An Asymptotically Optimal Strategy for Constrained Multi-armed Bandit Problems

This note considers the model of “constrained multi-armed bandit” (CMAB) that generalizes that of the classical stochastic MAB by adding a feasibility constraint for each action. The feasibility is in fact another (conflicting) objective that should be kept in order for a playing-strategy to achieve the optimality of the main objective. While the stochastic MAB model is a special case of the Markov decision process (MDP) model, the CMAB model is a special case of the constrained MDP model. For the asymptotic optimality measured by the probability of choosing an optimal feasible arm over infinite horizon, we show that the optimality is achievable by a simple strategy extended from the ϵt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _t$$\end{document}-greedy strategy used for unconstrained MAB problems. 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 {ϵt}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\epsilon _t\}$$\end{document} having the asymptotic convergence rate in the order of (1-1t)4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-\frac{1}{t})^4$$\end{document} that holds from a sufficiently large t is also discussed.