Dealing with distribution shifts is one of the central challenges for modern machine learning. One fundamental situation is the \emph{covariate shift}, where the input distributions of data change from training to testing stages while the input-conditional output distribution remains unchanged. In this paper, we initiate the study of a more challenging scenario -- \emph{continuous} covariate shift -- in which the test data appear sequentially, and their distributions can shift continuously. Our goal is to adaptively train the predictor such that its prediction risk accumulated over time can be minimized. Starting with the importance-weighted learning, we show the method works effectively if the time-varying density ratios of test and train inputs can be accurately estimated. However, existing density ratio estimation methods would fail due to data scarcity at each time step. To this end, we propose an online method that can appropriately reuse historical information. Our density ratio estimation method is proven to perform well by enjoying a dynamic regret bound, which finally leads to an excess risk guarantee for the predictor. Empirical results also validate the effectiveness.

翻译：处理分布偏移是现代机器学习面临的核心挑战之一。一个基本情形是\emph{协变量偏移}，即数据输入分布在训练阶段和测试阶段之间发生变化，而输入条件输出分布保持不变。本文针对一个更具挑战性的场景——\emph{连续}协变量偏移——展开研究，其中测试数据按顺序出现，且其分布可能连续变化。我们的目标是自适应地训练预测器，使其随时间累积的预测风险最小化。从重要性加权学习出发，我们证明若测试输入与训练输入随时间变化的密度比能被准确估计，该方法便能有效运作。然而，由于每个时间步的数据稀疏性，现有密度比估计方法将失效。为此，我们提出一种能恰当复用历史信息的在线方法。我们提出的密度比估计方法被证明具有良好的动态遗憾界，最终为预测器提供了超额风险保证。实验结果也验证了其有效性。