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Collaborative Estimation of Real Valued Function by Two Agents and a Fusion Center with Knowledge Exchange

Aneesh Raghavan, Karl H. Johansson

TL;DR

The paper addresses consistent collaborative estimation of a real-valued function over $\mathbb{R}$ using two agents and a fusion center, each operating in its own RKHS and exchanging models rather than data. It introduces an operator-based ColEst2L framework that decomposes estimation into agent-side problems, a fusion step, and a download step, with explicit transfer operators and a per-iteration operator $\mathbb{T}_n$ built from sub-operators. The main theoretical contribution proves strong consistency of the algorithm under conditions guaranteeing uniform boundedness of the operator sequence and valid input data, and shows invariance properties of the estimation process. A numerical example illustrates the approach and highlights practical considerations such as parameter sensitivity and conditioning, suggesting directions for extending the framework to infinite-dimensional spaces and inter-agent knowledge transfer.

Abstract

We consider a collaborative iterative algorithm with two agents and a fusion center for estimation of a real valued function (or ``model") on the set of real numbers. While the data collected by the agents is private, in every iteration of the algorithm, the models estimated by the agents are uploaded to the fusion center, fused, and, subsequently downloaded by the agents. We consider the estimation spaces at the agents and the fusion center to be Reproducing Kernel Hilbert Spaces (RKHS). Under suitable assumptions on these spaces, we prove that the algorithm is consistent, i.e., there exists a subsequence of the estimated models which converges to a model in the strong topology. To this end, we define estimation operators for the agents, fusion center, and, for every iteration of the algorithm constructively. We define valid input data sequences, study the asymptotic properties of the norm of the estimation operators, and, find sufficient conditions under which the estimation operator until any iteration is uniformly bounded. Using these results, we prove the existence of an estimation operator for the algorithm which implies the consistency of the considered estimation algorithm.

Collaborative Estimation of Real Valued Function by Two Agents and a Fusion Center with Knowledge Exchange

TL;DR

The paper addresses consistent collaborative estimation of a real-valued function over using two agents and a fusion center, each operating in its own RKHS and exchanging models rather than data. It introduces an operator-based ColEst2L framework that decomposes estimation into agent-side problems, a fusion step, and a download step, with explicit transfer operators and a per-iteration operator built from sub-operators. The main theoretical contribution proves strong consistency of the algorithm under conditions guaranteeing uniform boundedness of the operator sequence and valid input data, and shows invariance properties of the estimation process. A numerical example illustrates the approach and highlights practical considerations such as parameter sensitivity and conditioning, suggesting directions for extending the framework to infinite-dimensional spaces and inter-agent knowledge transfer.

Abstract

We consider a collaborative iterative algorithm with two agents and a fusion center for estimation of a real valued function (or ``model") on the set of real numbers. While the data collected by the agents is private, in every iteration of the algorithm, the models estimated by the agents are uploaded to the fusion center, fused, and, subsequently downloaded by the agents. We consider the estimation spaces at the agents and the fusion center to be Reproducing Kernel Hilbert Spaces (RKHS). Under suitable assumptions on these spaces, we prove that the algorithm is consistent, i.e., there exists a subsequence of the estimated models which converges to a model in the strong topology. To this end, we define estimation operators for the agents, fusion center, and, for every iteration of the algorithm constructively. We define valid input data sequences, study the asymptotic properties of the norm of the estimation operators, and, find sufficient conditions under which the estimation operator until any iteration is uniformly bounded. Using these results, we prove the existence of an estimation operator for the algorithm which implies the consistency of the considered estimation algorithm.
Paper Structure (27 sections, 20 theorems, 46 equations, 5 figures)

This paper contains 27 sections, 20 theorems, 46 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation and Related Work
  3. Objective and Contributions
  4. Outline
  5. Problem Formulation
  6. The Estimation Problem
  7. The Analysis Problems
  8. ColEst2L- The Estimation Algorithm
  9. Algorithm Description
  10. Estimation at Agents
  11. Fusion at Fusion Center
  12. Abstraction of ColEst2L Algorithm
  13. Operator for Estimation at The Agents
  14. The Multi-Agent Estimation and Upload Operator
  15. Operator for Fusion at The Fusion Center
  16. ...and 12 more sections

Key Result

Proposition 3.1

Let $\boldsymbol{\alpha^{i,*}_{n}} = ( \varrho^{i}_{n}\mathbf{K^{i}}+ \mathbf{\bar{K}^{i}}(x^{i}_n) \mathbf{\bar{K}^{i^T}}(x^{i}_n) )^{-1} \\ ( \mathbf{\bar{K}^{i}}(x^{i}_n) y^{i}_{n} + \varrho^{i}_{n}\mathbf{K^{i}} \boldsymbol{\bar{\alpha}^{i}_{n-1}})$. Then, $f^{i,*}_{n} = \boldsymbol{\alpha^{{i

Figures (5)

  • Figure 1: Schematic for the estimation architecture
  • Figure 2: ColEst2L - the estimation algorithm
  • Figure 3: The estimation operator at iteration $n$, $\mathbb{T}_{n}$, obtained through composition of operators, $\bar{T}, T, \hat{T}.$
  • Figure 4: True function and estimated functions at: (above) Agent 1 and (below) Agent 2.
  • Figure 5: True function and estimated functions at Fusion Center.

Theorems & Definitions (48)

  • Remark 1
  • Definition 1
  • Proposition 3.1
  • Proposition 3.2
  • Definition 2
  • Lemma 4.1
  • proof
  • Definition 3
  • Lemma 4.2
  • proof
  • ...and 38 more