  We view the Information Bottleneck Principle (IBP: Tishby et al., 1999;
Schwartz-Ziv and Tishby, 2017) and Predictive Information Bottleneck Principle
(PIBP: Still et al., 2007; Alemi, 2019) as special cases of a family of general
information bottleneck objectives (IBOs). Each IBO corresponds to a particular
constrained optimization problem where the constraints apply to: (a) the mutual
information between the training data and the learned model parameters or
extracted representation of the data, and (b) the mutual information between
the learned model parameters or extracted representation of the data and the
test data (if any). The heuristics behind the IBP and PIBP are shown to yield
different constraints in the corresponding constrained optimization problem
formulations. We show how other heuristics lead to a new IBO, different from
both the IBP and PIBP, and use the techniques from (Alemi, 2019) to derive and
optimize a variational upper bound on the new IBO.
  We then apply the theory of general IBOs to resolve the seeming contradiction
between, on the one hand, the recommendations of IBP and PIBP to maximize the
mutual information between the model parameters and test data, and on the
other, recent information-theoretic results (see Xu and Raginsky, 2017)
suggesting that this mutual information should be minimized. The key insight is
that the heuristics (and thus the constraints in the constrained optimization
problems) of IBP and PIBP are not applicable to the scenario analyzed by (Xu
and Raginsky, 2017) because the latter makes the additional assumption that the
parameters of the trained model have been selected to minimize the empirical
loss function. Aided by this insight, we formulate a new IBO that accounts for
this property of the parameters of the trained model, and derive and optimize a
variational bound on this IBO.
