This paper studies convergence rates for some value function approximations that arise in a collection of reproducing kernel Hilbert spaces (RKHS) $H(\Omega)$. By casting an optimal control problem in a specific class of native spaces, strong rates of convergence are derived for the operator equation that enables offline approximations that appear in policy iteration. Explicit upper bounds on error in value function approximations are derived in terms of power function $\Pwr_{H,N}$ for the space of finite dimensional approximants $H_N$ in the native space $H(\Omega)$. These bounds are geometric in nature and refine some well-known, now classical results concerning convergence of approximations of value functions.

翻译：本文研究了在再生核希尔伯特空间族 $H(\Omega)$ 中产生的某些价值函数逼近的收敛速率。通过将最优控制问题置于特定类别的原生空间中，推导出了策略迭代中出现的离线逼近所涉及的算子方程的强收敛速率。基于原生空间 $H(\Omega)$ 中有限维逼近空间 $H_N$ 的幂函数 $\Pwr_{H,N}$，本文给出了价值函数逼近误差的显式上界。这些上界具有几何性质，并改进了一些关于价值函数逼近收敛性的、现已经典的已知结果。