Achieving the Empirical Capacity Using Feedback Part I: 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 moduloadditive channel with alphabet X , where the noise sequence Z 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 |X | − Hemp(Z) can be universally attained, where Hemp(Z) is the empirical entropy of Z. 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. In part II of this work we demonstrate how even higher rates can be attained by using more elaborate models for channel actions, and by utilizing possible empirical dependencies in its behavior. Index Terms Feedback Communications, Universal Communications, Arbitrarily Varying Channels, Adversarial Channels, Individual Sequences O. Shayevitz is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. This research was supported in part by the Israel Science Foundation, grant no. 223/05.