  Directed information (DI) is a fundamental measure for the study and analysis
of sequential stochastic models. In particular, when optimized over input
distributions it characterizes the capacity of general communication channels.
However, analytic computation of DI is typically intractable and existing
optimization techniques over discrete input alphabets require knowledge of the
channel model, which renders them inapplicable when only samples are available.
To overcome these limitations, we propose a novel estimation-optimization
framework for DI over discrete input spaces. We formulate DI optimization as a
Markov decision process and leverage reinforcement learning techniques to
optimize a deep generative model of the input process probability mass function
(PMF). Combining this optimizer with the recently developed DI neural
estimator, we obtain an end-to-end estimation-optimization algorithm which is
applied to estimating the (feedforward and feedback) capacity of various
discrete channels with memory. Furthermore, we demonstrate how to use the
optimized PMF model to (i) obtain theoretical bounds on the feedback capacity
of unifilar finite-state channels; and (ii) perform probabilistic shaping of
constellations in the peak power-constrained additive white Gaussian noise
channel.
