We study bi-criteria combinatorial optimization under noisy function evaluations. While resilience and black-box offline-to-online reductions have been studied in single-objective settings, extending these ideas to bi-criteria problems introduces new challenges due to the coupled degradation of approximation guarantees for objectives and constraints. We introduce a notion of $(α,β,δ,\texttt{N})$-resilience for bi-criteria approximation algorithms, capturing how joint approximation guarantees degrade under bounded (possibly worst-case) oracle noise, and develop a general black-box framework that converts any resilient offline algorithm into an online algorithm for bi-criteria combinatorial multi-armed bandits with bandit feedback. The resulting online guarantees achieve sublinear regret and cumulative constraint violation of order $\tilde{O}(δ^{2/3}\texttt{N}^{1/3}T^{2/3})$ without requiring structural assumptions such as linearity, submodularity, or semi-bandit feedback on the noisy functions. We demonstrate the applicability of the framework by establishing resilience for several classical greedy algorithms in submodular optimization.

翻译：我们研究带有噪声函数评估下的双准则组合优化问题。尽管在单目标设置中已对弹性和黑箱离线到在线转化进行了研究，但由于目标和约束的近似保证存在耦合退化，将这些思想扩展到双准则问题带来了新的挑战。我们提出了双准则近似算法的$(α,β,δ,\texttt{N})$-弹性概念，用以刻画在有界（可能最坏情况）预言机噪声下联合近似保证的退化方式，并发展了一个通用的黑箱框架，可将任意弹性离线算法转化为用于双准则组合多臂老虎机（带Bandit反馈）的在线算法。所得在线保证实现了次线性遗憾和累计约束违反量，阶为$\tilde{O}(δ^{2/3}\texttt{N}^{1/3}T^{2/3})$，无需对噪声函数施加线性性、子模性或半Bandit反馈等结构假设。我们通过证明子模优化中若干经典贪心算法的弹性，展示了该框架的适用性。