Shift**: All values increased post-GRPO, with the lowest value rising from 1 to 51 and the highest from 38 to 98. 2. **Consistent Improvement**: Every cell shows improvement, with no outliers or declines. 3. **Color Gradient Alignment**: Darker colors (Before) correlate with lower values; lighter colors (After) correlate with higher values. 4. **Diagonal Patterns**: Both heatmaps exhibit diagonal trends, but the direction reverses post-GRPO. ### Interpretation The data suggests GRPO significantly enhances performance across all digit-length combinations. The most dramatic improvements occur in lower-digit pairs (e.g., 1x1, 2x2), where values nearly tripled (38→75). Higher-digit pairs (e.g., 10x10) show more modest gains (6→98), indicating GRPO’s impact may scale with problem complexity. The consistent upward trend implies GRPO optimizes the underlying process uniformly, though the exact mechanism (e.g., algorithmic efficiency, error reduction) requires further investigation. The absence of declines suggests no negative side effects in the measured metric.