Universal Compression, List Decoding, and Logarithmic Loss

Universal lossy source coding under the logarithmic loss (log-loss) criterion is studied. Bounds on the rate-redundancy of variable-length universal codes with respect to a family of distributions are derived. These bounds correspond to previously derived bounds on distortion-redundancy of fixed-length coding. The asymptotic behavior of the resulting optimization problem is studied for a family of i.i.d. sources with a finite alphabet size. As is the case with distortion-redundancy, rate-redundancy of memoryless sources is lower bounded by [k/2] log
n
, where
n
is the blocklength and
k
is the number of degrees of freedom in the parameter space. The impact of the distortion constraint is on the constant term: higher allowed distortion effectively reduces the volume of the parameter uncertainty set. In view of previously established connections between lossy variable-length coding under log-loss and compression with list decoding, the bounds derived in this work also apply to variable-length coding with list decoding.