Efficient Linear Bandits through Matrix Sketching

Ilja Kuzborskij, Leonardo Cella, Nicolò Cesa-Bianchi
Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, PMLR 89:177-185, 2019.

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.

Cite this Paper

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.} }
Endnote
%0 Conference Paper %T Efficient Linear Bandits through Matrix Sketching %A Ilja Kuzborskij %A Leonardo Cella %A Nicolò Cesa-Bianchi %B Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Masashi Sugiyama %F pmlr-v89-kuzborskij19a %I PMLR %P 177--185 %U https://proceedings.mlr.press/v89/kuzborskij19a.html %V 89 %X 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.
APA
Kuzborskij, I., Cella, L. & Cesa-Bianchi, N.. (2019). Efficient Linear Bandits through Matrix Sketching. Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 89:177-185 Available from https://proceedings.mlr.press/v89/kuzborskij19a.html.

Related Material