A Characterization of List Regression
This work extends existing list learning characterizations from classification to regression, addressing a theoretical gap in machine learning.
The paper tackles the problem of characterizing the sample complexity of list PAC regression, where a learning algorithm makes a list of predictions and requires one to be correct, by proposing two combinatorial dimensions that generalize known dimensions for standard regression to this setting.
There has been a recent interest in understanding and characterizing the sample complexity of list learning tasks, where the learning algorithm is allowed to make a short list of $k$ predictions, and we simply require one of the predictions to be correct. This includes recent works characterizing the PAC sample complexity of standard list classification and online list classification. Adding to this theme, in this work, we provide a complete characterization of list PAC regression. We propose two combinatorial dimensions, namely the $k$-OIG dimension and the $k$-fat-shattering dimension, and show that they characterize realizable and agnostic $k$-list regression respectively. These quantities generalize known dimensions for standard regression. Our work thus extends existing list learning characterizations from classification to regression.