arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新,休息一下吧,周末愉快~~
arXiv 2402.17375eess.SYcs.LGcs.SY
收藏

积分强化学习在连续时间控制中计算的影响

Impact of Computation in Integral Reinforcement Learning for Continuous-Time Control

University of Manchester(曼彻斯特大学);Tsinghua University(清华大学)

Wenhan Cao, Wei Pan

AI总结:

本文研究积分强化学习中求积规则的计算误差对控制性能的影响,将其类比为牛顿法中的误差项,并提出基于再生核希尔伯特空间的贝叶斯求积方法,给出局部收敛速率理论并实验验证。

摘要:

积分强化学习(IntRL)在其策略评估(PEV)阶段要求精确计算效用函数的积分。这通过求积规则实现,即对从离散时间获得的状态样本所评估的效用函数进行加权求和。我们的研究揭示了一个关键但尚未被充分探讨的现象:计算方法的选择——在本例中即求积规则——会显著影响控制性能。这种影响可追溯到以下事实:在PEV阶段引入的计算误差会影响策略迭代的收敛行为,进而影响所学到的控制器。为了阐明计算如何影响控制,我们将IntRL的策略迭代与应用于Hamilton-Jacobi-Bellman方程的牛顿法进行类比。从这一角度看,PEV中的计算误差表现为牛顿法每次迭代中的一个额外误差项,其误差上界与计算误差成正比。此外,我们证明当效用函数位于再生核希尔伯特空间(RKHS)中时,可以通过采用以RKHS诱导核函数进行的贝叶斯求积来实现最优求积。我们证明,使用梯形规则和带有Matérn核的贝叶斯求积时,IntRL的局部收敛速率分别为$O(N^{-2})$和$O(N^{-b})$,其中$N$是均匀间隔样本的数量,$b$是Matérn核的光滑性参数。这些理论发现最终通过两个典型控制任务得到了验证。

英文摘要:

Integral reinforcement learning (IntRL) demands the precise computation of the utility function's integral at its policy evaluation (PEV) stage. This is achieved through quadrature rules, which are weighted sums of utility functions evaluated from state samples obtained in discrete time. Our research reveals a critical yet underexplored phenomenon: the choice of the computational method -- in this case, the quadrature rule -- can significantly impact control performance. This impact is traced back to the fact that computational errors introduced in the PEV stage can affect the policy iteration's convergence behavior, which in turn affects the learned controller. To elucidate how computation impacts control, we draw a parallel between IntRL's policy iteration and Newton's method applied to the Hamilton-Jacobi-Bellman equation. In this light, computational error in PEV manifests as an extra error term in each iteration of Newton's method, with its upper bound proportional to the computational error. Further, we demonstrate that when the utility function resides in a reproducing kernel Hilbert space (RKHS), the optimal quadrature is achievable by employing Bayesian quadrature with the RKHS-inducing kernel function. We prove that the local convergence rates for IntRL using the trapezoidal rule and Bayesian quadrature with a Matérn kernel to be $O(N^{-2})$ and $O(N^{-b})$, where $N$ is the number of evenly-spaced samples and $b$ is the Matérn kernel's smoothness parameter. These theoretical findings are finally validated by two canonical control tasks.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

↑