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：本文提出了一种新的框架，用于审视和评估无损数据压缩的基本极限，重点强调真正的非渐近结果。将给定信源压缩的**样本复杂度**定义为：在特定约束速率下，能够以不超过指定超速率概率压缩该信源所需的最小数据块长度。该公式与统计学和计算机科学中的相关进展相呼应，并便于利用各种假设检验问题的现有样本复杂度结果。对于任意信源，研究表明通用变长压缩器的样本复杂度与前缀码和定长码的样本复杂度紧密相关。对于无记忆信源，结果显示样本复杂度并非由信源熵决定，而是由其阶数为$1/2$的Rényi熵刻画。本文获得了样本复杂度的非渐近界，并给出了显式常数。针对马尔可夫信源建立了推广结论，表明样本复杂度由信源的阶数为$1/2$的Rényi熵率决定。最后，针对无记忆信源族，本文推导了通用数据压缩样本复杂度的界，其中样本复杂度由该信源族内元素与均匀分布之间的阶数为$1/2$的Rényi散度的最小值表征。本文还探讨并讨论了该问题与身份检验及相关分离速率之间的联系。