Stability beyond Bounded Differences: Sharp Generalization Bounds under Finite $L_p$ Moments

Abstract: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.

Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)

arXiv:2606.06855 [stat.ML]

(or arXiv:2606.06855v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2606.06855 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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