Sample Complexity of Data-Driven Stochastic LQR with Multiplicative Uncertainty

TL;DR

This paper analyzes how the sample size affects the performance of data-driven stochastic LQR controllers with multiplicative noise, providing bounds on suboptimality that decrease as more data is collected.

Contribution

It establishes theoretical bounds on the suboptimality of covariance estimation-based stochastic LQR, extending to unknown means and distributionally robust settings.

Findings

Suboptimality decreases proportionally to 1/N with more samples

Methodology generalizes to unknown mean and robust cases

Provides bounds based on matrix perturbation analysis

Abstract

This paper studies the sample complexity of the stochastic Linear Quadratic Regulator when applied to systems with multiplicative noise. We assume that the covariance of the noise is unknown and estimate it using the sample covariance, which results in suboptimal behaviour. The main contribution of this paper is then to bound the suboptimality of the methodology and prove that it decreases with 1/N, where N denotes the amount of samples. The methodology easily generalizes to the case where the mean is unknown and to the distributionally robust case studied in a previous work of the authors. The analysis is mostly based on results from matrix function perturbation analysis.