Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finite-sample bounds for multi-output system identification

Léo Simpson, Katrin Baumgärtner, Johannes Köhler, Moritz Diehl

Abstract

This paper presents uniform-in-time finite-sample bounds for regularized linear regression with vector-valued outputs and conditionally zero-mean subgaussian noise. By revisiting classical self-normalized martingale arguments, we obtain bounds that apply directly to multi-output regression, unlike most of the prior work. Compared to the state of the art, the new results are more general and yield tighter bounds, even for scalar-valued outputs. The mild assumptions we use allow for unknown dependencies between regressors and past noise terms, typically induced by system dynamics or feedback mechanisms. Therefore, these novel finite-sample bounds can be applied to many affine-in-parameter system identification problems, including the identification of a linear time-invariant system from full-state measurements. These new results may lead to significant improvements in stochastic learning-based controllers for safety-critical applications.

Finite-sample bounds for multi-output system identification

Abstract

This paper presents uniform-in-time finite-sample bounds for regularized linear regression with vector-valued outputs and conditionally zero-mean subgaussian noise. By revisiting classical self-normalized martingale arguments, we obtain bounds that apply directly to multi-output regression, unlike most of the prior work. Compared to the state of the art, the new results are more general and yield tighter bounds, even for scalar-valued outputs. The mild assumptions we use allow for unknown dependencies between regressors and past noise terms, typically induced by system dynamics or feedback mechanisms. Therefore, these novel finite-sample bounds can be applied to many affine-in-parameter system identification problems, including the identification of a linear time-invariant system from full-state measurements. These new results may lead to significant improvements in stochastic learning-based controllers for safety-critical applications.
Paper Structure (18 sections, 8 theorems, 53 equations, 4 figures)

This paper contains 18 sections, 8 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The setup
  3. The linear regression model
  4. Finite-sample bounds
  5. Assumptions
  6. Main results
  7. Proof of Theorem \ref{['thm:main-result']}
  8. Consequences for confidence sets
  9. Pointwise bounds
  10. Discussion
  11. Asymptotic behavior of the bounds
  12. Relation to existing results
  13. Application to system identification
  14. Structure-agnostic identification of linear systems
  15. Numerical experiments
  16. ...and 3 more sections

Key Result

Theorem 1

Let Assumption assum:subgaussian hold. Let ${z} \in {\mathbb{R}}^{{n}}$ be a vector and $\bar{P} {\, \succ \, } 0$ be a positive definite matrix. Then, the following inequality holds with uniform-in-time probability at least $1 - \delta$: where $V_t \in {\mathbb{R}}^{{n} \times {n}}$ and $s_t \in {\mathbb{R}}^{{n}}$ are defined in eq:def-s-and-V.

Figures (4)

  • Figure 1: Output bounds \ref{['eq:output-bound']} corresponding to Theorem \ref{['thm:confidence-set']} for different realizations of example \ref{['eq:SO-example']}.
  • Figure 2: Left-hand side (blue) and right-hand sides $\beta_t$ (orange) and $\tilde{\beta}_t$ (purple) of the finite-sample bounds as a function of $c_\theta$.
  • Figure 3: Empirical violation frequency of the finite-sample bounds from Theorem \ref{['thm:confidence-set']} ("Novel bound") and abbasi2011improved ("Existing bound") as a function of $\delta$.
  • Figure 4: Left and right-hand sides of the bounds \ref{['eq:prediction-bound-max']} and \ref{['eq:prediction-bound-max-lti']} for the heat-transfer example \ref{['eq:heat-exchange']}.

Theorems & Definitions (17)

  • Definition 1: Subgaussian variables
  • Theorem 1: Self-normalized martingale bound
  • Lemma 1: A concentration inequality
  • Lemma 2: Supermartingale bound
  • proof : Proof of Theorem \ref{['thm:main-result']}
  • Theorem 2: Confidence set for $P {\, \succ \, } 0$
  • proof
  • Theorem 3: Confidence set for ${P} {\, \succcurlyeq \, } 0$
  • proof
  • Corollary 1: A pointwise bound
  • ...and 7 more