Geometry and Stability of Supervised Learning Problems
Facundo Mémoli Brantley Vose Robert C. Williamson
We introduce a notion of distance between supervised learning problems, which
we call the Risk distance. This optimal-transport-inspired distance facilitates
stability results; one can quantify how seriously issues like sampling bias,
noise, limited data, and approximations might change a given problem by
bounding how much these modifications can move the problem under the Risk
distance. With the distance established, we explore the geometry of the
resulting space of supervised learning problems, providing explicit geodesics
and proving that the set of classification problems is dense in a larger class
of problems. We also provide two variants of the Risk distance: one that
incorporates specified weights on a problem's predictors, and one that is more
sensitive to the contours of a problem's risk landscape.