Optimal Transport Perturbations for Safe Reinforcement Learning with Robustness Guarantees

TL;DR

This work introduces a safe reinforcement learning framework that incorporates robustness through the use of an optimal transport cost uncertainty set and provides an efficient implementation based on applying Optimal Transport Perturbations to construct worst-case virtual state transitions, which does not impact data collection during training and does not require detailed simulator access.

Abstract

Robustness and safety are critical for the trustworthy deployment of deep reinforcement learning. Real-world decision making applications require algorithms that can guarantee robust performance and safety in the presence of general environment disturbances, while making limited assumptions on the data collection process during training. In order to accomplish this goal, we introduce a safe reinforcement learning framework that incorporates robustness through the use of an optimal transport cost uncertainty set. We provide an efficient implementation based on applying Optimal Transport Perturbations to construct worst-case virtual state transitions, which does not impact data collection during training and does not require detailed simulator access. In experiments on continuous control tasks with safety constraints, our approach demonstrates robust performance while significantly improving safety at deployment time compared to standard safe reinforcement learning.

Figures (10)

• Figure 1: Illustration of Optimal Transport Perturbations. Left: Worst-case transition models in the optimal transport uncertainty set $\mathcal{P}_{s,a} \subseteq P(\mathcal{S})$ that correspond to the robust Bellman operators in \ref{['eq:bellman_r']} and \ref{['eq:bellman_c']}. Right: Tractable reformulation by applying Optimal Transport Perturbations directly in $\mathcal{S}$ to a given next state sample $\hat{s}' \sim \hat{p}_{s,a}$. The black arrow denotes the state transition observed in the nominal environment, and the dashed arrows denote virtual state transitions used only to calculate robust Bellman operators.
• Figure 2: Illustration of optimal transport cost between transition models $\hat{p}_{s,a}, p_{s,a} \in P(\mathcal{S})$. Informally, $\textnormal{OTC}_{d_{s,a}} ( \hat{p}_{s,a}, p_{s,a} )$ represents the cost of transporting the probability mass of $\hat{p}_{s,a}$ to $p_{s,a}$ using the optimal (i.e., minimum cost) transport plan $\nu^*$, where the transport cost is determined by $d_{s,a}$.
• Figure 3: Performance summary by task, aggregated across test environments. Performance of adversarial RL is evaluated without adversarial interventions. Algorithms to the right of the dotted lines require additional assumptions during data collection compared to standard safe RL (see Table \ref{['tab:experiments']} for details). Top: Average total reward, normalized relative to the average performance of standard safe RL for each test environment. Bottom: Percentage of policies that satisfy the safety constraint across all test environments.
• Figure 4: Comparison of OTP with standard safe RL across tasks and test environments. Shading denotes one standard error across policies. Vertical dotted lines represent nominal training environment. Top: Total reward. Bottom: Total cost, where horizontal dotted lines represent the safety budget and values below these lines represent safety constraint satisfaction.
• Figure 5: Average final training cost in the nominal training environment. Training cost of adversarial RL includes impact of adversarial interventions. Black bars denote one standard deviation across policies. Horizontal dotted line represents safety budget.

Theorems & Definitions (8)

• Remark 1
• Definition 1: Optimal transport cost
• Definition 2: Optimal transport uncertainty set
• Theorem 1
• proof
• Lemma 1
• proof
• proof