We consider communication over channels whose statistics are not known in full, but can be parameterized as a finite family of memoryless channels. A typical approach to address channel uncertainty is to design codes for the worst channel in the family, resulting in the well-known compound channel capacity. Although this approach is robust, it may suffer a significant loss of performance if the capacity-achieving distribution of the worst channel attains low rates over other channels. In this work, we cope with channel uncertainty through the lens of {\em competitive analysis}. The main idea is to optimize a relative metric that compares the performance of the designed code and a clairvoyant code that has access to the true channel. To allow communication rates that adapt to the channel at use, we consider rateless codes with a fixed number of message bits and random decoding times. We propose two competitive metrics: the competitive ratio between the expected rates of the two codes, and a regret defined as the difference between the expected rates. The competitive ratio, for instance, provides a percentage guarantee on the expected rate of the designed code when compared to the rate of the clairvoyant code that knows the channel at hand. Our main results are single-letter expressions for the optimal {\em competitive-ratio} and {\em regret}, expressed as a max-min or min-max optimization. Several examples illustrate the benefits of the competitive analysis approach to code design compared to the compound channel.

翻译：我们考虑信道统计特性不完全已知但可参数化为有限个无记忆信道族的情况。应对信道不确定性的典型方法是为该族中的最差信道设计编码，从而得到著名的复合信道容量。尽管该方法具有鲁棒性，但若最差信道的容量可达分布在其他信道上仅能实现较低速率，则可能造成显著的性能损失。本文通过竞争分析视角处理信道不确定性，核心思想是优化相对度量——将所设计编码的性能与具备真实信道信息的先验编码性能进行比较。为实现与使用中信道的自适应通信速率，我们考虑采用固定消息比特数和随机译码时间的无速率编码。提出两种竞争度量：两种编码期望速率的竞争比，以及期望速率差值的遗憾值。例如，竞争比能给出所设计编码相对于知晓当前信道先验编码速率的百分比保证。我们的主要结果是：最优竞争比与最优遗憾值的单字母表达式，表述为极大极小或极小极大优化问题。多个示例展示了相较于复合信道，竞争分析方法在编码设计中的优势。