Mathematical Analysis of Loss Functions in Classification and Regression

Classification and Regression are commonly used techniques in machine learning. The Loss functions are foundational to machine learning, serving as quantifiable measures of prediction error though which model optimization is lead. This study presents a comprehensive mathematical analysis of prominent loss functions utilized across classification and regression tasks. We present formal definitions and investigate inherent mathematical properties such as convexity and differentiability. Their implications for the learning process, specifically within gradient-based optimization frameworks. The analysis begins with fundamental regression losses such as Mean Squared Error and Mean Absolute Error. The analysis further extends to robust alternatives such as Huber and Quantile Loss. For classification, we explore binary and categorical Cross-Entropy, Hinge Loss, Exponential Loss by clarifying their characteristics. Also, we discuss practical considerations in selecting appropriate loss functions, importance of regularization, custom loss functions design, and the critical distinction between optimization objectives and evaluations metrics. The work aims to create understanding for readers with deeper theoretical understanding of how different loss functions shape model behaviour and performance in diverse machine learning applications.