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This paper provides upper and lower bounds on the optimal guessing moments ofa random variable taking values on a finite set when side information may beavailable. These moments quantify the number of guesses required for correctlyidentifying the unknown object and, similarly to Arikan's bounds, they areexpressed in terms of the Arimoto-R\'enyi conditional entropy. AlthoughArikan's bounds are asymptotically tight, the improvement of the bounds in thispaper is significant in the non-asymptotic regime. Relationships betweenmoments of the optimal guessing function and the MAP error probability are alsoestablished, characterizing the exact locus of their attainable values. Thebounds on optimal guessing moments serve to improve non-asymptotic bounds onthe cumulant generating function of the codeword lengths for fixed-to-variableoptimal lossless source coding without prefix constraints. Non-asymptoticbounds on the reliability function of discrete memoryless sources are derivedas well. Relying on these techniques, lower bounds on the cumulant generatingfunction of the codeword lengths are derived, by means of the smooth R\'enyientropy, for source codes that allow decoding errors.

Improved Bounds on Lossless Source Coding and Guessing Moments via R\'enyi Measures