# Tailoring Reproducing Kernels for Optimal Control via Policy Iteration

arXiv: https://arxiv.org/abs/2510.22494 · HTML: https://mlsanity.com/paper/2510.22494/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 4 of its 38 references that are in the map.
- First posted on arXiv 26 Oct 2025.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 4 of 4)

- [Robust feedback control of nonlinear PDEs by numerical approximation of high-dimensional Hamilton-Jacobi-Isaacs equations](https://mlsanity.com/paper/1905.06276/) (2019)
- [Policy iteration for Hamilton-Jacobi-Bellman equations with control constraints](https://mlsanity.com/paper/2004.03558/) (2020)
- [Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems](https://mlsanity.com/paper/2305.06122/) (2023)
- [Convergence Rates of Online Critic Value Function Approximation in Native Spaces](https://mlsanity.com/paper/2405.05887/) (2024)

## Abstract (arXiv)

This paper presents a novel approach to formulating the actor-critic method for optimal control by casting policy iteration in reproducing kernel Hilbert spaces (RKHSs -- also known as native spaces). By tailoring the reproducing kernel and RKHS to the dynamics of the nonlinear optimal control problem, we leverage recent advancements in characterizing error bounds from statistical and machine learning theory. These approximations define a general strategy to select the bases of the actor-critic networks, and we formally guarantee for the first time that this basis selection procedure leads to closed-form error bounds for the individual steps of policy iteration. These bounds often have a geometric and computable form, making them potentially useful for a priori or a posteriori evaluation of candidate collections of scattered bases. Numerical studies subsequently provide qualitative evidence of the practical performance achieved for the full recursion using the algorithms and theory developed for the single-step error bounds.