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Inverse Feasibility in Over-the-Air Federated Learning

Tomasz Piotrowski, Rafail Ismayilov, Matthias Frey, Renato L. G. Cavalcante

TL;DR

This work introduces inverse feasibility as a principled measure for the difficulty of reconstructing aggregated OTA FL signals from compressed wireless transmissions. By linking the reconstructability to the condition number of the forward operator, it analyzes the baseline OTA FL model and proposes optimizations—most notably a data-agnostic, well-conditioned compression map $W^\star$—to tighten inverse-feasibility bounds. It further incorporates a noisy, server-known perturbation model to reflect practical randomness and derives high-probability bounds on the forward-operator conditioning, while showing through MNIST experiments that the optimized design yields substantial stability and minimal reconstruction error under power control. The framework provides a complementary perspective to security and privacy in OTA FL and suggests extensions to nonlinear forward models for broader robustness and applicability.

Abstract

We introduce the concept of inverse feasibility for linear forward models as a tool to enhance OTA FL algorithms. Inverse feasibility is defined as an upper bound on the condition number of the forward operator as a function of its parameters. We analyze an existing OTA FL model using this definition, identify areas for improvement, and propose a new OTA FL model. Numerical experiments illustrate the main implications of the theoretical results. The proposed framework, which is based on inverse problem theory, can potentially complement existing notions of security and privacy by providing additional desirable characteristics to networks.

Inverse Feasibility in Over-the-Air Federated Learning

TL;DR

This work introduces inverse feasibility as a principled measure for the difficulty of reconstructing aggregated OTA FL signals from compressed wireless transmissions. By linking the reconstructability to the condition number of the forward operator, it analyzes the baseline OTA FL model and proposes optimizations—most notably a data-agnostic, well-conditioned compression map —to tighten inverse-feasibility bounds. It further incorporates a noisy, server-known perturbation model to reflect practical randomness and derives high-probability bounds on the forward-operator conditioning, while showing through MNIST experiments that the optimized design yields substantial stability and minimal reconstruction error under power control. The framework provides a complementary perspective to security and privacy in OTA FL and suggests extensions to nonlinear forward models for broader robustness and applicability.

Abstract

We introduce the concept of inverse feasibility for linear forward models as a tool to enhance OTA FL algorithms. Inverse feasibility is defined as an upper bound on the condition number of the forward operator as a function of its parameters. We analyze an existing OTA FL model using this definition, identify areas for improvement, and propose a new OTA FL model. Numerical experiments illustrate the main implications of the theoretical results. The proposed framework, which is based on inverse problem theory, can potentially complement existing notions of security and privacy by providing additional desirable characteristics to networks.
Paper Structure (14 sections, 2 theorems, 39 equations, 1 figure)

This paper contains 14 sections, 2 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. System Model
  3. Inverse Feasibility in Over-the-Air Federated Learning: Definition and Examples
  4. Definition of Inverse Feasibility
  5. Inverse Feasibility of OTA FL Models
  6. Power control
  7. Evaluation of Proposition \ref{['p3']}
  8. Parameter Estimation
  9. Numerical Evaluation
  10. Conclusion and final remarks
  11. Supplementary material
  12. Proof of Proposition 1
  13. Proof of Proposition 2
  14. Condition Number of the Operator in Eq. (18)

Key Result

Proposition 1

The model eq.model_original_received satisfies $F(\underbar{p})\stackrel{\mathclap{\tiny\hbox{df.}}}{=}\operatorname{cond}(A)\equiv\operatorname{cond}(\bm{L(p)}).$

Figures (1)

  • Figure 1: Reconstruction error of OTA FL models against the number of active nodes $M.$

Theorems & Definitions (7)

  • Definition 1: inverse feasibility
  • Remark 1
  • Proposition 1
  • Example 1
  • Proposition 2
  • Remark 2
  • Remark 3