In this thesis we consider the application of Fenchel's duality theory and gradient-based methods for the training and hyperparameter optimization of Support Vector Machines. We show that the dualization of convex training problems is possible theoretically in a rather general formulation. For training problems following a special structure (for instance, standard training problems) we find that the resulting optimality conditions can be interpreted concretely. This approach immediately leads to the well-known notion of support vectors and a formulation of the Representer Theorem. The proposed theory is applied to several examples such that dual formulations of training problems and associated optimality conditions can be derived straightforwardly. Furthermore, we consider different formulations of the primal training problem which are equivalent under certain conditions. We also argue that the relation of the corresponding solutions to the solution of the dual training problem is not always intuitive. Based on the previous findings, we consider the application of customized optimization methods to the primal and dual training problems. A particular realization of Newton's method is derived which could be used to solve the primal training problem accurately. Moreover, we introduce a general convergence framework covering different types of decomposition methods for the solution of the dual training problem. In doing so, we are able to generalize well-known convergence results for the SMO method. Additionally, a discussion of the complexity of the SMO method and a motivation for a shrinking strategy reducing the computational effort is provided. In a last theoretical part, we consider the problem of hyperparameter optimization. We argue that this problem can be handled efficiently by means of gradient-based methods if the training problems are formulated appropriately. Finally, we evaluate the theoretical results concerning the training and hyperparameter optimization approaches practically by means of several example training problems.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:87473 |
| Date | 18 October 2023 |
| Creators | Strasdat, Nico |
| Contributors | Fischer, Andreas, Sciandrone, Marco, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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