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Algorithms for Nonconvex Optimization Problems in Machine Learning and Statistics

Mohr, Robert

Abstract (englisch):
The purpose of this thesis is the design of algorithms that can be used to determine optimal solutions to nonconvex data approximation problems.

In Part I of this thesis, we consider a very general class of nonconvex and large-scale data approximation problems and devise an algorithm that efficiently computes locally optimal solutions to these problems. As a type of trust-region Newton-CG method, the algorithm can make use of directions of negative curvature to escape saddle points, which otherwise might slow down the optimization process when solving nonconvex problems. We present results of numerical experiments on convex and nonconvex problems which support our claim that our algorithm has significant advantages compared to methods like stochastic gradient descent and its variance-reduced versions.

In Part II we consider the univariate least-squares spline approximation problem with free knots, which is known to possess a large number of locally minimal points far from the globally optimal solution. Since in typical applications, neither the dimension of the decision variable nor the number of data points is particularly large, it is possible to make use of the specific problem structure in order to devise algorithmic approaches to approximate the globally optimal solution of problem instances of relevant sizes. ... mehr

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Volltext §
DOI: 10.5445/IR/1000119052
Veröffentlicht am 07.05.2020
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Operations Research (IOR)
Publikationstyp Hochschulschrift
Publikationsdatum 07.05.2020
Sprache Englisch
Identifikator KITopen-ID: 1000119052
Verlag Karlsruhe
Umfang X, 131 S.
Abschlussart Dissertation
Fakultät Fakultät für Wirtschaftswissenschaften (WIWI)
Institut Institut für Operations Research (IOR)
Prüfungsdatum 11.03.2020
Referent/Betreuer Prof. O. Stein
KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft
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