High-Dimensional Experimental Design and Kernel Bandits

TL;DR

This paper introduces a dimension-independent rounding method for experimental design in high-dimensional and infinite-dimensional spaces, improving kernel bandit algorithms for regret minimization and exploration.

Contribution

It proposes a novel rounding procedure that removes the dependence on dimension in experimental design, enabling effective bandit algorithms in high or infinite-dimensional settings.

Findings

Achieves near state-of-the-art regret bounds in kernel bandits.

Demonstrates robustness to model misspecification.

Outperforms lower-dimensional projection methods.

Abstract

In recent years methods from optimal linear experimental design have been leveraged to obtain state of the art results for linear bandits. A design returned from an objective such as $G$-optimal design is actually a probability distribution over a pool of potential measurement vectors. Consequently, one nuisance of the approach is the task of converting this continuous probability distribution into a discrete assignment of $N$ measurements. While sophisticated rounding techniques have been proposed, in $d$ dimensions they require $N$ to be at least $d$, $d \log(\log(d))$, or $d^2$ based on the sub-optimality of the solution. In this paper we are interested in settings where $N$ may be much less than $d$, such as in experimental design in an RKHS where $d$ may be effectively infinite. In this work, we propose a rounding procedure that frees $N$ of any dependence on the dimension $d$, while achieving nearly the same performance guarantees of existing rounding procedures. We evaluate the procedure against a baseline that projects the problem to a lower dimensional space and performs rounding which requires $N$ to just be at least a notion of the effective dimension. We also leverage our new approach in a new algorithm for kernelized bandits to obtain state of the art results for regret minimization and pure exploration. An advantage of our approach over existing UCB-like approaches is that our kernel bandit algorithms are also robust to model misspecification.