Performance of regression models as a function of experiment noise

A challenge in developing machine learning regression models is that it is difficult to know whether maximal performance has been reached on a particular dataset, or whether further model improvement is possible. In biology this problem is particularly pronounced as sample labels (response variables) are typically obtained through experiments and therefore have experiment noise associated with them. Such label noise puts a fundamental limit to the performance attainable by regression models. We address this challenge by deriving a theoretical upper bound for the coefficient of determination (R2) for regression models. This theoretical upper bound depends only on the noise associated with the response variable in a dataset as well as its variance. The upper bound estimate was validated via Monte Carlo simulations and then used as a tool to bootstrap performance of regression models trained on biological datasets, including protein sequence data, transcriptomic data, and genomic data. Although we study biological datasets in this work, the new upper bound estimates will hold true for regression models from any research field or application area where response variables have associated noise.