Distribution shift (DS) may have two levels: the distribution itself changes, and the support (i.e., the set where the probability density is non-zero) also changes. When considering the support change between the training and test distributions, there can be four cases: (i) they exactly match; (ii) the training support is wider (and thus covers the test support); (iii) the test support is wider; (iv) they partially overlap. Existing methods are good at cases (i) and (ii), while cases (iii) and (iv) are more common nowadays but still under-explored. In this paper, we generalize importance weighting (IW), a golden solver for cases (i) and (ii), to a universal solver for all cases. Specifically, we first investigate why IW may fail in cases (iii) and (iv); based on the findings, we propose generalized IW (GIW) that could handle cases (iii) and (iv) and would reduce to IW in cases (i) and (ii). In GIW, the test support is split into an in-training (IT) part and an out-of-training (OOT) part, and the expected risk is decomposed into a weighted classification term over the IT part and a standard classification term over the OOT part, which guarantees the risk consistency of GIW. Then, the implementation of GIW consists of three components: (a) the split of validation data is carried out by the one-class support vector machine, (b) the first term of the empirical risk can be handled by any IW algorithm given training data and IT validation data, and (c) the second term just involves OOT validation data. Experiments demonstrate that GIW is a universal solver for DS problems, outperforming IW methods in cases (iii) and (iv).

本文研究了分布转移问题（DS）中包含了四种情况的训练集和测试集的支持度不同的情况，通过普适的广义重要性加权（GIW）方法将IW方法进行了推广，使其不仅能适用于情况（i）和（ii）还能用于情（iii）和（iv），专注于考虑训练集和测试集支持度不匹配并且保证风险的一致性，实验表明GIW比IW方法在（iii）和（iv）情况下表现更好。

将重要性加权泛化为分布偏移问题的通用求解器