Adaptive Partitioning and Learning for Stochastic Control of Diffusion Processes

We study continuous-time and possibly high-dimensional stochastic control problems where drift coefficients and running reward functions are unknown. Due to these missing model primitives, we take the exploratory, reinforcement learning (RL) framework of Wang, Zariphopoulou, and Zhou(2020) with relaxed controls and entropy regularization. The objective is to develop theoretically grounded, efficient and scalable RL algorithms to learn both the optimal value functions (which also solve the exploratory HJB equation) and optimal exploratory feedback control policies. When the diffusion coefficients do not contain control, we employ probabilistic representations of both the optimal value function and its gradient based on an auxiliary state process depending only on the diffusion part of the original dynamics. With a delicate analysis on some properly defined mappings and their fixed points, this leads to the introduction of our policy iteration algorithms and their convergence. We demonstrate the performance of our algorithms through various numerical examples. Finally, we study a special control-dependent diffusion case where probability representation of the Hessian is called for.

Diffusion-based policies have recently emerged as powerful policy parameterizations for reinforcement learning, representing state-conditioned action distributions as terminal laws of diffusion processes with parameterized drifts. This terminal-law representation has shown substantial expressive flexibility in practice, enabling diffusion policies to model complex, multimodal, and highly non-Gaussian action distributions; however, it remains unclear what mathematically drives this expressivity and how to fully exploit it when the policy is learned from finite data. In this paper, we identify the drift Lipschitz budget $K$ as a central quantity governing the expressivity and statistical behavior of diffusion policies. We quantify expressivity through approximation: diffusion policies with $K$-Lipschitz drifts can concentrate near optimal deterministic policies and achieve value approximation error of order $1/K$; moreover, we prove a matching lower bound under nondegenerate diffusion noise. This increased expressivity comes with a statistical cost. When the drift is parameterized by neural networks, increasing $K$ improves approximation but increases statistical complexity. Balancing these two terms yields a finite-sample performance gap of order $\tilde{O}(n^{-2/(m+6)})$ for generic neural-network drifts, and a sharper rate $\tilde{O}(n^{-2/(m+4)})$ for one-sided dissipative drift classes, where $n$ is the sample size and $m$ is the dimension of the state space. Numerical experiments provide empirical evidence for the sample-dependent trade-off in $K$, supporting both theoretical regimes. Our framework also suggests a practical implementation principle: choose the diffusion budget $K$ according to the available sample size, and then select a neural-network architecture with the corresponding fixed Lipschitz coefficient.

Diffusion policies have recently emerged as a powerful paradigm for representing complex action distributions in reinforcement learning (RL). However, their application to online RL remains limited by the challenge of scalable training in the absence of ground-truth data, where standard optimization techniques such as score matching are not directly applicable. In this work, we introduce a highly efficient algorithm for optimizing diffusion policies by leveraging recent advances in stochastic optimal control. Our approach is based on adjoint matching, which enables simulation-free training and circumvents the need for explicit likelihood estimation or costly backpropagation through the diffusion process. Furthermore, we propose several extensions that improve the robustness and stability of the method in practical settings. Empirical results demonstrate that our approach achieves competitive performance while significantly reducing computational overhead, making diffusion policies more viable for online RL scenarios.