Soft Guessing Under Logarithmic Loss Allowing Errors and Variable-Length Source Coding
Shota Saito, Hamdi Joudeh
TL;DR
This work develops a unified theory of soft guessing under log-loss while allowing errors, deriving tight single-shot upper and lower bounds on the minimal guessing moments in terms of smooth Rényi entropies, and providing sharp i.i.d. asymptotics. The core result identifies an optimal list-based soft guessing strategy with probabilistic stopping, linking the guessing moment to the smooth entropy of a list-index variable $Z$ and recovering special cases such as guessing with errors and pure soft guessing. The analysis extends to side information via conditional smooth Rényi entropies and establishes a precise connection to variable-length lossy source coding under log-loss, bounding the normalized cumulant generating function of codeword lengths by the guessing moments. Collectively, the results unify several prior bounds, generalize them to error-allowing and side-information settings, and yield rate-distortion-type interpretations for log-loss in both guessing and coding contexts, with implications for sequential search and Bayesian reconstruction tasks.
Abstract
This paper considers the problem of soft guessing under a logarithmic loss distortion measure while allowing errors. We find an optimal guessing strategy, and derive single-shot upper and lower bounds for the minimal guessing moments as well as an asymptotic expansion for i.i.d. sources. These results are extended to the case where side information is available to the guesser. Furthermore, a connection between soft guessing allowing errors and variable-length lossy source coding under logarithmic loss is demonstrated. The Rényi entropy, the smooth Rényi entropy, and their conditional versions play an important role.