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Subgradient Method for System Identification with Non-Smooth Objectives

Baturalp Yalcin, Jihun Kim, Javad Lavaei

TL;DR

The paper tackles fast, provably convergent learning for non-smooth system identification of linear time-invariant systems under adversarial disturbances by proposing a time-evolving subgradient algorithm. It demonstrates linear convergence to the ground-truth matrix $\bar{A}$ for the best and Polyak step sizes after a burn-in, and sublinear convergence for constant and diminishing step sizes, all under a probabilistic disturbance sparsity model and persistent excitation. A thorough time-complexity comparison shows the subgradient approach offers substantial computational advantages over exact solvers, with supporting numerical experiments across varying system sizes and disturbance probabilities. The work lays a foundation for real-time, robust system identification with non-smooth objectives and highlights future directions such as stochastic/subsampled variants and theoretical analysis of backtracking steps.

Abstract

This paper investigates a subgradient-based algorithm to solve the system identification problem for linear time-invariant systems with non-smooth objectives. This is essential for robust system identification in safety-critical applications. While existing work provides theoretical exact recovery guarantees using optimization solvers, the design of fast learning algorithms with convergence guarantees for practical use remains unexplored. We analyze the subgradient method in this setting, where the optimization problems to be solved evolve over time as new measurements are collected, and we establish linear convergence to the ground-truth system for both the best and Polyak step sizes after a burn-in period. We further characterize sublinear convergence of the iterates under constant and diminishing step sizes, which require only minimal information and thus offer broad applicability. Finally, we compare the time complexity of standard solvers with the subgradient algorithm and support our findings with experimental results. This is the first work to analyze subgradient algorithms for system identification with non-smooth objectives.

Subgradient Method for System Identification with Non-Smooth Objectives

TL;DR

The paper tackles fast, provably convergent learning for non-smooth system identification of linear time-invariant systems under adversarial disturbances by proposing a time-evolving subgradient algorithm. It demonstrates linear convergence to the ground-truth matrix for the best and Polyak step sizes after a burn-in, and sublinear convergence for constant and diminishing step sizes, all under a probabilistic disturbance sparsity model and persistent excitation. A thorough time-complexity comparison shows the subgradient approach offers substantial computational advantages over exact solvers, with supporting numerical experiments across varying system sizes and disturbance probabilities. The work lays a foundation for real-time, robust system identification with non-smooth objectives and highlights future directions such as stochastic/subsampled variants and theoretical analysis of backtracking steps.

Abstract

This paper investigates a subgradient-based algorithm to solve the system identification problem for linear time-invariant systems with non-smooth objectives. This is essential for robust system identification in safety-critical applications. While existing work provides theoretical exact recovery guarantees using optimization solvers, the design of fast learning algorithms with convergence guarantees for practical use remains unexplored. We analyze the subgradient method in this setting, where the optimization problems to be solved evolve over time as new measurements are collected, and we establish linear convergence to the ground-truth system for both the best and Polyak step sizes after a burn-in period. We further characterize sublinear convergence of the iterates under constant and diminishing step sizes, which require only minimal information and thus offer broad applicability. Finally, we compare the time complexity of standard solvers with the subgradient algorithm and support our findings with experimental results. This is the first work to analyze subgradient algorithms for system identification with non-smooth objectives.

Paper Structure

This paper contains 14 sections, 11 theorems, 50 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Non-smooth Estimator and Disturbance Model
  3. Subgradient Method
  4. Subgradient Method with Best Step Size
  5. Subgradient Method with Polyak Step Size
  6. Subgradient Method with Constant and Diminishing Step Sizes
  7. Time Complexity Analysis
  8. Numerical Experiments
  9. Conclusion
  10. Exact Recovery under Disturbance-Restriction
  11. Proof of Theorem \ref{['thm:angle-cosine']}
  12. Proof of Theorem \ref{['thm:cosine-lower-bound']}
  13. Proof of Theorem \ref{['thm:polyak-result']}
  14. Proof of Theorems \ref{['thm:constant-step-size']} and \ref{['thm: diminishing-step-size']}

Key Result

lemma 1

The subdifferential of $f_T(\cdot)$ at a point $A$ is $\partial f_T(A) = \Bigr\{ - \sum_{t = 0}^{T-1} \partial \| x_{t+1} - A x_t \|_2 \cdot x_t^T \Bigr\},$ and the subdifferential of the $f_T(\cdot)$ at the point $\bar{A}$ is where $\mathcal{K} := \{t\in \{0, \dots, T-1\} ~|~ \bar{d}_t \neq 0\}$ is the set of time indices of nonzero disturbances, $\mathcal{K}^c :=\{0,\dots, T-1\}\setminus \

Figures (2)

  • Figure 1: Solution Gap and Loss Gap of Algorithm \ref{['alg']} with Different Step Sizes and Dimensions $n$.
  • Figure 2: Solution Gap and Loss Gap of Algorithm \ref{['alg']} with Best and Backtracking Step Sizes, Different Dimensions $n$ and Probabilistic Sparsity Parameters $p$.

Theorems & Definitions (19)

  • remark 1
  • definition 1
  • lemma 1
  • theorem 1
  • definition 2
  • theorem 2
  • corollary 1
  • theorem 3
  • theorem 4
  • theorem 5
  • ...and 9 more