Least Squares is widely utilized for numerical optimization and regression tasks in machine learning, primarily due to its ability to minimize Mean Squared Error (MSE). The L2 norm results in a smoother loss function compared to the L1 norm, enhancing its computational convenience. Least Squares is both the Best Linear Unbiased Estimator (BLUE) and equivalent to Maximum Likelihood Estimation under normal error conditions. However, it can become ineffective when data distributions deviate from assumptions, particularly in the presence of outliers, emphasizing the need for caution in its application.
The Least Squares approach is favored due to its computational convenience, as the square loss function enables easy differentiation and closed-form solutions for Linear Regression.
Ordinary Least-Squares (OLS) is renowned as the Best Linear Unbiased Estimator (BLUE), providing the lowest variance among unbiased estimators, solidifying its importance in statistics.
Utilizing Least-Squares for fitting models aligns with Maximum Likelihood Estimation under normally distributed errors, showcasing its broad applicability in various statistical contexts.
Despite its many advantages, Least Squares can become unreliable if theoretical assumptions are violated, particularly in the presence of outliers.
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