Optimizing Estimated Directed Information over Discrete Alphabets

Abstract

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.

Data-Driven Polar Codes for Unknown Channels With and Without Memory

“…This section claims that neural estimation methods for the estimation of the DI, as presented in [7], [8], may be used for the estimation of the channel embedding function independently from the NSC. The motivation for independent estimation of the channel embedding is demonstrated by memoryless channels.…”

“…The first trains the embedding and the NSC jointly. The second determines the parameters of the embedding E using neural estimation methods [7]- [9], and then, determines the parameters of the NSC while the parameters of E are fixed. After the training phase, the set of "clean" effective channels are determined by a Monte Carlo (MC) evaluation of the MI of the effective bit channels to complete the code design.…”

“…where (8). Then, for every ε > 0 there exists p ∈ N, compact Φ ⊂ R p and m ∈ N such that for M > m and i ∈ [N ], P − a.s.…”

“…Estimating the Capacity of Channels with Memory: Let W := W Y M N X M N be a binary-input channel with memory and let D MN ∼ P X ⊗M N ⊗ W Y N X N be a finite sample of its inputs-outputs pairs. The DINE algorithm [7], [8] estimates the DI rate from X to Y using the following formula:…”

“…The RVs Z N are auxiliary i.i.d. RVs distributed on Y N and independent of X N , Y N , that are used for the estimation of the DI as presented in [7]. The estimated maximizers of the first and second terms are denoted by T ΨXY and T ΨY , respectively.…”