We consider the task of predicting a response Y from a set of covariates X in settings where the conditional distribution of Y given X changes over time. For this to be feasible, assumptions on how the conditional distribution changes over time are required. Existing approaches assume, for example, that changes occur smoothly over time so that short-term prediction using only the recent past becomes feasible. In this work, we propose a novel invariance-based framework for linear conditionals, called Invariant Subspace Decomposition (ISD), that splits the conditional distribution into a time-invariant and a residual time-dependent component. As we show, this decomposition can be utilized both for zero-shot and time-adaptation prediction tasks, that is, settings where either no or a small amount of training data is available at the time points we want to predict Y at, respectively. We propose a practical estimation procedure, which automatically infers the decomposition using tools from approximate joint matrix diagonalization. Furthermore, we provide finite sample guarantees for the proposed estimator and demonstrate empirically that it indeed improves on approaches that do not use the additional invariant structure.

翻译：我们考虑在给定协变量X预测响应Y的任务，其中Y的条件分布随时间变化。为使该任务可行，需要对条件分布随时间变化的方式做出假设。现有方法假设，例如，变化随时间平滑发生，从而仅利用近期历史数据进行短期预测成为可能。本文提出了一种针对线性条件的新型不变性框架，称为不变子空间分解（Invariant Subspace Decomposition, ISD），它将条件分布分解为时间不变分量和残差时变分量。研究表明，该分解可同时用于零样本预测和时间适应预测任务，即在需要预测Y的时间点上分别无训练数据或仅有少量训练数据的情况。我们提出了一种实用的估计方法，利用近似联合矩阵对角化工具自动推断该分解。此外，我们给出了所提估计器的有限样本保证，并通过实证表明其确实优于未利用额外不变性结构的方法。