A new framework is introduced for examining and evaluating the fundamental limits of lossless data compression, that emphasizes genuinely non-asymptotic results. The {\em sample complexity} of compressing a given source is defined as the smallest blocklength at which it is possible to compress that source at a specifically constrained rate and to within a specified excess-rate probability. This formulation parallels corresponding developments in statistics and computer science, and it facilitates the use of existing results on the sample complexity of various hypothesis testing problems. For arbitrary sources, the sample complexity of general variable-length compressors is shown to be tightly coupled with the sample complexity of prefix-free codes and fixed-length codes. For memoryless sources, it is shown that the sample complexity is characterized not by the source entropy, but by its Rényi entropy of order~$1/2$. Nonasymptotic bounds on the sample complexity are obtained, with explicit constants. Generalizations to Markov sources are established, showing that the sample complexity is determined by the source's Rényi entropy rate of order~$1/2$. Finally, bounds on the sample complexity of universal data compression are developed for families of memoryless sources. There, the sample complexity is characterized by the minimum Rényi divergence of order~$1/2$ between elements of the family and the uniform distribution. The connection of this problem with identity testing and with the associated separation rates is explored and discussed.

翻译：引入了一个新框架用于考察和评估无损数据压缩的基本极限，该框架强调真正的非渐近结果。压缩给定信源的**样本复杂度**定义为能够以特定约束速率和指定超额速率概率压缩该信源的最小块长。这一表述与统计学和计算机科学中的相应发展相呼应，并便于利用各种假设检验问题中已有的样本复杂度结果。对于任意信源，通用变长编码器的样本复杂度被证明与前缀码和定长码的样本复杂度紧密耦合。对于无记忆信源，研究表明样本复杂度并非由信源熵刻画，而是由其1/2阶Rényi熵决定。获得了样本复杂度的非渐近界，并包含显式常数。建立了对马尔可夫信源的推广，证明样本复杂度由信源的1/2阶Rényi熵率决定。最后，针对无记忆信源族发展了通用数据压缩的样本复杂度界。在此情形下，样本复杂度由该信源族中元素与均匀分布之间的最小1/2阶Rényi散度刻画。本文还探讨并讨论了该问题与恒等性检验及相关分离速率之间的联系。