Kernels over Sets of Finite Sets using RKHS Embeddings, with Application to Bayesian (Combinatorial) Optimization
This addresses the problem of Bayesian optimization for combinatorial/structured inputs in fields like mechanical engineering and hydrogeology, representing an incremental improvement over existing set kernel methods.
The paper tackles kernel methods for set-valued inputs in Bayesian combinatorial optimization by introducing Deep Embedding (DE) kernels that overcome the lack of strict positive definiteness in existing Double Sum (DS) kernels, enabling optimization without jitter parameters. Experimental results on mechanical engineering and hydrogeology test cases demonstrate the applicability and merits of the approach.
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.