# Convergence of an actor-critic gradient flow for entropy regularised MDPs in general spaces

arXiv: https://arxiv.org/abs/2510.14898 · HTML: https://mlsanity.com/paper/2510.14898/

## 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 14 of its 25 references that are in the map.
- First posted on arXiv 16 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, 14 of 14)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [On the Global Convergence Rates of Softmax Policy Gradient Methods](https://mlsanity.com/paper/2005.06392/) (2020)
- [A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation](https://mlsanity.com/paper/1806.02450/) (2018)
- [Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes](https://mlsanity.com/paper/2102.00135/) (2021)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [Provable Benefits of Actor-Critic Methods for Offline Reinforcement Learning](https://mlsanity.com/paper/2108.08812/) (2021)
- [A Fisher-Rao gradient flow for entropy-regularised Markov decision processes in Polish spaces](https://mlsanity.com/paper/2310.02951/) (2023)
- [Finite-Time Analysis of Entropy-Regularized Neural Natural Actor-Critic Algorithm](https://mlsanity.com/paper/2206.00833/) (2022)
- [Analysis of a Target-Based Actor-Critic Algorithm with Linear Function Approximation](https://mlsanity.com/paper/2106.07472/) (2021)
- [Wasserstein Flow Meets Replicator Dynamics: A Mean-Field Analysis of Representation Learning in Actor-Critic](https://mlsanity.com/paper/2112.13530/) (2021)
- [Skill Transfer and Discovery for Sim-to-Real Learning: A Representation-Based Viewpoint](https://mlsanity.com/paper/2404.05051/) (2024)
- [Mirror Descent for Stochastic Control Problems with Measure-valued Controls](https://mlsanity.com/paper/2401.01198/) (2024)
- [Rethinking the Global Convergence of Softmax Policy Gradient with Linear Function Approximation: The Case of Multi-Armed Bandits](https://mlsanity.com/paper/2505.03155/) (2025)

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

- [Refined Analysis of Entropy-Regularized Actor-Critic](https://mlsanity.com/paper/2605.24357/) (2026)

## Abstract (arXiv)

We prove the stability and global convergence of a coupled actor-critic gradient flow for infinite-horizon and entropy-regularised Markov decision processes (MDPs) in continuous state and action space with linear function approximation under Q-function realisability. We consider a version of the actor critic gradient flow where the critic is updated using temporal difference (TD) learning while the policy is updated using a policy mirror descent method on a separate timescale. For general action spaces, the relative entropy regularizer is unbounded and thus it is not clear a priori that the actor-critc flow does not suffer from finite-time blow-up. Therefore we first demonstrate stability which in turn enables us obtain a convergence rate of the actor critic flow to the optimal regularised value function. The arguments presented show that timescale separation is crucial for stability and convergence in this setting.