Weighted empirical risk minimization is a common approach to prediction under distribution drift. This article studies its out-of-sample prediction error under nonstationarity. We provide a general decomposition of the excess risk into a learning term and an error term associated with distribution drift, and prove oracle inequalities for the learning error under mixing conditions. The learning bound holds uniformly over arbitrary weight classes and accounts for the effective sample size induced by the weight vector, the complexity of the weight and hypothesis classes, and potential data dependence. We illustrate the applicability and sharpness of our results in (auto-) regression problems with linear models, basis approximations, and neural networks, recovering minimax-optimal rates (up to logarithmic factors) when specialized to unweighted and stationary settings.

翻译：加权经验风险最小化是应对分布漂移预测问题的常用方法。本文研究其在非平稳条件下的样本外预测误差。我们将超额风险分解为学习项与分布漂移引起的误差项，并给出混合条件下学习误差的Oracle不等式。该学习界对任意权重类一致成立，且考虑了权重向量诱导的有效样本量、权重类与假设类的复杂度以及潜在的数据依赖性。通过在线性模型、基近似与神经网络的（自）回归问题中验证，我们展示了结果的适用性和最优性——当特化为无加权与平稳场景时，能恢复至极小极大最优收敛速率（对数因子除外）。