  Optimizing proper loss functions is popularly believed to yield predictors
with good calibration properties; the intuition being that for such losses, the
global optimum is to predict the ground-truth probabilities, which is indeed
calibrated. However, typical machine learning models are trained to
approximately minimize loss over restricted families of predictors, that are
unlikely to contain the ground truth. Under what circumstances does optimizing
proper loss over a restricted family yield calibrated models? What precise
calibration guarantees does it give? In this work, we provide a rigorous answer
to these questions. We replace the global optimality with a local optimality
condition stipulating that the (proper) loss of the predictor cannot be reduced
much by post-processing its predictions with a certain family of Lipschitz
functions. We show that any predictor with this local optimality satisfies
smooth calibration as defined in Kakade-Foster (2008), B{\l}asiok et al.
(2023). Local optimality is plausibly satisfied by well-trained DNNs, which
suggests an explanation for why they are calibrated from proper loss
minimization alone. Finally, we show that the connection between local
optimality and calibration error goes both ways: nearly calibrated predictors
are also nearly locally optimal.
