An Index-based Deterministic Asymptotically Optimal Algorithm for Constrained Multi-armed Bandit Problems

For the model of constrained multi-armed bandit, we show that by construction there exists an index-based deterministic asymptotically optimal algorithm. The optimality is achieved by the convergence of the probability of choosing an optimal feasible arm to one over infinite horizon. The algorithm is built upon Locatelli et al.'s "anytime parameter-free thresholding" algorithm under the assumption that the optimal value is known. We provide a finite-time bound to the probability of the asymptotic optimality given as 1-O(|A|Te^{-T}) where T is the horizon size and A is the set of the arms in the bandit. We then study a relaxed-version of the algorithm in a general form that estimates the optimal value and discuss the asymptotic optimality of the algorithm after a sufficiently large T with examples.

An Asymptotically Optimal Strategy for Constrained Multi-armed Bandit Problems

For the stochastic multi-armed bandit (MAB) problem from a constrained model that generalizes the classical one, we show that an asymptotic optimality is achievable by a simple strategy extended from the $\epsilon_t$-greedy strategy. We provide a finite-time lower bound on the probability of correct selection of an optimal near-feasible arm that holds for all time steps. Under some conditions, the bound approaches one as time $t$ goes to infinity. A particular example sequence of $\{\epsilon_t\}$ having the asymptotic convergence rate in the order of $(1-\frac{1}{t})^4$ that holds from a sufficiently large $t$ is also discussed.