## Heatmap: Distribution of Average Iterations ### Overview The heatmap shows the distribution of average iterations required for two different methods, "Ours" and "KL-only," as the sample complexity increases. The x-axis represents the sample complexity, and the y-axis represents the average iterations. ### Components/Axes - **x-axis**: Sample Complexity (k) - **y-axis**: Average Iterations - **Legend**: - "Ours" (blue) - "KL-only" (orange) ### Detailed Analysis or Content Details The heatmap illustrates that both methods require an increasing number of iterations as the sample complexity increases. The "Ours" method (blue) generally requires fewer iterations compared to the "KL-only" method (orange). However, the "Ours" method shows a more consistent increase in iterations, while the "KL-only" method experiences a peak before decreasing. The "Ours" method's iterations start lower and reach higher values as the sample complexity increases, indicating a potentially more efficient or robust approach. ### Key Observations - The "Ours" method consistently requires fewer iterations than the "KL-only" method across all sample complexities. - The "KL-only" method shows a peak in iterations at a sample complexity of around 22, after which the iterations decrease. - The "Ours" method's iterations start lower and reach higher values as the sample complexity increases. ### Interpretation The heatmap suggests that the "Ours" method is more efficient in terms of the number of iterations required as the sample complexity increases. This could indicate that the "Ours" method is better at handling larger datasets or is more robust to variations in sample complexity. The "KL-only" method, while requiring fewer iterations overall, has a higher peak in iterations, which might suggest it is more sensitive to changes in sample complexity. The "Ours" method's more consistent increase in iterations could also imply that it is more stable and reliable in its performance.

## Line Graph: Average Iterations vs. Sample Complexity (k) ### Overview The image is a line graph comparing the performance of two methods ("Ours" and "KL-only") across varying sample complexities (k). The x-axis represents sample complexity (k), ranging from 2 to 37, while the y-axis represents average iterations, scaled from 0 to 15. Two data series are plotted: a blue line labeled "Ours" and an orange line labeled "KL-only". ### Components/Axes - **X-axis (Sample Complexity, k)**: - Labels: "Sample Complexity (k)". - Ticks: 2, 7, 12, 17, 22, 27, 32, 37. - Scale: Linear increments of 5. - **Y-axis (Average Iterations)**: - Labels: "Avg. Iterations". - Ticks: 0, 5, 10, 15. - Scale: Linear increments of 5. - **Legend**: - Position: Top-right corner. - Entries: - Blue line: "Ours". - Orange line: "KL-only". ### Detailed Analysis - **Blue Line ("Ours")**: - Starts at approximately (2, 2) and increases steadily. - At k=37, reaches ~15 iterations. - Slope: Consistent upward trend with no plateaus or declines. - **Orange Line ("KL-only")**: - Starts similarly to "Ours" but peaks at ~10 iterations around k=22. - Declines sharply after k=22, reaching ~4 iterations by k=37. - Notable inflection point: k=22 (peak) followed by a steep drop. ### Key Observations 1. **Divergence at k=22**: The "KL-only" method peaks at k=22 (~10 iterations) and then declines, while "Ours" continues to rise. 2. **Performance Gap**: By k=37, "Ours" achieves ~15 iterations, whereas "KL-only" drops to ~4 iterations. 3. **Initial Similarity**: Both methods start with nearly identical performance at low k values (e.g., k=2–7). ### Interpretation The data suggests that the "Ours" method scales more effectively with increasing sample complexity compared to "KL-only". The "KL-only" method exhibits a performance ceiling, peaking at moderate k values before degrading, potentially due to overfitting or inefficiency at higher complexities. In contrast, "Ours" maintains a linear improvement, indicating robustness or adaptive optimization. This divergence highlights a critical threshold (k≈22) where "KL-only" becomes less viable, while "Ours" remains consistently effective. The results imply that "Ours" may be preferable for high-complexity scenarios, whereas "KL-only" might suffice for simpler tasks but fails to generalize.