Abstract
This paper initiates a systematic development of a theory of non-commutative optimization, a setting which greatly extends ordinary (Euclidean) convex optimization. It aims to unify and generalize a growing body of work from the past few years which developed and analyzed algorithms for natural geodesic convex optimization problems on Riemannian manifolds that arise from the symmetries of non-commutative groups. More specifically, these are algorithms to minimize the moment map (a non-commutative notion of the usual gradient), and to test membership in moment polytopes (a vast class of polytopes, typically of exponential vertex and facet complexity, which quite magically arise from this a-priori non-convex, non-linear setting). The importance of understanding this very general setting of geodesic optimization, as these works unveiled and powerfully demonstrate, is that it captures a diverse set of problems, many non-convex, in different areas of CS, math, and physics. Several of them were solved efficiently for the first time using non-commutative methods; the corresponding algorithms also lead to solutions of purely structural problems and to many new connections between disparate fields. In the spirit of standard convex optimization, we develop two general methods in the geodesic setting, a first order and a second order method, which respectively receive first and second order information on the 'derivatives' of the function to be optimized. These in particular subsume all past results. The main technical work, again unifying and extending much of the previous work, goes into identifying the key parameters of the underlying group actions which control convergence to the optimum in each of these methods. These non-commutative analogues of 'smoothness' in the commutative case are far more complex, and require significant algebraic and analytic machinery (much existing and some newly developed here). Despite this complexity, the way in which these parameters control convergence in both methods is quite simple and elegant. We also bound these parameters in several general cases. Our work points to intriguing open problems and suggests further research directions. We believe that extending this theory, namely understanding geodesic optimization better, is both mathematically and computationally fascinating; it provides a great meeting place for ideas and techniques from several very different research areas, and promises better algorithms for existing and yet unforeseen applications.
Original language | English |
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Title of host publication | Proceedings - 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019 |
Publisher | IEEE Computer Society |
Pages | 845-861 |
Number of pages | 17 |
ISBN (Electronic) | 9781728149523 |
DOIs | |
State | Published - Nov 2019 |
Externally published | Yes |
Event | 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019 - Baltimore, United States Duration: 9 Nov 2019 → 12 Nov 2019 |
Publication series
Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2019-November |
ISSN (Print) | 0272-5428 |
Conference
Conference | 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019 |
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Country/Territory | United States |
City | Baltimore |
Period | 9/11/19 → 12/11/19 |
Bibliographical note
Publisher Copyright:© 2019 IEEE.
Keywords
- computational complexity
- convex optimization
- geodesic convexity
- invariant theory
- moment polytopes
- non-commutative optimization
- null cone
- representation theory
- scaling algorithms