  Solving linear programs by using entropic penalization has recently attracted
new interest in the optimization community, since this strategy forms the basis
for the fastest-known algorithms for the optimal transport problem, with many
applications in modern large-scale machine learning. Crucial to these
applications has been an analysis of how quickly solutions to the penalized
program approach true optima to the original linear program. More than 20 years
ago, Cominetti and San Mart\'in showed that this convergence is exponentially
fast; however, their proof is asymptotic and does not give any indication of
how accurately the entropic program approximates the original program for any
particular choice of the penalization parameter. We close this long-standing
gap in the literature regarding entropic penalization by giving a new proof of
the exponential convergence, valid for any linear program. Our proof is
non-asymptotic, yields explicit constants, and has the virtue of being
extremely simple. We provide matching lower bounds and show that the entropic
approach does not lead to a near-linear time approximation scheme for the
linear assignment problem.
