In this work, we propose a multi-armed bandit-based framework for identifying a compact set of informative data instances (i.e., the prototypes) from a source dataset $S$ that best represents a given target set $T$. Prototypical examples of a given dataset offer interpretable insights into the underlying data distribution and assist in example-based reasoning, thereby influencing every sphere of human decision-making. Current state-of-the-art prototype selection approaches require $O(|S||T|)$ similarity comparisons between source and target data points, which becomes prohibitively expensive for large-scale settings. We propose to mitigate this limitation by employing stochastic greedy search in the space of prototypical examples and multi-armed bandits for reducing the number of similarity comparisons. Our randomized algorithm, ProtoBandit, identifies a set of $k$ prototypes incurring $O(k^3|S|)$ similarity comparisons, which is independent of the size of the target set. An interesting outcome of our analysis is for the $k$-medoids clustering problem $T = S$ setting) in which we show that our algorithm ProtoBandit approximates the BUILD step solution of the partitioning around medoids (PAM) method in $O(k^3|S|)$ complexity. Empirically, we observe that ProtoBandit reduces the number of similarity computation calls by several orders of magnitudes ($100-1000$ times) while obtaining solutions similar in quality to those from state-of-the-art approaches.

In this paper, we propose a constant word (RAM model) algorithm for regret minimisation for both finite and infinite Stochastic Multi-Armed Bandit (MAB) instances. Most of the existing regret minimisation algorithms need to remember the statistics of all the arms they encounter. This may become a problem for the cases where the number of available words of memory is limited. Designing an efficient regret minimisation algorithm that uses a constant number of words has long been interesting to the community. Some early attempts consider the number of arms to be infinite, and require the reward distribution of the arms to belong to some particular family. Recently, for finitely many-armed bandits an explore-then-commit based algorithm~\citep{Liau+PSY:2018} seems to escape such assumption. However, due to the underlying PAC-based elimination their method incurs a high regret. We present a conceptually simple, and efficient algorithm that needs to remember statistics of at most $M$ arms, and for any $K$-armed finite bandit instance it enjoys a $O(KM +K^{1.5}\sqrt{T\log (T/MK)}/M)$ upper-bound on regret. We extend it to achieve sub-linear \textit{quantile-regret}~\citep{RoyChaudhuri+K:2018} and empirically verify the efficiency of our algorithm via experiments.

We propose a PAC formulation for identifying an arm in an n-armed bandit whose mean is within a fixed tolerance of the m-th highest mean. This setup generalises a previous formulation with m = 1, and differs from yet another one which requires m such arms to be identified. The key implication of our proposed approach is the ability to derive upper bounds on the sample complexity that depend on n/m in place of n. Consequently, even when the number of arms is infinite, we only need a finite number of samples to identify an arm that compares favourably with a fixed reward quantile. This facility makes our approach attractive to applications such as drug discovery, wherein the number of arms (molecular configurations) may run into a few thousands. We present sampling algorithms for both the finite- and infinite-armed cases, and validate their efficiency through theoretical and experimental analysis.We also present a lower bound on the worst case sample complexity of PAC algorithms for our problem, which matches our upper bound up to a logarithmic factor.