We consider the problem of lossless data compression for codes with and without prefix constraints, when strict performance constraints are placed on the excess-rate probability. Sharp nonasymptotic bounds are derived for the best achievable compression rate when the excess-rate probability is required to be exponentially small in the blocklength. These are obtained via tools from large deviations and Gaussian approximation. When the source distribution is unknown, a universal achievability result is obtained with an explicit “price for universality” term. This is based on a fine combinatorial estimate on the number

of sequences with small empirical entropy, which might be of independent interest. Examples are shown indicating that the approximation to the fundamental performance limits suggested

by these bounds is significantly more accurate than previous approaches. The new bounds reinforce the crucial operational conclusion that, in applications where the blocklength is

relatively short and where stringent guarantees are required on the rate, the best achievable rate is no longer close to the entropy. Rather, it is an appropriate, more pragmatic rate,

determined by the inverse error exponent function and the blocklength. This is joint work with Andreas Theocharous and Lampros Gavalakis.