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分布漂移下预测的常用方法。本文研究了其在非平稳条件下的样本外预测误差。我们将超额风险一般性地分解为学习项和与分布漂移相关的误差项，并在混合条件下证明了学习误差的oracle不等式。该学习界在任意权重类上一致成立，并考虑了权重向量引致的有效样本量、权重类与假设类的复杂度以及潜在的数据依赖性。我们通过线性模型、基函数逼近和神经网络在（自）回归问题中展示了所得结果的适用性与紧致性，当特化为未加权平稳情形时，恢复了极小极大最优速率（至多对数因子）。