Convergence and Regret of the Policy Gradient for Multi-Armed Bandits in Diffusion Environment

AI 中文解读

核心亮点：这项研究首次证明了在复杂动态环境中，AI的“试错学习”策略能稳定找到最优选择，且学习效率有理论保障。通俗解读：想象一个多臂老虎机，但每个选项的奖励会随环境波动。过去AI在不确定环境中做选择容易陷入混乱，这次研究者用数学工具证明了，即便环境像水流一样持续变化，AI只要按固定节奏调整策略，最终一定能锁定最佳选项，而且学习速度不会太慢。他们找到了一个巧妙的“能量函数”来监控AI的学习过程，就像用温度计监测水温，让整个收敛过程变得清晰可控。实际影响：这项成果主要惠及算法设计和AI决策领域。未来用在智能推荐、广告投放或资源调度上，AI能在动态市场中更快适应变化，减少试错成本。对普通人而言，可能会感受到更精准的购物推荐、更高效的网约车派单，或者金融投资顾问给出更稳健的建议。虽然理论性较强，但它为AI在真实、多变环境中“稳住不慌”提供了扎实数学基础，是迈向更可靠智能系统的重要一步。

This paper studies the policy gradient update for a multi-arm bandit problem in diffusion environment that is described by a stochastic differential equation (SDE) under the continuous-time reinforcement learning framework by Wang et al. (2020), Jia and Zhou (2022b). With the logit parameterization for the stochastic policy, we show that it converges almost surely to the optimal arm under an arbitrary constant learning rate. Furthermore, we derive the non-asymptotic regret upper bound when the constant learning rate is below a time-invariant threshold; and the regret bound has order $O(\log T)$. We improve the analysis in Lattimore (2026a) for the same SDE by constructing a novel Lyapunov function and demonstrate the transparency of analyzing policy gradient using the tools in SDEs. In addition, the same Lyapunov function is also helpful in analyzing the discrete-time policy gradient algorithm.