\n ## Heatmap: Correlation Heatmap of Reward Functions ### Overview The image presents a correlation heatmap visualizing the relationships between three reward functions: R_LL, R_SE, and R_SD. The heatmap uses a color gradient to represent correlation coefficients, ranging from -1.00 (blue) to 1.00 (red). ### Components/Axes * **Title:** "Correlation Heatmap of Reward Functions" (top-center) * **X-axis Labels:** R_LL, R_SE, R_SD (bottom) * **Y-axis Labels:** R_LL, R_SE, R_SD (left) * **Color Legend:** Located in the top-right corner, ranging from blue (-1.00) to red (1.00), with intermediate values marked at -0.75, -0.50, -0.25, 0.25, 0.50, 0.75. ### Detailed Analysis The heatmap displays the correlation coefficients between each pair of reward functions. * **R_LL vs. R_LL:** 1.00 (dark red, top-left cell) - Perfect positive correlation. * **R_LL vs. R_SE:** 0.13 (light red, top-middle cell) - Weak positive correlation. * **R_LL vs. R_SD:** 0.02 (very light red, top-right cell) - Very weak positive correlation. * **R_SE vs. R_LL:** 0.13 (light red, middle-left cell) - Weak positive correlation. * **R_SE vs. R_SE:** 1.00 (dark red, middle-center cell) - Perfect positive correlation. * **R_SE vs. R_SD:** -0.10 (light blue, middle-right cell) - Weak negative correlation. * **R_SD vs. R_LL:** 0.02 (very light red, bottom-left cell) - Very weak positive correlation. * **R_SD vs. R_SE:** -0.10 (light blue, bottom-middle cell) - Weak negative correlation. * **R_SD vs. R_SD:** 1.00 (dark red, bottom-right cell) - Perfect positive correlation. ### Key Observations * Each reward function has a perfect positive correlation with itself (diagonal cells are all 1.00). * R_LL and R_SE exhibit a weak positive correlation. * R_SE and R_SD show a weak negative correlation. * The correlations between R_LL and R_SD are very close to zero, indicating almost no linear relationship. ### Interpretation The heatmap reveals the degree of linear association between the three reward functions. The strong positive correlations along the diagonal are expected, as each function is perfectly correlated with itself. The weak positive correlation between R_LL and R_SE suggests that when R_LL increases, R_SE tends to increase slightly, but the relationship is not strong. The weak negative correlation between R_SE and R_SD indicates that as R_SE increases, R_SD tends to decrease slightly. The near-zero correlation between R_LL and R_SD suggests these two reward functions are largely independent of each other in a linear sense. This information is valuable for understanding how these reward functions might interact when used in a reinforcement learning or optimization context. If the reward functions are highly correlated, they may provide redundant signals, while uncorrelated functions may offer more diverse information to the learning agent. The heatmap provides a concise visual summary of these relationships.