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Optimization in the Space of Measures : = New Techniques from Optimal Transport.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Optimization in the Space of Measures :/
其他題名:	New Techniques from Optimal Transport.
作者:	Kent, Carson Richard.
面頁冊數:	1 online resource (144 pages)
附註:	Source: Dissertations Abstracts International, Volume: 84-05, Section: B.
Contained By:	Dissertations Abstracts International84-05B.
標題:	Linear programming. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29756134click for full text (PQDT)
ISBN:	9798357503367

Optimization in the Space of Measures : = New Techniques from Optimal Transport.
Kent, Carson Richard.

Optimization in the Space of Measures :New Techniques from Optimal Transport. - 1 online resource (144 pages)

Source: Dissertations Abstracts International, Volume: 84-05, Section: B.

Thesis (Ph.D.)--Stanford University, 2021.

Includes bibliographical references

Optimal transport is a profound and elegant tool in probability theory which measures discrepancy between probability distributions. It is a legacy of more than 200 years of mathematical progress and unites disparate areas of science-- stretching from analytic geometry to physics. Because of this, optimal transport has become a primary mechanism for developing and understanding computational methods in robust optimization, machine learning, and artificial intelligence.In this work, we provide novel results which cement this link between optimal transport and data-driven, computational methods. Our first result connects the discrete optimal transport problem with recent developments in theoretical computer science. We exploit this connection to provide novel methods for computing solutions to the discrete optimal transport problem. Moreover, these procedures achieve computational complexities that are close to optimal. Our second result applies the geometry of optimal transport to develop a novel, infinite-dimensional, Frank-Wolfe method for a large class of problems in machine learning and artificial intelligence. Despite operating on an infinite-dimensional level, our procedure is concretely implementable (with finite sample guarantees) and has iteration complexities that mimic convergence rates of finite-dimensional, first-order methods. Using several canonical examples, we also demonstrate that practical implementations of our method possess attractive convergence properties. Finally, our third result develops a strong duality theory for non-parametric model uncertainty problems-- where model uncertainty is specified in terms of optimal transport discrepancy to a set of reference distributions. This duality theory extends a long line of previous results and, most importantly, establishes a computational basis for solving these (infinite-dimensional) problems. Specifically, we establish strong duality with a dual problem that can be solved computationally, using finite-dimensional optimization techniques.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2023

Mode of access: World Wide Web

ISBN: 9798357503367Subjects--Topical Terms:

560448
Linear programming.
Index Terms--Genre/Form:

542853
Electronic books.

Optimization in the Space of Measures : = New Techniques from Optimal Transport.
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502 $a Thesis (Ph.D.)--Stanford University, 2021.
504 $a Includes bibliographical references
520 $a Optimal transport is a profound and elegant tool in probability theory which measures discrepancy between probability distributions. It is a legacy of more than 200 years of mathematical progress and unites disparate areas of science-- stretching from analytic geometry to physics. Because of this, optimal transport has become a primary mechanism for developing and understanding computational methods in robust optimization, machine learning, and artificial intelligence.In this work, we provide novel results which cement this link between optimal transport and data-driven, computational methods. Our first result connects the discrete optimal transport problem with recent developments in theoretical computer science. We exploit this connection to provide novel methods for computing solutions to the discrete optimal transport problem. Moreover, these procedures achieve computational complexities that are close to optimal. Our second result applies the geometry of optimal transport to develop a novel, infinite-dimensional, Frank-Wolfe method for a large class of problems in machine learning and artificial intelligence. Despite operating on an infinite-dimensional level, our procedure is concretely implementable (with finite sample guarantees) and has iteration complexities that mimic convergence rates of finite-dimensional, first-order methods. Using several canonical examples, we also demonstrate that practical implementations of our method possess attractive convergence properties. Finally, our third result develops a strong duality theory for non-parametric model uncertainty problems-- where model uncertainty is specified in terms of optimal transport discrepancy to a set of reference distributions. This duality theory extends a long line of previous results and, most importantly, establishes a computational basis for solving these (infinite-dimensional) problems. Specifically, we establish strong duality with a dual problem that can be solved computationally, using finite-dimensional optimization techniques.
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650 4 $a Computer science. $3 523869
650 4 $a Algorithms. $3 536374
650 4 $a Geometry. $3 517251
650 4 $a Entropy. $3 546219
650 4 $a Optimization. $3 891104
650 4 $a Mathematics. $3 515831
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