Learning Continuous Functions through a New Linear Regression Method

Abstract

We revisit the linear regression problem in terms of a computational learning problem whose task is to identify a confidence region for a continuous function belonging in particular to the straight lines family. Within the Algorithmic Inference framework this function is deputed to explain a relation between pairs of variables that are observed through a limited sample. Hence it is a random item within the above family and we look for a partial order relation allowing us to state a cumulative distribution function over the function specifications, hence a pair of quantiles identifying the confidence region. The regions we compute in this way is theoretically and numerically attested to entirely contain the goal function with a given confidence. Its shape is quite different from the analogous region obtained through conventional methods as a collation of confidence intervals found for the expected value of the dependent variable as a function of the independent one.