Provably Efficient Reinforcement Learning with Linear Function Approximation

TL;DR

This paper introduces the first provably efficient reinforcement learning algorithm with polynomial runtime and sample complexity for linear function approximation, achieving regret bounds independent of state and action space sizes.

Contribution

It presents a novel optimistic LSVI algorithm with provable polynomial guarantees in linear RL settings, addressing key challenges in efficiency and exploration.

Findings

Achieves $ ilde{O}( oot{3}rom{d^3H^3T})$ regret bound.

Regret is independent of the number of states and actions.

First to provide polynomial guarantees without additional assumptions.

Abstract

Modern Reinforcement Learning (RL) is commonly applied to practical problems with an enormous number of states, where function approximation must be deployed to approximate either the value function or the policy. The introduction of function approximation raises a fundamental set of challenges involving computational and statistical efficiency, especially given the need to manage the exploration/exploitation tradeoff. As a result, a core RL question remains open: how can we design provably efficient RL algorithms that incorporate function approximation? This question persists even in a basic setting with linear dynamics and linear rewards, for which only linear function approximation is needed. This paper presents the first provable RL algorithm with both polynomial runtime and polynomial sample complexity in this linear setting, without requiring a "simulator" or additional assumptions. Concretely, we prove that an optimistic modification of Least-Squares Value Iteration (LSVI)---a classical algorithm frequently studied in the linear setting---achieves $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret, where $d$ is the ambient dimension of feature space, $H$ is the length of each episode, and $T$ is the total number of steps. Importantly, such regret is independent of the number of states and actions.