## Chart: Benchmark MATH500 ### Overview The image is a line chart comparing the validation scores of two models, GRPO and MEL, over training steps. The chart shows how the validation scores change as the models are trained. ### Components/Axes * **Title:** Benchmark: MATH500 * **X-axis:** Training Step, ranging from 0 to 140 in increments of 20. * **Y-axis:** Validation Score, ranging from 0.80 to 0.90 in increments of 0.02. * **Legend:** Located in the bottom-right corner. * GRPO (blue line with circle markers) * MEL (pink line with triangle markers) ### Detailed Analysis * **GRPO (blue line):** * Starts at approximately 0.80 at training step 0. * Increases to approximately 0.83 at training step 20. * Slightly decreases to approximately 0.825 at training step 40. * Increases to approximately 0.88 at training step 60. * Slightly decreases to approximately 0.87 at training step 80. * Decreases to approximately 0.87 at training step 100. * Decreases to approximately 0.86 at training step 120. * Increases to approximately 0.89 at training step 140. * **MEL (pink line):** * Starts at approximately 0.80 at training step 0. * Increases to approximately 0.83 at training step 20. * Increases to approximately 0.88 at training step 40. * Slightly decreases to approximately 0.875 at training step 60. * Increases to approximately 0.895 at training step 80. * Slightly decreases to approximately 0.885 at training step 100. * Slightly increases to approximately 0.89 at training step 120. * Increases to approximately 0.905 at training step 140. ### Key Observations * Both models start with the same validation score. * The MEL model generally has a higher validation score than the GRPO model after the initial training steps. * Both models show fluctuations in validation scores during training. * The MEL model shows a more consistent upward trend overall. ### Interpretation The chart compares the performance of two models, GRPO and MEL, on the MATH500 benchmark. The validation scores indicate how well the models generalize to unseen data during training. The MEL model appears to perform slightly better than the GRPO model, achieving higher validation scores throughout most of the training process. The fluctuations in validation scores suggest that both models experience some instability during training, but the MEL model seems to recover more effectively. The final validation scores at training step 140 suggest that the MEL model has a slightly better generalization capability on the MATH500 benchmark.

\n ## Line Chart: Validation Score vs. Training Step (Benchmark: MATH500) ### Overview This image presents a line chart illustrating the validation score of two models, GRPO and MEL, against the training step. The chart appears to track the performance of these models during a training process on the MATH500 benchmark. ### Components/Axes * **Title:** Benchmark: MATH500 (positioned at the top-center) * **X-axis:** Training Step (ranging from approximately 0 to 140, with markers at intervals of 20) * **Y-axis:** Validation Score (ranging from approximately 0.80 to 0.91, with markers at intervals of 0.02) * **Legend:** Located in the bottom-right corner. * GRPO (represented by a blue line with circular markers) * MEL (represented by a pink line with triangular markers) ### Detailed Analysis **GRPO (Blue Line):** The GRPO line initially slopes upward from approximately 0.81 at a training step of 0 to a peak of around 0.88 at a training step of 60. It then declines to approximately 0.86 at a training step of 80, fluctuates between 0.86 and 0.88 until a training step of 120, and then sharply decreases to approximately 0.85 at a training step of 140. * (0, 0.81) * (20, 0.83) * (40, 0.85) * (60, 0.88) * (80, 0.86) * (100, 0.87) * (120, 0.89) * (140, 0.85) **MEL (Pink Line):** The MEL line starts at approximately 0.80 at a training step of 0, rises to a peak of around 0.89 at a training step of 80, dips to approximately 0.88 at a training step of 100, and then increases to approximately 0.91 at a training step of 120. Finally, it decreases to approximately 0.89 at a training step of 140. * (0, 0.80) * (20, 0.83) * (40, 0.87) * (60, 0.88) * (80, 0.89) * (100, 0.88) * (120, 0.91) * (140, 0.89) ### Key Observations * Both models show an initial increase in validation score as training progresses. * MEL consistently achieves a higher validation score than GRPO throughout most of the training process. * GRPO experiences a more significant drop in validation score towards the end of the training process (at training step 140). * MEL reaches its peak performance at training step 120, while GRPO's performance plateaus before that point. ### Interpretation The chart suggests that MEL is a more robust model for the MATH500 benchmark, consistently outperforming GRPO. The initial increase in validation score for both models indicates that they are learning from the training data. The decline in GRPO's performance at the end of the training process could indicate overfitting or a need for adjustments to the training parameters. The peak performance of MEL at step 120 suggests an optimal training duration for this model on this benchmark. The difference in the final validation scores highlights the potential benefits of using MEL over GRPO for this specific task. The data suggests that continued training beyond step 120 may not be beneficial for MEL, and could even lead to a decrease in performance.

## Line Chart: Benchmark: MATH500 ### Overview The image displays a line chart comparing the validation score performance of two different methods, labeled "GAPO" and "MEL," over the course of training steps on the MATH500 benchmark. The chart tracks how the validation score for each method evolves as training progresses. ### Components/Axes * **Chart Title:** "Benchmark: MATH500" (centered at the top). * **X-Axis:** Labeled "Training Step." The axis is linear and marked with major ticks at intervals of 20, from 0 to 140. * **Y-Axis:** Labeled "Validation Score." The axis is linear and marked with major ticks at intervals of 0.02, from 0.80 to 0.90. * **Legend:** Located in the top-left corner of the plot area. It contains two entries: * A blue line with a circular marker labeled "GAPO". * A red line with a circular marker labeled "MEL". * **Data Series:** Two lines with markers at each data point. * **GAPO (Blue Line):** Represents one method's performance. * **MEL (Red Line):** Represents the second method's performance. ### Detailed Analysis **Trend Verification & Data Point Extraction:** * **GAPO (Blue Line) Trend:** The line shows an overall upward trend with moderate volatility. It starts low, rises quickly, experiences a period of fluctuation between steps 40-100, and then shows a final decline. * Step 0: ~0.80 * Step 10: ~0.825 * Step 20: ~0.83 * Step 30: ~0.83 * Step 40: ~0.83 * Step 50: ~0.855 * Step 60: ~0.88 (local peak) * Step 70: ~0.87 * Step 80: ~0.87 * Step 90: ~0.865 * Step 100: ~0.86 * Step 110: ~0.875 * Step 120: ~0.89 (global peak for GAPO) * Step 130: ~0.885 * Step 140: ~0.85 * **MEL (Red Line) Trend:** The line shows a more volatile but stronger overall upward trend, culminating in the highest score on the chart. It has a notable early dip. * Step 0: ~0.80 * Step 10: ~0.825 * Step 20: ~0.81 (significant dip) * Step 30: ~0.84 * Step 40: ~0.875 * Step 50: ~0.875 * Step 60: ~0.87 * Step 70: ~0.88 * Step 80: ~0.89 * Step 90: ~0.88 * Step 100: ~0.885 * Step 110: ~0.885 * Step 120: ~0.89 * Step 130: ~0.885 * Step 140: ~0.91 (global peak for the chart) ### Key Observations 1. **Final Performance:** At the final recorded step (140), MEL (~0.91) significantly outperforms GAPO (~0.85). 2. **Volatility:** MEL exhibits greater volatility, especially in the early training steps (sharp dip at step 20) and the final ascent. GAPO's path is somewhat smoother but still shows fluctuations. 3. **Peak Timing:** GAPO reaches its peak performance earlier (around step 120) before declining. MEL's performance is still climbing at the end of the charted range. 4. **Initial Phase:** Both methods start at the same point (~0.80) and perform similarly for the first 10 steps. They diverge significantly after step 20. 5. **Crossover Points:** The lines cross multiple times (e.g., near steps 10, 60, 70, 110, 120), indicating periods where one method temporarily surpasses the other before the lead changes again. ### Interpretation The chart suggests a trade-off between stability and peak performance for these two methods on the MATH500 benchmark. **GAPO** appears to be a more stable learner, avoiding major early setbacks but also failing to achieve the highest possible score, with performance degrading after step 120. This could indicate overfitting or a suboptimal learning rate schedule in later stages. **MEL**, in contrast, demonstrates a "high-risk, high-reward" profile. Its significant early dip suggests initial instability or sensitivity to early training conditions. However, it recovers strongly and ultimately achieves a superior final validation score, indicating it may have a higher capacity for learning or a better optimization trajectory in the long run. The fact that its score is still rising at step 140 implies that further training might yield even better results, whereas GAPO's performance has already peaked and begun to fall. The multiple crossover points highlight that the "better" method is not constant throughout training; the choice between them could depend on the available training budget (steps) or the need for consistent, predictable improvement versus chasing the absolute highest score.

## Line Graph: MATH500 Benchmark Validation Scores ### Overview The image displays a line graph comparing the validation scores of two optimization methods (GRPO and MEL) across 140 training steps on the MATH500 benchmark. The graph shows fluctuating performance trends with notable divergence between the two methods in later stages. ### Components/Axes - **X-axis**: Training Step (0 to 140, increments of 20) - **Y-axis**: Validation Score (0.80 to 0.90, increments of 0.02) - **Legend**: Located in bottom-right corner - Blue line: GRPO - Pink line: MEL - **Title**: "Benchmark: MATH500" (top-center) ### Detailed Analysis 1. **GRPO (Blue Line)**: - Starts at 0.80 (training step 0) - Peaks at 0.88 (training step 60) - Experiences a sharp dip to 0.85 (training step 100) - Final score: 0.85 (training step 140) - Notable volatility between steps 80-120 2. **MEL (Pink Line)**: - Starts at 0.80 (training step 0) - Rapid ascent to 0.88 (training step 40) - Maintains 0.88-0.89 range until step 120 - Final score: 0.90 (training step 140) - Steady upward trend after step 80 ### Key Observations - MEL consistently outperforms GRPO after training step 60 - GRPO shows significant instability between steps 80-100 - Both methods start with identical performance at step 0 - MEL achieves 0.05 higher validation score than GRPO by final step - GRPO's peak occurs earlier (step 60) vs MEL's sustained high performance ### Interpretation The data suggests MEL demonstrates superior optimization stability and final performance on the MATH500 benchmark. The GRPO method's mid-training dip (step 100) indicates potential overfitting or parameter instability, while MEL's gradual ascent suggests more robust convergence properties. The 0.05 score difference at final step highlights MEL's effectiveness in maintaining high validation performance through extended training. This pattern aligns with MEL's theoretical advantages in gradient estimation for complex optimization landscapes.