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Recursive Least Squares with Fading Regularization for Finite-Time Convergence without Persistent Excitation

Brian Lai, Dimitra Panagou, Dennis S. Bernstein

TL;DR

The paper extends Recursive Least Squares by incorporating time-varying fading regularization, enabling regularization to decay to zero and reducing reliance on weak persistent excitation. By deriving batch and recursive formulations, it shows that global (finite-time) attractivity of the parameter-estimation error can be achieved under milder conditions when the regularization fades, and that finite-time convergence occurs if the regularization vanishes after a finite time. Two algorithms, FR-RLS and Rank-1 FR-RLS (R1FR-RLS), are proposed to realize these properties with practical computational costs comparable to classical RLS; R1FR-RLS achieves this with rank-1 updates to the regularization, keeping complexity near that of standard RLS. Numerical experiments validate the theoretical guarantees, illustrating protection against over-regularization and demonstrating finite-time convergence in both noiseless and noisy scenarios, with the R1FR approach offering favorable computation times.

Abstract

This paper extends recursive least squares (RLS) to include time-varying regularization. This extension provides flexibility for updating the least squares regularization term in real time. Existing results with constant regularization imply that the parameter-estimation error dynamics of RLS are globally attractive to zero if and only the regressor is weakly persistently exciting. This work shows that, by extending classical RLS to include a time-varying (fading) regularization term that converges to zero, the parameter-estimation error dynamics are globally attractive to zero without weakly persistent excitation. Moreover, if the fading regularization term converges to zero in finite time, then the parameter estimation error also converges to zero in finite time. Finally, we propose rank-1 fading regularization (R1FR) RLS, a time-varying regularization algorithm with fading regularization that converges to zero, and which runs in the same computational complexity as classical RLS. Numerical examples are presented to validate theoretical guarantees and to show how R1FR-RLS can protect against over-regularization.

Recursive Least Squares with Fading Regularization for Finite-Time Convergence without Persistent Excitation

TL;DR

The paper extends Recursive Least Squares by incorporating time-varying fading regularization, enabling regularization to decay to zero and reducing reliance on weak persistent excitation. By deriving batch and recursive formulations, it shows that global (finite-time) attractivity of the parameter-estimation error can be achieved under milder conditions when the regularization fades, and that finite-time convergence occurs if the regularization vanishes after a finite time. Two algorithms, FR-RLS and Rank-1 FR-RLS (R1FR-RLS), are proposed to realize these properties with practical computational costs comparable to classical RLS; R1FR-RLS achieves this with rank-1 updates to the regularization, keeping complexity near that of standard RLS. Numerical experiments validate the theoretical guarantees, illustrating protection against over-regularization and demonstrating finite-time convergence in both noiseless and noisy scenarios, with the R1FR approach offering favorable computation times.

Abstract

This paper extends recursive least squares (RLS) to include time-varying regularization. This extension provides flexibility for updating the least squares regularization term in real time. Existing results with constant regularization imply that the parameter-estimation error dynamics of RLS are globally attractive to zero if and only the regressor is weakly persistently exciting. This work shows that, by extending classical RLS to include a time-varying (fading) regularization term that converges to zero, the parameter-estimation error dynamics are globally attractive to zero without weakly persistent excitation. Moreover, if the fading regularization term converges to zero in finite time, then the parameter estimation error also converges to zero in finite time. Finally, we propose rank-1 fading regularization (R1FR) RLS, a time-varying regularization algorithm with fading regularization that converges to zero, and which runs in the same computational complexity as classical RLS. Numerical examples are presented to validate theoretical guarantees and to show how R1FR-RLS can protect against over-regularization.
Paper Structure (8 sections, 4 theorems, 49 equations, 3 figures)

This paper contains 8 sections, 4 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Batch Least Squares with Time-Varying Regularization
  3. Recursive Least Squares with Time-Varying Regularization
  4. Global Attractivity
  5. Fading-Regularization (FR) RLS
  6. Rank-1 Fading-Regularization (R1FR) RLS
  7. Numerical Examples
  8. Conclusion

Key Result

Theorem 1

For all $k\in {\mathbb N}_0$, let $\phi_k\in {\mathbb R}^{p\times n}$, let $y_k \in{\mathbb R}^p$, and let $\Gamma_k \in {\mathbb R}^{p \times p}$ be positive semidefinite. Next, for all $k \ge 0$, let $\theta_{{\rm reg},k} \in {\mathbb R}^{n}$ and let $R_k \in {\mathbb R}^{n\times n}$ be positive s Then, for all $k \ge 0$, $J_{k} \colon {\mathbb R}^{n} \rightarrow {\mathbb R}$, defined in eqn: Jk

Figures (3)

  • Figure 1: Example 1: Parameter estimation error, $\Vert \theta_k - \theta \Vert$ versus time step $k$ for RLS, fading regularization RLS, and rank-1 fading regularization RLS. Solid lines indicate that data is persistently exciting, while dashed lines indicates that data is not persistently exciting for $k \ge 100$.
  • Figure 2: Exmaple 2: Parameter estimation error, $\Vert \theta_k - \theta \Vert$ versus time step $k$ for RLS, fading regularization, and rank-1 fading regularization, over 1000 independent trials. Solid line shows the mean of the 1000 trials and shaded shows the 95% confidence interval.
  • Figure 3: Example 2: Computation time per step in milliseconds of RLS, fading regularization, and rank-1 fading regularization. Fading regularization and rank-1 fading regularization are divided into $0 \le k \le 200$ (time-varying regularization), and $k > 200$ (constant zero regularization). Error bars give the 95% confidence intervals.

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof