Certainty Equivalence is Efficient for Linear Quadratic Control

TL;DR

This paper demonstrates that the certainty equivalent controller is highly efficient for linear quadratic control problems, achieving a quadratic rate of sub-optimality gap decay with respect to parameter error, even in partially observed settings.

Contribution

It provides the first sub-optimality guarantee for the partially observed LQG setting and improves existing bounds for fully observed LQR by establishing a quadratic error dependence.

Findings

Sub-optimality gap scales as the square of parameter error.

First guarantee for partially observed LQG control.

Improves upon previous linear error bounds for LQR.

Abstract

We study the performance of the certainty equivalent controller on Linear Quadratic (LQ) control problems with unknown transition dynamics. We show that for both the fully and partially observed settings, the sub-optimality gap between the cost incurred by playing the certainty equivalent controller on the true system and the cost incurred by using the optimal LQ controller enjoys a fast statistical rate, scaling as the square of the parameter error. To the best of our knowledge, our result is the first sub-optimality guarantee in the partially observed Linear Quadratic Gaussian (LQG) setting. Furthermore, in the fully observed Linear Quadratic Regulator (LQR), our result improves upon recent work by Dean et al. (2017), who present an algorithm achieving a sub-optimality gap linear in the parameter error. A key part of our analysis relies on perturbation bounds for discrete Riccati equations. We provide two new perturbation bounds, one that expands on an existing result from Konstantinov et al. (1993), and another based on a new elementary proof strategy.