## Line Chart: EGA vs. Environment Step for Different Alpha Values ### Overview The image is a line chart that plots the EGA (Expected Goal Achievement) on the y-axis against the Environment Step on the x-axis. There are four lines, each representing a different value of alpha (αs): 1, 2, 3, and 4. The chart illustrates how the EGA changes over time (environment steps) for each alpha value. The chart includes a legend in the bottom-right corner that identifies each line by its corresponding alpha value. ### Components/Axes * **X-axis:** Environment step, ranging from 0 to 3000, with major ticks at 0, 1000, 2000, and 3000. * **Y-axis:** EGA (Expected Goal Achievement), ranging from 0.2 to 1.0, with major ticks at 0.2, 0.4, 0.6, 0.8, and 1.0. * **Legend:** Located in the bottom-right corner, the legend identifies each line by its alpha value: * Dark Blue: αs = 1 * Orange: αs = 2 * Blue: αs = 3 * Green: αs = 4 ### Detailed Analysis * **αs = 1 (Dark Blue):** The EGA starts around 0.2, remains relatively flat until approximately environment step 500, then increases steadily until it reaches approximately 0.95 around environment step 1500, after which it plateaus. * **αs = 2 (Orange):** The EGA starts around 0.2, remains relatively flat until approximately environment step 500, then increases steadily until it reaches approximately 0.95 around environment step 1250, after which it plateaus. * **αs = 3 (Blue):** The EGA starts around 0.2, remains relatively flat until approximately environment step 500, then increases steadily until it reaches approximately 0.95 around environment step 1000, after which it plateaus. * **αs = 4 (Green):** The EGA starts around 0.15, remains relatively flat until approximately environment step 250, then increases steadily until it reaches approximately 0.97 around environment step 1000, after which it plateaus. ### Key Observations * All four lines show a similar trend: a period of low EGA followed by a rapid increase and then a plateau. * The alpha values affect the speed at which the EGA increases. Higher alpha values (3 and 4) reach the plateau faster than lower alpha values (1 and 2). * The final EGA value is approximately the same for all alpha values, around 0.95 to 0.97. * The shaded regions around each line likely represent the standard deviation or confidence interval, indicating the variability in the EGA for each alpha value. ### Interpretation The chart suggests that the alpha value influences the learning rate or the speed at which the agent achieves a high EGA. Higher alpha values lead to faster learning, as indicated by the steeper increase in EGA at earlier environment steps. However, the final EGA achieved is similar across all alpha values, suggesting that the alpha value primarily affects the learning speed rather than the ultimate performance. The shaded regions indicate the variability in the learning process, which is also influenced by the alpha value. The data demonstrates that increasing alpha beyond a certain point (likely between 3 and 4) provides diminishing returns in terms of learning speed, as the green line (αs = 4) plateaus only slightly earlier than the blue line (αs = 3).

\n ## Line Chart: EGA vs. Environment Step for Different Alpha Values ### Overview This image presents a line chart illustrating the relationship between EGA (presumably a performance metric) and the Environment Step, for four different values of αs (alpha_s). The chart shows how EGA evolves over time (Environment Step) for each αs value. ### Components/Axes * **X-axis:** "Environment step" ranging from approximately 0 to 3000. * **Y-axis:** "EGA" ranging from approximately 0.0 to 1.0. * **Legend:** Located in the top-right corner, it identifies four lines: * αs = 1 (Dark Blue) * αs = 2 (Gray) * αs = 3 (Orange) * αs = 4 (Green) * **Grid:** A light gray grid is present in the background to aid in reading values. ### Detailed Analysis The chart displays four distinct lines, each representing a different αs value. * **αs = 1 (Dark Blue):** The line starts at approximately 0.15 at Environment Step 0. It increases relatively slowly until around Environment Step 500, then accelerates, reaching approximately 0.95 at Environment Step 2500. * **αs = 2 (Gray):** This line begins at approximately 0.18 at Environment Step 0. It exhibits a similar trend to αs = 1, but with a slightly faster initial increase. It reaches approximately 0.98 at Environment Step 2500. * **αs = 3 (Orange):** Starting at approximately 0.17 at Environment Step 0, this line shows a rapid increase from the beginning. It reaches approximately 0.99 at Environment Step 1500 and plateaus. * **αs = 4 (Green):** This line starts at approximately 0.13 at Environment Step 0. It shows the fastest initial increase among all four lines, reaching approximately 0.97 at Environment Step 1200 and then plateaus. All lines converge towards a value of approximately 1.0 as the Environment Step increases. ### Key Observations * Higher αs values (3 and 4) lead to faster initial increases in EGA. * The lines representing αs = 3 and αs = 4 plateau earlier than those representing αs = 1 and αs = 2. * The difference in EGA values between the lines diminishes as the Environment Step increases, suggesting convergence. * There is a slight variation in the final EGA values achieved by each αs value, with αs = 2 reaching the highest value (approximately 0.98). ### Interpretation The data suggests that the parameter αs significantly influences the rate at which EGA increases with respect to the Environment Step. Higher values of αs result in faster initial gains in EGA, but also lead to earlier saturation. This could indicate that there's a trade-off between the speed of learning/adaptation (represented by EGA) and the potential for further improvement. The convergence of the lines at higher Environment Steps suggests that, regardless of the initial αs value, the system eventually reaches a similar level of performance. The slight difference in the final EGA values might be due to inherent stochasticity in the system or other factors not represented in this chart. The chart demonstrates a learning curve, where performance (EGA) improves with experience (Environment Step), and the learning rate is modulated by the αs parameter.

## Line Chart: EGA vs. Environment Step for Different αₛ Values ### Overview The image is a line chart plotting the performance metric "EGA" against "Environment step" for four different experimental conditions, labeled by the parameter αₛ. The chart shows learning curves where performance improves with more environment steps, eventually plateauing near a maximum value. ### Components/Axes * **X-Axis (Horizontal):** * **Label:** "Environment step" * **Scale:** Linear, ranging from 0 to 3000. * **Major Tick Marks:** 0, 1000, 2000, 3000. * **Y-Axis (Vertical):** * **Label:** "EGA" * **Scale:** Linear, ranging from 0.0 to 1.0. * **Major Tick Marks:** 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. * **Legend:** * **Position:** Bottom-right corner of the plot area. * **Content:** Four entries, each associating a colored line with a value of αₛ. * Dark Blue Line: αₛ = 1 * Orange Line: αₛ = 2 * Blue Line: αₛ = 3 * Green Line: αₛ = 4 * **Data Series:** Four lines, each with a shaded region (likely representing confidence intervals or standard deviation) around the central trend line. ### Detailed Analysis **Trend Verification & Data Points:** All four lines follow a similar sigmoidal (S-shaped) learning curve: a slow start, a period of rapid increase, and a final plateau. The primary difference is the speed of learning. 1. **αₛ = 4 (Green Line):** * **Trend:** This line shows the fastest initial learning. It rises most steeply from the start. * **Key Points:** Begins at EGA ≈ 0.15 at step 0. Crosses EGA = 0.6 at approximately step 600. Reaches a near-plateau (EGA ≈ 0.95) by step ~1200. It is the first to converge. 2. **αₛ = 3 (Blue Line):** * **Trend:** The second-fastest learning curve. It closely follows the green line initially but begins to lag slightly after step ~500. * **Key Points:** Similar starting point (≈0.15). Crosses EGA = 0.6 at approximately step 700. Reaches the plateau (EGA ≈ 0.95) by step ~1400. 3. **αₛ = 2 (Orange Line):** * **Trend:** Shows a moderate learning speed, clearly slower than αₛ=3 and 4. * **Key Points:** Starts at ≈0.15. Crosses EGA = 0.6 at approximately step 800. Reaches the plateau (EGA ≈ 0.95) by step ~1600. 4. **αₛ = 1 (Dark Blue Line):** * **Trend:** This is the slowest learning curve. It has the most gradual slope during the rapid learning phase. * **Key Points:** Starts at ≈0.15. Crosses EGA = 0.6 at approximately step 1000. Reaches the plateau (EGA ≈ 0.95) by step ~1800. **Convergence:** All four lines converge to the same final performance level, a plateau at an EGA value of approximately 0.95 to 0.97, by step 2000. From step 2000 to 3000, all lines are flat and overlapping at this maximum value. ### Key Observations 1. **Clear Performance Ordering:** There is a strict, monotonic relationship between the αₛ parameter and learning speed. Higher αₛ values (4 > 3 > 2 > 1) result in faster convergence to the maximum EGA. 2. **Identical Starting and Ending Points:** All conditions begin at the same low performance (EGA ≈ 0.15) and end at the same high performance (EGA ≈ 0.96). The parameter αₛ affects the *rate* of learning, not the initial state or final asymptotic performance. 3. **Convergence Point:** The lines begin to merge and become indistinguishable from each other starting around step 1500-1800, with full convergence by step 2000. 4. **Shaded Regions:** The shaded confidence bands are narrowest at the start and end of training and widest during the period of most rapid learning (between steps 500 and 1500), indicating higher variance in performance across runs during this phase. ### Interpretation This chart demonstrates the effect of the hyperparameter αₛ on the sample efficiency of a learning algorithm (likely in reinforcement learning, given the "Environment step" axis). The data suggests that **increasing αₛ accelerates the learning process** without affecting the final achievable performance. * **What it means:** A higher αₛ value allows the agent or model to extract more useful signal from each environment interaction, leading to faster skill acquisition. This could be due to αₛ controlling a learning rate, a target update frequency, or a similar mechanism that governs how aggressively the model updates its knowledge. * **Why it matters:** In practical applications, sample efficiency is critical. Reducing the number of environment steps needed to reach peak performance (e.g., from 1800 steps with αₛ=1 to 1200 steps with αₛ=4) translates to significantly lower computational cost and faster development cycles. * **Underlying Pattern:** The sigmoidal shape is characteristic of learning curves, where initial exploration is slow, followed by a phase of rapid insight and improvement, and finally a refinement phase as performance approaches its limit. The parameter αₛ effectively stretches or compresses this curve along the time (step) axis. The fact that all curves reach the same plateau indicates the task has a fixed maximum difficulty or performance ceiling that is achievable by all configurations, given enough time.

## Line Graph: EGA Convergence Across Environment Steps ### Overview The image depicts a line graph illustrating the convergence of Expected Goal Achievement (EGA) across different environment steps for four distinct α_s values (1, 2, 3, 4). The graph shows four colored lines with shaded confidence intervals, all starting near 0.2 and converging toward 1.0 as environment steps increase. ### Components/Axes - **X-axis**: "Environment step" (logarithmic scale, 0 to 3000) - **Y-axis**: "EGA" (linear scale, 0.0 to 1.0) - **Legend**: Located in the bottom-right corner, mapping colors to α_s values: - Black: α_s = 1 - Orange: α_s = 2 - Blue: α_s = 3 - Green: α_s = 4 - **Shaded Regions**: Represent variability/confidence intervals around each line. ### Detailed Analysis 1. **Line Trends**: - **α_s = 1 (Black)**: Starts at ~0.2, rises steeply to ~0.6 by 1000 steps, then plateaus. Confidence interval widest (~±0.15). - **α_s = 2 (Orange)**: Begins at ~0.2, surpasses α_s=1 by ~500 steps, reaches ~0.8 by 1500 steps. Confidence interval narrower (~±0.10). - **α_s = 3 (Blue)**: Starts at ~0.2, overtakes α_s=2 by ~1000 steps, reaches ~0.95 by 2000 steps. Confidence interval moderate (~±0.08). - **α_s = 4 (Green)**: Highest initial slope, reaches ~0.98 by 1000 steps, plateaus at 1.0. Confidence interval narrowest (~±0.05). 2. **Convergence Patterns**: - All lines converge to 1.0 by ~2000 steps, but α_s=4 achieves stability earliest (~1000 steps). - Variability decreases with higher α_s values (green line has minimal shading). ### Key Observations - Higher α_s values correlate with faster convergence and greater stability (narrower confidence intervals). - α_s=1 exhibits the slowest convergence and highest uncertainty. - Lines cross sequentially: α_s=2 > α_s=1 > α_s=3 > α_s=4 in early steps, but α_s=4 dominates after ~1000 steps. ### Interpretation The data suggests that increasing α_s improves both the speed and reliability of EGA convergence. The green line (α_s=4) demonstrates optimal performance, achieving near-perfect EGA with minimal variability. This implies α_s=4 is the most efficient parameter setting for the modeled system. The shaded regions highlight the trade-off between exploration (wider intervals) and exploitation (narrower intervals) in reinforcement learning contexts. The logarithmic x-axis emphasizes early-stage performance differences, which are critical for parameter tuning.