## 4 Results
### Effect of Regularization Parameter on Local Explainability Metrics  
Table 3 and Figure 3 show the impact of the explainability regularization parameter \(\lambda_{1}\) on the point fidelity, neighborhood fidelity, and stability metrics for the pre hoc framework on the ML-100k dataset. The results show a clear positive relationship between the regularization strength and the quality of the explanation. In the absence of regularization, the model achieves moderate fidelity scores (point fidelity of 0.8183 and neighborhood fidelity of 0.8050) but exhibits lower stability with a high total variation score of 0.3524. This indicates that without explainability regularization, the explanations are less consistent across different instances, even when the model achieves reasonable alignment with the explainer. As we train with the regularization parameter and gradually increase the \(\lambda_{1}\), we observe significant improvements across all metrics. With a minimal regularization of \(\lambda_{1}=0.01\), point fidelity improves to 0.8473 (3.5% increase), neighborhood fidelity increases to 0.8553 (6.2% increase), and stability improves dramatically with total variation decreasing to 0.1290 (63.4% reduction). At \(\lambda_{1}=0.25\), point fidelity reaches 0.9740 (19.0% increase from no regularization), neighborhood fidelity increases to 0.9903 (23.0% increase), and stability improves substantially with total variation reduced to 0.0189 (94.6% reduction). This indicates that moderate regularization significantly enhances both the alignment between the predictor and explainer models and the consistency of explanations across different instances. At higher regularization strengths (\(\lambda_{1}=0.5\) and \(\lambda_{1}=1.0\)), the metrics continue to improve but with decreasing returns. The point fidelity reaches its peak at 0.9951 with \(\lambda_{1}=1.0\), representing a 21.6% improvement over the non-regularized model. Similarly, neighborhood fidelity reaches 0.9953, a 23.6% improvement. The stability metric plateaus at 0.0078 for both \(\lambda_{1}=0.5\) and \(\lambda_{1}=1.0\), indicating that additional regularization beyond \(\lambda_{1}=0.5\) does not further improve the consistency of explanations. These results demonstrate that incorporating explainability regularization through the Jensen-Shannon divergence significantly enhances the quality of explanations generated by the pre hoc framework. Even 0.01 regularization provides substantial benefits, with optimal performance achieved at moderate to high regularization strengths (\(\lambda_{1}\geq 0.25\)). \begin{table}
\begin{tabular}{l c c c} \hline
**Explanation Method** & **Point Fidelity \(\uparrow\)** & **Neighborhood Fidelity \(\uparrow\)** & **Stability \(\downarrow\)** \\ \hline LIME & 0.6083 \(\pm\) 0.0050 & 0.6600 \(\pm\) 0.1939 & 0.2152 \(\pm\) 0.0175 \\ Pre hoc Framework & 0.8270 \(\pm\) 0.0260 & 0.9587 \(\pm\) 0.0766 & 0.0623 \(\pm\) 0.0110 \\ Co hoc Framework & 0.8300 \(\pm\) 0.0240 & 0.9647 \(\pm\) 0.0575 & 0.0502 \(\pm\) 0.0087 \\ \hline \end{tabular}
\end{table} 
Table 2: HELOC Dataset: Comparison with LIME based on neighborhood fidelity and stability results (\(\lambda=0.25\), \(k=10\)). \(\uparrow\) means higher score is better, \(\downarrow\) means lower score is better.