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.

翻译：现代决策场景通常涉及高维且富含高阶上下文信息的数据，而现有赌博机算法难以生成有效策略。为此，本文提出一种广义线性张量赌博机算法，通过引入低维张量结构应对这些挑战，并进一步构建了该算法的统一分析框架。具体而言，该框架采用带有弱可分解正则化器的凸优化方法，使其不仅能基于张量低秩结构假设获得更优结果，还可扩展至包含切片稀疏性与低秩性等其他低维结构的场景。理论分析表明，相较于现有低秩张量相关研究，本框架不仅提供了更紧的界，还具有更广泛的适用性。值得注意的是，在退化为低秩矩阵的特殊情形下，所得界在某些场景中仍具优势。