  The Information bottleneck (IB) method enables optimizing over the trade-off
between compression of data and prediction accuracy of learned representations,
and has successfully and robustly been applied to both supervised and
unsupervised representation learning problems. However, IB has several
limitations. First, the IB problem is hard to optimize. The IB Lagrangian
$\mathcal{L}_{IB}:=I(X;Z)-\beta I(Y;Z)$ is non-convex and existing solutions
guarantee only local convergence. As a result, the obtained solutions depend on
initialization. Second, the evaluation of a solution is also a challenging
task. Conventionally, it resorts to characterizing the information plane, that
is, plotting $I(Y;Z)$ versus $I(X;Z)$ for all solutions obtained from different
initial points. Furthermore, the IB Lagrangian has phase transitions while
varying the multiplier $\beta$. At phase transitions, both $I(X;Z)$ and
$I(Y;Z)$ increase abruptly and the rate of convergence becomes significantly
slow for existing solutions. Recent works with IB adopt variational surrogate
bounds to the IB Lagrangian. Although allowing efficient optimization, how
close are these surrogates to the IB Lagrangian is not clear. In this work, we
solve the IB Lagrangian using augmented Lagrangian methods. With augmented
variables, we show that the IB objective can be solved with the alternating
direction method of multipliers (ADMM). Different from prior works, we prove
that the proposed algorithm is consistently convergent, regardless of the value
of $\beta$. Empirically, our gradient-descent-based method results in
information plane points that are denser and comparable to those obtained
through the conventional Blahut-Arimoto-based solvers.
