The goal of machine learning is to find models that minimize prediction error on data that has not yet been seen. Its operational paradigm assumes access to a dataset $S$ and articulates a scheme for evaluating how well a given model performs on an arbitrary sample. The sample can be $S$ (in which case we speak of ``in-sample'' performance) or some entirely new $S'$ (in which case we speak of ``out-of-sample'' performance). Traditional analysis of generalization assumes that both in- and out-of-sample data are i.i.d.\ draws from an infinite population. However, these probabilistic assumptions cannot be verified even in principle. This paper presents an alternative view of generalization through the lens of sensitivity analysis of solutions of optimization problems to perturbations in the problem data. Under this framework, generalization bounds are obtained by purely deterministic means and take the form of variational principles that relate in-sample and out-of-sample evaluations through an error term that quantifies how close out-of-sample data are to in-sample data. Statistical assumptions can then be used \textit{ex post} to characterize the situations when this error term is small (either on average or with high probability).

翻译：机器学习的核心目标是寻找能最小化未见数据预测误差的模型。其操作范式假设可访问数据集$S$，并规定评估给定模型在任意样本上表现的方法。样本可以是$S$（此时讨论"样本内"表现），也可以是全新的$S'$（此时讨论"样本外"表现）。传统泛化分析假设样本内与样本外数据均来自无穷总体的独立同分布抽样，但这些概率假设在原则上无法验证。本文从优化问题解对问题数据扰动的敏感性分析视角，提出一种泛化分析的替代框架。在该框架下，泛化界通过纯确定性方法获得，并呈现为变分原理形式：通过一个量化样本外数据与样本内数据接近程度的误差项，建立样本内与样本外评估之间的关联。随后可事后利用统计假设，刻画该误差项何时较小（包括平均意义或高概率意义）。