Provably Efficient RL for Linear MDPs under Instantaneous Safety Constraints in Non-Convex Feature Spaces

TL;DR

This paper introduces provably efficient reinforcement learning algorithms for linear Markov decision processes with instantaneous safety constraints in non-convex feature spaces, achieving low regret and zero safety violation probability.

Contribution

It develops novel techniques for bounding the value function class's covering number and proposes a two-phase algorithm for non-star-convex cases, advancing safe RL in complex, non-convex environments.

Findings

Achieves regret bound of (igl(1 + 1/ auigr) \u221a{ ext{log}(1/ au) d^3 H^4 K})

Guarantees zero safety constraint violation with high probability

Demonstrates effectiveness through autonomous driving simulations

Abstract

In Reinforcement Learning (RL), tasks with instantaneous hard constraints present significant challenges, particularly when the decision space is non-convex or non-star-convex. This issue is especially relevant in domains like autonomous vehicles and robotics, where constraints such as collision avoidance often take a non-convex form. In this paper, we establish a regret bound of $\tilde{\mathcal{O}}\bigl(\bigl(1 + \tfrac{1}{\tau}\bigr) \sqrt{\log(\tfrac{1}{\tau}) d^3 H^4 K} \bigr)$, applicable to both star-convex and non-star-convex cases, where $d$ is the feature dimension, $H$ the episode length, $K$ the number of episodes, and $\tau$ the safety threshold. Moreover, the violation of safety constraints is zero with high probability throughout the learning process. A key technical challenge in these settings is bounding the covering number of the value-function class, which is essential for achieving value-aware uniform concentration in model-free function approximation. For the star-convex setting, we develop a novel technique called Objective Constraint-Decomposition (OCD) to properly bound the covering number. This result also resolves an error in a previous work on constrained RL. In non-star-convex scenarios, where the covering number can become infinitely large, we propose a two-phase algorithm, Non-Convex Safe Least Squares Value Iteration (NCS-LSVI), which first reduces uncertainty about the safe set by playing a known safe policy. After that, it carefully balances exploration and exploitation to achieve the regret bound. Finally, numerical simulations on an autonomous driving scenario demonstrate the effectiveness of NCS-LSVI.