Characterizing the invariances of learning algorithms using category theory
This provides a theoretical framework for understanding algorithm invariances, which is incremental as it applies existing mathematical concepts to a known problem in machine learning.
The paper formalizes the invariances of learning algorithms using category theory, characterizing how algorithms like linear and ridge regression transform when training data is altered.
Many learning algorithms have invariances: when their training data is transformed in certain ways, the function they learn transforms in a predictable manner. Here we formalize this notion using concepts from the mathematical field of category theory. The invariances that a supervised learning algorithm possesses are formalized by categories of predictor and target spaces, whose morphisms represent the algorithm's invariances, and an index category whose morphisms represent permutations of the training examples. An invariant learning algorithm is a natural transformation between two functors from the product of these categories to the category of sets, representing training datasets and learned functions respectively. We illustrate the framework by characterizing and contrasting the invariances of linear regression and ridge regression.