Efficient Online Lexicographic Generalized Low-Rank Matrix Bandits

This paper studies generalized low-rank matrix bandits with multiple prioritized objectives. At each round, the learner selects a matrix-valued arm and observes a vector-valued reward, whose components correspond to multiple objectives with different priority levels. Each objective is governed by an objective-specific generalized low-rank matrix model, and the learner evaluates arms according to a lexicographic preference order, prioritizing higher-level objectives before lower-level ones. We propose \textsc{Lexi-LowGLM}, an efficient online algorithm that first estimates objective-specific low-rank subspaces and then performs lexicographic learning in the reduced feature spaces. Unlike existing single-objective algorithms that repeatedly solve a batch generalized linear estimator using all historical observations, \textsc{Lexi-LowGLM} updates each objective-specific estimator via an online Newton step, reducing the estimator-update complexity over $T$ rounds from $O(T^2)$ to $O(T)$. We establish a regret bound of $\widetilde O\left(W_i^{\rm lex}\sqrt{m}\,(d_1+d_2)r\sqrt{T}\right)$ for each objective $i\in[m]$, where $r$ is an upper bound on the ranks of the objective-specific parameter matrices and $W_i^{\rm lex}$ characterizes the lexicographic trade-off effect. This bound depends on the effective low-rank dimension $(d_1+d_2)r$ rather than the ambient dimension $d_1d_2$. Numerical experiments further validate the effectiveness and computational efficiency of the proposed method.