Simplicial covering dimension of extremal concept classes
This provides a theoretical foundation for understanding replicability in machine learning, connecting topology to learning theory, but it is incremental as it adapts classical tools to a specific domain.
The paper tackles the problem of characterizing list replicability in PAC learning by adapting topological dimension theory to binary concept classes, proving that the simplicial covering dimension exactly equals the list replicability number for finite classes and enabling exact computation for extremal concept classes.
Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue covering) to binary concept classes. The topological space naturally associated with a concept class is its space of realizable distributions. The loss function and the class itself induce a simplicial structure on this space, with respect to which we define a simplicial covering dimension. We prove that for finite concept classes, this simplicial covering dimension exactly characterizes the list replicability number (equivalently, global stability) in PAC learning. This connection allows us to apply tools from classical dimension theory to compute the exact list replicability number of the broad family of extremal concept classes.