Sub-linear Regret Bounds for Bayesian Optimisation in Unknown Search Spaces

TL;DR

This paper introduces new Bayesian optimisation algorithms that adaptively expand the search space, achieving sub-linear regret growth and outperforming existing methods in unknown search space scenarios.

Contribution

The authors propose novel BO algorithms that dynamically expand the search space and scale to high dimensions, with proven sub-linear regret bounds.

Findings

Algorithms achieve sub-linear cumulative regret growth.

Experiments show superior performance over state-of-the-art methods.

Effective in both synthetic and real-world tasks.

Abstract

Bayesian optimisation is a popular method for efficient optimisation of expensive black-box functions. Traditionally, BO assumes that the search space is known. However, in many problems, this assumption does not hold. To this end, we propose a novel BO algorithm which expands (and shifts) the search space over iterations based on controlling the expansion rate thought a hyperharmonic series. Further, we propose another variant of our algorithm that scales to high dimensions. We show theoretically that for both our algorithms, the cumulative regret grows at sub-linear rates. Our experiments with synthetic and real-world optimisation tasks demonstrate the superiority of our algorithms over the current state-of-the-art methods for Bayesian optimisation in unknown search space.