Kumar Abhishek

Kumar Abhishek, Ganesh Ghalme, Sujit Gujar et al.

In this paper, we study an interesting combination of sleeping and combinatorial stochastic bandits. In the mixed model studied here, at each discrete time instant, an arbitrary \emph{availability set} is generated from a fixed set of \emph{base} arms. An algorithm can select a subset of arms from the \emph{availability set} (sleeping bandits) and receive the corresponding reward along with semi-bandit feedback (combinatorial bandits). We adapt the well-known CUCB algorithm in the sleeping combinatorial bandits setting and refer to it as \CSUCB. We prove -- under mild smoothness conditions -- that the \CSUCB\ algorithm achieves an $O(\log (T))$ instance-dependent regret guarantee. We further prove that (i) when the range of the rewards is bounded, the regret guarantee of \CSUCB\ algorithm is $O(\sqrt{T \log (T)})$ and (ii) the instance-independent regret is $O(\sqrt[3]{T^2 \log(T)})$ in a general setting. Our results are quite general and hold under general environments -- such as non-additive reward functions, volatile arm availability, a variable number of base-arms to be pulled -- arising in practical applications. We validate the proven theoretical guarantees through experiments.

Ayush Deva, Kumar Abhishek, Sujit Gujar

We explore the class of problems where a central planner needs to select a subset of agents, each with different quality and cost. The planner wants to maximize its utility while ensuring that the average quality of the selected agents is above a certain threshold. When the agents' quality is known, we formulate our problem as an integer linear program (ILP) and propose a deterministic algorithm, namely \dpss\ that provides an exact solution to our ILP. We then consider the setting when the qualities of the agents are unknown. We model this as a Multi-Arm Bandit (MAB) problem and propose \newalgo\ to learn the qualities over multiple rounds. We show that after a certain number of rounds, $τ$, \newalgo\ outputs a subset of agents that satisfy the average quality constraint with a high probability. Next, we provide bounds on $τ$ and prove that after $τ$ rounds, the algorithm incurs a regret of $O(\ln T)$, where $T$ is the total number of rounds. We further illustrate the efficacy of \newalgo\ through simulations. To overcome the computational limitations of \dpss, we propose a polynomial-time greedy algorithm, namely \greedy, that provides an approximate solution to our ILP. We also compare the performance of \dpss\ and \greedy\ through experiments.