Mae vs. MSE: more than just the mean vs. median debate (idlemachines.co.uk)

A recent analysis delves into the nuanced differences between Mean Squared Error (MSE) and Mean Absolute Error (MAE) as loss functions in machine learning. It highlights that MSE aligns with Gaussian noise, providing a predictive mean, whereas MAE corresponds to Laplace noise, emphasizing the conditional median. The significance of this distinction lies in how each loss function influences training dynamics, particularly regarding outlier sensitivity and the interpretation of residuals. MSE's sensitivity to large residuals can lead to inefficiencies in datasets where noise is heterogeneous or multimodal, whereas MAE offers robustness but at the cost of reduced statistical efficiency when working with Gaussian-like distributions. The discussion also introduces the concept of Huber loss, which combines the advantages of both MSE and MAE, being quadratic for small residuals and linear for larger ones. This hybrid approach aims to balance the robustness of MAE against the efficiency of MSE, catering to real-world scenarios where data distribution isn't uniform. Ultimately, the choice between MSE and MAE should be informed by the specific target statistic one aims to optimize and the operational costs associated with different types of errors in model predictions. This insight is crucial for AI/ML practitioners seeking to improve model performance and interpretability.