# Convex Regularization and Convergence of Policy Gradient Flows under Safety Constraints

arXiv: https://arxiv.org/abs/2411.19193 · HTML: https://mlsanity.com/paper/2411.19193/

## Key facts

- Cited by 1 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 33 of its 49 references that are in the map.
- First posted on arXiv 28 Nov 2024.

## 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, 30 of 33)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [End-to-End Safe Reinforcement Learning through Barrier Functions for Safety-Critical Continuous Control Tasks](https://mlsanity.com/paper/1903.08792/) (2019)
- [A Theory of Regularized Markov Decision Processes](https://mlsanity.com/paper/1901.11275/) (2019)
- [On the Global Convergence Rates of Softmax Policy Gradient Methods](https://mlsanity.com/paper/2005.06392/) (2020)
- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift](https://mlsanity.com/paper/1908.00261/) (2019)
- [Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization](https://mlsanity.com/paper/2007.06558/) (2020)
- [Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes](https://mlsanity.com/paper/2102.00135/) (2021)
- [Global Optimality Guarantees For Policy Gradient Methods](https://mlsanity.com/paper/1906.01786/) (2019)
- [Mean Field Analysis of Neural Networks: A Central Limit Theorem](https://mlsanity.com/paper/1808.09372/) (2018)
- [Policy Mirror Descent for Regularized Reinforcement Learning: A Generalized Framework with Linear Convergence](https://mlsanity.com/paper/2105.11066/) (2021)
- [Saute RL: Almost Surely Safe Reinforcement Learning Using State Augmentation](https://mlsanity.com/paper/2202.06558/) (2022)
- [Linear Convergence of Natural Policy Gradient Methods with Log-Linear Policies](https://mlsanity.com/paper/2210.01400/) (2022)
- [Leveraging Non-uniformity in First-order Non-convex Optimization](https://mlsanity.com/paper/2105.06072/) (2021)
- [A Survey of Constraint Formulations in Safe Reinforcement Learning](https://mlsanity.com/paper/2402.02025/) (2024)
- [Regularized Policies are Reward Robust](https://mlsanity.com/paper/2101.07012/) (2021)
- [Feasible Actor-Critic: Constrained Reinforcement Learning for Ensuring Statewise Safety](https://mlsanity.com/paper/2105.10682/) (2021)
- [Joint Synthesis of Safety Certificate and Safe Control Policy using Constrained Reinforcement Learning](https://mlsanity.com/paper/2111.07695/) (2021)
- [A Fisher-Rao gradient flow for entropy-regularised Markov decision processes in Polish spaces](https://mlsanity.com/paper/2310.02951/) (2023)
- [Convergence of Policy Gradient for Entropy Regularized MDPs with Neural Network Approximation in the Mean-Field Regime](https://mlsanity.com/paper/2201.07296/) (2022)
- [Global optimality of softmax policy gradient with single hidden layer neural networks in the mean-field regime](https://mlsanity.com/paper/2010.11858/) (2020)
- [Linear Convergence of Entropy-Regularized Natural Policy Gradient with Linear Function Approximation](https://mlsanity.com/paper/2106.04096/) (2021)
- [Control to flocking of the kinetic Cucker-Smale model](https://mlsanity.com/paper/1411.4687/) (2014)
- [Effects of Safety State Augmentation on Safe Exploration](https://mlsanity.com/paper/2206.02675/) (2022)
- [The Pontryagin Maximum Principle in the Wasserstein Space](https://mlsanity.com/paper/1711.07667/) (2017)

## Built on by (most cited first, 1 of 1)

- [Non-convex entropic mean-field optimization via Best Response flow](https://mlsanity.com/paper/2505.22760/) (2025)

## Abstract (arXiv)

This paper examines reinforcement learning (RL) in infinite-horizon decision processes with almost-sure safety constraints, crucial for applications like autonomous systems, finance, and resource management. We propose a doubly-regularized RL framework combining reward and parameter regularization to address safety constraints in continuous state-action spaces. The problem is formulated as a convex regularized objective with parametrized policies in the mean-field regime. Leveraging mean-field theory and Wasserstein gradient flows, policies are modeled on an infinite-dimensional statistical manifold, with updates governed by parameter distribution gradient flows. Key contributions include solvability conditions for safety-constrained problems, smooth bounded approximations for gradient flows, and exponential convergence guarantees under sufficient regularization. General regularization conditions, including entropy regularization, support practical particle method implementations. This framework provides robust theoretical insights and guarantees for safe RL in complex, high-dimensional settings.