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Kernel-based learning with guarantees for multi-agent applications

Krzysztof Kowalczyk, Paweł Wachel, Cristian R. Rojas

TL;DR

The paper tackles distributed learning of a latent multivariate nonlinear phenomenon observed locally by a network of agents. It introduces a kernel regression framework where each node builds a local estimator and then aggregates data via neighbor communication to form a distributed model, with non-asymptotic, high-probability error bounds that are robust to the input dimension. Key contributions include a finite-sample bound for single-agent kernel regression and its extension to a distributed data-aggregation scheme, along with a simple data-exchange protocol and convergence assurances. The results demonstrate that the distributed approach can achieve performance close to a centralized model, while requiring only mild regularity assumptions and offering explicit error guarantees useful for real-time multi-agent applications.

Abstract

This paper addresses a kernel-based learning problem for a network of agents locally observing a latent multidimensional, nonlinear phenomenon in a noisy environment. We propose a learning algorithm that requires only mild a priori knowledge about the phenomenon under investigation and delivers a model with corresponding non-asymptotic high probability error bounds. Both non-asymptotic analysis of the method and numerical simulation results are presented and discussed in the paper.

Kernel-based learning with guarantees for multi-agent applications

TL;DR

The paper tackles distributed learning of a latent multivariate nonlinear phenomenon observed locally by a network of agents. It introduces a kernel regression framework where each node builds a local estimator and then aggregates data via neighbor communication to form a distributed model, with non-asymptotic, high-probability error bounds that are robust to the input dimension. Key contributions include a finite-sample bound for single-agent kernel regression and its extension to a distributed data-aggregation scheme, along with a simple data-exchange protocol and convergence assurances. The results demonstrate that the distributed approach can achieve performance close to a centralized model, while requiring only mild regularity assumptions and offering explicit error guarantees useful for real-time multi-agent applications.

Abstract

This paper addresses a kernel-based learning problem for a network of agents locally observing a latent multidimensional, nonlinear phenomenon in a noisy environment. We propose a learning algorithm that requires only mild a priori knowledge about the phenomenon under investigation and delivers a model with corresponding non-asymptotic high probability error bounds. Both non-asymptotic analysis of the method and numerical simulation results are presented and discussed in the paper.
Paper Structure (8 sections, 4 theorems, 36 equations, 6 figures, 1 algorithm)

This paper contains 8 sections, 4 theorems, 36 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Network setup
  4. Phenomenon and observations
  5. Local agents' modelling
  6. Distributed modelling -- data aggregation
  7. Numerical experiments
  8. Conclusions

Key Result

Lemma 1

Let Assumptions 1--4 be in force. Consider the estimator $\hat{\mu}_{k,t}\in\mathbb{R}^d$ and fix a bandwidth parameter $h$. Let $x \in \mathcal{D}\subset\mathbb{R}^p$ be fixed or in general a measurable function of $\eta_{k,1:t-1}, \xi_{k,1:t}$ (e.g., $x=\xi_{k,t}$). Then, for every $0 <\delta <1$, and

Figures (6)

  • Figure 1: A network of distributed agents with highlighted neighbourhood of a selected node.
  • Figure 2: Example of a nonlinear phenomenon (blue) with marked agents' local domains of observations/measurements (grey).
  • Figure 3: Random topology network with 25 nodes.
  • Figure 4: A model of the estimated phenomenon provided by agent $k=13$ after $t=5000$ time steps, with cross-sections presenting its confidence bounds.
  • Figure 5: Comparison of the model provided by a single agent, with a global model, that uses all the agents' data.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • proof : Proof (of Lemma \ref{['L:local']}).
  • Lemma 2
  • proof
  • Lemma 3
  • proof