Statistical Agnostic Regression: a machine learning method to validate regression models

Regression analysis is a central topic in statistical modeling, aimed at estimating the relationships between a dependent variable, commonly referred to as the response variable, and one or more independent variables, i.e., explanatory variables. Linear regression is by far the most popular method for performing this task in various fields of research, such as data integration and predictive modeling when combining information from multiple sources. Classical methods for solving linear regression problems, such as Ordinary Least Squares (OLS), Ridge, or Lasso regressions, often form the foundation for more advanced machine learning (ML) techniques, which have been successfully applied, though without a formal definition of statistical significance. At most, permutation or analyses based on empirical measures (e.g., residuals or accuracy) have been conducted, leveraging the greater sensitivity of ML estimations for detection. In this paper, we introduce Statistical Agnostic Regression (SAR) for evaluating the statistical significance of ML-based linear regression models. This is achieved by analyzing concentration inequalities of the actual risk (expected loss) and considering the worst-case scenario. To this end, we define a threshold that ensures there is sufficient evidence, with a probability of at least $1-\eta$, to conclude the existence of a linear relationship in the population between the explanatory (feature) and the response (label) variables. Simulations demonstrate the ability of the proposed agnostic (non-parametric) test to provide an analysis of variance similar to the classical multivariate $F$-test for the slope parameter, without relying on the underlying assumptions of classical methods. Moreover, the residuals computed from this method represent a trade-off between those obtained from ML approaches and the classical OLS.