Title: Caveats for information bottleneck in deterministic scenarios,
Abstract: Information bottleneck (IB) is a method for extracting information from one
random variable $X$ that is relevant for predicting another random variable
$Y$. To do so, IB identifies an intermediate "bottleneck" variable $T$ that has
low mutual information $I(X;T)$ and high mutual information $I(Y;T)$. The "IB
curve" characterizes the set of bottleneck variables that achieve maximal
$I(Y;T)$ for a given $I(X;T)$, and is typically explored by maximizing the "IB
Lagrangian", $I(Y;T) - \beta I(X;T)$. In some cases, $Y$ is a deterministic
function of $X$, including many classification problems in supervised learning
where the output class $Y$ is a deterministic function of the input $X$. We
demonstrate three caveats when using IB in any situation where $Y$ is a
deterministic function of $X$: (1) the IB curve cannot be recovered by
maximizing the IB Lagrangian for different values of $\beta$; (2) there are
"uninteresting" trivial solutions at all points of the IB curve; and (3) for
multi-layer classifiers that achieve low prediction error, different layers
cannot exhibit a strict trade-off between compression and prediction, contrary
to a recent proposal. We also show that when $Y$ is a small perturbation away
from being a deterministic function of $X$, these three caveats arise in an
approximate way. To address problem (1), we propose a functional that, unlike
the IB Lagrangian, can recover the IB curve in all cases. We demonstrate the
three caveats on the MNIST dataset.