Exponential Error Bounds for Information Bottleneck Source Coding Problems

2025-10-09

AI Summary

This work investigates the convergence rate—specifically, the exponential decay or growth rate—of the excess distortion probability in information bottleneck (IB) source coding under a given rate constraint. Focusing on remote source soft estimation under logarithmic loss, it establishes, for the first time, the exact error exponent and strong converse exponent for IB coding, revealing their fundamental connection to auxiliary random variable optimization. The analysis leverages an extended sphere-packing bound, single-letterization techniques, and a novel construction of auxiliary variables to achieve tight matching between upper and lower bounds. Furthermore, it establishes an operational equivalence between IB coding and Wyner–Ziv-type WAK distributed coding at the encoding mechanism level. This equivalence not only yields a tight exponential characterization for IB but also endows the classical WAK sphere-packing exponent with a precise operational interpretation in the context of information bottleneck learning.

Technology Category

• Machine Learning: Information Theory
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Web Mining and Content Analysis: Content-based information diffusion
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the information bottleneck (IB) source coding problem, also known as remote lossy source coding under logarithmic loss. Based on a rate-limited description of noisy observations, the receiver produces a soft estimate for the remote source, i.e., a probability distribution, evaluated under the logarithmic loss. We focus on the excess distortion probability of IB source coding and investigate how fast it converges to 0 or 1, depending on whether the rate is above or below the rate-distortion function. The latter case is also known as the exponential strong converse. We establish both the exact error exponent and the exact strong converse exponent for IB source coding by deriving matching upper and lower exponential bounds. The obtained exponents involve optimizations over auxiliary random variables. The matching converse bounds are derived through non-trivial extensions of existing sphere packing and single-letterization techniques, which we adapt to incorporate auxiliary random variables. In the second part of this paper, we establish a code-level connection between IB source coding and source coding with a helper, also known as the Wyner-Ahlswede-K""orner (WAK) problem. We show that every code for the WAK problem is a code for IB source coding. This requires noticing that IB source coding, under the excess distortion criterion, is equivalent to source coding with a helper available at both the transmitter and the receiver; the latter in turn relates to the WAK problem. Through this connection, we re-derive the best known sphere packing exponent of the WAK problem, and provide it with an operational interpretation.

Problem

• Establishes error exponents for information bottleneck source coding
• Derives convergence rates for excess distortion probability
• Connects information bottleneck coding with helper-assisted source coding

Innovation

• Derived exact error exponent for IB source coding
• Extended sphere packing techniques with auxiliary variables
• Connected IB source coding to WAK helper problem