In contrast to the classic formulation of partial monitoring, linear partial monitoring can model infinite outcome spaces, while imposing a linear structure on both the losses and the observations. This setting can be viewed as a generalization of linear bandits where loss and feedback are decoupled in a flexible manner. In this work, we address a nonstochastic (adversarial), finite-actions version of the problem through a simple instance of the exploration-by-optimization method that is amenable to efficient implementation. We derive regret bounds that depend on the game structure in a more transparent manner than previous theoretical guarantees for this paradigm. Our bounds feature instance-specific quantities that reflect the degree of alignment between observations and losses, and resemble known guarantees in the stochastic setting. Notably, they achieve the standard $\sqrt{T}$ rate in easy (locally observable) games and $T^{2/3}$ in hard (globally observable) games, where $T$ is the time horizon. We instantiate these bounds in a selection of old and new partial information settings subsumed by this model, and illustrate that the achieved dependence on the game structure can be tight in interesting cases.

翻译：与经典的部分监测问题不同，线性部分监测能够建模无限的结果空间，同时对损失和观测施加线性结构。该设定可视为线性赌博机问题的一种推广，其中损失与反馈以灵活的方式解耦。在本工作中，我们通过一种易于高效实现的探索优化方法之简单实例，处理该问题的非随机（对抗性）有限动作版本。我们推导出的遗憾界以比先前该范式理论保证更透明的方式依赖于博弈结构。我们的界包含反映观测与损失之间对齐程度的实例特定量，且与随机设定中的已知保证相似。值得注意的是，它们在简单（局部可观测）博弈中达到标准的$\sqrt{T}$速率，在困难（全局可观测）博弈中达到$T^{2/3}$速率，其中$T$为时间范围。我们在该模型所涵盖的一系列新旧部分信息设定中实例化这些界，并说明所实现的博弈结构依赖性在有趣的情形下可能是紧的。