Learnability for the Information Bottleneck

Tailin Wu, Ian Fischer, Isaac L. Chuang, Max Tegmark
Proceedings of The 35th Uncertainty in Artificial Intelligence Conference, PMLR 115:1050-1060, 2020.

Abstract

The Information Bottleneck (IB) method (Tishby et al. (2000)) provides an insightful and principled approach for balancing compression and prediction for representation learning. The IB objective I (X ; Z ) − βI(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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v115-wu20b, title = {Learnability for the Information Bottleneck}, author = {Wu, Tailin and Fischer, Ian and Chuang, Isaac L. and Tegmark, Max}, booktitle = {Proceedings of The 35th Uncertainty in Artificial Intelligence Conference}, pages = {1050--1060}, year = {2020}, editor = {Adams, Ryan P. and Gogate, Vibhav}, volume = {115}, series = {Proceedings of Machine Learning Research}, month = {22--25 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v115/wu20b/wu20b.pdf}, url = {https://proceedings.mlr.press/v115/wu20b.html}, abstract = {The Information Bottleneck (IB) method (Tishby et al. (2000)) provides an insightful and principled approach for balancing compression and prediction for representation learning. The IB objective I (X ; Z ) − βI(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.} }
Endnote
%0 Conference Paper %T Learnability for the Information Bottleneck %A Tailin Wu %A Ian Fischer %A Isaac L. Chuang %A Max Tegmark %B Proceedings of The 35th Uncertainty in Artificial Intelligence Conference %C Proceedings of Machine Learning Research %D 2020 %E Ryan P. Adams %E Vibhav Gogate %F pmlr-v115-wu20b %I PMLR %P 1050--1060 %U https://proceedings.mlr.press/v115/wu20b.html %V 115 %X The Information Bottleneck (IB) method (Tishby et al. (2000)) provides an insightful and principled approach for balancing compression and prediction for representation learning. The IB objective I (X ; Z ) − βI(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.
APA
Wu, T., Fischer, I., Chuang, I.L. & Tegmark, M.. (2020). Learnability for the Information Bottleneck. Proceedings of The 35th Uncertainty in Artificial Intelligence Conference, in Proceedings of Machine Learning Research 115:1050-1060 Available from https://proceedings.mlr.press/v115/wu20b.html.

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