Computationally Efficient RL under Linear Bellman Completeness for Deterministic Dynamics

TL;DR

This paper introduces a computationally efficient reinforcement learning algorithm for the linear Bellman complete setting with deterministic dynamics, capable of handling large action spaces and random rewards, by using a novel noise injection technique.

Contribution

It presents the first computationally efficient RL algorithm for the linear Bellman complete setting with deterministic dynamics, utilizing a novel noise-injection method for optimistic value iteration.

Findings

Algorithm is computationally efficient and scalable.

Works for large action spaces and random rewards.

Ensures optimism through carefully designed noise injection.

Abstract

We study computationally and statistically efficient Reinforcement Learning algorithms for the linear Bellman Complete setting. This setting uses linear function approximation to capture value functions and unifies existing models like linear Markov Decision Processes (MDP) and Linear Quadratic Regulators (LQR). While it is known from the prior works that this setting is statistically tractable, it remained open whether a computationally efficient algorithm exists. Our work provides a computationally efficient algorithm for the linear Bellman complete setting that works for MDPs with large action spaces, random initial states, and random rewards but relies on the underlying dynamics to be deterministic. Our approach is based on randomization: we inject random noise into least squares regression problems to perform optimistic value iteration. Our key technical contribution is to carefully design the noise to only act in the null space of the training data to ensure optimism while circumventing a subtle error amplification issue.