突破传统稳定性假设，用有限Lp矩条件实现更宽松的泛化保证。
Stability beyond Bounded Differences: Sharp Generalization Bounds under Finite $L_p$ Moments
- 基于Lp矩约束构建新稳定性框架，无需有界或亚高斯假设。
- 在经验风险最小化等场景下，给出紧致的高概率泛化界。
- 适合处理重尾损失的现代学习任务，如异常值敏感的机器学习。
尽管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亚高斯/亚韦布尔尾部假设，这在具有重尾或无界损失的现代设置中可能过于严格。本文提出一个基于稳定性的新框架，仅需有限$L_p$矩条件。首个贡献是针对独立随机变量函数在$L_p$约束下的尖锐浓度不等式，将麦克迪尔米德的有界差异技术扩展至经典范式之外。利用这些结果，我们推导出经验风险最小化、转导回归和元学习等多种学习范式下的尖锐高概率泛化界。这些保证表明，即使在无界情况下，$L_p$稳定性仍足以实现鲁棒泛化，显著弱化了稳定性文献中的标准假设。
原文摘要 · Abstract (English)
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.
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