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Whom to Trust? Elective Learning for Distributed Gaussian Process Regression

Zewen Yang, Xiaobing Dai, Akshat Dubey, Sandra Hirche, Georges Hattab

TL;DR

The paper tackles distributed learning of a common function $f$ in a multi-agent system where agents hold heterogeneous priors $\hat{f}_i$ and local data. It proposes Pri-GP, a prior-error-based elective learning framework that lets agents selectively query predictions from trusted neighbors and balance prior information with optional posterior-variance cues through a tunable parameter $c$. A prior-error metric $e_i(x)=\hat{f}_i(x)-y(x)$ and its normalized form $\tilde{\varepsilon}_i(t_k)$ drive neighbor selection, while aggregation weights $\omega_{ij}$ are formed to enable fast, variance-aware or variance-free fusion. The approach provides a probabilistic guarantee in the form of a high-probability uniform error bound for per-agent predictions and demonstrates superior performance in function approximation and dynamical system identification, with reduced computational burden, making it well-suited for safety-critical MAS applications.

Abstract

This paper introduces an innovative approach to enhance distributed cooperative learning using Gaussian process (GP) regression in multi-agent systems (MASs). The key contribution of this work is the development of an elective learning algorithm, namely prior-aware elective distributed GP (Pri-GP), which empowers agents with the capability to selectively request predictions from neighboring agents based on their trustworthiness. The proposed Pri-GP effectively improves individual prediction accuracy, especially in cases where the prior knowledge of an agent is incorrect. Moreover, it eliminates the need for computationally intensive variance calculations for determining aggregation weights in distributed GP. Furthermore, we establish a prediction error bound within the Pri-GP framework, ensuring the reliability of predictions, which is regarded as a crucial property in safety-critical MAS applications.

Whom to Trust? Elective Learning for Distributed Gaussian Process Regression

TL;DR

The paper tackles distributed learning of a common function in a multi-agent system where agents hold heterogeneous priors and local data. It proposes Pri-GP, a prior-error-based elective learning framework that lets agents selectively query predictions from trusted neighbors and balance prior information with optional posterior-variance cues through a tunable parameter . A prior-error metric and its normalized form drive neighbor selection, while aggregation weights are formed to enable fast, variance-aware or variance-free fusion. The approach provides a probabilistic guarantee in the form of a high-probability uniform error bound for per-agent predictions and demonstrates superior performance in function approximation and dynamical system identification, with reduced computational burden, making it well-suited for safety-critical MAS applications.

Abstract

This paper introduces an innovative approach to enhance distributed cooperative learning using Gaussian process (GP) regression in multi-agent systems (MASs). The key contribution of this work is the development of an elective learning algorithm, namely prior-aware elective distributed GP (Pri-GP), which empowers agents with the capability to selectively request predictions from neighboring agents based on their trustworthiness. The proposed Pri-GP effectively improves individual prediction accuracy, especially in cases where the prior knowledge of an agent is incorrect. Moreover, it eliminates the need for computationally intensive variance calculations for determining aggregation weights in distributed GP. Furthermore, we establish a prediction error bound within the Pri-GP framework, ensuring the reliability of predictions, which is regarded as a crucial property in safety-critical MAS applications.
Paper Structure (13 sections, 3 theorems, 25 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 3 theorems, 25 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Setting
  3. Notation and Graph Theory
  4. Problem Description
  5. Gaussian Process Regression
  6. Elective Distributed Learning with Prior-Aware GPR
  7. Prior Error Quantification
  8. Prior-Aware Elective Cooperative Learning
  9. Prediction with Probabilistic Guarantee
  10. Numerical Evaluation
  11. Function Approximation
  12. Dynamical System Identification
  13. Conclusion

Key Result

Lemma 1

The variable $\varepsilon_{i}(t_{k})$ reflects the prediction error on the training data set $\mathbb{D}_i$, and the measurement error. In particular, $\varepsilon_{i}(t_{k})$ is written as where $\boldsymbol{G}_k = (\boldsymbol{I} + \sigma_{n}^{-2} \mathcal{K}(\boldsymbol{X}_i, \boldsymbol{X}_i) )$. The aggregated error denotes $\boldsymbol{\xi}_k = [\xi_0, \cdots, \xi_k]^T$ with $\xi_p = f(\bo

Figures (4)

  • Figure 1: True, prior and posterior curves.
  • Figure 2: Violin plots of prediction errors for different methods. The red line is the mean value and the top/bottom horizontal blue bar is the maximal/minimal value.
  • Figure 3: 3 Dimension Plots
  • Figure 4: Mean prediction error with standard deviation

Theorems & Definitions (6)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 2
  • Theorem 1