Sample Complexity Lower Bounds for Linear System Identification

Abstract

This paper establishes problem-specific sample complexity lower bounds for linear system identification problems. The sample complexity is defined in the PAC framework: it corresponds to the time it takes to identify the system parameters with prescribed accuracy and confidence levels. By problem-specific, we mean that the lower bound explicitly depends on the system to be identified (which contrasts with minimax lower bounds), and hence really captures the identification hardness specific to the system. We consider both uncontrolled and controlled systems. For uncontrolled systems, the lower bounds are valid for any linear system, stable or not, and only depend of the system finite-time controllability gramian. A simplified lower bound depending on the spectrum of the system only is also derived. In view of recent finitetime analysis of classical estimation methods (e.g. ordinary least squares), our sample complexity lower bounds are tight for many systems. For controlled systems, our lower bounds are not as explicit as in the case of uncontrolled systems, but could well provide interesting insights into the design of control policy with minimal sample complexity.

Regret Lower Bounds for Learning Linear Quadratic Gaussian Systems

“…Finally, we note in passing that non-singularity of the Fisher information is strongly related to the size of the smallest singular value of the covariates matrix, which has been the emphasis of some recent advances in linear system identification [FTM18, SMT + 18, SR19, JP20, WSJ21], and which actually quantifies the corresponding rate of convergence even in a non-asymptotic setting [JP19]. Interstingly, the lower bound by [JP19] reveals that the fundamental hardness of the problem is controlled by a controltheoretic quantity, namely the controllability gramian from noise to state. These ideas have been further developed in [TP21].…”

Non asymptotic estimation lower bounds for LTI state space models with Cramér-Rao and van Trees

“…From the previous discussion on the lower bound we note that even though the results of [9] are valid for all matrices not just scaled orthogonal ones as opposed to [17], the obtained bound does not account for the dimension d. The minimax rate obtained by [17] on the other hand while capturing the dimension factor, holds only for scaled orthogonal matrices which is a small subset of matrices. Furthermore, they do not indicate clearly the dependence of the minimax rate on the spectral properties of the matrix since all the eigenvalue of a scaled orthogonal matrix have the same magnitude.…”

“…This result also shows the existence of three different decay rates at least when estimating scaled orthogonal matrices: for stable matrices (ρ 1 − 1/N ) where A cannot be estimated faster than C N (d + log(1/δ))(1 − ρ 2 ), for limit stable matrices (ρ ∈ (1 − 1/N, 1 + 1/N )) where A cannot be estimated faster than C N 2 (d + log(1/δ)), and for unstable matrices (ρ 1 + 1/N ) where A cannot be estimated faster than C d+log(1/δ) N ρ 2N . • To extend these results beyond the class of scaled orthogonal matrices in the same setup, [9] provides high probability problem specific lower bounds in the sense that the bound specifically depends on the parameter matrix A. For the purpose of establishing such a bound, some conditions on the estimator have to be imposed.…”

On a Phase Transition of Regret in Linear Quadratic Control: The Memoryless Case

“…Another potential direction is to take a more directly information-theoretic route toward lower bounds as in [24] and as is traditional for bandits [20]. This was recently done for system identification in [25].…”