  In this work, we propose solving the Information bottleneck (IB) and Privacy
Funnel (PF) problems with Douglas-Rachford Splitting methods (DRS). We study a
general Markovian information-theoretic Lagrangian that includes IB and PF into
a unified framework. We prove the linear convergence of the proposed solvers
using the Kurdyka-{\L}ojasiewicz inequality. Moreover, our analysis is beyond
IB and PF and applies to any convex-weakly convex pair objectives. Based on the
results, we develop two types of linearly convergent IB solvers, with one
improves the performance of convergence over existing solvers while the other
can be independent to the relevance-compression trade-off. Moreover, our
results apply to PF, yielding a new class of linearly convergent PF solvers.
Empirically, the proposed IB solvers IB obtain solutions that are comparable to
the Blahut-Arimoto-based benchmark and is convergent for a wider range of the
penalty coefficient than existing solvers. For PF, our non-greedy solvers can
characterize the privacy-utility trade-off better than the clustering-based
greedy solvers.
