非线性蒙特卡洛方法与多项式运行时间的贝尔曼方程离散时间的高维度随机最佳控制问题
Christian Beck1, Arnulf Jentzen2,3, Konrad Kleinberg4
1Department of Mathematics, ETH Zurich, Zurich, Switzerland.
Applied mathematics and optimization
|February 7, 2025
概括
本研究介绍了新的非线性蒙特卡洛方法,用于在马尔科夫决策过程 (MDP) 中近似解决贝尔曼方程. 这些方法有效地克服了随机最佳控制问题的维度诅咒.
科学领域:
- 计算数学 计算数学 计算数学
- 人工智能的人工智能
- 运营研究 运营研究
背景情况:
- 随机最佳控制和马尔科夫决策过程 (MDP) 是不确定性和强化学习下的顺序决策的基础.
- 无限地平线MDP与一般状态空间的数值近似对于解决复杂的控制问题至关重要.
- 贝尔曼方程是MDPs中价值函数和最佳策略的特征的核心.
研究的目的:
- 开发和分析数值方法来近似解决无限地平线MDPs与一般状态空间的解决方案.
- 解决维度的诅咒在解决贝尔曼方程的挑战,以实现随机的最佳控制.
- 研究新型非线性蒙特卡洛方法的应用,其灵感来源于Q学习和多层次皮卡德近似.
主要方法:
- 结合了全历史递归多层次皮卡德近似方法与Q学习原理.
- 介绍了一类解决贝尔曼方程的非线性蒙特卡洛方法.
- 专注于离散时间的马尔科夫过程和无限地平线的最佳停止问题.
主要成果:
- 提出的非线性蒙特卡洛方法有效地近似解决贝尔曼方程.
- 证明这些方法没有受到维度的诅咒.
- 为离散时间随机最佳控制中的数值近似提供了一个强大的框架.
结论:
- 开发的非线性蒙特卡洛方法为解决复杂的MDP提供了一种计算效率高的方法.
- 这些发现推动了对随机最佳控制和强化学习问题的数值处理.
- 这些方法特别适用于具有无限地平线和一般状态空间的问题.
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