Learnability for the Information Bottleneck

TL;DR

This paper introduces the concept of IB-Learnability, revealing a phase transition in the Information Bottleneck method that guides optimal choice of the trade-off parameter , supported by theoretical analysis and empirical validation.

Contribution

It establishes the concept of IB-Learnability, identifies a phase transition in the IB method, and provides theoretical and practical guidelines for selecting .

Findings

A sharp phase transition between learnable and unlearnable regimes.

Theoretical conditions for IB-Learnability based on dataset subsets.

Practical algorithms for estimating optimal .

Abstract

The Information Bottleneck (IB) method (\cite{tishby2000information}) provides an insightful and principled approach for balancing compression and prediction for representation learning. The IB objective $I(X;Z)-\beta I(Y;Z)$ employs a Lagrange multiplier $\beta$ to tune this trade-off. However, in practice, not only is $\beta$ chosen empirically without theoretical guidance, there is also a lack of theoretical understanding between $\beta$, learnability, the intrinsic nature of the dataset and model capacity. In this paper, we show that if $\beta$ is improperly chosen, learning cannot happen -- the trivial representation $P(Z|X)=P(Z)$ becomes the global minimum of the IB objective. We show how this can be avoided, by identifying a sharp phase transition between the unlearnable and the learnable which arises as $\beta$ is varied. This phase transition defines the concept of IB-Learnability. We prove several sufficient conditions for IB-Learnability, which provides theoretical guidance for choosing a good $\beta$. We further show that IB-learnability is determined by the largest confident, typical, and imbalanced subset of the examples (the conspicuous subset), and discuss its relation with model capacity. We give practical algorithms to estimate the minimum $\beta$ for a given dataset. We also empirically demonstrate our theoretical conditions with analyses of synthetic datasets, MNIST, and CIFAR10.