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Shivaram Kalyanakrishnan, Ambuj Tewari, Peter Auer, and Peter
Stone. **PAC Subset Selection in Stochastic Multi-armed Bandits**. In *In proceedings of the 29th International Conference
on Machine Learning (ICML)*, pp. 655–662, Omnipress, New York, NY, USA, June-July 2012.

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We consider the problem of selecting, from among the arms of a stochastic $n$-armed bandit, a subset of size $m$ of those arms with the highest expected rewards, based on efficiently sampling the arms. This ``subset selection'' problem finds application in a variety of areas. In the authors' previous work \citepKalyanakrishnan+Stone:2010-bandit, this problem is framed under a PAC setting (denoted ``\textscExplore-$m$''), and corresponding sampling algorithms are analyzed. Whereas the formal analysis therein is restricted to the worst case sample complexity of algorithms, in this paper, we design and analyze an algorithm (``\textscLUCB'') with improved expected sample complexity. Interestingly \textscLUCB bears a close resemblance to the well-known \textscUCB algorithm for regret minimization. The expected sample complexity bound we show for \textscLUCB is novel even for single-arm selection (\textscExplore-$1$). We also give a lower bound on the worst case sample complexity of PAC algorithms for \textscExplore-$m$.

@InProceedings{ICML12-shivaram, author = {Shivaram Kalyanakrishnan and Ambuj Tewari and Peter Auer and Peter Stone}, title = {{PAC} Subset Selection in Stochastic Multi-armed Bandits}, booktitle = {In proceedings of the 29th International Conference on Machine Learning (ICML)}, location = {Edinburgh, UK}, month = {June-July}, year = {2012}, abstract = { We consider the problem of selecting, from among the arms of a stochastic $n$-armed bandit, a subset of size $m$ of those arms with the highest expected rewards, based on efficiently sampling the arms. This ``subset selection'' problem finds application in a variety of areas. In the authors' previous work~\citep{Kalyanakrishnan+Stone:2010-bandit}, this problem is framed under a PAC setting (denoted ``\textsc{Explore}-$m$''), and corresponding sampling algorithms are analyzed. Whereas the formal analysis therein is restricted to the worst case sample complexity of algorithms, in this paper, we design and analyze an algorithm (``\textsc{LUCB}'') with improved expected sample complexity. Interestingly \textsc{LUCB} bears a close resemblance to the well-known \textsc{UCB} algorithm for regret minimization. The expected sample complexity bound we show for \textsc{LUCB} is novel even for single-arm selection (\textsc{Explore}-$1$). We also give a lower bound on the worst case sample complexity of PAC algorithms for \textsc{Explore}-$m$. }, ISBN = {978-1-4503-1285-1}, editor = {John Langford and Joelle Pineau}, publisher = {Omnipress}, address = {New York, NY, USA}, pages = {655-662}, }

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