Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rate-Loss Regions for Polynomial Regression with Side Information

Jiahui Wei, Philippe Mary, Elsa Dupraz

TL;DR

This work analyzes how communication constraints affect learning performance when a source is compressed and decoded with side information at the receiver, focusing on polynomial regression. It derives both asymptotic and finite-blocklength rate-loss regions, showing that the minimum generalization error can be achieved at any positive rate using a Gaussian test-channel scheme and OLS estimation, with the generalization error converging to the optimum as the training length grows. A key finding is that, asymptotically, there is no trade-off between data reconstruction quality and polynomial regression performance under the considered model, and the Wyner–Ziv rate-distortion function aligns with the conditional rate-distortion function. The non-asymptotic analysis provides a dispersion-based bound for finite blocklengths, and numerical results illustrate how the rate-loss region expands with blocklength and allowable excess probability. Overall, the framework advances understanding of learning under communication constraints and suggests directions for extending to more general estimation tasks.

Abstract

In the context of goal-oriented communications, this paper addresses the achievable rate versus generalization error region of a learning task applied on compressed data. The study focuses on the distributed setup where a source is compressed and transmitted through a noiseless channel to a receiver performing polynomial regression, aided by side information available at the decoder. The paper provides the asymptotic rate generalization error region, and extends the analysis to the non-asymptotic regime.Additionally, it investigates the asymptotic trade-off between polynomial regression and data reconstruction under communication constraints. The proposed achievable scheme is shown to achieve the minimum generalization error as well as the optimal rate-distortion region.

Rate-Loss Regions for Polynomial Regression with Side Information

TL;DR

This work analyzes how communication constraints affect learning performance when a source is compressed and decoded with side information at the receiver, focusing on polynomial regression. It derives both asymptotic and finite-blocklength rate-loss regions, showing that the minimum generalization error can be achieved at any positive rate using a Gaussian test-channel scheme and OLS estimation, with the generalization error converging to the optimum as the training length grows. A key finding is that, asymptotically, there is no trade-off between data reconstruction quality and polynomial regression performance under the considered model, and the Wyner–Ziv rate-distortion function aligns with the conditional rate-distortion function. The non-asymptotic analysis provides a dispersion-based bound for finite blocklengths, and numerical results illustrate how the rate-loss region expands with blocklength and allowable excess probability. Overall, the framework advances understanding of learning under communication constraints and suggests directions for extending to more general estimation tasks.

Abstract

In the context of goal-oriented communications, this paper addresses the achievable rate versus generalization error region of a learning task applied on compressed data. The study focuses on the distributed setup where a source is compressed and transmitted through a noiseless channel to a receiver performing polynomial regression, aided by side information available at the decoder. The paper provides the asymptotic rate generalization error region, and extends the analysis to the non-asymptotic regime.Additionally, it investigates the asymptotic trade-off between polynomial regression and data reconstruction under communication constraints. The proposed achievable scheme is shown to achieve the minimum generalization error as well as the optimal rate-distortion region.
Paper Structure (13 sections, 4 theorems, 28 equations, 2 figures)

This paper contains 13 sections, 4 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Notation
  4. Source definitions
  5. Polynomial Regression
  6. Coding scheme
  7. Asymptotic Bound on the Rate-Loss function
  8. Rate-loss region
  9. Proof of Theorem \ref{['achievability_asymptotic']} : Achievability scheme
  10. Trade-off between data reconstruction and polynomial regression
  11. Rate-Loss non-asymptotic bound
  12. Numerical results
  13. Conclusion

Key Result

Theorem 1

Given any rate $R > 0$, the pair $(R, 0)$ is achievable for the polynomial regression scheme with squared loss, for sources $(X,Y)$ following the polynomial model pl_regression.

Figures (2)

  • Figure 1: Coding scheme for regression
  • Figure 2: Non-asymptotic rate-generalization error region labeled on the blocklength $n$ and the excess loss probability $\varepsilon$.

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Corollary 1