## Chart: Benchmark MATH500 Validation Score vs Training Step ### Overview The image is a line chart comparing the validation scores of two models, GRPO and MEL, over training steps. The x-axis represents the training step, and the y-axis represents the validation score. ### Components/Axes * **Title:** Benchmark: MATH500 * **X-axis:** Training Step, with markers at 0, 20, 40, 60, 80, 100, 120, and 140. * **Y-axis:** Validation Score, ranging from 0.78 to 0.86, with markers at 0.78, 0.80, 0.82, 0.84, and 0.86. * **Legend:** Located in the bottom-right corner. * GRPO (Blue) * MEL (Pink) ### Detailed Analysis * **GRPO (Blue):** * Trend: Generally increasing with fluctuations. * Data Points: * (0, 0.77) * (20, 0.79) * (40, 0.815) * (50, 0.805) * (60, 0.818) * (70, 0.848) * (80, 0.838) * (90, 0.848) * (100, 0.852) * (120, 0.85) * (130, 0.858) * (140, 0.844) * **MEL (Pink):** * Trend: Generally increasing with fluctuations, peaking around training step 100. * Data Points: * (0, 0.77) * (10, 0.812) * (20, 0.808) * (30, 0.844) * (40, 0.848) * (50, 0.838) * (60, 0.84) * (70, 0.854) * (80, 0.848) * (90, 0.858) * (100, 0.864) * (120, 0.854) * (130, 0.856) * (140, 0.864) ### Key Observations * Both models start with the same validation score at training step 0. * MEL generally outperforms GRPO until around training step 120, after which their performance becomes similar. * Both models exhibit fluctuations in their validation scores throughout the training process. ### Interpretation The chart illustrates the performance of two models, GRPO and MEL, on the MATH500 benchmark. The validation scores indicate how well each model generalizes to unseen data during training. MEL appears to have a slightly better initial learning curve and maintains a higher validation score for a significant portion of the training steps. However, towards the end of the training, the performance of the two models converges. The fluctuations in validation scores suggest that both models experience periods of overfitting or underfitting during training, which is common in machine learning. The data suggests that MEL might be a better choice for this particular benchmark, at least for the initial training phase.

\n ## Line Chart: Validation Score vs. Training Step (Benchmark: MATH500) ### Overview The image presents a line chart comparing the validation scores of two models, GPRO and MEL, against the training step. The chart appears to track the performance of these models 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 0.78 to 0.87, with markers at intervals of 0.02) * **Legend:** Located in the bottom-right corner. * GPRO (represented by a light blue line with circular markers) * MEL (represented by a light red line with triangular markers) * **Grid:** A faint grid is visible, aiding in reading values. ### Detailed Analysis **GPRO (Light Blue Line):** The GPRO line generally slopes upward from step 0 to approximately step 100, then plateaus and slightly declines. * Step 0: Approximately 0.78 * Step 20: Approximately 0.81 * Step 40: Approximately 0.81 * Step 60: Approximately 0.82 * Step 80: Approximately 0.84 * Step 100: Approximately 0.85 * Step 120: Approximately 0.86 * Step 140: Approximately 0.85 **MEL (Light Red Line):** The MEL line exhibits more fluctuation. It starts at approximately 0.78 at step 0, rises sharply, then experiences some dips before reaching a peak around step 100, and then declines slightly. * Step 0: Approximately 0.78 * Step 20: Approximately 0.82 * Step 40: Approximately 0.85 * Step 60: Approximately 0.84 * Step 80: Approximately 0.85 * Step 100: Approximately 0.87 * Step 120: Approximately 0.86 * Step 140: Approximately 0.86 ### Key Observations * Both models show an initial increase in validation score as training progresses. * MEL generally achieves a higher validation score than GPRO throughout most of the training process. * MEL's performance is more volatile, with larger fluctuations in validation score. * Both models appear to converge in performance towards the end of the training process (around step 120-140). ### Interpretation The chart demonstrates the learning curves of two models (GPRO and MEL) on the MATH500 benchmark. The validation score serves as a proxy for the model's generalization ability. The fact that both models' validation scores increase with training suggests that both are learning from the data. MEL's consistently higher scores indicate that it may be a more effective model for this particular benchmark. However, its higher volatility could also suggest that it is more sensitive to the training data or requires more careful hyperparameter tuning. The convergence of the two lines towards the end of the training process could indicate that GPRO is catching up to MEL, or that both models are reaching a point of diminishing returns. The data suggests that further training might not significantly improve the performance of either model. The benchmark MATH500 likely involves mathematical problem-solving, and the chart illustrates the progress of these models in mastering such tasks.

## Line Chart: Benchmark: MATH500 ### Overview The image is a line chart comparing the validation score performance of two different methods, labeled "GAPO" and "MEL," over the course of training steps on a benchmark dataset called "MATH500." The chart displays two fluctuating lines that generally trend upward, indicating improving performance with more training. ### 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, ranging from 0 to 140. * **Y-Axis:** Labeled "Validation Score." The axis is linear and marked with major ticks at intervals of 0.02, ranging from 0.78 to 0.86. * **Legend:** Located in the bottom-right corner of the plot area. It contains two entries: * A blue line with circular markers labeled "GAPO". * A red line with triangular markers labeled "MEL". ### Detailed Analysis **Data Series: GAPO (Blue line with circles)** * **Trend:** The line shows an overall upward trend with significant volatility. It starts low, rises sharply, experiences a notable dip, recovers, and reaches its peak towards the later steps before a final decline. * **Approximate Data Points:** * Step 0: ~0.78 * Step 10: ~0.80 * Step 20: ~0.81 * Step 30: ~0.82 * Step 40: ~0.82 * Step 50: ~0.805 (notable dip) * Step 60: ~0.845 * Step 70: ~0.835 * Step 80: ~0.85 * Step 90: ~0.855 * Step 100: ~0.85 * Step 110: ~0.855 * Step 120: ~0.86 (peak) * Step 130: ~0.855 * Step 140: ~0.845 **Data Series: MEL (Red line with triangles)** * **Trend:** The line also shows an overall upward trend but with a different pattern. It starts higher than GAPO, rises quickly to an early peak, fluctuates, reaches its maximum, and then shows a clear downward trend in the final steps. * **Approximate Data Points:** * Step 0: ~0.80 * Step 10: ~0.815 * Step 20: ~0.805 * Step 30: ~0.845 * Step 40: ~0.85 * Step 50: ~0.84 * Step 60: ~0.845 * Step 70: ~0.855 * Step 80: ~0.86 (peak) * Step 90: ~0.85 * Step 100: ~0.855 * Step 110: ~0.85 * Step 120: ~0.855 * Step 130: ~0.85 * Step 140: ~0.84 ### Key Observations 1. **Initial Performance:** MEL starts with a higher validation score (~0.80) than GAPO (~0.78) at step 0. 2. **Crossover Points:** The two lines intersect approximately at step 60 and again near step 100, indicating points where their performance was nearly identical. 3. **Peak Timing:** MEL reaches its peak performance (~0.86) earlier, around step 80. GAPO reaches its similar peak (~0.86) later, around step 120. 4. **Volatility:** Both methods show considerable step-to-step fluctuation, suggesting the training process or evaluation metric is noisy. 5. **Final Trend:** In the last 20 steps (120-140), the MEL line shows a more consistent downward trend, while the GAPO line's decline is less pronounced after its later peak. ### Interpretation This chart visualizes a comparative training run for two algorithms or model variants (GAPO and MEL) on the MATH500 benchmark. The "Validation Score" is the key performance metric. The data suggests that **MEL may learn faster initially**, achieving higher scores in the first third of the training steps. However, its performance peaks earlier and begins to degrade, which could be a sign of **overfitting** to the training data or instability in the later stages of optimization. In contrast, **GAPO shows a more sustained, albeit noisier, improvement** over a longer period. Its later peak suggests it might be more robust or continue to benefit from extended training. The significant dip around step 50 for GAPO is an anomaly that could correspond to a specific event in the training process, such as a learning rate change or a challenging batch of data. The overall takeaway is a trade-off: MEL offers quicker gains, while GAPO demonstrates potentially more stable long-term learning. The choice between them would depend on the available training budget (steps) and the importance of peak performance versus training stability. The volatility in both lines indicates that evaluating performance at a single step might be misleading; observing the trend over time is crucial.

## Line Graph: Benchmark MATH500 Validation Scores ### Overview The image displays a line graph comparing the validation scores of two models (GRPO and MEL) across training steps on the MATH500 benchmark. The x-axis represents training steps (0–140), and the y-axis represents validation scores (0.78–0.86). Two data series are plotted: GRPO (blue line) and MEL (pink line). ### Components/Axes - **X-axis**: "Training Step" (0–140, increments of 20) - **Y-axis**: "Validation Score" (0.78–0.86, increments of 0.02) - **Legend**: Located in the bottom-right corner, with: - Blue circle labeled "GRPO" - Pink triangle labeled "MEL" ### Detailed Analysis #### GRPO (Blue Line) - **Initial Phase**: Starts at 0.78 (step 0), rises sharply to 0.81 by step 20. - **Mid-Phase**: Dips to 0.80 at step 40, then climbs to 0.84 by step 60. - **Late Phase**: Peaks at 0.86 around step 100, stabilizes at ~0.85 by step 140. - **Key Values**: - Step 0: 0.78 - Step 20: 0.81 - Step 40: 0.80 - Step 60: 0.84 - Step 80: 0.85 - Step 100: 0.86 - Step 120: 0.85 - Step 140: 0.84 #### MEL (Pink Line) - **Initial Phase**: Begins at 0.77 (step 0), jumps to 0.81 by step 20. - **Mid-Phase**: Dips to 0.83 at step 40, rises to 0.85 by step 60. - **Late Phase**: Peaks at 0.87 around step 90, fluctuates between 0.85–0.87 until step 140. - **Key Values**: - Step 0: 0.77 - Step 20: 0.81 - Step 40: 0.83 - Step 60: 0.85 - Step 80: 0.84 - Step 90: 0.87 - Step 110: 0.86 - Step 130: 0.85 - Step 140: 0.87 ### Key Observations 1. **Trend Direction**: Both lines show an overall upward trend, but MEL exhibits greater volatility. 2. **Performance Gap**: MEL consistently outperforms GRPO in later stages (steps 80–140), with a maximum score of 0.87 vs. GRPO's 0.86. 3. **Volatility**: MEL's line has sharper peaks and troughs (e.g., drop from 0.87 at step 90 to 0.85 at step 110), while GRPO's trajectory is smoother. 4. **Convergence**: By step 140, both models plateau near 0.85–0.87, suggesting diminishing returns after extensive training. ### Interpretation The graph demonstrates that MEL achieves higher validation scores on the MATH500 benchmark but with greater instability compared to GRPO. The MEL model's volatility might indicate overfitting or sensitivity to training dynamics, whereas GRPO's steadier progression suggests robustness. The convergence at later steps implies both models reach similar performance ceilings, but MEL's higher peaks could justify its use in scenarios prioritizing maximum accuracy despite variability. The initial dip in GRPO's score at step 40 may reflect a learning phase adjustment, while MEL's early rise highlights rapid early learning. These patterns underscore trade-offs between stability and peak performance in model selection.