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Bottleneck problems are an important class of optimization problems that haverecently gained increasing attention in the domain of machine learning andinformation theory. They are widely used in generative models, fair machinelearning algorithms, design of privacy-assuring mechanisms, and appear asinformation-theoretic performance bounds in various multi-user communicationproblems. In this work, we propose a general family of optimization problems,termed as complexity-leakage-utility bottleneck (CLUB) model, which (i)provides a unified theoretical framework that generalizes most of thestate-of-the-art literature for the information-theoretic privacy models, (ii)establishes a new interpretation of the popular generative and discriminativemodels, (iii) constructs new insights to the generative compression models, and(iv) can be used in the fair generative models. We first formulate the CLUBmodel as a complexity-constrained privacy-utility optimization problem. We thenconnect it with the closely related bottleneck problems, namely informationbottleneck (IB), privacy funnel (PF), deterministic IB (DIB), conditionalentropy bottleneck (CEB), and conditional PF (CPF). We show that the CLUB modelgeneralizes all these problems as well as most other information-theoreticprivacy models. Then, we construct the deep variational CLUB (DVCLUB) models byemploying neural networks to parameterize variational approximations of theassociated information quantities. Building upon these information quantities,we present unified objectives of the supervised and unsupervised DVCLUB models.Leveraging the DVCLUB model in an unsupervised setup, we then connect it withstate-of-the-art generative models, such as variational auto-encoders (VAEs),generative adversarial networks (GANs), as well as the Wasserstein GAN (WGAN),Wasserstein auto-encoder (WAE), and adversarial auto-encoder (AAE) modelsthrough the optimal transport (OT) problem. We then show that the DVCLUB modelcan also be used in fair representation learning problems, where the goal is tomitigate the undesired bias during the training phase of a machine learningmodel. We conduct extensive quantitative experiments on colored-MNIST andCelebA datasets, with a public implementation available, to evaluate andanalyze the CLUB model.

Bottlenecks CLUB: Unifying Information-Theoretic Trade-offs Among Complexity, Leakage, and Utility