We study contextual bandits with nonlinear and path-dependent rewards through a novel signature-transform-based approach. Leveraging the universal nonlinearity property of signatures, we approximate continuous path-dependent reward functionals by linear functionals in the signature space. This representation enables the use of efficient linear contextual bandit methods while preserving expressive sequential structure. Building on this framework, we propose \texttt{DisSigUCB}, a signature-based disjoint upper confidence bound (UCB) algorithm. Under boundedness and non-degeneracy assumptions, we prove a high-probability data-dependent sublinear regret bound of order \(\tilde{\mathcal O}(\sqrt{(d+m)KT})\) where \(d\) is the context dimension and \(m\) is the signature feature dimension. Synthetic experiments and numerical applications on temperature sensor monitoring, sleep-stage classification, and hospital nurse staffing demonstrate that \texttt{DisSigUCB} consistently outperforms classical linear and kernelized contextual bandit baselines in nonlinear and path-dependent settings.

翻译：我们通过一种新颖的基于签名变换的方法，研究了具有非线性和路径依赖奖励的情境赌博机问题。利用签名的通用非线性性质，我们将连续路径依赖的奖励函数近似为签名空间中的线性泛函。这种表示在保留表达性序列结构的同时，使得高效的线性情境赌博机方法得以应用。基于此框架，我们提出了一种基于签名的分离上置信界算法\texttt{DisSigUCB}。在有界性和非退化假设下，我们证明了其具有阶为\(\tilde{\mathcal O}(\sqrt{(d+m)KT})\)的高概率数据依赖次线性遗憾界，其中\(d\)为情境维度，\(m\)为签名特征维度。在温度传感器监测、睡眠阶段分类及医院护士排班上的合成实验与数值应用表明，\texttt{DisSigUCB}在非线性和路径依赖场景中始终优于经典的线性和核化情境赌博机基线方法。