Finite-time Analysis of Approximate Policy Iteration for the Linear Quadratic Regulator

TL;DR

This paper provides a finite-time analysis of approximate policy iteration for the Linear Quadratic Regulator, highlighting the sample complexity of policy evaluation and proposing an adaptive exploration method with improved regret bounds.

Contribution

The paper quantifies the sample complexity of approximate policy iteration for LQR and introduces an adaptive $ extit{ extbf{ε}}$-greedy exploration procedure with better regret performance.

Findings

Policy evaluation dominates sample complexity with $(n+d)^3/ ext{ε}^2$ samples per step.

Only $ ext{log}(1/ ext{ε})$ policy improvement steps are needed.

The adaptive procedure achieves $T^{2/3}$ regret, improving previous results.

Abstract

We study the sample complexity of approximate policy iteration (PI) for the Linear Quadratic Regulator (LQR), building on a recent line of work using LQR as a testbed to understand the limits of reinforcement learning (RL) algorithms on continuous control tasks. Our analysis quantifies the tension between policy improvement and policy evaluation, and suggests that policy evaluation is the dominant factor in terms of sample complexity. Specifically, we show that to obtain a controller that is within $\varepsilon$ of the optimal LQR controller, each step of policy evaluation requires at most $(n+d)^3/\varepsilon^2$ samples, where $n$ is the dimension of the state vector and $d$ is the dimension of the input vector. On the other hand, only $\log(1/\varepsilon)$ policy improvement steps suffice, resulting in an overall sample complexity of $(n+d)^3 \varepsilon^{-2} \log(1/\varepsilon)$. We furthermore build on our analysis and construct a simple adaptive procedure based on $\varepsilon$-greedy exploration which relies on approximate PI as a sub-routine and obtains $T^{2/3}$ regret, improving upon a recent result of Abbasi-Yadkori et al.