Efficient Linear Bandits through Matrix Sketching
Ilja Kuzborskij, Leonardo Cella, Nicolò Cesa-Bianchi
Abstract
We prove that two popular linear contextual bandit algorithms, OFUL and Thompson Sampling, can be made efficient using Frequent Directions, a deterministic online sketching technique. More precisely, we show that a sketch of size $m$ allows a $\mathcal{O}(md)$ update time for both algorithms, as opposed to $\Omega(d^2)$ required by their non-sketched versions in general (where $d$ is the dimension of context vectors). This computational speedup is accompanied by regret bounds of order $(1+\varepsilon_m)^{3/2}d\sqrt{T}$ for OFUL and of order $\big((1+\varepsilon_m)d\big)^{3/2}\sqrt{T}$ for Thompson Sampling, where $\varepsilon_m$ is bounded by the sum of the tail eigenvalues not covered by the sketch. In particular, when the selected contexts span a subspace of dimension at most $m$, our algorithms have a regret bound matching that of their slower, non-sketched counterparts. Experiments on real-world datasets corroborate our theoretical results.
BibTeX
@InProceedings{pmlr-v89-kuzborskij19a,
title = {Efficient Linear Bandits through Matrix Sketching},
author = {Kuzborskij, Ilja and Cella, Leonardo and Cesa-Bianchi, Nicol\`{o}},
booktitle = {Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics},
pages = {177--185},
year = {2019},
editor = {Chaudhuri, Kamalika and Sugiyama, Masashi},
volume = {89},
series = {Proceedings of Machine Learning Research},
month = {16--18 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v89/kuzborskij19a/kuzborskij19a.pdf},
url = {https://proceedings.mlr.press/v89/kuzborskij19a.html},
abstract = {We prove that two popular linear contextual bandit algorithms, OFUL and Thompson Sampling, can be made efficient using Frequent Directions, a deterministic online sketching technique. More precisely, we show that a sketch of size $m$ allows a $\mathcal{O}(md)$ update time for both algorithms, as opposed to $\Omega(d^2)$ required by their non-sketched versions in general (where $d$ is the dimension of context vectors). This computational speedup is accompanied by regret bounds of order $(1+\varepsilon_m)^{3/2}d\sqrt{T}$ for OFUL and of order $\big((1+\varepsilon_m)d\big)^{3/2}\sqrt{T}$ for Thompson Sampling, where $\varepsilon_m$ is bounded by the sum of the tail eigenvalues not covered by the sketch. In particular, when the selected contexts span a subspace of dimension at most $m$, our algorithms have a regret bound matching that of their slower, non-sketched counterparts. Experiments on real-world datasets corroborate our theoretical results.}
}