# Matryoshka Policy Gradient for Entropy-Regularized RL: Convergence and Global Optimality

arXiv: https://arxiv.org/abs/2303.12785 · HTML: https://mlsanity.com/paper/2303.12785/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 62 references that are in the map.
- First posted on arXiv 22 Mar 2023.

## Builds on (most cited first, 30 of 40)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [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)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Soft Actor-Critic Algorithms and Applications](https://mlsanity.com/paper/1812.05905/) (2018)
- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [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)
- [Bridging the Gap Between Value and Policy Based Reinforcement Learning](https://mlsanity.com/paper/1702.08892/) (2017)
- [Understanding the impact of entropy on policy optimization](https://mlsanity.com/paper/1811.11214/) (2018)
- [Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization](https://mlsanity.com/paper/2007.06558/) (2020)
- [Maximum Entropy RL (Provably) Solves Some Robust RL Problems](https://mlsanity.com/paper/2103.06257/) (2021)
- [Equivalence Between Policy Gradients and Soft Q-Learning](https://mlsanity.com/paper/1704.06440/) (2017)
- [Neural Policy Gradient Methods: Global Optimality and Rates of Convergence](https://mlsanity.com/paper/1909.01150/) (2019)
- [Global Optimality Guarantees For Policy Gradient Methods](https://mlsanity.com/paper/1906.01786/) (2019)
- [Global Convergence of Policy Gradient Methods to (Almost) Locally Optimal Policies](https://mlsanity.com/paper/1906.08383/) (2019)
- [Sample Efficient Reinforcement Learning with REINFORCE](https://mlsanity.com/paper/2010.11364/) (2020)
- [Implicit Regularization of Random Feature Models](https://mlsanity.com/paper/2002.08404/) (2020)
- [On Linear Convergence of Policy Gradient Methods for Finite MDPs](https://mlsanity.com/paper/2007.11120/) (2020)
- [Policy Gradient Methods for the Noisy Linear Quadratic Regulator over a Finite Horizon](https://mlsanity.com/paper/2011.10300/) (2020)
- [Softmax Policy Gradient Methods Can Take Exponential Time to Converge](https://mlsanity.com/paper/2102.11270/) (2021)
- [Linear Convergence of Natural Policy Gradient Methods with Log-Linear Policies](https://mlsanity.com/paper/2210.01400/) (2022)

## Built on by (most cited first, 2 of 2)

- [Maximum Likelihood Reinforcement Learning](https://mlsanity.com/paper/2602.02710/) (2026)
- [Reinforcement Learning in the Real World: A Survey of Statistical Challenges and Future Directions](https://mlsanity.com/paper/2601.15353/) (2026)

## Abstract (arXiv)

A novel Policy Gradient (PG) algorithm, called textitMatryoshka Policy Gradient (MPG), is introduced and studied, in the context of fixed-horizon max-entropy reinforcement learning, where an agent aims at maximizing entropy bonuses additional to its cumulative rewards. In the linear function approximation setting with softmax policies, we prove uniqueness and characterize the optimal policy of the entropy regularized objective, together with global convergence of MPG. These results are proved in the case of continuous state and action space. MPG is intuitive, theoretically sound and we furthermore show that the optimal policy of the infinite horizon max-entropy objective can be approximated arbitrarily well by the optimal policy of the MPG framework. Finally, we provide a criterion for global optimality when the policy is parametrized by a neural network in terms of the neural tangent kernel at convergence. As a proof of concept, we evaluate numerically MPG on standard test benchmarks.