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：本文提出一种新的框架，用于考察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。给定信源的压缩样本复杂度定义为：在满足特定速率约束且不超过指定超额速率概率的条件下，能够对该信源进行压缩所需的最小分组长度。该表述与统计学和计算机科学中的对应发展相平行，并便于利用现有关于各类假设检验问题样本复杂度的研究成果。对于任意信源，研究表明通用变长压缩器的样本复杂度与无前缀码及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决定，而是由其阶数为$1/2$的Rényi熵决定。本文给出了带有显式常数的样本复杂度非渐近界，并建立了向马尔可夫信源的推广，证明其样本复杂度由信源阶数为$1/2$的Rényi熵率决定。最后，针对无记忆信源族，推导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阶数为$1/2$的最小Rényi散度表征。本文还探讨并讨论了该问题与恒等检验及关联分离速率之间的联系。