# Reinforcement Learning for Control with Probabilistic Stability Guarantee: A Finite-Sample Approach

arXiv: https://arxiv.org/abs/2603.00043 · HTML: https://mlsanity.com/paper/2603.00043/

## 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 17 of its 44 references that are in the map.
- First posted on arXiv 9 Feb 2026.

## 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, 17 of 17)

- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [A Lyapunov-based Approach to Safe Reinforcement Learning](https://mlsanity.com/paper/1805.07708/) (2018)
- [Lyapunov-based Safe Policy Optimization for Continuous Control](https://mlsanity.com/paper/1901.10031/) (2019)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems](https://mlsanity.com/paper/1808.00924/) (2018)
- [Sample Efficient Reinforcement Learning with REINFORCE](https://mlsanity.com/paper/2010.11364/) (2020)
- [Finite-Sample Analysis for SARSA with Linear Function Approximation](https://mlsanity.com/paper/1902.02234/) (2019)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Finite-Sample Analysis of Nonlinear Stochastic Approximation with Applications in Reinforcement Learning](https://mlsanity.com/paper/1905.11425/) (2019)
- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Actor-Critic Reinforcement Learning for Control with Stability Guarantee](https://mlsanity.com/paper/2004.14288/) (2020)
- [Stabilizing Neural Control Using Self-Learned Almost Lyapunov Critics](https://mlsanity.com/paper/2107.04989/) (2021)
- [Finite-Sample Analysis of Off-Policy TD-Learning via Generalized Bellman Operators](https://mlsanity.com/paper/2106.12729/) (2021)
- [Stochastic Approximation for Risk-aware Markov Decision Processes](https://mlsanity.com/paper/1805.04238/) (2018)
- [Learning Stabilization Control from Observations by Learning Lyapunov-like Proxy Models](https://mlsanity.com/paper/2303.02215/) (2023)
- [Global Adaptive Dynamic Programming for Continuous-Time Nonlinear Systems](https://mlsanity.com/paper/1401.0020/) (2013)

## Abstract (arXiv)

This paper presents a novel approach to reinforcement learning (RL) for control systems that provides probabilistic stability guarantees using finite data. Leveraging Lyapunov's method, we propose a probabilistic stability theorem that ensures mean square stability using only a finite number of sampled trajectories. The probability of stability increases with the number and length of trajectories, converging to certainty as data size grows. Additionally, we derive a policy gradient theorem for stabilizing policy learning and develop an RL algorithm, L-REINFORCE, that extends the classical REINFORCE algorithm to stabilization problems. The effectiveness of L-REINFORCE is demonstrated through simulations on a Cartpole task, where it outperforms the baseline in ensuring stability. This work bridges a critical gap between RL and control theory, enabling stability analysis and controller design in a model-free framework with finite data.