# Path-Space Mirror Descent for On-Policy Reinforcement Learning under the Generalized Schr"odinger Bridge

arXiv: https://arxiv.org/abs/2603.21621 · HTML: https://mlsanity.com/paper/2603.21621/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL with diffusion and flow models, one of 15 directions in Deep reinforcement learning methods.
- Builds on 44 of its 53 references that are in the map.
- First posted on arXiv 23 Mar 2026.

## Placed in

- Deep reinforcement learning methods / [RL with diffusion and flow models](https://mlsanity.com/deeprl/rl-with-diffusion-and-flow-models/)

## Builds on (most cited first, 30 of 44)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Isaac Gym: High Performance GPU-Based Physics Simulation For Robot Learning](https://mlsanity.com/paper/2108.10470/) (2021)
- [Flow-GRPO: Training Flow Matching Models via Online RL](https://mlsanity.com/paper/2505.05470/) (2025)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Diffusion Policies as an Expressive Policy Class for Offline Reinforcement Learning](https://mlsanity.com/paper/2208.06193/) (2022)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [IDQL: Implicit Q-Learning as an Actor-Critic Method with Diffusion Policies](https://mlsanity.com/paper/2304.10573/) (2023)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [A Theory of Regularized Markov Decision Processes](https://mlsanity.com/paper/1901.11275/) (2019)
- [Diffusion Policy Policy Optimization](https://mlsanity.com/paper/2409.00588/) (2024)
- [Flow Q-Learning](https://mlsanity.com/paper/2502.02538/) (2025)
- [Efficient Diffusion Policies for Offline Reinforcement Learning](https://mlsanity.com/paper/2305.20081/) (2023)
- [Learning a Diffusion Model Policy from Rewards via Q-Score Matching](https://mlsanity.com/paper/2312.11752/) (2023)
- [Diffusion-based Reinforcement Learning via Q-weighted Variational Policy Optimization](https://mlsanity.com/paper/2405.16173/) (2024)
- [Mirror Descent Policy Optimization](https://mlsanity.com/paper/2005.09814/) (2020)
- [Flow Matching Policy Gradients](https://mlsanity.com/paper/2507.21053/) (2025)
- [ReinFlow: Fine-tuning Flow Matching Policy with Online Reinforcement Learning](https://mlsanity.com/paper/2505.22094/) (2025)
- [Diffusion Actor-Critic with Entropy Regulator](https://mlsanity.com/paper/2405.15177/) (2024)

## Abstract (arXiv)

Classical on-policy algorithms such as PPO and mirror descent policy optimization provide stable proximal policy updates through tractable action likelihoods, but are typically instantiated with simple Gaussian policies whose expressiveness can be limited in complex continuous-control tasks. Generative policies based on diffusion and flow models provide more expressive action distributions, but they naturally define distributions over multi-step denoising paths whose terminal action density is often intractable, creating a mismatch with likelihood-based on-policy proximal updates. To address this mismatch, we introduce GSB-MDPO (Generalized Schr"odinger Bridge Mirror Descent Policy Optimization), which formulates on-policy generative policy optimization as a Generalized Schr"odinger Bridge problem over state-conditioned generation paths and instantiates the resulting path-measure update through mirror descent policy optimization. The key insight is that the GSB path-space KL plays the role of the proximal term in MDPO while upper-bounding the terminal action KL, enabling direct control of the executed action distribution without explicit terminal action likelihood evaluation. Experiments on 14 continuous-control tasks across Playground and Gym-MuJoCo demonstrate the empirical effectiveness of GSB-MDPO and support path-space regularization as a principled proximal update for multi-step generative policies.