Minimizers of the Empirical Risk and Risk Monotonicity
This work addresses a foundational issue in machine learning theory by revealing unexpected non-monotonic behavior in learning curves, which could impact how researchers analyze generalization.
The paper tackles the problem of understanding learning curves by introducing the concept of risk monotonicity, which expects risk not to worsen with more training data, and finds that standard empirical risk minimizers can violate this property across various tasks.
Plotting a learner's average performance against the number of training samples results in a learning curve. Studying such curves on one or more data sets is a way to get to a better understanding of the generalization properties of this learner. The behavior of learning curves is, however, not very well understood and can display (for most researchers) quite unexpected behavior. Our work introduces the formal notion of \emph{risk monotonicity}, which asks the risk to not deteriorate with increasing training set sizes in expectation over the training samples. We then present the surprising result that various standard learners, specifically those that minimize the empirical risk, can act \emph{non}monotonically irrespective of the training sample size. We provide a theoretical underpinning for specific instantiations from classification, regression, and density estimation. Altogether, the proposed monotonicity notion opens up a whole new direction of research.