High-dimensional Experimental Design and Kernel Bandits
Romain Camilleri, Kevin Jamieson, Julian Katz-Samuels
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 there, 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 provably robust to model misspecification.
BibTeX
@InProceedings{pmlr-v139-camilleri21a,
title = {High-dimensional Experimental Design and Kernel Bandits},
author = {Camilleri, Romain and Jamieson, Kevin and Katz-Samuels, Julian},
booktitle = {Proceedings of the 38th International Conference on Machine Learning},
pages = {1227--1237},
year = {2021},
editor = {Meila, Marina and Zhang, Tong},
volume = {139},
series = {Proceedings of Machine Learning Research},
month = {18--24 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v139/camilleri21a/camilleri21a.pdf},
url = {https://proceedings.mlr.press/v139/camilleri21a.html},
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 there, 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 provably robust to model misspecification.}
}