High-dimensional Experimental Design and Kernel Bandits

Romain Camilleri, Kevin Jamieson, Julian Katz-Samuels
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:1227-1237, 2021.

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.

Cite this Paper

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.} }
Endnote
%0 Conference Paper %T High-dimensional Experimental Design and Kernel Bandits %A Romain Camilleri %A Kevin Jamieson %A Julian Katz-Samuels %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-camilleri21a %I PMLR %P 1227--1237 %U https://proceedings.mlr.press/v139/camilleri21a.html %V 139 %X 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.
APA
Camilleri, R., Jamieson, K. & Katz-Samuels, J.. (2021). High-dimensional Experimental Design and Kernel Bandits. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:1227-1237 Available from https://proceedings.mlr.press/v139/camilleri21a.html.

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