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Influence Functions for Data Attribution in Linear System Identification and LQR Control

Jiachen Li, Shihao Li, Soovadeep Bakshi, Jiamin Xu, Dongmei Chen

Abstract

When a controller is designed from an identified model, its performance ultimately depends on the trajectories used for identification, but pinpointing which ones help or hurt remains an open problem. We bring influence functions, a data attribution tool from machine learning, into this setting by chaining two closed form sensitivity analyses across a regularized least squares identification and an infinite horizon LQR pipeline. On the identification side, the quadratic loss admits an exact leave one trajectory out parameter shift and a reusable first order approximation with a Neumann series error bound. On the control side, we implicitly differentiate through the DARE via its discrete Lyapunov structure and compress the cost gradient to a single adjoint Lyapunov solve. The resulting scores track true LOTO retraining with Pearson correlations above 0.99 and speedups of 7 to 60 times on linear systems of dimension 2 to 10.

Influence Functions for Data Attribution in Linear System Identification and LQR Control

Abstract

When a controller is designed from an identified model, its performance ultimately depends on the trajectories used for identification, but pinpointing which ones help or hurt remains an open problem. We bring influence functions, a data attribution tool from machine learning, into this setting by chaining two closed form sensitivity analyses across a regularized least squares identification and an infinite horizon LQR pipeline. On the identification side, the quadratic loss admits an exact leave one trajectory out parameter shift and a reusable first order approximation with a Neumann series error bound. On the control side, we implicitly differentiate through the DARE via its discrete Lyapunov structure and compress the cost gradient to a single adjoint Lyapunov solve. The resulting scores track true LOTO retraining with Pearson correlations above 0.99 and speedups of 7 to 60 times on linear systems of dimension 2 to 10.

Paper Structure

This paper contains 17 sections, 29 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. LOTO Analysis for Identification
  4. Exact Parameter Shift
  5. Reusable First-Order Approximation
  6. Identification-Level Influence Score
  7. Influence Propagation Through the DARE
  8. Setup and Notation
  9. Fréchet Derivative of the DARE Residual
  10. Riccati Sensitivity via Implicit Differentiation
  11. End-to-End Influence Score via Adjoint Sensitivity
  12. Approximation Error Analysis
  13. Empirical Evaluation
  14. LOTO Prediction Accuracy
  15. Computational Scaling
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Overview of the compositional influence-function framework.
  • Figure 2: Predicted versus true LOTO effects on the linear benchmarks (S1--S3).

Theorems & Definitions (3)

  • Remark 1: Sufficient condition for invertibility
  • Remark 2
  • Remark 3: Interpretation