## Line Graph: Accuracy Over Time (Acc_test vs. t) ### Overview The graph depicts the relationship between time (`t`) and test accuracy (`Acc_test`) for different parameter configurations. Six distinct lines represent combinations of learning rate (`λ`), regularization strength (`μ`), and model complexity (`K`). All lines exhibit a similar sigmoidal trend, peaking around `t=1` before declining. ### Components/Axes - **X-axis**: Time (`t`), ranging from 0.0 to 2.5 in increments of 0.5. - **Y-axis**: Test accuracy (`Acc_test`), ranging from 0.60 to 0.90 in increments of 0.05. - **Legend**: Located in the bottom-right corner, mapping colors to parameters: - Solid lines: Learning rate (`λ`) and regularization (`μ`) combinations. - Dashed lines: Model complexity (`K`) values. - **Line styles**: - Cyan: `λ=1.5, μ=3` - Blue: `λ=1, μ=2` - Dark blue: `λ=0.7, μ=1` - Magenta (dashed): `K=16` - Purple (dashed): `K=4` - Black (dashed): `K=2` ### Detailed Analysis 1. **`λ=1.5, μ=3` (cyan)**: - Peaks at `t=1` with `Acc_test ≈ 0.90`. - Declines steadily to `0.88` by `t=2.5`. - Highest accuracy among all lines. 2. **`λ=1, μ=2` (blue)**: - Peaks at `t=1` with `Acc_test ≈ 0.89`. - Declines to `0.87` by `t=2.5`. - Second-highest accuracy. 3. **`λ=0.7, μ=1` (dark blue)**: - Peaks at `t=1` with `Acc_test ≈ 0.88`. - Declines to `0.86` by `t=2.5`. - Third-highest accuracy. 4. **`K=16` (magenta, dashed)**: - Peaks at `t=1` with `Acc_test ≈ 0.89`. - Declines to `0.87` by `t=2.5`. - Matches the `λ=1, μ=2` line in accuracy. 5. **`K=4` (purple, dashed)**: - Peaks at `t=1` with `Acc_test ≈ 0.87`. - Declines to `0.85` by `t=2.5`. - Matches the `λ=0.7, μ=1` line in accuracy. 6. **`K=2` (black, dashed)**: - Peaks at `t=1` with `Acc_test ≈ 0.85`. - Declines to `0.83` by `t=2.5`. - Lowest accuracy among all lines. ### Key Observations - **Parameter correlation**: Higher `λ` and `μ` values correlate with higher peak accuracy (e.g., `λ=1.5, μ=3` outperforms `λ=0.7, μ=1`). - **Model complexity**: Larger `K` values (e.g., `K=16`) achieve accuracy comparable to moderate `λ`/`μ` combinations. - **Consistent trends**: All lines follow a sigmoidal pattern, suggesting diminishing returns after `t=1`. - **Color consistency**: Legend colors match line styles and parameter groupings exactly. ### Interpretation The graph demonstrates that: 1. **Learning rate and regularization** (`λ`, `μ`) significantly impact peak accuracy, with higher values yielding better performance. 2. **Model complexity** (`K`) acts as a proxy for capacity, where larger `K` values achieve similar accuracy to optimized `λ`/`μ` pairs. 3. The decline after `t=1` implies potential overfitting or saturation effects, as further training reduces generalization. 4. The `K=2` line (lowest accuracy) suggests underfitting due to insufficient model capacity. This analysis highlights the trade-offs between hyperparameter tuning and architectural complexity in optimizing test accuracy.