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Distributed Estimation by Two Agents with Different Feature Spaces

Aneesh Raghavan, Karl Henrik Johansson

TL;DR

This work considers the problem of estimation of a function by a system consisting of two agents and a fusion center, and presents the distributed estimation architecture that has been developed.

Abstract

We consider the problem of estimation of a function by a system consisting of two agents and a fusion center. The two agents collect data comprising of samples of an independent variable and the corresponding value of a dependent variable. The objective of the system is to collaboratively estimate the function without any exchange of data among the members of the system. To this end, we propose the following framework. The agents are given a set of features using which they construct suitable function spaces to formulate and solve the estimation problems locally. The estimated functions are uploaded to a fusion space where an optimization problem is solved to fuse the estimates (also known as meta-learning) to obtain the system estimate of the mapping. The fused function is then downloaded by the agents to gather knowledge about the other agents estimate of the function. With respect to the framework, we present the following: a systematic construction of fusion space given the features of the agents; the derivation of an uploading operator for the agents to upload their estimated functions to a fusion space; the derivation of a downloading operator for the fused function to be downloaded. Through an example on least squares regression, we illustrate the distributed estimation architecture that has been developed.

Distributed Estimation by Two Agents with Different Feature Spaces

TL;DR

This work considers the problem of estimation of a function by a system consisting of two agents and a fusion center, and presents the distributed estimation architecture that has been developed.

Abstract

We consider the problem of estimation of a function by a system consisting of two agents and a fusion center. The two agents collect data comprising of samples of an independent variable and the corresponding value of a dependent variable. The objective of the system is to collaboratively estimate the function without any exchange of data among the members of the system. To this end, we propose the following framework. The agents are given a set of features using which they construct suitable function spaces to formulate and solve the estimation problems locally. The estimated functions are uploaded to a fusion space where an optimization problem is solved to fuse the estimates (also known as meta-learning) to obtain the system estimate of the mapping. The fused function is then downloaded by the agents to gather knowledge about the other agents estimate of the function. With respect to the framework, we present the following: a systematic construction of fusion space given the features of the agents; the derivation of an uploading operator for the agents to upload their estimated functions to a fusion space; the derivation of a downloading operator for the fused function to be downloaded. Through an example on least squares regression, we illustrate the distributed estimation architecture that has been developed.
Paper Structure (14 sections, 7 theorems, 24 equations, 4 figures)

This paper contains 14 sections, 7 theorems, 24 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Problem Considered
  4. Contributions
  5. Novelty and Interpretation of Proposed Solution
  6. Construction of Knowledge Spaces
  7. Construction of Individual Knowledge Spaces for the Agents
  8. Construction of the Fusion Space
  9. Retrieval of Individual Knowledge Spaces from Fusion Space
  10. Regression and Fusion Problem
  11. Regression at the Agents
  12. Function Fusion Problem
  13. Example
  14. Conclusion and Future Work

Key Result

Theorem 2.1

If $K^{i}(\cdot,\cdot)$ is the reproducing kernel of Hilbert space $H^{i}$, with norm $||\cdot||_{H^i}$, then $K(x,y)=K^{1}(x,y) + K^{2}(x,y)$ is the reproducing kernel of the space $H = \{f: f= f^1 + f^2 | f^{i} \in H^{i}\}$ with the norm:

Figures (4)

  • Figure 1: Schematic for Distributed Estimation Architecture
  • Figure 2: True function, function uploaded by Agent 1, function downloaded by Agent 1.
  • Figure 3: True function, function uploaded by Agent 2, function downloaded by Agent 2.
  • Figure 4: True function, fused function, function obtained through centralized estimation at fusion center.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 6 more