Learnability for the Information Bottleneck
Name
entropy-21-00924-v2.pdf
Description
Published version
Size
2.33 MB
Format
Adobe PDF
Checksum (MD5)
537624952425d8d318a418845b2823a6
Author(s)
Wu, Tailin
Fischer, Ian
Chuang, Isaac L.
Tegmark, Max Erik
Date Issued
September 2019
Journal
Entropy
Publisher
MDPI AG
Citation
Wu, Tailin; Fischer, Ian; Chuang, Isaac L.; Tegmark, Max. "Learnability for the Information Bottleneck." Entropy 21,10 (2019): 924.
Version
Final published version
Abstract
The Information Bottleneck (IB) method provides an insightful and principled approach for balancing compression and prediction for representation learning. The IB objective I ( X ; Z ) - β I ( Y ; Z ) employs a Lagrange multiplier β to tune this trade-off. However, in practice, not only is β chosen empirically without theoretical guidance, there is also a lack of theoretical understanding between β , learnability, the intrinsic nature of the dataset and model capacity. In this paper, we show that if β 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 β 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 β . 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 β for a given dataset. We also empirically demonstrate our theoretical conditions with analyses of synthetic datasets, MNIST and CIFAR10.
MIT Department
Massachusetts Institute of Technology. Department of Physics
Terms of Use
Creative Commons Attribution 4.0 International license
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.3390/e21100924
