  We study the connection between multicalibration and boosting for squared
error regression. First we prove a useful characterization of multicalibration
in terms of a ``swap regret'' like condition on squared error. Using this
characterization, we give an exceedingly simple algorithm that can be analyzed
both as a boosting algorithm for regression and as a multicalibration algorithm
for a class H that makes use only of a standard squared error regression oracle
for H. We give a weak learning assumption on H that ensures convergence to
Bayes optimality without the need to make any realizability assumptions --
giving us an agnostic boosting algorithm for regression. We then show that our
weak learning assumption on H is both necessary and sufficient for
multicalibration with respect to H to imply Bayes optimality. We also show that
if H satisfies our weak learning condition relative to another class C then
multicalibration with respect to H implies multicalibration with respect to C.
Finally we investigate the empirical performance of our algorithm
experimentally using an open source implementation that we make available. Our
code repository can be found at
https://github.com/Declancharrison/Level-Set-Boosting.
