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Distributed Time-Varying Gaussian Regression via Kalman Filtering

Nicola Taddei, Riccardo Maggioni, Jaap Eising, Giulia De Pasquale, Florian Dorfler

TL;DR

The paper tackles distributed, real-time estimation of time-varying spatio-temporal functions in multi-agent control systems. It introduces DistKP, which reframes a time-varying Gaussian Process as a finite-dimensional Bayesian Linear Regression on Nyström features, with the coefficient vector θ(t) following a random-walk dynamic and observations y_i(t) = Φ(x_i(t))^T θ(t) + ν_i(t) being fused by a distributed Kalman filter with consensus. Key contributions include a Nyström-based kernel approximation enabling O(E^3) inference, a time-varying θ(t) model with hyperparameters E and σ_ω beyond standard GP, and a distributed, privacy-preserving regression framework that demonstrates robust tracking and uncertainty quantification on a UAV wind-field scenario. This work provides a scalable, robust approach for distributed learning of dynamic functions in safety-critical settings, with potential extensions to convergence guarantees and integration into planning pipelines.

Abstract

We consider the problem of learning time-varying functions in a distributed fashion, where agents collect local information to collaboratively achieve a shared estimate. This task is particularly relevant in control applications, whenever real-time and robust estimation of dynamic cost/reward functions in safety critical settings has to be performed. In this paper, we,adopt a finite-dimensional approximation of a Gaussian Process, corresponding to a Bayesian linear regression in an appropriate feature space, and propose a new algorithm, DistKP, to track the time-varying coefficients via a distributed Kalman filter. The proposed method works for arbitrary kernels and under weaker assumptions on the time-evolution of the function to learn compared to the literature. We validate our results using a simulation example in which a fleet of Unmanned Aerial Vehicles (UAVs) learns a dynamically changing wind field.

Distributed Time-Varying Gaussian Regression via Kalman Filtering

TL;DR

The paper tackles distributed, real-time estimation of time-varying spatio-temporal functions in multi-agent control systems. It introduces DistKP, which reframes a time-varying Gaussian Process as a finite-dimensional Bayesian Linear Regression on Nyström features, with the coefficient vector θ(t) following a random-walk dynamic and observations y_i(t) = Φ(x_i(t))^T θ(t) + ν_i(t) being fused by a distributed Kalman filter with consensus. Key contributions include a Nyström-based kernel approximation enabling O(E^3) inference, a time-varying θ(t) model with hyperparameters E and σ_ω beyond standard GP, and a distributed, privacy-preserving regression framework that demonstrates robust tracking and uncertainty quantification on a UAV wind-field scenario. This work provides a scalable, robust approach for distributed learning of dynamic functions in safety-critical settings, with potential extensions to convergence guarantees and integration into planning pipelines.

Abstract

We consider the problem of learning time-varying functions in a distributed fashion, where agents collect local information to collaboratively achieve a shared estimate. This task is particularly relevant in control applications, whenever real-time and robust estimation of dynamic cost/reward functions in safety critical settings has to be performed. In this paper, we,adopt a finite-dimensional approximation of a Gaussian Process, corresponding to a Bayesian linear regression in an appropriate feature space, and propose a new algorithm, DistKP, to track the time-varying coefficients via a distributed Kalman filter. The proposed method works for arbitrary kernels and under weaker assumptions on the time-evolution of the function to learn compared to the literature. We validate our results using a simulation example in which a fleet of Unmanned Aerial Vehicles (UAVs) learns a dynamically changing wind field.

Paper Structure

This paper contains 10 sections, 3 theorems, 23 equations, 3 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Problem Formulation
  3. Preliminaries on Gaussian Processes
  4. Technical adaptations
  5. Nyström approximation
  6. Time-varying function
  7. Distributed regression
  8. DistKP Algorithm
  9. Numerical Results
  10. Conclusions

Key Result

Lemma 1

Consider a GP with kernel $k(\cdot, \cdot)$ such that $k(\mathbf{x}, \mathbf{x}') = \Phi(\mathbf{x})^\top \Phi(\mathbf{x}')$. Then the GP regression is equivalent to a Bayesian Linear Regression on features given by $\Phi(\cdot)$.

Figures (3)

  • Figure 1: Comparison between measurement fusion with inverse variance weighting (left) and (centrally computed) BLUE (right). Sample locations are represented as red dots. We use $\sigma_\omega = 0$ since higher values make it difficult to compute the BLUE due to ill conditioned matrices.
  • Figure 2: Ground Truth (left), mean (center) and variance (right) estimates at $t=50$ (up), $t=300$ (middle), $t=600$ (bottom).
  • Figure 3: Comparison between our estimate at time $t=600$ (left), the estimate of a distributed GP with time-varying method from distributed_informative_planning (center) and the ground truth (right).

Theorems & Definitions (6)

  • Lemma 1: Equivalence of GP and BLR GP_in_ML
  • Definition 2: Radial Basis Function kernel
  • Definition 3: Laplace kernel
  • Theorem 4: Nyström approximation nystrom
  • Lemma 5: Nyström induced feature space
  • proof