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A Class of Convex Optimization-Based Recursive Algorithms for Identification of Stochastic Systems

Mingxia Ding, Wenxiao Zhao, Tianshi Chen

TL;DR

This work presents a unified convex optimization-based framework for identifying stochastic linear systems, encompassing LS, $L_l$, Huber, Log-cosh, and Quantile losses. It proves strong consistency of the minimizers and provides recursive online algorithms with expanding truncations that converge almost surely to the true parameter without requiring boundedness a priori. The approach enables online robust identification with lower computational burden than kernel-based methods and is supported by theoretical convergence results and numerical demonstrations. The methodology advances both theory and practice of online system identification by bridging convex ERM criteria with recursive stochastic approximation techniques.

Abstract

Focusing on identification, this paper develops a class of convex optimization-based criteria and correspondingly the recursive algorithms to estimate the parameter vector $θ^{*}$ of a stochastic dynamic system. Not only do the criteria include the classical least-squares estimator but also the $L_l=|\cdot|^l, l\geq 1$, the Huber, the Log-cosh, and the Quantile costs as special cases. First, we prove that the minimizers of the convex optimization-based criteria converge to $θ^{*}$ with probability one. Second, the recursive algorithms are proposed to find the estimates, which minimize the convex optimization-based criteria, and it is shown that these estimates also converge to the true parameter vector with probability one. Numerical examples are given, justifying the performance of the proposed algorithms including the strong consistency of the estimates, the robustness against outliers in the observations, and higher efficiency in online computation compared with the kernel-based regularization method due to the recursive nature.

A Class of Convex Optimization-Based Recursive Algorithms for Identification of Stochastic Systems

TL;DR

This work presents a unified convex optimization-based framework for identifying stochastic linear systems, encompassing LS, , Huber, Log-cosh, and Quantile losses. It proves strong consistency of the minimizers and provides recursive online algorithms with expanding truncations that converge almost surely to the true parameter without requiring boundedness a priori. The approach enables online robust identification with lower computational burden than kernel-based methods and is supported by theoretical convergence results and numerical demonstrations. The methodology advances both theory and practice of online system identification by bridging convex ERM criteria with recursive stochastic approximation techniques.

Abstract

Focusing on identification, this paper develops a class of convex optimization-based criteria and correspondingly the recursive algorithms to estimate the parameter vector of a stochastic dynamic system. Not only do the criteria include the classical least-squares estimator but also the , the Huber, the Log-cosh, and the Quantile costs as special cases. First, we prove that the minimizers of the convex optimization-based criteria converge to with probability one. Second, the recursive algorithms are proposed to find the estimates, which minimize the convex optimization-based criteria, and it is shown that these estimates also converge to the true parameter vector with probability one. Numerical examples are given, justifying the performance of the proposed algorithms including the strong consistency of the estimates, the robustness against outliers in the observations, and higher efficiency in online computation compared with the kernel-based regularization method due to the recursive nature.
Paper Structure (14 sections, 18 theorems, 127 equations, 4 figures, 2 tables, 4 algorithms)

This paper contains 14 sections, 18 theorems, 127 equations, 4 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Convex Optimization-Based Identification Criteria
  3. Recursive Identification Algorithms
  4. The function $\Phi(\cdot)$ with a continuous derivative $\varphi(\cdot)$
  5. The $L_1$ function $\Phi(t)=|t|$ and the Quantile function $\Phi_{\gamma}(x)$
  6. Numerical Examples
  7. Concluding Remarks
  8. General Convergence Theorem of SAAWET and Technical Lemmas
  9. Proof of Theorem \ref{['con']}
  10. Proof of Lemma \ref{['Nlim0']}
  11. Proof of Theorem \ref{['The2']}
  12. Proof of Proposition \ref{['Prop1']}
  13. Proof of Lemma \ref{['Le2']}
  14. Proof of Lemma \ref{['Lem4']}

Key Result

Theorem 1

If assumptions A1), A2), and A3) hold, then $R(\theta)$ is convex with respect to $\theta$ and $\theta^{*}$ is its unique minimizer.

Figures (4)

  • Figure 1: Box plot of identification errors
  • Figure 2: Box plot of identification errors with outliers
  • Figure 3: Box plot of identification errors with maximum number of recursion 10000
  • Figure 4: Recursive identification sequences $\theta_k^{i*}$ for $\theta^{i*},i=1,2,3,4$, $k=1,\cdots,10000$

Theorems & Definitions (41)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • ...and 31 more