This paper extends standard results from learning theory with independent data to sequences of dependent data. Contrary to most of the literature, we do not rely on mixing arguments or sequential measures of complexity and derive uniform risk bounds with classical proof patterns and capacity measures. In particular, we show that the standard classification risk bounds based on the VC-dimension hold in the exact same form for dependent data, and further provide Rademacher complexity-based bounds, that remain unchanged compared to the standard results for the identically and independently distributed case. Finally, we show how to apply these results in the context of scenario-based optimization in order to compute the sample complexity of random programs with dependent constraints.

翻译：本文将从独立数据的学习理论中得到的标准结果推广到依赖数据序列。与大多数文献不同，我们并未依赖混合论证或序列复杂性度量，而是沿用经典证明模式与容量度量推导出了均匀风险界。特别地，我们展示了基于VC维的标准分类风险界在依赖数据情形下仍保持完全相同的数学形式，并进一步提供了基于Rademacher复杂度的风险界——这些界与独立同分布情形下的标准结果相比也保持不变。最后，我们展示了如何将这些结果应用于基于场景的优化，以计算带有依赖约束的随机程序的样本复杂度。