Expectation Consistency Loss: Rethink Confidence Calibration under Covariate Shift

Confidence calibration for classification models is vital in safety-critical decision-making scenarios and has received extensive attention. General confidence calibration methods assume training and test data are independent and identically distributed ($i.i.d.$), limiting their effectiveness under covariate shifts. Previous calibration methods under covariate shift struggle with class-wise or canonical calibrations and often rely on unstable importance weighting when density ratios are large or unbounded. Given the above limitations, this paper rethinks confidence calibration under covariate shifts. First, we derive a necessary and sufficient condition for confidence calibration under covariate shifts, named Expectation consistency condition, which reveals covariate shifts do not necessarily lead to uncalibrated confidence and provides a weaker condition for confidence calibration than global covariate distribution alignment. Then, utilizing Expectation consistency condition, this paper proposes an unsupervised domain adaptation loss to calibrate confidence of the target domain, named Expectation consistency loss (ECL), which is compatible with canonical calibration, class-wise calibration, and top-label calibration. Third, we prove that computing ECL loss has the same sample complexity as Expected Calibration Error (ECE) and provide a theoretically grounded mini-batch trainable scheme for ECL loss. Finally, we validate the effectiveness of our method on both simulated and real-world covariate shift datasets.

Machine learning models are increasingly used in safety-critical decisions—such as diagnosing diseases or assessing medical risks—where being accurate isn't enough. They also need to honestly report how uncertain they are. For example, a model should know when to say "I'm only 60% sure" and defer to a human doctor. However, this self-awareness often breaks down under covariate shift—a scenario where the input data looks different but the underlying rules stay the same, such as when a model trained on one hospital's X-ray images is deployed in another with different equipment. Our research uncovered a surprising insight: you don't need to align entire data distributions to fix this. Instead, we proved that it is enough to ensure that at each confidence level, the model's expected accuracy stays consistent across environments—a much weaker and more practical condition. Building on this idea, we developed a training method that helps models maintain reliable confidence under covariate shift, without the instability issues of previous approaches. This could make AI systems safer and more trustworthy in real-world deployments where the data we face rarely matches the data we trained on.