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Regret Guarantees for Model-Free Cooperative Filtering under Asynchronous Observations

Jiachen Qian, Yang Zheng

TL;DR

This work derives an autoregressive representation that relates future local outputs to asynchronous past outputs and proposes an online least-squares algorithm to learn this autoregressive model for real-time prediction.

Abstract

Predicting the output of a dynamical system from streaming data is fundamental to real-time feedback control and decision-making. We first derive an autoregressive representation that relates future local outputs to asynchronous past outputs. Building on this structure, we propose an online least-squares algorithm to learn this autoregressive model for real-time prediction. We then establish a regret bound of O(log^3 N) relative to the optimal model-based predictor, which holds for marginally stable systems. Moreover, we provide a sufficient condition characterized via a symplectic matrix, under which the proposed cooperative online learning method provably outperforms the optimal model-based predictor that relies solely on local observations. From a technical standpoint, our analysis exploits the orthogonality of the innovation process under asynchronous data structure and the persistent excitation of the Gram matrix despite delay-induced asymmetries. Overall, these results offer both theoretical guarantees and practical algorithms for model-free cooperative prediction with asynchronous observations, thereby enriching the theory of online learning for dynamical systems.

Regret Guarantees for Model-Free Cooperative Filtering under Asynchronous Observations

TL;DR

This work derives an autoregressive representation that relates future local outputs to asynchronous past outputs and proposes an online least-squares algorithm to learn this autoregressive model for real-time prediction.

Abstract

Predicting the output of a dynamical system from streaming data is fundamental to real-time feedback control and decision-making. We first derive an autoregressive representation that relates future local outputs to asynchronous past outputs. Building on this structure, we propose an online least-squares algorithm to learn this autoregressive model for real-time prediction. We then establish a regret bound of O(log^3 N) relative to the optimal model-based predictor, which holds for marginally stable systems. Moreover, we provide a sufficient condition characterized via a symplectic matrix, under which the proposed cooperative online learning method provably outperforms the optimal model-based predictor that relies solely on local observations. From a technical standpoint, our analysis exploits the orthogonality of the innovation process under asynchronous data structure and the persistent excitation of the Gram matrix despite delay-induced asymmetries. Overall, these results offer both theoretical guarantees and practical algorithms for model-free cooperative prediction with asynchronous observations, thereby enriching the theory of online learning for dynamical systems.
Paper Structure (33 sections, 12 theorems, 170 equations, 3 figures, 1 algorithm)

This paper contains 33 sections, 12 theorems, 170 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related works
  4. Notations
  5. Preliminaries
  6. System model and centralized Kalman filter
  7. Online learning and regret
  8. Cooperative filtering with asynchronous observations and its regret
  9. Autoregressive modeling with asynchronous observations
  10. Optimal predictor with asynchronous observations
  11. Auto-regressive model and orthogonality of the innovation process
  12. Model-free Cooperative Filtering with Asynchronous Observations
  13. Model-free algorithm design
  14. Logarithmic regret against the optimal cooperative predictor
  15. Proof of Theorem \ref{['thm: regret']}
  16. ...and 18 more sections

Key Result

Proposition 3.1

Consider the linear stochastic system eq: LinearSystem in the presence of an external information source eq:external-source with i.i.d. Gaussian noises. The steady-state MMSE predictor $\hat{x}_{k+1}$ based on the local observations $y_0, \ldots, y_k$ and delayed external observations $y_0^{\mathrm{ where $\bar{x}_{k-d+1}\triangleq \mathbb{E}\left\{x_{k-d+1}\mid Y_{0:k-d}^{\mathrm{c}}\right\}$ fro

Figures (3)

  • Figure 1: Comparison between a standard local filter and our cooperative filter with delayed external observations. (a) Schematic illustration where our cooperative filter utilizes delayed external observations in \ref{['eq:external-source']}; (b) Numerical results in which accessing additional observations reduces the prediction error's covariance (see Example \ref{['example:comparison']}).
  • Figure 2: Logarithmic regret in Experiment 1. (a) Regret performance of the Ensemble-based Method in Section \ref{['subsec: paramterTuning']} for hyperparameter tuning in Theorem \ref{['thm: regret']}; (b) Performance improvement of our co-Filter in Algorithm \ref{['algPrediction']} with different time delays $d=3,5,\infty$ compared to a standard Kalman filter with only local observations. When $d=\infty$, our co-Filter is reduced to the online algorithm in tsiamis9894660; but with a finite time delay $d$, our co-Filter has a better performance than tsiamis9894660 and achieves negative regret against the standard Kalman filter with only local observations, which is a weaker benchmark.
  • Figure 3: Numerical experiment with real-life traffic trajectory data.

Theorems & Definitions (20)

  • Example 1
  • Proposition 3.1
  • Theorem 1
  • proof
  • Remark 3.1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Example 2
  • ...and 10 more