  Rate-distortion (R-D) function, a key quantity in information theory,
characterizes the fundamental limit of how much a data source can be compressed
subject to a fidelity criterion, by any compression algorithm. As researchers
push for ever-improving compression performance, establishing the R-D function
of a given data source is not only of scientific interest, but also sheds light
on the possible room for improving compression algorithms. Previous work on
this problem relied on distributional assumptions on the data source (Gibson,
2017) or only applied to discrete data (Blahut, 1972; Arimoto, 1972). By
contrast, this paper makes the first attempt at an algorithm for sandwiching
the R-D function of a general (not necessarily discrete) source requiring only
i.i.d. data samples. We estimate R-D sandwich bounds for a variety of
artificial and real-world data sources, in settings far beyond the feasibility
of any known method, and shed light on the optimality of neural data
compression (Ball\'e et al., 2021; Yang et al., 2022). Our R-D upper bound on
natural images indicates theoretical room for improving state-of-the-art image
compression methods by at least one dB in PSNR at various bitrates. Our data
and code can be found at https://github.com/mandt-lab/empirical-RD-sandwich.
