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

2025-07-06

AI Summary

This work introduces quantum computing to the Bandits with Knapsack (BwK) problem—a stochastic integer programming framework for resource-constrained online decision-making—marking the first such integration. We propose a novel algorithmic framework leveraging quantum oracles to access reward and resource consumption information, combined with an inexact quantum linear programming solver. Theoretically, under a problem-agnostic setting, our algorithm achieves a regret bound of $Oig(sqrt{T} + sqrt{T cdot B/mathrm{OPT}_{mathrm{LP}}}ig)$, yielding a $(1+sqrt{B/mathrm{OPT}_{mathrm{LP}}})$-factor speedup over the optimal classical regret bound. Under a problem-dependent setting, it attains quadratic improvement in regret and polynomial-time speedup scaling with problem dimension. This is the first work to establish rigorous quantum advantage guarantees for BwK, bridging quantum optimization and online learning under budget constraints.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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.

Problem

• Quantum algorithms for Bandits with Knapsacks (BwK) model
• Improving regret bounds in quantum BwK algorithms
• Achieving polynomial speedup in time complexity

Innovation

• Quantum oracles access reward and resource data
• Inexact quantum linear programming solver used
• Quadratic improvement in problem-dependent parameters