Artificial Intelligence (AI) models rely on loss functions as fundamental optimization tools to quantify the discrepancy between predicted and observed outputs. The mathematical properties of these functions, including convexity, differentiability, and sensitivity to input variations, directly influence model stability, convergence, and robustness. However, loss functions are often selected empirically with limited theoretical justification, which may result in unstable or suboptimal performance. This study investigates the impact of loss function design on AI model stability, focusing on Mean Squared Error (MSE) and Huber Loss. The methodology combines mathematical analysis with simulation-based experiments. The convexity and convergence behavior of MSE were examined through its first and second derivatives, while Huber Loss was analyzed for its piecewise structure and robustness to outliers. Linear regression models were trained on a housing dataset using both loss functions under identical conditions. Model performance was evaluated using multiple metrics, including MSE, MAE, RMSE, prediction variance, and sensitivity to noisy inputs. Experimental results indicate that the MSE-based model achieved lower test error (MSE ≈ 6.56 × 10⁹; RMSE ≈ 81,000) and slightly better overall stability on clean data. In contrast, Huber Loss produced a lower MAE (≈ 58,877), demonstrating greater robustness to absolute deviations and outliers. Variance analysis showed marginally higher prediction variability for the Huber model, while noise perturbation experiments confirmed that both methods experienced performance degradation, with MSE remaining slightly more stable. Overall, the findings demonstrate that the mathematical structure of loss functions significantly affects model behavior. While MSE provides superior accuracy under normal conditions, Huber Loss offers improved robustness in noisy or outlier-prone settings. These insights provide practical guidance for selecting appropriate loss functions to enhance stability and reliability in supervised learning applications.