Bi-Criteria Optimization for Combinatorial Bandits: Sublinear Regret and Constraint Violation under Bandit Feedback

• URL: http://arxiv.org/abs/2503.12285v1
• Date: Sat, 15 Mar 2025 22:52:27 GMT
• Title: Bi-Criteria Optimization for Combinatorial Bandits: Sublinear Regret and Constraint Violation under Bandit Feedback
• Authors: Vaneet Aggarwal, Shweta Jain, Subham Pokhriyal, Christopher John Quinn,
• Abstract summary: We study bi-criteria optimization for multi-armed bandits (CMAB) with bandit feedback.<n>We propose a general framework that transforms discrete bi-linear offline approximation algorithms into online algorithms with sublinear regret and cumulative constraint violation guarantees.<n>These applications highlight the framework's broad utility in adapting offline guarantees to online bi-criteria optimization under bandit feedback.
• Score: 27.613888121859393
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we study bi-criteria optimization for combinatorial multi-armed bandits (CMAB) with bandit feedback. We propose a general framework that transforms discrete bi-criteria offline approximation algorithms into online algorithms with sublinear regret and cumulative constraint violation (CCV) guarantees. Our framework requires the offline algorithm to provide an $(\alpha, \beta)$-bi-criteria approximation ratio with $\delta$-resilience and utilize $\texttt{N}$ oracle calls to evaluate the objective and constraint functions. We prove that the proposed framework achieves sub-linear regret and CCV, with both bounds scaling as ${O}\left(\delta^{2/3} \texttt{N}^{1/3}T^{2/3}\log^{1/3}(T)\right)$. Crucially, the framework treats the offline algorithm with $\delta$-resilience as a black box, enabling flexible integration of existing approximation algorithms into the CMAB setting. To demonstrate its versatility, we apply our framework to several combinatorial problems, including submodular cover, submodular cost covering, and fair submodular maximization. These applications highlight the framework's broad utility in adapting offline guarantees to online bi-criteria optimization under bandit feedback.

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