# Accuracy of Discretely Sampled Stochastic Policies in Continuous-time Reinforcement Learning

arXiv: https://arxiv.org/abs/2503.09981 · HTML: https://mlsanity.com/paper/2503.09981/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 18 of its 50 references that are in the map.
- First posted on arXiv 13 Mar 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, 18 of 18)

- [Policy Gradient and Actor-Critic Learning in Continuous Time and Space: Theory and Algorithms](https://mlsanity.com/paper/2111.11232/) (2021)
- [q-Learning in Continuous Time](https://mlsanity.com/paper/2207.00713/) (2022)
- [Policy Evaluation and Temporal-Difference Learning in Continuous Time and Space: A Martingale Approach](https://mlsanity.com/paper/2108.06655/) (2021)
- [Making Deep Q-learning methods robust to time discretization](https://mlsanity.com/paper/1901.09732/) (2019)
- [Entropy Regularization for Mean Field Games with Learning](https://mlsanity.com/paper/2010.00145/) (2020)
- [Continuous-Time Model-Based Reinforcement Learning](https://mlsanity.com/paper/2102.04764/) (2021)
- [Logarithmic regret for episodic continuous-time linear-quadratic reinforcement learning over a finite-time horizon](https://mlsanity.com/paper/2006.15316/) (2020)
- [Hamilton-Jacobi Deep Q-Learning for Deterministic Continuous-Time Systems with Lipschitz Continuous Controls](https://mlsanity.com/paper/2010.14087/) (2020)
- [Policy Optimization for Continuous Reinforcement Learning](https://mlsanity.com/paper/2305.18901/) (2023)
- [Convergence of policy gradient methods for finite-horizon exploratory linear-quadratic control problems](https://mlsanity.com/paper/2211.00617/) (2022)
- [Reinforcement learning for linear-convex models with jumps via stability analysis of feedback controls](https://mlsanity.com/paper/2104.09311/) (2021)
- [Actor-Critic learning for mean-field control in continuous time](https://mlsanity.com/paper/2303.06993/) (2023)
- [Continuous-time q-learning for mean-field control problems](https://mlsanity.com/paper/2306.16208/) (2023)
- [Entropy annealing for policy mirror descent in continuous time and space](https://mlsanity.com/paper/2405.20250/) (2024)
- [Continuous-time Risk-sensitive Reinforcement Learning via Quadratic Variation Penalty](https://mlsanity.com/paper/2404.12598/) (2024)
- [PhiBE: A PDE-based Bellman Equation for Continuous Time Policy Evaluation](https://mlsanity.com/paper/2405.12535/) (2024)
- [Convergence Analysis for Entropy-Regularized Control Problems: A Probabilistic Approach](https://mlsanity.com/paper/2406.10959/) (2024)
- [On the grid-sampling limit SDE](https://mlsanity.com/paper/2410.07778/) (2024)

## Built on by (most cited first, 13 of 13)

- [Deterministic Policy Gradient for Reinforcement Learning with Continuous Time and State](https://mlsanity.com/paper/2509.23711/) (2025)
- [Continuous-time q-learning for mean-field control with common noise, part-I: Theoretical foundations](https://mlsanity.com/paper/2604.27372/) (2026)
- [Policy Gradient for Continuous-Time Mean-Field Control](https://mlsanity.com/paper/2605.20718/) (2026)
- [Reconciling Discrete-Time Mixed Policies and Continuous-Time Relaxed Controls in Reinforcement Learning and Stochastic Control](https://mlsanity.com/paper/2504.21793/) (2025)
- [Deterministic Policy Gradient for Learning Equilibrium in Time-Inconsistent Control Problems](https://mlsanity.com/paper/2606.11798/) (2026)
- [Discretization error from regularized Reinforcement Learning to continuous-time stochastic control](https://mlsanity.com/paper/2604.21179/) (2026)
- [Mean-Field PhiBE: Continuous-Time Mean-Field Reinforcement Learning from Discrete-Time Data](https://mlsanity.com/paper/2606.26498/) (2026)
- [Actor-Critic Learning for Extended Mean Field Control with Deterministic Policies](https://mlsanity.com/paper/2607.11005/) (2026)
- [Continuous-time reinforcement learning: ellipticity enables model-free value function approximation](https://mlsanity.com/paper/2602.06930/) (2026)
- [An Actor-Critic Framework for Continuous-Time Jump-Diffusion Controls with Normalizing Flows](https://mlsanity.com/paper/2604.05398/) (2026)
- [Entropy Regularization Improves Policy Robustness in Continuous-Time Reinforcement Learning](https://mlsanity.com/paper/2607.03168/) (2026)
- [Convergence and Regret of the Policy Gradient for Multi-Armed Bandits in Diffusion Environment](https://mlsanity.com/paper/2607.29593/) (2026)
- [Amortized Score-Hamiltonian Policy Iteration: A Grid-Free Scheme for Relaxed Stochastic Control Problems](https://mlsanity.com/paper/2610.04285/) (2026)

## Abstract (arXiv)

Stochastic policies (also known as relaxed controls) are widely used in continuous-time reinforcement learning algorithms. However, executing a stochastic policy and evaluating its performance in a continuous-time environment remain open challenges. This work introduces and rigorously analyzes a policy execution framework that samples actions from a stochastic policy at discrete time points and implements them as piecewise constant controls. We prove that as the sampling mesh size tends to zero, the controlled state process converges weakly to the dynamics with coefficients aggregated according to the stochastic policy. We explicitly quantify the convergence rate based on the regularity of the coefficients and establish an optimal first-order convergence rate for sufficiently regular coefficients. Additionally, we prove a 1/2-order weak convergence rate that holds uniformly over the sampling noise with high probability, and establish a 1/2-order pathwise convergence for each realization of the system noise in the absence of volatility control. Building on these results, we analyze the bias and variance of various policy evaluation and policy gradient estimators based on discrete-time observations. Our results provide theoretical justification for the exploratory stochastic control framework in [H. Wang, T. Zariphopoulou, and X.Y. Zhou, J. Mach. Learn. Res., 21 (2020), pp. 1-34].