Astral Space: Convex Analysis at Infinity

Robert Schapire (Microsoft Research)
https://simons.berkeley.edu/talks/robert-schapire-microsoft-research-2026-05-27
The Role of TCS in Modern Machine Learning

Not all convex functions have finite minimizers; some can only be minimized by a sequence as it heads to infinity. In this work, we aim to develop a theory for understanding such minimizers at infinity. We study 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. Although not a vector space, nor even a metric space, astral space is nevertheless so well-structured as to allow useful and meaningful extensions of such concepts as convexity, conjugacy, and subdifferentials. We develop these concepts and analyze various properties of convex functions on astral space, including the detailed structure of their minimizers, exact characterizations of continuity, and convergence of descent algorithms.

This is joint work with Miro Dudík and Matus Telgarsky. For further reading, see aka.ms/astral.

Видео Astral Space: Convex Analysis at Infinity канала Simons Institute for the Theory of Computing