Phase Transitions for the Information Bottleneck in Representation Learning
This work addresses fundamental questions about representation learning dynamics for researchers in machine learning theory, though it is incremental in extending IB analysis.
The paper investigates phase transitions in the Information Bottleneck (IB) objective, showing that sudden jumps in prediction accuracy and derivative behavior correspond to learning new classes, and provides a theoretical formula and algorithm to predict these transitions, verified on datasets like MNIST and CIFAR10 with accurate predictions.
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: $\text{IB}_β[p(z|x)] = I(X; Z) - βI(Y; Z)$ defined on the encoding distribution p(z|x) for input $X$, target $Y$ and representation $Z$, where sudden jumps of $dI(Y; Z)/d β$ 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 second-order 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.