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Online Learning of Nonlinear Parametric Models under Non-smooth Regularization using EKF and ADMM

Lapo Frascati, Alberto Bemporad

TL;DR

This work tackles online identification of nonlinear parametric mappings under non-smooth penalties, including $R_f(N)$ and $R_c(N)$ regret terms, for time-varying systems.It proposes EKF-ADMM, a hybrid algorithm that interleaves EKF updates with ADMM iterations to realize fast online adaptation while accommodating general non-smooth regularizers.For linear time-varying models with convex regularizers, the paper proves sublinear regret bounds for both the prediction and constraint regrets and discusses computational efficiency suitable for embedded implementations.Simulation results on online LASSO, a static neural network with sparsity-promoting regularization, and time-varying neural-network adaptation demonstrate competitive performance against online-ADMM and offline solvers, highlighting practical relevance for embedded adaptive control.

Abstract

This paper proposes a novel combination of extended Kalman filtering (EKF) with the alternating direction method of multipliers (ADMM) for learning parametric nonlinear models online under non-smooth regularization terms, including l1 and l0 penalties and bound constraints on model parameters. For the case of linear time-varying models and non-smoothconvex regularization terms, we provide a sublinear regret bound that ensures the proper behavior of the online learning strategy. The approach is computationally efficient for a wide range of regularization terms, which makes it appealing for its use in embedded control applications for online model adaptation. We show the performance of the proposed method in three simulation examples, highlighting its effectiveness compared to other batch and online algorithms.

Online Learning of Nonlinear Parametric Models under Non-smooth Regularization using EKF and ADMM

TL;DR

This work tackles online identification of nonlinear parametric mappings under non-smooth penalties, including $R_f(N)$ and $R_c(N)$ regret terms, for time-varying systems.It proposes EKF-ADMM, a hybrid algorithm that interleaves EKF updates with ADMM iterations to realize fast online adaptation while accommodating general non-smooth regularizers.For linear time-varying models with convex regularizers, the paper proves sublinear regret bounds for both the prediction and constraint regrets and discusses computational efficiency suitable for embedded implementations.Simulation results on online LASSO, a static neural network with sparsity-promoting regularization, and time-varying neural-network adaptation demonstrate competitive performance against online-ADMM and offline solvers, highlighting practical relevance for embedded adaptive control.

Abstract

This paper proposes a novel combination of extended Kalman filtering (EKF) with the alternating direction method of multipliers (ADMM) for learning parametric nonlinear models online under non-smooth regularization terms, including l1 and l0 penalties and bound constraints on model parameters. For the case of linear time-varying models and non-smoothconvex regularization terms, we provide a sublinear regret bound that ensures the proper behavior of the online learning strategy. The approach is computationally efficient for a wide range of regularization terms, which makes it appealing for its use in embedded control applications for online model adaptation. We show the performance of the proposed method in three simulation examples, highlighting its effectiveness compared to other batch and online algorithms.

Paper Structure

This paper contains 13 sections, 2 theorems, 17 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. EKF for online model learning
  4. EKF under non-smooth regularization
  5. Computational complexity
  6. Regret analysis
  7. Simulation results
  8. Online learning of a static model
  9. $\ell_1$ regularization
  10. Bound constraints
  11. Online learning of a time-varying model
  12. Conclusions
  13. Proof of Theorem 1

Key Result

Theorem 4.1

Let $\{x_k, \nu_k,w_k\}_{k=0}^{N-1}$ be the sequence generated by Algorithm alg3 and let $x^{\star},\nu^{\star}$ be the best solution in hindsight, i.e. $x^{\star},\nu^{\star} = \mathop{\mathrm{arg\,min}}\limits _{x,\nu \in \mathcal{S}} \sum_{k=0}^{N-1} (f_k(x)+g(\nu))$. Let the following assumption Then, if $\eta = \frac{G_f \sqrt{N}}{D_x \sqrt{2\alpha}}$ and $\rho = \sqrt{N}$, the following subl

Figures (5)

  • Figure 1: Objective and constraint regret for online LASSO.
  • Figure 2: Online learning with $\ell_1$ regularization: Loss, Mse and Reg averaged over $20$ runs.
  • Figure 3: Online learning with $\ell_1$ regularization: regret and sample regret averaged over $20$ runs.
  • Figure 4: Online learning with bounds: Mse and constraints violation averaged over 20 runs.
  • Figure 5: Online learning with $\ell_0$ regularization: regret and sample regret averaged over $20$ runs.

Theorems & Definitions (2)

  • Theorem 4.1
  • Corollary 4.2