The rate-distortion saddle-point problem considered by Lapidoth (1997)
consists in finding the minimum rate to compress an arbitrary ergodic source
when one is constrained to use a random Gaussian codebook and minimum
(Euclidean) distance encoding is employed. We extend Lapidoth's analysis in
several directions in this paper. Firstly, we consider refined asymptotics. In
particular, when the source is stationary and memoryless, we establish the
second-order, moderate, and large deviation asymptotics of the problem.
Secondly, by "random Gaussian codebook", Lapidoth refers to a collection of
random codewords, each of which is drawn independently and uniformly from the
surface of an $n$-dimensional sphere. To be more precise, we term this as a
spherical Gaussian codebook. We also consider i.i.d.\ Gaussian codebooks in
which each random codeword is drawn independently from a product Gaussian
distribution. We derive the second-order, moderate, and large deviation
asymptotics when i.i.d.\ Gaussian codebooks are employed. Interestingly, in
contrast to the recent work on the channel coding counterpart by Scarlett, Tan
and Durisi (2017), the dispersions for spherical and i.i.d.\ Gaussian codebooks
are identical. Our bounds on the optimal error exponent for the spherical case
coincide on a non-empty interval of rates above the rate-distortion function.
The optimal error exponent for the i.i.d.\ case is established for all rates.