Kernels over Sets of Finite Sets using RKHS Embeddings, with Application to Bayesian (Combinatorial) Optimization

Buathong, Poompol; Ginsbourger, David; Krityakierne, Tipaluck (2020). Kernels over Sets of Finite Sets using RKHS Embeddings, with Application to Bayesian (Combinatorial) Optimization. In: Chiappa, Silvia; Calandra, Roberto (eds.) Twenty Third International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research: Vol. 108 (pp. 2731-2741). Online: PMLR

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We focus on kernel methods for set-valued inputs and their application to Bayesian set optimization, notably combinatorial optimization. We investigate two classes of set kernels that both rely on Reproducing Kernel Hilbert Space embeddings, namely the "Double Sum" (DS) kernels recently considered in Bayesian set optimization, and a class introduced here called "Deep Embedding" (DE) kernels that essentially consists in applying a radial kernel on Hilbert space on top of the canonical distance induced by another kernel such as a DS kernel. We establish in particular that while DS kernels typically suffer from a lack of strict positive definiteness, vast subclasses of DE kernels built upon DS kernels do possess this property, enabling in turn combinatorial optimization without requiring to introduce a jitter parameter. Proofs of theoretical results about considered kernels are complemented by a few practicalities regarding hyperparameter fitting. We furthermore demonstrate the applicability of our approach in prediction and optimization tasks, relying both on toy examples and on two test cases from mechanical engineering and hydrogeology, respectively. Experimental results highlight the applicability and compared merits of the considered approaches while opening new perspectives in prediction and sequential design with set inputs.

Item Type:

Conference or Workshop Item (Paper)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science

UniBE Contributor:

Ginsbourger, David

Subjects:

300 Social sciences, sociology & anthropology > 360 Social problems & social services
500 Science > 510 Mathematics
000 Computer science, knowledge & systems

Series:

Proceedings of Machine Learning Research

Publisher:

PMLR

Language:

English

Submitter:

David Ginsbourger

Date Deposited:

09 Sep 2020 12:18

Last Modified:

09 Sep 2020 12:24

BORIS DOI:

10.7892/boris.146439

URI:

https://boris.unibe.ch/id/eprint/146439

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