## Box Plot: Normalized MSE on LSR-Transform
### Overview
The image is a box plot comparing the performance of three different methods on a task called "LSR-Transform." Performance is measured by the logarithm (base 10) of the Normalized Mean Squared Error (MSE). Lower values indicate better performance. The plot includes a horizontal red dashed reference line.
### Components/Axes
* **Title:** "Normalized MSE on LSR-Transform"
* **Y-Axis:**
* **Label:** `log10(Normalized MSE)`
* **Scale:** Logarithmic. Major tick marks are visible at -15, -14, -13, -2.0, -1.5, 0, and 1. The axis has a break (indicated by diagonal hatching) between approximately -13 and -2.0, compressing this range.
* **X-Axis:**
* **Label:** `Method`
* **Categories (from left to right):**
1. `PySR` (Blue box)
2. `KeplerAgent @1` (Green box)
3. `KeplerAgent @3` (Red/Brown box)
* **Reference Line:** A red dashed horizontal line is positioned at `y = -2.0`.
* **Data Annotations:** Below each box, the median and mean values are provided in scientific notation.
### Detailed Analysis
**1. PySR (Blue Box, Left):**
* **Visual Trend:** The box spans from approximately -14.0 to -1.3 on the y-axis. The median line is near the bottom of the box. The upper whisker extends to near 0.8, and the lower whisker extends to near -14.5.
* **Annotated Values:**
* `median = 4.47×10⁻⁵` (This corresponds to `log10(4.47e-5) ≈ -4.35`)
* `mean = 2.82×10⁻¹` (This corresponds to `log10(0.282) ≈ -0.55`)
* **Interpretation:** The large discrepancy between the median (-4.35) and mean (-0.55) indicates a highly right-skewed distribution. Most runs have very low error (median), but a few runs with very high error pull the mean up significantly.
**2. KeplerAgent @1 (Green Box, Center):**
* **Visual Trend:** The box spans from approximately -14.0 to -1.4. The median line is slightly above the bottom of the box. The upper whisker extends to near 0.5, and the lower whisker extends to near -14.5.
* **Annotated Values:**
* `median = 1.40×10⁻⁴` (This corresponds to `log10(1.40e-4) ≈ -3.85`)
* `mean = 1.50×10⁻¹` (This corresponds to `log10(0.150) ≈ -0.82`)
* **Interpretation:** Similar to PySR, the distribution is right-skewed (mean > median). The median error is slightly higher than PySR's median.
**3. KeplerAgent @3 (Red/Brown Box, Right):**
* **Visual Trend:** The box spans from approximately -14.0 to -1.5. The median line is near the bottom of the box. The upper whisker extends to near 0.0, and the lower whisker extends to near -14.5.
* **Annotated Values:**
* `median = 1.94×10⁻⁵` (This corresponds to `log10(1.94e-5) ≈ -4.71`)
* `mean = 1.21×10⁻¹` (This corresponds to `log10(0.121) ≈ -0.92`)
* **Interpretation:** This method has the lowest median error of the three. It also shows right-skew, but its mean is the lowest among the three methods.
### Key Observations
1. **Performance Ranking (by Median):** KeplerAgent @3 (best, lowest median) > PySR > KeplerAgent @1 (worst, highest median).
2. **Performance Ranking (by Mean):** KeplerAgent @3 (best, lowest mean) > KeplerAgent @1 > PySR (worst, highest mean). The mean ranking differs from the median ranking due to the different skew magnitudes.
3. **Distribution Shape:** All three methods exhibit strongly right-skewed distributions of log10(Normalized MSE). This means that while the typical (median) performance is very good (errors around 10⁻⁴ to 10⁻⁵), there is a long tail of runs with much higher errors (up to ~10⁰ or 1).
4. **Spread:** The interquartile range (height of the boxes) and the whisker lengths are broadly similar across methods, indicating comparable variability in performance, aside from the skew.
5. **Reference Line:** The red dashed line at `log10(Normalized MSE) = -2.0` (Normalized MSE = 0.01) serves as a visual benchmark. The median of all three methods is well below this line, indicating that the central tendency of each method achieves an error less than 1% of the normalized scale.
### Interpretation
This chart evaluates symbolic regression or program synthesis methods (PySR and variants of KeplerAgent) on the LSR-Transform benchmark. The key takeaway is that **KeplerAgent @3 achieves the best median performance**, suggesting it is the most reliable method for producing low-error solutions on this task.
The pervasive right-skew across all methods is a critical finding. It indicates that while these algorithms often find excellent solutions, they are not perfectly robust; a subset of runs fails to converge well, resulting in high-error outliers. This could be due to random initialization, the stochastic nature of the search, or particular difficulty with certain sub-problems in the benchmark.
The comparison between `KeplerAgent @1` and `@3` suggests that the `@3` variant (which likely involves more computational resources, search depth, or ensemble size) provides a meaningful improvement in both median and mean error over the `@1` version. The fact that PySR's mean is the highest, despite having a median better than KeplerAgent @1, highlights how severely its performance is impacted by its worst-case runs.
In summary, for the LSR-Transform task, KeplerAgent @3 is the most accurate and reliable method on average, but all methods show a vulnerability to producing occasional high-error results. The red line at -2.0 provides a clear visual threshold that all medians comfortably surpass.
