Encoding Tasks and R\'enyi Entropy

A task is randomly drawn from a finite set of tasks and is described using a fixed number of bits. All the tasks that share its description must be performed. Upper and lower bounds on the minimum $\rho$-th moment of the number of performed tasks are derived. The case where a sequence of tasks is produced by a source and $n$ tasks are jointly described using $nR$ bits is considered. If $R$ is larger than the R\'enyi entropy rate of the source of order $1/(1+\rho)$ (provided it exists), then the $\rho$-th moment of the ratio of performed tasks to $n$ can be driven to one as $n$ tends to infinity. If $R$ is smaller than the R\'enyi entropy rate, this moment tends to infinity. The results are generalized to account for the presence of side-information. In this more general setting, the key quantity is a conditional version of R\'enyi entropy that was introduced by Arimoto. For IID sources two additional extensions are solved, one of a rate-distortion flavor and the other where different tasks may have different nonnegative costs. Finally, a divergence that was identified by Sundaresan as a mismatch penalty in the Massey-Arikan guessing problem is shown to play a similar role here.