  This work develops a new method for estimating and optimizing the directed
information rate between two jointly stationary and ergodic stochastic
processes. Building upon recent advances in machine learning, we propose a
recurrent neural network (RNN)-based estimator which is optimized via gradient
ascent over the RNN parameters. The estimator does not require prior knowledge
of the underlying joint and marginal distributions. The estimator is also
readily optimized over continuous input processes realized by a deep generative
model. We prove consistency of the proposed estimation and optimization methods
and combine them to obtain end-to-end performance guarantees. Applications for
channel capacity estimation of continuous channels with memory are explored,
and empirical results demonstrating the scalability and accuracy of our method
are provided. When the channel is memoryless, we investigate the mapping
learned by the optimized input generator.
