# Data-Driven Optimization of Directed Information over Discrete Alphabets

arXiv: https://arxiv.org/abs/2301.00621 · HTML: https://mlsanity.com/paper/2301.00621/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 68 references that are in the map.
- First posted on arXiv 2 Jan 2023.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Neural Estimation of Statistical Divergences](https://mlsanity.com/paper/2110.03652/) (2021)
- [Non-Asymptotic Performance Guarantees for Neural Estimation of mathsff-Divergences](https://mlsanity.com/paper/2103.06923/) (2021)
- [A Unified Bellman Equation for Causal Information and Value in Markov Decision Processes](https://mlsanity.com/paper/1703.01585/) (2017)
- [Neural Capacity Estimators: How Reliable Are They?](https://mlsanity.com/paper/2111.07401/) (2021)
- [Directed Information on Abstract Spaces: Properties and Variational Equalities](https://mlsanity.com/paper/1302.3971/) (2013)
- [On the existence of optimal stationary policies for average Markov decision processes with countable states](https://mlsanity.com/paper/2007.01602/) (2020)

## Built on by (most cited first, 2 of 2)

- [TREET: TRansfer Entropy Estimation via Transformers](https://mlsanity.com/paper/2402.06919/) (2024)
- [Code Rate Optimization via Neural Polar Decoders](https://mlsanity.com/paper/2506.15836/) (2025)

## Abstract (arXiv)

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.