Astral Space: Convex Analysis at Infinity
summary
Defines astral space as a compact extension of R^n that adds points at infinity so every linear functional extends continuously, and proves every convex function attains a minimum there; constructs points as leftward sums of at most n astrons (e.g. limits of sequences like (t,t) or (t,t^2)), defines leftward addition and extends convexity, conjugacy and subdifferentials, and uses that structure to show convergence of gradient and coordinate descent for problems (logistic, boosting, MLE). This is joint work with Miro Dudík and Matus Telgarsky.
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chapters (12)
- 0:00Introduction
- 1:28Minimizers at infinity motivation
- 5:51Astral space introduction
- 7:18Talk outline and notation
- 8:25Minimizers at infinity examples
- 13:28Building astral space from sequences
- 22:23Astral space definition and properties
- 25:19Points in astral space
- 30:46Extending functions to astral space
- 34:43Algorithm convergence applications
- 35:47Summary and future work
- 36:30Questions and discussion
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