Pragmatic Lossless Compression: Fundamental Limits and Universality

Abstract

The problem of variable-rate lossless data compression is considered, for codes with and without prefix constraints. Sharp bounds are derived for the best achievable compression rate of memoryless sources, when the excess-rate probability is required to be exponentially small in the blocklength. Accurate nonasymptotic expansions with explicit constants are obtained for the optimal rate, using 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, in the small excess-rate-probability regime, the approximation to the fundamental limit of the compression rate suggested by these bounds is significantly more accurate than the approximations provided by either normal approximation or error exponents. 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 excess-rate probability, the best achievable rate is no longer close to the entropy. Rather, it is an appropriate, more pragmatic rate, determined via the inverse error exponent function and the blocklength.