Quantum Algorithms for Bandits with Knapsacks with Improved Regret and Time Complexities

Bandits with knapsacks (BwK) constitute a fundamental model that combines aspects of stochastic integer programming with online learning. Classical algorithms for BwK with a time horizon $T$ achieve a problem-independent regret bound of ${O}(\sqrt{T})$ and a problem-dependent bound of ${O}(\log T)$. In this paper, we initiate the study of the BwK model in the setting of quantum computing, where both reward and resource consumption can be accessed via quantum oracles. We establish both problem-independent and problem-dependent regret bounds for quantum BwK algorithms. For the problem-independent case, we demonstrate that a quantum approach can improve the classical regret bound by a factor of $(1+\sqrt{B/\mathrm{OPT}_\mathrm{LP}})$, where $B$ is budget constraint in BwK and $\mathrm{OPT}_{\mathrm{LP}}$ denotes the optimal value of a linear programming relaxation of the BwK problem. For the problem-dependent setting, we develop a quantum algorithm using an inexact quantum linear programming solver. This algorithm achieves a quadratic improvement in terms of the problem-dependent parameters, as well as a polynomial speedup of time complexity on problem's dimensions compared to classical counterparts. Compared to previous works on quantum algorithms for multi-armed bandits, our study is the first to consider bandit models with resource constraints and hence shed light on operations research.