Astral Space: Convex Analysis at Infinity

From three of today’s top researchers in machine learning, a groundbreaking new theory for understanding convex minimization at infinityNumerous fields of study rely on methods for minimizing convex functions.Not all convex functions, however, have finite minimizers; some can only be minimized by a sequence as it heads to infinity, making it considerably more challenging to prove correct convergence to a minimizer. This book develops an expansive new theory for understanding such minimizers at infinity, introducing astral space, a compact extension of Euclidean space to which such points at infinity have been added.Astral space is constructed to be as small as possible while still ensuring that all linear functions can be continuously extended to the new space.These favorable properties make it especially compatible with standard convex analysis, whose key notions are systematically extended to the new space, providing the foundation for a more complete theory. Astral space includes Eu