On Best-Arm Identification with a Fixed Budget in Non-Parametric Multi-Armed Bandits

Antoine Barrier, Aurélien Garivier, Gilles Stoltz
Proceedings of The 34th International Conference on Algorithmic Learning Theory, PMLR 201:136-181, 2023.

Abstract

We lay the foundations of a non-parametric theory of best-arm identification in multi-armed bandits with a fixed budget $T$. We consider general, possibly non-parametric, models $\mathcal{D}$ for distributions over the arms; an overarching example is the model $\mathcal{D} = \mathcal{P}[0, 1]$ of all probability distributions over $[0,1]$. We propose upper bounds on the average log-probability of misidentifying the optimal arm based on information-theoretic quantities that we name $\mathcal{L}_{\inf}^{<}(\,\cdot\,,\nu)$ and $\mathcal{L}_{\inf}^{>}(\,\cdot\,,\nu)$ and that correspond to infima over Kullback-Leibler divergences between some distributions in $\mathcal{D}$ and a given distribution $\nu$. This is made possible by a refined analysis of the successive-rejects strategy of Audibert et al. (2010). We finally provide lower bounds on the same average log-probability, also in terms of the same new information-theoretic quantities; these lower bounds are larger when the (natural) assumptions on the considered strategies are stronger. All these new upper and lower bounds generalize existing bounds based, e.g., on gaps between distributions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v201-barrier23a, title = {On Best-Arm Identification with a Fixed Budget in Non-Parametric Multi-Armed Bandits}, author = {Barrier, Antoine and Garivier, Aur{\'e}lien and Stoltz, Gilles}, booktitle = {Proceedings of The 34th International Conference on Algorithmic Learning Theory}, pages = {136--181}, year = {2023}, editor = {Agrawal, Shipra and Orabona, Francesco}, volume = {201}, series = {Proceedings of Machine Learning Research}, month = {20 Feb--23 Feb}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v201/barrier23a/barrier23a.pdf}, url = {https://proceedings.mlr.press/v201/barrier23a.html}, abstract = {We lay the foundations of a non-parametric theory of best-arm identification in multi-armed bandits with a fixed budget $T$. We consider general, possibly non-parametric, models $\mathcal{D}$ for distributions over the arms; an overarching example is the model $\mathcal{D} = \mathcal{P}[0, 1]$ of all probability distributions over $[0,1]$. We propose upper bounds on the average log-probability of misidentifying the optimal arm based on information-theoretic quantities that we name $\mathcal{L}_{\inf}^{<}(\,\cdot\,,\nu)$ and $\mathcal{L}_{\inf}^{>}(\,\cdot\,,\nu)$ and that correspond to infima over Kullback-Leibler divergences between some distributions in $\mathcal{D}$ and a given distribution $\nu$. This is made possible by a refined analysis of the successive-rejects strategy of Audibert et al. (2010). We finally provide lower bounds on the same average log-probability, also in terms of the same new information-theoretic quantities; these lower bounds are larger when the (natural) assumptions on the considered strategies are stronger. All these new upper and lower bounds generalize existing bounds based, e.g., on gaps between distributions.} }
Endnote
%0 Conference Paper %T On Best-Arm Identification with a Fixed Budget in Non-Parametric Multi-Armed Bandits %A Antoine Barrier %A Aurélien Garivier %A Gilles Stoltz %B Proceedings of The 34th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2023 %E Shipra Agrawal %E Francesco Orabona %F pmlr-v201-barrier23a %I PMLR %P 136--181 %U https://proceedings.mlr.press/v201/barrier23a.html %V 201 %X We lay the foundations of a non-parametric theory of best-arm identification in multi-armed bandits with a fixed budget $T$. We consider general, possibly non-parametric, models $\mathcal{D}$ for distributions over the arms; an overarching example is the model $\mathcal{D} = \mathcal{P}[0, 1]$ of all probability distributions over $[0,1]$. We propose upper bounds on the average log-probability of misidentifying the optimal arm based on information-theoretic quantities that we name $\mathcal{L}_{\inf}^{<}(\,\cdot\,,\nu)$ and $\mathcal{L}_{\inf}^{>}(\,\cdot\,,\nu)$ and that correspond to infima over Kullback-Leibler divergences between some distributions in $\mathcal{D}$ and a given distribution $\nu$. This is made possible by a refined analysis of the successive-rejects strategy of Audibert et al. (2010). We finally provide lower bounds on the same average log-probability, also in terms of the same new information-theoretic quantities; these lower bounds are larger when the (natural) assumptions on the considered strategies are stronger. All these new upper and lower bounds generalize existing bounds based, e.g., on gaps between distributions.
APA
Barrier, A., Garivier, A. & Stoltz, G.. (2023). On Best-Arm Identification with a Fixed Budget in Non-Parametric Multi-Armed Bandits. Proceedings of The 34th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 201:136-181 Available from https://proceedings.mlr.press/v201/barrier23a.html.

Related Material