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Rao-Blackwellized Stein Gradient Descent for Joint State-Parameter Estimation

Milad Banitalebi Dehkordi, Manas Mejari, Dario Piga

Abstract

We present a filtering framework for online joint state estimation and parameter identification in nonlinear, time-varying systems. The algorithm uses Rao-Blackwellization technique to infer joint state-parameter posteriors efficiently. In particular, conditional state distributions are computed analytically via Kalman filtering, while model parameters including process and measurement noise covariances are approximated using particle-based Stein Variational Gradient Descent (SVGD), enabling stable real-time inference. We prove a theoretical consistency result by bounding the impact of the SVGD approximated parameter posterior on state estimates, relating the divergence between the true and approximate parameter posteriors to the total variation distance between the resulting state marginals. Performance of the proposed filter is validated on two case studies: a bioreactor with Haldane kinetics and a neural-network-augmented dynamic system. The latter demonstrates the filter's capacity for online neural network training within a dynamical model, showcasing its potential for fully adaptive, data-driven system identification.

Rao-Blackwellized Stein Gradient Descent for Joint State-Parameter Estimation

Abstract

We present a filtering framework for online joint state estimation and parameter identification in nonlinear, time-varying systems. The algorithm uses Rao-Blackwellization technique to infer joint state-parameter posteriors efficiently. In particular, conditional state distributions are computed analytically via Kalman filtering, while model parameters including process and measurement noise covariances are approximated using particle-based Stein Variational Gradient Descent (SVGD), enabling stable real-time inference. We prove a theoretical consistency result by bounding the impact of the SVGD approximated parameter posterior on state estimates, relating the divergence between the true and approximate parameter posteriors to the total variation distance between the resulting state marginals. Performance of the proposed filter is validated on two case studies: a bioreactor with Haldane kinetics and a neural-network-augmented dynamic system. The latter demonstrates the filter's capacity for online neural network training within a dynamical model, showcasing its potential for fully adaptive, data-driven system identification.
Paper Structure (12 sections, 4 theorems, 47 equations, 5 figures, 1 table, 3 algorithms)

This paper contains 12 sections, 4 theorems, 47 equations, 5 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Rao--Blackwellized filtering for joint state-parameter estimation
  4. Rao--Blackwellized Filtering with Stein Gradient Descent
  5. Rao--Blackwellized Stein Gradient Descent Filter
  6. Rao--Blackwellized Fisher Stein Gradient Descent Filter
  7. Theoretical Bounds on Posterior Approximation Error
  8. Case Studies
  9. Batch Bioreactor System
  10. Online training of neural networks
  11. Conclusion
  12. Proof of Proposition \ref{['thm:svgd_one_step_formal']}

Key Result

Lemma 1

Consider the transformation in eq:svgd_transform$: T_{\epsilon \phi}(\theta) = \theta + \epsilon \phi(\theta)$ and transformed particles $z = T_{\epsilon \phi}(\theta)$ with $\theta \sim q(\theta)$. Let $q_{[T_{\epsilon \phi}]}(z)$ denote the probability density of the particles after this transform i.e., $\phi_{q,p}^{\ast}$ is the direction of the steepest descent that maximizes the negative grad

Figures (5)

  • Figure 1: State estimation performance: true states (dotted black), filter means (solid lines) and 95% confidence intervals (shaded areas). State $X_t$ (top panels) and $S_t$ (bottom panels).
  • Figure 2: Mixing efficiency estimation: true $\eta_t$ (dotted black), filter means (solid lines) and uncertainty (shaded areas).
  • Figure 3: Filter performance on the Monte Carlo study: The left panel shows the log-scaled CRPS comparison for the RBFSGD filter, RBSGD filter, and RBPF across 50 independent realizations of $\eta_t$. The right panel shows the zoomed-in boxplot from the left panel for RBFSGD and RBSGD.
  • Figure 4: State estimation performance: true states (dotted black), RBFSGD filter mean (solid green) and uncertainty (shaded green), EKF mean (solid orange) and uncertainty (shaded orange).
  • Figure 5: Parameter estimation performance for the nonlinear system: true parameter (dotted black), RBFSGD filter mean estimate (solid green), estimated uncertainty (shaded green)

Theorems & Definitions (9)

  • Lemma 1
  • Proposition 1
  • Remark 1
  • Remark 2: MAP estimates
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 3