We develop the concept of exponential stochastic inequality (ESI), a novel notation that simultaneously captures high-probability and in-expectation statements. It is especially well suited to succinctly state, prove, and reason about excess-risk and generalization bounds in statistical learning, specifically, but not restricted to, the PAC-Bayesian type. We show that the ESI satisfies transitivity and other properties which allow us to use it like standard, nonstochastic inequalities. We substantially extend the original definition from Koolen et al. (2016) and show that general ESIs satisfy a host of useful additional properties, including a novel Markov-like inequality. We show how ESIs relate to, and clarify, PAC-Bayesian bounds, subcentered subgamma random variables and *fast-rate conditions* such as the central and Bernstein conditions. We also show how the ideas can be extended to random scaling factors (learning rates).

翻译：我们提出了指数随机不等式（ESI）这一概念，它是一种能够同时刻画高概率情形与期望情形的新型记号。该记号特别适用于简洁地表述、证明和推理统计学习中的超额风险与泛化界，尤其适用于PAC-贝叶斯类型（但不限于此）的界。我们证明了ESI满足传递性及其他性质，使其能够像标准非随机不等式一样使用。我们大幅拓展了Koolen等人（2016）提出的原始定义，并证明一般化的ESI具备一系列有用的附加性质，包括一种新颖的类似于马尔可夫不等式的不等式。我们展示了ESI与PAC-贝叶斯界、次中心化次伽马随机变量以及中心条件和伯恩斯坦条件等"快收敛率条件"之间的关联及其澄清作用。此外，我们还展示了如何将这些思想推广至随机缩放因子（学习率）的情形。