  We show through case studies that it is easier to estimate the fundamental
limits of data processing than to construct explicit algorithms to achieve
those limits. Focusing on binary classification, data compression, and
prediction under logarithmic loss, we show that in the finite space setting,
when it is possible to construct an estimator of the limits with vanishing
error with $n$ samples, it may require at least $n\ln n$ samples to construct
an explicit algorithm to achieve the limits.
