CovariateShift

Covariate shift, also known as covariate distribution shift, refers to a situation in supervised learning where the distribution over input features X changes between the training and test (or deployment) environments, while the conditional distribution of the target given the features, P(Y|X), remains the same. In formal terms, training data come from P_train(X,Y) = P_train(Y|X) P_train(X) and test data from P_test(X,Y) = P_test(Y|X) P_test(X), with P_train(Y|X) = P_test(Y|X) but P_train(X) ≠ P_test(X).

This assumption implies that the relationship between X and Y is stable, but the distribution of X itself differs across domains. If the covariate shift holds, predictive models can still generalize if the shift is properly accounted for, though naive application of a model trained on the source distribution may underperform on the target distribution.

Common approaches to mitigate covariate shift include reweighting training examples by importance weights w(x) = p_test(x)/p_train(x) and minimizing the weighted empirical risk. Estimating these density ratios can be done via density estimation, probabilistic classification, or methods such as kernel mean matching. Other strategies involve domain adaptation techniques that align feature distributions using unlabeled target data, including feature transformation, discrepancy minimization, and adversarial methods.

Covariate shift is distinct from label shift (prior probability change in Y, with P(X|Y) fixed) and concept drift (changes in P(Y|X) over time). It is a central concern in machine learning applications where training data and deployment data come from different environments, such as medical diagnostics, autonomous systems, and user-generated content. Limitations include difficulty in accurate density ratio estimation in high dimensions and potential residual mismatch if P(Y|X) is not perfectly invariant.