New Algorithm Boosts Multi-Objective Low-Rank Matrix Bandits.

This research addresses the challenge of online decision-making when faced with multiple objectives that have different priority levels, a scenario common in areas like recommendation systems or resource allocation. The problem is framed as generalized low-rank matrix bandits, where selecting an "arm" (e.g., an item or action) yields a vector of rewards corresponding to these objectives. The proposed algorithm, Lexi-LowGLM, is designed for efficiency. Instead of repeatedly solving complex batch estimators, it uses an online Newton step to update objective-specific estimators, drastically reducing the computational cost over time. This approach first identifies low-rank subspaces for each objective and then applies lexicographic learning within these reduced spaces. The algorithm achieves a favorable regret bound that depends on the effective low-rank dimension rather than the full ambient dimension, indicating better scalability. Numerical experiments further confirm both the effectiveness of the method in handling prioritized objectives and its computational advantages.