Unifying the stochastic and the adversarial Bandits with Knapsack
This work addresses a fundamental limitation in online learning for resource-constrained decision-making, though it is incremental by extending prior results to more general adversarial conditions.
The paper tackles the adversarial Bandits with Knapsack problem by proposing EXP3.BwK and EXP3++.BwK algorithms, which achieve order optimal regret in the general setting without the restriction that reward must exceed cost, and show that large costs can lead to worse regret bounds.
This paper investigates the adversarial Bandits with Knapsack (BwK) online learning problem, where a player repeatedly chooses to perform an action, pays the corresponding cost, and receives a reward associated with the action. The player is constrained by the maximum budget $B$ that can be spent to perform actions, and the rewards and the costs of the actions are assigned by an adversary. This problem has only been studied in the restricted setting where the reward of an action is greater than the cost of the action, while we provide a solution in the general setting. Namely, we propose EXP3.BwK, a novel algorithm that achieves order optimal regret. We also propose EXP3++.BwK, which is order optimal in the adversarial BwK setup, and incurs an almost optimal expected regret with an additional factor of $\log(B)$ in the stochastic BwK setup. Finally, we investigate the case of having large costs for the actions (i.e., they are comparable to the budget size $B$), and show that for the adversarial setting, achievable regret bounds can be significantly worse, compared to the case of having costs bounded by a constant, which is a common assumption within the BwK literature.