# Adaptive bandits: Towards the best history-dependent strategy

Odalric-Ambrym Maillard, Rémi Munos.
In Proceedings of the 14th international conference on Artificial Intelligence and Statistics,
AI&Statistics 2011, volume 15 of JMLR W&CP, 2011.

 Abstract: We consider multi-armed bandit games with possibly adaptive opponents. We introduce models Theta of constraints based on equivalence classes on the common history (information shared by the player and the opponent) which define two learning scenarios: (1) The opponent is constrained, i.e. she provides rewards that are stochastic functions of equivalence classes defined by some model theta* in Theta. The regret is measured with respect to (w.r.t.) the best history-dependent strategy. (2) The opponent is arbitrary and we measure the regret w.r.t. the best strategy among all mappings from classes to actions (i.e. the best history-class-based strategy) for the best model in  Theta. This allows to model opponents (case 1) or strategies (case 2) which handles finite memory, periodicity, standard stochastic bandits and other situations. When  Theta={theta}, i.e. only one model is considered, we derive tractable algorithms achieving a tight regret (at time T) bounded by  Õ(\sqrt{TAC}), where  C is the number of classes of  theta. Now, when many models are available, all known algorithms achieving a nice regret  O(\sqrt{T}) are unfortunately not tractable and scale poorly with the number of models  |Theta| . Our contribution here is to provide tractable algorithms with regret bounded by  T^{2/3}C^{1/3}log(|Theta|)^{1/2}.

You can dowload the paper from the JMLR webiste (here) or from the HAL online open depository* (here).

 Bibtex: @inproceedings{MaillardM11, author = {Odalric{-}Ambrym Maillard and R{\'{e}}mi Munos}, title = {Adaptive Bandits: Towards the best history-dependent strategy}, booktitle = {Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, {AISTATS} 2011, Fort Lauderdale, USA, April 11-13, 2011}, year = {2011}, pages = {570–578} editor = {Geoffrey J. Gordon and David B. Dunson and Miroslav Dud{\'{\i}}k}, series = {{JMLR} Proceedings}, year = {2011}, volume = {15} }
 Bibtex: @inproceedings{MaillardM10a, author = {{Odalric-Ambrym} Maillard and R\'{e}mi Munos}, title = {Online Learning in Adversarial Lipschitz Environments.}, booktitle = {Machine Learning and Knowledge Discovery in Databases, European Conference, {ECML} {PKDD} 2010, Barcelona, Spain, September 20-24, 2010, Proceedings, Part {II}}, year = {2010}, pages = {305–320}, editor = {Jos\'{e} L. Balc\{a}zar and Francesco Bonchi and Aristides Gionis and Mich\{e}le Sebag}, series = {Lecture Notes in Computer Science}, year = {2010}, volume = {6322}, publisher = {Springer} }