Paper:
In the Information Bottleneck (IB), when tuning the relative strength between compression and prediction terms, how do the two terms behave, and what's their relationship with the dataset and the learned representation? In this paper, we set out to answer these questions by studying multiple phase transitions in the IB objective: IB β [p(z|x)] = I(X; Z)−βI(Y ; Z) defined on the encoding distribution p(z|x) for input X, target Y and representation Z, where sudden jumps of dI(Y ;Z) dβ and prediction accuracy are observed with increasing β. We introduce a definition for IB phase transitions as a qualitative change of the IB loss landscape, and show that the transitions correspond to the onset of learning new classes. Using secondorder calculus of variations, we derive a formula that provides a practical condition for IB phase transitions, and draw its connection with the Fisher information matrix for parameterized models. We provide two perspectives to understand the formula, revealing that each IB phase transition is finding a component of maximum (nonlinear) correlation between X and Y orthogonal to the learned representation, in close analogy with canonical-correlation analysis (CCA) in linear settings. Based on the theory, we present an algorithm for discovering phase transition points. Finally, we verify that our theory and algorithm accurately predict phase transitions in categorical datasets, predict the onset of learning new classes and class difficulty in MNIST, and predict prominent phase transitions in CIFAR10.

Review:
This paper contributes theoretically to the information bottleneck (IB) principle. In particular, the author(s) provided theoretical reasoning on the phase transition phenomenon: when the beta parameter of IB varies, the generalization performance changes in a stepwise manner rather than continuously. The core result is given by theorem 1: the phase transition betas necessarily satisfy an equation, where the LHS is expressed in terms of an optimal perturbation of the encoding function X->Z.
Overall, the reviewer believes that this work is a solid contribution to the IB principle, and should be accepted. Remarkably, this work gives the first formulation on the phase transition of IB, which was an empirical observation. Furthermore, the author(s) give an algorithm to find the transition betas, which agrees with empirical studies on CIFAR and MNIST datasets.
The reasons for the weak acceptance instead of a strong acceptance are explained by the following weaknesses.
As a theoretical contribution, it is important to have good formulations and clear statements. The quality of the mathematical statements is not satisfactory and can be largely improved.
Definition 1: introduce the concept of relative perturbation first. As a definition, it must be self-contained. Therefore introduce the second-order variation first (don't point to the appendix).
Lemma 0.1: inf_r(z|x) \mathcal{G} \ge \beta, is equivalent to \forall r(z|x), \mathcal{G}\ge\beta, the latter is more elegant. What is \mathcal{G}, anyway?
Lemma 0.2:  State the condition on the scale of the $\Delta\theta$. The reviewer suspects that if $\Delta\theta$ is large enough, the statement can be violated.  This lemma feels like a digress. Why the Fisher information is needed for explaining the phase transition?
Theorem 1: the statement is fine. It is better to have all conditions/assumptions listed as this is the core theorem. To be more complete, the authors can discuss the number of roots after the theorem and what happens if there is no root. In general, there have to be a few remarks after theorem 1.
Definition 2: \rho_r(X,Y;Z) is defined with respect to what? For example, state "Given a joined distribution p(X,Y), the representation maximum correlation....". Use a different equivalent symbol for definition, such as ":=".
Theorem 2: Again, \forall{f(x,z)} is with respect to what? In the "if" part of the "if-then" statement, "\forall{f(x,z}}, it can be decomposed as .." is a false statement, change to "If f{x,z} can be decomposed"
In the abstract, avoid citations, and explain X, Y, Z, p(z|x), for completeness.
As the paper is over the recommended length,  the reviewer is asked to be harsher in the assessment.