High-Dimensional Bayesian Optimization via Nested Riemannian Manifolds

TL;DR

This paper introduces a geometry-aware Bayesian optimization method that leverages Riemannian manifolds to efficiently optimize high-dimensional functions by learning structure-preserving mappings to low-dimensional latent spaces.

Contribution

It proposes a novel Riemannian manifold-based approach that jointly learns embeddings and objective representations, improving high-dimensional Bayesian optimization performance.

Findings

Outperforms existing high-dimensional BO methods on benchmarks

Consistently finds better optima than geometry-unaware approaches

Effectively leverages domain geometry for sample-efficient optimization

Abstract

Despite the recent success of Bayesian optimization (BO) in a variety of applications where sample efficiency is imperative, its performance may be seriously compromised in settings characterized by high-dimensional parameter spaces. A solution to preserve the sample efficiency of BO in such problems is to introduce domain knowledge into its formulation. In this paper, we propose to exploit the geometry of non-Euclidean search spaces, which often arise in a variety of domains, to learn structure-preserving mappings and optimize the acquisition function of BO in low-dimensional latent spaces. Our approach, built on Riemannian manifolds theory, features geometry-aware Gaussian processes that jointly learn a nested-manifold embedding and a representation of the objective function in the latent space. We test our approach in several benchmark artificial landscapes and report that it not only outperforms other high-dimensional BO approaches in several settings, but consistently optimizes the objective functions, as opposed to geometry-unaware BO methods.