Lenient Regret for Multi-Armed Bandits
This work addresses the issue of over-exploration in large or brief-interaction bandit problems, offering a more practical approach for scenarios where finding optimal arms is infeasible.
The paper tackles the Multi-Armed Bandit problem by proposing a lenient regret criterion that ignores small suboptimality gaps, and presents a variant of Thompson Sampling called ε-TS, proving its asymptotic optimality and showing that its lenient regret is bounded by a constant under certain conditions.
We consider the Multi-Armed Bandit (MAB) problem, where an agent sequentially chooses actions and observes rewards for the actions it took. While the majority of algorithms try to minimize the regret, i.e., the cumulative difference between the reward of the best action and the agent's action, this criterion might lead to undesirable results. For example, in large problems, or when the interaction with the environment is brief, finding an optimal arm is infeasible, and regret-minimizing algorithms tend to over-explore. To overcome this issue, algorithms for such settings should instead focus on playing near-optimal arms. To this end, we suggest a new, more lenient, regret criterion that ignores suboptimality gaps smaller than some $ε$. We then present a variant of the Thompson Sampling (TS) algorithm, called $ε$-TS, and prove its asymptotic optimality in terms of the lenient regret. Importantly, we show that when the mean of the optimal arm is high enough, the lenient regret of $ε$-TS is bounded by a constant. Finally, we show that $ε$-TS can be applied to improve the performance when the agent knows a lower bound of the suboptimality gaps.