  We establish a new model-agnostic optimization framework for
out-of-distribution generalization via multicalibration, a criterion that
ensures a predictor is calibrated across a family of overlapping groups.
Multicalibration is shown to be associated with robustness of statistical
inference under covariate shift. We further establish a link between
multicalibration and robustness for prediction tasks both under and beyond
covariate shift. We accomplish this by extending multicalibration to
incorporate grouping functions that consider covariates and labels jointly.
This leads to an equivalence of the extended multicalibration and invariance,
an objective for robust learning in existence of concept shift. We show a
linear structure of the grouping function class spanned by density ratios,
resulting in a unifying framework for robust learning by designing specific
grouping functions. We propose MC-Pseudolabel, a post-processing algorithm to
achieve both extended multicalibration and out-of-distribution generalization.
The algorithm, with lightweight hyperparameters and optimization through a
series of supervised regression steps, achieves superior performance on
real-world datasets with distribution shift.
