# Equivalence of stochastic and deterministic policy gradients

arXiv: https://arxiv.org/abs/2505.23244 · HTML: https://mlsanity.com/paper/2505.23244/

## Key facts

- Cited by 2 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 4 of its 13 references that are in the map.
- First posted on arXiv 29 May 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, 4 of 4)

- [Global Optimality Guarantees For Policy Gradient Methods](https://mlsanity.com/paper/1906.01786/) (2019)
- [Expected Policy Gradients for Reinforcement Learning](https://mlsanity.com/paper/1801.03326/) (2018)
- [Wasserstein Policy Optimization](https://mlsanity.com/paper/2505.00663/) (2025)
- [On Policy Gradients](https://mlsanity.com/paper/1911.04817/) (2019)

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

- [Dynamical System Optimization](https://mlsanity.com/paper/2506.08340/) (2025)
- [Distributions as Actions: A Unified Framework for Diverse Action Spaces](https://mlsanity.com/paper/2506.16608/) (2025)

## Abstract (arXiv)

Policy gradients in continuous control have been derived for both stochastic and deterministic policies. Here we study the relationship between the two. In a widely-used family of MDPs involving Gaussian control noise and quadratic control costs, we show that the stochastic and deterministic policy gradients, natural gradients, and state value functions are identical; while the state-control value functions are different. We then develop a general procedure for constructing an MDP with deterministic policy that is equivalent to a given MDP with stochastic policy. The controls of this new MDP are the sufficient statistics of the stochastic policy in the original MDP. Our results suggest that policy gradient methods can be unified by approximating state value functions rather than state-control value functions.