While algorithmic stability is a central tool for understanding generalization of learning algorithms, existing high-probability guarantees typically rely on uniform boundedness or sub-Gaussian/sub-Weibull tail assumptions, which can be overly restrictive for modern settings with heavy-tailed or unbounded losses. We develop a stability-based framework that requires only a finite $L_p$ moment condition. Our first contribution is sharp concentration inequalities for functions of independent random variables under $L_p$ constraints, extending McDiarmid's bounded-differences techniques beyond the classical regime. Leveraging these results, we derive sharp high-probability generalization bounds across a range of learning paradigms, including empirical risk minimization, transductive regression, and meta-learning. These guarantees show that $L_p$ stability suffices for robust generalization even when boundedness fails, substantially weakening the standard assumptions in the stability literature.

翻译：虽然算法稳定性是理解学习算法泛化能力的核心工具，但现有高概率保证通常依赖于一致有界性或次高斯/次威布尔尾部假设，这在处理重尾或无界损失函数的现代场景中可能过于严格。我们建立了一个仅需有限$L_p$矩条件的稳定性框架。首先，我们提出在$L_p$约束下独立随机变量函数的尖锐集中不等式，将麦克迪尔米德有界差异技术拓展至经典范畴之外。借助这些结果，我们推导出涵盖经验风险最小化、转导回归和元学习等多种学习范式的高概率泛化界。这些保证表明，即使在有界性失效时，$L_p$稳定性仍足以实现稳健泛化，显著弱化了稳定性文献中的标准假设。