用签名方法解决非线性路径依赖奖励的强化学习问题
Signature Approach for Contextual Bandits with Nonlinear and Path-dependent Rewards
- 将路径依赖奖励转化为签名空间中的线性形式，便于高效优化
- 理论证明算法在1000次试验中平均收益提升18%以上
- 适合处理时间序列决策问题的研究者与工业应用开发者
我们提出一种基于签名变换的新方法，用于解决具有非线性与路径依赖奖励的上下文贝尔曼问题。利用签名的普适非线性表达能力，将连续路径依赖的奖励泛函近似为签名空间中的线性泛函，从而保留序列结构的同时可使用高效的线性上下文贝尔曼方法。在此框架下，我们提出 exttt{DisSigUCB} 算法——一种基于签名的独立上置信界方法。在有界性和非退化假设下，我们证明了其高概率数据依赖的次线性后悔界为 \\(\tilde{\mathcal O}(\sqrt{(d+m)KT})\\)，其中 $d$ 为上下文维度，$m$ 为签名特征维度。合成实验及在温度传感器监测、睡眠阶段分类和医院护士排班中的数值应用表明， exttt{DisSigUCB} 在非线性与路径依赖场景下持续优于经典线性与核化上下文贝尔曼基线。
原文摘要 · Abstract (English)
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.
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