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Solving Robust MDPs through No-Regret Dynamics

Etash Kumar Guha

TL;DR

The paper tackles robustness in reinforcement learning by formulating Robust MDPs as a two‑player no‑regret minimax game between a policy player and an environment player. It develops a flexible online‑learning framework that leverages gradient‑dominance to use a projected gradient descent oracle, achieving an overall robustness convergence rate of $O\left(\frac{1}{\sqrt{T}}\right)$ in the direct parameterization setting. The approach generalizes to faster rates under smoothness or strong gradient‑dominance and is validated empirically on GridWorld, showing competitive convergence with existing methods under various uncertainty shapes. By marrying no‑regret dynamics with robust MDPs, the work offers scalable, broadly applicable algorithms with concrete convergence guarantees.

Abstract

Reinforcement Learning is a powerful framework for training agents to navigate different situations, but it is susceptible to changes in environmental dynamics. However, solving Markov Decision Processes that are robust to changes is difficult due to nonconvexity and size of action or state spaces. While most works have analyzed this problem by taking different assumptions on the problem, a general and efficient theoretical analysis is still missing. However, we generate a simple framework for improving robustness by solving a minimax iterative optimization problem where a policy player and an environmental dynamics player are playing against each other. Leveraging recent results in online nonconvex learning and techniques from improving policy gradient methods, we yield an algorithm that maximizes the robustness of the Value Function on the order of $\mathcal{O}\left(\frac{1}{T^{\frac{1}{2}}}\right)$ where $T$ is the number of iterations of the algorithm.

Solving Robust MDPs through No-Regret Dynamics

TL;DR

The paper tackles robustness in reinforcement learning by formulating Robust MDPs as a two‑player no‑regret minimax game between a policy player and an environment player. It develops a flexible online‑learning framework that leverages gradient‑dominance to use a projected gradient descent oracle, achieving an overall robustness convergence rate of in the direct parameterization setting. The approach generalizes to faster rates under smoothness or strong gradient‑dominance and is validated empirically on GridWorld, showing competitive convergence with existing methods under various uncertainty shapes. By marrying no‑regret dynamics with robust MDPs, the work offers scalable, broadly applicable algorithms with concrete convergence guarantees.

Abstract

Reinforcement Learning is a powerful framework for training agents to navigate different situations, but it is susceptible to changes in environmental dynamics. However, solving Markov Decision Processes that are robust to changes is difficult due to nonconvexity and size of action or state spaces. While most works have analyzed this problem by taking different assumptions on the problem, a general and efficient theoretical analysis is still missing. However, we generate a simple framework for improving robustness by solving a minimax iterative optimization problem where a policy player and an environmental dynamics player are playing against each other. Leveraging recent results in online nonconvex learning and techniques from improving policy gradient methods, we yield an algorithm that maximizes the robustness of the Value Function on the order of where is the number of iterations of the algorithm.
Paper Structure (42 sections, 29 theorems, 97 equations, 3 figures, 1 table, 6 algorithms)

This paper contains 42 sections, 29 theorems, 97 equations, 3 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Works
  4. Robust MDPs
  5. No-Regret Learning
  6. Preliminaries
  7. Notation
  8. Robust Policies
  9. No Regret Dynamics
  10. The General Framework
  11. Online Learning Toolbox
  12. Online algorithms for the $\pi$-player
  13. Online algorithms for the $W$-player
  14. Constructing the Optimization Oracle
  15. Gradient-Dominance
  16. ...and 27 more sections

Key Result

Theorem 4.1

We have the difference between the robustnesses of the chosen policies and any policy $\bar{\pi}$ is upper bounded by the regret of the two players Here, $\text{Reg}_W$ and $\text{Reg}_{\pi}$ are the two average regrets of the two players $OL^W$ and $OL^{\pi}$.

Figures (3)

  • Figure 1: We plot the convergence of our algorithm over many different transition matrix uncertainty set shapes. We see that over all shapes, our algorithm converges in roughly the predicted $\frac{1}{\sqrt{T}}$ rate predicted by our results. The convergence curves of DRPG and our algorithms are very similar.
  • Figure : No-Regret RL
  • Figure : Projected Gradient Descent

Theorems & Definitions (48)

  • Definition 3.1
  • Definition 3.2
  • Theorem 4.1
  • Definition 4.1
  • Lemma 4.1
  • Lemma 4.2
  • Definition 5.1
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • ...and 38 more