Achieving the Empirical Capacity Using Feedback: Memoryless Additive Models

We address the problem of universal communications over an unknown channel with an instantaneous noiseless feedback, and show how rates corresponding to the empirical behavior of the channel can be attained, although no rate can be guaranteed in advance. First, we consider a discrete modulo-additive channel with alphabet ${\cal X}$, where the noise sequence $Z^n$ is arbitrary and unknown and may causally depend on the transmitted and received sequences and on the encoder's message, possibly in an adversarial fashion. Although the classical capacity of this channel is zero, we show that rates approaching the empirical capacity $\log{\vert {\cal X}\vert}-H_{\rm emp}(Z^n)$ can be universally attained, where $H_{\rm emp}(Z^n)$ is the empirical entropy of $Z^n$. For the more general setting, where the channel can map its input to an output in an arbitrary unknown fashion subject only to causality, we model the empirical channel actions as the modulo-addition of a realized noise sequence, and show that the same result applies if common randomness is available. The results are proved constructively, by providing a simple sequential transmission scheme approaching the empirical capacity.