  The multi-armed bandit (MAB) model has been widely adopted for studying many
practical optimization problems (network resource allocation, ad placement,
crowdsourcing, etc.) with unknown parameters. The goal of the player here is to
maximize the cumulative reward in the face of uncertainty. However, the basic
MAB model neglects several important factors of the system in many real-world
applications, where multiple arms can be simultaneously played and an arm could
sometimes be "sleeping". Besides, ensuring fairness is also a key design
concern in practice. To that end, we propose a new Combinatorial Sleeping MAB
model with Fairness constraints, called CSMAB-F, aiming to address the
aforementioned crucial modeling issues. The objective is now to maximize the
reward while satisfying the fairness requirement of a minimum selection
fraction for each individual arm. To tackle this new problem, we extend an
online learning algorithm, UCB, to deal with a critical tradeoff between
exploitation and exploration and employ the virtual queue technique to properly
handle the fairness constraints. By carefully integrating these two techniques,
we develop a new algorithm, called Learning with Fairness Guarantee (LFG), for
the CSMAB-F problem. Further, we rigorously prove that not only LFG is
feasibility-optimal, but it also has a time-average regret upper bounded by
$\frac{N}{2\eta}+\frac{\beta_1\sqrt{mNT\log{T}}+\beta_2 N}{T}$, where N is the
total number of arms, m is the maximum number of arms that can be
simultaneously played, T is the time horizon, $\beta_1$ and $\beta_2$ are
constants, and $\eta$ is a design parameter that we can tune. Finally, we
perform extensive simulations to corroborate the effectiveness of the proposed
algorithm. Interestingly, the simulation results reveal an important tradeoff
between the regret and the speed of convergence to a point satisfying the
fairness constraints.
