Information bottleneck is an information-theoretic principle of representation learning that aims to learn a maximally compressed representation that preserves as much information about labels as possible. Under this principle, two different methods have been proposed, i.e., information bottleneck (IB) and deterministic information bottleneck (DIB), and have gained significant progress in explaining the representation mechanisms of deep learning algorithms. However, these theoretical and empirical successes are only valid with the assumption that training and test data are drawn from the same distribution, which is clearly not satisfied in many real-world applications. In this paper, we study their generalization abilities within a transfer learning scenario, where the target error could be decomposed into three components, i.e., source empirical error, source generalization gap (SG), and representation discrepancy (RD). Comparing IB and DIB on these terms, we prove that DIB's SG bound is tighter than IB's while DIB's RD is larger than IB's. Therefore, it is difficult to tell which one is better. To balance the trade-off between SG and the RD, we propose an elastic information bottleneck (EIB) to interpolate between the IB and DIB regularizers, which guarantees a Pareto frontier within the IB framework. Additionally, simulations and real data experiments show that EIB has the ability to achieve better domain adaptation results than IB and DIB, which validates the correctness of our theories.

Lossy compression and clustering fundamentally involve a decision about what features are relevant and which are not. The information bottleneck method (IB) by Tishby, Pereira, and Bialek formalized this notion as an information-theoretic optimization problem and proposed an optimal tradeoff between throwing away as many bits as possible, and selectively keeping those that are most important. In the IB, compression is measure my mutual information. Here, we introduce an alternative formulation that replaces mutual information with entropy, which we call the deterministic information bottleneck (DIB), that we argue better captures this notion of compression. As suggested by its name, the solution to the DIB problem turns out to be a deterministic encoder, or hard clustering, as opposed to the stochastic encoder, or soft clustering, that is optimal under the IB. We compare the IB and DIB on synthetic data, showing that the IB and DIB perform similarly in terms of the IB cost function, but that the DIB significantly outperforms the IB in terms of the DIB cost function. We also empirically find that the DIB offers a considerable gain in computational efficiency over the IB, over a range of convergence parameters. Our derivation of the DIB also suggests a method for continuously interpolating between the soft clustering of the IB and the hard clustering of the DIB.