Budgeted Bayesian Bandits for Future Utilization

In multi-armed bandits, the most-explored arms are the most informative, while reward maximization typically pulls only the best arm. We study the tradeoff between identifying arm means accurately and accumulating reward, and present an algorithm with regret guarantees that interpolates between the two objectives. We provide both upper and lower bounds and validate empirically.

We study a stochastic multi-armed bandit problem where an agent is granted a free exploration budget before regret accumulates, a setting not captured by the classic regret minimization or pure exploration paradigms. The goal is to design an adaptive policy that strategically explores the bandit instance in the initial free exploration phase and minimizes the cumulative regret in the subsequent phase. We formalize this regret minimization with free exploration problem and identify an interesting regime where the free exploration budget scales logarithmically with the time horizon. To quantify the amount of regret saved with high probability as a result of the availability of the free exploration phase, we introduce a novel set of policies known as $(α,β)$-probably saving policies. We propose a two-phase, probably saving algorithm, UFE-KLUCB-H, which consists of a principled free exploration policy, UFE, and a history-aware regret minimization policy KLUCB-H. Instance-dependent upper bounds on UFE-KLUCB-H are derived, showing that UFE-KLUCB-H accumulates strictly less regret than policies that do not have access to a free exploration phase. Complementarily, we derive instance-dependent lower bounds based on novel multi-instance perturbation arguments tailored to the free-exploration setting, demonstrating the near-optimality of UFE-KLUCB-H for two-valued bandits. Our upper and lower bounds reveal sharp phase transitions in the accumulated regret depending on the amount of available free exploration. Simulations are conducted to demonstrate that forced exploration and adaptivity in the algorithm lead to greater regret savings.

Sample complexity is a widely used metric in sequential decision-making problems, defined as the number of suboptimal decisions during the interaction between the agent and an environment. We study the sample complexity of stochastic multi-armed bandit problems and introduce the expected sample complexity performance measure, analyzing it in a novel framework called approximately correct in expectation (ACE). We show that ACE guarantees imply almost sure convergence to the optimal expected reward, in contrast to high-probability guarantees found in other frameworks, and also show how to convert ACE guarantees into explicit expected regret bounds. We further show that, in contrast to existing measures, deterministic algorithms cannot obtain favorable ACE bounds, and analyze stochastic algorithms in two settings: when the allowed suboptimality level $ε$ is known to the algorithm and when it is unknown. In the former, we devise an explore-then-$ε$-greedy algorithm, and in the latter, we analyze the expected sample complexity of Thompson sampling. Finally, we establish nearly matching lower bounds for both settings, showing that the algorithms are tight in $ε$ and proving a performance separation between the two regimes.