Generalized Low-Rank Matrix Contextual Bandits with Graph Information

The matrix contextual bandit (CB), as an extension of the well-known multi-armed bandit, is a powerful framework that has been widely applied in sequential decision-making scenarios involving low-rank structure. In many real-world scenarios, such as online advertising and recommender systems, additional graph information often exists beyond the low-rank structure, that is, the similar relationships among users/items can be naturally captured through the connectivity among nodes in the corresponding graphs. However, existing matrix CB methods fail to explore such graph information, and thereby making them difficult to generate effective decision-making policies. To fill in this void, we propose in this paper a novel matrix CB algorithmic framework that builds upon the classical upper confidence bound (UCB) framework. This new framework can effectively integrate both the low-rank structure and graph information in a unified manner. Specifically, it involves first solving a joint nuclear norm and matrix Laplacian regularization problem, followed by the implementation of a graph-based generalized linear version of the UCB algorithm. Rigorous theoretical analysis demonstrates that our procedure outperforms several popular alternatives in terms of cumulative regret bound, owing to the effective utilization of graph information. A series of synthetic and real-world data experiments are conducted to further illustrate the merits of our procedure.

A Unified Regularization Approach to High-Dimensional Generalized Tensor Bandits

Modern decision-making scenarios often involve data that is both high-dimensional and rich in higher-order contextual information, where existing bandits algorithms fail to generate effective policies. In response, we propose in this paper a generalized linear tensor bandits algorithm designed to tackle these challenges by incorporating low-dimensional tensor structures, and further derive a unified analytical framework of the proposed algorithm. Specifically, our framework introduces a convex optimization approach with the weakly decomposable regularizers, enabling it to not only achieve better results based on the tensor low-rankness structure assumption but also extend to cases involving other low-dimensional structures such as slice sparsity and low-rankness. The theoretical analysis shows that, compared to existing low-rankness tensor result, our framework not only provides better bounds but also has a broader applicability. Notably, in the special case of degenerating to low-rank matrices, our bounds still offer advantages in certain scenarios.